diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzjsom" "b/data_all_eng_slimpj/shuffled/split2/finalzzjsom" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzjsom" @@ -0,0 +1,5 @@ +{"text":"\\section{\\@startsection {section}{1}{\\z@}%\n {-3.5ex \\@plus -1ex \\@minus -.2ex\n {2.3ex \\@plus.2ex}%\n {\\normalfont\\large\\bfseries}}\n\n\\renewcommand\\subsection{\\@startsection{subsection}{2}{\\z@}%\n {-3.25ex\\@plus -1ex \\@minus -.2ex}%\n {1.5ex \\@plus .2ex}%\n {\\normalfont\\bfseries}}\n\n\n\\renewcommand{\\H}{\\mathcal{H}}\n\\newcommand{\\mbox{SU}}{\\mbox{SU}}\n\\newcommand{\\chi^{{\\rm U}(\\infty)}}{\\chi^{{\\rm U}(\\infty)}}\n\\newcommand{\\rm f}{\\rm f}\n\\linespread{1.3}\n\n\\newcommand{\\td}[2]{\\frac{d #1}{d #2}}\n\\newcommand{\\pd}[2]{\\frac{\\partial #1}{\\partial #2}}\n\\newcommand{\\avg}[1]{\\left< #1 \\right>}\n\\newcommand{\\bra}[1]{\\left< #1 \\right|}\n\\newcommand{\\ket}[1]{\\left| #1 \\right>}\n\\newcommand{\\braket}[2]{\\left< #1 \\right| \\left. #2 \\right>}\n\\newcommand{\\tdd}[2]{\\frac{d^2 #1}{d #2^2}}\n\\newcommand{\\pdd}[2]{\\frac{\\partial^2 #1}{\\partial #2^2}}\n\n\n\\newcommand{\\rom}[1]{\\mathrm{#1}}\n\\newcommand{\\begin{equation}}{\\begin{equation}}\n\\newcommand{\\end{equation}}{\\end{equation}}\n\\newcommand{\\begin{eqnarray}}{\\begin{eqnarray}}\n\\newcommand{\\end{eqnarray}}{\\end{eqnarray}}\n\\newcommand{\\begin{eqnarray*}}{\\begin{eqnarray*}}\n\\newcommand{\\alpha}{\\alpha}\n\\newcommand{\\mc}[1]{\\mathcal{#1}}\n\\newcommand{\\bar{c}}{\\bar{c}}\n\\newcommand{\\partial}{\\partial}\n\\newcommand{\\kappa}{\\kappa}\n\\newcommand{\\mathcal{O}}{\\mathcal{O}}\n\\newcommand{\\text{\\begin{footnotesize}$~{}^\\circ_\\circ~$\\end{footnotesize}}}{\\text{\\begin{footnotesize}$~{}^\\circ_\\circ~$\\end{footnotesize}}}\n\n\\newcommand{\\labell}[1]{\\label{#1}}\n\\newcommand{\\labels}[1]{\\label{#1}}\n\\newcommand{\\reef}[1]{(\\ref{#1})}\n\\newcommand{\\scriptscriptstyle}{\\scriptscriptstyle}\n\\newcommand{{e.g.}\\ }{{e.g.}\\ }\n\\newcommand{{ i.e.}\\ }{{ i.e.}\\ }\n\\newcommand{\\textrm{Tr}}{\\textrm{Tr}}\n\\newcommand{\\textrm{diag}}{\\textrm{diag}}\n\\newcommand{\\bar{\\psi}}{\\bar{\\psi}}\n\\newcommand{\\bar{\\phi}}{\\bar{\\phi}}\n\\newcommand{\\bar{\\theta}}{\\bar{\\theta}}\n\\newcommand{\\textrm{d}}{\\textrm{d}}\n\\newcommand{\\bar{\\mu}}{\\bar{\\mu}}\n\\newcommand{{\\hat {\\cal A}}}{{\\hat {\\cal A}}}\n\\newcommand{{\\cal L}}{{\\cal L}}\n\\newcommand{{\\cal D}}{{\\cal D}}\n\\newcommand{{\\cal M}}{{\\cal M}}\n\\newcommand{{\\varepsilon}}{{\\varepsilon}}\n\\newcommand{{\\mathbb R}}{{\\mathbb R}}\n\\newcommand{{\\text d}}{{\\text d}}\n\\newcommand{{\\cal F}}{{\\cal F}}\n\\newcommand{{\\cal A}}{{\\cal A}}\n\\def\\[{\\[}\n\\def\\]{\\]}\n\\def{\\text{ct}}{{\\text{ct}}}\n\\newcommand{{\\cal C}}{{\\cal C}}\n\n\\def\\partial{\\partial}\n\\newcommand{\\unarray}[2]{\\left(\\!\\!\\begin{array}{c}#1\\\\#2\\end{array}\\!\\!\\right)}\n\\newcommand{\\matrixarray}[4]{\\left(\\!\\!\\begin{array}{cc}#1\\\\#3\\end{array}\\!\\!\\right)}\n\\def\\boldsymbol{p}{\\boldsymbol{p}}\n\\newcommand{\\innerproductwith}[1]{\\left<#1\\right|}\n\\def\\mathcal{V}{\\mathcal{V}}\n\\def{\\mathbb C}{{\\mathbb C}}\n\\def{\\mathbb R}{{\\mathbb R}}\n\\def{\\mathbb Z}{{\\mathbb Z}}\n\n\n\\newcommand{\\bd}[1]{\\begin{fmffile}{#1}\\begin{fmfgraph*}}\n\\newcommand{\\end{fmfgraph*}\\end{fmffile}}{\\end{fmfgraph*}\\end{fmffile}}\n\n\\newcommand{\\mathbb D_{\\textrm{hol}}^{(2g-4,2n)}}{\\mathbb D_{\\textrm{hol}}^{(2g-4,2n)}}\n\\newcommand{\\mathfrak}{\\mathfrak}\n\n\\newcommand{\\mathcal{N}}{\\mathcal{N}}\n\\newcommand{\\mathbb}{\\mathbb}\n\\newcommand{\\textrm{Re}\\,}{\\textrm{Re}\\,}\n\\newcommand{\\textrm{Im}\\,}{\\textrm{Im}\\,}\n\\newcommand{\\mN=2^\\star}{\\mathcal{N}=2^\\star}\n\n\n\\begin{document}\n\n\n\\begin{titlepage}\n\n\\begin{flushright}\n\tCERN-PH-TH\/2015-263\n\\end{flushright}\n\n\\unitlength = 1mm~\\\\\n\\vskip 1cm\n\\begin{center}\n\n{\\LARGE{\\textsc{$\\mc{N}=2^\\star$ from Topological Amplitudes \\\\[0.3cm] in String Theory}}}\n\n\n\n\\vspace{0.8cm}\nIoannis Florakis\\,{}\\footnote{\\tt ioannis.florakis@cern.ch} and Ahmad Zein Assi\\,{}\\footnote{\\tt azein\\_as@ictp.it}\n\n\n\\vspace{1cm}\n\n{\\it ${}^1$ Theory Division - CERN, CH-1211 Geneva 23, Switzerland \\\\\n\t${}^2$ High Energy Section - ICTP, Strada Costiera, 11-34014 Trieste, Italy \n\n}\n\n\\vspace{0.8cm}\n\n\\begin{abstract}\nIn this paper, we explicitly construct string theory backgrounds that realise the so-called $\\mathcal{N}=2^\\star$ gauge theory. We prove the consistency of our models by calculating their partition function and obtaining the correct gauge theory spectrum. We further provide arguments in favour of the universality of our construction which covers a wide class of models all of which engineer the same gauge theory. We reproduce the corresponding Nekrasov partition function once the $\\Omega$-deformation is included and the appropriate field theory limit taken. This is achieved by calculating the topological amplitudes $F_g$ in the string models. In addition to heterotic and type II constructions, we also realise the mass deformation in type I theory, thus leading to a natural way of uplifting the result to the instanton sector.\n\n\\end{abstract}\n\n\n\n\\vspace{1.0cm}\n\n\n\\end{center}\n\n\n\\end{titlepage}\n\n\n\\pagestyle{empty}\n\\pagestyle{plain}\n\n\\def{\\vec x}{{\\vec x}}\n\\def\\partial{\\partial}\n\\def$\\cal P_O${$\\cal P_O$}\n\n\\pagenumbering{arabic}\n\n\\tableofcontents\n\\bibliographystyle{utphys}\n\n\n\\section{Introduction}\n\n\nDuring the last decades, the interplay between string theory and supersymmetric gauge theories has been the driving force for many discoveries in both fields. One of the most striking examples is the connection between topological string theory \\cite{TopoString} and the $\\mathcal{N}=2$ gauge theory with the $\\Omega$-deformation \\cite{Losev:1997bz,Nekrasov:2002qd}. Indeed, it has been realised that the partition function of the topological string reduces, in the field theory limit, to the free energy of the $\\Omega$-deformed $\\mathcal{N}=2$ gauge theory, often referred to as the Nekrasov partition function. This correspondence is valid in the so-called topological limit of the $\\Omega$-background, in which one of its parameters is set to zero, while the other is identified with the topological string coupling $g_{\\rm s}$. Therefore, what plays the role of a regularisation parameter in gauge theory acquires physical significance once uplifted to string theory. The extension of this connection to the general setup including two deformation parameters is a programme called \\emph{refinement} and has led to many fruitful discoveries \\cite{Hollowood,Topvertex,MatrixTheory,Hellerman:2011mv}. In particular, a worldsheet realisation of the $\\Omega$-background has been carried out in \\cite{AFHNZ,Antoniadis:2013mna}, even though the explicit definition of the twisted, topological theory is still lacking.\n\nFrom the point of view of the physical string in this approach, the $\\Omega$-deformation boils down to a background of anti-self-dual graviphotons \\cite{Bershadsky:1993cx,Antoniadis:1993ze} and self-dual gauge field strenghts \\cite{AFHNZ,Antoniadis:2013mna}. The gauge theory partition function descends from a class of BPS amplitudes which has been studied both in the $\\mathcal{N}=2$ \\cite{Ooguri:1995cp,Antoniadis:1996qg,Antoniadis:2009nv,Antoniadis:2010iq} as well as in the $\\mathcal{N}=4$ case \\cite{Antoniadis:2006mr,Antoniadis:2007cw}. Their BPS nature translates itself into the holomorphic moduli dependence of the corresponding coupling in the string effective action. However, this property is broken at the string level due to boundary effects as expressed, for instance, by the holomorphic anomaly equation \\cite{Bershadsky:1993cx}. The generalisation of the latter to the refined case has been analysed in \\cite{Huang:2010kf} (see also \\cite{Krefl:2010fm}) and more recently in \\cite{NewRecRel} from the worldsheet perspective.\n\nIn the present work, we study a particular deformation of $\\mathcal{N}=4$ Super Yang-Mills theory, commonly referred to as $\\mN=2^\\star$. It corresponds to a mass deformation of the former under which the $\\mathcal{N}=2$ adjoint hypermultiplet acquires a mass. Hence, analyticity of the mass parameter renders this theory an interpolation between the pure $\\mathcal{N}=4$ and $\\mathcal{N}=2$ theories, the latter being recovered as particular limits of zero and infinite mass, respectively. In addition, the $\\mN=2^\\star$ theory is a flagship example in the context of the AGT conjecture \\cite{Alday:2009aq} which relates it to the two-dimensional Liouville theory on a torus with one puncture playing the role of the massive hypermultiplet. In this correspondence, the Nekrasov partition function is mapped to the Liouville theory conformal block. More general connections can be established by obtaining four-dimensional gauge theories from the $(2,0)$ theory compactified on a genus $g$ Riemann surface with $n$ punctures \\cite{Gaiotto:2009we}, and considering Liouville theory on the Riemann surface. For instance, a torus with four punctures leads to the $\\mathcal{N}=2$, $\\rm SU(2)$ gauge theory with four flavours.\n\nSince a worldsheet description of $\\mN=2^\\star$ in string theory is lacking, our goal is to fill this gap by studying string theory with spontaneous breaking of supersymmetry from $\\mathcal{N}=4$ to $\\mathcal{N}=2$, in such a way that the adjoint hypermultiplet acquires a moduli dependent mass. This is achieved by considering a freely-acting orbifold of the $T^6$ torus, implementing the uplift of the Scherk-Schwarz mechanism to string theory \\cite{Ferrara:1987qp,Kounnas:1988ye,Ferrara:1988jx,Kiritsis:1997ca,Antoniadis:1998ep,Condeescu:2012sp}. In this construction, certain states acquire a mass that is inversely proportional to the volume of some cycle of the internal space and the unbroken supersymmetric model is recovered in the limit of zero mass. We first present a heterotic asymmetric orbifold construction which turns out to be the most natural way of realising $\\mathcal{N}=2^\\star$ in perturbative closed string theory. We show that it indeed provides mass to the adjoint hypermultiplet and correctly reproduces the $\\mN=2^\\star$ spectrum without modifying the gauge group. More generally, we present a correspondence between a wide class of freely-acting orbifolds of $\\mathcal{N}=4$ compactifications and the $\\mN=2^\\star$ gauge theory. We elaborate on the universality of this construction and provide evidence supporting the correspondence. In particular, we compute topological amplitudes for these $\\mN=2^\\star$ theories, for symmetric and asymmetric freely acting orbifolds, and confirm that they correctly reduce, in the field theory limit, to the $\\Omega$-deformed partition function \\cite{Nekrasov:2003rj,Billo:2013fi,Pestun:2007rz} of the $\\mN=2^\\star$ gauge theory. This class of BPS amplitudes, which has proven useful in the study of string dualities, is also analysed in type I theory. The latter turns out to be a natural framework for incorporating gauge theory instanton corrections.\n\nThe paper is structured as follows. In Section \\ref{AsymOrb}, we begin by explicitly constructing a worldsheet realisation of $\\mN=2^\\star$ as an asymmetric freely-acting orbifold in heterotic string theory, analyse its string partition function and show the accordance with the spectrum of the $\\mN=2^\\star$ gauge theory. We further confirm our result by calculating the topological couplings $F_g$ at the one-loop level in string perturbation theory and show that they correctly reproduce the perturbative part of the mass-deformed Nekrasov partition function. In Section \\ref {StringModel}, we formulate a general correspondence between certain classes of freely-acting orbifolds in string theory and the $\\mN=2^\\star$ gauge theory, and illustrate our construction in terms of an explicit symmetric orbifold model with a type IIA dual. Moreover, in Section \\ref{TypeOne}, we realise the mass deformation in type I orientifolds and derive the Nekrasov partition function in a specific model. Finally, Section \\ref{Conclusion} summarises our conclusions. \n\n\n\n\\section{\\texorpdfstring{$\\mN=2^\\star$ realisation in string theory}{N=2 star realisation in string theory}}\\label{AsymOrb}\n\n\\subsection{An asymmetric orbifold model}\n\nIn order to realise the $\\mathcal{N}=2^\\star$ theory in the heterotic string, we begin with the ${\\rm E}_8\\times {\\rm E}_8$ heterotic model\\footnote{A similar asymmetric construction can also be performed for the ${\\rm SO}(32)$ heterotic string.} on $T^6$ and identify a suitable orbifold action that gives mass to part of the $\\mathcal{N}=4$ vector multiplet and to two gravitini. As shown below, the resulting string theory has eight unbroken supercharges and a massive adjoint hypermultiplet. In order for the orbifold not to project out non-invariant states but rather give them non-trivial masses, the orbifold in question must act freely on the $T^6$ coordinates, {\\it i.e.} without fixed points. The simplest possibility is to consider a $\\mathbb Z_2$ orbifold which couples rotations on two of the complexified coordinates parametrising the $T^6$ with a translation along the third complexified coordinate. Hence, we obtain a freely-acting variation of the standard orbifold realisation of the K3 surface as $T^4\/\\mathbb Z_2$. \n\nTo identify the orbifold action, we start by constructing the vertex operators of the fields in the $\\mathcal{N}=4$ vector multiplet. For simplicity, we work in the light-cone gauge. Let us denote by $Z^1$, $Z^2$, $Z^3$ the complexified coordinates on $T^6$ and by $\\Psi^1$, $\\Psi^2$, $\\Psi^3$ their fermionic worldsheet superpartners. The latter can be bosonised in terms of three chiral bosons $\\Phi^j(z)$ as\n\\begin{equation}\n\t\\Psi^j(z) = e^{i\\sqrt{2}\\Phi^j} \\ ,\\qquad \\bar\\Psi^j(z) = e^{-i\\sqrt{2}\\Phi^j} \\ ,\\qquad j=1,2,3 \\ ,\n\\end{equation}\nwhere we have conventionally set $\\alpha'=1$. In order for the resulting theory to enjoy an unbroken $\\mathcal{N}=2$ supersymmetry, the $\\mathbb Z_2$ orbifold must rotate the fermionic coordinates in the planes $j=2,3$ with opposite angles $\\pi$, $-\\pi$\n\\begin{equation}\n\t\\begin{split}\n\t&\\Psi^2 \\to e^{i\\pi} \\Psi^2 \\ ,\\qquad ~\\, \\Phi^2 \\to \\Phi^2 +\\tfrac{\\pi}{\\sqrt{2}} \\,, \\\\\n\t&\\Psi^3 \\to e^{-i\\pi} \\Psi^3 \\ , \\qquad \\Phi^3 \\to \\Phi^3 - \\tfrac{\\pi}{\\sqrt{2}} \\,.\n\t\\end{split}\n\\label{orbFerm}\n\\end{equation}\nThe presence of the $\\sqrt{2}$ factors in the above expressions is consistent with the fact that bosonisation of the worldsheet fermions occurs at the fermionic radius $r=1\/\\sqrt{2}$. Note also that the above action on the fermions induces the breaking of the ${\\rm SO}(6)$ R-symmetry group of $\\mathcal{N}=4$ down to ${\\rm SO}(2)\\times {\\rm SO}(4)$.\n\nThe worldsheet fermions $\\psi^\\mu$ of the ten-dimensional theory transform under the ${\\rm SO}(8)$ little group. Upon toroidally compactifying down to four dimensions, one may decompose ${\\rm SO}(8)\\to {\\rm SO}(4)\\times {\\rm SO}(4)$, where the first ${\\rm SO}(4)$ factor corresponds to the transverse spacetime fermions $\\psi^\\mu$ together with the complex fermion $\\Psi^1$ associated to the internal $T^2$ that is left unrotated by the $\\mathbb Z_2$ orbifold. The second ${\\rm SO}(4)$ factor then corresponds to the complex fermions $\\Psi^2$, $\\Psi^3$ in the $T^4$ directions that are rotated by the orbifold action.\nSimilarly, we define spin fields for the fermions associated to the two ${\\rm SO}(4)$ factors via\n\\begin{equation}\n\t\\begin{split}\n\tS_{\\alpha} &= e^{\\pm\\frac{i}{\\sqrt{2}}( \\Phi^0+\\Phi^1)}\\ , \\,\\qquad S_{\\dot\\alpha} = e^{\\pm\\frac{i}{\\sqrt{2}}( \\Phi^0-\\Phi^1)} \\,,\\\\\n\t\\Sigma_{A} &= e^{\\pm\\frac{i}{\\sqrt{2}}( \\Phi^2+\\Phi^3)}\\ ,\\qquad \\Sigma_{\\dot A} = e^{\\pm\\frac{i}{\\sqrt{2}}( \\Phi^2-\\Phi^3)} \\,.\n\t\\end{split}\n\\end{equation}\nThe indices $\\alpha, \\dot\\alpha$ correspond to the (Weyl) spinor and conjugate spinor representations of the first ${\\rm SO}(4)$ factor and, similarly, $A,\\dot A$ label the two Weyl spinor representations associated to the second ${\\rm SO}(4)$ factor.\n\nIn the above notation, the $\\mathcal{N}=4$ vector multiplet vertex operators may be written explicitly as\n\\begin{equation}\n\t\\begin{split}\n\t\t{\\rm a vector\\,boson} \\quad &: \\quad \\psi^\\mu(z) \\, \\bar J(\\bar z) \\,, \\\\\n\t\t{\\rm gaugini} \\quad &: \\quad S_{\\dot\\alpha}\\,\\Sigma_A(z)\\, \\bar J(\\bar z) \\ ,\\quad S_\\alpha \\, \\Sigma_{\\dot A}(z) \\, \\bar J(\\bar z) \\, ,\\\\\n\t\t{\\rm scalars} \\quad &: \\quad e^{\\pm i\\sqrt{2}\\Phi^j(z)} \\, \\bar J(\\bar z)\\,, \\qquad\\ j=1,2,3 \\,,\n\t\\end{split}\n\\end{equation}\nwhere $\\bar J(\\bar z)$ is a right-moving current in the Kac Moody algebra of the gauge group. It is now straightforward to see that under the $\\mathbb Z_2$ action \\eqref{orbFerm} on the fermions, the left-moving operators \n\\begin{equation}\n\tS_\\alpha\\,\\Sigma_{\\dot A}(z) \\ ,\\qquad e^{\\pm i\\sqrt{2}\\Phi^2} \\ ,\\qquad e^{\\pm i\\sqrt{2} \\Phi^3} \\,,\n\\end{equation}\ntransform with a minus sign, which implies the decomposition of the $\\mathcal{N}=4$ vector multiplet into an $\\mathcal{N}=2$ vector multiplet:\n\\begin{equation}\n\t\\begin{split}\n\t\t{\\rm vector\\,boson} \\quad &: \\quad \\psi^\\mu(z) \\, \\bar J(\\bar z) \\,, \\\\\n\t\t{\\rm gaugini} \\quad &: \\quad S_{\\dot\\alpha}\\,\\Sigma_A(z)\\, \\bar J(\\bar z) \\, ,\\\\\n\t\t{\\rm scalars} \\quad &: \\quad e^{\\pm i\\sqrt{2} \\Phi^1(z)} \\, \\bar J(\\bar z) \\,,\n\t\\end{split}\n\\end{equation}\nand an $\\mathcal{N}=2$ adjoint hypermultiplet \n\\begin{equation}\n\t\\begin{split}\n\t\t{\\rm fermion} \\quad &: \\quad S_\\alpha \\, \\Sigma_{\\dot A}(z) \\, \\bar J(\\bar z) \\, ,\\\\\n\t\t{\\rm scalars} \\quad &: \\quad e^{\\pm i\\sqrt{2}\\Phi^j(z)} \\, \\bar J(\\bar z) \\ , \\qquad j=2,3 \\,.\n\t\\end{split}\n\\end{equation}\nIn order to preserve an unbroken local $\\mathcal{N}=(1,0)$ worldsheet supersymmetry, the action \\eqref{orbFerm} of the $\\mathbb Z_2$ orbifold on the fermions needs to be supplemented with the corresponding $\\mathbb Z_2$ action on the left-moving worldsheet bosons\n\\begin{equation}\n\tZ_L^2 \\to e^{i\\pi}\\, Z_L^2 \\ ,\\qquad Z_L^3\\to e^{-i\\pi}\\,Z_L^3 \\,.\n\\label{orbBos}\n\\end{equation}\nIf the $\\mathbb Z_2$ rotation is consistently coupled to a translation along some direction in the first plane $Z^1$, states that are charged under the action of the rotation are no longer projected out in the orbifolded theory, but rather acquire a Scherk-Schwarz mass proportional to their charge.\n\n It is then straightforward to see that in order to render the $\\mathcal{N}=2$ hypermultiplet massive via the Scherk-Schwarz mechanism without affecting the gauge group, the orbifold action on the right moving currents $\\bar J$ must be trivial. In particular, this requires that the right moving $T^6$ coordinates $Z^i_R(\\bar z)$ remain unrotated and the $\\mathbb Z_2$ orbifold is identified as an asymmetric, freely acting orbifold generated by\n \\begin{equation}\n\t\tg = e^{i\\pi Q_L}\\,\\delta \\,.\n \\end{equation}\nHere, $\\exp(i\\pi Q_L)$ is the operator rotating the worldsheet super-coordinates in the $j=2,3$ complex planes by opposite angles $\\pm\\pi$ as in eqs. \\eqref{orbFerm},\\eqref{orbBos}, and $\\delta$ is an order two (momentum) shift along the first cycle of the $j=1$ complex plane\n\\begin{equation}\n\t\\delta\\ : \\ Z^1\\to Z^1+i\\pi \\sqrt{\\tfrac{T_2}{U_2}} \\,,\n\\label{shiftdelta}\n\\end{equation}\nwhere $T,U$ are the K\\\"ahler and complex structure moduli of the $T^2$ parametrised by $Z^1=X^2+iX^1$. In the special case of a rectangular 2-torus without $B$ field, corresponding to the factorisation limit $T^2=S^1\\times S^1$, the above shift is nothing but a translation by half the circumference of the first circle $X^1$. \n\n\\subsubsection{Spectrum of the theory}\n\nThe one loop partition function of the theory reads\n\\begin{equation}\n\tZ=\\frac{1}{2^2\\,\\eta^{12}\\,\\bar\\eta^{24}}\\sum_{\\genfrac{}{}{0pt}{}{a,b=0,1}{H,G=0,1}}(-1)^{a+b+ab+HG}\\,\\vartheta\\bigr[^{a}_{b}\\bigr]^2 \\vartheta\\bigr[^{a+H}_{b+G}\\bigr]\\vartheta\\bigr[^{a-H}_{b-G}\\bigr]\\,\\Gamma_{4,4}\\bigr[^H_G\\bigr]\\,\\Gamma_{2,2}\\bigr[^H_G\\bigr]\\,\\Gamma_{{\\rm E}_8}^2(\\bar\\tau) \\,,\n\\label{partFunc}\n\\end{equation}\nwhere $a,b$ are summed over the spin structures on the worldsheet torus, $H$ labels the orbifold sectors and the sum over $G$ imposes the $\\mathbb Z_2$ invariance projection. $\\Gamma_{4,4}\\bigr[^H_G\\bigr]$ is the asymmetrically twisted lattice partition function of the $j=2,3$ planes, $\\Gamma_{2,2}\\bigr[^H_G\\bigr](T,U)$ is the shifted Narain lattice associated to the $j=1$ plane and $\\Gamma_{\\rm E_8}(\\bar\\tau)$ is the anti-chiral $\\rm E_8$ lattice partition function. $\\vartheta\\bigr[^a_b\\bigr](\\tau)$ stands for the one-loop Jacobi theta constant with characteristics and $\\eta(\\tau)$ is the Dedekind eta function.\n\nDue to the asymmetric nature of the orbifold rotation, the K3 lattice is taken to lie at the fermionic factorised point, where the $T^4$ parametrised by $Z^2$, $Z^3$ can be viewed as the product of four circles with all radii equal to $r=1\/\\sqrt{2}$ and its moduli are stabilised at the minimum of the tree level scalar potential generated by the Scherk-Schwarz gauging of $\\mathcal{N}=4$ supergravity:\n\\begin{equation}\n\tV(\\chi_\\alpha)\\sim \\frac{1}{S_2 T_2 U_2}\\sum_{\\alpha} \\frac{|1+\\chi_{\\alpha}^2|^2}{ {\\rm Im}(\\chi_{\\alpha})} \\,,\n\\end{equation}\nwhere $S_2$ is the imaginary part of the heterotic dilaton and $\\chi_\\alpha$ are the moduli of $T^4$.\n\n As a result, its partition function can be entirely expressed in terms of one-loop Jacobi theta constants\n\\begin{equation}\n\t\\Gamma_{4,4}\\bigr[^H_G\\bigr] = \\frac{1}{2}\\sum_{\\gamma,\\delta=0,1}\\vartheta\\bigr[^\\gamma_\\delta\\bigr]^2\\,\\vartheta\\bigr[^{\\gamma+H}_{\\delta+G}\\bigr]\\,\\vartheta\\bigr[^{\\gamma-H}_{\\delta-G}\\bigr]\\,\\bar\\vartheta\\bigr[^\\gamma_\\delta\\bigr]^4 \\,.\n\\end{equation}\nOn the other hand, the $T,U$ moduli of the $T^2$ parametrised by $Z^1$ remain massless and correspond to the no-scale moduli of the partial supersymmetry breaking. The contribution of the shifted Narain lattice reads\n\\begin{equation}\n\t\\Gamma_{2,2}\\bigr[^H_G\\bigr](T,U) = \\tau_2 \\sum_{m,n} (-1)^{m_1 G}\\, q^{\\frac{1}{4}|P_L|^2}\\,\\bar q^{\\frac{1}{4}|P_R|^2} \\,,\n\\end{equation}\nwith $P_L$, $P_R$ being the complexified lattice momenta\n\\begin{equation}\n\tP_L = \\frac{m_2-\\bar U m_1 + \\bar T (n^1+\\frac{H}{2}+\\bar U n^2)}{\\sqrt{T_2 U_2}} \\ ,\\qquad P_R = \\frac{m_2-\\bar U m_1 + T (n^1+\\frac{H}{2}+\\bar U n^2)}{\\sqrt{T_2 U_2}} \\,.\n\\end{equation}\nIn particular, $P_L$ is identified as the central charge of the $\\mathcal{N}=2$ superalgebra.\n\nThe one-loop partition function of the theory is consistent with unitarity and modular invariance at all genera, as guaranteed by the theorems of fermionic \\cite{Antoniadis:1986rn} and asymmetric orbifold \\cite{Narain:1986qm,Narain:1990mw} constructions. It is convenient to rewrite \\eqref{partFunc} in terms of ${\\rm SO}(2n)$ characters\n\\begin{equation}\n\t\\begin{split}\n\t& O_{2n} = \\frac{1}{2\\eta^n}(\\vartheta_3^n+\\vartheta_4^n) \\ , \\qquad \\quad \\ V_{2n} = \\frac{1}{2\\eta^n}(\\vartheta_3^n-\\vartheta_4^n) \\,, \\\\\n\t& S_{2n} = \\frac{1}{2\\eta^n}(\\vartheta_2^n+i^{-n}\\vartheta_1^n) \\ , \\qquad C_{2n} = \\frac{1}{2\\eta^n}(\\vartheta_2^n-i^{-n}\\vartheta_1^n) \\,,\n\t\\end{split}\n\\end{equation}\nfrom which the spectrum may be easily read. The expansion explicitly reads\n\\begin{equation}\n\t\\begin{split}\n\t\\eta^2\\bar\\eta^2\\,Z =&(V_4 O_4-S_4 S_4)\\bigr(\\Gamma_{4,4}\\bigr[^{\\,0}_+\\bigr]\\Gamma_{2,2}\\bigr[^{\\,0}_+\\bigr]+\\Gamma_{4,4}\\bigr[^{\\,0}_-\\bigr]\\Gamma_{2,2}\\bigr[^{\\,0}_-\\bigr]\\bigr)\\,(\\bar O_{16}+\\bar C_{16})^2 \\\\\n\t+&(O_4 V_4-C_4 C_4)\\bigr(\\Gamma_{4,4}\\bigr[^{\\,0}_-\\bigr]\\Gamma_{2,2}\\bigr[^{\\,0}_+\\bigr]+\\Gamma_{4,4}\\bigr[^{\\,0}_+\\bigr]\\Gamma_{2,2}\\bigr[^{\\,0}_-\\bigr]\\bigr)\\,(\\bar O_{16}+\\bar C_{16})^2 \\\\\n\t+&(V_4 C_4-S_4 V_4)\\bigr(\\Gamma_{4,4}\\bigr[^{\\,1}_+\\bigr]\\Gamma_{2,2}\\bigr[^{\\,1}_+\\bigr]+\\Gamma_{4,4}\\bigr[^{\\,1}_-\\bigr]\\Gamma_{2,2}\\bigr[^{\\,1}_-\\bigr]\\bigr)\\,(\\bar O_{16}+\\bar C_{16})^2 \\\\\n\t+&(O_4 S_4-C_4 O_4)\\bigr(\\Gamma_{4,4}\\bigr[^{\\,1}_+\\bigr]\\Gamma_{2,2}\\bigr[^{\\,1}_-\\bigr]+\\Gamma_{4,4}\\bigr[^{\\,1}_-\\bigr]\\Gamma_{2,2}\\bigr[^{\\,1}_+\\bigr]\\bigr)\\,(\\bar O_{16}+\\bar C_{16})^2 \\,,\n\t\\end{split}\n\\label{partFuncChar}\n\\end{equation}\nwhere we decomposed the lattices into irreducible representations\n\\begin{equation}\n\t\\Gamma_{d,d}\\bigr[^H_{\\pm}\\bigr]=\\frac{1}{2\\eta^d\\bar\\eta^d}\\,\\bigr( \\Gamma_{d,d}\\bigr[^H_{\\,0}\\bigr]\\pm \\Gamma_{d,d}\\bigr[^H_{\\,1}\\bigr] \\bigr) \\,,\n\\end{equation}\nso that the $\\pm$ sign indicates the eigenvalue of the object with respect to the $\\mathbb Z_2$ action, and we included also the associated oscillator contributions encoded in the Dedekind functions. \n\nIt is clear that the shifted lattice with $H=1$ carries non-trivial winding charge and the corresponding states decouple in the low energy limit. We hence focus our attention on the untwisted sector $H=0$. In this sector, the $\\mathbb Z_2$ invariant shifted lattice $\\Gamma_{2,2}\\bigr[^{\\,0}_{+}\\bigr]$ projects to the $m_1=0\\,{\\rm mod}\\, 2$ subsector and gives rise to massless states with vanishing momentum and winding numbers. On the other hand, the $\\mathbb Z_2$ charged lattice $\\Gamma_{2,2}\\bigr[^{\\,0}_{-}\\bigr]$ instead projects to the $m_1=1\\,{\\rm mod}\\,2$ subsector and is responsible for attributing Scherk-Schwarz masses to charged states.\n\nThe twisted lattice contribution to the conformal weights can be read off from its explicit expansion around the cusp $\\tau=\\infty$:\n\\begin{equation}\n\t\\begin{split}\n\t&\\Gamma_{4,4}\\bigr[^{\\,0}_{+}\\bigr] = (q\\bar q)^{1\/6}\\left(1 +12\\,q+28\\, \\bar q+96\\, q^{1\/2}\\,\\bar q^{1\/2}+\\ldots\\right) \\,, \\\\\n\t&\\Gamma_{4,4}\\bigr[^{\\,0}_{-}\\bigr] = (q\\bar q)^{1\/6}\\left(96\\,q^{1\/2}\\,\\bar q^{1\/2}+16\\,q+\\ldots\\right) \\,.\n\t\\end{split}\n\\end{equation}\nNotice, in particular, that the non-invariant combination $\\Gamma_{4,4}\\bigr[^{\\,0}_{-}\\bigr]$ always contributes at least $1\/2$ unit of left-moving conformal weight, hence giving rise to states with string scale masses, which can be similarly ignored in a low energy analysis of the spectrum.\n\nThe $\\mathcal{N}=2$ vector multiplets arise from the $V_4 O_4-S_4 S_4$ part in eq. \\eqref{partFuncChar} and the matter gauge group of the theory at a generic point in the $T,U$ moduli space is identified as ${\\rm G}={\\rm E}_8\\times {\\rm E}_8\\times {\\rm SO}(8)\\times {\\rm U}(1)^2$. The $\\rm U(1)$ factors arise from the dimensional reduction on the unrotated $T^2$ parametrised by $Z^1$ and correspond to the currents $\\bar J(\\bar z) = i\\bar\\partial X^1$ or $i\\bar\\partial X^2$. On the other hand, the ${\\rm SO}(8)$ factor is a consequence of the enhancement ${\\rm U}(1)^4\\to {\\rm SO}(8)$, due to the compactification of the $T^4$ directions $Z^2=X^4+iX^3$, $Z^3=X^6+iX^5$ at the fermionic radii. The ${\\rm SO}(8)$ Kac Moody algebra is then realised by the currents\n\\begin{equation}\n\t\\bar J(\\bar z) = \\{ \\ i\\bar\\partial X^a \\ , \\quad e^{\\pm i\\sqrt{2}X_R^b(\\bar z) \\pm i\\sqrt{2} X_R^c(\\bar z)} \\ \\} \\ ,\\quad b\\neq c \\,,\n\\end{equation}\nwith $a,b,c=1,\\ldots, 4$.\n\nThe hypermultiplets arise from the $O_4 V_4-C_4 C_4$ part of eq. \\eqref{partFuncChar} and are massive, by construction. This can be seen by noticing that these states carry odd momentum charge along the direction of the shift.\n\nThe low lying states of interest, which become massless in the limit where $\\mathcal{N}=4$ supersymmetry is recovered, are given by $m_1=\\pm 1$ and $n^1=0$. The first possibility is to take $m_2=n^2=0$, in which case the corresponding vertex operators are\n\\begin{equation}\n\t\\left.\\begin{array}{l}\n\tS_\\alpha \\,\\Sigma_{\\dot A} (z) \\,\\bar J(\\bar z)\\\\ \n\te^{\\pm i\\sqrt{2}\\Phi^j(z)}\\,\\bar J(\\bar z)\\\\\n\t\\end{array}\\right\\} \\times \\exp\\left[\\pm i\\,\\frac{UZ^1(z,\\bar z)+\\bar U \\bar Z^1(z,\\bar z)}{2\\sqrt{T_2 U_2}}\\right] \\,.\n\\label{HyperE8}\n\\end{equation}\nNote that they are invariant under the orbifold action, since the shift in $Z^1$ eq. \\eqref{shiftdelta} precisely cancels the the minus sign from the left-movers.\nTheir physical mass is given by\n\\begin{equation}\n\tm_{\\rm hyp,1}^2 = \\frac{|U|^2}{T_2 U_2} \\,,\n\\end{equation}\nand is further identified as the scale $m_{3\/2}^2$ of the $\\mathcal{N}=4\\to \\mathcal{N}=2$ supersymmetry breaking. The presence of $\\bar J$ in the vertex operators ensures that the hypers indeed transform in the adjoint of the gauge group ${\\rm G}$, as expected. \n\nThe second possibility is to consider $m_2=n^2=\\pm 1$, in which case the corresponding vertex operators are\n\\begin{equation}\n\t\\left.\\begin{array}{l}\n\tS_\\alpha \\,\\Sigma_{\\dot A} (z) \\\\ \n\te^{\\pm i\\sqrt{2}\\Phi^j(z)}\\\\\n\t\\end{array}\\right\\} \\times \\exp\\left[ \\frac{i}{2}\\bigr( P_L \\bar Z_L^1+\\bar P_L Z_L^1+P_R \\bar Z_R^1+\\bar P_R Z_R^1\\bigr)\\right] \\,,\n\t\\label{Hyper2}\n\\end{equation}\nwith the lattice momenta given by\n\\begin{equation}\n\tP_L=\\frac{\\pm(1+\\bar T\\bar U)\\pm \\bar U}{\\sqrt{T_2 U_2}}\\ , \\quad P_R = \\frac{\\pm(1+T\\bar U)\\pm\\bar U}{\\sqrt{T_2 U_2}} \\,.\n\\label{HyperSU2}\n\\end{equation}\nThese states are again massive, with their physical mass being given by\n\\begin{equation}\n\tm^2_{\\rm hyp,2} = \\frac{|1+TU\\pm U|^2}{T_2 U_2} \\,.\n\\end{equation}\nThey correspond to states charged under the adjoint of an $\\rm SU(2)$ gauge group that arises as a result of enhancement, as we discuss below.\n\n\\subsubsection{Enhanced gauge symmetry}\n\nWe next investigate the possibility of enhanced gauge symmetry at special points in the moduli space ${\\rm SO}(2,2)\/{\\rm SO}(2)\\times{\\rm SO}(2)$ parametrised by the $T,U$ moduli of the unrotated $T^2$ torus. Clearly, this can only arise from the sector\n\\begin{equation}\n\t(V_4 O_4-S_4 S_4)\\,\\Gamma_{2,2}\\bigr[^{\\,0}_{+}\\bigr] \\,.\n\\end{equation}\nIn order to create a level matched state, we demand that the lattice momenta satisfy $|P_L|^2-|P_R|^2=-4(m_1 n^1+m_2 n^2)=-4$. Since in this sector $m_1=0\\,{\\rm mod}\\,2$, it is sufficient to choose the state $m_1=n^1=0$ and $m_2=n^2=\\pm 1$, in which case the vertex operators of the extra charged states read\n\\begin{equation}\n\t\\left.\\begin{array}{l}\n\t\\psi^\\mu(z) \\\\\n\tS_{\\dot\\alpha} \\,\\Sigma_{ A} (z)\\\\ \n\te^{\\pm i\\sqrt{2}\\Phi^1(z)}\\\\\n\t\\end{array}\\right\\} \\times \\exp\\left[ \\frac{i}{2}\\bigr( P_L \\bar Z_L^1+\\bar P_L Z_L^1+P_R \\bar Z_R^1+\\bar P_R Z_R^1\\bigr)\\right] \\,,\n\\end{equation}\nwhere\n\\begin{equation}\n\tP_L = \\pm \\frac{1+\\bar T\\bar U}{\\sqrt{T_2 U_2}}\\ , \\qquad P_R = \\pm \\frac{1+T\\bar U}{\\sqrt{T_2 U_2}} \\,.\n\\end{equation}\nand their physical mass is\n\\begin{equation}\n\tm^2_{\\rm vec} = \\frac{|1+TU|^2}{T_2 U_2} \\,.\n\\end{equation}\nIn particular, they become massless at the point $T=-1\/U$. Note that, contrary to the $\\mathcal{N}=2$ string theory obtained by compactifying the heterotic string on ${\\rm K}3\\times T^2$, in the present $\\mathcal{N}=2^\\star$ case, the T-duality group is no longer the full ${\\rm SL}(2;\\mathbb Z)_T \\times {\\rm SL}(2;\\mathbb Z)_U \\ltimes \\mathbb Z_2$ but is actually broken to its subgroup $\\Gamma^0(2)_T\\times \\Gamma_0(2)_U\\ltimes\\mathbb Z_2$. As a result, this point of gauge symmetry enhancement is not equivalent to the $T=U$ one, since $\\binom{0~-1}{1~~0}\\notin\\Gamma_0(2)$. This is an effect originating from the freely acting nature of the $\\mathbb Z_2$ orbifold.\n\nAt the enhancement point $T=-1\/U$, the ${\\rm U}(1)\\times {\\rm U}(1)$ gauge symmetry becomes enhanced to ${\\rm SU}(2)\\times {\\rm U}(1)$. Explicitly, the vertex operators of the massless vector multiplets transforming under the ${\\rm SU}(2)\\times {\\rm U}(1)$ become\n\\begin{equation}\n\t\\left.\\begin{array}{l}\n\t\\psi^\\mu(z) \\\\\n\tS_{\\dot\\alpha} \\,\\Sigma_{ A} (z)\\\\ \n\te^{\\pm i\\sqrt{2}\\Phi^1(z)}\\\\\n\t\\end{array}\\right\\} \\times \\{ \\ \\bar J_0(\\bar z)\\ , \\ \\bar J_{\\pm}(\\bar z)\\ , \\ \\hat{\\bar J}_0(\\bar z) \\ \\} \\,,\n\\end{equation}\nwhere\n\\begin{equation}\n\t\\bar J_0(\\bar z) = i\\, \\frac{U_1\\,\\bar\\partial X^1+U_2\\,\\bar\\partial X^2}{|U|} \\ ,\\qquad \\bar J_\\pm(\\bar z) = \\exp\\left[\\pm 2i\\,\\frac{U_1 X_R^1(\\bar z)+U_2 X_R^2(\\bar z)}{|U|}\\right] \\,,\n\\end{equation}\nrealise the ${\\rm SU}(2)$ Kac Moody algebra at level $k=1$, whereas the remaining ${\\rm U}(1)$ is generated by the orthogonal combination\n\\begin{equation}\n\t\\hat{\\bar J}_0(\\bar z) = i\\, \\frac{U_2\\,\\bar\\partial X^1-U_1\\,\\bar\\partial X^2}{|U|} \\,.\n\\end{equation}\nThis U(1) is the one used in the Scherk-Schwarz mechanism in order to give masses to all hypermultiplets of the theory. \n\nAt the $T=-1\/U$ enhancement point, the hypermultiplets transforming in the adjoint of ${\\rm E}_8\\times{\\rm E}_8\\times {\\rm SO}(8)$ and those transforming in the adjoint of ${\\rm SU}(2)$ given in eqs.\\eqref{HyperE8} and \\eqref{HyperSU2}, respectively, become degenerate in mass, $m^2_{\\rm hyp,1}=m^2_{\\rm hyp,2}=|U|^4\/U_2^2$. This degeneracy can be lifted by turning on non-trivial Wilson lines along the unrotated $T^2$ for the ${\\rm E}_8\\times{\\rm E_8}\\times{\\rm SO}(8)$ gauge group factors, breaking them to ${\\rm U}(1)^{24}$. The latter can be accomplished by adding a marginal deformation of the form $A_i^a\\, \\partial X^i\\bar J^a(\\bar z)$ to the worldsheet Lagrangian. The effect of the Wilson line is to modify the masses of the states so that the above enhancement point is now modified to $TU-W^2\/2=-1$. The hypermultiplet masses are also modified:\n\\begin{equation}\n\tm^2_{\\rm hyp,1}= \\frac{|Q\\cdot W+U|^2}{T_2 U_2-\\frac{1}{2}W_2^2}\\ ,\\quad m^2_{\\rm hyp,2}=\\frac{|1+TU-\\frac{1}{2}W^2\\pm U|^2}{T_2 U_2-\\frac{1}{2}W_2^2} \\,,\n\\end{equation}\nwith $Q^a$ being the associated Cartan charges associated to the roots of ${\\rm E}_8\\times{\\rm E}_8\\times{\\rm SO}(8)$, normalised to length $Q^2=2$, and $W^a=Y^a_2-U Y^a_1$ being the complexified Wilson line. Notice that, indeed, at the $TU-W^2\/2=-1$ enhancement point, the hypermultiplet masses $m^2_{\\rm hyp,1}$ with $Q\\neq 0$ and $m^2_{\\rm hyp,2}$ are no longer degenerate. This hierarchy of masses permits one to focus on the field theory limit of various gauge group factors in ${\\rm G}$ by consistently decoupling the degrees of freedom of the others. For simplicity, we henceforth focus on the ${\\rm SU}(2)$ gauge factor. The case $Q=0$, on the other hand, corresponds to the case where $\\bar J$ in \\eqref{HyperE8} is associated with the Cartan current $\\bar J_0$ of SU(2). At the enhancement point, this state becomes degenerate with the hypermultiplets \\eqref{Hyper2} of mass $m_{\\rm hyp,2}^2$ and completes the adjoint representation of SU(2).\n\n\n\n\\subsection{\\texorpdfstring{Topological amplitudes in $\\mathcal{N}=2^\\star$}{Topological amplitude in N=2*}}\\label{TopoAmp}\n\nA well-defined check of our proposal for the realisation of $\\mN=2^\\star$ in terms of the asymmetric orbifold model discussed in the previous section is the computation of the partition function of the mass deformed gauge theory on $\\mathbb R^4\\times S^1$ twisted by equivariant rotation parameters $\\epsilon_{1,2}$. More precisely, exploit the relation of the Nekrasov partition function of gauge theory to the free energy of topological string theory or, equivalently, to physical string amplitudes involving insertions of anti-self-dual graviphoton field strengths. Indeed, the field theory limit of the genus $g$ partition function $F_g$ of topological string theory on a Calabi-Yau manifold, is related to the Nekrasov partition function $Z_{\\rm Nek}$ via\n\\begin{align}\n\t\\sum_{g\\geq 0}\\epsilon^{2g-2}\\,F_g \\Bigr|_{\\rm F.T.}= \\log\\, Z_{\\rm Nek}(\\epsilon_1=-\\epsilon_2\\equiv \\epsilon)\\,.\n\\end{align}\nOn the other hand, $F_g$ computes physical, genus $g$ string amplitudes of the (untwisted) type II theory, involving two insertions of anti-self-dual Riemann tensors and $2g-2$ insertions of anti-self-dual graviphoton field strengths. In the dual heterotic theory, these couplings start receiving corrections at genus one while being protected against higher perturbative corrections. As a result, in order to extract the perturbative part of the gauge theory partition function corresponding to our asymmetric orbifold model, it is sufficient to compute heterotic topological amplitudes at genus one and expand their field theory limit around a suitable point of enhanced gauge symmetry. Specifically, we focus on the SU(2) enhancement arising at the $TU-W^2\/2=-1$ point.\n\n\n\\subsubsection{Setup and derivation of the amplitude}\n\nThe topological amplitudes of interest calculate higher derivative terms in the string effective action of the form\n\\begin{equation}\n\t\\int d^4\\theta\\, F_g(X)\\,W^{2g} = F_g(\\phi)\\,R^2_{-}\\,F^{2g-2}_{-}+\\ldots\\,.\n\t\\label{ActionTopAmpl}\n\\end{equation}\nHere, $R_{-}$, $F_{-}$ are the anti self-dual Riemann tensor and graviphoton field strength, respectively, which arise as the bosonic components of the Weyl superfield $W$. Furthermore, $X$ collectively denotes the chiral superfields of the theory with scalar component $\\phi$.\n\nIn heterotic string perturbation theory, these amplitudes receive contributions at one loop in $g_{\\rm s}$ and are otherwise protected against higher perturbative corrections. Denoting by $Z^4, Z^5$ the complexified spacetime coordinates and by $\\Psi^4,\\Psi^5$ their fermionic superpartners, one may write the relevant vertex operators in a convenient kinematic configuration as\n\\begin{equation}\n\t\\begin{split}\n\tV_{ R_-}(p_4) = (\\partial Z^5-ip_4\\Psi^4\\Psi^5)\\,\\bar\\partial Z^5\\,e^{ip_4 \\,Z^4}\\,,\\\\\n\tV_{ R_-}(\\bar p_4) = (\\partial \\bar Z^5-i\\bar p_4\\bar\\Psi^4\\bar\\Psi^5)\\,\\bar\\partial \\bar Z^5\\,e^{i\\bar p_4 \\,\\bar Z^4}\\,,\\\\\n\tV_{ F_-}(p_4) = (\\partial Z^1-ip_4\\Psi^4\\Psi^1)\\,\\bar\\partial Z^5\\,e^{ip_4 \\,Z^4}\\,,\\\\\n\tV_{ F_-}(\\bar p_4) = (\\partial Z^1-i\\bar p_4\\bar\\Psi^4\\Psi^1)\\,\\bar\\partial \\bar Z^5\\,e^{i\\bar p_4 \\,\\bar Z^4}\\,.\n\t\\end{split}\n\\end{equation}\nIn calculating the correlator $\\langle R^2_{-}F_{-}^{2g-2}\\rangle$, one notices that the effect of the two Riemann tensor insertions is simply to soak up the $\\Psi^4,\\Psi^5$ zero modes, as reflected in \\eqref{ActionTopAmpl}. In addition, the worldsheet fermions $\\Psi^1$ in the graviphoton vertex operators do not contract and their one loop contribution cancels against that of the superghost. The correlator reads\n\\begin{equation}\n\t(p_4\\bar p_4)^{g-1} \\bigr\\langle \\prod_i^{g-1}\\int dx_i \\,\\partial Z^1 (Z^4\\bar\\partial Z^5)(x_i)\\,\\prod_j^{g-1}\\int dy_j\\,\\partial Z^1 (\\bar Z^4\\bar\\partial\\bar Z^5)(y_j)\\bigr\\rangle\\,.\n\\end{equation}\nThis can be straightforwardly calculated by taking suitable derivatives of the generating function\n\\begin{equation}\n\t\\mathcal F(\\epsilon)=\\sum_{g}\\frac{\\epsilon^{2g}}{(g!)^2}\\,F_g \n\t\t\t\t\t= \\biggr\\langle \\exp\\Bigr[-\\epsilon\\int d^2 z\\,\\partial Z^1(Z^4\\bar\\partial Z^5+\\bar Z^5\\bar\\partial \\bar Z^4) \\Bigr] \\biggr\\rangle \\,,\n\\end{equation}\nwhere $\\epsilon$ can be physically viewed as a vacuum expectation value for the graviphoton field strength.\nSince $Z^1$ contributes only through its zero mode, the path integral over $Z^4,Z^5$ becomes effectively gaussian and is explicitly evaluated to be\n\\begin{equation}\n\tG_{\\rm bos}(\\epsilon)= \\frac{(2\\pi i\\epsilon)^2 \\bar\\eta^6}{\\bar\\vartheta_1(\\tilde\\epsilon)^2}\\,e^{-\\frac{\\pi \\tilde\\epsilon^2}{\\tau_2}} \\,,\n\\end{equation}\nwhere $\\tilde\\epsilon=\\epsilon\\tau_2 P_L\/\\sqrt{T_2 U_2-W_2^2\/2}$.\n\nThis is to be supplemented by the path integral over the internal worldsheet fields. In particular, the contribution of the worldsheet fermion correlators after appropriate summation over the spin structures yields \n\\begin{equation}\n\t\\frac{1}{2}\\sum_{a,b=0,1}(-1)^{a+b} \\bigr(\\langle \\Psi(z)\\bar\\Psi(0)\\rangle[^a_b]\\bigr)^2 \\vartheta[^a_b]^2\\,\\vartheta[^{a+H}_{b+G}]\\,\\vartheta[^{a-H}_{b-G}] = [\\vartheta_1'(0)]^2 \\,\\vartheta[^{1+H}_{1+G}]\\,\\vartheta[^{1-H}_{1-G}]\\,.\n\\end{equation}\nThis expression vanishes for $H=G=0$, reflecting the $\\mathcal{N}=4$ subsector of the theory. The contribution of the twisted worldsheet fermions $\\Psi^2, \\Psi^3$ exactly cancels against that of their twisted bosonic superpartners $Z^2,Z^3$, as can be seen from the identity\n\\begin{equation}\n\t\\Gamma_{4,4}[^H_G] = \\frac{4\\eta^6}{\\vartheta[^{1+H}_{1+G}]^2}\\,\\left(\\tfrac{1}{2}\\sum_{\\gamma,\\delta=0,1}(-1)^{G(\\gamma+H)}\\,\\bar\\vartheta[^\\gamma_\\delta]^4-\\tfrac{(-)^G}{2}\\,\\bar\\vartheta[^{1+H}_{1+G}]^4\\right) \\,,\n\\end{equation}\nvalid for $(H,G)\\neq(0,0)$. Furthermore, the contribution of $[\\vartheta_1'(0)]^2=(2\\pi\\eta^3)^2$, together with the $\\eta^6$ arising from the twisted lattice exactly cancels against the remaining $\\eta^{-12}$ contribution of the left moving oscillators and the amplitude takes the manifestly BPS form\n\\begin{equation}\n\t\\begin{split}\n\t\\mc F(\\epsilon) &= \\int_{\\mc F}\\frac{d^2\\tau}{\\tau_2^2} \\, \\frac{G_{\\rm bos}(\\epsilon)}{4\\bar\\Delta}\\sum_{\\genfrac{}{}{0pt}{}{H,G=0,1}{(H,G)\\neq(0,0)}} \\sum_{\\genfrac{}{}{0pt}{}{\\gamma,\\delta=0,1}{(\\gamma,\\delta)\\neq (1,1)}}(-1)^{G(\\gamma+1)}\\,\\bar\\vartheta[^{\\gamma+H}_{\\delta+G}]^4 \\,\\Gamma_{2,18}[^H_G](T,U,W) \\,,\n\t\\end{split}\n\\end{equation}\nwhere $\\Delta(\\tau)=\\eta^{24}(\\tau)$ is the modular discriminant and $\\mathcal F$ is the fundamental domain of $\\rm{SL}(2;\\mathbb Z)$.\n\n\n\\subsubsection{Field theory limit}\n\nWe can now extract the field theory limit of the amplitude expanded around the ${\\rm SU}(2)$ enhancement point $TU-W^2\/2=-1$. Since states in the $H=1$ sector are characterised by non-trivial winding along the $X^1$ direction, the only relevant contribution stems from the sector $H=0, G=1$:\n\\begin{equation}\n\t\\mc F(\\epsilon)\\bigr|_{\\rm F.T.} =\\tfrac{1}{2}\\pi^2 \\epsilon^2\\sum_{m, n\\atop Q=0}\\int\\frac{dt}{t} \\,\\frac{(-1)^{m_1}}{\\sin^2 \\epsilon\\,t}\\,e^{-t \\sqrt{T_2 U_2-\\frac{1}{2}W_2^2}\\bar P_L} \\,.\n\\end{equation}\nAround the enhancement point, the amplitude exhibits a singularity due to the extra massless vectormultiplet states and includes the effect of the hypermultiplets charged under $\\rm SU(2)$. For these contributions, the field theory limit of the amplitude takes the form\n\\begin{equation}\n\t\\frac{1}{(2\\pi\\epsilon)^2}\\,\\mc F(\\epsilon)\\bigr|_{\\rm F.T.} = \\sum_{k=0,\\pm 1}\\left[\\gamma_{\\hbar}(k\\mu)-\\gamma_{\\hbar}(k\\mu+ m_{\\rm h}) \\right] \\,,\n\\label{FieldTheorLimitNstar}\n\\end{equation}\nwhere $\\gamma_\\epsilon(a)=\\log \\Gamma_2(a|\\epsilon,-\\epsilon)$ is given in terms of Barnes' double gamma function, and we have set $\\hbar= 2\\pi i \\epsilon$. In addition, $\\mu=1+ T U- W^2\/2$ and $m_{\\rm h}= U$ are the BPS mass parameters associated to the vector- and hyper- multiplets, respectively. As anticipated, our amplitude exactly reproduces the $\\Omega$-deformed partition function of the $\\mathcal{N}=2^\\star$ gauge theory \\cite{Nekrasov:2002qd,Pestun:2007rz}.\n\n\nThe topological amplitudes $F_g$ evaluated at the one loop level in the heterotic string coupling, may be used to extract information about the dual type IIA compactification. More precisely, from the work of \\cite{Bershadsky:1993cx,Bershadsky:1993ta}, these objects encode interesting topological information about the dual Calabi-Yau since they correspond, in the standard case, to the genus $g$ topological string free energy. The heterotic dilaton $S$ lies in a vector multiplet, which implies that the vector multiplet moduli space receives perturbative and non-perturbative corrections, whereas the hypermultiplet one is exact. In type IIA constructions based on Calabi-Yau manifolds, the dilaton lies in a hypermultiplet and the vector moduli space receives no corrections. A peculiarity of the realisation of $\\mN=2^\\star$ in terms of the freely acting asymmetric orbifold of Section \\ref{AsymOrb} is that, since all hypermultiplets have acquired Scherk-Schwarz masses and the corresponding moduli are stabilised at points $T_i,U_i\\sim 1$, the would-be dual IIA dilaton on a Calabi-Yau is massive and stabilised at $g_{\\rm II}\\sim 1$. In other words, a perturbative type IIA description, dual to our asymmetric heterotic construction, does not exist. Nevertheless, the heterotic amplitudes $F_g$ are expected to probe topological information of a type II dual theory obtained as a Scherk-Schwarz reduction of M-theory.\n\n\n\n\\section{Geometrically engineering \\texorpdfstring{$\\mathcal{N}=2^\\star$ in String Theory}{N=2* realisation in String Theory}}\\label{StringModel}\n\nWe have seen that the heterotic asymmetric orbifold realisation of the previous section does not admit a perturbative type II dual. It is, however, possible to generalise our construction in order to geometrically engineer $\\mN=2^\\star$ for more general orbifolds, including symmetric ones for which explicit dual type II descriptions are known.\n\n\n\n\\subsection{The general correspondence}\n\nIn order to construct a string theory uplift of the mass deformation in gauge theory, it is necessary to first elaborate on the properties of the latter since these are crucial in establishing our main correspondence. Indeed, as mentioned above, the mass deformation is responsible for the breaking of the $\\mathcal{N}=4$ supersymmetry to $\\mathcal{N}=2$ in such a way that the $\\mathcal{N}=2$ hypermultiplet inside the $\\mathcal{N}=4$ vectormultiplet acquires a mass. Clearly, the resulting theory -- the $\\mathcal{N}=2^\\star$ gauge theory -- interpolates between the $\\mathcal{N}=4$ and the $\\mathcal{N}=2$ theories, in the zero and infinite mass limits, respectively.\n\nThe D-brane description turns out to be very useful in order to establish the correspondence. More precisely, the pure $\\mathcal{N}=4$ gauge theory can be realised as the theory of massless excitations of a stack of parallel D3-branes in which the degrees of freedom of the open strings stretching between the branes lead to the $\\mathcal{N}=4$ vector multiplet. In this picture, the mass deformation correponds to tilting the branes in some internal directions, in such a way that the mass be proportional to the angle. Recalling that branes at angles can alternatively be described by magnetic fluxes in the original background, one may, by analogy, expect that the mass deformation of gauge theory can also be described in string theory by a freely acting orbifold of the original $\\mathcal{N}=4$ compactification. It is very striking that this picture is quite generic and is due to the universality of the low-energy behaviour of a wide class of string models. Therefore, we propose a general correspondence for which we extensively argue below: \\emph{freely acting orbifolds of $\\mathcal{N}=4$ compactifications realise $\\mathcal{N}=2^\\star$ in string theory.} More general constructions, such as flux compactifications corresponding to gaugings of $\\mathcal{N}=4$ supergravity are also expected to provide candidates for realising $\\mN=2^\\star$ in string theory. However, they do not generically admit an exact worldsheet description.\n\nWe insist that, as explained below, the details of the specific construction of the freely-acting model do not impinge on the correspondence since the $\\mathcal{N}=2^\\star$ features turn out ot be quite generic in the string moduli space, provided that its action preserves eight supercharges. This is what we refer to as \\emph{universality}. In addition, the original $\\mathcal{N}=4$ theory is recovered in a standard fashion as a decompactification limit of the Scherk-Schwarz cycle.\n\nIn the present work, we mainly focus on compact internal spaces, in which case the gauge groups that can be described in the field theory are only subgroups of the stringy-allowed ones. Equivalently, anomaly cancellation restricts the number of allowed D-branes and, therefore, the maximum rank of the gauge group. However, one could consider non-compact internal spaces like $\\mathbb C^2\/\\mathbb Z_k$ for arbitrary $k$ such that arbitrary ranks are permitted. Additionally, this leads to a natural way of creating a hierarchy of masses by taking the large $k$ limit while keeping the product of the Scherk-Schwarz radius $R$ with $k$ fixed, since the adjoint mass is then proportional to $1\/kR$.\n\nIn what follows, we elaborate on the correspondence by analysing the properties of a broad class of freely-acting orbifolds. In particular, we show that the resulting spectrum is the $\\mathcal{N}=2^\\star$ one.\n\n\\subsection{The class of models and their spectrum}\\label{correspSubsec}\n\nWithout loss of generality, we focus on $\\mathbb Z_2$ freely-acting orbifolds of $\\mathcal{N}=4$ heterotic compactifications for which we specify only the pertinent part of the orbifold action in order to keep the discussion generic, while deferring the discussion of specific models to subsequent sections. Starting from a $T^6$ compactification, the action of the orbifold should at least rotate the left-movers of two of the complex toroidal coordinates while shifting the third one, in such a way that the action does not have any fixed point. The partition function of this theory is\n\n\\begin{equation}\n\tZ=\\frac{1}{2^2\\,\\eta^{12}\\,\\bar\\eta^{24}}\\sum_{\\genfrac{}{}{0pt}{}{a,b=0,1}{H,G=0,1}}(-1)^{a+b+ab}\\,\\vartheta\\bigr[^{a}_{b}\\bigr]^2 \\vartheta\\bigr[^{a+H}_{b+G}\\bigr]\\vartheta\\bigr[^{a-H}_{b-G}\\bigr]\\,\\Gamma_{4,4}\\bigr[^H_G\\bigr] \\,\\Gamma_{2,18}\\bigr[^H_G\\bigr]\\,,\n\\label{PartFnGen}\n\\end{equation}\nwhere $a,b$ are summed over the spin structures on the worldsheet torus, $H$ labels the orbifold sectors and the sum over $G$ imposes the $\\mathbb Z_2$ invariant projection. $\\Gamma_{4,4}\\bigr[^H_G\\bigr]$ is the twisted lattice partition function and $\\Gamma_{2,18}\\bigr[^H_G\\bigr](T,U)$ is the shifted Narain lattice associated to the $T^2$ together with the gauge lattice directions. For convenience, we join the latter into a single lattice partition function $Z_{6,22}\\bigr[^H_G\\bigr]$. We further absorb potential phase factors depending on the specific model and ensuring modular invariance into the definition of $\\Gamma_{2,18}$. It is clear that the orbifold action on the left movers is identical as that of the asymmetric model of Section \\ref{AsymOrb} and preserves eight supercharges. Contrary to the asymmetric model, however, we now allow for a non-trivial action on the right-movers as well, incorporating also the case of symmetric actions on the $T^4$ coordinates.\n\nThe spectrum of the theory can be organised into contributions of the gravity, vector and hyper multiplets by expressing the partition function \\eqref{PartFnGen} in terms of SO$(2n)$ characters:\n\\begin{equation}\n\t\\begin{split}\n\t\\eta^4\\bar\\eta^4\\,Z =&(V_4 O_4-S_4 S_4)Z_{6,22}\\bigr[^{\\,0}_+\\bigr]+(O_4 V_4-C_4 C_4)Z_{6,22}\\bigr[^{\\,0}_-\\bigr]\\\\\n\t+&(V_4 C_4-S_4 V_4)Z_{6,22}\\bigr[^{\\,1}_\\mp\\bigr]+(O_4 S_4-C_4 O_4)Z_{6,22}\\bigr[^{\\,1}_\\pm\\bigr]\\,.\n\t\\end{split}\n\\label{PartFnGenChar}\n\\end{equation}\nIn the twisted sector $H=1$, the parity under the orbifold action of the lattice block $Z_{6,22}$ depends on whether the orbifold is asymmetric or symmetric. In addition, it can be straightforwardly expanded in terms of the toroidal and gauge lattices. For instance,\n\\begin{equation}\n Z_{6,22}\\bigr[^{H}_+\\bigr]=\\Gamma_{4,4}\\bigr[^{H}_+\\bigr]\\Gamma_{2,18}\\bigr[^{H}_+\\bigr]+\\Gamma_{4,4}\\bigr[^{H}_-\\bigr]\\Gamma_{2,18}\\bigr[^{H}_-\\bigr]\\,,\\textrm{ etc.}\n\\end{equation}\nAs in the asymmetric orbifold case, the $H=1$ sector involves states with non-trivial winding and generically decouple in the low energy limit, provided the characteristic radius of the Scherk-Schwarz cycle is much larger than the string scale, $T_2\/U_2\\gg 1$. We hence focus our attention on the untwisted sector $H=0$.\n\nClearly, the states of interest around the SU(2) enhancement point $TU-W^2\/2=-1$ are neutral with respect to the gauge bundle, $Q=0$, and arise only from the sectors:\n\\begin{equation}\n\t\\begin{split}\n\t\t\t\t& (V_4 O_4-S_4 S_4)\\,\\Gamma_{4,4}\\bigr[^{\\,0}_+\\bigr]\\Gamma_{2,18}\\bigr[^{\\,0}_+\\bigr] \\\\\n\t\t\t\t& (O_4 V_4-C_4 C_4) \\,\\Gamma_{4,4}\\bigr[^{\\,0}_+\\bigr]\\Gamma_{2,18}\\bigr[^{\\,0}_-\\bigr] \\,. \n\t\\end{split}\n\\end{equation}\nIndeed, one obtains the vector multiplet associated to the SU(2) gauge bosons, and the associated massive adjoint hypermultiplet. The quantum numbers, vertex operators, and masses of these states are precisely identical to those appearing in the asymmetric orbifold construction of Section \\ref{AsymOrb}. This is a consequence of the fact that the SU(2) enhancement is realised by the Kaluza-Klein and winding states of the $T^2$ torus alone. In other words, they arise from a universal sector that is not affected by the gauge bundle, as can be seen by turning on generic Wilson lines.\n\nTherefore, these models correctly engineer the $\\mathcal{N}=2^\\star$ gauge theory in the field theory limit. This picture may be explicitly confirmed by calculating the partition function of the $\\Omega$-deformation of the latter, as a field theory limit of the string theoretic graviphoton amplitude \\eqref{ActionTopAmpl} in these general freely-acting orbifold backgrounds. Indeed, denoting by $\\Phi[^H_G]$ the path integral over the internal worldsheet field (that depends on the specific details of the model), the amplitude takes form\n\\begin{equation}\n\t\\begin{split}\n\t\\mc F(\\epsilon) &= \\int_{\\mc F}\\frac{d^2\\tau}{\\tau_2^2} \\,G_{\\rm bos}(\\epsilon)\\sum_{\\genfrac{}{}{0pt}{}{H,G=0,1}{}} \\Phi[^H_G]\\,\\Gamma_{2,2}[^H_G] \\,,\n\t\\end{split}\n\\end{equation}\nand one indeed recovers \\eqref{FieldTheorLimitNstar}.\n\n\n\\subsection{An example heterotic\/type II dual pair}\n\nIn order to illustrate our previous considerations, we consider an explicit example of heterotic\/type II dual pair with $h^{1,1}=h^{2,1}=11$ discussed in \\cite{Ferrara:1995yx,Gregori:1998fz}. We briefly review their construction here. The starting point is the familiar heterotic\/type II duality in six spacetime dimensions, with the type II theory living on ${\\rm K3}\\simeq T^4\/\\mathbb Z_2$ and the heterotic one on $T^4$. Upon toroidal compactification of each theory on a $T^2$, one obtains a four dimensional theory with $\\mathcal{N}=4$ supersymmetry. Subsequently, one introduces a freely acting $\\mathbb Z_2'$ orbifold on both theories, which is responsible for the spontaneous breaking of $\\mathcal{N}=4\\to \\mathcal{N}=2$. On the type IIA side, this free action generates an elliptic fibration of the Enriques surface over $\\mathbb P^1\\simeq T^2\/\\mathbb Z_2$. Explicitly, if we denote by $Z^1$ the complex coordinate of $T^2$, and $Z^2, Z^3$ the coordinates of $T^4$ in the type IIA theory, the orbifold action $g\\in\\mathbb Z_2$ generating the singular limit of K3 and the free action $G\\in\\mathbb Z_2' $ generating the elliptic fibration can be chosen to act as\n\\begin{equation}\n\t\\begin{split}\n\t\tg: &\\quad Z^1\\to Z^1 \\quad,\\quad Z^2\\to -Z^2 \\quad,\\quad Z^3\\to -Z^3 \\,,\\\\\n\t\tG: &\\quad Z^1\\to -Z^1 \\quad,\\quad Z^2\\to Z^2+i\\pi\\sqrt{\\tfrac{T_2^{(2)}}{U_2^{(2)}} } \\quad,\\quad Z^3\\to -Z^3+i\\pi \\sqrt{\\tfrac{T_2^{(3)}}{U_2^{(3)}} }\\,.\n\t\\end{split}\n\\end{equation}\nHere, $T^{(i)}$ and $U^{(i)}$ are the K\\\"ahler and complex structure moduli of the $T^2\\times T^2\\times T^2$ decomposition of the six dimensional internal space. By considering the six dimensional duality, it is straightforward to identify $T^{(1)}$ as the dual of the heterotic dilaton modulus $S$. Similarly, the K\\\"ahler moduli of the K3 directions $T^{(2)}, T^{(3)}$ belonging to type IIA vector multiplets are mapped to the K\\\"ahler and complex structure moduli $T,U$ of the dual heterotic theory.\n\nThe action of $\\mathbb Z_2'$ has no fixed points in the total internal space and, hence, there are no additional massless states arising from the twisted sectors. In addition to the $\\mathcal{N}=2$ gravity multiplet, one obtains an equal number $h^{1,1}=h^{2,1}=11$ of vector and hyper multiplets, as well as the tensor multiplet of the type IIA dilaton. The partition function of the theory explicitly reads\n\\begin{equation}\n\t\\begin{split}\n\tZ_{\\rm IIA} = \\frac{1}{2^2\\,\\eta^{12}\\bar\\eta^{12}}\\sum_{h,g=0,1\\atop H,G=0,1}&\\frac{1}{2}\\sum_{a,b=0,1}(-1)^{a+b}\\, \\vartheta[^a_b]\\,\\vartheta[^{a+H}_{b+G}]\\,\\vartheta[^{a+h}_{b+g}]\\,\\vartheta[^{a-h-H}_{b-g-G}] \\\\\n\t\\times &\\frac{1}{2}\\sum_{\\bar a,\\bar b=0,1}(-1)^{\\bar a+\\bar b+\\bar a\\bar b} \\,\\bar\\vartheta[^{\\bar a}_{\\bar b}]\\,\\bar\\vartheta[^{\\bar a+H}_{\\bar b+G}]\\,\\bar\\vartheta[^{\\bar a+h}_{\\bar b+g}]\\,\\bar\\vartheta[^{\\bar a-h-H}_{\\bar b-g-G}] \\\\\n\t\\times & \\ \\Gamma_{2,2}^{(1)}\\bigr[^{0}_{0}\\bigr|^{H}_{G}\\bigr] \\ \\Gamma_{2,2}^{(2)}\\bigr[^{H}_{G}\\bigr|^{h}_{g}\\bigr]\\ \\Gamma_{2,2}^{(3)}\\bigr[^{H}_{G}\\bigr|^{h+H}_{g+G}\\bigr] \\,.\n\t\\end{split}\n\\end{equation}\nHere, $h,H=0,1$ label the orbifold sectors of $\\mathbb Z_2\\times \\mathbb Z_2'$, and the sum over $g,G=0,1$ enforces the corresponding orbifold projections. The notation employed for the lattice sums associated to the three $T^2$ planes is as follows:\n\\begin{equation}\n\t\\Gamma_{2,2}^{(j)}\\bigr[^{h_1}_{g_1}\\bigr|^{h_2}_{g_2}\\bigr] = \\left\\{\n\t\t\t\t\\begin{split}\n\t\t\t\t\t\t\t\\frac{4\\eta^3 \\bar\\eta^3}{\\bigr|\\vartheta\\bigr[^{1+h_2}_{1+g_2}\\bigr]\\,\\vartheta\\bigr[^{1-h_2}_{1-g_2}\\bigr]\\bigr|} \\quad &,\\ {\\rm for}\\ (h_2,g_2)\\neq(0,0)\\ {\\rm and}\\ (h_1,g_1)\\in\\{(0,0),(h_2,g_2)\\} \\,,\\\\\n\t\t\t\t\t\t\t\\Gamma_{2,2}^{(j),{\\rm shift}}\\bigr[^{h_1}_{g_1}\\bigr] ~~\\quad\\quad &, \\ {\\rm for}\\ (h_2,g_2)=(0,0) \\,,\\\\\n\t\t\t\t\t\t\t 0 \\quad\\qquad\\qquad &, \\ {\\rm otherwise} \\,.\n\t\t\t\t\\end{split}\n\t\\right.\n\\end{equation}\nIn the above notation for the shifted\/twisted (2,2) lattice, the first column $[^{h_1}_{g_1}]$ denotes a shift of the two-torus along its first cycle, whereas the second column $[^{h_2}_{g_2}]$ denotes a twist. The vanishing components of the shifted\/twisted lattice can be understood, for instance, from the fact that a lattice in the twisted sector with respect to the translation (shift) orbifold $(h_1,h_2)=(1,0)$ necessarily involves states with non-trivial momentum and\/or winding numbers. On the other hand, a simultaneous projection with respect to the rotation (twist) orbifold $(g_1,g_2)=(0,1)$ only receives contributions from states with both momentum and winding numbers vanishing. By modular transformations, one shows the vanishing of the remaining sectors.\n\nThe dual theory may be constructed by consistently translating the free action of $\\mathbb Z_2'$ on the heterotic degrees of freedom. It is clear that the action of $\\mathbb Z_2'$ on the left movers must be identical to that of the freely acting asymmetric orbifold of Section \\ref{AsymOrb}, and is responsible for the spontaneous breaking of $\\mathcal{N}=4\\to \\mathcal{N}=2$. Since the type IIA model does have massless hyper multiplets, one is led to consider instead a symmetric action of $\\mathbb Z_2'$ on the $T^6$ internal space of the ${\\rm E}_8\\times {\\rm E}_8$ heterotic string:\n\\begin{equation}\n\t\\mathbb Z_2' : \\quad Z^1 \\to Z^1+i\\pi \\sqrt{\\tfrac{T_2}{U_2}} \\quad,\\quad Z^2\\to -Z^2 \\quad,\\quad Z^3\\to -Z^3 \\,.\n\\end{equation}\nDetermining the orbifold action on the heterotic gauge degrees of freedom is more subtle and requires a careful mapping of the $\\mathbb Z_2'$ action on the 16 RR gauge fields arising from the twisted sector of the type IIA theory on ${\\rm K3}\\times T^2$. There, the shift action of $\\mathbb Z_2'$ has the effect of permuting the 16 fixed points, giving rise to 8 positive and 8 negative eigenvalues and reducing the rank of the ${\\rm U}(1)^{16}$ gauge group by a factor of 2. On the heterotic side, this should be similarly mapped into an action on the ${\\rm U}(1)^{16}$ Cartan generators of ${\\rm E}_8\\times {\\rm E}_8$ with 8 positive and 8 negative eigenvalues.\n\nFollowing \\cite{Gregori:1998fz}, we consider a point in the moduli space where the ${\\rm E}_8\\times {\\rm E_8}$ gauge group is broken to $\\left({\\rm SU}(2)\\times {\\rm SU}(2)\\right)^8_{k=1} \\simeq {\\rm SO}(4)^8_{k=1}$. It is clear that a pairwise interchange of the two ${\\rm SU}(2)$ gauge group factors introduces the correct number of negative eigenvalues, as required from the type IIA action of $\\mathbb Z_2'$. Let us see how this is realised in the heterotic theory at the level of the partition function.\n\nWe first introduce three additional $\\mathbb Z_2$ shift orbifolds, which correspond to discrete Wilson lines around the K3 directions. Each one of these is labeled by $h_i,g_i=0,1$ with $i=1,2,3$ and acts as a momentum shift along a separate $T^4$ direction, supplemented by an appropriate action on the gauge bundle degrees of freedom. It is most convenient to employ the fermionic realisation of ${\\rm E_8\\times E_8}$. The action of the $h_i,g_i$ is such that it breaks the gauge group down to ${\\rm SO}(4)^8_{k=1}$ and its partition function contribution reads\n\\begin{equation}\n\t\\begin{split}\n\t\\Gamma_{8+8}\\bigr[^{h_1,h_2,h_3}_{g_1,g_2,g_3}\\bigr] = \\tfrac{1}{2}\\sum_{\\gamma,\\delta=0,1} &\\bar\\vartheta\\bigr[^\\gamma_\\delta\\bigr]^2\\,\\bar\\vartheta\\bigr[^{\\gamma+h_1}_{\\delta+g_1}\\bigr]^2\\,\\bar\\vartheta\\bigr[^{\\gamma+h_2}_{\\delta+g_2}\\bigr]^2\\,\\bar\\vartheta\\bigr[^{\\gamma+h_3}_{\\delta+g_3}\\bigr]^2\\\\\n\t\\times\\\t&\\bar\\vartheta\\bigr[^{\\gamma+h_1-h_2}_{\\delta+g_1-h_2}\\bigr]^2\\,\\bar\\vartheta\\bigr[^{\\gamma+h_2-h_3}_{\\delta+g_2-g_3}\\bigr]^2\\,\\bar\\vartheta\\bigr[^{\\gamma+h_3-h_1}_{\\delta+g_3-g_1}\\bigr]^2\\,\\bar\\vartheta\\bigr[^{\\gamma-h_1-h_2-h_3}_{\\delta-g_1-g_2-g_3}\\bigr]^2 \\,.\n\t\\end{split}\n\\end{equation}\nThis describes a conformal system of 32 real fermions $\\tilde\\psi^a(\\bar z)$ which are assembled into eight sets of four real fermions each. All fermions within a given set are assigned the same boundary conditions. Due to the different boundary conditions assigned to each of the eight sets, all Kac-Moody currents $\\tilde J_{ab}(\\bar z)=i\\tilde\\psi^a \\tilde\\psi^b$ are twisted except for those currents $\\tilde J_{ab}$ formed out of real fermions belonging to the same set. Hence, each set of four real fermions realises an ${\\rm SO}(4)_{k=1}$ gauge group, consistently with the fact that the central charge of ${\\rm SO}(4)_k$ is given by $c=6k\/(k+2)$. The level one realisation of this current algebra is obtained in terms of four real fermions $\\tilde\\psi^1, \\tilde\\psi^2, \\tilde\\psi^3, \\tilde\\psi^4$ by forming the ${\\rm SU}(2)_{+}\\times {\\rm SU}(2)_{-}$ currents\n\\begin{equation}\n\t\\tilde J^a_{+}(\\bar z) = \\frac{i}{2}\\left( \\tfrac{1}{2}\\,\\epsilon^{abc}\\tilde\\psi^b\\,\\tilde\\psi^c + \\tilde\\psi^a\\,\\tilde\\psi^4\\right) \\quad,\\quad \\tilde J^a_{-}(\\bar z) = \\frac{i}{2}\\left( \\tfrac{1}{2}\\,\\epsilon^{abc}\\tilde\\psi^b\\,\\tilde\\psi^c - \\tilde\\psi^a\\,\\tilde\\psi^4\\right) \\,,\n\\end{equation}\nwhere $a,b,c=1,2,3$. We now need to define the action of $\\mathbb Z_2'$ on these fermionic variables, such that the two ${\\rm SU}(2)_{k=1}$ factors are exchanged. It is clear that this can occur in two ways, either by twisting one of the four real fermions\n\\begin{equation}\n\t{\\rm (i)} :\\qquad \\tilde\\psi^a \\to \\tilde\\psi^a \\quad,\\quad \\tilde\\psi^4 \\to -\\psi^4 \\,,\n\\end{equation}\nor by twisting three out of four\n\\begin{equation}\n\t{\\rm (ii)} :\\qquad\t\\tilde\\psi^a \\to -\\tilde\\psi^a \\quad,\\quad \\tilde\\psi^4 \\to \\psi^4 \\,.\n\\end{equation}\nMoreover, in order to generate 8 negative eigenvalues, as required by the action of $\\mathbb Z_2'$ on the type IIA side, the twisting of fermions must be repeated for all eight sets, such that all eight ${\\rm SU}(2)\\times {\\rm SU}(2)$ factors are exchanged under it. This modifies the partition function contribution to\n\\begin{equation}\n\t\\begin{split}\n\t\\Gamma_{8+8}\\bigr[^{h_i\\,,\\,H}_{g_i\\,,\\,G}\\bigr] = \\tfrac{1}{2}\\sum_{\\gamma,\\delta=0,1}\\ &\\bar\\vartheta\\bigr[^{\\gamma+H}_{\\delta+G}\\bigr]^{3\/2}\\,\\bar\\vartheta\\bigr[^{\\gamma+H+h_1}_{\\delta+G+g_1}\\bigr]^{3\/2}\\,\\bar\\vartheta\\bigr[^{\\gamma+h_2}_{\\delta+g_2}\\bigr]^{3\/2}\\,\\bar\\vartheta\\bigr[^{\\gamma+h_3}_{\\delta+g_3}\\bigr]^{3\/2} \\,(-1)^{G(\\gamma+H)}\\\\\n\t\\times\\\t&\\bar\\vartheta\\bigr[^{\\gamma+h_1-h_2}_{\\delta+g_1-g_2}\\bigr]^{3\/2}\\,\\bar\\vartheta\\bigr[^{\\gamma+h_2-h_3}_{\\delta+g_2-g_3}\\bigr]^{3\/2}\\,\\bar\\vartheta\\bigr[^{\\gamma+h_3-h_1}_{\\delta+g_3-g_1}\\bigr]^{3\/2}\\,\\bar\\vartheta\\bigr[^{\\gamma-h_1-h_2-h_3}_{\\delta-g_1-g_2-g_3}\\bigr]^{3\/2} \\\\\n\t\\times\\ & \\bar\\vartheta\\bigr[^\\gamma_\\delta\\bigr]^{1\/2}\\,\\bar\\vartheta\\bigr[^{\\gamma+h_1}_{\\delta+g_1}\\bigr]^{1\/2} \n\t\t\\bar\\vartheta\\bigr[^{\\gamma+H+h_2}_{\\delta+G+g_2}\\bigr]^{1\/2}\\,\\bar\\vartheta\\bigr[^{\\gamma+H+h_3}_{\\delta+G+g_3}\\bigr]^{1\/2}\\\\\n\t\\times\\\t&\\bar\\vartheta\\bigr[^{\\gamma+H+h_1-h_2}_{\\delta+G+g_1-g_2}\\bigr]^{1\/2}\\,\\bar\\vartheta\\bigr[^{\\gamma+H+h_2-h_3}_{\\delta+G+g_2-g_3}\\bigr]^{1\/2}\\,\\bar\\vartheta\\bigr[^{\\gamma+H+h_3-h_1}_{\\delta+G+g_3-g_1}\\bigr]^{1\/2}\\,\\bar\\vartheta\\bigr[^{\\gamma+H-h_1-h_2-h_3}_{\\delta+G-g_1-g_2-g_3}\\bigr]^{1\/2} \\,.\n\t\\end{split}\n\t\\label{LatticeTwistSU2x8}\n\\end{equation}\nFor two of the sets, we have chosen to twist three real fermions, whereas for the remaining six sets we twisted one real fermion. The above partition function displays the reduction of the rank of the gauge group from 16 to 8 due to the $\\mathbb Z_2'$ action. Indeed, only the diagonal ${\\rm SU}(2)^8$ remains after the twisting, and each ${\\rm SU}(2)_{k=2}$ factor can be realised at level two by three real fermions, $\\tilde J^{ab}=i\\tilde\\psi^a \\tilde\\psi^b$, with $a,b=1,2,3$.\n\nOf course, in order to compare with the perturbative spectrum of the type IIA construction, the heterotic ${\\rm SU}(2)^8$ gauge group must be broken down to its abelian subgroup ${\\rm U}(1)^{8}$. This is achieved by introducing an additional discrete Wilson line, parametrised by $h_4,g_4=0,1$ acting as a momentum shift along the fourth direction of $T^4$ and responsible for twisting one of the real fermions that realise each ${\\rm SU}(2)$. Simultaneously, it further twists the eight real fermions corresponding to the $\\bar\\vartheta^{1\/2}$ of the last two lines of \\eqref{LatticeTwistSU2x8}. One then obtains the following contribution to the gauge bundle partition function\n\\begin{equation}\n\t\\begin{split}\n\t\\Gamma_{8+8}\\bigr[^{h_i\\,,\\,H}_{g_i\\,,\\,G}\\bigr] = &\\, \\tfrac{1}{2}\\sum_{\\gamma,\\delta=0,1} \\bar\\vartheta\\bigr[^{\\gamma-H}_{\\delta-G}\\bigr]\\,\\bar\\vartheta\\bigr[^{\\gamma+H+h_1}_{\\delta+G+g_1}\\bigr]\\,\\bar\\vartheta\\bigr[^{\\gamma+h_2}_{\\delta+g_2}\\bigr]\\,\\bar\\vartheta\\bigr[^{\\gamma+h_3}_{\\delta+g_3}\\bigr] \\\\\n\t\\times\\\t&\\bar\\vartheta\\bigr[^{\\gamma+h_1-h_2}_{\\delta+g_1-g_2}\\bigr]\\,\\bar\\vartheta\\bigr[^{\\gamma+h_2-h_3}_{\\delta+g_2-g_3}\\bigr]\\,\\bar\\vartheta\\bigr[^{\\gamma+h_3-h_1}_{\\delta+g_3-g_1}\\bigr]\\,\\bar\\vartheta\\bigr[^{\\gamma-h_1-h_2-h_3}_{\\delta-g_1-g_2-g_3}\\bigr] \\\\\n\t\\times\\ & \\bar\\vartheta\\bigr[^{\\gamma+h_4}_{\\delta+g_4}\\bigr]^{1\/2}\\,\\bar\\vartheta\\bigr[^{\\gamma+h_4+h_1}_{\\delta+g_4+g_1}\\bigr]^{1\/2} \n\\bar\\vartheta\\bigr[^{\\gamma+h_4+h_2}_{\\delta+g_4+g_2}\\bigr]^{1\/2}\\,\\bar\\vartheta\\bigr[^{\\gamma+h_4+h_3}_{\\delta+g_4+g_3}\\bigr]^{1\/2} \\\\\n\t\\times\\\t&\\bar\\vartheta\\bigr[^{\\gamma+h_4+h_1-h_2}_{\\delta+g_4+g_1-g_2}\\bigr]^{1\/2}\\,\\bar\\vartheta\\bigr[^{\\gamma+h_4+h_2-h_3}_{\\delta+g_4+g_2-g_3}\\bigr]^{1\/2}\\,\\bar\\vartheta\\bigr[^{\\gamma+h_4+h_3-h_1}_{\\delta+g_4+g_3-g_1}\\bigr]^{1\/2}\\,\\bar\\vartheta\\bigr[^{\\gamma+h_4-h_1-h_2-h_3}_{\\delta+g_4-g_1-g_2-g_3}\\bigr]^{1\/2}\\\\\t\t\n\t\\times\\\t\t&\\bar\\vartheta\\bigr[^{\\gamma+h_4+H}_{\\delta+g_4+G}\\bigr]^{1\/2}\\,\\bar\\vartheta\\bigr[^{\\gamma+h_4+H+h_1}_{\\delta+g_4+G+g_1}\\bigr]^{1\/2}\\,\\bar\\vartheta\\bigr[^{\\gamma+h_4+H+h_2}_{\\delta+g_4+G+g_2}\\bigr]^{1\/2}\\,\\bar\\vartheta\\bigr[^{\\gamma+h_4+H+h_3}_{\\delta+g_4+G+g_3}\\bigr]^{1\/2}\\\\\n\t\\times\\\t&\\bar\\vartheta\\bigr[^{\\gamma+h_4+H+h_1-h_2}_{\\delta+g_4+G+g_1-g_2}\\bigr]^{1\/2}\\,\\bar\\vartheta\\bigr[^{\\gamma+h_4+H+h_2-h_3}_{\\delta+g_4+G+g_2-g_3}\\bigr]^{1\/2}\\,\\bar\\vartheta\\bigr[^{\\gamma+h_4+H+h_3-h_1}_{\\delta+g_4+G+g_3-g_1}\\bigr]^{1\/2}\\,\\bar\\vartheta\\bigr[^{\\gamma+h_4+H-h_1-h_2-h_3}_{\\delta+g_4+G-g_1-g_2-g_3}\\bigr]^{1\/2} \\,.\n\t\t\\end{split}\n\\end{equation}\nThe total partition function of the heterotic dual theory then reads\n\\begin{equation}\n\t\\begin{split}\n\t\tZ_{\\rm het} = \\frac{1}{2^5\\,\\eta^{12}\\,\\bar\\eta^{24}}\\sum_{H,G=0,1 \\atop h_i,g_i=0,1} \\frac{1}{2}\\sum_{a,b=0,1}(-1)^{a+b+ab}\\,\\vartheta\\bigr[^a_b\\bigr]^2\\,\\vartheta\\bigr[^{a+H}_{b+G}\\bigr]\\,\\vartheta\\bigr[^{a-H}_{b-G}\\bigr]\\,\\Gamma_{2,2}\\bigr[^{H\\,,\\,h_1}_{G\\,,\\,g_1}\\bigr]\\,\\Gamma_{4,4}\\bigr[^{h_i}_{g_i}\\bigr|^H_G\\bigr]\\,\\Gamma_{8+8}\\bigr[^{h_i\\,,\\,H}_{g_i\\,,\\,G}\\bigr]\\,,\n\t\\end{split}\n\\end{equation}\nwhere $(H,G)$ acts on the (2,2) lattice as a momentum shift along the first cycle of $T^2$, $(h_1,g_1)$ acts on the (2,2) lattice as a winding shift along the same cycle, and the four $(h_i,g_i)$ orbifolds act on the (4,4) lattice as momentum shifts along four distinct cycles of $T^4$. Because of the simultaneous twist by $(H,G)$ and the shifts by $(h_i,g_i)$ along the four directions of $T^4$, the $\\mathcal{N}=2$ subsectors $(H,G)\\neq(0,0)$ only receive contributions from $(h_i,g_i)=(0,0)$ or $(h_i,g_i)=(H,G)$ for each $i=1,2,3,4$. Notice that a momentum shift due to $(H,G)$ together with a winding shift due to $(h_1,g_1)$ along the same direction of $T^2$ introduces a level matching asymmetry in the (2,2) lattice which, in turn, ensures that the level matching constraints of the entire theory are properly satisfied and the partition function $Z_{\\rm het}$ is modular invariant.\n\nUsing the explicit form of their respective partition functions, it is straightforward to show that the massless spectra of the type IIA and heterotic models indeed match. Furthermore, it is easy to show that the vector multiplets arising from the $T^2$ gauge group, including the ${\\rm SU}(2)$ enhancement at $TU=-1$, together with the massive hypermultiplets charged under it, survive the projections and the form of their corresponding vertex operators is identical to those in the asymmetric model of Section \\ref{AsymOrb}. The fact that this orbifold construction correctly reproduces \\eqref{FieldTheorLimitNstar} in the field theory limit and, hence, geometrically engineers $\\mN=2^\\star$ gauge theory, is a natural consequence of the arguments given in Section \\ref{correspSubsec}.\n\nThe recovery of $\\mathcal{N}=4$ supersymmetry occurs in the decompactification limit of $T^2$ where the Scherk-Schwarz radius is taken to infinity, $R_1\\to\\infty$, whereas in the limit $R_1\\to 0$, one recovers a standard $\\mathcal{N}=2$ theory as in a non-freely acting orbifold. This property is precisely reproduced in the type IIA dual, by considering the large and small volume limits in the $T^{(2)}, T^{(3)}$ moduli. This implies that the map between the heterotic and type IIA moduli is, in fact, linear and the correspondence can be extracted explicitly \\cite{Gregori:1998fz} by a careful analysis of gravitational threshold corrections on both sides, yielding $S=T^{(1)}\/8\\pi$, $T=T^{(2)}$ and $U=T^{(3)}$.\n\nTo further our check of this realisation of $\\mN=2^\\star$, we again compute the topological amplitudes $F_g$ of eq. \\eqref{ActionTopAmpl} at genus one in heterotic perturbation theory:\n\\begin{equation}\n\t\\begin{split}\n\t\\mc F(\\epsilon) &= \\int_{\\mc F}\\frac{d^2\\tau}{\\tau_2^2} \\, \\frac{G_{\\rm bos}(\\epsilon)}{\\bar\\Delta}\\sum_{\\genfrac{}{}{0pt}{}{H,G=0,1}{(H,G)\\neq(0,0)}} \\,\\left( \\Gamma_{2,2}^{\\lambda=0}\\bigr[^H_G\\bigr]\\,\\bar\\Phi^{(0)}\\bigr[^H_G\\bigr] + \\Gamma_{2,2}^{\\lambda=1}\\bigr[^H_G\\bigr]\\,\\bar\\Phi^{(1)}\\bigr[^H_G\\bigr] \\right)\\,,\n\t\\end{split}\n\\end{equation}\nwhere the forms $\\Phi^{(\\lambda)}$ are given by\n\\begin{align}\\nonumber\n\t\\Phi^{(0)}\\bigr[^0_1\\bigr] &=\\tfrac{1}{2}\\,(\\vartheta_3^4+\\vartheta_4^4)\\,\\vartheta_3^8\\,\\vartheta_4^8 \\,, \\\\ \n\t\\Phi^{(0)}\\bigr[^1_0\\bigr] &=-\\tfrac{1}{2}\\,(\\vartheta_2^4+\\vartheta_3^4)\\,\\vartheta_2^8\\,\\vartheta_3^8 \\,, \\\\ \\nonumber\n\t\\Phi^{(0)}\\bigr[^1_1\\bigr] &=\\tfrac{1}{2}\\,(\\vartheta_2^4-\\vartheta_4^4)\\,\\vartheta_2^8\\,\\vartheta_4^8 \\,,\\\\ \n\\intertext{and} \\nonumber\n\t\\Phi^{(1)}\\bigr[^0_1\\bigr] &=\\vartheta_3^{10}\\,\\vartheta_4^{10} \\,, \\\\ \n\t\\Phi^{(1)}\\bigr[^1_0\\bigr] &=-\\vartheta_2^{10}\\,\\vartheta_3^{10} \\,, \\\\ \\nonumber\n\t\\Phi^{(1)}\\bigr[^1_1\\bigr] &=-\\vartheta_2^{10}\\,\\vartheta_4^{10} \\,. \n\\end{align}\nThe notation $\\Gamma_{2,2}^{\\lambda}[^H_G]$ for the $T^2$ lattice denotes a momentum shift along the first cycle for $\\lambda=0$, whereas for $\\lambda=1$ it stands for a simultaneous momentum and winding shift along the same cycle. The contribution involving the $\\lambda=1$ lattice decouples in the field theory limit since $\\Gamma^{\\lambda=1}$ involves states which simultaneously carry non-trivial momentum and winding numbers. Their masses are of the form $\\sim 1\/R^2+R^2$ and are therefore of the order of the string scale. Consequently, only the $\\lambda=0$ sector leads to a non trivial field theory limit. Repeating the analysis of Section \\ref{AsymOrb}, one similarly recovers the correct gauge theory partition function \\eqref{FieldTheorLimitNstar} at the $SU(2)$ symmetry enhancement point.\n\n\n\n\n\\section{Type I description of the mass deformation}\\label{TypeOne}\n\nSo far, we have presented a heterotic realisation of $\\mN=2^\\star$ in terms of a freely-acting asymmetric orbifold in Section \\ref{AsymOrb} and its symmetric generalisations in Section \\ref{StringModel}, the latter naturally admitting dual type II descriptions on K3 fibered Calabi-Yau manifolds. In this section, we discuss the analogous realisation of $\\mN=2^\\star$ by performing an orientifold of the type IIB theory on a freely acting version of ${\\rm K3}\\times T^2$ implementing the Scherk Schwarz breaking $\\mathcal{N}=4\\to\\mathcal{N}=2$.\n\nIn the heterotic construction of Section \\ref{AsymOrb}, the main ingredients were: {\\it (i)} the spontaneous nature of the breaking of $\\mathcal{N}=4\\to\\mathcal{N}=2$ via a Scherk-Schwarz flux, {\\it (ii)} the gauge group being unaffected by the orbifold action, and {\\it (iii)} rendering adjoint hypermultiplet states massive. To this end, the simplest prototype type I vacuum realising $\\mN=2^\\star$ may be naturally identified with a freely acting $\\mathbb Z_2$ orbifold of the standard $\\mathcal{N}=4$ type I theory on $T^6$ with ${\\rm SO}(32)$ gauge group. This model was first constructed in \\cite{Antoniadis:1998ep} in the context of partial supersymmetry breaking in type I theory.\n\nThe $\\mathbb Z_2$ orbifold acts as a momentum shift along one of the complexified string coordinates, supplemented by a reflection of the remaining two:\n\\begin{equation}\\label{shifttwistaction}\n\tZ^1 \\to Z^1 + i\\pi \\sqrt{\\tfrac{T_2}{U_2}} \\quad,\\quad Z^2 \\to - Z^2 \\quad,\\quad Z^3\\to - Z^3 \\,,\n\\end{equation}\nand the torus partition function reads\n\\begin{equation}\n\t\\begin{split}\n\t\\mathcal T= \\frac{1}{2\\,\\eta^{12}\\,\\bar\\eta^{12}}\\sum_{H,G=0,1} &\\frac{1}{2}\\sum_{a,b} (-1)^{a+b+ab}\\,\\vartheta\\bigr[^a_b\\bigr]^2\\,\\vartheta\\bigr[^{a+H}_{b+G}\\bigr]\\,\\vartheta\\bigr[^{a-H}_{b-G}\\bigr]\\ \\\\\n\t\\times & \\frac{1}{2}\\sum_{\\bar a,\\bar b} (-1)^{\\bar a+\\bar b+\\bar a\\bar b}\\,\\bar\\vartheta\\bigr[^{\\bar a}_{\\bar b}\\bigr]^2\\,\\bar\\vartheta\\bigr[^{\\bar a+H}_{\\bar b+G}\\bigr]\\,\\bar\\vartheta\\bigr[^{\\bar a-H}_{\\bar b-G}\\bigr]\\ \\\\\n\t\\times & \\Gamma_{2,2}\\bigr[^H_G\\bigr](T,U) \\ \\Gamma_{4,4}\\bigr[^H_G\\bigr] \\,,\n\t\\end{split}\n\\end{equation}\nwhere the (2,2) lattice is shifted, while the (4,4) one is twisted as indicated in \\eqref{shifttwistaction}. Clearly, the perturbative closed string sector enjoys unbroken $\\mathcal N=(2,2)$ supersymmetry and, hence, cannot give rise to non-abelian gauge symmetry. It is irrelevant for the purposes of this discussion, and we henceforth focus entirely on the open string sector. The relevant diagrams here are the annulus and the M\\\"obius strip.\n\nFollowing \\cite{Antoniadis:1998ep}, one may construct a consistent vacuum with $N=32$ D9 branes, where the $\\mathbb Z_2$ orbifold acts trivially on the Chan-Paton degrees of freedom. The annulus and M\\\"obius strip amplitudes in the direct channel read\n\\begin{equation}\\label{Annul}\n\t\\mathcal A = \\frac{N^2}{4\\,\\eta^{12}} \\sum_{G=0,1} \\frac{1}{2}\\sum_{a,b} (-1)^{a+b+ab}\\,\\vartheta\\bigr[^a_b\\bigr]^2\\,\\vartheta\\bigr[^{\\ \\,a}_{b+G}\\bigr]\\,\\vartheta\\bigr[^{\\ \\,a}_{b-G}\\bigr]\\ \\Gamma_{2}^{(1)}\\bigr[^{\\,0}_G\\bigr] \\,\\Gamma_{4}\\bigr[^{\\,1}_G\\bigr] \\,,\n\\end{equation}\n\n\\begin{equation}\\label{Moebi}\n\t\\mathcal M = -\\frac{N}{4\\,\\hat\\eta^{12}} \\sum_{G=0,1} \\frac{1}{2}\\sum_{a,b} (-1)^{a+b+ab}\\,\\hat\\vartheta\\bigr[^a_b\\bigr]^2\\,\\hat\\vartheta\\bigr[^{\\ \\,a}_{b+G}\\bigr]\\,\\hat\\vartheta\\bigr[^{\\ \\,a}_{b-G}\\bigr]\\ \\Gamma_{2}^{(1)}\\bigr[^{\\,0}_G\\bigr] \\,\\hat\\Gamma_{4}\\bigr[^{\\,1}_G\\bigr] \\,,\n\\end{equation}\nwith $\\Gamma_2$ being the shifted $T^2$ lattice partition function\n\\begin{equation}\n\t\\Gamma_2^{(1)}\\bigr[^{\\,0}_G\\bigr] =\\sum_{m_1,m_2\\in \\mathbb Z} (-1)^{ m_1 G}\\,e^{-\\pi t|P|^2} \\,,\n\\end{equation}\ndefined in terms of the real Schwinger parameter $t$ and the lattice momenta\n\\begin{equation}\n\tP = \\frac{ m_2-U m_1 }{\\sqrt{T_2 U_2} }\\,.\n\\end{equation}\nFurthermore, $\\Gamma_4$ is the twisted $T^4$ partition function\n\\begin{equation}\n\t\\Gamma_4 \\bigr[^{\\, 0}_G\\bigr] = \\left\\{ \\begin{split}\n\t\t\t\t\t\t\t\t\t\t\t\t\\Gamma_2^{(2)}\\bigr[^0_0\\bigr] \\,\\Gamma_2^{(3)}\\bigr[^0_0\\bigr] \\quad,\\quad G=0\\\\\n\t\t\t\t\t\t\t\t\t\t\t\t\\left(\\frac{2\\eta^3}{\\vartheta_2}\\right)^2 \\quad\\quad,\\quad G=1\n\t\t\t\t\t\t\t\t\\end{split} \\right. \\,.\n\\end{equation}\nThe arguments of the Dedekind $\\eta(\\tau)$ and Jacobi $\\vartheta\\bigr(\\tau)$ functions in the annulus and M\\\"obius amplitudes above are understood to be $\\tau=\\frac{it}{2}$ and $\\tau=\\frac{it}{2}+\\frac{1}{2}$, respectively, and the hatted characters denote the real basis for the M\\\"obius amplitude \\cite{Bianchi:1990yu,Bianchi:1990tb}. The Chan-Paton factors $N^2$ and $N$ simply reflect the fact that each of the string endpoints is attached to one of the $N$ D9's in the case of the annulus, or to the fact that the string is stretched between one of the $N$ D9's and an O9 plane, in the case of the M\\\"obius.\n\nTo extract the low lying spectrum, it is convenient to decompose the partition functions in terms of characters:\n\\begin{equation}\n\t\\mathcal A=\\frac{1}{4\\eta^8}\\,N^2 \\,\\sum_{m_1,m_2\\in\\mathbb Z}\\left[ (Q_o+Q_v)\\,\\Gamma^{(1)}_{m_1,m_2}\\,\\Gamma_2^{(2)}\\,\\Gamma_2^{(3)}+(Q_o-Q_v)\\,\\left(\\frac{2\\eta^3}{\\theta_2}\\right)^2\\,(-1)^{m_1}\\,\\Gamma^{(1)}_{m_1,m_2}\\right] \\,,\n\\end{equation}\n\n\\begin{equation}\n\t\\mathcal M=-\\frac{1}{4\\hat\\eta^8}\\,N \\,\\sum_{m_1,m_2\\in\\mathbb Z}\\left[ (\\hat Q_o+\\hat Q_v)\\,\\Gamma^{(1)}_{m_1,m_2}\\,\\Gamma^{(2)}\\,\\Gamma^{(3)}+(\\hat Q_o-\\hat Q_v)\\,\\left(\\frac{2\\hat\\eta^3}{\\hat\\theta_2}\\right)^2\\,(-1)^{m_1}\\,\\Gamma^{(1)}_{m_1,m_2}\\right] \\,.\n\\end{equation}\nWe employ here the traditional ${\\rm SO}(2n)$ character notation $Q_o = V_4 O_4 -S_4 S_4$ and $Q_v = O_4 V_4-C_4 C_4$, corresponding to the $\\mathcal{N} =2$ vector and hyper multiplets, respectively. Furthermore, we have set $\\Gamma_{m_1,m_2}^{(1)} = e^{-\\pi t |P|^2}$. \n\nIt is straightforward to see that the theory has ${\\rm SO}(32)$ gauge symmetry with all hypers transforming in the adjoint representation of ${\\rm SO}(32)$, and carrying non-trivial Scherk-Schwarz mass. For lowest lying such states, the mass reads $m^2 = |U|^2\/T_2 U_2$. Moreover, by Poisson resumming the $T^2$ lattice momenta of $\\Gamma_2^{(1)}$, it is easy to see that the limit $T_2 \\to \\infty$ in \\eqref{Annul} and \\eqref{Moebi} projects onto the $G=0$ sector and, hence, one recovers $\\mathcal{N}=4$ supersymmetry. In fact, the type I vacuum discussed above is in several ways similar to the asymmetric orbifold construction of Section \\ref{AsymOrb} and possesses all the properties that one expects from a string theoretic realisation of the $\\mN=2^\\star$ gauge theory.\n\nBefore we discuss the calculation of topological amplitudes in this background, it is necessary to consider first a deformation of this theory in which the ${\\rm SO}(32)$ gauge symmetry is broken to the Coulomb branch ${\\rm U}(1)^{16}$ by turning on generic Wilson lines around the $T^2$. This deformation then acts as a natural parameter controlling the enhancement when we consider the topological amplitudes. A standard parametrisation of the Wilson line background for each boundary is\n\\begin{equation}\n\tU = \\bigoplus_{i=1}^{16} \\, e^{2\\pi i W^i \\sigma_3}\\,,\n\\end{equation}\nwith $\\sigma_3$ being the Pauli matrix and $W_i$ being the Wilson lines in the Cartan sub-algebra of the ${\\rm SO}(32)$ gauge group. Taking into account that, in the transverse channel, the Chan-Paton factors for the annulus and M\\\"obius become $({\\rm Tr}\\, U)^2$ and ${\\rm Tr}\\, (U^2)$, respectively, one finds in the direct channel\n\\begin{equation}\n\t\\mathcal A = \\frac{1}{4\\,\\eta^{12}} \\sum_{G=0,1} \\frac{1}{2}\\sum_{a,b} (-1)^{a+b+ab}\\,\\vartheta\\bigr[^a_b\\bigr]^2\\,\\vartheta\\bigr[^{\\ \\,a}_{b+G}\\bigr]\\,\\vartheta\\bigr[^{\\ \\,a}_{b-G}\\bigr]\\ \\tilde\\Gamma_{2}^{(1)}\\bigr[^{\\,0}_G\\bigr] \\,\\Gamma_{4}\\bigr[^{\\,1}_G\\bigr] \\,.\n\\end{equation}\nFor later convenience, the Chan-Paton factors have been entirely absorbed into the modified lattice sum\n\\begin{equation}\n\t\\tilde \\Gamma_2^{(1)} \\bigr[^{\\, 0}_G\\bigr] =\\sum_{m_1,m_2\\in \\mathbb Z} \\ \\sum_{i,j=1}^{16}\\ \\sum_{q_1,q_2=\\pm 1}(-1)^{m_1 G}\\exp\\left[-\\pi t\\,\\frac{|m_2-Um_1+q_1 W^i+q_2 W^j |^2}{T_2 U_2-\\frac{1}{2}W_2^2}\\right] \\,.\n\\end{equation}\nSimilarly, for the M\\\"obius strip one finds\n\\begin{equation}\n\t\\mathcal M = -\\frac{1}{4\\,\\hat\\eta^{12}} \\sum_{G=0,1} \\frac{1}{2}\\sum_{a,b} (-1)^{a+b+ab}\\,\\hat\\vartheta\\bigr[^a_b\\bigr]^2\\,\\hat\\vartheta\\bigr[^{\\ \\,a}_{b+G}\\bigr]\\,\\hat\\vartheta\\bigr[^{\\ \\,a}_{b-G}\\bigr]\\ \\check\\Gamma_{2}^{(1)}\\bigr[^{\\,0}_G\\bigr] \\,\\hat\\Gamma_{4}\\bigr[^{\\,1}_G\\bigr] \\,,\n\\end{equation}\nwith \n\\begin{equation}\n\t\\check\\Gamma_{2}^{(1)}\\bigr[^{\\,0}_G\\bigr] =\\sum_{m_1,m_2\\in \\mathbb Z} \\ \\sum_{i=1}^{16}\\ \\sum_{q_1=\\pm 1}(-1)^{m_1 G}\\exp\\left[-\\pi t\\,\\frac{|m_2-Um_1+2q_1 W^i |^2}{T_2 U_2-\\frac{1}{2}W_2^2}\\right] \\,.\n\\end{equation}\n\nThe spectrum of the theory contains 16 massless vector multiplets, corresponding to the Cartan sub-algebra ${\\rm U}(1)^{16}$. The charged vectors, corresponding to the roots $\\rho$ of ${\\rm SO}(32)$ are instead massive. For the low lying states without Kaluza-Klein momenta, the mass reads\n\\begin{equation}\n\tm_\\rho^2 = \\frac{ |\\rho\\cdot W|^2}{T_2 U_2-\\frac{1}{2}W_2^2} \\,.\n\\end{equation}\nThe hyper multiplets, on the other hand, are massive, as they carry odd momentum number $m_1 \\in 2\\mathbb Z+1$, and the low lying states have\n\\begin{equation}\\label{hypmassform}\n\tm_Q^2 = \\frac{ | U+Q\\cdot W |^2 }{T_2 U_2-\\frac{1}{2}W_2^2} \\,.\n\\end{equation}\nHere, $Q^i$ is the charge vector with respect to the Cartan subalgebra of ${\\rm SO}(32)$ associated to the adjoint representation. For simplicity, we consider an ${\\rm SU}(2)$ enhancement point obtained by choosing $\\mu \\equiv W_1 -W_2\\to 0$. Then, clearly, the charged vectors with $Q=Q_\\pm\\equiv \\pm(1,-1,0,\\ldots,0)$ become massless and enhance to an ${\\rm SU}(2)$. Similarly, the hyper multiplets transforming in the adjoint of ${\\rm SU}(2)$ carry masses given by \\eqref{hypmassform} for $Q=Q_\\pm$ and $Q=(0,\\ldots,0)$.\n\nWe would like to confirm the result above by calculating the topological amplitude $F_g$ perturbatively in this theory. In order to do so, we recall the vertex operator of the graviphoton in the orientifold theory:\n\\begin{align}\nV(p,\\epsilon)=\\epsilon_\\mu &\\bigg[\\left(\\partial Z^1+i(p\\cdot \\Psi) \\Psi^1\\right) \\left(\\bar{\\partial} Z^{\\mu}+i(p\\cdot\\tilde{\\Psi}) \n\\tilde{\\Psi}^\\mu\\right)+(\\rm{left} \\leftrightarrow \\rm{right})\\nonumber\\\\\n-&e^{-\\tfrac{1}{2}(\\varphi+\\tilde{\\varphi})} p_\\nu S^\\alpha {(\\sigma^{\\mu\\nu})_\\alpha}^\\beta \\tilde{S}_\\beta\\,e^{\\frac{i}{2}(\\Phi_1+\\tilde{\\Phi}_1)}\\, \n\\epsilon^{AB}\\Sigma_A \\tilde{\\Sigma}_B\\bigg] \\,e^{ip\\cdot Z}\\,.\\label{TypeIVertexGTtype}\n\\end{align}\nHere, $\\epsilon$ and $p$ are the polarisation and momentum vectors of the operator, respectively, satisfying $\\epsilon\\cdot p=0$, and $\\varphi$ is the ghost field. In addition, the tilded fields are the right-moving degrees of freedom. \nSince \\eqref{TypeIVertexGTtype} leads solely to deformations of the worldsheet sigma model in the space-time directions, the calculation goes along the same lines as in \\cite{Gava:1996hr,AFHNZ}. More precisely, the path integral over the space-time directions can be obtained through the generating function\n\\begin{align}\nG_{\\text{bos}}(\\epsilon)=\\left\\langle \\exp\\Biggr[\\frac{\\hat\\epsilon}{t}\\int d^2\\sigma\n\\left(Z^4(\\bar\\partial-\\partial) Z^5+ \\bar Z^4(\\bar\\partial-\\partial)\\bar Z^5\\right)\\Biggr]\\right\\rangle ~,\\label{TypeIBosSpaceTime}\n\\end{align}\nfor the bosonic fields and\n\\begin{align}\\label{SpacetimeFermDef}\n G_{\\text{ferm}}(\\epsilon)= \\left\\langle \\exp\\Biggr[\\frac{\\hat\\epsilon}{t}\\int \nd^2\\sigma \\left[ (\\Psi^4-\\tilde\\Psi^4)(\\Psi^5-\\tilde\\Psi^5)+(\\bar\\Psi^4-\\tilde{\\bar{\\Psi}}^4)(\\bar\\Psi^5-\\tilde{\\bar{\\Psi}}^5)\n\\right] \\Biggr]\\right\\rangle~,\n\\end{align}\nfor the fermionic ones. Here, we have introduced the notation $\\hat\\epsilon=\\epsilon P t\/\\sqrt{T_2 U_2-W_2^2\/2}$. As argued above, we are only interested in the contributions of the annulus and the M\\\"obius strip. To this end, the mode expansions for the worldsheet fields are\n\\begin{align}\\label{ModesZ}\n& Z^i = \\sum\\limits_{n,m} Z^i_{n,m}\\cos(\\pi n\\sigma_2)e^{2\\pi i m \\sigma_1}\\,,&& \\bar Z^i = \\sum\\limits_{n,m} \n\\bar{Z}^i_{n,m}\\cos(\\pi n\\sigma_2)e^{2\\pi i m \\sigma_1}\\,,\\\\\n&\\Psi^i = \\sum\\limits_{n,m}\\Psi^i_{n,m}\\, e^{\\pi i(n\\sigma_2+2m \\sigma_1)}~, &&\\tilde\\Psi^i = \\sum\\limits_{n,m}\\Psi^i_{n,m}\\, \ne^{\\pi i(-n\\sigma_2+2m\\sigma_1)}\\,,\\\\\n&\\bar\\Psi^i = \\sum\\limits_{n,m}\\bar\\Psi^i_{n,m}\\, e^{\\pi i (n\\sigma_2+2m \\sigma_1)}~, &&\\tilde{\\bar{\\Psi}}^i = \n\\sum\\limits_{n,m}\\bar\\Psi^i_{n,m}\\, e^{\\pi i(-n\\sigma_2+2m \\sigma_1)}\\,.\n\\end{align}\nfor the annulus, whereas for the M\\\"obius strip we use the same expansion with the constraint $m+n\\in2\\mathbb Z$ and replace $t$ by $2t$. As a consequence of the BPS property, after summing over spin structures, the fermionic path integral yields a vanishing contribution for $G=0$ (as expected from an $\\mathcal{N}=4$ sector), whereas for $G=1$ the bosonic and fermionic path integrals for both the annulus and M\\\"obius worldsheets cancel each other except for the zero modes $n=0$ for which only the bosonic fields contribute:\n\\begin{equation}\nG^{\\mc A,\\mc M}(\\epsilon)=\\frac{\\pi^2\\hat\\epsilon^2}{\\sin^2(\\pi\\hat\\epsilon)}\\,.\n\\end{equation}\nNotice that, despite the different boundary conditions between the annulus and the M\\\"obius strip, both correlators are equal. The reason is that for the $n=0$ modes, the fact that $m$ is even for the strip is compensated by the doubling of the Teichm\\\"uller parameter. Finally, the path integral over the internal degrees of freedom is trivial and, hence, the total contribution to $F_g$ is\n\\begin{equation}\n\\mc F^{\\mc A+\\mc M}(\\epsilon)=\\frac{1}{4}\\int \\frac{dt}{t}G^{\\mc A}(\\epsilon)\\left(\\tilde\\Gamma_{2}^{(1)}\\bigr[^{0}_1\\bigr]-\\check\\Gamma_{2}^{(1)}\\bigr[^{0}_1\\bigr]\\right)\\,.\n\\label{topAmplTypeI}\n\\end{equation}\nNote, however, that this lattice combination yields precisely the BPS mass formula\n\\begin{equation}\n\t\\frac{1}{2}\\left(\\tilde\\Gamma_{2}^{(1)}\\bigr[^{0}_1\\bigr]-\\check\\Gamma_{2}^{(1)}\\bigr[^{0}_1\\bigr]\\right) = \\sum_{m_1,m_2\\in \\mathbb Z} \\ \\sum_{Q\\in {\\rm Adj} } (-1)^{m_1}\\exp\\left[-\\pi t\\,\\frac{|m_2-Um_1+Q\\cdot W |^2}{T_2 U_2-\\frac{1}{2}W_2^2}\\right] \\,,\n\\end{equation}\nwith the Cartan charges $Q$ being now summed over the adjoint representation of ${\\rm SO}(32)$.\n\nIt is now straightforward to extract the field theory limit of the type I amplitude. Indeed, around the $SU(2)$ enhancement point, only the nearly massless vectormultiplet states survive in the field theory limit, together with the massive adjoint hypermultiplet states. Therefore, we recover the same result as in the heterotic and type II models:\n\\begin{equation}\n\t\\frac{1}{(2\\pi\\epsilon)^2}\\,\\mc F(\\epsilon)\\bigr|_{\\rm F.T.} = \\sum_{k=0,\\pm 1}\\left[\\gamma_{\\hbar}(k\\mu)-\\gamma_{\\hbar}(k\\mu+ m_{\\rm h}) \\right] \\,,\n\t\t\\label{NpartFuncTypI}\n\\end{equation}\nwhere $\\mu=W_1-W_2$ and $m_{\\rm h}= U$ are the BPS mass parameters associated to the vector- and hyper- multiplets, respectively. As anticipated, the amplitude exactly reproduces the $\\Omega$-deformed partition function of the $\\mathcal{N}=2^\\star$ gauge theory \\cite{Nekrasov:2002qd,Pestun:2007rz}.\n\n\nIt is possible to give a higher dimensional version of the $\\mN=2^\\star$ partition function. This can be viewed as a generalisation of the Nekrasov-Okounkov formula \\cite{Nekrasov:2003rj}, which describes the $\\Omega$-deformed partition function of a five dimensional $\\mathcal{N}=2$ gauge theory on $\\mathbb R^4\\times S^1$, by further including the mass deformation. To this end, we first consider the large volume limit of $T^2$, in which case the Kaluza-Klein spectrum becomes dense and the sum over momenta has to be Poisson resummed in \\eqref{topAmplTypeI}, while keeping $U$ and $W$ fixed. By explicitly performing the Schwinger integral over $t$, one finds\n\\begin{equation}\n\tF_g \\sim \\frac{(2g-1)\\,B_{2g}}{(2g)!} \\sum_{Q\\in{\\rm Adj}} \\left[ \\mathcal E(2-g,2g-2;2U,Q\\cdot W)-\\frac{1}{2}\\,\\mathcal E(2-g,2g-2;U,Q\\cdot W) \\right] \\,,\n\t\\label{NO6d}\n\\end{equation}\nwhere $\\mathcal E(s,w;\\tau,z)$ is the Kronecker-Eisenstein series of weight $w$ and is defined by\n\\begin{equation}\n\t\\mathcal E(s,w;\\tau,z) = {\\sum_{m,n\\in\\mathbb Z}}' \\,(m\\tau+n)^{-w}\\, \\frac{\\tau_2^{s-\\frac{w}{2}} }{|m\\tau+n|^{2s-w}} \\, e^{2\\pi i (n\\xi-m\\eta)} \\,,\n\\end{equation}\nwith $\\xi,\\eta\\in \\mathbb R$ and $s\\in \\mathbb C$. The Jacobi parameter is identified as $z=\\eta+\\tau \\xi$. It should be stressed that \\eqref{NO6d} exhibits modular covariance under the Jacobi group with weight $2g-2$. It is to be seen as a deformation of the partition function \\eqref{NpartFuncTypI}, regarded as a compactification of a 6d theory on $T^2$. Considering a square $T^2$ with $U=iR_2\/R_1$ and ${\\rm Re}(W)=0$, and Fourier expanding the above expression, it is straightforward to obtain\n\\begin{equation}\n\tF_g = (2\\pi R_2)^{2g-2}\\,\\frac{(2g-1)\\,B_{2g}}{(2g)!} \\sum_{Q\\in{\\rm Adj}}\\sum_{n\\in \\mathbb Z}\\left[ {\\rm Li}_{3-2g}\\left( q_U^{2n}\\,e^{2\\pi i Q\\cdot W}\\right)-{\\rm Li}_{3-2g}\\left(q_U^{2n+1}\\,e^{2\\pi i Q\\cdot W}\\right)\\right] \\,.\n\\end{equation}\nIndeed, this is the generalisation of the Nekrasov-Okounkov formula \\cite{Nekrasov:2003rj} as can be seen by re-expressing it in the form\n\\begin{equation}\n\tF_g = \\sum_{Q\\in{\\rm Adj}}\\sum_{n\\in \\mathbb Z}\\left[ \\gamma_g\\left( \\frac{Q\\cdot Y +2n}{R_1} ; \\beta \\right)- \\gamma_g\\left( \\frac{Q\\cdot Y +2n}{R_1}+m_{\\rm h} ; \\beta\\right) \\right] \\,, \n\\end{equation}\nwhere $Y$ is a real Wilson line, $\\beta=2\\pi R_2$ and $m_{\\rm h} = 1\/R_1$.\n\n\n\n\n\\section{Conclusions}\\label{Conclusion}\n\nIn this work, we have elucidated a universal structure underlying the string theory realisation of the mass deformed $\\mc N=2$ gauge theory. Indeed, we have shown that freely acting orbifolds of $\\mc N=4$ string compactifications, spontaneously breaking supersymmetry to $\\mc N=2$, are the relevant models uplifting the $\\mN=2^\\star$ gauge theory to string theory. As we have argued, the low-lying states of these models precisely match those of the mass deformed gauge theory with the adjoint mass arising from the Scherk-Schwarz deformation of the string theory. The non-abelian group of the gauge theory, possibly including exceptional groups, can be recovered at points of enhanced symmetry in the string moduli space. These properties were explicitly demonstrated in various explicit models, with symmmetric and asymmetric orbifold actions, in type I, II and heterotic string theories, therefore supporting our proposal.\n\nFurthermore, we have performed a non-trivial test of the correspondence by calculating topological amplitudes in the aforementioned string backgrounds. More precisely, the graviphoton amplitude $F_g$ is known to reduce, in the field theory limit, to the pure gauge theory partition function in the standard $\\mc N=2$ case. Here, we have extended this property by proving, perturbatively, that the point particle limit of the graviphoton amplitude leads, in our general class of models, to the mass-deformed partition function of the supersymmetric gauge theory.\n\nOne may wonder about the fate of our correspondence at the full non-perturbative level. This is best seen from the type I or type II point of view by studying instanton corrections to the models under consideration. In addition, since in the pure $\\mc N=2$ case $F_g$ is the topological string partition function, it would be interesting to understand the implications of this unveiled universality for the topological string. \n\n\\section*{Acknowledgements}\n\nWe would like to thank C. Condeescu and N. Mekareeya for useful discussions. I.F. wishes to thank the ICTP, Trieste and\nA.Z.A. would like to thank the CERN Theory Department for their warm hospitality during the accomplishment of this work.\n\n\n\n\n\n\n\n\n\n\\bibliographystyle{utphys}\n\\providecommand{\\href}[2]{#2}\\begingroup\\raggedright","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nIn \\cite{Voigt2016} the separated fragment (SF) of first-order logic with equality is introduced. \nIts defining principle is that universally and existentially quantified variables may not occur together in atoms.\n(Topmost existential quantifier blocks are exempt from this rule.)\nSF properly generalizes both the Bernays--Sch\\\"onfinkel--Ramsey (BSR) fragment ($\\exists^*\\forall^*$-sentences with equality) and the relational monadic fragment without equality (MFO).\nStill, the satisfiability problem for SF is decidable.\n\nIn computational logic formulas are often classified based on the shape of quantifier prefixes.\nThere is a wealth of results that separate decidable first-order formulas from undecidable ones in this fashion, see \\cite{Borger1997} for references.\nThe definition of the BSR fragment is only one example.\nIn the context of computational complexity, hierarchies are defined, such as the polynomial hierarchy, where the hardness of problems is assumed to grow with the number of quantifier alternations that are allowed to occur.\n\nAlthough the definition of SF breaks with the paradigm of restricting quantifier prefixes, the known upper bound on the complexity of SF-satisfiability is again based on quantifier prefixes:\nDeciding whether an SF sentence $\\varphi := \\exists \\vz \\,\\forall \\vx_1 \\exists \\vy_1 \\ldots \\forall \\vx_n \\exists \\vy_n. \\psi$ with quantifier-free $\\psi$ is satisfiable requires a nondeterministic computing time that is at most $n$-fold exponential in the length of $\\varphi$ (cf.\\ Theorem~17 in \\cite{Voigt2016}).\nOn the one hand, we complement this result with a corresponding lower bound in the present paper.\nThat is, we show that SF-satisfiability is indeed non-elementary.\nOn the other hand, we derive a refined upper bound that is based on the \\emph{degree $\\degree$ of interaction of existential variables}.\nAn overview of the resulting hierarchy of complete problems is depicted in Figure~\\ref{figure:ComplexityOfSF}.\nIntuitively, $\\varphi$ exhibits a degree $\\degree_\\varphi = k$, if variables from $k$ distinct existential quantifier blocks interact. \nWe say that two variables $x, y$ interact, if they occur together in at least one atom or if there is a third variable $z$ that interacts with both $x$ and $y$ (i.e.\\\nthe property is transitive). \nFor instance, in the SF sentence \n\t$ \\forall x_1 \\exists y_1 v_1 \\forall x_2 \\exists y_2 v_2 \\forall x_3 \\exists y_3 v_3.\\; \\bigl(P(x_1, x_2, x_3) \\wedge \\neg Q(y_1, y_3)\\bigr) \\vee P(y_2, v_2, v_3) \\vee \\neg Q(y_3, v_1) $ \nthe sets $\\{y_1, y_3, v_1\\}$ and $\\{y_2, v_2, v_3\\}$ form the maximal sets of interacting existential variables.\nSince each of these sets contains variables from at most two distinct quantifier blocks, the formula exhibits a degree $\\degree = 2$.\n\nIn Section~\\ref{section:DegreeOfInteraction}, and in particular in Theorem~\\ref{theorem:SFComplexityDependingOnDepth}, we observe that the satisfiability problem for $\\SF_{\\degree \\leq k}$---the set of all SF sentences $\\varphi$ with $\\degree_\\varphi \\leq k$---lies in $k$-\\NEXPTIME.\nIt is worth mentioning that this result adequately accounts for the known complexity of MFO-satisfiability.\nFor every MFO sentence $\\varphi$ we trivially have $\\degree_\\varphi = 1$, since all occurring predicate symbols have an arity of at most one.\nTheorem~\\ref{theorem:SFComplexityDependingOnDepth} entails that MFO-satisfiability is in \\NEXPTIME, which is well known.\nStill, this bound is not reproducible with the analysis of the complexity of SF-satisfiability conducted in \\cite{Voigt2016}.\n\\begin{figure}[hb]\n\t\\begin{center}\n\t\t\\begin{picture}(190, 190)\n\t\t\t\\put(0,0){\\framebox(166,185){}}\n\t\t\n\t\t\t\\put(3,0){\\cbezier(0,0)(10,185)(150,185)(160,0)}\n\t\t\t\\put(3,0){\\cbezier(10,0)(20,133)(140,133)(150,0)}\n\t\t\t\\put(3,0){\\cbezier(20,0)(30,80)(130,80)(140,0)}\n\t\t\t\\put(3,0){\\cbezier(30,0)(40,50)(120,50)(130,0)}\n\t\t\t\\put(3,0){\\cbezier(40,0)(50,20)(110,20)(120,0)}\n\t\t\t\n\t\t\t\\put(52,170){\\begin{tabular}{c} \\textsc{Primitive}\\\\ \\textsc{Recursive}\\end{tabular}}\n\t\t\t\\put(81,149){$\\vdots$}\n\t\t\t\\put(54,120){\\textsc{Elementary}}\n\t\t\t\\put(81,105){$\\vdots$}\n\t\t\t\\put(55,82){$k$-\\textsc{NExpTime}}\n\t\t\t\\put(81,66){$\\vdots$}\n\t\t\t\\put(55,44){\\textsc{2-NExpTime}}\n\t\t\t\\put(59,23){\\textsc{NExpTime}}\n\t\t\t\\put(77,3){\\textsc{NP}}\n\t\t\t\n\t\t\t\\put(83,141.5){\\circle*{3}}\n\t\t\t\\put(83,141.5){\\line(1,0){91}}\n\t\t\t\\put(83,98.5){\\circle*{3}}\n\t\t\t\\put(83,98.5){\\line(1,0){91}}\n\t\t\t\\put(83,59){\\circle*{3}}\n\t\t\t\\put(83,59){\\line(1,0){91}}\n\t\t\t\\put(83,36.5){\\circle*{3}}\n\t\t\t\\put(83,36.5){\\line(1,0){91}}\n\t\t\t\\put(83,14){\\circle*{3}}\n\t\t\t\\put(83,14){\\line(1,0){91}}\n\t\t\n\t\t\t\\put(175,138){\\mbox{SF}}\n\t\t\t\\put(175,95){\\mbox{$\\SFd{k}$}}\n\t\t\t\\put(175,56){\\mbox{$\\SFd{2}$}}\n\t\t\t\\put(175,33.5){\\mbox{$\\SFd{1}$}}\n\t\t\t\\put(175,11){\\mbox{$\\SFd{0}$}}\n\t\t\\end{picture}\n\t\t\\caption{The subfragments of SF scale over the major nondeterministic complexity classes in \\textsc{Elementary}, while SF itself goes even beyond. }\n\t\t\\label{figure:ComplexityOfSF}\n\t\\end{center}\n\\end{figure}\nApparently, non-elementary satisfiability problems are not very widespread among the decidable fragments of classical fist-order logic known today.\nWe show in Section~\\ref{section:ComputationalLowerBounds} that SF falls into this category.\nTo the present author's knowledge, the only known companion in this respect is the fluted fragment (FL). \nIndeed, Pratt-Hartmann, Szwast, and Tendera show in \\cite{PrattHartmann2016} that satisfiability of fluted sentences with at most $2k$ variables is $k$-\\NEXPTIME-hard.\nMoreover, they argue that satisfiability of fluted sentences with at most $k$ variables lies in $k$-\\NEXPTIME.\nAlthough a significant gap between these lower and upper bounds remains to be closed, the fluted fragment seems to comprise a similar hierarchy of hard problems as SF does.\n\nAnother characteristic of SF is that it enjoys a \\emph{small model property}.\nMore precisely, given an SF sentence $\\varphi$, one can compute a positive integer $n$ that depends on the degree $\\degree_\\varphi$ and the length of $\\varphi$ such that, if there is a model of $\\varphi$ at all, then there also is a model whose domain contains at most $n$ elements.\nMany first-order fragments are known to enjoy a small model property. The BSR fragment and MFO are among the classical ones (see \\cite{Borger1997} for references). More recently defined fragments include the two-variable fragment (FO$_2$) \\cite{Mortimer1975}, \\cite{Gradel1997}, the fluted fragment (FL) \\cite{Quine1969}, \\cite{Quine1976}, \\cite{PrattHartmann2016}, the guarded fragment (GF) \\cite{Andreka1998}, \\cite{Gradel1999}, the guarded negation fragment (GNF) \\cite{Barany2015}, and the uniform one-dimensional fragment (UF$_1$) \\cite{Kieronski2014}.\nWhile GNF and UF$_1$ are incomparable, GNF extends GF, and UF$_1$ can be considered as a generalization of FO$_2$.\nGuarded fragments and the two-variable fragment have received quite some attention due to the fact that modal logics have natural translations into them.\nAs a continuation of that theme, we shall see in Section~\\ref{section:ExpressivenessOfSF} how classes of sentences enjoying a small model property can be effectively translated into subclasses of SF.\nDuring the translation process the length of formulas increases by a factor that is logarithmic in the size of small models of the original.\nOne benefit of translating non-SF sentences into SF sentences is that in SF one can natively express concepts such as transitivity and basic counting quantifiers (Proposition~\\ref{proposition:CountingQuantifiers}). \nThis is not always possible in other fragments enjoying a small model property. For example, transitivity cannot be expressed in FO$_2$, GF, and FL. \n\nSumming up, the main contributions are:\\newline\n(i) Based on the novel concept of the degree of interaction of existential variables, we substantially refine the existing analysis of the time required to decide SF-satisfiability.\n\tMore concretely, we show that a satisfiable SF sentence $\\varphi$ with $\\degree_\\varphi = k$ has a model whose domain is of a size that is at most $k$-fold exponential in the length of $\\varphi$ (Section~\\ref{section:TranslationSFintoBSR}, Theorem~\\ref{theorem:SFComplexityDependingOnDepth}).\n\tWith this refined approach we can close the complexity gap for the class of \\emph{strongly separated} sentences (Corollary~\\ref{corollary:ComplexitySSF}) that was left open in~\\cite{Voigt2016}. Moreover, the complexity of MFO can be explained in the refined framework. \n\t\\newline\t\n(ii) We complement the complexity analysis with corresponding lower bounds in two respects.\n\tWe first derive a lower bound on the length of shortest BSR sentences that are equivalent to a given SF sentence (Section~\\ref{section:LowerBoundsSmallestBSRsentences}, Theorem~\\ref{theorem:LengthSmallestBSRsentences}).\n\n\tIn Section~\\ref{section:ComputationalLowerBounds}, we prove $k$-\\NEXPTIME-hardness of satisfiability for the class of SF sentences $\\varphi$ with $\\degree_\\varphi = k$ (Theorem~\\ref{theorem:ComputationalLowerBoundForSF}). \n\tSince SF is in general defined without restrictions on the degree $\\degree_\\varphi$, our result implies that SF-satisfiability is non-elementary.\n\t\\newline\n(iii) We devise a simple translation from classes of first-order sentences that enjoy a small model property into SF (Proposition~\\ref{proposition:TranslationFMPintoSF}).\n\tMoreover, we argue that SF can express basic counting quantifiers (Proposition~\\ref{proposition:CountingQuantifiers}).\n\t\n\nIn order to facilitate smooth reading, most proofs are only sketched in the main text and presented in full in the appendix. \nThe present paper is the full version of the extended abstract~\\cite{Voigt2017}. \n\n\n\n\\section{Preliminaries}\\label{section:Preliminaries}\n\nWe mainly reuse the basic notions from \\cite{Voigt2016}. We repeat the definition of necessary concepts and notation for the sake of completeness.\n\nWe consider first-order logic formulas. The underlying signature shall not be mentioned explicitly, but will become clear from the current context. For the distinguished \\emph{equality} predicate we use $\\approx$.\nWe follow the convention that negation binds strongest, that conjunction binds stronger than disjunction, and that all of the aforementioned bind stronger than implication. The scope of quantifiers shall stretch as far to the right as possible.\nBy $\\len(\\cdot)$ we denote a reasonable measure of the length of formulas satisfying $\\len( \\varphi \\rightarrow \\psi ) = \\len( \\neg \\varphi \\vee \\psi )$ and $\\len( \\varphi \\leftrightarrow \\psi ) = \\len( (\\neg \\varphi \\vee \\psi) \\wedge (\\varphi \\vee \\neg \\psi) )$.\n\nWe write $\\varphi(x_1, \\ldots, x_m)$ to denote a formula $\\varphi$ whose free variables form a subset of $\\{x_1, \\ldots, x_m\\}$.\nIn all formulas we tacitly assume that no variable occurs freely and bound at the same time and that no variable is bound by two different occurrences of quantifiers, unless explicitly stated otherwise.\nFor convenience, we sometimes identify tuples $\\vx$ of variables with the set containing all the variables that occur in $\\vx$.\nWe write $\\vars(\\varphi)$ to address the set of all variable symbols that occur in $\\varphi$. Similarly, $\\consts(\\varphi)$ denotes the set of all constant symbols in $\\varphi$.\nWe denote \\emph{substitution} by $\\varphi\\subst{x\/t}$ if every free occurrence of $x$ in $\\varphi$ is to be substituted with the term $t$.\n\nA \\emph{literal} is an atom or a negated atom, and a \\emph{clause} is a disjunction of literals.\nWe say that a formula is in \\emph{conjunctive normal form (CNF)}, if it is a conjunction of clauses, possibly preceded by a quantifier prefix.\nA formula in CNF is \\emph{Horn} if every clause contains at most one non-negated literal.\nIt is \\emph{Krom} if every clause contains at most two literals at all.\n\nA sentence $\\varphi := \\forall \\vx_1 \\exists \\vy_1 \\ldots \\forall \\vx_n \\exists \\vy_n. \\psi$ is in \\emph{standard form}, if it is in \\emph{negation normal form} (i.e.\\ every negation symbol occurs directly before an atom) and $\\psi$ is quantifier free, contains exclusively the Boolean connectives $\\wedge, \\vee, \\neg$, and does not contain non-constant function symbols.\nThe tuples $\\vx_1$ and $\\vy_n$ may be empty, i.e.\\ the quantifier prefix does not have to start with a universal quantifier, and it does not have to end with an existential quantifier.\nMoreover, we require that every variable occurring in the quantifier prefix does also occur in $\\psi$.\n\nAs usual, we interpret a formula $\\varphi$ with respect to given structures. A \\emph{structure} $\\cA$ consists of a nonempty \\emph{universe} $\\fU_\\cA$ (also: \\emph{domain}) and interpretations $f^\\cA$ and $P^\\cA$ of all considered function and predicate symbols, respectively, in the usual way. \nGiven a formula $\\varphi$, a structure $\\cA$, and a variable assignment $\\beta$, we write $\\cA, \\beta \\models \\varphi$ if $\\varphi$ evaluates to \\emph{true} under $\\cA$ and $\\beta$.\nWe write $\\cA \\models \\varphi$ if $\\cA, \\beta \\models \\varphi$ holds for every $\\beta$. \nThe symbol $\\semequiv$ denotes \\emph{(semantic) equivalence} of formulas, i.e.\\ $\\varphi \\semequiv \\psi$ holds whenever for every structure $\\cA$ and every variable assignment $\\beta$ we have $\\cA,\\beta \\models \\varphi$ if and only if $\\cA,\\beta \\models \\psi$. \nWe call two sentences $\\varphi$ and $\\psi$ \\emph{equisatisfiable} if $\\varphi$ has a model if and only if $\\psi$ has one.\n\nA structure $\\cA$ is a \\emph{substructure} of a structure $\\cB$ (over the same signature) if (1) $\\fU_\\cA \\subseteq \\fU_\\cB$, (2) $c^\\cA = c^\\cB$ for every constant symbol $c$, (3) $P^\\cA = P^\\cB \\cap \\fU_\\cA^m$ for every $m$-ary predicate symbol $P$, and (4) $f^\\cA(\\fa_1, \\ldots, \\fa_m) = f^\\cB(\\fa_1, \\ldots, \\fa_m)$ for every $m$-ary function symbol $f$ and every $m$-tuple $\\<\\fa_1, \\ldots, \\fa_m\\> \\in \\fU_\\cA^m$. \nThe following is a standard lemma, see, e.g., \\cite{Ebbinghaus1994} for a proof.\n\n\\begin{lemma}[Substructure lemma]\n\tLet $\\varphi$ be a first-order sentence in prenex normal form without existential quantifiers and let $\\cA$ be a substructure of $\\cB$.\n\n\t$\\cB \\models \\varphi$ entails $\\cA \\models \\varphi$.\n\\end{lemma}\n\n\n\\begin{lemma}[Miniscoping]\\label{lemma:BasicQuantifierEquivalences}\n\tLet $\\varphi, \\psi, \\chi$ be arbitrary first-order formulas, and assume that $x$ does not occur freely in $\\chi$.\\\\\n\t\\centerline{$\n\t\t\\begin{array}{lcl}\n\t\t\t\\exists x. (\\varphi \\vee \\psi) \t&\t\\semequiv\t&\t(\\exists x_1. \\varphi) \\vee (\\exists x_2. \\psi) ~,\\\\\n\t\t\t\\exists x. (\\varphi \\circ \\chi) \t&\t\\semequiv\t&\t(\\exists x. \\varphi) \\circ \\chi ~,\\\\\n\t\t\t\\forall x. (\\varphi \\wedge \\psi) \t&\t\\semequiv\t&\t(\\forall x_1. \\varphi) \\wedge (\\forall x_2. \\psi) ~,\\\\\n\t\t\t\\forall x. (\\varphi \\circ \\chi) \t&\t\\semequiv\t&\t(\\forall x. \\varphi) \\circ \\chi ~,\n\t\t\\end{array}\n\t$}\t\n\twhere $\\circ \\in \\{\\wedge, \\vee\\}$.\n\\end{lemma}\n\nWe use the notation $[k]$ to abbreviate the set $\\{1, \\ldots, k\\}$ for any positive integer $k$.\nMoreover, $\\fP$ shall be used as the power set operator, i.e.\\ $\\fP S$ denotes the set of all subsets of a given set $S$.\nFinally, we need some notation for the \\emph{tetration operation}. We define $\\twoup{k}{m}$ inductively: $\\twoup{0}{m} := m$ and $\\twoup{k+1}{m} := 2^{\\left(\\twoup{k}{m}\\right)}$.\n\n\n\n\n\\section{The separated fragment}\n\n\nLet $\\varphi$ be a first-order formula. We call two disjoint sets of variables $X$ and $Y$ \\emph{separated in $\\varphi$} if and only if for every atom $A$ occurring in $\\varphi$ we have $\\vars(A) \\cap X = \\emptyset$ or $\\vars(A) \\cap Y = \\emptyset$.\n\\begin{definition}[Separated fragment (SF), \\cite{Voigt2016}]\\label{definition:SeparatedFragment}\n\tThe \\emph{separated fragment (SF)} of first-order logic consists of all existential closures of prenex formulas without non-constant function symbols in which existentially quantified variables are separated from universally quantified ones. \n\n\tMore precisely, SF consists of all first-order sentences with equality but without non-constant function symbols that are of the form $\\exists \\vz\\, \\forall\\vx_1 \\exists\\vy_1 \\ldots \\forall\\vx_n \\exists\\vy_n.\\, \\psi$, in which $\\psi$ is quantifier-free, and in which the sets $\\vx_1 \\cup \\ldots \\cup \\vx_n$ and $\\vy_1 \\cup \\ldots \\cup \\vy_n$ are separated.\n\t\n\tThe tuples $\\vz$ and $\\vy_n$ may be empty, i.e.\\ the quantifier prefix does not have to start with an existential quantifier and it does not have to end with an existential quantifier either.\n\\end{definition}\nNotice that the variables in $\\vz$ are not subject to any restriction concerning their occurrences.\n\nIn \\cite{Voigt2016} the authors show that the satisfiability problem for SF sentences (\\emph{SF-satisfiability}) is decidable.\nBefore we start investigating the complexity issues related to SF-satisfiability, we elaborate on the expressiveness of SF.\n\n\n\n\\subsection{Expressiveness}\\label{section:ExpressivenessOfSF}\n\nEvery SF sentence is equivalent to a BSR sentence (\\cite{Voigt2016}, Lemma~6). \nWe shall outline in Section~\\ref{section:TranslationSFintoBSR} how to analyze the blow-up that we have to incur during this translation process and how it depends on the degree of interaction of existential variables.\nSince the BSR fragment enjoys a small model property (cf.\\ Proposition~\\ref{proposition:SmallModelsBSR}), SF inherits the small model property from BSR.\nHowever, regarding the size of minimal models of satisfiable formulas, SF sentences are much more compact.\nWhile satisfiable BSR sentences have models whose domain is linear in the length of the formula, satisfiable SF sentences can enforce domains of a size that cannot be bounded from above by a finite tower of exponentials. \nWe provide first evidence for this fact in Theorem~\\ref{theorem:LengthSmallestBSRsentences}, where we give a non-elementary lower bound on the length of equivalent BSR sentences. \nThis lower bound even applies to the SF-Horn-Krom subfragment of SF.\nMoreover, we exploit the capability of SF sentences $\\varphi$ to enforce models of $\\degree_\\varphi$-fold exponential size in the proof of the $k$-\\NEXPTIME-hardness of SF-satisfiability (for every $k \\geq 1$).\n\nApart from compactness of representation, and from the perspective of satisfiability, all first-order fragments that enjoy small model properties share a common ground of expressiveness.\nNeglecting efficiency, every sentence $\\varphi$ from such a fragment can be effectively translated into a (finite) propositional formula $\\phi$ in such a way that from a satisfying variable assignment for $\\phi$ one can straightforwardly reconstruct a (Herbrand) model of $\\varphi$.\nThe reason is simply that universal quantification can then be understood as finite conjunction (over a finite domain) and existential quantification can be conceived as finite disjunction.\n\nThe following proposition illustrates why SF is to some extent prototypical for first-order fragments that enjoy a small model property. \n\\begin{proposition}\\label{proposition:TranslationFMPintoSF}\n\tConsider any nonempty class $\\cC$ of first-order formulas without non-constant function symbols for which we know a computable mapping $f : \\cC \\to \\Nat$ such that every satisfiable $\\varphi$ in $\\cC$ has a model of size at most $f(\\varphi)$. Then there exists an effective translation $T$ from $\\cC$ into SF such that for every $\\varphi \\in \\cC$\n\t\t\t(a) every model of $T(\\varphi)$ is also a model of $\\varphi$, \n\t\t\t(b) every model of $\\varphi$ whose size is at most $f(\\varphi)$ can be extended to a model of $T(\\varphi)$ over the same domain, and\n\t\t\t(c) the length of $T(\\varphi)$ lies in $\\cO\\bigl( \\len(\\varphi) \\cdot \\log f(\\varphi) \\cdot \\log \\log f(\\varphi) \\bigr)$.\n\\end{proposition}\t\n\\begin{proof}\n\tWe outline the translation $T$ for some given input sentence $\\varphi$, which we assume to be in negation normal form (without loss of generality). \n\tLet $m := \\lceil \\log_2 f(\\varphi) \\rceil$ and let $Q_1, \\ldots, Q_m$ be unary predicate symbols that do not occur in $\\varphi$.\n\tFor all terms $s,t$ we define $s \\happrox t$ as abbreviation of $\\bigwedge_{i = 1}^m Q_i(s) \\leftrightarrow Q_i(t)$.\n\n\tIn order to restrict the domain to $2^m$ elements, we conjoin the formula\n\t\t$\\AxFin := \\forall x y.\\, x \\!\\happrox\\! y \\rightarrow x \\!\\approx\\! y$.\n\tSince in any structure $\\cA$ there are at most $2^m$ domain elements that can be distinguished by their membership in the sets $Q_1^\\cA, \\ldots, Q_m^\\cA$, it is clear that $\\cA \\models \\AxFin$ entails $|\\fU_\\cA| \\leq 2^m$.\n\n\tMoreover, we observe the following property.\n\t\\begin{itemize}\n\t\t\\item[($*$)] \tLet $\\cA$ be any structure, let $\\beta$ be any variable assignment over $\\cA$'s domain, and let $s, t$ be two terms. \n\t\t\tIf $\\cA \\models \\AxFin$ holds, then we get $\\cA, \\beta \\models s \\happrox t$ if and only if $\\cA, \\beta \\models s \\approx t$.\n\t\\end{itemize}\t\n\tThis means, if we restrict our attention to domains with at most $2^m$ domain elements, we can now use a separated form of equality.\n\t\n\t\\begin{itemize}\n\t\t\\item [($**$)] Let $\\psi$ be any first-order formula and let $v$ be some variable that does not occur in $\\psi$. Then $\\psi$ is equivalent to $\\forall v.\\, u \\approx v$ $\\rightarrow\\; \\psi\\subst{u\/v}$. \n\t\\end{itemize}\n\tWe can transform $\\varphi$ into an equivalent sentence $\\varphi'$ by consecutively replacing each subformula of the form $\\exists y. \\psi$ in $\\varphi$ with $\\exists y \\forall v.\\; y \\!\\approx\\! v \\rightarrow \\psi\\subst{y\/v}$, where we assume $v$ to be fresh (one fresh variable for every replaced subformula).\n\tConsequently, every atom in $\\varphi'$ that is not an equation contains exclusively universally quantified variables.\n\tMoreover, ($**$) implies that $\\varphi$ and $\\varphi'$ are equivalent.\n\t\n\tLet $\\varphi''$ be the result of replacing all equations $y \\approx v, v \\approx y$ in $\\varphi'$ in which $y$ is existentially quantified and $v$ universally quantified with the formula $y \\happrox v$.\n\tWe then set $\\varphi_\\SF := \\AxFin \\wedge \\varphi''$.\n\tBy ($*$), any model of $\\varphi_\\SF$ is also a model of $\\varphi$.\n\tConversely, any model $\\cA$ of $\\varphi$ that has at most $2^m$ domain elements can be converted into a model $\\cB$ of $\\varphi_\\SF$ by defining the relations $Q_1^\\cB, \\ldots, Q_m^\\cB$ in an appropriate way.\n\\end{proof}\n\t The $\\varphi_\\SF$ in the above proof belongs to a subfragment of SF that we call \\emph{strongly separated} (cf.\\ Definition~\\ref{definition:StronglySeparatedFragment}) and whose satisfiability problem is complete for \\NEXPTIME\\ (cf.\\ Corollary~\\ref{corollary:ComplexitySSF}).\n\n\nUnfortunately, the translation methodology of Proposition~\\ref{proposition:TranslationFMPintoSF} does not help in the quest for new decidable first-order fragments. \nThe reason is simply that we already need arguments leading to a small model property before we can start the translation process, as we need information about the size of the models that have to be considered. \nNevertheless, such translations can be useful in view of the expressiveness of SF that other first-order fragments, such as FO$_2$, the fluted fragment, and GF, lack.\nFor instance, SF sentences can naturally express the axioms of equivalence, most prominently, transitivity.\nHence, fundamental and interesting properties of predicates that have to be assumed at the meta-level when dealing with less expressive logics can be formalized directly in SF.\nMoreover, basic counting quantifiers can be defined natively in SF and do not have to be introduced via special operators.\nMore precisely, given any formula $\\exists^{\\geq n} y.\\, \\varphi$ with positive $n$ and without non-constant function symbols, its standard translation \n\t$\\exists y_1 \\ldots y_n. \\bigwedge_{i=1}^n \\varphi\\subst{y\/y_i} \\wedge \\bigwedge_{i < j} y_i \\!\\not\\approx\\! y_j$ \nis not in conflict with the separateness conditions of SF's definition, if the set $\\{y_1, \\ldots, y_n\\}$ is separated in $\\varphi$ from the set of universally quantified variables. \n\\begin{proposition}\\label{proposition:CountingQuantifiers}\n\tCounting quantifiers $\\exists^{\\geq n}$ with positive integer $n$ are expressible in SF.\n\\end{proposition}\n\n\n\n\n\\subsection{Basic complexity considerations}\\label{section:ComplexityBasics}\nWe first recall the well-known small model properties of SF's subfragments BSR and MFO (see \\cite{Borger1997} for references).\n\\begin{proposition}\\label{proposition:SmallModelsBSR}\n\tLet $\\varphi := \\exists \\vz\\, \\forall \\vx. \\psi$ be a satisfiable BSR sentence, i.e.\\ $\\psi$ is quantifier free and does not contain non-constant function symbols.\n\tThere is a model $\\cA \\models \\varphi$ such that $|\\fU_\\cA| \\leq \\max \\bigl( |\\vz| + |\\consts(\\varphi)|, 1 \\bigr)$.\n\\end{proposition}\nWe make use of this property when we derive an upper bound on the size of small models for satisfiable SF sentences, as our approach will be based on an effective translation of SF sentences into equivalent BSR sentences.\n\n\\begin{proposition}\\label{proposition:SmallModelsMFO}\n\tLet $\\varphi := \\exists \\vz\\, \\forall\\vx_1 \\exists\\vy_1 \\ldots$ $\\forall\\vx_n \\exists\\vy_n. \\psi$ be a satisfiable monadic sentence without equality and without non-constant function symbols, i.e.\\ all predicate symbols in $\\varphi$ are of arity $1$.\n\tMoreover, assume that $\\varphi$ contains $k$ distinct predicate symbols.\n\tThere is a model $\\cA \\models \\varphi$ such that $|\\fU_\\cA| \\leq 2^{k}$.\n\\end{proposition}\nNotice that the shape of the quantifier prefix does not contribute to the upper bound.\n\nThe following lemma links bounds on the size of models with the computing time that is required to decide satisfiability.\n\\begin{lemma}[cf.\\ \\cite{Borger1997}, Proposition~6.0.4]\n\t\\label{lemma:ComplexityWithSmallModelProperty}\n\tLet $\\varphi$ be a first-order sentence in prenex normal form containing $n$ universally quantified variables.\n\tThe question whether $\\varphi$ has a model of cardinality $m$ can be decided nondeterministically in time $p\\bigl(m^n \\cdot \\len(\\varphi)\\bigr)$ for some polynomial $p$.\n\\end{lemma}\nWith this lemma at hand, it is enough to prove a small model property for a given class of first-order sentences, in order to bound the worst-case time complexity of the corresponding satisfiability problem from above. This is exactly what the authors of \\cite{Voigt2016} have done for SF.\n\n\\begin{proposition}[\\cite{Voigt2016}, Theorem~17]\\label{theorem:SFComplexityOld}\n\tLet $\\varphi := \\exists\\vz\\, \\forall \\vx_1 \\exists \\vy_1 \\ldots \\forall \\vx_n \\exists \\vy_n. \\psi$ be an SF sentence for some quantifier-free $\\psi$.\n\tThere is some equivalent BSR sentence $\\exists \\vu\\, \\forall \\vv. \\psi'$ in which the number of occurring constant symbols plus the number of existential quantifiers is at most $\\len(\\varphi) + n \\cdot \\len(\\varphi) \\cdot \\bigl( \\twoup{n}{\\len(\\varphi)} \\bigr)^n$. \n\tAs a result, satisfiability of $\\varphi$ can be decided nondeterministically in time that is at most $n$-fold exponential in $\\len(\\varphi)$.\n\\end{proposition}\nClearly, applying this result to an MFO sentence substantially overshoots the actual worst-case time requirements.\nTo stress it again, the notion of the degree of interaction is a remedy to this sort of inaccuracies, as we shall see in Section~\\ref{section:TranslationSFintoBSR}.\n\nA special case that is worth considering, before we investigate the complexity of full SF, is the class of SF sentences that do not contain universal quantifiers.\nThis species of formulas coincides with the purely existential fragment of first-order logic without non-constant function symbols, and it is a close relative of propositional logic.\nRecall that SAT is NP-complete \\cite{Cook1971}, Horn-SAT is P-complete \\cite{Kasif1986, Plaisted1984}, and 2SAT is NL-complete \\cite{Jones1976}.\n\\begin{proposition}\\label{proposition:ComplexityExistentialSF}~\n\t\\begin{enumerate}[label=(\\roman{*}), ref=(\\roman{*})]\n\t\t\\item\\label{enum:ComplexityExistentialSF:I} Satisfiability for the class of SF sentences without universal quantifiers is NP-complete.\n\t\t\\item\\label{enum:ComplexityExistentialSF:II} Satisfiability for the class of SF-Horn sentences without universal quantifiers is P-complete.\n\t\t\\item\\label{enum:ComplexityExistentialSF:III} Satisfiability for the class of SF-Krom sentences without universal quantifiers and without equality is NL-complete.\n\t\\end{enumerate}\n\\end{proposition}\n\\begin{proof}[Proof sketch]\n\tThe proof of \\ref{enum:ComplexityExistentialSF:I} -- \\ref{enum:ComplexityExistentialSF:III} proceeds by reductions to the corresponding satisfiability problems for propositional logic and back.\n\tThis is straightforwardly done by Skolemization as long as we consider only SF sentences without equality (cf.\\ Lemma~\\ref{lemma:ComplexitySAT} in the appendix).\n\t\n\tIn the latter cases, we first Skolemize exhaustively, producing $\\varphi_\\gnd$, which is ground and contains only Skolem constants and no non-constant function symbols.\n\tThen we use the standard trick to eliminate the equality predicate $\\approx$.\n\tWe introduce a fresh binary relation symbols $E$ and replace\n\tevery equation $c \\approx d$ with an atom $E(c,d)$.\n\n\tMoreover, we add axioms for reflexivity, symmetry, transitivity, and congruence. \n\tOf course, we do not use the universally quantified axioms but rather add their instances with respect to all the constant symbols that occur in $\\varphi_\\gnd$.\n\tTo avoid an exponential blow-up in the case of the congruence axioms, we only add the instances that affect non-equational atoms which really occur in $\\varphi_\\gnd$.\n\t\n\tLet $\\phi$ be the propositional formula that results from $\\varphi_\\gnd$ by replacing every ground atom $A$ with the propositional variable $p_A$. \n\tWe observe that $\\len(\\phi) \\in \\cO\\bigl( \\len(\\varphi)^3 \\bigr )$.\n\tMoreover, if $\\varphi$ is a Horn formula, then $\\phi$ is Horn.\n\tNotice that the outlined elimination of equality does not preserve the Krom property.\n\\end{proof}\n\n\n\n\n\\section{Translation of SF sentences into BSR sentences}\\label{section:TranslationSFintoBSR}\n\nIn this section, we analyze the transformation process from SF into the BSR fragment from the perspective of the \\emph{degree of interaction of existential variables}.\nOur aim is to derive upper and lower bounds on the length of the resulting BSR-formulas. \nRoughly speaking, in the first phase of the translation process all quantifiers are moved inwards as far as possible (cf.\\ the proof of Lemma~\\ref{lemma:TranslationSFintoBSRunderDegree}).\nIn order to do so, we first transform the sentence in question into a formula in CNF. \nAfter that, we employ the well-known rules of miniscoping (cf.\\ Lemma~\\ref{lemma:BasicQuantifierEquivalences}), supplemented by the rule formulated in the following lemma.\n\n\\begin{lemma}\\label{lemma:AdvancedMiniscoping:refined}\n\tLet $I$ and $K_i$, $i \\in I$, be sets that are finite, nonempty, and pairwise disjoint. \n\tThe elements of these sets serve as indices.\n\tLet \n\t\\[ \\varphi := \\exists \\vy. \\bigwedge_{i \\in I} \\Bigl( \\chi_i(\\vz) \\vee \\bigvee_{k \\in K_i} \\eta_k(\\vy, \\vz) \\Bigr) \\]\n\tbe some first-order formula where the $\\chi_i$ and the $\\eta_k$ denote arbitrary subformulas that we treat as indivisible units in what follows.\n\n\tWe say that $f : I \\to \\bigcup_{i \\in I} K_i$ is a \\emph{selection function} \n\tif for every $i \\in I$ we have $f(i) \\in K_i$.\n\tWe denote the set of all selection functions of this form by $\\cF$.\n\t\n\tThen $\\varphi$ is equivalent to $\\varphi' :=$\n\t\t\\[ \\bigwedge_{\\text{\\scriptsize $\\begin{array}{c} S \\subseteq I \\\\ S \\neq \\emptyset \\end{array}$ \\normalsize}} \\Bigl( \\bigvee_{i \\in S} \\chi_i(\\vz) \\Bigr) \\vee \\bigvee_{f \\in \\cF} \\Bigl( \\exists \\vy. \\bigwedge_{i \\in S} \\eta_{f(i)} (\\vy, \\vz) \\Bigr) ~. \\]\n\\end{lemma}\n\\begin{proof}[Proof sketch]\nThe proof of this lemma follows a conceptually simple strategy.\nUsing the distributivity of $\\wedge$ over $\\vee$, we first transform $\\varphi$ into a disjunction of conjunctions of the indivisible units $\\chi_i(\\vz)$ and $\\eta_k(\\vy, \\vz)$.\nThen, exploiting the equivalences in Lemma~\\ref{lemma:BasicQuantifierEquivalences}, we push the existential quantifier block $\\exists \\vy$ inwards such that it only binds conjunctions of units $\\eta_k(\\vy,\\vz)$. This is possible, because none of the variables in $\\vy$ occurs in any of the $\\chi_i(\\vz)$.\nFrom this point on, we treat the newly emerged subformulas $\\exists \\vy. \\bigwedge_{k'} \\eta_{k'}(\\vy,\\vz)$ as if they were indivisible.\nWe then transform the formula back into a conjunction of disjunctions of indivisible units, this time using the distributivity of $\\vee$ over $\\wedge$.\nIt then remains to show that the result of this transformation exhibits a highly redundant structure and is actually equivalent to $\\varphi'$.\n\\end{proof}\n\n\n\\subsection{Degree of interaction of existential variables and the size of small models}\\label{section:DegreeOfInteraction}\n\nConsider the formula $\\varphi := \\exists \\vz\\, \\forall \\vx_1 \\exists \\vy_1 \\ldots$ $\\forall \\vx_n \\exists \\vy_n. \\psi$ in standard form in which $\\psi$ is quantifier free and in which the sets $\\vx := \\vx_1 \\cup \\ldots \\cup \\vx_n$ and $\\vy := \\vy_1 \\cup \\ldots \\cup \\vy_n$ are separated. In addition, we assume that $\\vx_1$ and $\\vy_1$ are nonempty. The tuple $\\vz$, on the other hand, may be empty.\n\nFor any $j \\in [n]$ and any variable $y \\in \\vy_j$ we say that \\emph{$y$ is a level-$j$ variable}, denoted $\\lvl(y) = j$.\nFor any nonempty set $Y \\subseteq \\vy$ of existentially quantified variables and any positive integer $k$ we say that \\emph{$Y$ has degree $k$ in $\\varphi$}, denoted $\\degree_{Y,\\varphi} = k$, if $k$ is the maximal number of distinct variables $y_1, \\ldots, y_k \\in Y$ that belong to different levels in $\\varphi$, i.e.\\ $\\lvl(y_1) < \\ldots < \\lvl(y_k)$.\nWe say that \\emph{$\\varphi$'s degree of interaction of existential variables} (short: \\emph{degree}) is $k$, denoted $\\degree_\\varphi = k$, if $k$ is the smallest positive integer such that we can partition $\\vy$ into $m > 0$ parts $Y_1, \\ldots, Y_m$ that are all pairwise separated in $\\varphi$ and for which $k = \\max\\bigl\\{ k_j \\bigm| \\degree_{Y_j, \\varphi} = k_j, 1 \\leq j \\leq m \\bigr\\}$.\nSentences $\\varphi := \\exists \\vz\\, \\forall \\vx.\\, \\psi$ in standard form with quantifier-free $\\psi$ are said to have \\emph{degree} zero, i.e.\\ $\\degree_\\varphi = 0$, if $\\vx$ is empty and we define $\\degree_\\varphi = 1$ if $\\vx$ is nonempty.\n\n\\begin{lemma}\\label{lemma:TranslationSFintoBSRunderDegree}\n\tLet $\\varphi := \\exists \\vz\\, \\forall \\vx_1 \\exists \\vy_1 \\ldots \\forall \\vx_n \\exists \\vy_n. \\psi$ be an SF sentence of positive degree $\\degree_\\varphi$ in standard form.\n\tLet $\\cL_\\varphi (\\vy)$ denote the set of all literals in $\\varphi$ that contain at least one variable $y \\in \\vy := \\vy_1 \\cup \\ldots \\cup \\vy_n$.\n\tThere exists a sentence $\\varphi_{\\text{BSR}} = \\exists \\vz\\, \\exists \\vu\\, \\forall \\vv. \\psi_{\\text{BSR}}$ in standard form with quantifier-free $\\psi_{\\text{BSR}}$ that is equivalent to $\\varphi$ and contains at most $|\\vz| + |\\vy| \\cdot \\degree_\\varphi \\cdot \\bigl( \\twoup{\\degree_\\varphi}{|\\cL_\\varphi (\\vy)|} \\bigr)^{\\degree_\\varphi}$ leading existential quantifiers.\n\\end{lemma}\n\\begin{proof}[Proof sketch]\n\tLet $\\vx := \\vx_1 \\cup \\ldots \\cup \\vx_n$.\n\n\tWe transform $\\varphi$ into CNF and then apply Lemma~\\ref{lemma:AdvancedMiniscoping:refined} and the rules of miniscoping given in Lemma~\\ref{lemma:BasicQuantifierEquivalences} to push all quantifier blocks inwards. \n\tSince the sets $\\vx$ and $\\vy$ are separated in $\\varphi$, these operations can be performed in such a way that in the resulting formula $\\varphi'$ no universal quantifier lies within the scope of any existential quantifier (other than the ones in $\\exists \\vz$) and vice versa.\n\tAfter removing redundant parts from $\\varphi'$, the depth of nestings of existential quantifier blocks (interspersed with conjunctive connectives in $\\varphi'$'s syntax tree) can be upper bounded by $\\degree_\\varphi$.\n\tAs a consequence, $\\varphi'$ contains at most $\\twoup{\\degree_\\varphi}{|\\cL_\\varphi(\\vy)|}$ distinct subformulas that are of the form $\\exists y. \\psi'$ and do not lie within the scope of any quantifier.\n\n\tAfter further transformations, we obtain a formula $\\varphi'' := \\bigvee_{k} \\bigl( \\chi_k(\\vx) \\wedge \\bigwedge_{r_k} \\eta_{r_k}(\\vy) \\bigr)$ where the $r_k$ range over at most $\\twoup{\\degree_\\varphi}{|\\cL_\\varphi(\\vy)|}$ indices.\n\tMoreover, every constituent $\\bigwedge_{r_k} \\eta_{r_k}$ in $\\varphi''$ contains at most $|\\vy| \\cdot \\sum_{k'=1}^{\\degree_\\varphi} \\prod_{d = k'}^{\\degree_\\varphi} \\twoup{d}{|\\cL_\\varphi(\\vy)|}$ occurrences of existential quantifiers.\n\n\tSince these existential quantifiers distribute over the topmost disjunction when we move them outwards to the front of the sentence $\\varphi''$, and since the universal quantifiers in the $\\chi_k$ may also be moved back outwards, one can show that $\\varphi$ is equivalent to some BSR sentence with at most $|\\vy| \\cdot \\degree_\\varphi \\cdot \\bigl( \\twoup{\\degree_\\varphi}{|\\cL_\\varphi(\\vy)|} \\bigl)^{\\degree_\\varphi}$ leading existential quantifiers.\n\\end{proof}\n\nProposition~\\ref{proposition:SmallModelsBSR} now entails that any satisfiable SF-sentence $\\varphi$ has a model of size at most \n\t\\begin{equation}\\label{eqn:ModelSizeDependingOnDepth}\n\t\t\\len(\\varphi) + \\len(\\varphi) \\cdot \\degree_\\varphi \\cdot \\bigl(\\twoup{\\degree_\\varphi}{\\len(\\varphi)}\\bigr)^{\\degree_\\varphi} ~.\n\t\\end{equation}\t\n\\begin{theorem}\\label{theorem:SFComplexityDependingOnDepth}\n\tLet $k$ be any positive integer.\n\tThe satisfiability problem for the class of SF sentences $\\varphi$ in standard form with degree $\\degree_\\varphi \\!\\leq\\! k$ can be decided in nondeterministic $k$-fold exponential time.\t\n\\end{theorem}\nTogether with Proposition~\\ref{proposition:ComplexityExistentialSF}\\ref{enum:ComplexityExistentialSF:I}, this establishes the upper bounds depicted in Figure~\\ref{figure:ComplexityOfSF}.\n\nIn cases where $\\degree_\\varphi = 1$, Expression~(\\ref{eqn:ModelSizeDependingOnDepth}) simplifies to $\\len(\\varphi) + \\len(\\varphi) \\cdot 2^{\\len(\\varphi)}$.\nThe syntactic class of sentences satisfying this property is called \\emph{strongly separated} in \\cite{Voigt2016}.\n\\begin{definition}[\\cite{Voigt2016}]\\label{definition:StronglySeparatedFragment}\n\tLet $\\varphi := \\forall \\vx_1 \\exists \\vy_1 \\ldots$ $\\forall \\vx_n \\exists \\vy_n. \\psi$ be an SF sentence and assume that $\\psi$ is quantifier free.\n\n\tWe say that $\\varphi$ belongs to the \\emph{strongly separated fragment (SSF)} if and only if the sets $\\vx := \\vx_1 \\cup \\ldots \\cup \\vx_n$ and $\\vy_1, \\ldots, \\vy_n$ are all pairwise separated in $\\varphi$.\t\n\\end{definition}\nSince MFO and BSR sentences fall into this syntactic category, and since their decision problem is known to be \\NEXPTIME-hard, we obtain the following corollary.\n\n\\begin{corollary}\\label{corollary:ComplexitySSF}\n\tThe satisfiability problem for SSF is \\NEXPTIME-complete.\n\\end{corollary}\n\n\n\nNotice that the presented method can explain the asymptotic complexity of MFO-satisfiability and yields a reasonable upper bound on the size of small models of satisfiable MFO sentences.\nThis works in spite of the fact that monadic sentences may contain arbitrarily nested alternating quantifiers.\nThis is a considerable improvement compared to the methods used in~\\cite{Voigt2016}.\n\nLet $\\varphi$ be any SF sentence with the maximally possible degree $\\degree_\\varphi = n$, where $n$ is the number of occurring $\\forall\\exists$-alternations. Then the upper bound shown in Expression~(\\ref{eqn:ModelSizeDependingOnDepth}) regarding the number of elements in small models fits the corresponding result entailed by Proposition~\\ref{theorem:SFComplexityOld}. \nAs one consequence, Theorem~\\ref{theorem:SFComplexityDependingOnDepth} in the present paper subsumes Theorem~17 in~\\cite{Voigt2016}.\nMoreover, Corollary~\\ref{corollary:ComplexitySSF} improves the double exponential upper bound on SSF-satisfiability given in Theorem~15 in~\\cite{Voigt2016}.\nFinally, it is worth noticing that all SF sentences with the quantifier prefix $\\exists^* \\forall^* \\exists^* \\forall^*$ belong to the strongly separated fragment. \nHence, Corollary~\\ref{corollary:ComplexitySSF} subsumes Theorem~14 in \\cite{Voigt2016}. The latter stipulates \\NEXPTIME-completeness of SF sentences with quantifier prefix $\\exists^* \\forall^* \\exists^*$.\nClearly, the refined analysis based on the degree of interaction of existential variables, rather than the number of quantifier alternations, yields significantly tighter results in many cases.\n\n\n\n\n\\subsection{Lower bounds on the length of equivalent BSR formulas}\\label{section:LowerBoundsSmallestBSRsentences}\n\nBefore we derive lower bounds on the time that is required to decide SF-satisfiability in the worst case, we establish lower bounds on the length of the results of the translation from SF into the BSR fragment.\n\\begin{theorem}\\label{theorem:LengthSmallestBSRsentences}\n\tThere is a class of SF sentences that are Horn and Krom such that for every positive integer $n$ the class contains a sentence $\\varphi$ of degree $\\degree_\\varphi = n$ and with a length linear in $n$ for which any equivalent BSR sentence contains at least $\\sum_{k=1}^{n}\\twoup{k}{n}$ leading existential quantifiers.\n\\end{theorem}\n\\begin{proof}[Proof sketch]\n\tRecall that $[n]$ abbreviates the set $\\{1, \\ldots, n\\}$ and that $\\fP S$ denotes the power set of a given set $S$. \n\tLet $n \\geq 1$ be some positive integer.\n\tConsider the following first-order sentence in which the sets $\\{x_1, \\ldots, x_n\\}$ and $\\{y_1, \\ldots, y_n\\}$ are separated:\n\t\t\\[ \\varphi := \\forall x_n \\exists y_n \\ldots \\forall x_1 \\exists y_1. \\bigwedge_{i=1}^{4n} \\bigl( P_i(x_1, \\ldots, x_n) \\leftrightarrow Q_i(y_1, \\ldots, y_n) \\bigr) ~.\\]\n\tNotice that we change the orientation of the indices in the quantifier prefix in this proof.\n\t\n\tIn order to construct a particular model of $\\varphi$, we inductively define the following sets:\n\t\t$\\cS_1 := \\bigl\\{ S \\subseteq [4n] \\bigm| |S| = 2n \\bigr\\}$, $\\cS_{k+1} := \\bigl\\{ S \\in \\fP \\cS_k \\bigm| |S| = \\tfrac{1}{2} \\cdot |\\cS_k| \\bigr\\}$ for every $k$, $1 < k \\leq n$.\n\tHence, we observe\\\\\n\t\t$|\\cS_1| = {{4n} \\choose {2n}} \\geq \\bigl( \\frac{4n}{2n} \\bigr)^{2n} = 2^{2n}$,\\\\\n\t\t$|\\cS_2| = {{|\\cS_1|} \\choose {|\\cS_1|\/2}} \\geq \\bigl( \\frac{|\\cS_1|}{|\\cS_1|\/2} \\bigr)^{|\\cS_1|\/2} \\geq 2^{2^{2n-1}}$,\\\\\n\t\t$\\strut\\qquad\\vdots$\\\\\n\t\t$|\\cS_n| = {{|\\cS_{n-1}|} \\choose {|\\cS_{n-1}|\/2}} \\geq 2^{2^{2^{\\vdots^{2^{2n-1}-1}}-1}} \\geq \\twoup{n}{n+1}$.\\\\\n\tWe now define the structure $\\cA$ as follows:\t\n\t\t\\begin{itemize}\n\t\t\t\\item $\\fU_\\cA := \\bigcup_{k = 1}^{n} \\bigl\\{ \\fa^{(k)}_{S}, \\fb^{(k)}_{S} \\bigm| S \\in \\cS_k \\bigr\\}$, \n\t\t\t\\item $P_i^\\cA := \\bigl\\{ \\<\\fa^{(1)}_{S_1}, \\ldots, \\fa^{(n)}_{S_n}\\> \\in \\fU_\\cA^n \\bigm| i \\in S_1 \\in S_2 \\in \\ldots \\in S_n \\bigr\\}$ for $i = 1, \\ldots, 4n$, and\n\t\t\t\\item $Q_i^\\cA := \\bigl\\{ \\<\\fb^{(1)}_{S_1}, \\ldots, \\fb^{(n)}_{S_n}\\> \\in \\fU_\\cA^n \\bigm| i \\in S_1 \\in S_2 \\in \\ldots \\in S_n \\bigr\\}$ for $i = 1, \\ldots, 4n$.\n\t\t\\end{itemize}\t\t\n\tOne can easily show that $\\cA$ is a model of $\\varphi$.\n\tMoreover, employing a game-theoretic argument, one can show the following property:\n\t\\begin{itemize}\n\t\t\\item[($*$)] the substructure induced by $\\cA$'s domain after removing at least one of the $\\fb^{(k)}_{S}$ does not satisfy $\\varphi$.\n\t\\end{itemize}\n\t\n\tWe know that $\\fU_\\cA$ contains at least\n\t\t$\\sum_{k=1}^{n}\\twoup{k}{n}$\n\telements of the form $\\fb^{(k)}_{S}$.\n\t\t\n\tUsing ($*$) and the substructure lemma, one can argue that any BSR sentence $\\varphi_*$ that is semantically equivalent to $\\varphi$ must contain at least $\\sum_{k=1}^{n}\\twoup{k}{n}$ leading existential quantifiers.\n\t\n\tThe key idea is that $\\varphi_*$, which is satisfied by $\\cA$, must contain one existential quantifier for each and every $\\fb^{(k)}_{S}$.\n\tOtherwise, there would be one $\\fb^{(k)}_{S}$, call it $\\fb_*$, such that we could remove $\\fb_*$ from $\\cA$'s domain and any tuple $\\< \\ldots, \\fb_*, \\ldots \\>$ from the sets $Q_i^\\cA$, and the resulting structure would then still be a model of $\\varphi_*$.\n\tBut this would contradict ($*$).\n\t\t\n\tSince every atom $Q_i(y_1, \\ldots, y_n)$ contains $n$ variables from existential quantifier blocks that are separated by universal ones, the degree $\\degree_\\varphi$ of $\\varphi$ is $n$. \n\tMoreover, $\\varphi$ can easily be transformed into a CNF that is Horn and Krom at the same time.\n\t\n\tHence, the theorem holds.\n\\end{proof}\n\n\nTheorem~\\ref{theorem:LengthSmallestBSRsentences} entails that there is no elementary upper bound on the length of the BSR sentences that result from an equivalence-preserving transformation of SF sentences into BSR. On the other hand, by Lemma~\\ref{lemma:TranslationSFintoBSRunderDegree}, there is an elementary upper bound, if we only consider SF sentences up to a certain degree.\n\n\n\n\n\\section{Lower bounds on the computational complexity of SF-satisfiability}\\label{section:ComputationalLowerBounds}\n\n\nIn this section we establish lower bounds on the worst-case time complexity of SF-satisfiability.\nOur arguments will be based on a particular form of bounded domino (or tiling) problems developed by Gr\\\"adel (see \\cite{Gradel1990b} and \\cite{Borger1997}, Section~6.1.1).\nBy $\\Int_t$ we denote the set of integers $\\{0, \\ldots, t-1\\}$ for any positive $t \\geq 1$.\n\\begin{definition}[\\cite{Borger1997}, Definition 6.1.1]\n\tA \\emph{domino system} $\\fD := \\<\\cD, \\cH, \\cV\\>$ is a triple where $\\cD$ is a finite set of tiles and $\\cH, \\cV \\subseteq \\cD \\times \\cD$ are binary relations determining the allowed horizontal and vertical neighbors of tiles, respectively. Consider the torus $\\Int_t^2 := \\Int_t \\times \\Int_t$ and let $\\bD := D_0 \\ldots D_{n-1}$ be a word over $\\cD$ of length $n \\leq t$. The letters of $\\bD$ represent tiles.\n\tWe say that $\\fD$ \\emph{tiles the torus $\\Int_t^2$ with initial condition $\\bD$} if and only if there exists a mapping $\\tau : \\Int_t^2 \\to \\cD$ such that for every $\\ \\in \\Int_t^2$ the following conditions hold, where ``$+1$'' denotes increment modulo $t$.\n\t\\begin{enumerate}[label=(\\alph{*}), ref=(\\alph{*})]\n\t\t\\item If $\\tau(x,y) = D$ and $\\tau(x+1, y) = D'$, then $\\ \\in \\cH$.\n\t\t\\item If $\\tau(x,y) = D$ and $\\tau(x, y+1) = D'$, then $\\ \\in \\cV$.\n\t\t\\item $\\tau(i,0) = D_{i}$ for $i = 0, \\ldots, n-1$.\n\t\\end{enumerate}\n\\end{definition}\n\n\\begin{definition}[\\cite{Borger1997}, Definition 6.1.5]\n\tLet $T : \\Nat \\to \\Nat$ be a function and let $\\fD := \\<\\cD, \\cH, \\cV\\>$ be a domino system.\n\tThe problem $\\Domino(\\fD, T(n))$ is the set of those words $\\bD$ over the alphabet $\\cD$ for which $\\fD$ tiles $\\Int_{T(|\\bD|)}^2$ with initial condition $\\bD$.\n\\end{definition}\n\nDomino problems provide a convenient way of deriving lower bounds via reductions. \nSuppose we are given some well-behaved time bound $T(n)$ that grows sufficiently fast.\nFurther assume there is a reasonable translation from $\\Domino(\\fD, T(n))$ into some problem $\\cL$ where the length of the results is upper bounded by a function $g(n)$.\nIt follows that the time required to solve the hardest instances of $\\cL$ lies in $\\Omega \\bigl( T(h(n)) \\bigr)$, where $h(n)$ may be conceived as an inverse of $g(n)$ from an asymptotic point of view.\nThe next proposition formalizes this observation.\n\\begin{proposition}[\\cite{Borger1997}, Theorem 6.1.8]\\label{theorem:ComputationalLowerBoundViaPolynomialReduction}\n\tLet $T: \\Nat\\!\\to\\!\\Nat$ be a time-constructible function with $T(c' n)^2 \\in o(T(n))$ for some constant $c' > 0$ and let $\\cL$ be a problem such that for every domino system $\\fD$ we have $\\Domino(\\fD,T(n)) \\leq_{g(n)} \\cL$, i.e.\\ $\\Domino(\\fD,T(n))$ is \\emph{polynomially reducible} to $\\cL$ via length order g(n) (cf.\\ Definition 6.1.7 in \\cite{Borger1997}).\n\tMoreover, let $h: \\Nat \\to \\Nat$ be a function such that $h(d \\cdot g(n)) \\in \\cO(n)$ for any positive $d$.\n\tThere exists a positive constant $c > 0$ such that $\\cL \\not\\in \\Ntime(T(c \\cdot h(n)))$.\n\\end{proposition}\n\n\nSubsections~\\ref{section:EnforcingLargeModels} and \\ref{section:FormalizingTheTiling} are devoted to the purpose of outlining the following reductions.\n\\begin{lemma}~\n\t\\begin{enumerate}[label=(\\roman{*})]\n\t\t\\item\n\t\t\tFix some positive integer $k>0$ and let $\\fD$ be an arbitrary domino system. \n\t\t\tLet $\\SatSFdk$ be the set containing all satisfiable SF sentences whose degree $\\degree$ is at most $k$.\n\t\t\n\t\t\tWe have $\\Domino\\bigl( \\fD, \\twoup{k}{n} \\bigr) \\leq_{n \\cdot \\log n} \\SatSFdk$.\n\t\t\n\t\t\\item \n\t\t\tFix some positive integer $m > 1$ and let $\\fD$ be an arbitrary domino system. \n\t\t\tLet $\\SatSF$ be the set containing all satisfiable SF sentences.\n\t\t\n\t\t\tWe have $\\Domino(\\fD, \\twoup{n}{m}) \\leq_{n^2 \\cdot \\log n} \\SatSF$.\n\t\\end{enumerate}\t\n\\end{lemma}\nHaving these reduction results at hand, Proposition~\\ref{theorem:ComputationalLowerBoundViaPolynomialReduction} implies the sought lower bounds on SF-satisfiability for classes of sentences with bounded degree and the class of unbounded SF sentences.\n\\begin{theorem}\\label{theorem:ComputationalLowerBoundForSF}\n\tThere are positive constants $c, d > 0$ for which\n\t\t\\[ \\SatSFdk \\not\\in \\Ntime \\bigl( \\twoup{k}{c n \/ \\log n} \\bigr) \\]\n\tand\n\t\t\\[ \\SatSF \\not\\in \\Ntime \\bigl( \\twoup{d \\cdot \\sqrt{n \/ \\log n}}{2} \\bigr) ~. \\]\n\\end{theorem}\nThese lower bounds also hold if we do not allow equality in SF, see Section~\\ref{section:ReplaceEqualityInLowerBoundProof}.\nThe remainder of Section~\\ref{section:ComputationalLowerBounds} is devoted to the formalization of sufficiently large tori in SF and to the translation from a given domino system $\\fD = \\<\\cD, \\cH, \\cV\\>$ (for nonempty $\\cD, \\cH, \\cV$) plus an initial condition $\\bD$ into an SF sentence $\\varphi$ such that $\\varphi$ is satisfiable if and only if $\\bD \\in \\Domino(\\fD, T_i(|\\bD|))$ with $T_1(n) = \\twoup{\\kappa}{n}$ for any given $\\kappa > 0$ and $T_2(n) = \\twoup{n}{\\mu}$ for any given $\\mu>1$.\n\n\n\n\\subsection{Enforcing a large domain}\\label{section:EnforcingLargeModels}\n\nThe following description gives a somewhat simplified view. Technical details will follow.\n\nA crucial part in the reduction is that a grid of size $t \\times t$ has to be encoded, where $t$ defines the required computing time and we assume $t := \\twoup{\\kappa}{\\mu}$ for positive integers $\\kappa$ and $\\mu > 1$ that we consider as parameters of the construction.\n\nEvery point $p$ on the grid is represented by a pair $p = \\$, where each of the coordinates $x$ and $y$ is represented by a bit string of length $\\log \\bigl(\\twoup{\\kappa}{\\mu}\\bigr) = \\twoup{\\kappa-1}{\\mu}$.\nGiven a bit string $\\overline{b}$, we represent the $j$-th bit $b_j$ by the truth value of the atom $J(\\ul{\\kappa}, \\overline{b}, j)$, where $\\ul{\\kappa}$ is the constant used to address the topmost level of a hierarchy of $\\kappa+1$ sets of indices.\nThe crux of our approach is that we have to enforce the existence of sufficiently many indices $j$, namely $\\twoup{\\kappa-1}{\\mu}$ many, to address the single bits of $\\overline{b}$.\nAgain, we address each of these indices as a bit string, this time of length $\\twoup{\\kappa-2}{\\mu}$.\n\nThus, we proceed in an inductive fashion, building up a hierarchy of indices with $\\kappa+1$ levels.\nThe lowest level, level zero, is inhabited by $\\mu$ indices, which we represent as constants with distinct values.\nFor every $\\ell \\geq 1$ any index $j$ on the $\\ell$-th level is represented by a bit string consisting of $\\twoup{\\ell-1}{\\mu}$ bits, i.e.\\ the $\\ell$-th level of the index hierarchy contains $\\twoup{\\ell}{\\mu}$ indices. The $i$-th bit of an $\\ell$-th-level index $j$ corresponds to the truth value of the atom $J(\\ul{\\ell}, j,i)$.\n\n\\begin{example}\n\tAssume $\\mu = 2$ and $\\kappa = 3$.\\\\\n\t\\centerline{\n\t\t\\begin{tabular}{c|c|c}\n\t\t\tindex\t& \tset of \t\t\t\t\t\t\t\t\t\t& \tnumber\\\\\n\t\t\t level\t&\tindices\t\t\t\t\t\t\t\t\t&\tof indices\t\\\\\n\t\t\t\\hline\n\t\t\t0\t\t&\t$\\{\\fc_1, \\fc_2\\} \\hfill$\t\t\t\t\t\t&\t2\t\t\t\\\\\n\t\t\t1\t\t&\t$\\{00,01,10,11\\} \\hfill$\t\t\t\t\t\t\t&\t4\t\t\t\\\\\n\t\t\t2\t\t&\t$\\{0000, 0001, \\ldots, 1111\\} \\hfill$\t\t\t&\t16\t\t\t\\\\\n\t\t\t3\t\t&\t$\\{0, 1\\}^{16}$\t\t\t\t\t\t\t\t&\t$65536$\t\t\n\t\t\\end{tabular}\n\t}\t\n\tOn every index level, the bits of one index are indexed by the indices from the previous level. We illustrate this for the word $1010$ on all levels from 2 down to 0. The bits of 1010 on level two are indexed by bit strings from level one, each of them having a length of two. The bits of the indices of level one are themselves indexed by objects of level zero which are some values $\\fc_1, \\fc_2$ assigned to the constants $c_1, c_2$. To improve readability, we connect the bits of words by dashes.\\\\[0.5ex]\n\t\\centerline{\n\t\t\\begin{tabular}{l@{\\hspace{1ex}}l@{\\hspace{-0.75ex}}c@{}l@{}c@{}l@{\\hspace{-0.75ex}}c@{}l@{}c@{}l@{\\hspace{-0.75ex}}c@{}l@{}c@{}l@{\\hspace{-0.75ex}}c@{}l}\n\t\t\tlevel 2:\t&\t1\t\t&|&\t||\t\t&|& \t0 \t\t&|&\t||\t\t&|&\t1\t\t&|&\t||\t\t&|&\t0\t\t\t\\\\\n\t\t\t\t\t&\t$\\uparrow$\t&&\t\t\t&&\t$\\uparrow$\t&&\t\t\t&&\t$\\uparrow$\t&&\t\t\t&&\t$\\uparrow$\t\t\\\\\n\t\t\tlevel 1:\t&\t0\t\t&|&\t0\t\t&&\t0\t\t&|&\t1\t\t&&\t1\t\t&|&\t0\t\t&&\t1\t\t&|&\t1\t\\\\\t\n\t\t\t\t\t&\t$\\uparrow$\t&&\t$\\uparrow$\t&&\t$\\uparrow$\t&&\t$\\uparrow$\t&&\t$\\uparrow$\t&&\t$\\uparrow$\t&&\t$\\uparrow$\t&&\t$\\uparrow$\t\\\\\n\t\t\tlevel 0:\t&\t$\\fc_1$\t&&\t$\\fc_2$\t&&\t$\\fc_1$\t&&\t$\\fc_2$\t&&\t$\\fc_1$\t&&\t$\\fc_2$\t&&\t$\\fc_1$\t&&\t$\\fc_2$\t\n\t\t\\end{tabular}\n\t}\t\n\\end{example}\nFor technical reasons the number of indices per level grows slightly slower than described above (cf.\\ Lemma~\\ref{lemma:IndexSetsProperties}).\nThe described index hierarchies can be encoded by SF formulas with the quantifier prefix $\\exists^*(\\forall\\exists)^{\\kappa}$ that have a length that is polynomial in $\\kappa$ and $\\mu$.\nWe use the following constant and predicate symbols with the indicated meaning:\\\\\n\\centerline{\n\t\\begin{tabular}{ll}\n\t\t$\\ul{0}, \\ul{1}, \\ldots, \\ul{\\kappa}$\t&\tconstants denoting the levels from $0$ to $\\kappa$, \\\\\n\t\t$c_1, \\ldots, c_\\mu$\t\t\t&\tdenote the indices at level 0, \\\\\n\t\t$d_1, \\ldots, d_\\kappa$\t\t\t&\t$d_\\ell$ is the min.\\ index at level $\\ell$, \\\\\n\t\t$e_1, \\ldots, e_\\kappa$\t\t\t&\t$e_\\ell$ is the max.\\ index at level $\\ell$, \\\\\n\t\t$L(\\ul{\\ell}, j)$\t\t\t\t&\tindex $j$ belongs to level $\\ell$, \\\\\n\t\t$\\MinIdx(\\ul{\\ell}, j)$\t\t\t& \t$j$ is a min.\\ index at level $\\ell$, \\\\\n\t\t$\\MaxIdx(\\ul{\\ell}, j)$\t\t\t& \t$j$ is a max.\\ index at level $\\ell$, \\\\\n\t\t$J(\\ul{\\ell}, j, i, b)$\t\t\t\t&\tthe $i$-th bit of the index $j$ at level $\\ell$ is $b$, \\\\\n\t\t$J^*(\\ul{\\ell}, j, i, b)$\t\t\t&\t$b=1$ indicates that all the bits of the index $j$ that \\\\\n\t\t\t\t\t\t\t\t&\tare less significant than $j$'s $i$-th bit are 1, \\\\ \n\t\t$\\Succ(\\ul{\\ell}, j, j')$\t\t\t&\t$j'$ is the successor index of $j$ at level $\\ell$.\n\t\\end{tabular}\n}\nOn every level we establish an ordering over the indices of that level.\nWe use the usual ordering on natural numbers encoded in binary.\nMoreover, we formalize the usual successor relation on these numbers by the predicate $\\Succ$.\n\nOne difficulty that we encounter is that we cannot assert the existence of successors simply by adding $\\forall j \\exists j'.\\, \\Succ(\\ul{\\ell}, j, j')$, as $j$ and $j'$ would not be separated.\nTherefore, we fall back on a trick: we start from the equivalent formula $\\forall j \\exists \\tj j'.\\, j \\!\\approx\\! \\tj \\wedge \\Succ(\\ul{\\ell}, \\tj, j')$, and replace the atom $j \\!\\approx\\! \\tj$ by a subformula $\\eq{\\ell}_{j,\\tj}$ in which $j$ and $\\tj$ are separated and which expresses a certain similarity between $j$ and $\\tj$ instead of identity.\nHowever, we specify the hierarchy of indices in a sufficiently strong way such that the similarity expressed by $\\eq{\\ell}_{j,\\tj}$ actually coincides with identity.\n\nNext, we formalize the described index hierarchies in $\\SF_{\\degree \\leq \\kappa}$.\nEvery formula is accompanied by a brief description of its purpose.\nWe shall try to use as few non-Horn sentences as possible.\n\n\\begin{align*}\n\t\\psi_1 \\!:=\n\t\t& \\bigwedge_{\\ell = 0}^{\\kappa} \\bigwedge_{\\text{\\scriptsize $\\begin{array}{c} \\ell' = 0 \\\\ \\ell' \\neq \\ell \\end{array}$ \\normalsize}}^{\\kappa} \\forall j.\\; L(\\ul{\\ell}, j) \\rightarrow \\neg L(\\ul{\\ell}', j) \\\\\n\t&\\hspace{-6ex}\\text{Every index belongs to at most one level.}\\\\\n\t\\psi_2 \\!:=\n\t\t& \\bigwedge_{\\ell = 0}^{\\kappa} \\bigl( \\forall j.\\; \\MinIdx(\\ul{\\ell}, j) \\rightarrow L(\\ul{\\ell}, j) \\bigr) \\;\\;\\wedge\\;\\; \\bigl( \\forall j j'.\\; \\MinIdx(\\ul{\\ell}, j) \\rightarrow \\neg \\Succ(\\ul{\\ell}, j', j) \\bigr) \\\\\n\t&\\hspace{-6ex}\\text{A min.\\ index of level $\\ell$ belongs to level $\\ell$. A min.\\ index does not have a predecessor.} \\\\\n\t\\psi_3 \\!:=\n\t\t& \\bigwedge_{\\ell = 0}^{\\kappa} \\MinIdx(\\ul{\\ell}, d_\\ell) \\;\\;\\wedge\\;\\; \\bigl( \\forall j.\\; \\MinIdx(\\ul{\\ell}, j) \\rightarrow j \\approx d_\\ell \\bigr) \\\\\n\t&\\hspace{-6ex}\\text{There is a unique min.\\ index on every level.}\\\\\n\t\\psi_4 \\!:=\n\t\t& \\bigwedge_{\\ell = 0}^{\\kappa} \\bigl( \\forall j.\\; \\MaxIdx(\\ul{\\ell}, j) \\rightarrow L(\\ul{\\ell}, j) \\bigr) \\;\\;\\wedge\\;\\; \\bigl( \\forall j j'.\\; \\MaxIdx(\\ul{\\ell}, j) \\rightarrow \\neg \\Succ(\\ul{\\ell}, j, j') \\bigr) \\\\\n\t&\\hspace{-6ex}\\text{A max.\\ index of level $\\ell$ belongs to level $\\ell$. A max.\\ index does not have a successor.}\n\\end{align*}\n\\begin{align*}\t\t\n\t\\psi_5 \\!:=\n\t\t& \\bigwedge_{\\ell = 0}^{\\kappa} \\MaxIdx(\\ul{\\ell}, e_\\ell) \\;\\;\\wedge\\;\\; \\bigl( \\forall j.\\; \\MaxIdx(\\ul{\\ell}, j) \\rightarrow j \\approx e_\\ell \\bigr) \\\\\n\t&\\hspace{-6ex}\\text{There is a unique max.\\ index on every level.} \\\\\n\t\\psi_6 \\!:=\n\t\t& \\bigwedge_{\\ell = 0}^{\\kappa} \\forall j j'.\\; \\Succ(\\ul{\\ell}, j, j') \\;\\;\\rightarrow\\;\\; L(\\ul{\\ell}, j) \\wedge L(\\ul{\\ell}, j') \\\\\n\t&\\hspace{-6ex}\\text{If $j'$ is the successor of $j$ at level $\\ell$, then both $j$ and $j'$ belong to level $\\ell$.}\\\\\n\t\\psi_7 \\!:=\n\t\t&\\bigwedge_{\\ell = 0}^{\\kappa} \\forall j j' j''.\\; \\neg \\Succ(\\ul{\\ell}, j, j) \\;\\;\\wedge\\;\\; \\bigl( \\Succ(\\ul{\\ell}, j, j') \\wedge \\Succ(\\ul{\\ell}, j, j'') \\rightarrow j' \\approx j'' \\bigr) \\\\\n\t\t&\\hspace{26ex}\\wedge\\;\\; \\bigl( \\Succ(\\ul{\\ell}, j', j) \\wedge \\Succ(\\ul{\\ell}, j'', j) \\rightarrow j' \\approx j'' \\bigr) \\\\\n\t&\\hspace{-6ex}\\text{The successor relation is irreflexive. Every index $j$ has at most one successor and at most}\\\\[-1ex]\n\t&\\hspace{-6ex}\\text{one predecessor.} \\\\\n\t\\psi_8 \\!:=\\;\n\t\t& \\MinIdx(\\ul{0}, c_1) \\wedge \\MaxIdx(\\ul{0}, c_\\mu) \\;\\;\\wedge\\;\\; \\bigwedge_{i=1}^{\\mu-1} \\Succ(\\ul{0}, c_i, c_{i+1}) \\\\\n\t&\\hspace{-6ex}\\text{At level zero we have the sequence $c_1, \\ldots, c_\\mu$ of successors, where $c_1$ is min.\\ and $c_\\mu$ max.} \\\\\n\t\\psi_9 \\!:=\\;\n\t\t&\\bigwedge_{\\ell = 1}^{\\kappa} \\forall j j' i.\\; \\Succ(\\ul{\\ell}, j, j') \\wedge L(\\ul{\\ell\\!-\\!1}, i) \\;\\;\\rightarrow\\;\\; \\Bigl( \\bigl( J^*(\\ul{\\ell}, j, i, 1) \\wedge J(\\ul{\\ell}, j, i, 1) \\rightarrow J(\\ul{\\ell}, j', i, 0) \\bigr) \\\\[-0.5ex]\n\t\t&\\hspace{41ex}\t\t\t\t\\wedge \\bigl( J^*(\\ul{\\ell}, j, i, 1) \\wedge J(\\ul{\\ell}, j, i, 0) \\rightarrow J(\\ul{\\ell}, j', i, 1) \\bigr) \\\\\n\t\t&\\hspace{41ex}\t\t\t\t\\wedge \\bigl( J^*(\\ul{\\ell}, j, i, 0) \\wedge J(\\ul{\\ell}, j, i, 1) \\rightarrow J(\\ul{\\ell}, j', i, 1) \\bigr) \\\\\n\t\t&\\hspace{41ex}\t\t\t\t\\wedge \\bigl( J^*(\\ul{\\ell}, j, i, 0) \\wedge J(\\ul{\\ell}, j, i, 0) \\rightarrow J(\\ul{\\ell}, j', i, 0) \\bigr) \\Bigr) \\\\\n\t&\\hspace{-6ex}\\text{Define what it means to be a successor at level $\\ell$, $\\ell > 0$, in terms of the binary increment}\\\\[-1ex]\n\t&\\hspace{-6ex}\\text{operation modulo $\\twoup{\\ell}{\\mu}$.}\\\\\n\t\\psi_{10} \\!:=\\;\n\t\t&\\bigwedge_{\\ell = 1}^{\\kappa} \\forall j i.\\; \\MinIdx(\\ul{\\ell}, j) \\wedge L(\\ul{\\ell\\!-\\!1}, i) \\;\\;\\rightarrow\\;\\; J(\\ul{\\ell}, j, i, 0) \\\\\n\t&\\hspace{-6ex}\\text{All bits of a minimal index $j$ are 0.}\\\\\n\t\\psi_{11} \\!:=\n\t\t& \\bigwedge_{\\ell = 1}^{\\kappa} \\forall j i.\\; \\MaxIdx(\\ul{\\ell}, j) \\wedge \\MaxIdx(\\ul{\\ell\\!-\\!1}, i) \\;\\;\\rightarrow\\;\\; J(\\ul{\\ell}, j, i, 1) \\\\\n\t&\\hspace{-6ex}\\text{Define what it means to be max.\\ (part 1): the most significant bit is 1.} \\\\\n\t\\psi_{12} \\!:=\n\t\t& \\bigwedge_{\\ell = 1}^{\\kappa} \\forall j i.\\; L(\\ul{\\ell}, j) \\wedge \\MaxIdx(\\ul{\\ell\\!-\\!1}, i) \\wedge J(\\ul{\\ell}, j, i, 1) \\;\\;\\rightarrow\\;\\; \\MaxIdx(\\ul{\\ell}, j) \\\\\n\t&\\hspace{-6ex}\\text{Define what it means to be max.\\ (part 2): any index with 1 as its most significant bit is max.}\\\\\n\t\\psi_{13} \\!:=\n\t\t& \\bigwedge_{\\ell = 1}^{\\kappa} \\forall j i.\\; L(\\ul{\\ell}, j) \\wedge L(\\ul{\\ell\\!-\\!1}, i) \\;\\;\\rightarrow\\;\\; \\bigl( J(\\ul{\\ell}, j, i, 0) \\rightarrow \\neg J(\\ul{\\ell}, j, i, 1) \\bigr) \\\\[-0.5ex]\n\t\t&\\hspace{32ex}\\wedge \\bigl( J^*(\\ul{\\ell}, j, i, 0) \\rightarrow \\neg J^*(\\ul{\\ell}, j, i, 1) \\bigr) \\\\\n\t&\\hspace{-6ex}\\text{No bit of an index is $0$ and $1$ at the same time. An analogous requirement is stipulated for $J^*$.} \\\\\n\t\\psi_{14} \\!:=\n\t\t& \\bigwedge_{\\ell = 1}^{\\kappa} \\forall j i.\\; L(\\ul{\\ell}, j) \\wedge \\MinIdx(\\ul{\\ell\\!-\\!1}, i) \\;\\;\\rightarrow\\;\\; J^*(\\ul{\\ell}, j, i, 1) \\\\\n\t&\\hspace{-6ex}\\text{$J^*(\\ul{\\ell}, j, d_{\\ell-1}, 1)$ holds for every index $j$.}\n\\end{align*}\n\\begin{align*}\n\t\\psi_{15} \\!:=\n\t\t&\\bigwedge_{\\ell = 1}^{\\kappa} \\forall j i i'.\\; L(\\ul{\\ell}, j) \\wedge \\Succ(\\ul{\\ell\\!-\\!1}, i, i') \\;\\;\\rightarrow\\;\\; \\bigl( J^*(\\ul{\\ell}, j, i', 1) \\leftrightarrow \\bigl( J^*(\\ul{\\ell}, j, i, 1) \\wedge J(\\ul{\\ell}, j, i, 1) \\bigr) \\bigr) \\\\[-0.5ex]\n\t\t &\\hspace{40ex}\t\\wedge \\bigl( J(\\ul{\\ell}, j, i, 0) \\rightarrow J^*(\\ul{\\ell}, j, i', 0) \\bigr) \\\\\n\t\t &\\hspace{40ex} \t\\wedge \\bigl( J^*(\\ul{\\ell}, j, i, 0) \\rightarrow J^*(\\ul{\\ell}, j, i', 0) \\bigr) \\\\ \n\t&\\hspace{-6ex}\\text{Define the semantics of $J^*$ as indicating that all bits strictly less significant than the $i$-th}\\\\[-1ex]\n\t&\\hspace{-6ex}\\text{bit are 1.} \\\\\n\t\\eq{1}_{j,\\tj} \\!:=\\;\n\t\t& L(\\ul{1}, j) \\wedge L(\\ul{1}, \\tj) \\;\\;\\wedge\\;\\; \\bigwedge_{i = 1}^\\mu \\bigl( J(\\ul{1}, j, c_i, 0) \\leftrightarrow J(\\ul{1}, \\tj, c_i, 0) \\bigr) \\wedge \\bigl( J(\\ul{1}, j, c_i, 1) \\leftrightarrow J(\\ul{1}, \\tj, c_i, 1) \\bigr) \\\\\n\t&\\hspace{-6ex}\\text{Base case of equality of indices.}\\\\\n\t\\eq{\\ell}_{j,\\tj} \\!:=\\;\n\t\t&L(\\ul{\\ell}, j) \\wedge L(\\ul{\\ell}, \\tj) \\wedge \\forall i.\\; L(\\ul{\\ell\\!-\\!1}, i) \\;\\;\\rightarrow\\;\\; \\exists \\ti.\\; L(\\ul{\\ell\\!-\\!1}, \\ti) \\wedge \\eq{\\ell\\!-\\!1}_{i,\\ti}\n\t\t\t\t \\wedge \\bigl( J(\\ul{\\ell}, j, i, 0) \\leftrightarrow J(\\ul{\\ell}, \\tj, \\ti, 0) \\bigr) \\\\[-0.5ex]\n\t\t&\\hspace{57.4ex} \\wedge \\bigl( J(\\ul{\\ell}, j, i, 1) \\leftrightarrow J(\\ul{\\ell}, \\tj, \\ti, 1) \\bigr) \\\\\n\t&\\hspace{-6ex}\\text{Inductive case of equality of indices for $\\ell > 1$.}\\\\\n\t\\psi_{16} \\!:=\\;\n\t\t&\\bigwedge_{\\ell = 1}^{\\kappa} \\forall j i.\\; L(\\ul{\\ell}, j) \\wedge \\MaxIdx(\\ul{\\ell-1}, i) \\wedge J(\\ul{\\ell}, j, i, 0) \\;\\;\\rightarrow\\;\\; \\exists \\tj\\, \\tj'.\\; \\eq{\\ell}_{j, \\tj} \\wedge \\Succ(\\ul{\\ell}, \\tj, \\tj')\\\\\n\t&\\hspace{-6ex}\\text{For every index at level $\\ell$ that is not maximal, i.e. whose most significant bit is 0, there}\\\\[-1ex]\n\t&\\hspace{-6ex}\\text{ exists a successor index.}\n\\end{align*}\n\nUntil now, we have only introduced sentences that can easily be transformed into SF sentences in Horn form, existential variables are separated from universal ones, as all quantifiers occur with positive polarity, and consequents of implications are (conjunctions of) literals.\n\nRegarding the length of the above sentences, we observe the following:\n\t\\begin{itemize}\n\t\t\\item $\\len(\\psi_1) \\in \\cO(\\kappa^2 \\log \\kappa)$\n\t\t\\item $\\len(\\psi_2), \\ldots, \\len(\\psi_7), \\len(\\psi_9), \\ldots, \\len(\\psi_{15}) \\in \\cO(\\kappa \\log \\kappa)$\n\t\t\\item $\\len(\\psi_8) \\in \\cO\\bigl( \\mu (\\log \\kappa + \\log \\mu) \\bigr)$\n\t\t\\item $\\len\\bigl( \\eq{1}_{j,j'} \\bigr) \\in \\cO\\bigl( \\mu (\\log \\kappa + \\log \\mu) \\bigr)$\n\t\t\\item $\\len\\bigl( \\eq{\\ell}_{j,j'} \\bigr) \\in \\cO(\\log \\kappa) + \\len\\bigl( \\eq{1}_{j,j'} \\bigr)$\n\t\t\\item $\\len(\\psi_{16}) \\in \\cO\\bigl( \\kappa^2 \\log \\kappa + \\kappa \\mu (\\log \\kappa + \\log \\mu) \\bigr)$\n\t\\end{itemize}\nIn total, this yields $\\len(\\psi_1 \\wedge \\ldots$ $\\wedge \\psi_{16}) \\in \\cO\\bigl( \\kappa^2 \\log \\kappa + \\kappa \\mu (\\log \\kappa + \\log \\mu) \\bigr)$.\n\nThe following three sentences do not produce Horn formulas when transformed into CNF. \nThey serve the purpose of removing spurious elements from the model.\nIn particular, $\\chi_3$ is essential to enforce large models for $\\kappa \\geq 2$.\n\\begin{align*}\n\t\\chi_1 \\!:=\\;\n\t\t& \\forall j.\\; L(\\ul{0}, j) \\rightarrow \\bigvee_{i=1}^\\mu j \\approx c_i \\\\\n\t\t&\\hspace{-6ex}\\text{On level 0 there are no indices but $c_1, \\ldots, c_\\mu$.}\\\\\n\t\\chi_2 \\!:=\n\t\t& \\bigwedge_{\\ell = 1}^{\\kappa} \\forall j i.\\; L(\\ul{\\ell}, j) \\wedge L(\\ul{\\ell-1}, i) \\;\\;\\rightarrow\\;\\; J(\\ul{\\ell}, j, i, 0) \\vee J(\\ul{\\ell}, j, i, 1) \\\\\n\t\t&\\hspace{-6ex}\\text{We stipulate totality for the predicate $J$.}\\\\\n\n\t\\chi_3 \\!:=\n\t\t&\\bigwedge_{\\ell = 1}^{\\kappa} \\forall j j'.\\; L(\\ul{\\ell}, j) \\wedge L(\\ul{\\ell}, j') \\rightarrow \\exists \\tj\\, \\tj'.\\; \\eq{\\ell}_{j, \\tj} \\wedge \\eq{\\ell}_{j', \\tj'} \\\\\n\t\t&\\hspace{30ex}\t\\wedge \\Bigl( \\Bigl( \\forall \\ti.\\; L(\\ul{\\ell-1}, \\ti) \\rightarrow \\bigl( J(\\ul{\\ell}, \\tj, \\ti, 0) \\leftrightarrow J(\\ul{\\ell}, \\tj', \\ti, 0) \\bigr) \\Bigr) \\rightarrow j \\approx j' \\Bigr) \\\\ \n\t\t&\\hspace{-6ex}\\text{Two indices at the same level that agree on all of their bits are required to be identical.}\t\n\\end{align*}\nNotice that $\\chi_3$ is (almost) an SF sentence, since the $\\forall \\ti$ turns into a $\\exists \\ti$ as soon as we bring the sentence into prenex normal form.\nRegarding the length of $\\chi_1, \\chi_2, \\chi_3$, we observe $\\len(\\chi_1) \\in \\cO(\\log \\kappa + \\mu \\log \\mu)$, $\\len(\\chi_2) \\in \\cO(\\kappa \\log \\kappa)$, and $\\len(\\chi_3) \\in \\cO\\bigl( \\kappa^2 \\log \\kappa + \\kappa \\mu (\\log \\kappa + \\log \\mu) \\bigr)$.\nHence, we overall have $\\len(\\chi_1 \\wedge \\chi_2 \\wedge \\chi_3) \\in \\cO\\bigl( \\kappa^2 \\log \\kappa + \\kappa \\mu (\\log \\kappa + \\log \\mu) \\bigr)$.\n\nConsider any model $\\cA$ of $\\psi_{1} \\wedge \\ldots$ $\\wedge \\psi_{16}\\wedge \\chi_1 \\wedge \\chi_2 \\wedge \\chi_3$.\n\\begin{definition}\\label{definition:IndexSetsAndRelatedNotation}\n\tWe define the following sets and relations:\n\t\t$\\cI_{\\ell} := \\bigl\\{ \\fa \\in \\fU_\\cA \\mid \\cA, [j \\Mapsto \\fa] \\models L(\\ul{\\ell}, j) \\bigr\\}$ for every $\\ell = 0, \\ldots, \\kappa$;\n\t\t${\\prec_\\ell} \\subseteq \\cI_{\\ell} \\times \\cI_{\\ell}$ for every $\\ell = 0, \\ldots, \\kappa$ such that $\\fa \\prec_\\ell \\fa'$ holds if and only if $\\cA, [j \\Mapsto \\fa, j' \\Mapsto \\fa'] \\models \\Succ(\\ul{\\ell}, j, j')$.\t\n\\end{definition}\n\n\\begin{lemma}\\label{lemma:IndexSetsProperties}\n\tFor every $\\ell = 1, \\ldots, \\kappa$ we have $|\\cI_{\\ell}| = p$ where $p := 2^{|\\cI_{\\ell-1}|-1}+1 = \\twoup{\\ell}{\\mu-1}+1$.\n\tMoreover, there is a unique chain $\\fa_1 \\prec_\\ell \\ldots \\prec_\\ell \\fa_p$ comprising all elements in $\\cI_\\ell$.\n\\end{lemma}\n\nLeaving out the non-Horn parts $\\chi_1, \\chi_2, \\chi_3$ renders the lemma invalid for $\\ell > 1$. \nOn the other hand, for $\\kappa = 1$ the sentence $\\psi_1 \\wedge \\ldots \\wedge \\psi_{16}$---which can be transformed into an equivalent Horn sentence---has only models $\\cA$ for which $\\cI_1$ contains at least $2^{\\mu-1}+1$ elements.\nNotice that this could be used to derive \\EXPTIME-hardness of satisfiability for the class of Horn SF sentences of degree $1$.\nBut such lower bounds are already known for the Horn subfragments of MFO and of the BSR fragment, which are proper subsets of SF's Horn subfragment.\n\n\\subsection{Formalizing a tiling of the torus}\\label{section:FormalizingTheTiling}\n\nIn order to formalize a given domino problem $\\fD = \\<\\cD, \\cH, \\cV\\>$ and an initial condition $\\bD$, we introduce the following constant and predicate symbols:\\\\\n\\centerline{\n\t\\begin{tabular}{ll}\n\t\t$H(x, y, x', y')$\t\t\t& \t$\\$ is the horiz.\\ neighbor of $\\$, \\\\\n\t\t\t\t\t\t\t&\ti.e.\\ $x' = x + 1$ (mod $\\twoup{\\kappa}{\\mu-1}+1$) and $y' = y$, \\\\\n\t\t$V(x, y, x', y')$\t\t\t& \t$\\$ is the vert.\\ neighbor of $\\$, \\\\\n\t\t$\\ul{D}(x,y)$\t\t\t&\t$\\$ is tiled with $D \\in \\cD$, \\\\\n\t\t$f_1, \\ldots, f_{|\\bD|}$\t\t&\tconstants addressing points $\\<0,0\\>, \\ldots, \\<|\\bD|-1,0\\>$.\n\t\\end{tabular}\n}\nWith the ideas we have seen when formalizing the index hierarchy, it is now fairly simple to formalize the torus.\nFor instance, the following sentence makes sure that every point that is not on the ``edge'' of the torus has a horizontal neighbor.\n\\begin{align*}\t\n\t\\eta_{3} \\!:=\\;\t\n\t\t& \\forall x y i.\\; L(\\ul{\\kappa}, x)\\wedge L(\\ul{\\kappa}, y) \\wedge \\MaxIdx(\\ul{\\kappa-1}, i) \\wedge J(\\ul{\\kappa}, x, i, 0) \\\\\n\t\t&\\hspace{10ex} \\rightarrow\\;\\; \\exists \\tx\\, \\ty\\, \\tx'.\\; \\eq{\\kappa}_{x, \\tx} \\wedge \\eq{\\kappa}_{y, \\ty} \\wedge \\Bigl( \\bigwedge_{D \\in \\cD} \\ul{D}(x,y) \\leftrightarrow \\ul{D}(\\tx, \\ty) \\Bigr) \\wedge H(\\tx, \\ty, \\tx', \\ty)\n\\end{align*}\nThe next sentence, on the other hand, makes sure that the rules of the domino system $\\fD$ are obeyed.\n\\begin{align*}\n\t\t\\eta_{15} \\!:=\\;\n\t\t& \\forall x x' y.\\; H(x, y, x', y) \\;\\;\\rightarrow\\;\\; \\bigvee_{\\ \\in \\cH} \\ul{D}(x,y) \\wedge \\ul{D}'(x',y)\n\\end{align*}\nProceeding this way, the formalization $\\eta$ of a domino system in SF requires a length in $\\cO \\bigl( \\widehat{n} \\log \\widehat{n} \\bigr)$, where $\\widehat{n} := \\max \\{ \\kappa, \\mu, |\\bD|, |\\cD|^2 \\}$.\n\n\n\\begin{lemma}\n\tAssume that $\\cD$, $\\cH$, and $\\cV$ are nonempty and\n\tlet $\\cA$ be a model of the sentence $\\psi_{1} \\wedge \\ldots \\wedge \\psi_{16}\\wedge \\chi_1 \\wedge \\chi_2 \\wedge \\chi_3 \\wedge \\eta$.\n\t$\\cA$ induces a tiling $\\tau$ of $\\Int_t^2$ with initial condition $\\bD := D_1, \\ldots, D_n$, where $t := \\twoup{\\kappa}{\\mu-1}+1$.\n\\end{lemma}\n\n\n\n\n\n\n\\subsection{Replacing the equality predicate}\\label{section:ReplaceEqualityInLowerBoundProof}\n\nSince SF can express reflexivity, symmetry, transitivity, and congruence properties, it is easy to formulate an SF sentence without equality that is equisatisfiable to $\\psi_1 \\wedge \\ldots \\wedge \\psi_{16} \\wedge \\chi_1 \\wedge \\chi_2 \\wedge \\chi_3 \\wedge \\eta$ and uses atoms $E(s,t)$ instead of $s \\approx t$.\nIn addition to replacing equational atoms as indicated, we add the usual axioms concerning the fresh predicate symbol $E$.\nOverall, the additional formulas have a length that lies in $\\cO \\bigl( \\kappa \\log \\kappa + |\\cD| \\log |\\cD| \\bigr)$.\n\nConsequently, the hardness result that we have obtained for SF with equality can be directly transferred to SF without equality.\nMoreover, notice that all the above formulas can be transformed into Horn form. \nHence, one could also replace $\\approx$ by $E$ in hardness proofs for the Horn subfragment of SF.\n\n\n\n\n\\section{Conclusion}\n\nWe stress in this paper that an analysis of the computational complexity of satisfiability problems can greatly benefit from an analysis of how variables occur together in atoms instead of exclusively considering the number of occurring quantifier alternations.\nWhat we have not yet taken into account is the Boolean structure of sentences.\nThis may widen the scope of our methods considerably and may moreover help understand where the hardness of satisfiability problems stems from.\n\nConsider a quantified Boolean formula $\\varphi := \\forall \\vx_1 \\exists \\vy_1 \\ldots \\forall \\vx_n \\exists \\vy_n . \\psi$ with quantifier-free $\\psi$.\nAll satisfiable formulas of this shape together form a hard problem residing on the $n$-th level of the polynomial hierarchy.\nBut what if, for instance, $\\psi$ has the form $\\bigl( \\bigwedge_i K_i \\bigr) \\wedge \\bigl( \\bigvee_j L_j \\bigr)$, where the $K_i$ and the $L_j$ are literals and none of the existential variables in $\\bigwedge_i K_i$ occurs in $\\bigvee_j L_j$?\nSince Boolean variables cannot jointly occur in atoms, $\\varphi$ can be transformed into the equivalent formula $\\exists \\vy_1 \\ldots \\vy_n \\forall \\vx_1 \\ldots \\vx_n . \\psi$ by application of the rules of miniscoping (cf.\\ Lemma~\\ref{lemma:BasicQuantifierEquivalences}). Apparently, $\\varphi$ belongs to a class of QBF sentences that resides on the first level of the polynomial hierarchy rather than on the \\mbox{$n$-th}.\n\nPerhaps it is time to reconsider some of the definitions that are based on the shape of quantifier prefixes alone.\n\n\n\n\n\\section*{Acknowledgments}\nThe author would like to thank the anonymous reviewers for valuable hints and suggestions.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{introduction}\n\nSignificant technological advances in the astronomical instrumentation during the last four decades enabled measurements of the\nfar-infrared thermal dust emission (usually optically thin in that wavelength range) and hence estimates of the masses of dusty\nobjects. Fitting the far-infrared and submillimeter flux or intensity distributions of optically thin sources can give their\naverage temperatures and masses \\citep{Hildebrand1983}. This simple method has become standard in studies of Galactic star\nformation and a major source of our knowledge of the physical properties and evolution of self-gravitating cores and protostars.\nAlthough there are more sophisticated approaches \\citep[e.g.,][]{Kelly_etal2012}, a simple fitting of the observed spectral shapes\nremains the most widely used method in the observational studies of star formation \\citep[e.g.,][]{{Ko\"nyves_etal2015}}. Its\ninaccuracies, biases, and limitations need to be carefully investigated before reliable conclusions can be made on the physical\nproperties and evolution of the observed objects.\n\nMass derivation from fitting total fluxes or pixel intensities involves a strong assumption of a constant temperature within an\nobject. In addition to such poorly known parameters as the distance, the far-infrared opacity and its power-law slope, and the\ndust-to-gas mass ratio, the most problematic assumption is that a single color temperature obtained from the fitting is a good\napproximation of the mass-averaged physical dust temperatures. This may be true for only the simplest case of the lowest density\nstarless cores, transparent in the visible wavelength range and thus practically isothermal, but it is clearly invalid for the\nprotostellar envelopes that are centrally heated by accretion luminosity. Very sensitive dependence of the emission of dust grains\non their temperature warrants careful investigation of the effects of nonuniform temperatures. There are papers that have\ninvestigated some aspects of the problem, notably the correlation between the estimated temperatures and power-law opacity slopes\n\\citep[e.g.,][and references therein]{Shetty_etal2009a,Shetty_etal2009b,JuvelaYsard2012a} and the inaccuracies of mass derivation\nand their effect on the resulting core mass function \\citep[][]{Malinen_etal2011}.\n\nThe present purely model-based study simplifies the problem by removing the ``observational layer'' between the physical reality\nand observers. To investigate the intrinsic effects related to the physical objects, all observational intricacies (the complex\nfilamentary backgrounds, instrumental noise, calibration errors, different angular resolutions across wavebands, etc.) were\nexplicitly ignored. Measurement errors in intensities and fluxes are assumed to be nonexistent and the radiation emitted by the\nmodel objects is known to a high precision, limited only by their numerical accuracy. A grid of radiative transfer models of\nstarless cores and protostellar envelopes was computed and their total fluxes and image intensities were fitted to derive the model\nmasses. Known true values of the numerical models allow us to assess the qualities of the methods and fitting models, as well as\nthe effects of nonuniform temperatures, far-infrared opacity slope, selected subsets of wavelengths, background subtraction, and\nangular resolutions. The main goal was to quantify how much the mass derivation methods are affected, what the realistic\nuncertainties of the temperatures and masses are, and what one could possibly do to improve the estimates. Although this study is\ncompletely independent of the instruments and wavebands used in actual observations, it employs six \\emph{Herschel} wavebands\n\\citep[70, 100, 160, 250, 350, and 500\\,{${\\mu}$m};][]{Pilbratt_etal2010}, for which a wealth of recent results in star formation\nhas been obtained \\citep[e.g.,][and references therein]{Ko\"nyves_etal2015}.\n\n\\begin{figure} \n\\centering\n\\centerline{\\resizebox{0.695\\hsize}{!}{\\includegraphics{loo.cfg.trim.pdf}}}\n\\caption{\nOpacities of grains (per gram of dust) and model radial optical depths. Subscripts on the curve labels ($n_{30}$ to $n_{0.03}$) \nindicate the model mass $M$ (in $M_{\\sun}$). The wavelength dependence of $\\kappa_{\\rm sca}$, $\\kappa_{\\rm abs}$, and $\\kappa_{\\rm \next}$ is shown by thick solid lines (see Sect.~\\ref{dust.properties} for details). The extinction optical depths $\\tau_{\\rm ext}$ \nof the protostellar envelopes ($L_{\\star}\\,{=}\\,0.3\\,L_{\\sun}$) and starless cores are indicated respectively by the three sets of \ndashed blue and red lines. \n} \n\\label{loo}\n\\end{figure}\n\nThe radiative transfer models of starless cores and protostellar envelopes are presented in Sect.~\\ref{rtmodels}, the methods of\nmass derivation from fitting far-infrared and submillimeter observations are introduced in Sect.~\\ref{fitting.methods}, the results\nof this work are presented in Sect.~\\ref{results} and discussed in Sect.~\\ref{discussion}, the conclusions are outlined in\nSect.~\\ref{conclusions}, and further details are found in Appendices \\ref{AppendixA}\\,{--}\\,\\ref{AppendixE}.\n\n\n\\section{Radiative transfer models}\n\\label{rtmodels}\n\nThe models were computed with the 3D Monte Carlo radiative transfer code \\textsl{RADMC-3D} by C.\\,Dullemond\\footnote{\n\\url{http:\/\/www.ita.uni-heidelberg.de\/~dullemond\/software\/radmc-3d}}. Spherical model geometry was chosen to simplify the problem\nby reducing the number of free parameters involved in the study: asymmetries in model density distribution would introduce\ndependence on viewing angle \\citep[e.g.,][]{Men'shchikovHenning1997,Men'shchikov_etal1999,Stamatellos_etal2004} and hence increase\nthe uncertainties of derived parameters. Isotropic scattering by dust grains was considered.\n\nGrids of models for starless cores and protostellar envelopes were constructed, covering the ranges of masses $M$ ($0.03\\,{-}\\,30$\n$M_{\\sun}$) and luminosities $L$ ($0.03\\,{-}\\,30$\\,$L_{\\sun}$) relevant for both low- and intermediate-mass star formation. The\nmasses and luminosities were sampled at the values of $0.0316, 0.1, 0.316, 1, 3.16, 10, 31.6$ (separated by a factor of\n$\\sqrt{10}$); for simplicity, they will be referred to as $0.03, 0.1, 0.3, 1, 3, 10, 30$ ($M_{\\sun}$, $L_{\\sun}$). Although the\nluminosity of an accreting protostar depends on its mass, the goal is to separate the effects of masses and luminosities.\n\nIn addition to isolated models, their embedded variants were constructed by implanting the isolated models into the centers of\nlarger spherical background shells of uniform densities, in order to simulate the fact that stars form within their dense parental\nclouds that shield the embedded objects from the interstellar radiation field. All models were put at a distance $D{\\,=\\,}140$ pc\nof the nearest star-forming regions.\n\n\\begin{figure}\n\\centering\n\\centerline{\\resizebox{0.695\\hsize}{!}{\\includegraphics{drp.bes.pro.cfg.trim.pdf}}}\n\\caption{\nDensity structure of the model starless cores and protostellar envelopes. Subscripts on the curve labels ($n_{30}$ to $n_{0.03}$)\nindicate the model mass $M$ (in $M_{\\sun}$). The dashed vertical line shows the outer boundary radius $R\\,{=}\\,10^{4}$\\,AU for all\nmodels. Embedded models are implanted in larger uniform-density clouds with an outer boundary at $3\\,{\\times}\\,10^{4}$\\,AU. The\ndashed horizontal lines continue the densities of starless cores within the innermost radial zone. The dashed diagonal lines \ncontinue the densities of protostellar envelopes towards the radius of the inner dust-free cavity $R_{0}$, whose size depends on \nthe model temperature profile $T_{\\rm d}(r)$ (cf. Fig.~\\ref{trp.bes.pro}) and adopted dust sublimation temperature ($T_{\\rm \nS}\\,{=}$ $\\,10^{3}$\\,K). In other words, the dashed diagonal lines visualize the range of densities and radial distances over which\nthe inner boundary $R_{0}$ is located for $L_{\\star}$ spanning the entire range $0.03\\,{-}\\,30$\\,$L_{\\sun}$ (see \nEq.~(\\ref{inner.boundary})).\n} \n\\label{drp.bes.pro}\n\\end{figure}\n\n\\subsection{Dust properties}\n\\label{dust.properties}\n\nProperties of the real astrophysical dust grains are poorly known and they are unlikely to be universal in the different\nstar-forming regions observed. The standard mass derivation methods ignore many complications related to the cosmic dust grains,\nassuming just a simple power-law opacity across all bands being fitted. For example, the presence of very small, stochastically\nheated grains is neglected \\citep[e.g.,][]{Desert_etal1990}; the contribution of these grains to the emission of starless cores and\nprotostellar envelopes can become significant at $\\lambda\\,{\\la}\\,100$\\,{${\\mu}$m}\n\\citep[e.g.,][]{Bernard_etal1992,Siebenmorgen_etal1992}. For consistency with the mass derivation methods and previous studies of\nstar formation, this model study adopts tabulated absorption opacities $\\kappa_{\\rm abs}$ for grains with thin ice mantles\n\\citep{OssenkopfHenning1994}, corresponding to coagulation time $t\\,{=}\\,10^5$\\,yr and number density $n_{\\rm\nH}\\,{=}\\,10^6$\\,cm$^{-3}$ (Fig.~\\ref{drp.bes.pro}).\n\nThe opacity values at long wavelengths $\\lambda\\,{>}\\,70$\\,{${\\mu}$m} were replaced with a power law\n$\\kappa_{\\lambda}\\,{\\propto}\\,\\lambda^{-2}$; the modification aimed at testing the widely used assumption on the power-law\nfar-infrared opacities $\\kappa_{\\lambda}\\,{=}\\,\\kappa_{0}\\left(\\lambda_0\/\\lambda\\right)^{\\,\\beta}$. At short wavelengths\n(0.1$\\,{<}\\,\\lambda\\,{<}\\,1$\\,{${\\mu}$m}), the opacities were extrapolated with a power law\n$\\kappa_{\\lambda}\\,{\\propto}\\,\\lambda^{-0.87}$ based on the last tabulated values. Although dust scattering is unimportant in the\nfar-infrared, scattering opacities were constructed to resemble the values and wavelength dependence $\\kappa_{\\rm\nsca}\\,{\\propto}\\,\\lambda^{-4}$ of typical dust grains. The resulting dust opacity at $\\lambda\\,{>}\\,70$\\,{${\\mu}$m} was\nparameterized by $\\kappa_0\\,{=}\\,$9.31\\,cm$^{2}$g$^{-1}$ (per gram of dust), $\\lambda_0\\,{=}\\,$300\\,{${\\mu}$m}, and\n$\\beta\\,{=}\\,2$, with the maximum opacities limited by $10^5$\\,cm$^{2}$g$^{-1}$ (Fig.~\\ref{loo}).\n\n\\begin{figure*}\n\\centering\n\\centerline{\\resizebox{0.3386\\hsize}{!}{\\includegraphics{trp.bes.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{trp.bes.emb.cfg.trim.pdf}}}\n\\centerline{\\resizebox{0.3386\\hsize}{!}{\\includegraphics{trp.pro.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{trp.pro.emb.cfg.trim.pdf}}}\n\\caption{\nRadiative-equilibrium dust temperature profiles of starless cores (\\emph{upper}) and protostellar envelopes (\\emph{lower}) for the\nisolated models (\\emph{left}) and their embedded variants (\\emph{right}). Subscripts of the curve labels ($T_{30}$ to $T_{0.03}$)\nindicate the model mass $M$ (in $M_{\\sun}$). The dashed horizontal lines in the upper panels continue the profiles of starless cores\nwithin the innermost radial zone. The dashed vertical line shows the outer boundary radius $R\\,{=}\\,10^{4}$\\,AU of all models. The\nmaximum temperature and the inner boundary of the dusty protostellar envelopes are defined by the adopted dust sublimation\ntemperature $T_{\\rm S}\\,{=}\\,10^{3}$\\,K. Three dashed horizontal lines in the lower panels indicate the range of radial distances\nover which the boundaries $R_{0}$ of the dust-free cavities are located for the model luminosities in the entire range of\n$0.03\\,{-}\\,30$\\,$L_{\\sun}$ (see Fig.~\\ref{drp.bes.pro}). For protostellar envelopes with the same $M$, the ``hotter'' profiles\ncorrespond to higher $L_\\star$, larger $R_{0}$ (cf. Eq.~(\\ref{inner.boundary})), and lower radial optical depths\n$\\tau_{\\lambda}\\,{\\propto}\\,L_{\\star}^{-1\/3}$. Two dashed lines bracketing the profiles of protostellar envelopes indicate the\nslopes $T_{\\rm d}(r)\\,{\\propto}\\,r^{-0.88}$ and ${\\propto}\\,r^{-1\/3}$, the latter describing the temperatures of a\ntransparent dusty envelope with the adopted grain properties (see also Appendix~\\ref{AppendixA}). Vertically aligned filled circles\nindicate the mass-averaged temperature $T_{M}$ for each model, defined by Eq.~(\\ref{mass.averaged}). The slight wiggling of some\nprofiles around their minima reflects the discrete nature of the Monte Carlo radiative transfer method.\n} \n\\label{trp.bes.pro}\n\\end{figure*}\n\n\\subsection{Density distributions}\n\\label{density.profiles}\n\nThe density structure of starless cores was approximated by an isothermal Bonnor-Ebert sphere \\citep{Bonnor1956} with a temperature\nof $7$\\,{K} and a central density of $5.2\\,{\\times}\\,10^{-18}$\\,g\\,cm$^{-3}$. This somewhat arbitrary choice of $\\rho(r)$ gives\njust a simple and convenient functional form (Fig.~\\ref{drp.bes.pro}) resembling the observed flat-topped density profiles of\nstarless cores \\citep[e.g.,][]{Alves_etal2001,Evans_etal2001}. The issue of the gravitational instability (or stability) of the\nmodel cores is irrelevant for this study of the mass derivation methods. Protostellar envelopes were modeled as infalling spherical\nenvelopes with the power-law densities $\\rho(r)\\,{\\propto}\\,r^{-2}$ \\citep[e.g.,][]{Larson1969,Shu_1977} around a central source of\naccretion energy (Fig.~\\ref{drp.bes.pro}).\n\nModel dust densities were scaled to obtain the desired grid of masses $0.03, 0.1, 0.3, 1, 3, 10$, and $30$\\,$M_{\\sun}$ using the\nstandard dust-to-gas mass ratio $\\eta\\,{=}\\,0.01$. The outer boundary of all the models was placed at the same distance of\n$R\\,{=}\\,10^{4}$\\,AU, beyond which their density either changed to zero (isolated models) or remained constant until $R_{\\rm\nE}\\,{=}\\,3\\,{\\times}\\,R$ (embedded models). The embedding cloud density was set equal to $\\rho(R)$ (Fig.~\\ref{drp.bes.pro}), which\ncorresponds to the denser models (i.e., more massive) being formed in a denser environment. Most of the mass of the model starless\ncores and protostellar envelopes ($96{\\%}$ and $90{\\%}$, respectively) is contained in their outer parts ($0.1\\,R\\,{<}\\,r\\,{<}\\,R$).\n\nFor the starless cores, the inner boundary was arbitrarily set to $R_{0}\\,{=}\\,50$\\,AU, as their densities are essentially constant\nand hence do not need to be resolved at smaller radii. The inner boundary of the dusty protostellar envelopes is defined by the\ndust sublimation temperature $T_{\\rm S}\\,{\\sim}\\,10^{3}$\\,K. An exact value of $T_{\\rm S}$ depends on the chemical composition and\nsizes of dust grains and so does the radius $R_{0}$ of the inner dust-free cavity. For the purpose of this study, it is adequate to\nadopt a single value $T_{\\rm S}\\,{=}\\,10^{3}$\\,K. With the model $\\kappa_{\\nu}$ and $\\rho(r)$ (Sect.~\\ref{dust.properties}), the\nresulting radiative-equilibrium temperatures (Fig.~\\ref{trp.bes.pro}) lead to the inner boundaries of the dusty protostellar\nenvelopes that are fairly accurately described by a simple formula,\n\\begin{equation}\nR_{0} = 2 \\left[\\left(M\/M_{\\sun}\\right) \\left(L_{\\star}\/L_{\\sun}\\right)\\right]^{1\/3} {\\rm AU}.\n\\label{inner.boundary}\n\\end{equation}\n\nThe model space between the inner and outer boundaries was discretized by nonuniform grids with the relative zone sizes\n$\\delta\\log{r}$ that smoothly varied from $0.6$ to $0.02$ ($100\\,{-}\\,150$ zones) for starless cores and from $0.002$ to $0.06$\n($200\\,{-}\\,300$ zones) for protostellar envelopes.\n\n\\begin{figure*}\n\\centering\n\\centerline{\\resizebox{0.3327\\hsize}{!}{\\includegraphics{sed.bes.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{sed.bes.emb.cfg.trim.pdf}}}\n\\centerline{\\resizebox{0.3327\\hsize}{!}{\\includegraphics{sed.pro.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{sed.pro.emb.cfg.trim.pdf}}}\n\\caption{\nSpectral energy distributions starless cores (\\emph{upper}) and protostellar envelopes (\\emph{lower}). Shown are the\nbackground-subtracted fluxes for the \\emph{isolated} models (\\emph{left}) and their \\emph{embedded} variants (\\emph{right}).\nSubscripts of the curve labels ($F_{30}$ to $F_{0.03}$) indicate the model mass (in $M_{\\sun}$). For protostellar envelopes of the \nsame mass, the SEDs with higher fluxes correspond to higher accretion luminosities $L_\\star$ (0.03, 0.1, 0.3, 1, 3, 10, \n30\\,$L_{\\sun}$). Dashed lines indicate the fluxes of the ISRF that were integrated over the projected area of either the isolated \nmodels or the embedding clouds.\n} \n\\label{sed.bes.pro}\n\\end{figure*}\n\n\\subsection{Radiation sources and optical depths}\n\\label{radiation.odepths}\n\nAll models were illuminated from the outside by an isotropic interstellar radiation field \\citep{Black1994} with the ``strength''\nparameter $G_{0}\\,{=}\\,1$ \\citep[e.g.,][]{Parravano_etal2003}. The bolometric luminosity of the interstellar radiation field (ISRF)\nentering the isolated models at $R$ amounted to $L_{\\rm ISRF}\\,{=}\\,1$\\,$L_{\\sun}$, whereas that crossing the boundary of embedding\nclouds at $R_{\\rm E}$ was $9$\\,$L_{\\sun}$.\n\nIn addition to the external radiation field, the models of protostellar envelopes were assumed to be heated at their centers by a\nblackbody source of luminosity $L_{\\star}$ of $0.03, 0.1, 0.3, 1, 3, 10$, and $30$\\,$L_{\\sun}$ with an effective temperature of\n$T_{\\star}{=}\\,5770$\\,K. Actual values of $T_{\\star}$ are unimportant, as the sources of accretion energy are surrounded by the\ncompletely opaque dusty envelopes reprocessing the hot radiation to $T\\,{\\la}\\,10^{3}$\\,K very deep in their interiors.\n\nDistribution of optical depths within dusty envelopes is one of the main parameters (along with the density structure) for the\ntransfer of radiation and resulting radiative-equilibrium temperatures. All models are quite opaque at visible wavelengths, with\nradial optical depths $\\tau_{V}\\,{\\approx}\\,3\\,{-}\\,3\\,{\\times}\\,10^{3}$ for starless cores and\n$\\tau_{V}\\,{\\approx}\\,500\\,{-}\\,6\\,{\\times}\\,10^{5}$ for protostellar envelopes of different masses and luminosities\n(Fig.~\\ref{loo}). At the far-infrared wavelength of $100$\\,{${\\mu}$m}, starless cores with $M\\,{<}\\,3$\\,$M_{\\sun}$ are transparent,\nwhereas the ones with $M\\,{\\ga}\\,3$\\,$M_{\\sun}$ are optically thick towards their centers. All protostellar envelopes are optically\nthick at $100$\\,{${\\mu}$m} and some of them (with $M\\,{\\ga}\\,0.3\\,M_{\\sun}$) are even opaque at $500$\\,{${\\mu}$m} towards their\ncenters. Sizes of the dust-free cavities of protostellar envelopes increase with the luminosity of their central energy sources\n(cf. Eq.~(\\ref{inner.boundary}), Fig.~\\ref{drp.bes.pro}), thus the optical depths of the enve\\-lopes decrease, approximately as\n$\\tau_{\\lambda}\\,{\\propto}\\,L_{\\star}^{-1\/3}$.\n\nHigh far-infrared optical depths of the model starless cores and protostellar envelopes are localized within relatively small\nspherical zones around their centers. Angular radii of the opaque dusty zones in the protostellar models can be described (at \n$70\\,{-}\\,500$\\,{${\\mu}$m}) by a simple empirical expression\n\\begin{equation}\n\\vartheta \\approx 1.6{\\arcsec} \\left(M\/M_{\\sun}\\right) \\left(\\kappa_{\\lambda}\/\\kappa_{70}\\right),\n\\label{opaque.zone}\n\\end{equation}\nwhich can also be used (within a factor of $1.5\\,{-}\\,2$) for the high-mass models of starless cores of $3, 10$, and\n$30$\\,$M_{\\sun}$, in which the opaque zone exists only at ${\\lambda}\\,{\\le}\\,70, 160$, and $250$\\,{${\\mu}$m}, respectively. \n\nThe density profiles $\\rho(r)\\,{\\propto}\\,r^{-2}$ of the protostellar envelopes are similar to those of the starless cores for\n$r\\,{\\ga}\\,0.1\\,R$ (Fig.~\\ref{drp.bes.pro}). Therefore, whenever an inner opaque zone exists in the objects, its mass obeys \n$m(\\vartheta)\\,{\\propto}\\,\\vartheta$, hence the fractional mass is the fractional radius and, using Eq.~(\\ref{opaque.zone}), can \nbe written as\n\\begin{equation}\nm(\\vartheta)\/M \\approx {\\vartheta}D\/R \\approx 0.022{\\arcsec} \\left(M\/M_{\\sun}\\right) \\left(\\kappa_{\\lambda}\/\\kappa_{70}\\right),\n\\label{frac.mass}\n\\end{equation}\nwhere $\\vartheta$, $D$, and $R$ are in units of arcsec, pc, and AU, respectively. At $\\lambda \\,{\\la}\\, 100$\\,{${\\mu}$m}, the\nopaque zone of high-mass objects extends over a large fraction of their mass. This means that the standard assumption of the\nfar-infrared transparency is severely violated for massive objects. Protostellar envelopes with $M\\,{<}\\,3$\\,$M_{\\sun}$ have small\nopaque zones that contain little mass and thus they cannot substantially affect the standard methods of mass derivation.\n\n\\subsection{Temperature distributions}\n\\label{temperatures}\n\nThe models of starless cores and protostellar envelopes acquire radiative-equilibrium dust temperatures $T_{\\rm d}(r)$ shown in\nFig.~\\ref{trp.bes.pro}. In the adopted isotropic ISRF, the radiative-equilibrium temperature of dust grains with the model\nopacities from Fig.~\\ref{loo} is $T_{\\rm d}\\,{=}\\,17.4$\\,K, the value that the isolated models and embedding clouds acquire at\ntheir outer boundaries in the limit $\\tau_{\\lambda}\\,{\\rightarrow}\\,0$. The lower mass models of starless cores are transparent and\nthus almost isothermal. Their higher mass counterparts develop steeper temperature gradients under the outer boundaries of the\nisolated models and embedding clouds and lower temperatures in their interiors (Fig.~\\ref{trp.bes.pro}).\n\nDisplaying the same behavior under their outer boundaries, protostellar envelopes of all masses develop steep temperature gradients\ntowards the inner boundary (Fig.~\\ref{trp.bes.pro}). Higher accretion luminosities make the dust hotter and thus, with the adopted\ndust sublimation temperature $T_{\\rm S}\\,{=}\\,10^{3}$\\,K, the boundary of the inner dust-free cavity shifts towards larger radial\ndistances (cf. Eq.~(\\ref{inner.boundary})). An analytical approximation of the profiles $T_{\\rm d}(r)$ for protostellar envelopes\ncan be found in Appendix~\\ref{AppendixA}.\n\nDifferences between the isolated and embedded models are highlighted by their different temperature distributions at the outer\nmodel boundary (Fig.~\\ref{trp.bes.pro}). The temperatures of embedded models at $r\\,{=}\\,R$ are significantly lower than those of\nthe isolated models, owing to the absorption of ISRF in the embedding clouds ($R\\,{<}\\,r\\,{\\le}\\,R_{\\rm E}$). The denser the\nembedding cloud is, the lower $T_{\\rm d}(R)$ is and the greater the contrast to the isolated model (Fig.~\\ref{trp.bes.pro}). As the\nbulk of the mass of the models is contained in the outer parts, the differences in the temperature profiles between the isolated\nand embedded models can greatly affect their observational properties, such as the images and total (integrated) fluxes.\n\n\\subsection{Spectral energy distributions}\n\\label{seds}\n\nAfter computing the self-consistent radiative-equilibrium dust temperature distributions $T_{\\rm d}(r)$ from the radiative transfer\nmodels, observables -- such as the intensity maps $\\mathcal{I}_{\\nu}$ and total fluxes $F_{\\nu}$ -- were obtained by a ray-tracing\nalgorithm in separate runs of \\textsl{RADMC-3D}. Effects of the Monte Carlo noise on $F_{\\nu}$, evaluated from the standard\ndeviations about the azimuthally averaged intensity profiles $I_{\\nu}(\\vartheta)$, are below $0.003{\\%}$ and $3{\\%}$ for the\nstarless cores and protostellar envelopes, respectively, in all models and wavebands.\n\nTo emulate the standard observational procedure of flux measurements, $F_{\\nu}$ were integrated from the background-subtracted\nmodel images $\\mathcal{I}_{\\nu}$. The model background $I^{\\rm B}_{\\nu}$ was evaluated as an average intensity\n$\\bar{I_{\\nu}}(R^{\\prime})$ within a ${\\delta}R^{\\prime}$-wide annulus placed just outside the outer model boundary\n($R^{\\prime}\\,{>}\\,R$). In practice, the annulus was one pixel in width (${\\delta}R^{\\prime}\\,{=}\\,0.47\\arcsec$) and it was\ndetached from the boundary by one additional pixel. For the isolated models, $I^{\\rm B}_{\\nu}\\,{=}\\,I^{\\rm ISRF}_{\\nu}$ is the\nintensity of the isotropic ISRF, whereas for the embedded models, $I^{\\rm B}_{\\nu}$ is determined by both $I^{\\rm ISRF}_{\\nu}$ and\nthe transfer of radiation in the background cloud ($R^{\\prime}\\,{\\le}\\,r\\,{\\le}\\,R_{\\rm E}$) along the rays passing through the\nannulus. Inaccuracies inherent in the standard algorithm of background subtraction are discussed in Sect.~\\ref{bg.subtraction} and\nAppendix \\ref{AppendixB}.\n\nSpectral energy distributions (SEDs) of the models of starless cores and protostars are shown in Fig.~\\ref{sed.bes.pro}. The SED\nshapes depend on the density and temperature distributions (Figs.~\\ref{drp.bes.pro} and \\ref{trp.bes.pro}). Large differences\nbetween the SEDs for the isolated and embedded cores are mainly caused by differences in their temperature profiles near the model\nboundary. The SEDs of protostellar envelopes are affected by the same effects to a much lesser degree as their density profiles\nare centrally peaked and their temperature profiles are dominated by the internal radiation source. The SEDs of the models of\ndifferent masses and luminosities show a large variety of shapes in the far-infrared domain (Fig.~\\ref{sed.bes.pro}) due to \nvarying optical depths and temperatures.\n\nProviding a useful reference in our analysis, additional ray-tracing runs of \\textsl{RADMC-3D} computed the fluxes of isothermal\nmodels. These are the same models described above (Fig.~\\ref{drp.bes.pro}), in which self-consistent temperature profiles\n(Fig.~\\ref{trp.bes.pro}) have been replaced with their mass-averaged values:\n\\begin{equation}\n\\begin{split}\nT_{M} &= M^{\\,-1}\\!\\int{T_{\\rm d}(x,y,z)\\,\\rho(x,y,z)\\,\\mathrm{d}{x}\\,\\mathrm{d}{y}\\,\\mathrm{d}{z}}\\\\\n &= M^{\\,-1}\\,4\\pi\\!\\int{T_{\\rm d}(r)\\,\\rho(r)\\,r^{2}\\,\\mathrm{d}{r}}.\n\\end{split}\n\\label{mass.averaged}\n\\end{equation}\nThe resulting total fluxes of the isothermal models are denoted $F_{\\nu}(T_{M})$.\n\n\n\\section{Fitting source fluxes and intensities}\n\\label{fitting.methods}\n\nIn observational studies, after obtaining multiwavelength images $\\mathcal{I}_{\\nu}$ and integrating background-subtracted (and\ndeblended) fluxes $F_{\\nu}$ of extracted sources, their spectral distributions need to be fitted to derive fundamental physical\nparameters, such as the source mass and luminosity.\n\nThe standard technique uses the well-known formal solution of the radiative transfer equation that can be written as\n\\begin{equation}\nI_{\\nu} = B_{\\nu}(T)\\left(1 - \\exp\\left(-\\tau_{\\nu}\\right)\\right),\n\\label{rtsolution}\n\\end{equation}\nwhere $I_{\\nu}$ is the observed specific intensity, $T$ is the homogeneous temperature of an object, $B_{\\nu}(T)$ is the blackbody\nintensity, and $\\tau_{\\nu}$ is the optical depth of the object. After obtaining an image\n$\\mathcal{I}_{\\nu}\\,{\\equiv}\\,I_{\\nu\\,ij}$, the total flux $F_{\\nu}\\,{=}\\,\\!\\int{\\mathcal{I}_{\\nu}\\,\\mathrm{d}{\\Omega}}$ can be\nintegrated over the solid angle $\\Omega$ subtended by the object. For constant intensity, it reduces to\n$F_{\\nu}\\,{=}\\,I_{\\nu}\\,\\Omega$. A critical assumption used in the derivation of Eq.~(\\ref{rtsolution}) is that the object is\nhomogeneous in temperature, whereas the temperatures of the astrophysical objects are actually nonuniform (cf.\nFig.~\\ref{trp.bes.pro}).\n\nTwo methods and two fitting models were explored in this work that have been used in observational studies of star formation to\nestimate source temperatures and masses.\n\n\\subsection{Fitting total fluxes $F_{\\nu}$}\n\\label{fitting.fluxes}\n\n\nIn this method, the total fluxes $F_{\\nu}$ are integrated from background-subtracted and deblended images $\\mathcal{I}_{\\nu}$ of\nsource intensities and then are fitted to estimate source mass as one of the fitting parameters. With the adopted parameterization\nof the power-law opacity $\\kappa_{\\nu}\\,{=}\\,\\kappa_{0}\\left(\\nu\/\\nu_{0}\\right)^{\\,\\beta}$, it is possible to write \nEq.~(\\ref{rtsolution}) in the form\n\\begin{equation}\nF_{\\nu} = B_{\\nu}(T)\\left(1 - \\exp\\left(-\\kappa_{0}\\left(\\nu\/\\nu_{0}\\right)^{\\,\\beta\\!} \\eta M D^{-2} \\Omega^{-1}\\right)\\right) \n\\Omega,\n\\label{modbody}\n\\end{equation}\nwhere $\\eta$ is the dust-to-gas mass ratio and $D$ is the source distance. The fitting model of Eq.~(\\ref{modbody}) with five\nparameters ($T$, $M$, $\\beta$, $D$, $\\Omega$) is referred to as \\textsl{modbody} in this paper. After fitting $F_{\\nu}$ and\nestimating the model parameters, the average column density $N_{\\rm H_2}$ can be obtained from $M\\,{=}\\,\\mu m_{\\rm H} N_{\\rm H_2}\nD^{2} \\Omega$, where $\\mu\\,{=}\\,2.8$ is the mean molecular weight per H$_2$ molecule and $m_{\\rm H}$ is the hydrogen mass.\n\nWith an additional assumption that measured fluxes $F_{\\nu}$ represent optically thin emission\\footnote{Far-infrared transparency\nis an important assumption that is wrong for high-mass objects (Sect.~\\ref{radiation.odepths}).}, Eq.~(\\ref{modbody}) can be \nwritten as\n\\begin{equation}\nF_{\\nu} = B_{\\nu}(T) \\,\\kappa_{0}\\left(\\nu\/\\nu_{0}\\right)^{\\,\\beta} \\eta M D^{-2}.\n\\label{thinbody}\n\\end{equation}\nThe fitting model of Eq.~(\\ref{thinbody}) with four parameters ($T$, $M$, $\\beta$, $D$) is referred to as \\textsl{thinbody} in this\npaper. By the definition ($\\tau_{\\nu}\\,{\\ll}\\,1$), it produces only fits with the modified blackbody shapes\n${\\kappa_{\\nu}\\,B_{\\nu}(T)}$ that are scaled up or down, depending on $M$. Obviously, the \\textsl{modbody} fits with\n$\\tau_{\\nu}\\,{\\ll}\\,1$ produce the same shapes as the \\textsl{thinbody} model does, whereas the \\textsl{modbody} fits with\n$\\tau_{\\nu}\\,{\\gg}\\,1$ resemble a blackbody $B_{\\nu}(T)$. In the intermediate (semi-opaque) cases, the short-wavelength parts of\nthe fitted curves can be described by $B_{\\nu}(T)$ while morphing into ${\\kappa_{\\nu}\\,B_{\\nu}(T)}$ at long wavelengths where the\nradiation becomes optically thin. With more flexible shapes, \\textsl{modbody} can give better fits of the data, but it does not\nnecessarily lead to good estimates of temperatures and masses.\n\nAfter fitting fluxes with a \\textsl{modbody} or \\textsl{thinbody} model, an estimate of $T_{F}$ and the corresponding mass $M_{F}$\nare obtained. For the realistic objects with strongly nonuniform temperatures $T_{\\rm d}(r)$ (Fig.~\\ref{trp.bes.pro}), emerging\nfluxes $F_{\\nu}$ are heavily distorted from the simple shapes of the fitting models, hence these models are inadequate and an\nestimate of $T_{F}$ does not guarantee that $M_{F}$ is close to the true mass $M$. For the purpose of obtaining accurate\n$M_{F}\\,{\\approx}\\,M$, it is necessary (but not sufficient) to have $T_{F}\\,{\\approx}\\,T_{M}$, i.e., it is possible to interpret\n$T$ in Eq.~(\\ref{thinbody}) as the mass-averaged $T_{M}$ from Eq.~(\\ref{mass.averaged}). In fact, assuming $\\tau_{\\nu}\\,{\\ll}\\,1$\nin the far-infrared, the observed fluxes contain emission of all dust grains, which is proportional to the mass of dust at \ndifferent $T_{\\rm d}$ in the entire volume of an object:\n\\begin{equation}\nF_{\\nu} = \\kappa_{\\nu} \\eta D^{-2}\\!\\int{B_{\\nu}(T_{\\rm d}(x,y,z))\\,\\rho(x,y,z)\\,\\mathrm{d}{x}\\,\\mathrm{d}{y}\\,\\mathrm{d}{z}}. \n\\label{emission}\n\\end{equation}\nEquations~(\\ref{thinbody}) and (\\ref{emission}) can immediately be combined into a definition of the mass-averaged intensity\n$B_{\\nu M}$. Since $B_{\\nu}(T)\\,{\\propto}\\,T$ in the Rayleigh-Jeans domain, the equations are also readily converted into $T_{M}$\nfrom Eq.~(\\ref{mass.averaged}). In the model objects studied here, differences between $B_{\\nu M}$ and $B_{\\nu}(T_{M})$ quickly\nbecome negligible beyond the peak wavelength of the latter ($\\lambda$ $\\ga$ $2\\,\\lambda_{\\rm peak}$). Therefore, $T_{M}$ is fully\nconsistent with the fitting models at long wavelengths.\n\n\\subsection{Fitting image intensities $I_{\\nu}$}\n\\label{fitting.intens}\n\nIn this method, it is possible to fit pixel intensity distributions $I_{\\nu\\,ij}$ of the background-subtracted and deblended images\n$\\mathcal{I}_{\\nu}$ of a source\\footnote{In an alternative approach, multiwavelength images of an entire field can be fitted to\nderive its $\\mathcal{N}_{\\rm H_2}$ image, then to identify (extract) the sources and to integrate their masses. Both approaches are\nequivalent in this model-based study, hence the alternative method was not used.}, to derive a map of its column densities\n$\\mathcal{N}_{\\rm H_2}\\,{\\equiv}\\,N_{{\\rm H_2} ij}$ and then the source mass $M\\,{=}\\,\\mu m_{\\rm H} D^{2} \\Omega \\sum_{}{N_{{\\rm\nH_2} ij}}$. It is convenient to express the \\textsl{modbody} and \\textsl{thinbody} models from Eqs.~(\\ref{modbody}) and\n(\\ref{thinbody}) as functions of the pixel column density $N_{\\rm H_2}$:\n\\begin{eqnarray}\nI_{\\nu}&\\!\\!\\!\\!\\!=\\!\\!\\!\\!\\!&B_{\\nu}(T)\\left(1 - \\exp\\left(-\\kappa_{0}\\left(\\nu\/\\nu_0\\right)^{\\,\\beta} \\eta \\mu m_{\\rm H} \nN_{\\rm H_2}\\right)\\right), \\label{modbody.intens}\\\\\nI_{\\nu}&\\!\\!\\!\\!\\!=\\!\\!\\!\\!\\!&B_{\\nu}(T) \\,\\kappa_{0}\\left(\\nu\/\\nu_0\\right)^{\\,\\beta} \\eta \\mu m_{\\rm H} \nN_{\\rm H_2}. \\label{thinbody.intens}\n\\end{eqnarray}\nIn this formulation, both models have only three fitting parameters ($T$, $N_{\\rm H_2}$, $\\beta$) in contrast to the case where\ntotal fluxes are fitted (five parameters for \\textsl{modbody} in Eq.~(\\ref{modbody}) and four parameters for \\textsl{thinbody} in \nEq.~(\\ref{thinbody}); see Sect.~\\ref{fitting.fluxes}). Furthermore, limited angular resolutions of real images makes the results of \nfitting $I_{\\nu}$ depend sensitively on the degree to which a source is resolved.\n\nFor fully resolved sources, such as the model objects used in this work, relatively small pixels sample completely independent\nintensities from different rays. For progressively lower angular resolutions, intensities within a beam become increasingly blended\ntogether. Radiation with different temperatures gets mixed not only along the line of sight, but also in the transverse directions,\nin the plane of the sky. For unresolved objects with intrinsic temperature gradients, radiation from the entire object becomes\nheavily blended, leading to strong distortions of their spectral intensity distributions.\n\nAn important assumption used in the derivation of $N_{{\\rm H_2} ij}$ is a constant temperature $T_{\\rm d}(x,y,z)$ along the lines\nof sight within a certain radial distance from a pixel $(i,j)$. The distance depends on the angular resolution of images: for less\nresolved sources, temperatures from a larger environment of the pixel contribute to its intensity. With low optical depths\n$\\tau_{\\nu}\\,{\\ll}\\,1$ in the far-infrared, emission is observed from the entire column of dust grains at $(i,j)$ with different\ntemperatures $T_{\\rm d}(z)$ along the line of sight. The reasoning associated with Eq.~(\\ref{emission}) can be applied to show that\n$T$ in Eq.~(\\ref{thinbody.intens}) is consistent with the column-averaged temperature\n\\begin{equation}\nT_{N ij} = N_{{\\rm H_2} ij}^{\\,-1}\\!\\int{T_{\\rm d}(x,y,z)\\,\\rho(x,y,z)\\,\\mathrm{d}{z}}.\n\\label{coldens.averaged}\n\\end{equation}\nA mass-averaged temperature, equivalent to that from Eq.~(\\ref{mass.averaged}), can be obtained as \n$T_{M}\\,{=}\\,M^{-1} \\mu m_{\\rm H} D^{2} \\Omega \\sum_{}{T_{N ij}\\,N_{{\\rm H_2} ij}}$.\n\n\\subsection{Variable and fixed parameters}\n\\label{variable.fixed}\n\nIn most studies, the opacity slope $\\beta$ has been kept fixed in the fitting process to reduce the number of free parameters and\nimprove the robustness of derived parameters. Following this practice, Sect.~\\ref{results} presents and discusses only the results\nof fitting with a fixed opacity slope. When fitting intensities $I_{\\nu}$ with $\\beta$ fixed, the number of free variable\nparameters becomes $\\gamma\\,{=}\\,2$ for both \\textsl{thinbody} and \\textsl{modbody} models ($T$, $N_{\\rm H_2}$). When fitting\nfluxes $F_{\\nu}$, distance $D$ is also assigned a fixed value to further reduce the degrees of freedom, although astronomical\ndistances are poorly known. The number of free variable parameters is thus $\\gamma\\,{=}\\,2$ for \\textsl{thinbody} ($T$, $M$) and\n$\\gamma\\,{=}\\,3$ for \\textsl{modbody} ($T$, $M$, $\\Omega$).\n\nIn practice, after measuring $F_{\\nu}\\,{=}\\,\\!\\int{I_{\\nu}\\,\\mathrm{d}{\\Omega}}$, the solid angle $\\Omega$ over which $I_{\\nu}$\nwere integrated is known\\footnote{In real observations, images $\\mathcal{I}_{\\nu}$ usually have different angular resolutions and\nthe flux integration area is wavelength dependent.} and its value can be fixed, reducing $\\gamma$ for \\textsl{modbody} to two free\nvariables ($T$, $M$). In this model study, one could also keep $\\Omega\\,{=}\\,{\\pi\\,}(R\/D)^{2}$ constant, as the true values of $R$\nand $D$ are known; however, \\textsl{modbody} would then become completely equivalent to \\textsl{thinbody}. Indeed, fixing $\\Omega$\nof transparent objects at accurate (or even overestimated) values means that the optical depths in Eq.~(\\ref{modbody}) are very\nsmall ($\\tau_{\\nu}\\,{=}\\,\\kappa_{\\nu} \\eta M D^{-2} \\Omega^{-1}{\\ll}\\,1$), which effectively converts \\textsl{modbody} into\n\\textsl{thinbody}. Only when fixing strongly underestimated values $\\Omega\\,{\\ll}\\,{\\pi}\\,(R\/D)^{2}$, the far-infrared $\\tau_{\\nu}$\nbecome large enough to produce any noticeable differences between \\textsl{modbody} and \\textsl{thinbody}. This work investigates\nqualities of two \\emph{different} models, hence $\\Omega$ was allowed to vary in all \\textsl{modbody} fits of $F_{\\nu}$.\n\n\\begin{figure*}\n\\centering\n\\centerline{\\resizebox{0.3400\\hsize}{!}{\\includegraphics{fits.bes.thin.cfg.trim.pdf}}\n \\resizebox{0.3400\\hsize}{!}{\\includegraphics{fits.bes.gray.cfg.trim.pdf}}}\n\\centerline{\\resizebox{0.3400\\hsize}{!}{\\includegraphics{fits.pro.thin.cfg.trim.pdf}}\n \\resizebox{0.3400\\hsize}{!}{\\includegraphics{fits.pro.gray.cfg.trim.pdf}}}\n\\caption{\nFluxes of the \\emph{isolated} starless cores (\\emph{upper}) and protostellar envelopes (\\emph{lower}) fitted with the\n\\textsl{thinbody} (\\emph{left}) and \\textsl{modbody} (\\emph{right}) models. The fluxes for the 0.03, 0.3, and 3\\,$M_{\\sun}$ cores\nand envelopes (with $L_{\\star}\\,{=}\\,1\\,L_{\\sun}$) from Fig.~\\ref{sed.bes.pro} are shown as black triangles, squares, and\ncircles, respectively. Successful fits (Sect.~\\ref{data.subsets}) using different subsets $\\Phi_{n}$ of fluxes are indicated by\nthe solid and dashed lines of different widths and colors. The thin black curves show the flux distributions for the\n\\emph{isothermal} models where the actual model temperatures (Fig.~\\ref{trp.bes.pro}) were replaced with their mass-averaged \nvalues: $T_{\\rm d}(r)\\,{=}\\,T_{M}$ ($16.3, 13.2, 9.2$\\,K for starless cores and $18.1, 15.6, 12.3$\\,K for protostellar envelopes).\n} \n\\label{fits.examples}\n\\end{figure*}\n\nWhen fitting pixel intensities $I_{\\nu}$ instead of $F_{\\nu}$, the far-infrared $\\tau_{\\nu\\,ij}$ within most of the image pixels\n$(i,j)$ are small, even for perfectly resolved sources. For poorly resolved or unresolved sources, radiation within the beams gets\ndiluted and maximum values of $\\tau_{\\nu\\,ij}$ in the images become smaller. All models of starless cores and protostellar\nenvelopes contain $96{\\%}$ and $90{\\%}$ of their masses, respectively, in their outer parts ($r\\,{\\ga}\\,0.1\\,R$,\nFig.~\\ref{drp.bes.pro}). Intensities in the outer parts of the source images come mostly from the pixel columns of dust with\n$\\tau_{\\nu\\,ij}{\\,\\ll\\,}1$ in the far-infrared. Only in the models with $M\\,{\\ga}\\,3$\\,$M_{\\sun}$ do they become substantially\naffected by the radiation from the central opaque zone (Sect.~\\ref{radiation.odepths}). As a result, the masses derived from\nfitting $I_{\\nu}$ are almost the same (within ${\\sim}\\,20{\\%}$) for both fitting models and hence only the \\textsl{thinbody}\nresults are presented for this method.\n\n\\subsection{Data points and their subsets}\n\\label{data.subsets}\n\nFitting was executed for a set of the total model fluxes $\\left\\{ F_{\\lambda_{i}} \\right\\}$ or pixel intensities $\\left\\{\nI_{\\lambda_{i}} \\right\\}$ $(i\\,{=}\\,1, 2, \\dots, 6)$ at the \\emph{Herschel} wavelengths $\\lambda_{i}$ of 70, 100, 160, 250, 350,\nand 500\\,{${\\mu}$m}. In this model-based study, the intensities and fluxes of numerical models have essentially no measurement\nerrors. It makes sense, however, to make their uncertainties resemble typical observational values, hence to get an idea of\nrealistic inaccuracies of the estimated parameters (masses, temperatures). Before the fitting, the model intensities and fluxes\nwere assigned an additional (optimistic) uncertainty of $15{\\%}$, a value similar to the levels of calibration errors in real\nobservations (e.g., with \\emph{Herschel}). The above uncertainties were associated with the exact data points to see how typical\ndata uncertainties translate into the resulting error bars of the derived parameters. Extra uncertainties come from the fact that\nthe dust-to-gas ratio $\\eta$, reference opacity $\\kappa_0$, and distance $D$, which are used in the fitting models\n(Sects.~\\ref{fitting.fluxes}, \\ref{fitting.intens}) but held constant, are actually poorly known. Conservatively assuming that the\nquantities have random and independent uncertainties of $20{\\%}$, the latter were added in quadrature to those of the derived\nmasses, for the same purpose of obtaining the total resulting mass uncertainties.\n\nTo isolate the effects of temperature gradients in starless cores and protostellar envelopes (Fig.~\\ref{trp.bes.pro}), the fitting\nwas done for several subsets of data, removing some (or none) of the shortest-wavelength points from the fitting process. The data\nsubsets are denoted $\\Phi_{n}\\,{=}\\,\\left\\{Y_{\\lambda_{i}}\\right\\}$, where $Y_{\\lambda_{i}}$ is either $F_{\\lambda_{i}}$ or\n$I_{\\lambda_{i}}$ and $n$ is the number of the longest wavelengths used in the fitting\\footnote{ $\\Phi_{6}\\,{=}\\,\\{Y_{70}, Y_{100},\nY_{160}, Y_{250}, Y_{350}, Y_{500}\\}$, $\\Phi_{5}\\,{=}\\,\\{Y_{100}, Y_{160}, Y_{250}, Y_{350}, Y_{500}\\}$, $\\Phi_{4}\\,{=}\\,\\{Y_{160},\nY_{250}, Y_{350}, Y_{500}\\}$, $\\Phi_{3}\\,{=}\\,\\{Y_{250}, Y_{350}, Y_{500}\\}$, $\\Phi_{2}\\,{=}\\,\\{Y_{350}, Y_{500}\\}$.}. Fits of\ntotal fluxes were considered successful (acceptable) and their results are shown below, if $\\chi^{2}\\,{\\le}\\,{n-\\gamma+1}$, with\nthe last term added to allow testing $\\chi^{2}$ for zero degrees of freedom ($n\\,{=}\\,\\gamma$). Fits of image intensities were\nconsidered successful, if the same goodness condition was fulfilled in \\emph{all} pixels within an object. These results, as well\nas the somewhat less reliable results with ${n-\\gamma+1}\\,{<}\\,\\chi^{2}\\,{<}\\,10$ in \\emph{some} pixels are presented below.\nDetails of the fitting algorithm can be found in Appendix \\ref{AppendixC}.\n\n\n\\section{Results}\n\\label{results}\n\nThis section describes derived parameters for both starless cores and protostellar envelopes, obtained from acceptable fits for all\nsubsets $\\Phi_{n}$ (Sect.~\\ref{data.subsets}) for both \\textsl{modbody} and \\textsl{thinbody} (Sect.~\\ref{fitting.fluxes}). Results\nfor the isothermal models are presented in Appendix \\ref{AppendixD}. To evaluate the effects of the un\\-certain far-infrared\nopacity slope, results are shown for $\\beta\\,{=}\\,2$ used in the radiative transfer modeling and for two other $\\beta$ values\n($1.67$, $2.4$), differing from the true value by a factor of $1.2$. Results obtained with variable fitting parameter $\\beta$ are\ndescribed in Appendix~\\ref{AppendixE}.\n\nMasses derived from fitting images $\\mathcal{I}_{\\nu}$ of objects with temperature gradients must depend on their angular\nresolutions (Sect.~\\ref{fitting.intens}). To investigate this effect, the model images with pixels of $0.47{\\arcsec}$ were\nconvolved with Gaussian beams of $1$, $36$, and $144{\\arcsec}$ (FWHM) and then resampled to $1$, $12$, and $48{\\arcsec}$ pixels,\nrespectively. For the objects with diameters of $142{\\arcsec}$ ($2\\,{\\times}\\,10^{4}$\\,AU, Fig.~\\ref{drp.bes.pro}), the three\nvariants represent resolved, partially resolved, and unresolved cases. \n\nIn this paper, the term \\emph{uncertainties} refers to the error bars of measured or derived quantities, the term\n\\emph{inaccuracies} (sometimes simply \\emph{errors}) refers to the deviations of the derived quantities from their model values,\nand the term \\emph{biases} denotes variable systematic dependences of inaccuracies across the ranges of model parameters ($M$, $L$,\n$T_{M}$).\n\n\\subsection{Selected examples}\n\\label{selected.examples}\n\nExamples of the fits of $F_{\\nu}$ for the isolated starless cores and protostellar envelopes with masses of $0.03$, $0.3$, and\n$3\\,M_{\\sun}$ are shown in Fig.~\\ref{fits.examples}. Although the qualitatively similar plots for embedded models are not\npresented, their derived parameters and uncertainties are described in Sect.~\\ref{derived.properties}.\n\n\\begin{figure*}\n\\centering\n\\centerline{\\resizebox{0.3327\\hsize}{!}{\\includegraphics{bes.TT.b1p7.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{bes.TT.b2p0.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{bes.TT.b2p4.cfg.trim.pdf}}}\n\\centerline{\\resizebox{0.3327\\hsize}{!}{\\includegraphics{bes.MM.b1p7.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{bes.MM.b2p0.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{bes.MM.b2p4.cfg.trim.pdf}}}\n\\centerline{\\resizebox{0.3327\\hsize}{!}{\\includegraphics{bes.emb.TT.b1p7.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{bes.emb.TT.b2p0.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{bes.emb.TT.b2p4.cfg.trim.pdf}}}\n\\centerline{\\resizebox{0.3327\\hsize}{!}{\\includegraphics{bes.emb.MM.b1p7.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{bes.emb.MM.b2p0.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{bes.emb.MM.b2p4.cfg.trim.pdf}}}\n\\caption{\nTemperatures $T_{F}$ and masses $M_{F}$ derived from fitting $F_{\\nu}$ of both \\emph{isolated} and \\emph{embedded} starless cores \nvs. the true model values of $T_{M}$ and $M$ for three $\\beta$ values (1.67, 2, 2.4). For various subsets $\\Phi_{n}$ of fluxes, \nresults from successful \\textsl{thinbody} and \\textsl{modbody} fits (Sect.~\\ref{data.subsets}) are displayed by the colored and gray \nlines, respectively. Error bars represent the $1\\,{\\times}\\,\\sigma$ uncertainties of the derived parameters returned by the fitting \nroutine combined with the assumed $\\pm\\,20{\\%}$ uncertainties of $\\eta$, $\\kappa_{0}$, and $D$ (Sect.~\\ref{data.subsets}). The black \nsolid lines show the locations where $T_{F}$ and $M_{F}$ are equal to the true values. To preserve clarity of the plots, \\emph{much} \nless accurate \\textsl{modbody} results are displayed only for correct $\\beta\\,{=}\\,2$ and without error bars.\n} \n\\label{temp.mass.bes}\n\\end{figure*}\n\n\\begin{figure*}\n\\centering\n\\centerline{\\resizebox{0.3327\\hsize}{!}{\\includegraphics{pro.TT.b1p7.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{pro.TT.b2p0.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{pro.TT.b2p4.cfg.trim.pdf}}}\n\\centerline{\\resizebox{0.3327\\hsize}{!}{\\includegraphics{pro.LM.b1p7.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{pro.LM.b2p0.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{pro.LM.b2p4.cfg.trim.pdf}}}\n\\centerline{\\resizebox{0.3327\\hsize}{!}{\\includegraphics{pro.emb.TT.b1p7.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{pro.emb.TT.b2p0.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{pro.emb.TT.b2p4.cfg.trim.pdf}}}\n\\centerline{\\resizebox{0.3327\\hsize}{!}{\\includegraphics{pro.emb.LM.b1p7.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{pro.emb.LM.b2p0.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{pro.emb.LM.b2p4.cfg.trim.pdf}}}\n\\caption{\nTemperatures $T_{F}$ and masses $M_{F}$ derived from fitting $F_{\\nu}$ of both \\emph{isolated} and \\emph{embedded} protostellar\nenvelopes (with the true masses $M$ of 0.03, 0.3, and 3$\\,M_{\\sun}$) vs. the true model values of $T_{M}$ and $L$ for three\n$\\beta$ values (1.67, 2, 2.4). Results from successful \\textsl{thinbody} and \\textsl{modbody} fits for various subsets $\\Phi_{n}$ \nof fluxes (Sect.~\\ref{data.subsets}) are displayed by the colored and gray lines, respectively. See Fig.~\\ref{temp.mass.bes} for \nmore details.\n} \n\\label{temp.mass.pro}\n\\end{figure*}\n\nFlux distributions of the isolated starless cores (Fig.~\\ref{sed.bes.pro}) are similar to those of the modified blackbodies\n$\\kappa_{\\nu}\\,B_{\\nu}(T_{M})$. The fits for a low-mass core with $M\\,{=}\\,0.03\\,M_{\\sun}$ shown in Fig.~\\ref{fits.examples} are\nidentical for all subsets $\\Phi_{n}$ since the core is nearly isothermal, with $T_{\\rm d}(r)$ very similar to its\n$T_{M}\\,{=}\\,16.3$\\,K. Fluxes of the higher-mass cores of $0.3$ and $3\\,M_{\\sun}$ display larger deviations from the fluxes\n$F_{\\nu}(T_{M})$ of isothermal models for larger subsets $\\Phi_{n}$ ($n\\,{=}\\,3\\,{\\rightarrow}\\,6$). The shapes of $F_{\\nu}$ become\n``hotter'' because of the steeper temperature profiles (Sect.~\\ref{temperatures}) at the outer boundary (Fig.~\\ref{trp.bes.pro}).\nFor a massive core of $3\\,M_{\\sun}$, discrepancies between $F_{\\nu}$ and $F_{\\nu}(T_{M})$ at $\\lambda\\,{\\la}\\,160$\\,{${\\mu}$m}\nreach factors ${\\ga}\\,5$.\n\nFlux distributions of an isolated protostellar envelope with $M\\,{=}\\,0.03\\,M_{\\sun}$ (Fig.~\\ref{sed.bes.pro}) display various\nshapes that are quite different from those of $\\kappa_{\\nu}\\,B_{\\nu}(T_{M})$, whereas for a more opaque envelope of $3\\,M_{\\sun}$\nthey become similar to the modified blackbody shapes. The protostellar fits (Fig.~\\ref{fits.examples}) show greater deviations for\nlarger subsets $\\Phi_{n}$ ($n\\,{=}\\,3\\,{\\rightarrow}\\,6$), much larger than those of starless cores. Differences between $F_{\\nu}$\nand $F_{\\nu}(T_{M})$ reach orders of magnitude at $\\lambda\\,{\\la}\\,100$\\,{${\\mu}$m}. The shapes appear much ``hotter'' owing to\n$T_{\\rm d}\\,{\\sim}\\,100{-}10^{3}$\\,K (Sect.~\\ref{temperatures}) deep inside the envelopes (Fig.~\\ref{trp.bes.pro}). The\nlower-mass protostellar envelopes are more transparent and the hot emission greatly distorts $F_{\\nu}$ at\n$\\lambda\\,{\\la}\\,250$\\,{${\\mu}$m}.\n\n\\subsection{Properties derived from fitting fluxes $F_{\\nu}$}\n\\label{derived.properties}\n\nIsolated starless cores, $\\beta\\,{=}\\,2$ (Fig.~\\ref{temp.mass.bes}). For the low-mass, transparent cores\n($M\\,{\\rightarrow}\\,0.03\\,M_{\\sun}$), quite accurate values $T_{F}\\,{\\approx}\\,T_{M}$ and $M_{F}\\,{\\approx}\\,M$ are derived for all\nsubsets $\\Phi_{n}$. For the denser, more opaque cores ($M\\,{\\rightarrow}\\,30\\,M_{\\sun}$), derived $T_{F}$ and $M_{F}$ become more\nover- and underestimated, respectively, as the spectral shapes of $F_{\\nu}$ become much wider and distorted towards shorter\nwavelengths (Fig.~\\ref{sed.bes.pro}). The biases and inaccuracy of the estimates depend on the subset $\\Phi_{n}$, with the least\ninaccurate $T_{F}$ and $M_{F}$ obtained for the \\textsl{thinbody} fits of $\\Phi_{2}$. However, the biases of the parameters across\nthe entire mass range remains fairly strong. Derived masses of the starless cores are underestimated within a factor of $2$ for\n$1\\,{<}\\,M\\,{\\le}\\,3\\,M_{\\sun}$ and factor of $5$ for $3\\,{<}\\,M\\,{\\le}\\,30\\,M_{\\sun}$.\n\nEmbedded starless cores, $\\beta\\,{=}\\,2$ (Fig.~\\ref{temp.mass.bes}). For the low-mass, transparent cores\n($M\\,{\\rightarrow}\\,0.03\\,M_{\\sun}$), $M_{F}$ are underestimated by a factor of $1.35$ for all subsets $\\Phi_{n}$, although $T_{F}$\nare quite accurate because the standard observational procedure of background subtraction ignores the fact that embedding\nbackgrounds tend to be rim-brightened at their outer boundary $R$ (Appendix \\ref{AppendixB}, Sect.~\\ref{bg.subtraction}). The\nembedded cores have $T_{\\rm d}(r)$ that are quite flat across their boundary for all masses (Fig.~\\ref{trp.bes.pro}). Having no\nflux distortions caused by nonuniform temperatures (Fig.~\\ref{sed.bes.pro}), the $F_{\\nu}$ peaks of the most massive cores\n($M\\,{\\rightarrow}\\,30\\,M_{\\sun}$) move towards the longest wavelength ($\\lambda_{6}\\,{=}\\,500$\\,{${\\mu}$m}), which leads to\n$T_{F}$ and $M_{F}$ that are under- and overestimated, respectively.\n\nIsolated protostellar envelopes, $\\beta\\,{=}\\,2$ (Fig.~\\ref{temp.mass.pro}). Emission of the hot dust with $T_{\\rm\nd}{\\sim}\\,100{-}10^{3}$\\,K greatly skews their $F_{\\nu}$ towards shorter wavelengths (Fig.~\\ref{sed.bes.pro}). This becomes\nespecially significant for the lower mass, more transparent envelopes ($M\\,{\\rightarrow}\\,0.03\\,M_{\\sun}$,\n$L_{\\star}\\,{\\rightarrow}\\,30\\,L_{\\sun}$) that produce hotter dust over a much larger volume (Fig.~\\ref{trp.bes.pro}). The\n\\textsl{thinbody} fits of larger subsets $\\Phi_{n}$ ($n\\,{=}\\,3\\,{\\rightarrow}\\,6$), lead to errors in $T_{F}$ and $M_{F}$ that\nreach factors of $1.6$ and $3$, respectively. The smallest subset $\\Phi_{2}$ is unaffected by the hot emission and it produces\nfairly accurate \\textsl{thinbody} estimates of $T_{F}$ and $M_{F}$ (for all $M$ and $L_{\\star}$) within factors of $1.1$ and\n$1.3$, respectively.\n\nEmbedded protostellar envelopes, $\\beta\\,{=}\\,2$ (Fig.~\\ref{temp.mass.pro}). Results are qualitatively similar to those of the\nisolated envelopes, although with larger inaccuracies. Derived $M_{F}$ are underestimated by at least a factor of $1.5$, mostly due\nto over-subtraction of the rim-brightened embedding background (Appendix \\ref{AppendixB}, Sect.~\\ref{bg.subtraction}). Although the\nenvelopes have $T_{\\rm d}(r)$ that are quite flat across their boundaries (Fig.~\\ref{trp.bes.pro}), their derived parameters are\ngreatly affected by the skewed $F_{\\nu}$ owing to the hot dust deep in their interiors. The \\textsl{thinbody} fits of large subsets\n$\\Phi_{n}$ ($n\\,{=}\\,3\\,{\\rightarrow}\\,6$) lead to inaccuracies in $T_{F}$ and $M_{F}$ as large as factors of $1.4{-}1.8$ and\n$3{-}5$, respectively. The most accurate $T_{F}$ and $M_{F}$, obtained for the smallest subset $\\Phi_{2}$, are underestimated\nwithin factors of $1.2$ and $2$.\n\nEffects of the adopted opacity slope $\\beta$ on the estimated parameters are similar for both starless cores\n(Fig.~\\ref{temp.mass.bes}) and protostellar envelopes (Fig.~\\ref{temp.mass.pro}). Although detailed behavior of the differences\nwith respect to the above results for true $\\beta\\,{=}\\,2$ depends on the subset $\\Phi_{n}$, clear general trends can be seen.\nShallower slopes ($\\beta\\,{=}\\,1.67$) lead to an increase in $T_{F}$ and thus $M_{F}$ becomes smaller, whereas steeper slopes\n($\\beta\\,{=}\\,2.4$) lead to a decrease in $T_{F}$ and hence $M_{F}$ becomes larger, in both cases by a factor of approximately $2$.\n\nThe \\textsl{thinbody} fitting model produces much better overall results than \\textsl{modbody} does. Parameters estimated with\n\\textsl{modbody} become so incorrect that they may be considered completely unusable. The importance of estimating accurate\nmass-averaged temperatures $T_{M}$ for deriving correct masses $M_{F}$ is illustrated by the isothermal models presented in\nAppendix \\ref{AppendixD}.\n\n\\subsection{Properties derived from fitting images $\\mathcal{I}_{\\nu}$}\n\\label{coldens.properties}\n\nThis section presents results for both starless cores and protostellar envelopes, obtained from successful fits of the\nbackground-subtracted $\\mathcal{I}_{\\nu}$ for all subsets $\\Phi_{n}$, for only the \\textsl{thinbody} fitting model. Derived\n\\textsl{modbody} masses are practically the same as the \\textsl{thinbody} masses, because the bulk of the model mass is in\noptically thin regions (Sect.~\\ref{variable.fixed}). Effects of the adopted far-infrared opacity slopes are the same as when\nfitting $F_{\\nu}$ (Sect.~\\ref{derived.properties}): under- or overestimating $\\beta$ by a factor of $1.2$ gives masses\n$M_{\\mathcal{I}}$ that are systematically under- or overestimated by a factor of $2$. The method of fitting images\n$\\mathcal{I}_{\\nu}$, thereby deriving $\\mathcal{N}_{\\rm H_2}$, and afterwards integrating source mass $M_{\\mathcal{I}}$ brings\nclear benefits for well-resolved starless cores with nonuniform temperatures, compared to the other method (Sect.\n\\ref{derived.properties}) of first integrating total fluxes $F_{\\nu}$ from $\\mathcal{I}_{\\nu}$ (losing all spatial information) and\nthen estimating $M_{F}$ from the fitting model.\n\nIsolated starless cores, $\\beta\\,{=}\\,2$ (Fig.~\\ref{coldens.bes}). For the fully resolved models, derived $T_{\\mathcal{I}}$ and\n$M_{\\mathcal{I}}$ have fairly good accuracy and little bias for acceptable fits, although the range of the latter for larger\n$\\Phi_{n}$ ($n\\,{=}\\,3\\,{\\rightarrow}\\,6$) shrinks to the lowest masses. As the transparent low-mass cores\n($M\\,{\\rightarrow}\\,0.03\\,M_{\\sun}$) are almost isothermal, derived $T_{\\mathcal{I}}$ and $M_{\\mathcal{I}}$ perfectly agree with\n$T_{M}$ and $M$ for any subset $\\Phi_{n}$. Massive cores with more variable $T_{\\rm d}(r)$ (Fig.~\\ref{trp.bes.pro}) also have\nsignificant variations of $T_{\\rm d}(z)$ along the line of sight at pixel $(i,j)$. Emission of hot dust skews the spectral shapes\nof $I_{\\nu\\,ij}$ towards shorter wavelengths, even more so at the high-mass end. The most accurate masses are obtained for\n$\\Phi_{2}$, whereas larger $\\Phi_{n}$ ($n\\,{=}\\,3\\,{\\rightarrow}\\,6$) give increasingly incorrect $T_{\\mathcal{I}}$ and\n$M_{\\mathcal{I}}$. With degrading angular resolutions, the inaccuracies and biases increase, especially for\n$M\\,{\\rightarrow}\\,30\\,M_{\\sun}$ and larger $\\Phi_{n}$ ($n\\,{=}\\,3\\,{\\rightarrow}\\,6$). As expected, in the limiting case of\nunresolved objects the results approach those obtained with the method of fitting fluxes $F_{\\nu}$ (Fig.~\\ref{temp.mass.bes}).\n\nEmbedded starless cores, $\\beta\\,{=}\\,2$ (Fig.~\\ref{coldens.bes}). For the fully resolved models, derived $T_{\\mathcal{I}}$ have\nfairly good accuracy and little bias for the acceptable fits, although the range of the latter for larger $\\Phi_{n}$\n($n\\,{=}\\,3\\,{\\rightarrow}\\,6$) shrinks to even lower masses than for the isolated models. Showing no particularly large bias over\nalmost the entire range of model masses, $M_{\\mathcal{I}}$ are underestimated by a factor of $1.3$ owing to the standard\nobservational procedure of background subtraction (Appendix \\ref{AppendixB}, Sect.~\\ref{bg.subtraction}). Derived parameters of the\nmodels do not depend on angular resolutions, as they have relatively flat $T_{\\rm d}(r)$ across their boundaries\n(Fig.~\\ref{trp.bes.pro}), hence the spectral distortions of $I_{\\nu\\,ij}$ are negligible.\n\n\\begin{figure*}\n\\centering\n\\centerline{\\resizebox{0.3327\\hsize}{!}{\\includegraphics{bes.TT.b2p0.coldens.c1as.p1as.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{bes.TT.b2p0.coldens.c36as.p12as.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{bes.TT.b2p0.coldens.c144as.p48as.cfg.trim.pdf}}}\n\\centerline{\\resizebox{0.3327\\hsize}{!}{\\includegraphics{bes.MM.b2p0.coldens.c1as.p1as.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{bes.MM.b2p0.coldens.c36as.p12as.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{bes.MM.b2p0.coldens.c144as.p48as.cfg.trim.pdf}}}\n\\centerline{\\resizebox{0.3327\\hsize}{!}{\\includegraphics{bes.emb.TT.b2p0.coldens.c1as.p1as.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{bes.emb.TT.b2p0.coldens.c36as.p12as.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{bes.emb.TT.b2p0.coldens.c144as.p48as.cfg.trim.pdf}}}\n\\centerline{\\resizebox{0.3327\\hsize}{!}{\\includegraphics{bes.emb.MM.b2p0.coldens.c1as.p1as.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{bes.emb.MM.b2p0.coldens.c36as.p12as.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{bes.emb.MM.b2p0.coldens.c144as.p48as.cfg.trim.pdf}}}\n\\caption{\nTemperatures $T_{\\mathcal{I}}$ and masses $M_{\\mathcal{I}}$ derived from fitting images $\\mathcal{I}_{\\nu}$ of both \\emph{isolated} \nand \\emph{embedded} starless cores vs. the true model values of $T_{M}$ and $M$ for correct $\\beta\\,{=}\\,2$. The three columns of \npanels present results for three angular resolutions (resolved, partially resolved, and unresolved cases) and for various subsets \n$\\Phi_{n}$ of pixel intensities. Error bars represent the $1\\,{\\times}\\,\\sigma$ uncertainties of the derived $T_{\\mathcal{I}}$ and \n$M_{\\mathcal{I}}$ (computed over all pixels as the $N_{\\rm H_2}$-averaged errors of $T_{N ij}$ and integrated errors of \n$N_{\\rm H_2}$, correspondingly), combined with the assumed $\\pm\\,20{\\%}$ uncertainties of $\\eta$, $\\kappa_{0}$, and $D$ \n(Sect.~\\ref{data.subsets}). Less reliable results (${n-\\gamma+1}\\,{<}\\,\\chi^{2}\\,{<}\\,10$ in some pixels) are shown without \nerror bars. See Fig.~\\ref{temp.mass.bes} for more details.\n}\n\\label{coldens.bes}\n\\end{figure*}\n\n\\begin{figure*}\n\\centering\n\\centerline{\\resizebox{0.3327\\hsize}{!}{\\includegraphics{pro.TT.b2p0.coldens.c1as.p1as.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{pro.TT.b2p0.coldens.c36as.p12as.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{pro.TT.b2p0.coldens.c144as.p48as.cfg.trim.pdf}}}\n\\centerline{\\resizebox{0.3327\\hsize}{!}{\\includegraphics{pro.LM.b2p0.coldens.c1as.p1as.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{pro.LM.b2p0.coldens.c36as.p12as.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{pro.LM.b2p0.coldens.c144as.p48as.cfg.trim.pdf}}}\n\\centerline{\\resizebox{0.3327\\hsize}{!}{\\includegraphics{pro.emb.TT.b2p0.coldens.c1as.p1as.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{pro.emb.TT.b2p0.coldens.c36as.p12as.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{pro.emb.TT.b2p0.coldens.c144as.p48as.cfg.trim.pdf}}}\n\\centerline{\\resizebox{0.3327\\hsize}{!}{\\includegraphics{pro.emb.LM.b2p0.coldens.c1as.p1as.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{pro.emb.LM.b2p0.coldens.c36as.p12as.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{pro.emb.LM.b2p0.coldens.c144as.p48as.cfg.trim.pdf}}}\n\\caption{\nTemperatures $T_{\\mathcal{I}}$ and masses $M_{\\mathcal{I}}$ derived from fitting images $\\mathcal{I}_{\\nu}$ of both \\emph{isolated} \nand \\emph{embedded} protostellar envelopes (with the true masses $M$ of $0.03, 0.3$, and $3\\,M_{\\sun}$) vs. the true model values \nof $T_{M}$ and $L$ for correct $\\beta\\,{=}\\,2$. The three columns of panels present results for three angular resolutions indicated \n(resolved, partially resolved, and unresolved cases). See Fig.~\\ref{coldens.bes} for more details.\n} \n\\label{coldens.pro}\n\\end{figure*}\n\nIsolated protostellar envelopes, $\\beta\\,{=}\\,2$ (Fig.~\\ref{coldens.pro}). For the fully resolved models, derived $T_{\\mathcal{I}}$\nand $M_{\\mathcal{I}}$ are very accurate across all masses and luminosities. With degrading angular resolutions and for larger\n$\\Phi_{n}$ ($n\\,{=}\\,3\\,{\\rightarrow}\\,6$) the inaccuracies and biases increase quite considerably. The accretion energy released\nin the envelopes centers heats the dust to $T_{\\rm S}\\,{\\sim}\\,10^{3}$\\,K, making $T_{\\rm d}(r)$ strongly nonuniform. For the lines\nof sight passing through the inner radial zones, the hot emission skews the $I_{\\nu\\,ij}$ shapes towards shorter wavelengths. For\nthe unresolved envelopes, the results become similar to those obtained with the method of fitting fluxes $F_{\\nu}$\n(Fig.~\\ref{temp.mass.pro}).\n \nEmbedded protostellar envelopes, $\\beta\\,{=}\\,2$ (Fig.~\\ref{coldens.pro}). For the fully-re\\-solved models, derived\n$T_{\\mathcal{I}}$ are slightly less accurate for the acceptable fits than in the case of the isolated envelopes. The range of the\nlatter in more massive models for larger $\\Phi_{n}$ ($n\\,{=}\\,3\\,{\\rightarrow}\\,6$) shrinks towards higher $L_{\\star}$. The most\naccurate $M_{\\mathcal{I}}$, obtained for $\\Phi_{2}$, are underestimated by a factor of $1.45$, mostly because of the\nover-subtraction of the rim-brightened background (Appendix \\ref{AppendixB}, Sect.~\\ref{bg.subtraction}). For the\npartially resolved and unresolved envelopes, the most accurate $M_{\\mathcal{I}}$ (for $\\Phi_{2}$) are underestimated by factors of\n$1.5\\,{-}\\,2$, whereas fitting larger $\\Phi_{n}$ ($n\\,{=}\\,3\\,{\\rightarrow}\\,6$) leads to errors by factors of $4\\,{-}\\,5$.\n\n\n\\section{Discussion}\n\\label{discussion}\n\nSpectral flux and intensity distributions of the radiative transfer models of the starless cores and protostellar envelopes\n($0.03$\\,{--}\\,$30\\,M_{\\sun}, L_{\\sun}$) were fitted using the \\textsl{modbody} and \\textsl{thinbody} models. Derived values of the\nfitting parameters were then compared to their true values to quantify the qualities of the mass derivation methods, fitting\nmodels, and various sources of errors.\n\nAs shown in Sect.~\\ref{results}, large \\emph{intrinsic} inaccuracies and biases need to be taken into account when applying the\nmethods of mass derivation to the observed sources. In addition to being affected by nonuniform temperatures, estimated masses are\nalso affected by the adopted value of $\\beta$ and subset of data points $\\Phi_{n}$, as well as by the removal algorithm of the\nbackground emission of an embedding cloud. In the method of fitting fluxes $F_{\\nu}$, the masses depend on the fitting model,\nwhereas in the method of fitting images $\\mathcal{I}_{\\nu}$, they depend on the angular resolution.\n\nThe results of this purely model-based work discussed below may be directly applicable \\emph{only} to sources with \\emph{very}\naccurate measurements (with negligible errors). Real observations deal with images of relatively faint, crowded sources on strong\nand variable backgrounds, obtained with quite different angular resolutions, and thus they carry much larger measurement errors.\nObservations are substantially affected by various statistical and systematic errors, depending on the adopted source extraction\nmethod \\citep[e.g.,][]{Men'shchikov_etal2012, Men'shchikov2013} and especially the treatment of background subtraction and\ndeblending. Implications for the real-life studies are considered below, whenever possible.\n\n\\subsection{Mass derivation methods}\n\\label{the.methods}\n\nIn the first method, source fluxes $F_{\\nu}$ are integrated from the images $\\mathcal{I}_{\\nu}$, their spectral distribution is\nfitted, and source mass $M_{F}$ is estimated from the fitting model. In the second method, the pixel spectral shapes $I_{\\nu}$ of\nthe images $\\mathcal{I}_{\\nu}$ are fitted and the source mass $M_{\\mathcal{I}}$ is integrated from the resulting image\n$\\mathcal{N}_{\\rm H_2}$ of column densities. For unresolved sources and the \\textsl{thinbody} fitting model, the methods give very\nsimilar levels of inaccuracy, whereas for resolved images, the methods differ quite substantially.\n\nWhen fitting $F_{\\nu}$, the observed source emission from its entire volume is blended in the spatially integrated fluxes that\nretain no spatial information. For the models with strongly nonuniform $T_{\\rm d}(r)$ (Fig.~\\ref{trp.bes.pro}), resulting heavy\ndistortions of the spectral shapes of $F_{\\nu}$ (Fig.~\\ref{sed.bes.pro}) from those of the fitting models lead to large systematic\nerrors in estimated parameters (Figs.~\\ref{temp.mass.bes} and \\ref{temp.mass.pro}).\n\nWhen fitting $\\mathcal{I}_{\\nu}$, it is very beneficial to have a higher angular resolution. For fully resolved objects, pixels\n$(i,j)$ sample independent $I_{\\nu\\,ij}$ from different columns of dust. For the transparent lower mass models\n($M\\,{\\la}\\,0.3\\,M_{\\sun}$), derived $M_{\\mathcal{I}}$ are quite accurate (Figs.~\\ref{coldens.bes},\\,\\ref{coldens.pro}). For lower\nresolutions, the intensity of each pixel $(i,j)$ blends with that of its larger surroundings within the beam, not only along the\nline of sight. The contamination of $I_{\\nu\\,ij}$ by the more distant areas, leads to a substantial degradation of\n$T_{\\mathcal{I}}$ and $M_{\\mathcal{I}}$, especially when fitting large $\\Phi_{n}$ ($n\\,{=}\\,3\\,{\\rightarrow}\\,6$). Thus, the\nbenefits of this method are vanishing with decreasing angular resolutions.\n\nMultiwavelength \\emph{Herschel} images have been used to reconstruct radial temperature and density profiles of well-resolved\nsources \\citep[][]{Roy_etal2014}. Whenever such reconstructed densities are accurate enough, they can be used to obtain masses of\nthe nearby sources. Results of this study demonstrate, however, that the simple method of fitting images $\\mathcal{I}_{\\nu}$ is\nable to deliver accurate masses for spatially resolved sources (Sect.~\\ref{coldens.properties}).\n\n\\subsection{Background subtraction}\n\\label{bg.subtraction}\n\nStars form in the densest parts of interstellar clouds, hence the embedded models of starless cores and protostellar envelopes must\nbe more realistic than the isolated models. Although the spherical uniform-density embedding clouds are idealized, in a first\napproximation they account for the absorption and re-emission of ISRF, leading to realistic temperature profiles within the model\nobjects. However, the presence of surrounding material makes it necessary to subtract its contribution to study the properties of\nthe starless cores and protostellar envelopes alone. In observational practice, backgrounds are estimated by an average intensity\nin a narrow annulus placed just outside a source (cf. Sect.~\\ref{seds}). Subtraction of such a flat background is not quite\naccurate as a transparent embedding cloud around any object always tends to be rim-brightened and resembles a crater, in contrast\nto a distant, physically unrelated back- or foreground. This effect is discussed in detail in Appendix \\ref{AppendixB}.\n\nThe actual observable depths of the background craters may be shallower, when the local (filamentary) background itself is embedded\nin a less dense but more extended cloud or is seen in projection onto a distant, physically unrelated back- or foreground. The\nrim-brightening effect gets diluted, if the column densities of the source-embedding background and of the other unrelated clouds\nare similar. Poorer angular resolutions also tend to smear out the effect for less resolved sources. Realistic temperature\ngradients within the embedding backgrounds (Fig.~\\ref{trp.bes.pro}) can either reduce or increase the crater depths by\n${\\sim}\\,10{\\%}$ for starless cores and protostellar envelopes, respectively (Appendix \\ref{AppendixB}).\n\nFor unresolved sources, the observational algorithm of background subtraction is likely to overestimate fluxes as stars are born\nwithin the gravitationally unstable densest peaks of the parent clouds. Large beams blend the object's emission with that of its\nmountain-like environment, spreading the mix downhill, towards the valleys of lower cloud densities. The real background under an\nunresolved source must be hill-like, whereas the background values from an annulus tend to come from more distant valleys. The\nproblem is aggravated in crowded regions, where no local source-free annuli around overlapping sources can be found and where one\nneeds to deblend sources. Angular resolution degrades with wavelengths, hence the degree of flux overestimation becomes strongly\nbiased towards longer wavelengths.\n\n\\subsection{Nonuniform temperatures}\n\\label{nonuniform.temps}\n \nBoth fitting models make a sensitive assumption that the objects have a uniform temperature $T$, which seems to make them\ninadequate for the applications to starless cores and protostellar envelopes with nonuniform $T_{\\rm d}(r)$. For the purpose of the\nderivation of accurate masses, however, the uniform $T$ can be interpreted as an appropriate average quantity. In the methods of\nfitting $F_{\\nu}$ and $\\mathcal{I}_{\\nu}$, the temperature is consistent with $T_{M}$ and $T_{N ij}$ from\nEqs.~(\\ref{mass.averaged}) and (\\ref{coldens.averaged}), respectively (cf. Sects.~\\ref{fitting.fluxes} and \\ref{fitting.intens}).\nIn other words, to estimate masses $M_{F}$ or $M_{\\mathcal{I}}$ that are accurate (${\\approx}\\,M$), it is necessary that the\nfitting returns $T_{F}$ or $T_{\\mathcal{I} ij}$ as close as possible to the average values $T_{M}$ or $T_{N ij}$, respectively.\nThis is clearly demonstrated in Appendix \\ref{AppendixD} by the accurate masses obtained for the isothermal models with $T_{\\rm\nd}(r)\\,{=}\\,T_{M}$.\n\nThe inhomogeneous temperatures tend to distort the spectral shapes of $F_{\\nu}$ and $I_{\\nu\\,ij}$ of the objects towards shorter\nwavelengths (Figs.~\\ref{sed.bes.pro} and \\ref{fits.examples}). With a strong dependence of the dust emission peak on temperature\n(${\\kappa_{\\nu_{\\rm P}} B_{\\nu_{\\rm P}}(T)}\\,{\\propto}\\,T^{5}$), the radial zones with higher $T_{\\rm d}(r)$ make a much greater\ncontribution to the observed spectral shapes. Therefore, the shapes are skewed mainly owing to the emission of those parts of the\nobjects that have $T_{\\rm d}(r)\\,{>}\\,T_{M}$ or $T_{\\rm d}(r)\\,{>}\\,T_{N ij}$. In other words, distortions of the spectral shapes\nare caused by the dust with \\emph{excess} temperatures above the average values.\n\nThis is further demonstrated by additional ray-tracing observations of the models, in which the excess temperatures were removed:\n$T_{\\rm d}(r){\\rightarrow}\\min\\left\\{T_{\\rm d}(r),T_{M}\\right\\}$. Derived masses of these mostly isothermal models (not shown) are\nalmost as accurate as those of the fully isothermal models (Appendix \\ref{AppendixD}), only within a few percent lower. As is\nexpected, there is almost no dependence on the subsets $\\Phi_{n}$ ($n{\\,=\\,}2{\\,\\rightarrow\\,}6$), which indicates that the\nspectral shapes are indeed not distorted.\n\n\\subsection{Fitting models}\n\\label{choice.model}\n\nWhen fitting images $\\mathcal{I}_{\\nu}$, both fitting models are equivalent and estimated parameters are indistinguishable\n(Sect.~\\ref{variable.fixed}). When fitting fluxes $F_{\\nu}$, the results of this work show that \\textsl{thinbody} generally returns\nfar more accurate masses than \\textsl{modbody} does for both isolated and embedded variants of starless cores and protostellar\nenvelopes (Figs.~\\ref{temp.mass.bes} and \\ref{temp.mass.pro}).\n\nAlthough the \\textsl{modbody} fits often look better (i.e., they have smaller $\\chi^{2}$ values), they generally bring parameters\nthat are much more inaccurate. Indeed, the spectral shapes of $F_{\\nu}$ are skewed towards short wavelengths by emission from their\nhotter parts. With more free parameters, \\textsl{modbody} describes more flexible shapes, between $B_{\\nu}(T)$ and\n${\\kappa_{\\nu}\\,B_{\\nu}(T)}$. It is able to produce better fits of the distorted spectral shapes of objects with nonuniform $T_{\\rm\nd}(r)$ and hence it always tends to produce significantly over- and underestimated $T_{F}$ and $M_{F}$, respectively\n(Sect.~\\ref{derived.properties}). Furthermore, most of the \\textsl{modbody} fits have $\\tau_{\\nu}\\,{\\sim}\\,1$ even in the\nfar-infrared, which is fundamentally inconsistent with the radiative transfer models whose fluxes $F_{\\nu}$ represent optically\nthin emission ($\\tau_{\\nu}\\,{\\ll}\\,1$).\n\nThe \\textsl{thinbody} model produces the best overall results and smallest biases and inaccuracies in derived $T_{F}$ and $M_{F}$\nfor the isolated and embedded starless cores and protostellar envelopes (Figs.~\\ref{temp.mass.bes}\\,{--}\\,\\ref{coldens.pro}). The\n\\textsl{thinbody} fits are, by definition, optically thin in the far-infrared and thus consistent with the radiative transfer\nmodels. Only two variable fitting parameters of \\textsl{thinbody} contribute to better robustness of $T_{F}$ and $M_{F}$, compared\nto \\textsl{modbody} with one extra free parameter.\n\nContrary to what is usually assumed in observational studies, the results show that it must be counterproductive to aim at precise\nfitting of the peaks and shorter wavelength parts of $I_{\\nu}$ and $F_{\\nu}$. When the distorted shapes are reproduced more\naccurately, the estimates of the temperatures and masses are less accurate.\n\n\\subsection{Opacity slopes}\n\\label{choice.beta}\n\nThe standard methods of mass derivation ignore the presence of very small stochastically heated dust particles, assuming just a\nsimple power-law opacity across all bands, and so do the radiative transfer models in this study. Emission of such very small\ngrains within the real objects could enhance fluxes at $70$ and $100\\,{\\mu}$m and, in effect, skew their spectral shapes farther\ntowards short wavelengths, leading to more heavily overestimated temperatures and underestimated masses.\n\nVarious compositional and structural properties of real cosmic dust grains in different environments may lead to far-infrared\nopacity slopes that are different from $\\beta\\,{\\approx}\\,2$ (expected for small compact spherical grains) and even to\nwavelength-de\\-pendent $\\beta_{\\lambda}$. This study explored three constant values ($1.67$, $2.0$, $2.4$) to probe their influence\non the accuracy of derived masses. Fixing $\\beta$ in the fitting process reduces the number of free parameters and improves the\nconsistency (reduces biases) of derived parameters for objects with different physical properties ($M$, $L_{\\star}$).\n\nWith the correct $\\beta$ value, masses derived with the \\textsl{thinbody} fits are generally off the true mass $M$, the magnitude\nof discrepancy depending on how much and in what direction derived temperature deviates from $T_{M}$. When $\\beta$ is over- or\nunderestimated by a factor of $1.2$, derived masses become over- or underestimated within a factor of $2$ with respect to the\nmasses obtained using the true value $\\beta{\\,=\\,}2$. This is a direct consequence of the temperatures being under- or\noverestimated, correspondingly, a behavior that is easy to understand. In contrast to the \\textsl{thinbody} fits, no clear trends\nwith respect to the inaccuracies in the adopted $\\beta$ value can be found for \\textsl{modbody}, except that it generally returns\ngreatly over- and underestimated $T_{F}$ and $M_{F}$.\n \nTo quantify the effects of freedom in this fitting parameter, additional fits with variable $\\beta$ were performed\n(Appendix~\\ref{AppendixE}). As is expected, they showed much greater biases and inaccuracies in derived parameters\n(Figs.~\\ref{beta.bes.pro} and \\ref{beta.bes.pro.coldens}), as the extra degree of freedom also makes the resulting $\\beta$ values\nincorrect (Fig.~\\ref{beta.beta}), the magnitude of error depending on the true values of $M$ and $L_{\\star}$. It is possible to\ncompare these results with those obtained in previous studies focused on the relationships between the derived $\\beta_{F}$ and\n$T_{F}$ \\citep[][and references therein]{Shetty_etal2009a,Shetty_etal2009b,JuvelaYsard2012a}. The present models of starless cores\nand protostellar envelopes show that the correlations of the two quantities may be both positive and negative\n(Fig.~\\ref{tem.beta.bes.pro}), with almost no correlation in the case of isothermal models. They must be induced by deviations of\nthe spectral shapes of $F_{\\nu}$ (Fig.~\\ref{sed.bes.pro}) from $F_{\\nu}(T_{M})$ (Fig.~\\ref{sed.tmav.bes.pro}), caused by the\nnonuniform $T_{\\rm d}(r)$ (Fig.~\\ref{trp.bes.pro}). For the protostellar envelopes, the correlations are non-monotonic and they may\neither be strongly negative or positive, depending on the luminosity.\n\n\\subsection{Data subsets}\n\\label{choice.subset}\n\nFor nonuniform profiles $T_{\\rm d}(r)$ of starless cores and protostellar envelopes (Fig.~\\ref{trp.bes.pro}), better parameters are\nestimated with \\textsl{thinbody} when using smaller subsets of data ($n\\,{=}\\,6\\,{\\rightarrow}\\,2$) as the latter are less affected\nby the skewed spectral shapes. The most accurate masses are obtained by fitting just two of the longest wavelength data points; in\nmost cases, however, a subset $\\Phi_{3}$ produces very similar results. Larger subsets $\\Phi_{n}$ ($n\\,{=}\\,2\\,{\\rightarrow}\\,6$)\nmay give slightly better $M_{F}$ only when fixing an incorrect $\\beta$ value for the lower-mass starless cores\n($M\\,{\\la}\\,1\\,M_{\\sun}$, Fig.~\\ref{temp.mass.bes}). Using the inadequate fitting model with an incorrect $\\beta$, larger\n$\\Phi_{n}$ can constrain $T_{F}$ to better resemble $T_{M}$. For overestimated $\\beta$, derived $M_{F}$ always shift to higher\nvalues (Fig.~\\ref{temp.mass.pro}), which offsets the general opposite trend to underestimate $M_{F}$ and thus may give more\naccurate results.\n\nInaccuracies of the data points in real observations are usually more substantial than those assumed in this work, aggravated by\nthe systematic uncertainties that may lead to both over- and underestimated $F_{\\nu}$ (Sect.~\\ref{bg.subtraction}).\nBackground-subtracted and deblended $I_{\\nu}$ at each wavelength with different angular resolutions have independent and different\nsystematic errors. The latter must be large and uncertain on the bright and structured backgrounds in star-forming regions.\nMoreover, unresolved sources are likely to include emission from \\emph{clusters} of objects.\n\nResults of this model study are directly relevant to real observations \\emph{only} in the simplest case (which is rare) of accurate\nmeasurements with negligible errors. A blind application of the findings to real complex images may lead to incorrect results if\nthe above caution is ignored and small subsets $\\Phi_{2}$ of data points with large and independent measurement errors are fitted.\nFor such data, it would be safer and more appropriate to fit a larger subset of the longest wavelength data ($\\Phi_{3}$ or\n$\\Phi_{4}$, depending on the quality of measurements). Distortions of the observed spectral shapes towards shorter wavelengths is\nan intrinsic property of both starless cores and protostellar envelopes, affecting all sources in star-forming regions,\nindependently of the level of measurement errors.\n\nIt beyond the scope of this model-based work to give general recipes to observers on how to select data points to fit. This study\nhighlights the intrinsic behavior of the mass derivation methods by eliminating the ``observational layer'' (with all its\ncomplications and uncertainties) between the objects and the observer. It is important to realize that the peak and shorter\nwavelength shapes of $I_{\\nu}$ and $F_{\\nu}$ are most skewed by the temperature excesses within objects\n(Sect.~\\ref{nonuniform.temps}) and their influence has to be minimized to obtain accurate results. In view of the strong dependence\nof the results on $\\Phi_{n}$, it is advisable to examine fits of \\emph{all} subsets of data points for each observed source to\nestimate the robustness of the results and to possibly choose the fits giving the best mass estimate.\n\n\\subsection{Mass uncertainties}\n\\label{uncert.masses}\n\nTo make a bridge between this purely model-based study with no measurement errors and actual observational studies and see how\ntypical statistical errors in the input data would translate into those of the derived masses, this work assigned (fairly\noptimistically) $\\pm\\,15{\\%}$ errors to the model intensities and fluxes, and adopted $\\pm\\,20{\\%}$ errors in $\\eta$, \n$\\kappa_0$, and $D$.\n\nThe uncertainties in derived masses returned by the fitting algorithm are $40\\,{-}\\,70{\\%}$, depending on the subset $\\Phi_{n}$ of\nfluxes (Figs.~\\ref{temp.mass.bes}\\,{--}\\,\\ref{coldens.pro}, \\ref{mass.bes.pro.tmav}, \\ref{coldens.bes.pro.tmav}). For the\nacceptable fits of larger subsets ($n{\\,=\\,}2{\\,\\rightarrow\\,}6$), the derived mass uncertainty is dominated by the $20{\\%}$ errors\nof the parameters $\\eta$, $\\kappa_0$, and $D$, because the effect of the $15{\\%}$ measurement errors becomes smaller for the fits\nconstrained by a larger number of independent data points. For smaller subsets ($n{\\,=\\,}6{\\,\\rightarrow\\,}2$), the fits are less\nconstrained, hence the contribution of the $15{\\%}$ error bars to the derived mass uncertainty becomes larger. Different subsets\n$\\Phi_{n}$ give very similar results only for fully resolved sources with the method of fitting images $\\mathcal{I}_{\\nu}$\n(Figs.~\\ref{coldens.bes} and \\ref{coldens.pro}).\n\nIn real observations, statistical measurement uncertainties in $I_{\\nu}$ and $F_{\\nu}$ are larger than the $\\pm\\,15{\\%}$ errors\nassumed in this study. Furthermore, it would be more realistic to adopt uncertainties of $\\eta$, $\\kappa_0$, and $D$ of at least\n$\\pm\\,50{\\%}$, which would raise the derived mass uncertainties well beyond $100{\\%}$. By including the mass inaccuracies (of a\nfactor of $2$) induced by a $20{\\%}$ uncertainty in $\\beta$ and systematic errors (of factors of at least $2$) caused by the\nnonuniform temperatures within the observed sources, it is clear that the absolute values of masses derived from fitting are\ninaccurate and uncertain (within a factor of at least $2\\,{-}\\,3$). It is possible to neglect the uncertainties in $\\eta$,\n$\\kappa_0$, and $D$, if the focus is on studying relative properties of a population of objects all at roughly the same distance\nwithin a certain star-forming cloud with homogeneous dust properties. Apart from this, however, one has to derive accurate\n\\emph{absolute} values of the most fundamental parameters to make correct and physically meaningful conclusions.\n\nIt is quite important to carefully estimate mass uncertainties: without realistic error bars, derived masses are meaningless and\ncorrect conclusions are unlikely. To go one step further and obtain an idea of the \\emph{actual} errors of derived masses, it is\npossible to construct radiative transfer models of the observed population of sources, distribute the model sources over the\nobserved images and extract them, and finally derive their masses. Comparing derived masses with the fully known model properties,\nreasonable estimates of the actual errors in derived masses are obtained.\n\n\n\\section{Conclusions}\n\\label{conclusions}\n\nThis paper presented a model-based study of the uncertainties and biases of the standard methods of mass derivation (fitting fluxes\n$F_{\\nu}$ and images $\\mathcal{I}_{\\nu}$), widely applied in observational studies of the low- and intermediate star formation. To\nfocus on the intrinsic effects related to the physical objects, all observational complications leading to additional flux or\nintensity errors (filamentary and fluctuating backgrounds, instrumental noise, calibration errors, different resolutions, blending\nwith nearby sources, etc.) were assumed to be nonexistent. As a consequence, results of this work are directly relevant \\emph{only}\nfor the simplest case of bright isolated sources on faint backgrounds with negligible measurement errors. The real mass\nuncertainties for starless cores and protostellar envelopes are likely to be larger than those found in this work.\n \nBackground subtraction. Embedding backgrounds of physical objects are rim-brightened (i.e., they tend to resemble craters), their\ndepths depend on the sizes of the object and embedding cloud. The standard observational procedure of flat background subtraction\nmay give systematically underestimated $I_{\\nu}$ and $F_{\\nu}$, and hence masses for resolved sources. Poorer angular resolutions\nat longer wavelengths tend to systematically overestimate $I_{\\nu}$ and $F_{\\nu}$, and hence masses for unresolved objects, as\ntheir emission gets blended with that of the mountain-like background and possibly with other objects within the same beam.\n\nNonuniform temperatures. Temperature excesses above average values $T_{M}$ and $T_{N ij}$ is the primary reason for the skewness of\nthe spectral shapes of $F_{\\nu}$ and $I_{\\nu\\,ij}$ towards shorter wavelengths. Depending on $M$, $L_{\\star}$, $\\beta$, $\\Phi_{n}$,\nfitting model, and angular resolution, they lead to overestimated temperatures and various biases. With the method of fitting\n$F_{\\nu}$, masses become underestimated by factors $2\\,{-}\\,5$. When fitting $\\mathcal{I}_{\\nu}$, similarly large inaccuracies are\nfound only for unresolved objects, whereas with better angular resolutions they decrease and become very small for well-resolved\nobjects.\n\nFitting models. When fitting $\\mathcal{I}_{\\nu}$, both models are equivalent and estimated $M_{\\mathcal{I}}$ are indistinguishable.\nWhen fitting $F_{\\nu}$, \\textsl{thinbody} gives far more accurate $M_{F}$ than \\textsl{modbody} does. The latter causes such great\nbiases and inaccuracies in $T_{F}$ and $M_{F}$ that \\textsl{modbody} must be considered unusable.\n\nOpacity slopes. Fixing $\\beta$ reduces biases in derived parameters. When $\\beta$ is too high or low by a factor of $1.2$, derived\nmasses become over- or underestimated by a factor of $2$ with respect to those obtained using the true $\\beta\\,{=}\\,2$.\nQualitatively, this behavior is caused by the natural tendency of steeper $\\beta$ to produce lower temperatures, hence higher\nmasses. Quantitatively, the factors are approximate and they may depend on some of the assumptions used in this study. Mass\nderivation with a free variable $\\beta$ should be avoided, as it tends to lead to very strong biases and erroneous masses.\n\nData subsets. Derived masses strongly depend on the subsets $\\Phi_{n}$ of data points, except when fitting images\n$\\mathcal{I}_{\\nu}$ of fully resolved sources. Given the nonuniform $T_{\\rm d}(r)$ of the model objects, the most accurate masses\nare estimated with \\textsl{thinbody} using subsets that are as small as possible ($n\\,{=}\\,6\\,{\\rightarrow}\\,2$). In real\nobservations with substantial independent errors in different wavebands, it should be much safer and more accurate to fit slightly\nlarger subsets ($\\Phi_{3}$ or even $\\Phi_{4}$). Those data points that are on the peak of their spectral distribution or on the\nshort-wavelength side should be ignored, whenever possible, to improve the accuracy of derived masses. In practice, it is advisable\nto investigate fits of \\emph{all} subsets of data for each observed source, to verify robustness of the results and to possibly\nchoose the best mass estimate.\n\nDerived masses. Dividing the mass range of $0.03\\,{-}\\,30\\,M_{\\sun}$ at $1\\,M_{\\sun}$ into the low- and high-mass objects and\nconsidering unresolved or poorly resolved sources with $\\beta\\,{=}\\,2$, the following conclusions can be drawn. Masses of the\nisolated low- and high-mass starless cores are underestimated by factors $1\\,{-}\\,1.3$ and $1.3\\,{-}\\,4$, respectively. The mass\ninaccuracies increase towards the high-mass end and for larger subsets $\\Phi_{n}$ ($n\\,{=}\\,2\\,{\\rightarrow}\\,6$). Masses of the\nembedded low-mass cores are underestimated by a factor of $1.4$. They are more biased towards the high-mass end, changing from\nunder- to overestimated within a similar factor. Masses of the protostellar envelopes are considerably biased over the range of\n$0.03\\,{-}\\,30\\,L_{\\sun}$ and their inaccuracies strongly increase for larger subsets of $\\Phi_{n}$\n($n\\,{=}\\,2\\,{\\rightarrow}\\,6$). Masses of the isolated and embedded envelopes become underestimated by factors $2\\,{-}\\,3$ and\n$3\\,{-}\\,5$, respectively. Masses of the low-mass starless cores are likely to be determined much more accurately than those of\nprotostellar envelopes.\n\nMass uncertainties. Adopting statistical errors of $15{\\%}$ for model intensities (fluxes) and optimistically assuming that $\\eta$,\n$\\kappa_0$, and $D$ were known to within $20{\\%}$, typical mass uncertainties returned by the fitting algorithm are\n$40\\,{-}\\,70{\\%}$, depending on $\\Phi_{n}$. If more realistic statistical errors in the measurements and parameters of at least\n$50{\\%}$ are adopted, the mass uncertainties increase well beyond $100{\\%}$. Larger subsets $\\Phi_{n}$\n($n\\,{=}\\,2\\,{\\rightarrow}\\,6$) of independent data points are beneficial in somewhat reducing the resulting mass uncertainties. On\nthe other hand, the larger subsets are also highly undesirable, because they escalate the systematic mass inaccuracies by at least\na factor of $2$ as a result of nonuniform temperatures. Smaller subsets $\\Phi_{n}$ ($n\\,{=}\\,6\\,{\\rightarrow}\\,2$) are able to\nminimize the systematic errors caused by the temperature variations, but they increase the chances of getting incorrect masses in\nthe case of inaccurate data measurements in real observations.\n\nGlobal inaccuracies. Without extremely accurate flux measurements and knowledge of the free parameters ($\\eta, \\kappa_0, \\beta,\nD$), and without radiative transfer simulations to have an idea of the actual mass errors, it would be reasonable to assume that\nthe \\emph{absolute} values of masses of the unresolved or poorly resolved objects are inaccurate to within \\emph{at least} a factor\nof $2\\,{-}\\,3$. This may be less problematic, if the relative properties are studied of a population of objects within a\nstar-forming cloud, hopefully with the same distance and dust opacities. Ultimately, however, accurate absolute masses are\nnecessary to make correct, physically meaningful conclusions.\n \nAccuracy is paramount. There are several ways to improve mass estimates: (1) using a multiwavelength source extraction method\nmeasuring the most accurate, least biased background-subtracted and deblended fluxes across all wavebands; (2) selecting the best\nsources from the extraction catalogs, with the most accurately and consistently measured fluxes over at least three longest\nwavelengths; (3) using the \\textsl{thinbody} fitting model for the purposes of temperature or mass derivation; (4) estimating the\nmodel parameters $\\eta$, $\\kappa_0$, $\\beta$, and $D$ as accurately as possible and always performing fitting with $\\beta$ fixed;\n(5) for resolved sources, fitting their background-subtracted (and deblended) images and integrating source masses from column\ndensities; (6) fitting all subsets of data points for each source and choosing the smallest possible subset ($\\Phi_{3}$ or\n$\\Phi_{4}$) that gives the most accurate temperatures and masses; (7) using radiative transfer models to simulate observed images,\nextracting the model sources, deriving their masses, and comparing them to the true model values to have an idea of the\n\\emph{actual} errors for the derived masses of observed sources.\n\n\n\\begin{acknowledgements}\nThis study employed \\textsl{SAOImage DS9} (by William Joye) developed at the Smithsonian Astrophysical Observatory (USA), the\n\\textsl{CFITSIO} library (by William D. Pence) developed at HEASARC NASA (USA), and \\textsl{SWarp} (by Emmanuel Bertin) developed\nat Institut d'Astrophysique de Paris (France). Radiative transfer code \\textsl{MC3D-sph} version 3.12 \\citep[by Sebastian \nWolf,][]{Wolf2003} was used to compute the first generation of the models in this work. The \\textsl{plot} utility and\n\\textsl{ps12d} library used in this work to draw figures directly in the \\textsl{PostScript} language were written by the author\nusing the \\textsl{PSPLOT} library (by Kevin E. Kohler) developed at Nova Southeastern University Oceanographic Center (USA) and the\nplotting subroutines from the \\textsl{AZEuS} MHD code (by David A. Clarke and the author) developed at Saint Mary's University\n(Canada). Collaborative work within the \\emph{Herschel} Gould Belt and HOBYS key projects was very beneficial. Useful comments on a\ndraft made by Pierre Didelon, Arabindo Roy, Alana Rivera-Ingraham, Sarah Sadavoy, Philippe Andr{\\'e}, and by the anonymous referee \nhelped improve this paper.\n\\end{acknowledgements}\n\n\n\\begin{appendix}\n\n\\section{Protostellar envelopes temperatures}\n\\label{AppendixA}\n\nWith the adopted $\\kappa_{\\nu}$ and $\\rho(r)$ (Sects.~\\ref{dust.properties} and \\ref{density.profiles}), the radial temperature\nprofiles of protostellar envelopes (Fig.~\\ref{trp.bes.pro}) can be approximated by a combination of two power laws:\n\\begin{equation}\nT_{\\rm d}(r) = A\\,r^{-1.44} + B\\,r^{-1\/3},\n\\label{approximation}\n\\end{equation}\nwhere $r$ is in AU and parameters $A$ and $B$ depend on mass and accretion luminosity:\n\\begin{eqnarray*}\nA&\\!\\!\\!\\!\\!=\\!\\!\\!\\!\\!&2.93^{C\\,} 900 \\left(M\/M_{\\sun}\\right)^{1\/2},\\\\\nB&\\!\\!\\!\\!\\!=\\!\\!\\!\\!\\!&1.63^{C\\,} 84 \\left(1.33+0.22\\log{\\left(M\/M_{\\sun}\\right)}\\right)^{-1},\\\\\nC&\\!\\!\\!\\!\\!=\\!\\!\\!\\!\\!&1.5+\\log{\\left(L_{\\star}\/L_{\\sun}\\right)}.\n\\end{eqnarray*}\nEquation (\\ref{approximation}) describes the temperatures induced by the central accretion energy source (ignoring ISRF), valid for\n$T_{\\rm d}\\,{\\la}\\,300$\\,K. The first term in Eq.~(\\ref{approximation}) approximates the steepest profiles in the inner semi-opaque\nregion, the second term represents temperatures in the transparent outer part of the envelopes. An approximate borderline between \nthe two regimes can be estimated directly from Fig.~\\ref{trp.bes.pro} as\n\\begin{equation}\n\\begin{split}\n\\hat{T}\\,&{=}\\,0.65\\,(M\/M_{\\sun})^{-0.606}\\,\\hat{R}^{\\,0.714},\\\\\n\\hat{R}\\,&{=}\\,175\\left(M\/M_{\\sun}\\right)^{1\/2}\\left(L_{\\star}\/L_{\\sun}\\right)^{1\/5}\\,{\\rm AU},\\\\\n\\hat{T}\\,&{=}\\,26\\left(M\/M_{\\sun}\\right)^{-1\/4}\\left(L_{\\star}\/L_{\\sun}\\right)^{1\/7}\\,{\\rm K}.\n\\end{split}\n\\label{boundary}\n\\end{equation}\n\n\\section{Rim-brightened backgrounds}\n\\label{AppendixB}\n\n\\begin{figure*}\n\\centering\n\\centerline{\\resizebox{0.3250\\hsize}{!}{\\includegraphics{rim.bright.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{rim.bright.bes.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{rim.bright.pro.cfg.trim.pdf}}}\n\\caption{\nBackground rim brightening in the spherical uniform-density embedding shells, either isothermal (\\emph{left}) or with the\nradiative-equilibrium $T_{\\rm d}(r)$ from the models (Sect.~\\ref{rtmodels}) of embedded starless cores (\\emph{middle}) and\nprotostellar envelopes with $L_{\\star}\\,{=}\\,30\\,L_{\\sun}$ (\\emph{right}). The intensity profiles on the left are labeled by the\nsize ratios $R_{\\rm E}\/R\\,{=}\\,\\{30, 3, 1.3\\}$ of the embedding cloud and the central cavity, whereas the intensity profiles at\n$500\\,{\\mu}$m in the middle and on the right indicate the mass (in $M_{\\sun}$) of the embedded models. For uniform temperatures \n(\\emph{left}), the brightening factors $f_{\\rm S}$ are $1.03, 1.41$, and $2.77$ are the values given by Eq.~(\\ref{background}). For \nnonuniform temperatures (and $R_{\\rm E}\/R\\,{=}\\,3$, $R\\,{=}\\,10^{4}$ AU), they range from $1.27$ to $1.40$ (\\emph{middle}) and\nfrom $1.44$ to $1.53$ (\\emph{right}).\n} \n\\label{rim.bright}\n\\end{figure*}\n\nIn contrast to the emission of the distant and physically unrelated backgrounds or foregrounds, embedding backgrounds resemble\ncraters. The central spherical region occupied by an object ($r\\,{\\le}\\,R$) does not belong to the embedding cloud\n($R\\,{<}\\,r\\,{\\le}\\,R_{\\rm E}$) and thus must be considered empty when determining the object's background. Rim brightening for\nuniform-density transparent isothermal clouds with such a cavity depends only on their relative radial thickness $(R_{\\rm\nE}\\,{-}\\,R)\/R$. It can be quantified by the ratio $f_{\\rm S}$ of intensities (or column densities) along the lines of sight passing\nthrough the rim and the center of the cavity:\n\\begin{equation}\nf_{\\rm S} = \\left(1 + \\frac{2\\,R}{R_{\\rm E} - R}\\right)^{1\/2}.\n\\label{background}\n\\end{equation}\n\nAccording to Eq.~(\\ref{background}), the background under embedded objects can be overestimated from a few percent to a factor of\nseveral, hence background-subtracted values and masses may become substantially underestimated. For the present models with\n$R\\,{=}\\,10^{4}\\,$AU and $R_{\\rm E}\/R\\,{=}\\,3$, the background and masses are over- and underestimated by $f_{\\rm S}\\,{=}\\,1.41$,\nrespectively. The value is the discrepancy of derived masses seen for the embedded starless cores and protostellar envelopes\n(Figs.~\\ref{temp.mass.bes}\\,{--}\\,\\ref{coldens.pro}, \\ref{mass.bes.pro.tmav}, \\ref{coldens.bes.pro.tmav}). The effect becomes much\nstronger for very thin shell-like embedding clouds, whereas it vanishes for extended background clouds. For the size ratios $R_{\\rm\nE}\/R$ of $1.3$ and $30$, the factor $f_{\\rm S}$ takes the values of $2.77$ and $1.03$, respectively. Numerical examples of the\nrim-brightened backgrounds for both isothermal and non-isothermal spherical embedding clouds are shown in Fig.~\\ref{rim.bright}.\n\nRealistic temperature profiles in the embedding clouds bring only minor quantitative changes, not altering qualitatively the rim \nbrightening effect (Fig.~\\ref{rim.bright}). Steep positive or negative temperature gradients in the dense shells around embedded\nstarless cores and protostellar envelopes (Fig.~\\ref{trp.bes.pro}) tend to slightly reduce or increase the brightening effect (by\n${\\sim\\,}10{\\%}$), respectively.\n\nActual geometry of the real background clouds is of minor importance, the only relevant assumption is that the embedded object\n(hence, its background cavity) has a convex shape. For instance, assuming a plane-parallel embedding cloud with thickness \n$2\\,R_{\\rm E}$ along the line of sight, it is possible to obtain a slightly different expression than Eq.~(\\ref{background}) for \nthe rim brightening factor:\n\\begin{equation}\nf_{\\rm P} = \\frac{1}{2} \\left(f_{\\rm S} + \\frac{R_{\\rm E}}{R_{\\rm E} - R}\\right).\n\\label{planeparallel}\n\\end{equation}\nPlane-parallel geometry makes the brightening factors $f_{\\rm P}$ somewhat larger than $f_{\\rm S}$ from Eq.~(\\ref{background}), \nwith the difference being stronger for thinner shell-like clouds. For instance, the size ratios $R_{\\rm E}\/R$ of $30, 3$, and \n$1.3$, correspond to the brightening factors $f_{\\rm P}$ of $1.03, 1.46$, and $3.55$, respectively. \n\nDepths of the background craters may be quite dissimilar for different objects in real observations. The observations show that the\ninterstellar medium is strongly filamentary and that stars tend to form in narrow, very dense filaments\n\\citep[e.g.,][]{Men'shchikov_etal2010,Andre_etal2014}. For an object embedded in a cylindrical filament of radius $R_{\\rm E}$ in\nthe plane of the sky, the brightening factor $f_{\\rm C}$ is intermediate between $f_{\\rm S}$ and $f_{\\rm P}$, depending on the\nposition angle of the radius-vector from the center of the object to its outer boundary $R$. It is easy to see that $f_{\\rm\nC}\\,{=}\\,f_{\\rm P}$ along the filament's axis, whereas $f_{\\rm C}\\,{=}\\,f_{\\rm S}$ in the orthogonal direction, across the filament.\n\nThe widths of the embedding filaments appear to be similar to the sizes of embedded objects \\citep[][]{Men'shchikov_etal2010}.\nAssuming their cylindrical geometry, the embedding filaments of starless cores and protostellar envelopes are likely to have\n$R_{\\rm E}$ only a factor of about $2\\,{-}\\,3$ larger than $R$. For such narrow filamentary backgrounds of resolved objects, the\nrim brightening effect is quantified by factors $f_{\\rm C}\\,{\\approx}\\,1.8{-}1.4$.\n\nBackground rim brightening may be observable only when imaging the nearby resolved sources unaffected by other distant back- or\nforegrounds. With this effect in action, the standard observational algorithm of background subtraction underestimates $F_{\\nu}$ by\nfactors similar to $f_{\\rm S}$ or $f_{\\rm P}$. With poorer angular resolutions, the rim of the background crater gets smeared out\nand thus the brightening effect eventually vanishes for unresolved sources which also have an opposite trend to produce\nunderestimated background and hence overestimated $I_{\\nu}$ and $F_{\\nu}$ (cf. Sect.~\\ref{bg.subtraction}).\n\n\\section{Fitting procedure}\n\\label{AppendixC}\n\nThe nonlinear least-squares fitting algorithm used in this work employs the Le\\-ven\\-berg-Marquardt method \\citep{Press_etal1992}\nthat minimizes $\\chi^{2}$ residuals between the model and data. The method requires a user to provide initial guesses for model\nparameters. Tests have shown that an arbitrary choice of the initial values of the fitting models (Sects.~\\ref{fitting.fluxes},\n\\ref{fitting.intens}) does not guarantee convergence to the global $\\chi^{2}$ minimum. A fully automated fitting procedure has been\ndesigned to overcome this problem and avoid any need to make arbitrary initial guesses.\n\nThe algorithm explores the multidimensional parameter space of the model with a large number of trial fittings of the spectral\ndistributions of data points. The parameter space is discretized in logarithmically equidistant steps $\\delta\\log{p}$, covering all\nrelevant initial values of temperature $T$ ($4\\,{-}\\,10^{3}$\\,K), mass $M$\n($3\\,{\\times}\\,10^{-3}\\,{-}\\,3\\,{\\times}\\,10^{2}\\,M_{\\sun}$), column density $N_{\\rm H_2}$ ($10^{17}\\,{-}\\,10^{27}$\\,cm$^{-2}$),\nand solid angle $\\Omega$ ($1.85\\,{\\times}\\,10^{-13}\\,{-}\\,1.85\\,{\\times}\\,10^{-5}$\\,sr). Large initial discretization steps\n$\\delta\\log{p}\\,{\\sim}\\,10$ and the above ranges of parameters are adaptively refined in an iterative binary search down to\n$\\delta\\log{p}\\,{\\approx}\\,0.1$, accelerating the algorithm in finding the global minimum. Data points are fitted using all\ncombinations of the model parameters\\footnote{Although the number of trial fittings for a spectral shape of $F_{\\nu}$ or\n$I_{\\nu\\,ij}$ may reach ${\\sim\\,}10^{3}$ in some cases, computation time is never an issue as all of the fits are completed within\na second. Fitting of an entire set of six images with $10^{6}$ pixels each may take a couple of hours.} in the adaptively refined\nparameter space and their initial values are found that converge to the globally smallest $\\chi^{2}$ in a fully automated procedure.\n\nAlgorithms described in this paper were written as a versatile and robust FORTRAN utility \\textsl{fitfluxes} that efficiently\nestimates \\textsl{modbody} or \\textsl{thinbody} parameters from fitting either total fluxes of cataloged sources or intensities\nof multiwavelength images. The code is easy to install and use and it is freely available from the author upon request.\n\n\\section{Results for isothermal models}\n\\label{AppendixD}\n\n\\begin{figure*}\n\\centering\n\\centerline{\\resizebox{0.3327\\hsize}{!}{\\includegraphics{sed.bes.tmav.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{sed.bes.emb.tmav.cfg.trim.pdf}}}\n\\centerline{\\resizebox{0.3327\\hsize}{!}{\\includegraphics{sed.pro.tmav.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{sed.pro.emb.tmav.cfg.trim.pdf}}}\n\\caption{\nSpectral energy distributions of the \\emph{isothermal} models of starless cores (\\emph{upper}) and protostellar envelopes\n(\\emph{lower}). Shown are the background-subtracted fluxes of the \\emph{isolated} models (\\emph{left}) and of their \\emph{embedded}\nvariants (\\emph{right}). For the cores of increasing masses, $T_{M}\\,{=}\\,$\\{16.3, 15.0, 13.2, 11.2, 9.25, 7.66, 6.42 K\\}. For the\nenvelopes of increasing luminosities and masses, $T_{M}\\,{=}\\,$\\{16.6, 16.8, 17.3, 18.1, 19.5, 21.9, 25.9 K\\},\n$T_{M}\\,{=}\\,$\\{13.8, 14.1, 14.7, 15.6 17.1, 19.4, 23.0 K\\}, $T_{M}\\,{=}\\,$\\{10.0, 10.4, 11.1, 12.3, 14.1, 16.8, 20.7 K\\}. See\nFig.~\\ref{sed.bes.pro} for more details.\n} \n\\label{sed.tmav.bes.pro}\n\\end{figure*}\n\n\\begin{figure*}\n\\centering\n\\centerline{\\resizebox{0.3327\\hsize}{!}{\\includegraphics{bes.tmav.MM.b1p7.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{bes.tmav.MM.b2p0.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{bes.tmav.MM.b2p4.cfg.trim.pdf}}}\n\\centerline{\\resizebox{0.3327\\hsize}{!}{\\includegraphics{bes.emb.tmav.MM.b1p7.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{bes.emb.tmav.MM.b2p0.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{bes.emb.tmav.MM.b2p4.cfg.trim.pdf}}}\n\\centerline{\\resizebox{0.3327\\hsize}{!}{\\includegraphics{pro.tmav.LM.b1p7.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{pro.tmav.LM.b2p0.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{pro.tmav.LM.b2p4.cfg.trim.pdf}}}\n\\centerline{\\resizebox{0.3327\\hsize}{!}{\\includegraphics{pro.emb.tmav.LM.b1p7.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{pro.emb.tmav.LM.b2p0.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{pro.emb.tmav.LM.b2p4.cfg.trim.pdf}}}\n\\caption{\nMasses $M_{F}$ derived from fitting $F_{\\nu}$ for the \\emph{isothermal} models of both \\emph{isolated} and \\emph{embedded} starless\ncores and protostellar envelopes. Compared to the results for isolated models, all derived masses of embedded models are\nunderestimated by approximately a factor of $1.3$, owing to background over-subtraction (Sect.~\\ref{bg.subtraction}). See \nFig.~\\ref{sed.tmav.bes.pro} for SEDs and Fig.~\\ref{temp.mass.bes} for more details.\n} \n\\label{mass.bes.pro.tmav}\n\\end{figure*}\n\nThe importance of estimating $T_{F}$ that approaches the mass-ave\\-raged temperature $T_{M}$ from Eq.~(\\ref{mass.averaged}) for\nderiving accurate masses is shown by isothermal models, those described in Sect.~\\ref{rtmodels} and used throughout this paper\nwhere self-consistent (radiative-equilibrium) profiles $T_{\\rm d}(r)$ were replaced with $T_{M}$. The isothermal models were then\nobserved and imaged in a ray-tracing run of the radiative transfer code. The resulting SEDs (Fig.~\\ref{sed.tmav.bes.pro}) are\nessentially the modified blackbody shapes $\\kappa_{\\nu}\\,B_{\\nu}(T_{M})\\,{\\nu}$ and the same is true for the spectral shapes of\nimage pixels. The temperature excesses above $T_{M}$ (Sect.~\\ref{nonuniform.temps}) that greatly distorted the model SEDs\n(Fig.~\\ref{sed.bes.pro}) towards shorter wavelengths do not exist in the isothermal models. Consequently, the isothermal shapes of\n$F_{\\nu}$ and $I_{\\nu\\,ij}$ bring much more expected and accurate results.\n\n\\begin{figure*}\n\\centering\n\\centerline{\\resizebox{0.3327\\hsize}{!}{\\includegraphics{bes.tmav.MM.b2p0.coldens.c1as.p1as.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{bes.tmav.MM.b2p0.coldens.c144as.p48as.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{bes.emb.tmav.MM.b2p0.coldens.c144as.p48as.cfg.trim.pdf}}}\n\\centerline{\\resizebox{0.3327\\hsize}{!}{\\includegraphics{pro.tmav.LM.b2p0.coldens.c1as.p1as.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{pro.tmav.LM.b2p0.coldens.c144as.p48as.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{pro.emb.tmav.LM.b2p0.coldens.c144as.p48as.cfg.trim.pdf}}}\n\\caption{\nMasses $M_{\\mathcal{I}}$ derived from fitting images $\\mathcal{I}_{\\nu}$ of the \\emph{isothermal} models of both \\emph{isolated}\nand \\emph{embedded} starless cores and protostellar envelopes for the correct value of $\\beta\\,{=}\\,2$. Two columns of panels\n(\\emph{left}, \\emph{middle}) display the masses derived for resolved and unresolved images of the isolated models, whereas the\nthird column of panels (\\emph{right}) present results for unresolved embedded models. Angular resolutions do not make any \ndifference for the isothermal models. See Fig.~\\ref{mass.bes.pro.tmav} and Fig.~\\ref{coldens.bes} for more details.\n} \n\\label{coldens.bes.pro.tmav}\n\\end{figure*}\n\nWith the correct value $\\beta\\,{=}\\,2$, derived $M_{F}$ of the isolated starless cores and protostellar envelopes derived with\n\\textsl{thinbody} agree with the true masses $M$ (Fig.~\\ref{mass.bes.pro.tmav}). The same results are obtained for\n\\textsl{modbody}, with the exception of the minimal subset $\\Phi_{3}$ for some models. An inspection of the problematic fits show\nthat $T_{F}$ is somewhat overestimated because of an additional degree of freedom in \\textsl{modbody} and a formal search for a\nglobally best fit in its parameter space. The fits in question have the globally lowest $\\chi^{2}$ value, but they correspond to\nsmall values of $\\Omega$ and, therefore, to high optical depth $\\tau_{\\nu}{\\,\\sim\\,}1$. However, there are also other \\emph{very}\ngood fits with somewhat larger $\\chi^{2}$ that do produce accurate $T_{F}\\,{=}\\,T_{M}$ with $\\tau_{\\nu}{\\,\\ll\\,}1$, consistent with\nthe models. The problem seems to be just a simple consequence of the finite accuracy of the numerical models and their fluxes.\n\nWith the same $\\beta\\,{=}\\,2$, the derived masses of the embedded models (Fig.~\\ref{mass.bes.pro.tmav}) are almost uniformly\nunderestimated by a factor of approximately $1.3$. The reason for the difference with respect to the isolated models is the\nconventional approach to background subtraction. Emission of a transparent cloud embedding a physical object tends to be rim\nbrightened (Appendix~\\ref{AppendixB}, Sect.~\\ref{bg.subtraction}). An average intensity in an annulus may overestimate background,\nfrom a few percent to a factor of several, hence may underestimate the background-subtracted intensities $I_{\\nu}$, fluxes\n$F_{\\nu}$, and derived masses (see Sects.~\\ref{derived.properties} and \\ref{coldens.properties}).\n\nFor an inadequate fitting model, skewed by $\\beta$ values fixed below or above its correct value, the fits are obviously biased to\nover- or underestimate $T_{F}$ and hence to under- or overestimate $M_{F}$, correspondingly (Fig.~\\ref{mass.bes.pro.tmav}). The\ninaccuracy is within a factor of $2$, independently of whether $\\beta$ is a factor of $1.2$ lower or higher. Fitting larger subsets\n$\\Phi_{n}$ ($n\\,{=}\\,2\\,{\\rightarrow}\\,6$) for the isothermal models may give somewhat better derived parameters, as they better\nconstrain $T_{F}$ with an incorrect slope $\\beta$.\n\nMass derivation from images $\\mathcal{I}_{\\nu}$ of the isothermal models delivers results that are similar to those described above\nfor both isolated and embedded variants (Fig.~\\ref{coldens.bes.pro.tmav}). Dependence on the adopted $\\beta$ is the same as\ndescribed above, hence only the results with correct $\\beta\\,{=}\\,2$ are presented. With no temperature deviations from $T_{M}$ in\nthe isothermal models, derived $M_{\\mathcal{I}}$ are very accurate for all angular resolutions, in contrast to the results with the\nself-consistent $T_{\\rm d}(r)$ (Figs.~\\ref{coldens.bes} and \\ref{coldens.pro}). Derived $M_{\\mathcal{I}}$ for the embedded models\nare practically identical to $M_{F}$, being underestimated by a factor of $1.3$ owing to the background rim-brightening effect\n(Appendix \\ref{AppendixB}, Sect.~\\ref{bg.subtraction}).\n\n\\section{Results for free variable $\\beta$}\n\\label{AppendixE}\n\n\\begin{figure*}\n\\centering\n\\centerline{\\resizebox{0.3327\\hsize}{!}{\\includegraphics{bes.TT.beta.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{bes.emb.TT.beta.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{bes.tmav.TT.beta.cfg.trim.pdf}}}\n\\centerline{\\resizebox{0.3327\\hsize}{!}{\\includegraphics{bes.MM.beta.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{bes.emb.MM.beta.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{bes.tmav.MM.beta.cfg.trim.pdf}}}\n\\centerline{\\resizebox{0.3327\\hsize}{!}{\\includegraphics{pro.TT.beta.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{pro.emb.TT.beta.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{pro.tmav.TT.beta.cfg.trim.pdf}}}\n\\centerline{\\resizebox{0.3327\\hsize}{!}{\\includegraphics{pro.LM.beta.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{pro.emb.LM.beta.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{pro.tmav.LM.beta.cfg.trim.pdf}}}\n\\caption{\nTemperatures $T_{F}$ and masses $M_{F}$ derived from fitting $F_{\\nu}$ of \\emph{isolated}, \\emph{embedded}, and \\emph{isothermal} \nmodels (fits with free variable $\\beta$) for starless cores (\\emph{upper}) and protostellar envelopes (\\emph{lower}). See \nFig.~\\ref{temp.mass.bes} for more details.\n} \n\\label{beta.bes.pro}\n\\end{figure*}\n\n\\begin{figure*}\n\\centering\n\\centerline{\\resizebox{0.3327\\hsize}{!}{\\includegraphics{bes.MB.beta.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{bes.emb.MB.beta.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{bes.tmav.MB.beta.cfg.trim.pdf}}}\n\\centerline{\\resizebox{0.3327\\hsize}{!}{\\includegraphics{pro.m3p00.LB.beta.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{pro.m3p00.emb.LB.beta.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{pro.m3p00.tmav.LB.beta.cfg.trim.pdf}}}\n\\centerline{\\resizebox{0.3327\\hsize}{!}{\\includegraphics{pro.m0p30.LB.beta.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{pro.m0p30.emb.LB.beta.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{pro.m0p30.tmav.LB.beta.cfg.trim.pdf}}}\n\\centerline{\\resizebox{0.3327\\hsize}{!}{\\includegraphics{pro.m0p03.LB.beta.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{pro.m0p03.emb.LB.beta.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{pro.m0p03.tmav.LB.beta.cfg.trim.pdf}}}\n\\caption{\nOpacity slope $\\beta_{F}$ derived from fitting $F_{\\nu}$ of \\emph{isolated}, \\emph{embedded}, and \\emph{isothermal} models (fits \nwith free variable $\\beta$) for starless cores (\\emph{upper}) and protostellar envelopes of selected masses ($3$, $0.3$, and \n$0.03\\,M_{\\sun}$, \\emph{lower}). See Fig.~\\ref{temp.mass.bes} for more details.\n} \n\\label{beta.beta}\n\\end{figure*}\n\n\\begin{figure*}\n\\centering\n\\centerline{\\resizebox{0.3327\\hsize}{!}{\\includegraphics{bes.MM.beta.coldens.c1as.p1as.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{bes.MM.beta.coldens.c144as.p48as.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{bes.emb.MM.beta.coldens.c144as.p48as.cfg.trim.pdf}}}\n\\centerline{\\resizebox{0.3327\\hsize}{!}{\\includegraphics{pro.LM.beta.coldens.c1as.p1as.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{pro.LM.beta.coldens.c144as.p48as.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{pro.emb.LM.beta.coldens.c144as.p48as.cfg.trim.pdf}}}\n\\caption{\nMasses $M_{\\mathcal{I}}$ derived from fitting images $\\mathcal{I}_{\\nu}$ of the \\emph{isolated} and \\emph{embedded} starless cores\nand protostellar envelopes (fits with free variable $\\beta$). Two columns of panels (\\emph{left}, \\emph{middle}) display the\nmasses derived for the resolved and unresolved images of the isolated models, whereas the third column of panels (\\emph{right})\npresents results for the unresolved embedded models. Derived masses of the latter are underestimated by a factor of approximately\n$1.3$ as a result of the conventional procedure of background subtraction (Sect.~\\ref{bg.subtraction}). See Fig.~\\ref{coldens.bes} \nfor more details.\n} \n\\label{beta.bes.pro.coldens}\n\\end{figure*}\n\n\\begin{figure*}\n\\centering\n\\centerline{\\resizebox{0.3216\\hsize}{!}{\\includegraphics{bes.TB.beta.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{pro.TB.beta.cfg.trim.pdf}}\n \\resizebox{0.3204\\hsize}{!}{\\includegraphics{bes.pro.tmav.TB.beta.cfg.trim.pdf}}}\n\\caption{\nRelationships between $T_{F}$ and $\\beta_{F}$ derived from fitting fluxes $F_{\\nu}$ of starless cores and protostellar envelopes \n(fits with free variable $\\beta$). Curves plotted with squares (dark colors) and circles (bright colors) in the left and middle \npanels correspond to the \\emph{isolated} and \\emph{embedded} models, whereas in the right panel they correspond to the \n\\emph{isothermal} versions of the \\emph{isolated} starless cores and protostellar envelopes, respectively. Results for the starless \ncores are plotted for all masses $0.03{-}30\\,M_{\\sun}$. Results for the protostellar envelopes, shown for only selected masses \n($0.03, 0.3$, and $3\\,M_{\\sun}$, three sets of identical lines), span the entire range of luminosities $0.03{-}30\\,L_{\\sun}$. Thick \nhorizontal lines indicate the true value $\\beta\\,{=}\\,2$. For clarity of the plots, error bars are not shown.\n} \n\\label{tem.beta.bes.pro}\n\\end{figure*}\n\nIn some applications of the mass derivation methods, the opacity slope $\\beta$ has been allowed to vary along with the other\nfitting parameters ($T$, $M$, $\\Omega$). To quantify effects of the extra degree of freedom, additional fits with variable $\\beta$\nwere performed in this study. Although the parameters for both fitting models were derived and analyzed, only the much less\nincorrect \\textsl{thinbody} results are presented and discussed here.\n\nFigure~\\ref{beta.bes.pro} compares derived $T_{F}$ and $M_{F}$ of the isolated, embedded, and isothermal variants of starless cores\nand protostellar envelopes with the true model values ($T_{M}$, $M$). Although the isolated cores and envelopes display behavior\nthat is qualitatively similar to the $\\beta\\,{=}\\,2$ case (Sect.~\\ref{derived.properties}), the biases in $T_{F}$ and $M_{F}$\ntowards denser (more massive) cores and envelopes, as well as over $L_{\\star}$ for the latter, become much stronger. For example,\nfor the isolated starless cores with $M\\,{\\ga}\\,10\\,M_{\\sun}$, derived $T_{F}$ are overestimated by a factor of $2$, whereas\n$M_{F}$ are underestimated by a factor of $15$. For the embedded protostellar envelopes of $M\\,{=}\\,3\\,M_{\\sun}$ with\n$L_{\\star}\\,{\\la}\\,1\\,L_{\\sun}$, temperatures are overestimated by a similarly large factor and $M_{F}$ underestimated by a factor\nof $8$.\n\nSuch errors are caused by the derived $\\beta_{F}$ whose values for starless cores are systematically lowered towards higher mass\nmodels, and are underestimated by a factor of $4$ (Fig.~\\ref{beta.beta}). In the case of the embedded protostellar envelopes, the\nvalues of $\\beta_{F}$ are progressively underestimated towards lower luminosities, up to a factor of $2$ (Fig.~\\ref{beta.beta}).\nThe very large errors in $\\beta_{F}$ are, in turn, caused by the temperature excesses over $T_{M}$ (Sect.~\\ref{nonuniform.temps}),\nwhich is highlighted in Fig.~\\ref{beta.bes.pro} by the accurate results for the isothermal models. As in the fixed $\\beta$ case,\nerrors in derived parameters for the embedded starless cores are smaller than those for the isolated cores, whereas the behavior is\nopposite for the embedded protostellar envelopes (cf. Figs.~\\ref{temp.mass.bes}, \\ref{temp.mass.pro}). However, the biases over the\nmasses and luminosities in Fig.~\\ref{beta.bes.pro} become stronger than in the fixed $\\beta$ case, which again is attributed to the\nadditional biases in the derived $\\beta_{F}$ values (Fig.~\\ref{beta.beta}).\n\nThe method of fitting images $\\mathcal{I}_{\\nu}$ delivers results that are similar to those described above, for both isolated and\nembedded variants (Fig.~\\ref{beta.bes.pro.coldens}) of partially resolved and unresolved objects. Derived masses for the\nfully resolved objects are much more accurate and they do not depend on the subset of data points $\\Phi_{n}$, for the reasons\ndiscussed in Sects.~\\ref{coldens.properties} and \\ref{the.methods}. With degrading angular resolutions, accuracy of the estimated\nparameters deteriorates to the levels obtained from the method of fitting total fluxes $F_{\\nu}$ (Fig.~\\ref{beta.bes.pro}). Larger\nbeams heavily blend emission with nonuniform temperatures from different pixels, distorting their spectral distribution towards\nshorter wavelengths. In both methods, derived masses become systematically much less accurate (greatly underestimated) when fitting\nlarger subsets $\\Phi_{n}$ ($n\\,{=}\\,3\\,{\\rightarrow}\\,6$). However, even the smallest subsets $\\Phi_{3}$ show very significant\ninaccuracies and different biases that depend on the mass and luminosity of an object.\n\nThe above results obtained with free fitting parameter $\\beta$ can be compared with those from the previous studies focused on the\nrelationship between derived temperature and opacity slope \\citep[cf.][and references\ntherein]{Shetty_etal2009a,Shetty_etal2009b,JuvelaYsard2012a}. Figure~\\ref{tem.beta.bes.pro} shows the intrinsic dependencies\nbetween $\\beta_{F}$ and $T_{F}$ for the isolated and embedded starless cores and protostellar envelopes, and for the isothermal\nversions of the isolated models. The correlations between $\\beta_{F}$ and $T_{F}$ take variety of shapes, from a strongly positive\nto a strongly negative correlation, with practically no correlation for the isothermal models. This suggests that they must be\ncaused by different kinds of deviations of the spectral shapes of $F_{\\nu}$ (Fig.~\\ref{sed.bes.pro}) produced by the nonuniform\n$T_{\\rm d}(r)$ (Fig.~\\ref{trp.bes.pro}) from $F_{\\nu}(T_{M})$ (Fig.~\\ref{sed.tmav.bes.pro}). Smaller subsets $\\Phi_{n}$\n($n\\,{=}\\,6\\,{\\rightarrow}\\,3$) bring less correlated $\\beta_{F}$ and $T_{F}$ than the large subsets do. For the protostellar\nenvelopes, the correlations are non-monotonic and they may either be strongly negative or positive, depending on the luminosity of\nthe central energy source.\n\n\\end{appendix}\n\n\n\\bibliographystyle{aa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nThere has recently been considerable interest in the Galactic \nevolution of the abundances of the light elements Li, Be and B \n(Feltzing \\& Gustafsson 1994; Reeves 1994; Cass\\'e et al. 1995; \nFields et al. 1994,1995; Ramaty et al. 1996,1997; Vangioni-Flam et \nal. 1998). The Be abundance is particularly interesting because this \nelement is thought to be produced exclusively by spallation reactions \ninvolving collisions between nuclei of the CNO group of elements and \nprotons or alpha particles at energies greater than about $30\\,\\rm \nMeV$ per nucleon (MeV\/n). Thus the evolution of the Be abundance \ncontains information about the particle acceleration and cosmic ray \nhistory of the Galaxy.\n\nThe evolution of the Be abundance, and indeed the evolution of all \nelemental abundances, has to be deduced from observations of the \nfossil abundances preserved in the oldest halo stars. Advances in \nspectroscopy over the last decade have greatly improved the quality of \nthe data available (Duncan et al. 1992,1997; Edvardsson et al. 1994; \nGilmore et al. 1992; Kiselman \\& Carlsson 1996; Molaro et al. 1997; \nRyan et al. 1994) and the main result is easily summarized~: in old \nhalo stars of low metallicity, the ratio of the Be abundance to the \nIron (Fe) abundance appears constant, that is to say the Be abundance \nrises {\\em linearly} with the Fe abundance.\n\nThis has been a surprising result. Naively one had expected that, \nbecause Be is a secondary product produced from the primary CNO \nnuclei, its abundance should vary {\\em quadratically} as a function of \nthe primary abundances at low metallicities. Indeed, considering that \nthe cosmic rays (CRs) responsible for the Be production are somehow \nrelated to the explosion of supernovae (SNe) in the Galaxy, it is \nnatural to assume that their flux is proportional to the SN rate, $\\mathrm{d} \nN_{\\mathrm{SN}}\/\\mathrm{d} t$. Now since the number of CNO nuclei present in \nthe Galaxy at time $t$ is proportional to the total number of SN \nhaving already exploded, $N_{\\mathrm{SN}}(t)$, the Be production rate \nhas to be proportional to $N_{\\mathrm{SN}}\\mathrm{d} N_{\\mathrm{SN}}\/\\mathrm{d} t$. \nTherefore, the integrated amount of Be grows as $N_{\\mathrm{SN}}^{2}$, \nthat is quadratically with respect to the ambient metallicity (C,N,O \nor Fe, assumed to be more or less proportional to one another).\n\nThe above reasoning, however, relies on two basic assumptions that \nneed not be fulfilled~: i) the CRs recently accelerated interact with \nall the CNO nuclei already produced and dispersed in the entire Galaxy \nand ii) the CRs are made of the ambient material, dominated by H and \nHe nuclei. Instead, it might be i) that the proton rich CRs recently \naccelerated interact predominantly with the freshly synthesized CNO \nnuclei near the explosion site and ii) that a significant fraction of \nthe CNO rich SN ejecta are also accelerated. In both cases, a linear \ngrowth of the Be abundance with respect to Fe or O would arise very \nnaturally, since the number of Be-producing spallation reactions \ninduced by each individual supernova would be directly linked to its \nlocal, individual CNO supply, independently of the accumulated amount \nof CNO in the Galaxy.\n\nIn fact, as emphasised by Ramaty et al. (1997), the simplest \nexplanation of the observational data is to assume that each \ncore-collapse supernova produces on average 0.1~\\hbox{$M_{\\odot}$}~of Fe, one to \nfew \\hbox{$M_{\\odot}$}~of the CNO elements and $2.8\\times10^{-8}\\,\\hbox{$M_{\\odot}$}$ of Be, \n{\\em with no metallicity dependence}. Clearly if this is the case and \nthe production of Be is directly linked to that of the main primary \nelements, the observed linear relation between Be and Fe will be \nreproduced whatever the complications of infall, mixing and outflow \nrequired by the Galactic evolution models. On the other hand, \nalthough the simplest explanation of the data is clearly to suppose a \nprimary behaviour for the Be production, it is possible that this \ncould be an artifact of the evolutionary models (as argued, e.g., by \nCasuso \\& Beckman 1997).\n\nMost work in this area has attempted to deduce information about \ncosmic ray (or other accelerated particle populations) in the early \ngalaxy by working backwards from the abundance observations. While \nperfectly legitimate, our feeling is that the observational errors and \nthe uncertainties relating to Galactic evolution in general make this \na very difficult task. We have chosen to approach the problem from \nthe other direction and ask what currently favoured models for \nparticle acceleration in supernova remnants (SNRs) imply for light \nelement production. This is in the general spirit of recent \ncalculations of the $\\pi^0$-decay gamma-ray luminosity of SNRs (Drury \net~al. 1994) and the detailed chemical composition of SNR shock \naccelerated particles (Ellison, Drury \\& Meyer, 1997) where we look \nfor potentially observable consequences of theoretical models for \ncosmic ray production in SNRs.\n\nInterestingly enough, the study of particle acceleration in SNRs \nsuggests that both alternatives to the naive scenario mentioned above \ndo occur in practice, as demonstrated qualitatively in \nSect.~\\ref{PartAccInSNRs}. The first of these alternatives, namely \nthe local interaction of newly accelerated cosmic rays in the vicinity \nof SN explosion sites, has already been called upon by Feltzing \\& \nGustaffson (1994), as well as the second, the acceleration of enriched \nejecta through a SN reverse shock, by Ramaty et al. (1997). However, \nno careful calculations have yet been done, taking the dynamics of the \nprocess into account, notably the dilution of SN ejecta and the \nadiabatic losses. Yet we show below that they have a crucial \ninfluence on the total amount of Be produced, and that a \ntime-dependent treatment is required. Indeed, the evolution of a SNR \nis essentially a dynamical problem in which the acceleration rate as \nwell as the chemical composition inside the remnant are functions of \ntime. The results of the full calculation of both processes and the \ndiscussion of their implications for the chemical evolution of the \nGalaxy will be found in an associated paper (Parizot \\&~Drury 1999). \nHere we present simple analytical calculations which provide an \naccurate understanding of the dynamics of light element production in \nSNRs and elucidates the role and influence of the different \nparameters, notably the ambient density.\n\nAlthough Li and B are also produced in the processes under study, we \nshall choose here Be as our `typical' light element, because nuclear \nspallation of CNO is thought to be its only production mechanism, \nwhile Li is also (and actually mainly) produced through $\\alpha + \n\\alpha$ reactions, $^{7}$Li may be produced partly in AGB stars (Abia \net~al. 1993), and $^{11}$B neutrino spallation may be important as \nseems to be required by chemical evolution analysis (Vangioni-Flam \net~al. 1996). In order to compare our results with the observations, \nwe simply note that, as emphasized in Ramaty et al. (1997), the data \nrelating to the Galactic Be evolution as a function of [Fe\/H] indicate \nthat $\\sim 1.6\\,10^{-6}$ nuclei of Be must be produced in the early \nGalaxy for each Fe nucleus. Therefore, if Be production is indeed \ninduced, directly or indirectly, by SNe explosions, and since the \naverage SN yield in Fe is thought to be $\\sim 0.11\\,\\hbox{$M_{\\odot}$}$, each \nsupernova must lead to an average production of $\\sim 3.8\\,10^{48}$ \nnuclei (or $\\sim 2.8\\,10^{-8}\\,\\hbox{$M_{\\odot}$}$) of Be, with an uncertainty of \nabout a factor of~2 (Ramaty et al. 1997). We adopt this value as the \n`standard needed number' of Be per supernova explosion. To state this \nagain in a different way, for an average SN yield in CNO of, say, \n$\\sim 1\\,\\hbox{$M_{\\odot}$}$, the required spallation rate per CNO atom is $\\sim \n3\\,10^{-8}$.\n\n\\section{Particle acceleration in SNRs}\n\\label{PartAccInSNRs}\n\nIt is generally believed that cosmic ray production in SNRs occurs\nthrough the process of diffusive shock acceleration operating at the\nstrong shock waves generated by the interaction between the ejecta\nfrom the supernova explosion and the surrounding medium. Significant\neffort has been put into developing dynamical models of SNR evolution\nwhich incorporate, at varying levels of detail, this basic\nacceleration and injection process (one of the major advantages of\nshock acceleration is that it does not require a separate injection\nprocess). Qualitatively the main features can be crudely summarised as\nfollows.\n\nIn a core collapse SN the collapse releases roughly the gravitational\nbinding energy of a neutron star, some $10^{53}\\,\\rm erg$, but most of\nthis is radiated away in neutrinos. About $\\hbox{${E_{\\rm SN}}$} = 10^{51}\\,\\rm erg$ is\ntransferred, by processes which are still somewhat obscure, to the\nouter layers of the progenitor star which are then ejected at\nvelocities of a few percent of the speed of light. Initially the\nexplosion energy is almost entirely in the form of kinetic energy of\nthese fast-moving ejecta. As the ejecta interact with the surrounding\ncircumstellar and interstellar material they drive a strong shock\nahead into the surrounding medium. The region of very hot high\npressure shocked material behind this forward shock also drives a\nweaker shock backwards into the ejecta giving rise to a characteristic\nforward reverse shock pair separated by a rather unstable contact\ndiscontinuity.\n\nThis initial phase of the remnant evolution lasts until the amount of\nambient matter swept up by the remnant is roughly equal to the\noriginal ejecta mass. At this so-called sweep-up time, \\hbox{${t_{\\rm SW}}$}, the\nenergy flux through the shocks is at its highest, the expansion of the\nremnant begins to slow down, and a significant part of the explosion\nenergy has been converted from kinetic energy associated with the bulk\nexpansion to thermal (and non-thermal) energy associated with \nmicroscopic degrees of freedom of the system. The remnant now enters\nthe second, and main, phase of its evolution in which there is rough\nequipartition between the microscopic and macroscopic energy\ndensities. The evolution in this phase is approximately self-similar\nand resembles the exact solution obtained by Sedov for a strong point\nexplosion in a cold gas. \n\nIt is important to realise that the approximate equality of the energy\nassociated with the macroscopic and microscopic degrees of freedom in\nthe Sedov-like phase is not a static equilibrium but is generated\ndynamically by two competing processes. As long as the remnant is\ncompact the energy density, and thus pressure, of the microscopic\ndegrees of freedom is very much greater than that of the external\nmedium. This strong pressure gradient drives an expansion of the\nremnant which adiabatically reduces the microscopic degrees of freedom\nof the medium inside the remnant and converts the energy back into\nbulk kinetic energy of expansion. At the same time the strong shock\nwhich marks the boundary of the remnant converts this macroscopic\nkinetic energy of expansion back into microscopic internal form. Thus\nthere is a continuous recycling of the original explosion energy\nbetween the micro and macro scales. This continues until either the\nexternal pressure is no longer negligible compared to the internal, or\nthe time-scales become so long that radiative cooling becomes\nimportant. The time scales for the conversion of kinetic energy to\ninternal energy and vice versa are roughly equal and of order the\ndynamical time scale of the remnant which is of order the age of the\nremnant, hence the approximately self-similar evolution. \n\nIn terms of particle acceleration the theory assumes that strong \ncollisionless shocks in a tenuous plasma automatically and \ninevitably generate an approximately power law distribution of \naccelerated particles which connects smoothly to the shock-heated \nparticle distribution at `the\\-rmal' energies and extends up to a \nmaximum energy constrained by the shock size, speed, age and magnetic \nfield. The acceleration mechanism is a variant of Fermi acceleration \nbased on scattering from magnetic field structures on both sides of \nthe shock. A key point is that these scattering structures are not \nthose responsible for general scattering on the ISM, but strongly \namplified local structures generated in a non-linear bootstrap process \nby the accelerated particles themselves. As long as the shock is \nstrong it will be associated with strong magnetic turbulence which \ndrives the effective local diffusion coefficient down to values close \nto the Bohm value. As pointed out by Achterberg et al. (1994) the \nextreme sharpness of the radio rims of some shell type SNRs can be \ninterpreted as observational evidence for this type of effect. The \nsource of free energy for the wave excitation is of course the strong \ngradient in the energetic particle distribution at the edge. Thus in \nthe interior of the remnant, where the gradients are absent or much \nweaker, we do not expect such low values of the diffusion coefficient.\n\nThe net effect is that the edge of the remnant, as far as the\naccelerated particles are concerned, is both a self-generated diffusion\nbarrier and a source of freshly accelerated particles. Except at the\nvery highest energies the particles produced at the shock are\nconvected with the post-shock flow into the interior of the remnant\nand effectively trapped there until the shock weakens to the point\nwhere the self-generated wave field around the shock can no longer be\nsustained. At this point the diffusion barrier collapses and the\ntrapped particle population is free to diffuse out into the general\nISM.\n\nIn terms of bulk energetics, the total energy of the accelerated \nparticle population is low during the first ballistic phase of the \nexpansion (because little of the explosion energy has been processed \nthrough the shocks) but rises rapidly as $t\\approx\\hbox{${t_{\\rm SW}}$}$. During the \nsedov-like phase the total energy in accelerated particles is roughly \nconstant at a significant fraction of the explosion energy (0.1 to 0.5 \ntypically). However, this is because of the dynamic recycling \ndescribed above. Any individual particle is subject to adiabatic \nlosses on the dynamical time-scale of the remnant, while the \nenergy lost this way goes into driving the shock and thus generating \nnew particles, distributed over the whole energy spectrum.\n\n\n\\section{Spallation reactions within SNRs}\n\n\\subsection{Qualitative overview}\n\nWe now turn to the production of Li, Be and B (LiBeB) by spallation \nreactions within a SNR. As emphasized above, there are two obvious \nmechanisms. One is the irradiation of the CNO ejecta by accelerated \nprotons and alphas. It is clear that the fresh CNO nuclei produced by \nthe SN will, for the lifetime of the SNR, be exposed to a flux of \nenergetic particles (EPs) very much higher than the average \ninterstellar flux, and this must lead to some spallation production of \nlight elements. This process starts at about \\hbox{${t_{\\rm SW}}$}~with a very intense \nradiation field and continues with an intensity decreasing roughly as \n$R^{-3}\\propto t^{-6\/5}$ (where $R$ is the radius of the SNR) until \nthe remnant dies.\n\nThe second process is that some of the CNO nuclei from the ejecta are \naccelerated, either by the reverse shock in its brief powerful phase \nat $t\\approx\\hbox{${t_{\\rm SW}}$}$ or by some of this material managing to get ahead of \nthe forward shock. This later possibility is not impossible, but \nseems unlikely to be as important as acceleration by the reverse \nshock. Calculations of the Raleigh-Taylor instability of the contact \ndiscontinuity do suggest that some fast-moving blobs of ejecta can \npunch through the forward shock at about \\hbox{${t_{\\rm SW}}$} (Jun \\& Norman 1996), \nand in addition Ramaty and coworkers have suggested that fast moving \ndust grains could condense in the ejecta at $t<\\hbox{${t_{\\rm SW}}$}$ and then \npenetrate through into the region ahead of the main shock. In all \nthese pictures acceleration of CNO nuclei takes place only at about \n\\hbox{${t_{\\rm SW}}$}~and the energy deposited in these accelerated particles is \ncertainly less than the explosion energy \\hbox{${E_{\\rm SN}}$}, although it might \noptimistically reach some significant fraction of that value (say $\\la \n10\\%$). Crucially the accelerated\nCNO nuclei are then confined to the interior of the SNR and will thus \nbe adiabatically cooled on a rather rapid time-scale, initially of \norder \\hbox{${t_{\\rm SW}}$}.\n\n\n\\subsection{Evaluation of the first process (forward shock)}\n\nFrom the above arguments, it is clear that SNe do induce some Be \nproduction. Now the question is~: how much? Let us first consider \nthe irradiation of the ejecta by particles (H and He nuclei) \naccelerated at the forward shock during the Sedov-like phase -- \nprocess~1. We have already indicated that detailed studies of \nacceleration in SNRs show that the fraction of the explosion energy \ngiven to the EPs is roughly constant during the Sedov-like phase and \nof order 0.1 to 0.5 or so. Let $\\theta_{1}$ be that fraction. Since \nthe EPs are distributed more or less uniformly throughout the interior \nof the remnant, the energy density can be estimated as\n\\begin{equation}\n\t\\hbox{${\\cal E}_{\\rm CR}$} \\approx \\frac{3\\theta_{1}\\hbox{${E_{\\rm SN}}$}}{4\\pi R^{3}}\n\t\\label{EdCR}\n\\end{equation}\nwhere $R$ is the radius of the remnant and \\hbox{${E_{\\rm SN}}$}~is the explosion \nenergy.\n\nTo derive a spallation rate from this we need to assume some form for \nthe spectrum of the accelerated particles. Shock acceleration \nsuggests that the distribution function should be close to the \ntest-particle form $f(p)\\propto p^{-4}$ and extend from an injection \nmomentum close to `thermal' values to a cut-off momentum at about \n$p_{\\mathrm{max}} = 10^{5}\\,{\\rm GeV}\/c$. The spallation rate per \ntarget CNO atom to produce a Be atom is then obtained by integrating \nthe cross sections\n\\begin{equation}\n\t\\nu_{\\rm spall} = \\int_{\\hbox{${p_{\\rm th}}$}}^{p_{\\mathrm{max}}}\n\t\\sigma v f(p) \\mathrm{d} p,\n\\end{equation}\nwith the normalisation $\\int E(p)f(p)4\\pi p^{2}\\mathrm{d} p = \n\\hbox{${\\cal E}_{\\rm CR}$}$. Looking at graphs of the spallation cross-sections for Be (as \ngiven, e.g., in Ramaty et al. 1997), it is clear that these \ncross-sections can be well approximated as zero below a threshold at \nabout 30--40~MeV\/n and a constant value $\\sigma_0\\simeq 5\\times \n10^{-27}\\,\\mathrm{cm}^{2}$ above it. One then obtains roughly~:\n\\begin{equation}\n\t\\nu_{\\rm spall} \\approx \\frac{\\sigma_{0}}{mc} \\hbox{${\\cal E}_{\\rm CR}$}\n\t\\frac{1-\\ln(\\hbox{${p_{\\rm th}}$}\/mc)}{1 + \\ln(p_{\\rm max}\/mc)}\n\t\\simeq 0.2 \\sigma_0 c \\frac{\\hbox{${\\cal E}_{\\rm CR}$}}{m c^2},\n\\end{equation}\nwhere $\\hbox{${p_{\\rm th}}$} \\simeq mc\/5$ is the momentum corresponding to the \nspallation threshold and $m$ refers to the proton mass. Fortunately, \nfor this form of the spectrum the upper cut-off and the spallation \nthreshold only enter logarithmically. A softer spectrum would lead to \nhigher spallation yields and a stronger dependence on the spallation \nthreshold.\n\nUsing Eq.~(\\ref{EdCR}) and the adiabatic expansion law for the forward \nshock radius, $R = R_{\\mathrm{SW}}(t\/\\hbox{${t_{\\rm SW}}$})^{2\/5}$, we can now \nestimate the total fraction of the CNO nuclei which will be converted \nto Be during the Sedov-like phase as~:\n\\begin{eqnarray}\n\t\\phi_{1} &=& \\int_{\\hbox{${t_{\\rm SW}}$}}^{\\hbox{${t_{\\rm end}}$}} 0.2 \\sigma_0 c \n\t\\frac{\\theta_{1}\\hbox{${E_{\\rm SN}}$}}{mc^{2}} \\frac{3}{4\\pi R^3} \\mathrm{d} t\\\\\n\t&=& 0.2 \\sigma_0 c \\frac{\\theta_{1}\\hbox{${E_{\\rm SN}}$}}{mc^{2}} \\frac{3}{4\\pi\n\t\\hbox{${R_{\\rm SW}}$}^3} \\int_{\\hbox{${t_{\\rm SW}}$}}^{\\hbox{${t_{\\rm end}}$}} \\left(\\frac{t}{\\hbox{${t_{\\rm SW}}$}}\\right)^{-6\/5} dt,\n\t\\label{phi1Interm}\n\\end{eqnarray}\nor\n\\begin{equation}\n\t\\phi_{1} = \\sigma_0 c \\frac{\\theta_{1}\\hbox{${E_{\\rm SN}}$}}{mc^{2}}\n\t\\frac{\\rho_{0}}{\\hbox{${M_{\\rm ej}}$}} \\hbox{${t_{\\rm SW}}$}\n\t\\left[1 - \\left(\\frac{\\hbox{${t_{\\rm SW}}$}}{\\hbox{${t_{\\rm end}}$}}\\right)^{1\/5}\\right]\n\t\\label{phi1}\n\\end{equation}\nwhere as usual $\\rho_{0}$ denotes the density of the ambient medium \ninto which the SNR is expanding and \\hbox{${M_{\\rm ej}}$}~is the total mass of the SNR \nejecta. We now recall that the sweep-up time is given in terms of the \nSN parameters and the ambient number density, $n_{0} \\approx \n\\rho_{0}\/m$, as\n\\begin{equation}\n\t\\hbox{${t_{\\rm SW}}$} = \\frac{n_{0}^{-1\/3}}{v_{\\mathrm{ej}}}\n\t\\left(\\frac{3}{4\\pi}\\frac{M_{\\mathrm{ej}}}{m}\\right)^{1\/3},\n\\end{equation}\nwhere $v_{\\mathrm{ej}}\\approx (2\\hbox{${E_{\\rm SN}}$}\/\\hbox{${M_{\\rm ej}}$})^{1\/2}$ is the velocity of \nthe ejecta, or numerically~:\n\\begin{equation}\n\t\\hbox{${t_{\\rm SW}}$} = (1.4\\,10^{3}\\,\\mathrm{yr})\n\t\\left(\\frac{M_{\\mathrm{ej}}}{10\\hbox{$M_{\\odot}$}}\\right)^{\\hspace{-2pt}\\frac{5}{6}}\n\t\\hspace{-4pt}\n\t\\left(\\frac{E_{\\mathrm{SN}}}{10^{51}\\mathrm{erg}}\\right)^{\\hspace{-2pt}-\\frac{1}{2}}\n\t\\hspace{-4pt}\n\t\\left(\\frac{n_{0}}{1\\mathrm{cm}^{-3}}\\right)^{\\hspace{-2pt}-\\frac{1}{3}}.\n\t\\label{SweepUpTime}\n\\end{equation}\n\nReplacing in Eq.~(\\ref{SweepUpTime}) and using canonical values of $\\hbox{${E_{\\rm SN}}$} = \n10^{51}\\,\\rm erg$ and $\\hbox{${M_{\\rm ej}}$} = 10\\,\\hbox{$M_{\\odot}$}$, we finally get~:\n\n\\begin{equation}\n\t\\phi_{1} \\simeq 4\\times10^{-10}\\,\\theta_{1} \n\t\\left(\\frac{n_{0}}{1\\,\\mathrm{cm}^{-3}}\\right)^{2\/3} \\left[1 - \n\t\\left(\\frac{\\hbox{${t_{\\rm SW}}$}}{\\hbox{${t_{\\rm end}}$}}\\right)^{1\/5}\\right].\n\t\\label{phi1Num}\n\\end{equation}\n\nClearly this falls short of the value of order $10^{-8}$ required to \nexplain the observations, even for values of $\\theta_{1}$ as high as \n$0.5$. It might seem from Eq.~(\\ref{phi1Num}) that very high ambient \ndensities could help to make the spallation yields closer to the \nneeded value. This is however not the case. First, the above \nestimate does not take energy losses into account, while both \nionisation and adiabatic losses act to lower the genuine production \nrates. Second, and more significantly, the ratio $\\hbox{${t_{\\rm SW}}$}\/\\hbox{${t_{\\rm end}}$}$ (and \n\\emph{a fortiori} its fifth root) becomes very close to 1 in dense \nenvironments, lowering $\\phi_{1}$ quite notably (see \nFig.~\\ref{BeYield}). In fact, it turns out that there is no \nSedov-like phase at all in media with densities of order \n$10^{4}\\,\\mathrm{cm}^{-3}$, the physical reason being that the \nradiative losses then act on a much shorter time-scale, eventually \nshorter than the sweep-up time.\n\n\n\\subsection{Evaluation of the second process (reverse shock)}\n\\label{SecondProcess}\n\nLet us now turn to the second process, namely the spallation of \nenergetic CNO nuclei accelerated at the reverse shock from the SN \nejecta and interacting within the SNR with swept-up ambient material. \nWe have argued above that this reverse shock acceleration is only \nplausible at times around \\hbox{${t_{\\rm SW}}$}~and certainly the amount of energy \ntransferred to CNO nuclei cannot be more than a fraction of \\hbox{${E_{\\rm SN}}$}. Let \n$\\theta_2$ be the fraction of the explosion energy that goes into \naccelerating the ejecta at or around \\hbox{${t_{\\rm SW}}$}, and $\\theta_{\\mathrm{CNO}}$ \nthe fraction of that energy that is indeed transferred to CNO nuclei. \nThese particles are then confined to the interior of the remnant where \nthey undergo spallation reactions as well as adiabatic losses. Let us \nagain assume that the spectrum is of the form $f(p)\\propto p^{-4}$. \nThen the production rate of Be atoms per unit volume is approximately\n\\begin{equation}\n0.2 {n_{0} \\sigma_0\\over m c} {\\hbox{${\\cal E}_{\\rm CR}$}\\over 14}\n\\label{ProdRate2}\n\\end{equation}\nwhere \\hbox{${\\cal E}_{\\rm CR}$}~now refers to the accelerated CNO nuclei, the factor~14 \ncomes from the mean number of nucleons per CNO nucleus and the factor \n0.2, as before, from the $f(p)\\propto p^{-4}$ spectral shape (assuming \nthe same upper cut-off position, but this only enters \nlogarithmically). Integrating over the remnant volume, we obtain the \nspallation rate at \\hbox{${t_{\\rm SW}}$}~:\n\\begin{equation}\n\t\\frac{\\mathrm{d}\\mathcal{N}_{\\mathrm{Be}}}{\\mathrm{d} t} \\approx 0.2 \\sigma_{0}c\n\t\\frac{\\theta_{\\mathrm{CNO}}\\theta_{2}E_{\\mathrm{SN}}}{14 m c^{2}}.\n\t\\label{ProdRate2bis}\n\\end{equation}\n\nNow the adiabatic losses need to be evaluated rather carefully. It is \ngenerally argued that they act so that the momentum of the particles \nscales as the inverse of the linear dimensions of the volume occupied. \nAccordingly, in the expanding spherical SNR the EPs should lose \nmomentum at a rate $\\dot{p}\/p = - \\dot{R}\/R$, reminiscent, \nincidentally, of the way photons behave in the expanding universe. In \nour case, however, the situation is complicated by the fact that the \nEPs do not push directly against the `walls' limiting the volume of \nconfinement, which move at the expansion velocity, $V = \\dot{R}$, but \nare reflected off the diffusion barrier consisting of magnetic waves \nand turbulence at rest with respect to the downstream flow, and thus \nexpanding at velocity $\\frac{3}{4}\\dot{R}$.\n\nTo see how this influences the actual adiabatic loss rate, it is safer \nto go back to basic physical laws. Adiabatic losses must arise \nbecause the EPs are more or less isotropised within the SNR and \ntherefore participate to the pressure. Now this pressure, $P$, works \npositively while the remnant expands, implying an energy loss rate \nequal to the power contributed, given by~:\n\\begin{equation}\n\t\\frac{\\mathrm{d} U}{\\mathrm{d} t} = - \\int\\hspace{-4pt}\\int_{\\mathcal{S}}\\vec{F}\\cdot\\vec{v}\n\t= - \\int\\hspace{-4pt}\\int_{\\mathcal{S}}P\\mathrm{d} S\\times\\frac{3}{4}\\dot{R}\n\t= - 3\\pi R^{2}\\dot{R}P,\n\t\\label{KinEnTh}\n\\end{equation}\nwhere $U = \\frac{4}{3}\\pi R^{3}\\epsilon$ is the total kinetic energy \nof the particles. Considering that $P = \\frac{2}{3}\\epsilon$ in the \nnon-relativistic limit (NR) and $P = \\frac{1}{3}\\epsilon$ in the \nultra-relativistic limit (UR), Eq.~(\\ref{KinEnTh}) can be re-writen \nas~:\n\\begin{equation}\n\\begin{split}\n\t\\frac{\\mathrm{d}\\epsilon}{\\mathrm{d} t} = -\\frac{9}{4}P\\frac{\\dot{R}}{R}\n\t&= - \\frac{3}{2}\\epsilon\\frac{\\dot{R}}{R}\\quad(\\mathrm{NR})\\\\\n\t&= - \\frac{3}{4}\\epsilon\\frac{\\dot{R}}{R}\\quad(\\mathrm{UR}).\n\\end{split}\n\\end{equation}\n\nFinally, dividing both sides by the space density of the EPs and \nnoting that $E = p^{2}\/2m$ in the NR limit, and $E = pc$ in the UR \nlimit, we obtain the momentum loss rate for individual particles, \nvalid in any velocity range~:\n\n\\begin{equation}\n\t\\frac{\\dot{p}}{p} = -\\frac{3}{4}\\frac{\\dot{R}}{R}.\n\t\\label{AdiabLossRate}\n\\end{equation}\n\nFrom this one deduces that at the time when the remnant has expanded \nto radius $R$, only those particles whose {\\em initial} momenta at \n\\hbox{${t_{\\rm SW}}$}~were more than $(R\/\\hbox{${R_{\\rm SW}}$})^{3\/4}\\hbox{${p_{\\rm th}}$}$ are still above the \nspallation threshold. For a $p^{-4}$ distribution function the \nintegral number spectrum decreases as $p^{-1}$ and thus the number of \naccelerated nuclei still capable of spallation reactions decreases as \n$R^{-3\/4}\\propto t^{-3\/10}$. For a softer accelerated spectrum the \neffect would be even stronger because there are proportionally fewer \nparticles at high initial momenta.\n\nThis being established, we can integrate Eq.~(\\ref{ProdRate2bis}) over \ntime, to obtain the total production of Be atoms~:\n\\begin{equation}\n\t\\mathcal{N}_{\\mathrm{Be}} = 0.2 n \\sigma_{0} c\n\t\\frac{\\theta_{\\mathrm{CNO}}\\theta_{2}\\hbox{${E_{\\rm SN}}$}}{14 mc^{2}}\n\t\\int_{\\hbox{${t_{\\rm SW}}$}}^{\\hbox{${t_{\\rm end}}$}} \\left(\\frac{t}{\\hbox{${t_{\\rm SW}}$}}\\right)^{-3\/10}\\,\\mathrm{d} t\n\t\\label{phi2Interm}\n\\end{equation}\nthat is~:\n\\begin{equation}\n\t\\mathcal{N}_{\\mathrm{Be}} = \\frac{2}{7} n \\sigma_{0} c\n\t\\frac{\\theta_{\\mathrm{CNO}}\\theta_{2}\\hbox{${E_{\\rm SN}}$}}{14 mc^{2}} \\hbox{${t_{\\rm SW}}$}\n\t\\left[\\left(\\frac{\\hbox{${t_{\\rm end}}$}}{\\hbox{${t_{\\rm SW}}$}}\\right)^{7\/10} - 1\\right].\n\\end{equation}\n\nDividing by the total number of CNO nuclei in the ejecta, \n$N_{\\mathrm{ej,CNO}} = (\\theta_{\\mathrm{CNO}}\/14) N_{\\mathrm{ej,tot}} \n\\simeq (\\theta_{\\mathrm{CNO}} M_{\\mathrm{ej}}\/14 m)$, we get the final \nresult~:\n\n\\begin{equation}\n\t\\phi_{2} = \\sigma_{0}c \\frac{\\theta_{2}E_{\\mathrm{SN}}}{m \n\tc^{2}}\\frac{\\rho_{0}}{M_{\\mathrm{ej}}}\\hbox{${t_{\\rm SW}}$}\n\t\\frac{2}{7}\\left(\\frac{\\hbox{${t_{\\rm end}}$}}{\\hbox{${t_{\\rm SW}}$}}\\right)^{7\/10}\n\t\\left[1 - \\left(\\frac{\\hbox{${t_{\\rm SW}}$}}{\\hbox{${t_{\\rm end}}$}}\\right)^{7\/10}\\right].\n\t\\label{phi2}\n\\end{equation}\nNote that we assumed that the mass fraction of CNO in the ejecta is \nthe same as the energy fraction of CNO in the EPs (which was the \noriginal meaning of $\\theta_{\\mathrm{CNO}}$). Considering that all \nnuclear species have the same spectrum in MeV\/n, and thus a total \nenergy proportional to their mass number, this simply means that the \nacceleration process is not chemically selective, in the sense that \nthe composition of the EPs is just the same as that of the material \npassing through the shock.\n\nNumerically, again with $\\hbox{${E_{\\rm SN}}$}=10^{51}\\,\\rm erg$ and $\\hbox{${M_{\\rm ej}}$} = \n10\\,\\hbox{$M_{\\odot}$}$, we finally obtain~:\n\n\\begin{equation}\n\\begin{split}\n\t\\phi_2 \\simeq 1\\times 10^{-10}&\\theta_{2}\n\t\\left(n_0\\over 1\\,\\rm cm^{-3}\\right)^{2\/3}\\\\\n\t&\\times\\left(\\frac{\\hbox{${t_{\\rm end}}$}}{\\hbox{${t_{\\rm SW}}$}}\\right)^{7\/10}\n\t\\left[1 - \\left(\\frac{\\hbox{${t_{\\rm SW}}$}}{\\hbox{${t_{\\rm end}}$}}\\right)^{7\/10}\\right].\n\t\\label{phi2Num}\n\\end{split}\n\\end{equation}\n\n\\subsection{Relative contribution of the two processes}\n\\label{RelativeContribution}\n\nIt is worth emphasizing the similarity between expressions \n(\\ref{phi1}) and (\\ref{phi2}) that we obtained for the spallation \nrates per CNO nuclei by the two processes considered here. This \nformal analogy allows us to write down their relative contributions \nstraightforwardly~:\n\\begin{equation}\n\t\\frac{\\phi_{2}}{\\phi_{1}} = \\frac{\\theta_{2}}{\\theta_{1}}\n\t\\times \\frac{2}{7}\\left(\\hbox{${t_{\\rm end}}$}\/\\hbox{${t_{\\rm SW}}$}\\right)^{7\/10}\n\t\\frac{\\left[1 - \\left(\\hbox{${t_{\\rm SW}}$}\/\\hbox{${t_{\\rm end}}$}\\right)^{7\/10}\\right]}\n\t {\\left[1 - \\left(\\hbox{${t_{\\rm SW}}$}\/\\hbox{${t_{\\rm end}}$}\\right)^{1\/5}\\right]}.\n\t\\label{phi2\/phi1}\n\\end{equation}\n\nAs is often the case, this similarity is not fortuitous and has a \nphysical meaning. The two processes may indeed be regarded as `dual' \nprocesses, the first consisting of the irradiation of the SN ejecta by \nthe ambient medium, and the second of the ambient medium by the SN \nejecta. The `symmetry' is only broken by the dynamical aspect of the \nprocesses. First, of course, the energy imparted to the EPs in both \ncases needs not be the same, for it depends on the acceleration \nefficiency as well as the total energy of the shock involved (forward \nor reverse). This is expressed by the expected ratio \n$\\theta_{2}\/\\theta_{1}$. And secondly, in the first process one has \nto fight against the dilution of the ejecta -- integration of \n$(t\/\\hbox{${t_{\\rm SW}}$})^{-6\/5}$, see Eq.~(\\ref{phi1Interm}) -- while in the second \nprocess one fights against the adiabatic losses -- integration of \n$(t\/\\hbox{${t_{\\rm SW}}$})^{-3\/10}$, see Eq.~(\\ref{phi2Interm}). This is expressed by \nthe last factor in Eq.~(\\ref{phi2\/phi1}).\n\nClearly the latter decrease of the production rates is the least \ndramatic, and the reverse shock process must dominate the LiBeB \nproduction in supernova remnants. However, this conclusion still \ndepends on the genuine efficiency of reverse shock acceleration, and \nonce the relative acceleration efficiency $\\theta_{2}\/\\theta_{1}$ is \ngiven, the weight of the first process relative to the second still \ndepends on the total duration of the Sedov-like phase, appearing \nnumerically in Eq.~(\\ref{phi2\/phi1}) through the ratio \\hbox{${t_{\\rm end}}$}\/\\hbox{${t_{\\rm SW}}$}, \nwhich in turn depends on the ambient density, $n_{0}$. The expression \nof \\hbox{${t_{\\rm SW}}$}~as a function of the parameters has been given in \nEq.~(\\ref{SweepUpTime}), so we are left with the evaluation of the time, \n\\hbox{${t_{\\rm end}}$}, when the magnetic turbulence collapses and the EPs leave the \nSNR, putting an end to Be production. We argued above that \n\\hbox{${t_{\\rm end}}$}~should correspond to the end the Sedov-like phase, when the \nshock induced by the SN explosion becomes radiative, that is when the \ncooling time of the post-shock gas becomes of the same order as the \ndynamical time.\n\nIn principle, the cooling rate can be derived from the so-called \ncooling function, $\\Lambda(T) (\\mathrm{erg~cm}^{3}\\mathrm{s}^{-1}$), \nwhich depends on the physical properties of the post-shock material, \nnotably on its temperature, $T$, and metallicity, $Z$~:\n\\begin{equation}\n\t\\tau_{\\mathrm{cool}}\\approx\\frac{\\frac{3}{2}k_{\\mathrm{B}}T}{n\\Lambda(T)},\n\t\\label{tauCool}\n\\end{equation}\nwhere $n$ is the post-shock density, equal to $4n_{0}$ if the \ncompression ratio is that of an ideal strong shock (nonlinear effects \nprobably act to increase the compression ratio to values larger \nthan~4). As for the dynamical time, we simply write\n\\begin{equation}\n\t\\tau_{\\mathrm{dyn}} \\approx \\frac{\\dot{R}}{R} \\approx \\frac{5}{2}\\,t.\n\t\\label{tauDyn}\n\\end{equation}\n\nTo obtain \\hbox{${t_{\\rm end}}$}, we then need to solve the following equation in the \nvariable $t$, obtained by equating $\\tau_{\\mathrm{cool}}$ and \n$\\tau_{\\mathrm{dyn}}$ given above~:\n\\begin{equation}\n\tt \\approx \\frac{3k_{\\mathrm{B}}T}{20n_{0}\\Lambda(T)},\n\t\\label{tEndEq}\n\\end{equation}\nwhere it should be clear that the right hand side also depends on time \nthrough the temperature, $T$, and thus indirectly through the cooling \nfunction too. In the non-radiative SNR expansion phase, the function \n$T(t)$ is obtained directly from the hydrodynamical jump conditions at \nthe shock discontinuity~:\n\\begin{equation}\n\tT \\approx \\frac{3m}{8k_{\\mathrm{B}}}V^{2},\n\\end{equation}\nor numerically~:\n\\begin{equation}\n\tT \\approx (2\\times 10^{5}\\,\\mathrm{K})\n\t\\left(\\frac{E_{\\mathrm{SN}}}{10^{51}\\mathrm{erg}}\\right)^{\\hspace{-2pt}\\frac{2}{5}}\n\t\\hspace{-4pt}\n\t\\left(\\frac{n_{0}}{1\\mathrm{cm}^{-3}}\\right)^{\\hspace{-2pt}-\\frac{2}{5}}\n\t\\hspace{-4pt}\n\t\\left(\\frac{t}{10^{5}yr}\\right)^{\\hspace{-2pt}-\\frac{6}{5}}.\n\t\\label{Temperature}\n\\end{equation}\n\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics{fig1.eps}} \n\\caption{Cooling function (bold line) as a function of the temperature \nfor a medium with metallicity lower than $\\sim 10^{-2}Z_{\\odot}$. The \ndashed lines illustrate the graphical determination of \\hbox{${t_{\\rm end}}$}, for an \nambient density $n_{0} = 10\\,\\mathrm{cm}^{-3}$ (see text).}\n\\label{CoolingFunction}\n\\end{figure}\n\nTo solve Eq.~(\\ref{tEndEq}), we still need to know the cooling \nfunction $\\Lambda(T)$. In the range of temperatures corresponding to \nthe end of the Sedov-like phase, $10^{5}\\,\\mathrm{K}\\la T\\la \n10^{7}\\,\\mathrm{K}$, it happens to depend significantly on \nmetallicity, with differences up to two orders of magnitude for \nmetallicities from $Z = 0$ to $Z = 2 Z_{\\odot}$ (B\\\"oh\\-rin\\-ger \\& \nHens\\-ler 1989). Because we focus on Be production in the early \nGalaxy, we adopt the cooling function corresponding to zero \nmetallicity, represented in Fig.~\\ref{CoolingFunction} (adapted from \nB\\\"oh\\-rin\\-ger \\& Hens\\-ler 1989), which holds for values of $Z$ up \nto $\\sim 10^{-2}Z_{\\odot}$.\n\nFor high enough ambient densities, the shock will become radiative \nearly in the SNR evolution, when the temperature is still very high, \nsay above $T\\ga 2\\,10^{6}$~K. In this case, the cooling function is \ndominated by Bremsstrahlung emission and can be written analytically \nas~:\n\\begin{equation}\n\t\\Lambda_{\\mathrm{Br}}(T) \\approx \n\t(2.4\\,10^{-23}\\,\\mathrm{erg~cm}^{3}\\mathrm{s}^{-1})\n\t\\left(\\frac{T}{10^{8}\\,\\mathrm{K}}\\right)^{1\/2}.\n\t\\label{LambdaBr}\n\\end{equation}\nSubstituting from (\\ref{Temperature}) and (\\ref{LambdaBr}) in \nEq.~(\\ref{tEndEq}) and solving for $t$, we find~:\n\\begin{equation}\n\t\\hbox{${t_{\\rm end}}$} = (1.1\\,10^{5}\\,\\mathrm{yr})\n\t\\left(\\frac{E_{\\mathrm{SN}}}{10^{51}\\mathrm{erg}}\\right)^{1\/8}\n\t\\left(\\frac{n_{0}}{1\\mathrm{cm}^{-3}}\\right)^{-3\/4}.\n\t\\label{tEnd}\n\\end{equation}\n\nTo check the consistency of our assumption $\\Lambda \\approx \n\\Lambda_{\\mathrm{Br}}$ (i.e. $T\\ga 2\\,10^{6}$~K), let us now report \nEq.~(\\ref{tEnd}) in (\\ref{Temperature}) and write down the temperature \n$T_{\\mathrm{End}}$ at the end of the Sedov-like phase~:\n\\begin{equation}\n\tT_{\\mathrm{End}} \\approx (2\\times 10^{5}\\,\\mathrm{K})\n\t\\left(\\frac{E_{\\mathrm{SN}}}{10^{51}\\mathrm{erg}}\\right)^{1\/4}\n\t\\left(\\frac{n_{0}}{1\\mathrm{cm}^{-3}}\\right)^{1\/2},\n\\end{equation}\nwhich means that the above analytical treatment is valid only for \nambient densities greater than about $100\\,\\mathrm{cm}^{-3}$. For \nlower densities, we must solve Eq.~(\\ref{tEndEq}) graphically. First, \nwe invert Eq.~(\\ref{Temperature}) to express $t$ as a function of \ntemperature, then we plot the function $f(T)\\equiv \n3k_{\\mathrm{B}}T\/(20n_{0}t)$ on the same graph as $\\Lambda$ (see \nFig.~\\ref{CoolingFunction} for an example), find the value of $T$ at \nintersection, and finally convert this value into the sought time \n\\hbox{${t_{\\rm end}}$}~making use again of Eq.~(\\ref{Temperature}). The results, \nshowing \\hbox{${t_{\\rm end}}$}~as a function of the ambient density, are shown in \nFig.~\\ref{TimeScales}.\n\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics{fig2.eps}} \n\\caption{Comparison of \\hbox{${t_{\\rm end}}$}~and \\hbox{${t_{\\rm SW}}$}~as a function of the ambient \ndensity, $n_{0}$, for two different values of the mass ejected by the \nsupernova (10 and 30~\\hbox{$M_{\\odot}$}). The dashed line shows the asymptotic \nanalytic estimate of Eq.~(\\ref{tEnd}).}\n\\label{TimeScales}\n\\end{figure}\n\nWe now have all the ingredients to plot the efficiency ratio of the \ntwo processes calculated above. Figure~\\ref{Phi2\/Phi1} shows the \nratio $\\phi_{2}\/\\phi_{1}$ given in Eq.~(\\ref{phi2\/phi1}) as a function \nof the ambient density, assuming that $\\theta_{1} = \\theta_{2}$. Two \ndifferent values of the ejected mass have been used, corresponding to \ndifferent progenitor masses ($\\sim 10-40\\,\\hbox{$M_{\\odot}$}$). It can be seen \nthat low densities are more favourable to the reverse shock \nacceleration process. This is due to $\\hbox{${t_{\\rm end}}$}\/\\hbox{${t_{\\rm SW}}$}$ being larger, \nimplying a larger dilution of the ejecta (process~1 less efficient) \nand smaller adiabatic losses, which indeed decrease as $t^{-1}$ \n(process~2 more efficient). The part of the plot corresponding to \n$\\phi_{2}\/\\phi_{1}\\le 1$ is not physical, because it requires \n$\\hbox{${t_{\\rm end}}$}\\le\\hbox{${t_{\\rm SW}}$}$, which simply means that the Sedov-like phase no longer \nexists and the whole calculation becomes groundless. Note however \nthat in Fig.~\\ref{Phi2\/Phi1} the energy imparted to the EPs has been \nassumed equal for both processes, which is most certainly not the \ncase. Actually, if $\\theta_{2}\/\\theta_{1} = 0.1$ (e.g. $\\theta_{1} = \n10\\%$ and $\\theta_{1} = 1\\%$), then process~1 is found to dominate Be \nproduction during the Sedov-like phase, regardless of the ambient \ndensity.\n\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics{fig3.eps}} \n\\caption{Comparison of the Be (and B) production efficiency through \nthe forward and reverse shock acceleration processes, for two values \nof the ejected mass (10~and 30~\\hbox{$M_{\\odot}$}). The ratio $\\phi_{2}\/\\phi_{1}$ \n(see text) is plotted as a function of the ambient density, assuming \nthat both processes impart the same total energy to the EPs \n($\\theta_{1} = \\theta_{2}$).}\n\\label{Phi2\/Phi1}\n\\end{figure}\n\n\n\\section{Spallation reactions after the Sedov-like phase}\n\nAt the end of the Sedov-like phase, the EPs are no longer confined and \nleave the SNR to diffuse across the Galaxy. At the stage of chemical \nevolution we are considering here, there are no or few metals in the \ninterstellar medium (ISM), so that energetic protons and $\\alpha$ \nparticles accelerated at the forward shock will not produce any \nsignificant amount of Be after \\hbox{${t_{\\rm end}}$}~(although Li production will \nstill be going on through $\\alpha + \\alpha$ reactions). In the case \nof the second process, however, the EPs contain CNO nuclei which just \ncannot avoid being spalled while interacting with the ambient H and He \nnuclei at rest in the Galaxy. This may be regarded as a third process \nfor Be production, which lasts until either the EPs are slowed down by \nCoulombian interactions to subnuclear energies (i.e. below the \nspallation thresholds) or they simply diffuse out of the Galaxy. \nSince the confinement time of cosmic rays in the early Galaxy is \nvirtually unknown, we shall assume here that the Galaxy acts as a \nthick target for the EPs leaving the SNR, an assumption which actually \nprovides us with an upper limit on the spallation yields.\n\nUnlike the first two processes evaluated above, this third process is \nessentially independent of dynamics. Thus, time-dependent \ncalculations are no longer needed and, from this stage on, the \ncalculations made by Ramaty et~al. (1997) or any steady-state \ncalculation is perfectly valid. In particular, the ambient density \nhas no influence on light element production, since a greater number \nof reactions per second, as would result from a greater density, \nimplies an equal increase of both the spallation rates and the energy \nloss rate. Once integrated over time, both effects cancel out \nexactly, and in fact, given the energy spectrum of the EPs, the \nefficiency of Be production (and Li, and B), expressed as the number \nof nuclei produced per erg injected in the form of EPs, depends only \non their chemical composition.\n\nResults are shown in Fig.~\\ref{Be\/erg} for different values of the \nsource abondance ratios, $\\mathrm{H}\/\\mathrm{He}$ and \n$(\\mathrm{H}+\\mathrm{He})\/(\\mathrm{C}+\\mathrm{O})$, allowing one to \nderive the spallation efficiency for any composition. Two-steps \nprocesses (such as $^{12}\\mathrm{C} + ^{1}\\mathrm{H} \\longrightarrow \n^{10}\\mathrm{B}$ followed by $^{10}\\mathrm{B} + ^{1}\\mathrm{H} \n\\longrightarrow ^{9}\\mathrm{Be}$) have been taken into account. Test \nruns show good agreement with the results of Ramaty et~al. (1997).\n\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics{fig4.eps}} \n\\caption{Production efficiency of Be, as a function of the EP \ncomposition (all abundances are by number). The ordinate is the \nnumber of Be nuclei produced by spallation reactions per erg injected \nin the form of EPs. A thick target has been assumed, with zero \nmetallicity. Carbon and Oxygen abundances were set equal in the EP \ncomposition.}\n\\label{Be\/erg}\n\\end{figure}\n\nAs can be seen on Fig.~\\ref{Be\/erg}, pure Carbon and Oxygen have a \nproduction efficiency of about $0.22\\,\\mathrm{nuclei}\/\\mathrm{erg}$, \nwhile this efficiency decreases by at least a factor of~10 for \ncompositions with hundred times more H and He than metals (or about \nten times more by mass). According to models of explosions for SN \nwith low metallicity progenitors, the average \n$(\\mathrm{H}+\\mathrm{He})\/(\\mathrm{C}+\\mathrm{O})$ ratio among the EPs \nshould indeed be expected to be $\\ga 200$, unless selective \nacceleration occurs to enhance the abundance of the metals. As a \nconsequence, efficiencies greater than $\\sim \n10^{-2}\\,\\mathrm{nuclei}\/\\mathrm{erg}$ should not be expected, so that \na production of $\\sim 4\\,10^{48}$~atoms of Be requires an energy of \n$\\sim 4\\,10^{50}$~erg to be imparted to the EPs. This seems very \nunlikely considering that the total energy available in the reverse \nshock (the source of the EPs) should be of order one tenth of the SN \nexplosion energy, not to mention the acceleration efficiency. \nMoreover, a significant fraction of the energy originally imparted to \nthe EPs has been lost during the Sedov-like phase of the SNR evolution \nthrough adiabatic losses.\n\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics{fig5.eps}} \n\\caption{Fraction of the energy imparted to the EPs at time \\hbox{${t_{\\rm SW}}$}~which \nis still available at \\hbox{${t_{\\rm end}}$}, after the Sedov-like phase, as a function \nof othe ambient density.}\n\\label{EnergyReduction}\n\\end{figure}\n\nTo evaluate the `surviving' fraction of energy, it sufficies to go \nback to Eq.~(\\ref{AdiabLossRate}), which indicates that when the \nradius of the shock is multiplied by a factor $\\eta$, the momentum $p$ \nof all the particles is multiplied by a factor $\\eta^{-3\/4}$. It is \nworthwhile noting that, because of their specific momentum dependence, \nadiabatic losses do not modify the shape of the EP energy spectrum. \nIn our case, $f(p)\\propto p^{-4}$, so that when all momenta $p$ are \ndivided by a factor $\\zeta$, the distribution function $f(p)$ is \ndivided by the same factor $\\zeta$. To see that, the easiest way is \nto work out the number of particules between momenta $p$ and $p + \\mathrm{d} \np$ after the momentum scaling. This number writes $\\mathrm{d} N^{\\prime} = \nf^{\\prime}(p)4\\pi p^{2}\\mathrm{d} p$, where $f^{\\prime}(p)$ is the new \ndistribution function. Now $\\mathrm{d} N^{\\prime}$ must be equal to the \nnumber of particles that had momentum between $\\zeta p$ and $\\zeta(p + \n\\mathrm{d} p)$, which is, by definition, $\\mathrm{d} N = f(\\zeta p)4\\pi(\\zeta \np)^{2}\\zeta\\mathrm{d} p$. Equating $\\mathrm{d} N$ and $\\mathrm{d} N^{\\prime}$ yields the \nresult $f^{\\prime}(p) = \\zeta^{-1}f(p)$.\n\nPutting all pieces together, we find that when the shock radius $R$ is \nmultiplied by a factor $\\eta$, the distribution function and, thus, \nthe total energy of the EPs are multiplied by $\\eta^{-3\/4}$. Now \nconsidering that $R$ increases as $t^{2\/5}$ during the Sedov-like \nphase, we find that the total energy of the EPs decreases as \n$t^{-3\/10}$. Note that this is nothing but an other way to work out \nthe decrease of the spallation rates for our second process during the \nSedov-like phase (cf. Sect.~\\ref{SecondProcess}). Finally, we find \nthat a fraction $(\\hbox{${t_{\\rm end}}$}\/\\hbox{${t_{\\rm SW}}$})^{-3\/10}$ of the initial energy imparted \nto the EPs is still available for spallation at the end of the \nSedov-like phase. This factor is plotted on \nFig.~\\ref{EnergyReduction}, as a function of the ambient density. It \ncan be seen that for $n_{0} = 1\\,\\mathrm{cm}^{-3}$, the energy \navailable to power our third process of light element production has \nbeen reduced by adiabatic losses to not more than one third of its \ninitial value, and less than one half for densities up to \n$100\\,\\mathrm{cm}^{-3}$. Clearly, high densities are favoured \n(energetically) because they tend to shorten the Sedov-like phase, and \ntherefore merely avoid the adiabatic losses.\n\n\n\\section{Discussion}\n\nSince light element production in the interstellar medium obviously \nrequires a lot of energy in the form of supernuclear particles (i.e. \nwith energies above the nuclear thresholds) as well as metals \n(especially C and O), it is quite natural to consider SNe as possible \nsources of the LiBeB observed in halo stars. We have analysed in \ndetail the spallation nucleosynthesis induced by a SN explosion on the \nbasis of known physics and theoretical results relating to particle \nshock acceleration. Two major processes can be identified, depending \non whether the ISM or the ejecta are accelerated, respectively at the \nforward and reverse shocks. In the first case, the EPs consist mostly \nof protons and alpha particles and must therefore interact with C and \nO nuclei, which are much more numerous within the SNR than in the \nsurrounding medium (especially at early stages of Galactic evolution). \nThe process will thus last as long as the EPs stay confined in the \nSNR, i.e. approximately during the Sedov-like phase, but not more. \nIn the second case, freshly synthesized CNO nuclei are accelerated, \nand Be production occurs through interaction with ambient H and He \nnuclei. The process is then divided into two, one stretching over the \nSedov-like phase, with the particles suffering adiabatic losses, and \nthe other one occuring outside the remnant, with only Coulombian \nlosses playing a role.\n\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics{fig6.eps}} \n\\caption{Spallation efficiency of CNO during the Sedov-like phase, as \na function of the ambient density. The fraction of freshly \nsynthesized CNO nuclei being spalled to Be by processes~1 ($\\phi_{1}$) \nand~2 ($\\phi_{2}$) is obtained from Eqs.~(\\ref{phi1Num}) \nand~(\\ref{phi2Num}) and the values of \\hbox{${t_{\\rm end}}$}~derived in \nSect.~\\ref{RelativeContribution}, for two values of the ejected mass \n(10~and 30~\\hbox{$M_{\\odot}$}).}\n\\label{BeYield}\n\\end{figure}\n\nWe have calculated the total Be production in these three processes, \ntaking the dynamics of the SNR evolution into account (dilution of the \nejecta by metal-poor material and adiabatic losses). The results are \nshown in Fig.~\\ref{BeYield} for processes 1 and~2 (from \nEqs.~(\\ref{phi1Num}) and~(\\ref{phi2Num})). We find that with \ncanonical values of $\\theta_{1} = 0.1$, $\\theta_{2} = 0.01$, \n$M_{\\mathrm{ej}} = 10\\,\\hbox{$M_{\\odot}$}$ and a mean ambient density $n_{0} = \n10\\,\\mathrm{cm}^{-3}$, the fraction of freshly synthesized CNO nuclei \nspalled into Be in these processes is $\\phi_{1} \\sim 8\\,10^{-11}$ and \n$\\phi_{2} \\sim 3\\,10^{-11}$, respectively, which is very much less \nthan the value `required' by the observations, discussed in the \nintroduction ($\\phi_{\\mathrm{obs}}\\sim 3\\,10^{-8}$). Even allowing \nfor unreasonably high values of the acceleration efficiency, \n$\\theta_{1}\\sim \\theta_{2}\\la 1$, the total Be production by processes \n1 and~2 would still be more than one order of magnitude below the \nobserved value.\n\nAs suggested by Fig.~\\ref{BeYield} and our analytical study, higher \ndensities improve the situation. However, even with $n_{0} = \n10^{3}\\,\\mathrm{cm}^{-3}$ and acceleration efficiencies equal to~1, \nthe Be yield is still unsufficient. Moreover, it should be noted that \nour calculations did not consider Coulombian energy losses (because \nthey are negligeable as compared to adiabatic losses for usual \ndensities), which become important as the density increases and \ntherefore make the Be yield smaller. Finally, since we are trying to \naccount for the mean abundance of Be in halo stars, as compared to Fe, \nwe have to evaluate the Be production for an ambient density \ncorresponding to the mean density encountered around explosion sites \nin the early Galaxy, which is very unlikely to be as high as \n$10^{3}\\,\\mathrm{cm}^{-3}$. It could even be argued that although the \ngas density might have been higher in the past than it is now (hence \nour `canonical value' $n_{0} = 10\\,\\mathrm{cm}^{-3}$), the actual mean \ndensity about SN explosion sites could be lower than \n$1\\,\\mathrm{cm}^{-3}$, because most SNe may explode within superbubble \ninteriors, where the density is much less than in the mean ISM.\n\nThus, our conclusion is that processes 1 and~2 both fail in accounting \nfor the Be observed in metal-poor stars in the halo of our Galaxy. \nConcerning the third process, adopting canonical values for the \nparameters again leads to unsufficient Be production, as noted in the \nprevious section. While higher densities improve the situation by \navoiding the adiabatic energy losses, one should nevertheless expect \nat least half of the EP energy to be lost in this way, for any \nreasonable density. This means that even if 10\\%\nof the explosion energy is imparted to EPs accelerated at the reverse \nshock, which is certainly a generous upper limit, the required number \nof $\\sim 4\\,10^{48}$ nuclei of Be per SN implies a spallation \nefficiency of $\\sim 0.1$~nucleus\/erg. Now Fig.~\\ref{Be\/erg} shows \nthat this requires an EP composition in which at least one particle \nout of ten is a CNO nucleus. In other words, the ejected mass of CNO \nmust be of the order of that of H and He together. None of the SN \nexplosion models published so far can reproduce such a requirement, \nand so there is clearly a problem with Be production in the early \nGalaxy.\n\nThe results presented here are in fact interesting in many regards. \nFirst, they show that it is definitely very difficult to account for \nthe amount of Be found in halo stars. Consequently, we feel that the \nmain problem to be addressed in this field of research is probably not \nthe chemical evolution of Be (and Li and B) in the Galaxy, as given by \nthe ensemble of the data points in the abundance vs metallicity \ndiagrams (e.g. whether Be is proportional to Fe or to its square) \nbut, to begin with, the position of any of these points. Are we able \nto describe in some detail one process which could explain the amount \nof Be (relative to Fe) present in any of the stars in which it is \nobserved? The answer, we are afraid, seems to be no at this stage. \nIt is however instructive to ask why the processes investigated here \nhave failed. Concerning process~1 (acceleration of ISM, interaction \nwith fresh CNO within the SNR), the main reason is that the CNO rich \nejecta are `too much diluted' by the swept-up material as the SNR \nexpands, so that the spallation efficiency is too low (or the \navailable energy is too small). However, it seems rather hard to \nthink of any region in the Galaxy where the concentration in CNO is \nhigher than inside a SNR during the Sedov-like phase (especially in \nthe first stages of chemical evolution)! So the conclusion that \nprocess~1 cannot work, even with a 100\\% acceleration\nefficiency, seems to rule out any other process based on the \nacceleration of the ISM, initially devoided of metals.\n\nThe other solution is then of course to accelerate CNO nuclei \nthemselves, which provides the maximum possible spallation efficiency, \nindependently of the ambient metallicity. Every energetic CNO will \nlead to the production of as much Be as possible given the spallation \ncross sections and the energy loss rates. The latter cannot \nphysically be smaller than the Coulombian loss rate in a neutral \nmedium, and this leads to the efficiency plotted in Fig.~\\ref{Be\/erg}. \nUnfortunately, a significant amount of the CNO rich ejecta of an \nisolated SN can only be accelerated at the reverse shock at a time \naround the sweep-up time, \\hbox{${t_{\\rm SW}}$}. This means that i) the total amount of \nenergy available is smaller than the explosion energy (probably of \norder 10\\%, i.e. $\\sim 10^{50}$~erg), and ii) the accelerated nuclei\nwill suffer adiabatic losses during the Sedov-like phase, reducing \ntheir energy by a factor of 2 or~3. As shown above, this makes \nprocess~2-3 incapable of producing enough Be, as long as the EPs have \na composition reflecting that of the SN ejecta.\n\nThis suggest that a solution to the problem could be that the reverse \nshock accelerates preferentially CNO nuclei rather than H and He. For \nexample, recent calculations have shown that such a selective \nacceleration arises naturally if the metals are mostly condensed in \ngrains (Ellison et~al. 1997). The proposition by Ramaty et~al. \n(1997) that grains condense in the ejecta before being accelerated \ncould then help to increase the abundance of CNO in the EPs. However, \nwe have to keep in mind that any selective process called upon must be \nvery efficient indeed, since as we indicated above, the data require \nthat the EP composition be as rich as one CNO nuclei out of ten EPs, \nwhich means that CNO nuclei must be accelerated at least ten times \nmore efficiently than H and He. This would have to be increased by \nanother factor of ten if the energy initially imparted to the EPs by \nthe acceleration process were only a factor 2 or~3 lower (i.e. $\\sim \n3\\,10^{49}$~erg, which is more reasonable from the point of view of \nparticle acceleration theory). Clearly, more work is needed in this \nfield before one can safely invoke a solution in terms of selective \nacceleration.\n\nAs can be seen, playing with the composition to increase the \nspallation efficiency has its own limits, and in any case, \nFig.~\\ref{Be\/erg} gives an unescapable upper limit, obtained with pure \nCarbon and Oxygen (at least for the canonical spectrum considered here \n- other spectra were also investigated, as in Ramaty et al. (1997), \nleaving the main conclusions unchanged). This would then suggest that \nanother source of energy should be sought. However, the constraint \nthat it should be more energetic than SNe is rather strong.\n\nAnother interesting line of investigations could be the study of the \ncollective effects of SNe. Most of the massive stars and SN \nprogenitors are believed to be born (and indeed observed, Melnik \n\\&~Efremov 1995) in associations, and their joint explosions lead to \nthe formation of superbubbles which may provide a very favourable \nenvironment for particle acceleration (Bykov \\&~Fleishman 1992). \nParizot et~al. (1998) have proposed that these superbubbles could be \nthe source of most of the CNO-rich EPs, and Parizot \\&~Knoedlseder \n(1998) further investigated the gamma-ray lines induced by such an \nenergetic component. The most interesting features of a scenario in \nwhich Be-producing EPs are accelerated in superbubbles is that i) when \na new SN explodes, the CNO nuclei ejected by the previous SNe are \naccelerated at the \\emph{forward shock}, instead of the reverse shock \nin the case of an isolated SN, which implies a greater energy, and ii) \nno significant adiabatic losses occur, because of the dimensions and \nlow expansion velocity of the superbubble. This makes the superbubble \nscenario very appealing, and it will be investigated in detail in a \nforthcoming paper.\n\nHowever that may be, we should also keep in mind that when we say that \na process does not produce enough Be, it always means that it does not \nproduce enough Be \\emph{as compared to Fe}. Now it could also be that \nSN explosion models actually produce too much Fe. The point is that \nBe is compared to Fe in the observations, while it has no direct \nphysical link with it. Indeed, Be is not made out of Fe, but of C \nand~O. So to be really conclusive, the studies of spallative \nnucleosynthesis should compare theoretical Be\/O yields to the \ncorresponding abundance ratio in metal-poor stars. Unfortunately, the \ndata are much more patchy for Be as a function of [O\/H] than as a \nfunction of [Fe\/H], especially in very low metallicity stars. The \nusually assumed proportionality between O and Fe could turn out to be \nonly approximate, as recent observational works possibly indicate \n(Israelian et~al. 1998; Boesgaard et~al. 1998; these observations, \nhowever, still ?? need to be confirmed by an independent method, all \nthe more that they come into conflict with several theoretical and \nobservational results; cf. Vangioni-Flam et~al. 1998b). We shall \naddress this question in greater detail in the attending paper \n(Parizot and Drury, 1999, Paper~II).\n\nFinally, we wish to stress that the calculations presented in this \npaper rely on a careful account of the dynamics of the problem. More \ngenerally, time-dependent calculations are required to properly \nevaluate the spallation processes in environments where compositions \nand energy densities are evolving. In particular, as argued in \nParizot (1998), no variation with density can be obtained with a \nstationary model, since an increase in the density induces an \nequivalent and cancelling increase in the spallation rates and the \nenergy loss rates. By contrast, we have shown that all three of the \nprocesses considered here are more efficient at higher density -- a \nresult which could not have been found otherwise. Detailed, numerical \ntime-dependent calculations will be presented in paper~II, with \nconclusions similar to those demonstrated here.\n\n\n\\begin{acknowledgements}\nThis work was supported by the TMR programme of the European Union \nunder contract FMRX-CT98-0168. It was initiated during a visit by LD \nto the Service d'Astrophysique, CEA Saclay, whose hospitality is \ngratefully acknowledged. We wish to thank M. Cass\\'e and Elisabeth \nVangioni-Flam for stimulating discussions of these and related topics.\n\\end{acknowledgements}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section*{Introduction}\n\nA two-dimensional operator $A$ in $SL(2,\\mathbb Z)$ is said to be {\\it\nhyperbolic}, if its eigenvalues are real and distinct. In the\npresent paper we study the connection between periods of\ngeometric continued fraction in the sense of Klein and reduced\noperators, described by J.~Lewis and D.~Zagier in~\\cite{DZg}.\nActually, a determination of a period of the geometric continued\nfraction for the operator $A$ is equivalent to a calculation of a\nperiod of the ordinary continued fraction for the tangent of an\nangle between any eigen straight line of $A$ and the horizontal\naxis.\n\nIn present paper for any period of a geometric continued fraction\nin the sense of Klein we make an explicit construction of a\nreduced hyperbolic operator in $SL(2,\\mathbb Z)$ with the given period\nfor the geometric continued fraction (Theorem~\\ref{th}). It turns\nout that reduced operators naturally forms one-parametric\nfamilies. Further we describe an algorithm to construct a period\nfor an arbitrary operator of $SL(2,\\mathbb Z)$. The base part of the\nalgorithm is to determine a reduced operator that is conjugate to\nthe given (the Gauss Reduction Theory).\n\nIn 1993 V.~I.~Arnold formulated some questions on periods of\ncontinued fractions related to the eigenvectors of the\n$SL(2,\\mathbb Z)$-operators, see for instance in~\\cite{ArnProb}\nand~\\cite{Arn2}. The first studies of these problems are made in\nthe article~\\cite{MPav} by M.~Pavlovskaya, in which the author\nexperimentally investigates statistical questions on geometrical\ncontinued fractions (such as verification of the average length of\nperiods of continued fractions). The questions on the\ndistribution of positive integers in minimal periods of quadratic\ncontinued fractions were studied by M.~O.~Avdeeva and\nB.~A.~Bykovskii in the works~\\cite{Avd1} and~\\cite{Avd2}. In his\nwork~\\cite{ArnAr} V.~I.~Arnold investigates palindromic\nproperties of continued fraction periods and the connection to\nthe integer forms (the complete proof of the palindromic property\nis given by F.~Aicardi and M.~Pavlovskaya). The relation between\n``$-$''-continued fractions of hyperbolic operators and Fuchian\ngroups, and a few words on the algorithm of integer conjugacy\ncheck for the operators in $SL(2,\\mathbb Z)$ is described in the work of\nS.~Katok~\\cite{Kat}. Some estimates of the period lengths for\nordinary continued fractions for $\\sqrt{d}$ are given\nin~\\cite{Hic} by D.~R.~Hickerson.\n\nIn the works~\\cite{Kle1} and~\\cite{Kle2} F.~Klein generalized\ngeometric continued fraction to the multidimensional case of\n$SL(n,\\mathbb Z)$-operators. For more information see the works of\nV.~I.~Arnold~\\cite{Arn2}, E.~I.~Korkina~\\cite{Kor2},\nG.~Lachaud~\\cite{Lac1}, J.-O.~Moussafir~\\cite{Mou2}, the\nauthor~\\cite{Kar1}, etc.\n\nThe work is organized as follows. In the first section we give\ndefinitions of ordinary continued fractions and geometric\ncontinued fractions in the sense of Klein, we show a duality of\nsails of a geometric continued fraction. Further in the second\nsection we construct families of operators possessing the\ngeometric continued fractions (actually the corresponding\nLLS-sequences) with the period $(a_1,\\ldots,a_{2n-1},t)$, where\n$t$ is the parameter of families. Further we give an algorithm of\nperiod calculation for geometric continued fractions. In the\nthird section we formulate some questions and show the results of\na few experiments related to the algorithm.\n\nThe author is grateful to V.~I.~Arnold for constant attention to\nthis work\nand Mathematisch Instituut of Universiteit Leiden for the\nhospitality and excellent working conditions.\n\n\\section{Geometric continued fractions in the sense of Klein}\n\n{\\bf Ordinary continued fractions.} Consider an arbitrary finite\nsequence of integers $(a_0, a_1, \\ldots ,a_n)$, where $a_0$ is\nany integer and $a_i>0$ for $i>0$. The rational\n$$\na_0+1\/(a_1+1\/(a_2+\\ldots)\\ldots))\n$$\nis called the {\\it ordinary continued fraction} and denoted by\n$[a_0: a_1; \\ldots ;a_n]$.\n\nA continued fraction with even (odd) number of elements is called\nan {\\it even $($odd$)$} ordinary continued fraction respectively.\n\n\n\\begin{proposition}\nFor any rational there exist a unique even and a unique odd\ncontinued fractions. \\qed\n\\end{proposition}\n\nFor instance the even and odd ordinary continued fractions for a\nrational $47\/39$ are $[1{:}4;1;7]$ and $[1{:}4;1;6;1]$\nrespectively.\n\n\\vspace{3mm}\n\n{\\bf Sails of hyperbolic operator.} By $[[a,c][b,d]]$ we denote\nthe operator\n$$\\left(\n\\begin{array}{cc}\na&c\\\\\nb&d\\\\\n\\end{array}\n\\right).\n$$\n\nAn operator in $SL(2,\\mathbb R)$ with two distinct real eigenvalues is\ncalled {\\it hyperbolic}.\n\nA vector (segment) is said to be {\\it integer} if it (its\nendpoints) has integer coordinates.\n\nConsider a hyperbolic operator $A$ with no integer eigenvectors.\nThe operator $A$ has exactly two distinct eigen straight lines.\nThese lines do not contain integer points of the lattice distinct\nto the origin. The complement to the union of the lines consists\nof four piece-wise connected components, each of which is an open\noctant. Consider one of these octants. The boundary of the convex\nhull of all integer points except the origin in the closure of\nthe octant is called a {\\it sail} of the operator $A$. The set of\nall sails is called the {\\it geometric continued fraction in the\nsense of Klein} for the operator $A$ (see also the works of\nF.~Klein~\\cite{Kle1}, E.~I.~Korkina~\\cite{Kor2}, and\nV.~I.~Arnold~\\cite{Arn2}). Two sails are said to be {\\it\nequivalent} if there exists a linear lattice-preserving\ntransformation, taking one of the sails to another. Geometric\ncontinued fractions with equivalent sails are called {\\it\nequivalent}.\n\n\\vspace{1mm}\n\n{\\it Remark.} The majority of constructions of this article can\nbe naturally generalized to the case of operators that have\ninteger eigenvectors. For simplicity of the exposition we do not\nconsider such operators in the article.\n\n\\vspace{3mm}\n\n{\\bf LLS-sequence.} An {\\it integer length} of an integer segment\n$PQ$ is the quantity of inner integer points of the segment plus\none, it is denoted by $\\il(PQ)$. Let integer segments $PQ$ and\n$PR$ do not lie on the same straight line. An {\\it integer sine}\nof the angle $QPR$ is an index of the sublattice generated by the\ninteger vectors of the straight lines $PQ$ and $PR$ in the\nlattice of all integer vectors (we denote it by $\\isin (QPR)$).\nAn integer sine of an angle contained in some straight line is\nsupposed to be equivalent to zero.\n\n\nAny sail of the hyperbolic operator $A$ with no integer eigen\nvectors is a two-sides broken line with infinite in both sides\nnumber of segments.\n\n\\begin{definition}\nLet $\\ldots V_{-2}V_{-1}V_0V_1V_2 \\ldots$ be a sail of some\noperator. The infinite in both sides sequence of positive integers\n$$\n(\\ldots,\\il(V_{-2}V_{-1}),\\isin\\angle V_{-2}V_{-1}V_0,\n\\il(V_{-1}V_{0}),\\isin\\angle V_{-1}V_{0}V_1,\n\\il(V_{0}V_{1}),\\isin\\angle V_{0}V_{1}V_2, \\ldots )\n$$\nis called the {\\it LLS-sequence} of the sail (see also\nin~\\cite{KarPrep}).\n\\end{definition}\n\nTwo LLS-sequences are called {\\it equivalent}, if one can be\nobtained from another by shifting on a finite number of elements\nand\/or reversing the order of all elements. The LLS-sequences of\nequivalent sails are equivalent, since all integer lengths and\nsines are invariants under integer-linear transformations.\n\n\\begin{figure}\n$$\\epsfbox{cf.1}$$\n\\caption{Geometric continued fraction of the operator\n$[[7,18][5,13]]$.}\\label{cf.1}\n\\end{figure}\n\nOn Figure~\\ref{cf.1} we show the geometric continued fraction for\nthe operator $[[7,18][5,13]]$. Integer lengths of edges are\ndenoted by black digits, and integer sines --- by white. The\nLLS-sequences of all four sails are equivalent to\n$(\\ldots,2,1,1,3,2,1,1,\\ldots)$.\n\n\\vspace{3mm}\n\n\n{\\bf Duality of sails.} Two sails are called {\\it dual} with\nrespect to each other if their LLS-sequences coincide, the\nsequence of integer lengths of the first sail coincides with the\nsequence of integer sines of the second, and the sequence of\ninteger sines of the first sail coincides with the sequence of\ninteger lengths of the second. (On the duality for\nmultidimensional continued fractions in the sense of Klein see in\nthe book by G.~Lachaud~\\cite{Lac1}.)\n\n\n\\begin{proposition}\\label{st1}\nLet $A$ be a hyperbolic operator with no integer eigenvectors.\nThen the LLS-sequences of all his four octants coincide. Moreover\nthe sails of the opposite octants are equivalent. The sails of the\nadjacent octants are dual.\n\\end{proposition}\n\nIn the proof of Proposition~\\ref{st1} we use the following\nproperties of an integer sine (for further information see\nin~\\cite{KarPrep}):\n$$\n\\isin(PQR)=\\isin(RQP); \\quad \\isin(PQR) = \\isin (\\pi -PQR),\n$$\nwhere by $\\pi {-}PQR$ we denote the angle adjacent to the angle\n$PQR$.\n\nThe statement of Proposition~\\ref{st1} is known to the author from\nV.~I.~Arnold and E.~I.~Korkina and supposed to be classical, still\nits formulation and proof seem to be missing in the literature.\nTo avoid further questions we give the proof here.\n\n\n\\begin{proof}\nThe sails of the opposite octants (and so their LLS-sequences)\nare equivalent, since they are symmetric about the origin.\n\nConsider now the case of adjacent sails. Let now $S$ and $S'$ be\nadjacent sails of the operator $A$, where $S'$ is counterclockwise\nwith respect to $S$ while turning around the origin $O$. Let $S$\nbe a broken line $\\ldots V_{-1}V_0V_{1}\\ldots$ with\ncounterclockwise order of vertices with respect to the origin.\n\n\nLet us show a natural bijection between the edges of the sail $S$\nand vertices of the sail $S'$. Consider an edge $V_{i}V_{i+1}$ of\nthe sail $S$. Denote by $V$ the closest integer point in the\nsegment $V_{i}V_{i+1}$ to the endpoint $V_{i+1}$ and distinct to\n$V_{i+1}$. Denote the integer point $O{+}\\bar{VV_{i+1}}$ by\n$V_i'$. Notice that the point $V_i'$ is in the convex hull of\n$S'$.\n\nSince the triangle $OVV_{i+1}$ does not contain integer points\ninside, the triangle $OV_i'V_{i+1}$ does not contain integer\npoints inside. Hence there is no integer points between the\nparallel lines $OV$ and $V_i'V_{i+1}$. Therefore, $V_i'$ is a\nvertex of the sail $S'$ (see on Figure~\\ref{proof.123}a).\n\n\\begin{figure}\n$$\n\\begin{array}{ccc}\n\\epsfbox{proof.1}&\\epsfbox{proof.2}&\\epsfbox{proof.3}\\\\\n\\hbox{a). Construction}&\\hbox{b). Construction }&\\hbox{c). Calculation}\\\\\n\\hbox{of the vertex $V'_i$.}&\\hbox{of the edge.}&\\hbox{of the sine.}\\\\\n\\end{array}\n$$\n\\caption{Duality of adjacent angles.}\\label{proof.123}\n\\end{figure}\n\n\nFrom the other side $V'$ is a vertex of the sail $S'$. Denote by\n$l$ the closest to the line $OV'$ parallel line containing\ninteger points and intersecting $W$. The line $l$ intersects with\nthe octant containing $S'$ in a ray. Denote by $W'$ the closest to\nthe vertex of the ray integer point of this ray. Then the point\n$V'{+}\\bar{W'V'}$ is in the octant opposite to the octant with\nthe sail $S$ (otherwise $V'$ is not a vertex). Hence the point\n$W'{+}2\\bar{V'O}$ symmetric about the origin to the point\n$V'{+}\\bar{W'V'}$ lies in the octant of the sail $S$. Therefore,\nthe integer points $W'{+}\\bar{V'O}$ and $W'{+}2\\bar{V'O}$ lie in\nthe octant containing $S$. Since $l$ is the closest to the line\n$XV'$ parallel line containing integer points and intersecting\n$S$, the segment with the endpoints $W'{+}\\bar{V'O}$ and\n$W'{+}2\\bar{V'O}$ in contained in the sail (see on\nFigure~\\ref{proof.123}b).\n\nTherefore the above correspondence between the edges\n$V_{i}V_{h+2}$ of the sail $S$ and vertices $V_i'$ of the sail\n$S'$ is a bijection. Moreover the order of vertices $V_i'$ is\nclockwise.\n\nLet us prove now that\n$\\il(V_iV_{i+1})=\\isin(V_{i-1}'V_i'V_{i+1}')$. Denote the point\n$V_i{+}\\bar{OV_i'}$ by~$W$. By construction, the point $W$\nbelongs to the segment $V_iV_{i+1}$ (see\nFigure~\\ref{proof.123}c). Then\n$$\n\\isin(V_{i-1}'V_i'V_{i+1}')=\\isin(\\pi-WV_i'V_{i+1}')=\\isin(WV_i'V_{i+1}')\n=\\il(WV_{i+1}')=\\il(V_iV_{i+1}).\n$$\n\nBy the analogous argumentation the following equalities also hold:\n$$\n\\il(V_i'V_{i+1}')=\\isin(V_{i-1}V_iV_{i+1}).\n$$\n\nTherefore, the LLS-sequences of the sails $S$ and $S'$ coincide,\nmoreover the sequence of integer lengths (sines) of $S$ coincides\nwith the sequence of integer sines (lengths) of $S'$. Hence $S$\nand $S'$ are dual. Proposition~\\ref{st1} is proved.\n\\end{proof}\n\n\\vspace{3mm}\n\n{\\bf Existence and uniqueness of the equivalence classes of\ncontinued fractions with a given LLS-sequences.}\n\n\\begin{definition}\nThe {\\it LLS-sequence} of an operator $A$ is the LLS-sequence for\nany of its sails.\n\\end{definition}\n\n\\begin{proposition}~\\label{st2}\ni$)$. For any infinite in two sides sequence of integers there\nexists a hyperbolic operator whose LLS-sequence coincides with the\ngiven.\n\nii$)$. Two sails with coinciding LLS-sequences are either\nequivalent or dual. All sails dual to the given are equivalent to\neach other.\n\\end{proposition}\n\n\\begin{proof}\nBoth statements follows directly from Corollary~5.11\nof~\\cite{KarPrep} (see also in the work of\nE.~I.~Korkina~\\cite{Kor2}).\n\\end{proof}\n\n\\section{Algebraic sails.}\n\n{\\bf Construction of the operator with the given period.}\nConsider now an algebraic case of hyperbolic operators of the\ngroup $SL(2,\\mathbb Z)$ with irreducible over rationals characteristic\npolynomial. Let $A$ be an integer hyperbolic operator of\n$SL(2,\\mathbb Z)$. Denote by $\\Xi(A)$ the group of operators in all\n$SL(2,\\mathbb Z)$ commuting with $A$ and having positive real\neigenvalues. By Dirichlet unity theorem~\\cite{BSh} the group\n$\\Xi(A)$ is isomorphic to $\\mathbb Z$. Any sail of the operator $A$ is\ninvariant under the action of the group $\\Xi(A)$, moreover the\noperators of the group $\\Xi(A)$ act on the sails by shifting the\nedges of the broken line along thy broken line. The sails of the\noperator $A$ are called {\\it algebraic}.\n\nTherefore LLS-sequences of hyperbolic algebraic operators are\nperiodic. The converse is also true (see Corollary~\\ref{st3}\nbelow).\n\n\\vspace{1mm}\n\n{\\it Remark.} On Figure~\\ref{cf.1} we show the sails of\nhyperbolic algebraic operator $[[7,18][5,13]]$ with a period of\nthe LLS-sequence equals $(2,1,1,3)$.\n\n\\vspace{1mm}\n\n\\begin{theorem}\\label{th}\nConsider an $ST(2,\\mathbb Z)$-operator $[[a,c+\\lambda a][b,d+\\lambda\nb]]$.\n\\\\\ni$)$. Let $a=0$, $b=1$, $d=1$. If $\\lambda>2$ then the operator\n$A$ is hyperbolic and its sails are algebraic. One of the periods\nof the LLS-sequence of the operator $A$ is\n$$\n(1,\\lambda-1).\n$$\nii$)$. Let $b>a \\ge 1$, $0a \\ge 0$ there exists a couple of integers $(c,d)$,\nsatisfying $00$ the\nperiods equal\n$$\n(a'_0,a'_1,\\ldots,a'_{2m},|\\lambda|-2),\n$$\nwhere $[a'_0{:}a'_1;\\ldots;a'_{2m}]$ --- is the odd ordinary\ncontinued fraction for $b\/(b-a)$.\n\n\\vspace{1mm}\n\n\\begin{proof}\nThe discriminant of the characteristic polynomial of the operator\n$A$ equals $((a+\\lambda b+d)^2-4$. Since $\\lambda\\ge 1$, $b\\ge\n1$, $d\\ge 1$, and $a\\ge 0$, the discriminant is nonnegative.\nBesides it equals zero in the exceptional case $a=0$, $b=1$,\n$\\lambda=1$, $d=1$. Therefore the operator $A$ is hyperbolic in\nall cases. Since $t^2-4$ for integer $t>2$ is not a square of some\ninteger, the sails of the operator $A$ are algebraic.\n\nLet us now construct a period for the LLS-sequence. Note that\nboth eigenvalues of the operator $A$:\n$$\n\\frac{a+\\lambda b+d\\pm \\sqrt{((a+\\lambda b+d)^2-4}}{2}\n$$\nare positive, and thus the operator $A$ takes each sail to\nitself. Consider the sail $S$, whose convex hull contains the\npoint $P=(1,0)$.\n\n\\vspace{1mm}\n\nSuppose the operator $A$ is of series i). Then the set of the\nvertices for one of its sails coincides with the set of points\n$A^n(1,0)$ with an integer parameter $n$. Simple calculations\nlead to the result of the theorem.\n\n\\vspace{1mm}\n\nLet now an operator $A$ be an operator of series ii), i.~e.\n$b>a>0$.\n\nDenote by $\\alpha$ the closed convex angle with vertex at the\norigin and edges passing through the points $P$ and $A(P)$.\nConsider the boundary of the convex hull of all integer points\ninside $\\alpha$ except the origin. The boundary consists of two\nrays and a finite broken line. We call the finite broken line in\nthe boundary the {\\it sail of the angle $\\alpha$} and denote it by\n$S_\\alpha$.\n\nNow we show that the sail of the angle $\\alpha$ is completely\ncontained in one of the sails of the operator $A$. Denote by\n$S_\\alpha^\\infty$ the following infinite broken line:\n$$\n\\bigcup\\limits_{i=-\\infty}^{+\\infty} \\Big(A^i(S_\\alpha)\\Big).\n$$\nBy the construction the convex hull of $S_\\alpha^\\infty$ coincides\nwith the convex hull of the sail $S$. It remains to verify if\n$S_\\alpha^\\infty$ coincides with the boundary of its convex hull.\nThe broken line $S_\\alpha^\\infty$ is the boundary of the convex\nhull if the convex angles generated by adjacent edges of the\nbroken line do not contain the origin. To check this it is\nsufficient to study all the angles of one of the periods of the\nbroken line $S_\\alpha^\\infty$, for instance all the angles with\nvertices at vertices of $S_\\alpha$ except the point $A(P)$. All\nconvex angles generated by couples of adjacent edges at inner\nvertices of the sail $S_\\alpha$ do not contain the origin by\ndefinition. It remains to check the angle with vertex at\n$P=(1,0)$.\n\nSince $b\/a>1$, the first edge is parallel to the vector $(0,1)$.\nConsider the second edge $PQ$. Since the triangle $OPQ$ does not\ncontain integer points distinct to the integer points of the\nsegment $PQ$ and the vertex $O$, the segment $PQ$ contains the\npoint with coordinates $(x,-1)$. By the convexity of finite\nbroken line $A^{-1}(S_\\alpha)$ the value of $x$ is determined by\nthe eigen direction:\n$$\n\\left(\\frac{-a+\\lambda b+d+ \\sqrt{((a+\\lambda\nb+d)^2-4}}{2b},-1\\right),\n$$\nnamely,\n$$\nx=\\left\\lfloor\\frac{-a+\\lambda b+d+ \\sqrt{((a+\\lambda\nb+d)^2-4}}{2b}\\right\\rfloor+1,\n$$\nwhere $\\lfloor t\\rfloor$ denotes the maximal integer that does not\nexceeding $t$. As one can show, $x$ is contained in the open\ninterval $\\big( \\lambda{+}1{+}(d{-}1)\/b,\\lambda{+}1{+}d\/b \\big)$.\nBy condition $b\\ge d>0$ we have\n$$\nx=\\lambda+1.\n$$\nHence for $\\lambda>1$ the convex angle at vertex $P$ does not\ncontain the origin. Therefore, the broken line $S_\\alpha^\\infty$\ncoincides with the sail.\n\n\nIn the paper~\\cite{KarPrep} it is shown that the sail $S_\\alpha$\nconsists of $n{+}1$ segments. The integer lengths of the\nconsecutive segments equal $a_0,a_2,\\ldots, a_{2n}$, and the\ninteger sines of the corresponding angles equal $a_1,a_3,\\ldots,\na_{2n-1}$ respectively. Now note, that from the explicit value of\n$x$ it follows that the integer sine for the angle at point $P$\nequals $\\lambda$. Hence the LLS-sequence of the sail $S$ has a\nperiod\n$$\n(a_0,a_1,\\ldots,a_{2n},\\lambda).\n$$\nTherefore the LLS-sequence of the operator $A$ has the prescribed\nperiod.\n\\end{proof}\n\n\n\\begin{corollary}\\label{st3}\nA sail with the periodic LLS-sequence is algebraic $($i.~e. a\nsail of some algebraic hyperbolic operator$)$.\n\\end{corollary}\n\n\\begin{proof}\nIn Theorem~\\ref{th} we constructed the algebraic operators for\nall finite sequences as periods. Then in Proposition~\\ref{st2} we\nshowed that the sails with equivalent LLS-sequences are either\nequivalent or dual. Therefore any sail with periodic LLS-sequence\nis algebraic.\n\\end{proof}\n\n\\vspace{1mm}\n\n{\\it Remark.} Consider some sail with periodic LLS-sequence. Let a\nminimal period of LLS-sequence is even and consists of $2n$\nelements. Then there exists an $SL(2,\\mathbb Z)$-operator $A$ with\npositive eigenvalues, that makes an $n$-edge shift of the sail\nalong the sail. Precisely this operator generates the group\n$\\Xi(A)$ of the sail shifts (see above). Let a minimal period of\nLLS-sequence is odd and consists of $2n{+}1$ elements (in\nparticular this implies that the sail is equivalent to any dual\nsail). Then there exists a $GL(2,\\mathbb Z)$-operator $B$ with negative\ndiscriminant, whose square makes an $(2n{+}1)$-edge shift of the\nsail along the sail. Moreover, the operator $B^2$ generates the\ngroup $\\Xi(T)$.\n\n\\vspace{1mm}\n\n{\\it Remark.} Let us say a few words about non-hyperbolic\noperators in $SL(2,\\mathbb Z)$. It turns out that each of such operators\nis equivalent to exactly one of the operators of the following\nlist:\n\n--- $[[1,1][-1,0]]$;\n\n--- $[[0,1][-1,0]]$;\n\n--- $[[0,1][-1,-1]]$;\n\n--- $[[1,n][0,1]]$, where $n$ is any integer.\n\n\n\n\n\\vspace{3mm}\n\n{\\bf Algorithm of finding a period of the LLS-sequence for a\nhyperbolic algebraic operator.}\n\n\\begin{definition}\nAn operator $[[a,c][b,d]]$ in $SL(2,\\mathbb Z)$ is said to be {\\it\nreduced}, if the following holds: $d> b\\ge a\\ge 0$.\n\\end{definition}\n\n{\\it Remark.} The definition of a reduced operator is slightly\ndifferent to one given in the works~\\cite{DZg} and~\\cite{Man}:\n{\\it an operator in $SL(2,\\mathbb Z)$ is reduced iff it has non-negative\nentries which are non-decreasing downwards and to the right}.\n\nThe main idea of the calculation of the period is to find a\nreduced operator with the sails equivalent to the sails of the\ngiven one. Then it remains to calculate the period of the reduced\noperator by Theorem~\\ref{th}.\n\n\\vspace{1mm}\n\n{\\it Data.} Suppose we know the integer entries of an operator\n$[[a,c][b,d]]$ with unit determinant and positive discriminant.\nLet also the characteristic polynomial does not has roots $\\pm 1$.\nFrom the listed conditions it follows that the operator\n$[[a,c][b,d]]$ is hyperbolic operator in $SL(2,\\mathbb Z)$ with\nirreducible characteristic polynomial.\n\n\\vspace{1mm}\n\n{\\it It is requested} to construct one of the periods of the\nLLS-sequence of the hyperbolic algebraic operator $[[a,c][b,d]]$.\n\n\\vspace{1mm}\n\n{\\bf Description of the algorithm.}\n\n{\\it Step 1.} If $b<0$, then we multiply the operator\n$[[a,c][b,d]]$ by $[[-1,0][0,-1]]$. The LLS-sequence does not\nchange at that.\n\n{\\it Step 2.} So, now the element $b$ is positive. Conjugate the\noperator $[[a,c][b,d]]$ by the operator $[[1,-\\lfloor a\/b\n\\rfloor][0,1]]$. We obtain the operator $[[a',b'][c',d']]$, where\n$0\\le a' \\le b'$.\n\n{\\it Step 3.1.} Suppose $b'=1$, then $a'=0$, $c'=-1$. Moreover we\nhave $|d|>2$, since otherwise the original operator is not\nalgebraic. Therefore a period of the LLS-sequence equals\n$(3,|d|-2)$.\n\n{\\it Step 3.2.1.} Suppose, $b'>1$. If $d'>b'$, then we have found\na reduced operator, now we go to Step~4.\n\n{\\it Step 3.2.2.} Suppose, $b'>1$. If $d'<-b'$, then we conjugate\nby the operator $[[-1,1][0,1]]$. Finally we have the operator\n$[[a'',c''][b'',d'']]$ with $b''=b'$, $a''=b'-a'$, and\n$d''=-b'-d'>0$, further we should go to Step~3.2.1, or to\nStep~3.2.3.\n\n{\\it Step 3.2.3.} Suppose, $b'>1$. The case $|d'|\\le |b'|$. Note\nthat the absolute values of $b'$ and $d'$ do not coincide since\nthe determinant of the operator does not have divisors distinct\nto the unity. Therefore it remains the case $|d|<|b|$. In this\ncase we have:\n$$\n|c'|=\\left|\\frac{a'd'-1}{b'}\\right|\\le\\frac{(b'-1)^2+1}{b'}\\le\nb'-1.\n$$\nWe conjugate the operator $[[a',c'][b',d']]$ with the operator\n$[[0,-1][-1,0]]$ and obtain $[[d',b'][c',a']]$, where $|c'|<|b'|$.\nNow we return back to Step~1 with the obtained operator\n$[[d',b'][c',a']]$.\n\n{\\it Step 4.} We obtained a reduced operator $[[\\tilde a,\\tilde\nc][\\tilde b,\\tilde d]]$, $\\tilde b > 1$ with the LLS-sequence\nequivalent to the LLS-sequence of the original operator. By\nTheorem~\\ref{th} to construct one of the periods of the\nLLS-sequences of the reduced operator we should construct the odd\nordinary continued fraction for $\\tilde b\/\\tilde a$, and find the\ninteger $\\big\\lfloor(\\tilde d{-}1)\/\\tilde b \\big\\rfloor$.\n\n\\section{Some questions and examples}\n\n\n{\\bf A question on complexity of the minimal period.} Note that\nfor any operator there exist finitely many reduced operators with\nthe same trace and LLS-sequence. If we study the reduced operators\nthat make shifts of sails on a minimal possible period, then the\nnumber of such operators coincides with the length of the minimal\nperiod (see also in~\\cite{DZg}). Let $[[a,c][b,d]]$ be a reduced\noperator. We call the integer $b$ --- its {\\it complexity}.\n\n\\begin{problem}\nStudy the minimal complexity for reduced operators with\nLLS-sequence having a length $n$ period $(a_1,\\ldots, a_n)$.\n\\end{problem}\n\n{\\it Remark.} The minimal complexity coincides with the minimal\npositive value of the integer sine of the angles $POQ$, where $O$\nis the origin, $P=(x,y)$ is an arbitrary integer point distinct to\n$O$, and $Q=A(P)$. Therefore the minimal complexity, considered\nas the minimal possible integer sine, is well defined for all\noperators and it is invariant under conjugations.\n\nIf $n$ is even, then the problem is equivalent to finding the\nminimal numerator among the numerators of the rationals:\n$$\n[a_1{:}\\ldots;a_{n-1}], \\quad [a_2{:}\\ldots;a_{n}], \\quad\n[a_3{:}\\ldots;a_{n};a_1], \\quad \\ldots \\quad\n,[a_n{:}a_1;\\ldots;a_{n-2}].\n$$\n\n\\begin{example} Let the period contains two elements: $(a,b)$, where $aa$\nexcept for the case $a=b=c=d$. Then the rational with the minimal\nnumerator can be found from the following table.\n\n\\begin{center}\n\\begin{tabular}{|c||c|}\n\\hline\n$(a,b,c,d)$ & Rational with \\\\\n$d\\ge a,b,c$ &the minimal numerator \\\\\n\\hline \\hline\n$d> a,b,c$ &$[a{:}b;c]$ \\\\\n$d=c;ba,c$ &$[a{:}b;c]$ and $[c{:}d;a]$ \\\\\n$d=c=b;d>a$ &$[a{:}b;c]$ and $[c{:}d;a]$ \\\\\n$d=c=b=a$ &all \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{example}\n\nIf $n$ is odd, then the problem is equivalent to finding the\nminimal numerator among the numerators of the rationals (we define\n$a_{n+k}=a_k$):\n$$\n[a_1{:}\\ldots;a_{2n-1}], \\quad [a_2{:}\\ldots;a_{2n}], \\quad\n[a_3{:}\\ldots;a_{2n};a_1], \\quad \\ldots \\quad\n,[a_{2n}{:}a_1;\\ldots;a_{2n-2}].\n$$\n\n\\begin{example}\nLet the period consists of three elements: $(a,b,c)$, where $c\\ge\na,b$. Then the fraction $[a{:}b;c;a;b]$ has the minimal numerator\n(or of one of some equivalent minimal numerators in the case of\n$a=c$ or $b=c$).\n\\end{example}\n\n\\vspace{1mm}\n\nOne can suppose that we should skip one of the maximal elements\nof the period, but that is not true for the six element sequence:\n$(1,4,5,4,1,4)$. The minimum of the numerators is attained at the\nfraction $[1:4;5;4;1]$, and not at the fraction $[4:1;4;1;4]$.\n\n\\vspace{3mm}\n\n{\\bf On frequencies of occurrences of the reduced operators.}\nFirst we describe a proper probabilistic space. Let\n$P=(a_1,a_2,\\ldots,a_{2n-1},a_{2n})$ be some period. Denote by\n$\\#_N(P)$ the quantity of all operators satisfying the following\nconditions:\\\\\n{\\it i}). The absolute value of any entry of the operator does not\nexceed $N$.\n\\\\\n{\\it ii}). The sequence $P$ is one of the periods of SL-sequence\nfor the operator.\n\\\\\n{\\it iii}). Starting from the operator, the algorithm of the\nprevious section constructs the reduced operator $[[a,b][c,d]]$,\nwhere $(a,b)=(0,1)$ for the case $P=(1,a_2)$; and\n$$\nb\/a=[a_1:a_2;\\ldots;a_{2n-1}]\n$$\nin the remaining cases.\n\nThen one studies the relative statistics of $\\#_N(P)$ while $N$\ntends to infinity. The following questions are of interest.\n\n\\begin{problem}\n{\\bf a).} Which one of the reduced operators with a given trace\n(or with a fixed LLS-sequence) is the most frequent as a result of\nthe algorithm of the previous section?\\\\\n{\\bf b).} What is the probability of that? \\\\\n{\\bf c).} Is it true that the maximal possible probability is\nattained at reduced operators with minimal complexity?\n\\end{problem}\n\n\nIn Table~1 we give some results of calculation of $\\#_{25000}(P)$\nfor the operators with small absolute value of the trace. We\nremind that the minimal absolute value of the trace of hyperbolic\n$SL(2,\\mathbb Z)$-operator equals $3$.\n\nIt is interesting to note that $SL(2,\\mathbb Z)$-operators corresponding\nto $P=(1,2)$ are more frequent than the $SL(2,\\mathbb Z)$-operators\ncorresponding to $P=(1,1)$. This occurs since the sails whose\nLLS-sequences has the period $(1,1)$ are equivalent to their\nduals. If we enumerate the operators with multiplicities\nequivalent to the number of equivalent sails for the operators\nthen we get:\n$$\n4\\#_{25000}(1,1)>2\\#_{25000}(1,2)+2\\#_{25000}(2,1).\n$$\n\n\n\\begin{table}\\label{Tab25000}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\nAbsolute value &Notation for classes &Period & Operator & Value of \\\\\nof the trace &of equivalent operators &$P$ &$[[a,b][c,d]]$&$\\#_{25000}(P)$ \\\\\n\\hline \\hline\n3 & $L_3$& $(1,1)$ & $[[0,1][-1,3]]$ & $663160$\\\\\n\\hline\n4 & $L_{4}$ & $(1,2)$ & $[[0,1][-1,4]]$ & $834328$\\\\\n & & $(2,1)$ & $[[1,2][1,3]]$ & $304776$\\\\\n\\hline\n5 & $L_{5}$ & $(1,3)$ & $[[0,1][-1,5]]$ & $818200$\\\\\n & & $(3,1)$ & $[[1,3][1,4]]$ & $194528$\\\\\n\\hline\n6 & $L_{6,1}$& $(1,4)$ & $[[0,1][-1,6]]$ & $777128$ \\\\\n & & $(4,1)$ & $[[1,4][1,5]]$ & $141784$ \\\\\n\\cline{2-5}\n & $L_{6,2}$& $(2,2)$ & $[[1,2][2,5]]$ & $446432$\\\\\n\\hline\n7 & $L_{7,1}$& $(1,5)$ & $[[0,1][-1,7]]$ & $734904$\\\\\n & & $(5,1)$ & $[[1,5][1,6]]$ & $110848$\\\\\n\\cline{2-5}\n & $L_{7,2}$& $(1,1,1,1)$ & $[[2,3][3,5]]$& $201744$\\\\\n\\hline\n8 & $L_{8,1}$ &$(1,6)$ & $[[0,1][-1,8]]$ & $695560$\\\\\n & &$(6,1)$ & $[[1,6][1,7]]$ & $90688$ \\\\\n\\cline{2-5}\n & $L_{8,2}$ &$(2,3)$ & $[[1,2][3,7]]$ & $435472$\\\\\n & &$(3,2)$ & $[[1,3][2,7]]$ & $310872$\\\\\n\\hline\n9 & $L_{9}$ &$(1,7)$ & $[[0,1][-1,9]]$ & $660984$\\\\\n & &$(7,1)$ & $[[1,7][1,8]]$ & $76552$ \\\\\n\\hline\n10 & $L_{10,1}$ & $(1,8)$ & $[[0,1][-1,10]]$ & $630592$\\\\\n & & $(8,1)$ & $[[1,8][1,9]]$ & $66064$ \\\\\n\\cline{2-5}\n & $L_{10,2}$ & $(2,4)$ & $[[1,2][4,9]]$ & $408216$\\\\\n & & $(4,2)$ & $[[1,4][2,9]]$ & $239712$\\\\\n\\cline{2-5}\n & $L_{10,3}$ &$(1,1,1,2)$ & $[[2,3][5,8]]$ & $260872$\\\\\n & &$(2,1,1,1)$ & $[[2,5][3,8]]$ & $114084$\\\\\n & &$(1,2,1,1)$ & $[[3,4][5,7]]$ & $149832$\\\\\n & &$(1,1,2,1)$ & $[[3,5][4,7]]$ & $114084$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Values of $\\#_{25000}(P)$ for the operators with small\nabsolute values of the traces.}\n\\end{table}\n\nIn conclusion we formulate the following question. Denote by\n$GK(P)$ the probability of the sequence\n$P=(a_1,a_2,\\ldots,a_{2n-1})$ in the sense of Gauss-Kuzmin:\n$$\nGK(P)=\\frac{1}{\\ln(2)}\\ln \\left(\n\\frac{(\\alpha_1+1)\\alpha_2}{\\alpha_1 (\\alpha_2+1)}\\right),\n$$\nwhere $\\qquad \\alpha_1=[a_1{:}a_2;\\ldots ;\na_{2n-2};a_{2n-1}],\\qquad \\alpha_2=[a_1{:}a_2;\\ldots ;\na_{2n-2};a_{2n-1}{+}1]$.\n\n\\begin{problem}\nLet\n$$\n\\begin{array}{ll}\nP_1=(a_1,a_2,\\ldots a_{2n-1},a_{2n}),& P_1'=(a_1,a_2,\\ldots\na_{2n-1}),\\\\\nP_2=(a_2,a_3,\\ldots a_{2n},a_{1}),& P_2'=(a_2,a_3,\\ldots a_{2n}).\\\\\n\\end{array}\n$$\n{\\it Is the following true}:\n$$\n\\lim\\limits_{n\\to\n\\infty}\\frac{\\#_n(P_1)}{\\#_n(P_2)}=\\frac{GK(P_1')}{GK(P_2')}\\hbox{\n?}\n$$\n\\end{problem}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}