diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzmdwe" "b/data_all_eng_slimpj/shuffled/split2/finalzzmdwe" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzmdwe" @@ -0,0 +1,5 @@ +{"text":"\\section*{Introduction}\nB\\'ezier curves are parametrized curves extensively used in computer aided geometric design (CAGD) \\cite{Mortenson}. B\\'ezier curves are related to Berenstein polynomials, they have very interesting mathematical properties and the literature contains many proposals of algorithms for their algebraic construction and their visualization \\cite{Farin,winkel}. In this work, we will focus on quadratic B\\'ezier curves. Since a quadratic B\\'ezier curve may be described as the envelope of a family of segments whose endpoints move along two straight lines, then it is conceivable to replace this family of lines with another suitable family of lines whose corresponding endpoints are close to each other. When the quadratic B\\'ezier curve is relative to the points $(p,q)$, $(0,0)$ and $(q,p)$, it turns out that such replacement lines may be characterized as having as endpoints pairs of B\\'ezout coefficients of coprime numbers close to the points $(p,q)$ and $(q,p)$, respectively. The main observation that leads to the verification of this statement is fairly simple: Denote $P_0=(p,q)$, $P_1=(0,0)$ and $P_2=(q,p)$ and suppose that the coprime pair of positive numbers $(r,s)$ is close to the pair $(p,q)$. Then, the B\\'ezout coefficients $Q_1$ and $Q_2$ of the two coprime pairs $(r,s)$ and $(s,r)$, respectively, are close to their projections $R_1$ and $R_2$ on the lines $P_0P_1$ and $P_1P_2$, respectively. Moreover, the distance from $R_1$ to $P_0$ is approximately the same as the distance from $R_2$ to $P_1$. Hence, the line with endpoints $Q_1$ and $Q_2$ is the candidate to replace a line in the envelope of the quadratic B\\'ezier curve associated to $P_0$, $P_1$ and $P_2$. When the norm of $(p,q)$ is sufficiently large, there will be many coprime pairs in a neighbourhood of $(p,q)$. Thus, when plotting all such lines with endpoints $Q_1$ and $Q_2$, they will approximate the quadratic B\\'ezier curve corresponding to $P_0$, $P_1$ and $P_2$.\n\n\n\n\n\nThis work is divided in two sections. In Section 1 we establish the notation and state the elementary results needed in the rest of the paper and Section 2 contains the main results. \n\n\\section{B\\'ezier Quadratics}\n\nThe linear B\\'ezier curve for two points ${P}_0$ and ${P}_1$ in the plane, is defined as just the parametrized line $t\\mapsto (1-t) {P}_0+t {P}_1$ from $P_0$ to $P_1$.\nGiven three different points $ {P}_0$, $ {P}_1$ and $ {P}_2$ in the plane, the quadratic B\\'ezier curve is a parametrized curve in terms of the linear B\\'ezier curves from $ {P}_0$ to $ {P}_1$ and from $ {P}_1$ to $ {P}_2$, that is,\nthe quadratic B\\'ezier curve for the three points $ {P}_0$, $ {P}_1$ and ${P}_2$ in the Cartesian plane, is the parametric curve: \n\\begin{align*}\nt&\\mapsto (1-t)\\left((1-t) {P}_0+t {P}_1\\right)+t\\left((1-t) {P}_1+t {P}_2\\right) \n=(1-t)^{2} {P} _{0}+2(1-t)t {P} _{1}+t^{2} {P} _{2}.\n\\end{align*}\n\n\\begin{definition}\\label{Bezier}\nGiven a pair of nonnegative integers $(p,q)$, we will denote\nby $\\alpha_{p,q}$ and $\\beta_{p,q}$ the linear B\\'ezier curves corresponding to the pairs of points $(p,q)$, $(0,0)$ and $(0,0)$, $(q,p)$, respectively. That is\n\\begin{equation*}\n \\alpha_{p,q}(t)=(1-t)(p,q)\\hspace*{1cm}\\text{ and }\\hspace*{1cm}\n \\beta_{p,q}(t)=t(q,p).\n\\end{equation*}\nIn addition, for each $0\\leq s \\leq 1$ we will denote by $\\gamma_{s}$ the parametric line through the points $\\alpha_{p,q}(s)$ and $\\beta_{p,q}(s)$. Therefore\n\\begin{equation*}\n \\gamma_s(t)=(1-t)\\alpha_{p,q}(s)+t\\beta_{p,q}(s).\n\\end{equation*}\nThe quadratic B\\'ezier curve \n$c_{p,q}$ corresponding to $P_0=(p,q)$, $P_1=(0,0)$ and $ P_2=(q,p)$, is then defined as\n\\begin{equation*}\n c_{p,q}(t)=(1-t)\\alpha_{p,q}(t)+t\\beta_{p,q}(t)= (1-t)^2(p,q)+t^2(q,p){\\mbox{ , }}0\\leq t\\leq 1.\n\\end{equation*}\n\\end{definition}\n\nFor each $0< t_0< 1$, it is well known that a parametrization of the straight line tangent to $c_{p,q}(t)$ at the point $c_{p,q}(t_0)$, when $t=t_0$, is the line $\\gamma_{t_0}$. Indeed, a parametrization of such a tangent line is:\n\\begin{align*}\n c_{p,q}(t_0)+\\dfrac{1}{2}(t-t_0)c^{\\prime}_{p,q}(t_0)&= \n c_{p,q}(t_0)+(t-t_0)\\left(-(1-t_0)(p,q)+t_0(q,p)\\right)\\\\\n &=(1-t_0)\\left((1-t_0)-(t-t_0)\\right)(p,q)+t_0\\left(t_0+(t-t_0)\\right)(q,p)\\\\\n &=(1-t)\\gamma_{p,q}(t_0)+t\\beta_{p,q}(t_0)\\\\\n &=\\gamma_{t_0}(t).\n\\end{align*}\n\nIn fact, $c_{p,q}$ may be described as the envelope of the family of lines $\\gamma_s$.\n\n\n\nThe following notion was introduced as Definition~1\nin \\cite{ILM}.\n\\begin{definition}\\label{defbezout}\nGiven a pair of positive coprime integers $(p,q)$, we define the {\\em B\\'ezout coefficients} of $(p,q)$ \nas the unique pair of coprime numbers, denoted $\\mathcal B(p,q)=(a,b)$, such that $00$, then $\\|\\gamma_1(t)-\\gamma_2(t)\\|<\\epsilon$, for every $0\\leq t\\leq 1$.\n\\end{proposition}\n\\begin{proof}\n Assume that $\\mathrm{dist}(L_1,L_2)<\\epsilon$. Let $0\\leq t\\leq 1$. Then\n \\begin{align*}\n \\|\\gamma_1(t)-\\gamma_2(t)\\|&\\leq\n (1-t)\\|A_1-A_2\\|+t\\|B_1-B_2\\|\\\\\n &=(1-t)\\min\\{ \\|A_1-A_2\\|, \\|A_1-B_2\\|\\} +\n t\\min\\{ \\|B_1-A_2\\|,\\|B_1-B_2\\|\\}\\\\\n &\\leq (1-t)\\mathrm{dist}(L_1,L_2)\n +t\\, \\mathrm{dist}(L_1,L_2)\\\\\n &<\\epsilon.\n \\end{align*}\n\\end{proof}\n\n\\section{Approximating a quadratic B\\'ezier curve}\n\nIn this section we prove our main result, namely, that a quadratic B\\'ezier curve $c_{p,q}$ as in Definition~\\ref{Bezier} can be approximated through a family of B\\'ezier-B\\'ezout line segments given\nin Definition~\\ref{bezoutlines}.\n\n\n\n\n\n\nFor integers $p>3$ and $0\\leq q 3$ and $0\\leq q3$ and $0\\leq q\\tilde{T}$ and $\\tilde{m}(T)=\\bar{m} \\tilde{T}$ for $T<\\tilde{T}$ and where $\\tilde{T}$ is the temperature where the thermal mass should be linear ($\\tilde{T}\/T_{\\text{c}} \\sim 2-3$). Consequently we obtain\n\\begin{eqnarray}\n\t\\partial_{T\/T_{\\text{c}}}(e_0 T^{-4})|_{T\/T_{\\text{c}}\\simeq 1}&\\le& \\left.\\frac{d}{2\\pi^2}M \\bar{m} \\frac{\\tilde{T}}{T_{\\text{c}}} \\left(\\frac{T_{\\text{c}}}{T}\\right)^2\\right|_{T\/T_{\\text{c}}\\simeq 1} \\nonumber \\\\\n\t&\\lesssim& 1.3 \\frac{\\tilde{T}}{T_{\\text{c}}} \\lesssim 4,\n\\end{eqnarray}\nsince $T\/T_{\\text{c}}\\simeq 1$, $\\tilde{T}\/T_{\\text{c}} \\simeq 2-3 $, $\\bar{m}\\simeq 1$ and $M \\simeq 1.6$. This upper bound on the derivative of the energy is much lower than the value given by the lattice data. Large values for this derivative can only be obtained if $\\partial_T m(T)< 0$ near the critical temperature.\n\nIn consequence, we have analytically shown that an agreement with lattice QCD can be obtained provided that \n\\begin{equation}\\label{manaly}\n\t\\partial_T m(T\\gtrsim T_{\\text{c}})<0,\\quad{\\rm and}\\quad m(T\\gg T_{\\text{c}})=0.973\\, T.\n\\end{equation}\n\n\\subsection{Numerical results}\n\\label{ssec:fitm}\n\nIn the previous section, we have shown that the shape of $m(T)$ is rather constrained by the lattice data within the frameworks of Models 1 and 2. Let us now explicitly extract numerically $m(T)$ from these data. \n\nWe begin by considering Model 1, where the form of the energy density is preserved and consequently given by Eq.~(\\ref{energy0c}) in which the dispersion relation~(\\ref{disper}) is chosen. At a given temperature, $T^*$, the thermal gluon mass, $m(T^*)$, can be obtained by numerically solving the equation $e^{(1)}(T^*,m(T^*))=e_0(T^*,m(T^*))=e^{{\\rm lat}}(T^*)$, where $e^{{\\rm lat}}(T^*)$ is the lattice energy density at the considered temperature. The computed gluon mass is plotted in Fig.~\\ref{fig2} and can be well fitted by the following form\n\\begin{subequations}\\label{mfit1}\n\\begin{equation}\\label{mmodel1}\n\t\\frac{m^{(1)}(T)}{T_{\\text{c}}}=m_0\\, \\frac{T}{T_{\\text{c}}}+\\frac{m_1}{(T\/T_{\\text{c}}-m_2)^{m_3}},\n\\end{equation}\nwith\n\\begin{eqnarray}\n\tm_0&=&0.873,\\quad m_1=0.612,\\nonumber\\\\\n\tm_2&=&0.983,\\quad m_3=0.411.\n\\end{eqnarray}\n\\end{subequations}\n\n\\begin{figure}[b]\n\\includegraphics*[width=\\columnwidth]{fig2.eps}\n\\caption{(Color online) Thermal gluon masses obtained by fitting Model 1 (circles) and Model 2 (triangles) to the lattice data of Ref.~\\cite{boyd95}, see Fig.~\\ref{fig01}. Models 1 and 2 are defined by Eqs.~(\\ref{model1}) and (\\ref{model2}) respectively, with the dispersion relation~(\\ref{disper}). The fitted forms~(\\ref{mfit1}) and (\\ref{mfit2}) are also plotted (solid lines).}\n\\label{fig2}\n\\end{figure}\n\nThe observation of Fig.~\\ref{fig2} and of the fitted form~(\\ref{mfit1}) clearly shows the different behaviors predicted in Sec.~\\ref{ssec:genem}. First, the linear increase of $m(T)$ is obvious for $T\/T_{\\text{c}}\\geq 2.5$, and corresponds to the region III in Fig.~\\ref{fig2}. However, the slope $m_0$ differs from the asymptotic value of $0.973$ predicted in the previous section by about 10\\%. This can be understood by remarking that $m_3$ is rather small while $m_1$ is of the same order of magnitude than $m_0$: The term supplementing the linear one in Eq.~(\\ref{mmodel1}) still brings a non negligible contribution at large temperature, causing the fitted slope $m_0$ to be smaller than in the case of a genuine linearly rising mass, see Eq.~(\\ref{manaly}). Second, $m(T)$ strongly decreases for $T\/T_{\\text{c}}\\simeq 1.0-1.2$, corresponding to region I in Fig.~\\ref{fig2}. The fitted form we get is actually singular near the critical temperature, the parameter $m_3$ playing the role of a critical exponent. Third, there exists an intermediate zone between the singular and the linear behaviors, in which $m(T)$ reaches a minimum. This zone corresponds to region II in Fig.~\\ref{fig2}. \n\nIn Model 2, the form of the energy density is no longer preserved as in Model 1, but the entropy density is so. That is why the thermal gluon mass at a given temperature, $T^*$, can be numerically computed in a very similar way by solving the equation $s^{(2)}(T^*,m(T^*))=s_0(T^*,m(T^*))=s^{{\\rm lat}}(T^*)$. We recall that $s_0$ is given by Eq.~(\\ref{entropy0c}). The computed thermal gluon mass is plotted in Fig.~\\ref{fig2} and can be accurately fitted by the following form \n\\begin{subequations}\\label{mfit2}\n\\begin{equation}\\label{mmodel2}\n\t\\frac{m^{(2)}(T)}{T_{\\text{c}}}=k_0\\, \\frac{T}{T_{\\text{c}}}+\\frac{k_1}{(T\/T_{\\text{c}}-k_2)^{k_3}},\n\\end{equation}\nwith\n\\begin{eqnarray}\n\tk_0&=&0.724,\\quad k_1=0.982,\\nonumber\\\\\n\tk_2&=&0.973,\\quad k_3=0.345.\n\\end{eqnarray}\n\\end{subequations}\nEquation~(\\ref{mfit2}) is formally equivalent to (\\ref{mfit1}); only the values of the numerical coefficients are slightly different. Thus the same comments as for Model 1 can be done. It is worth noting that the regions where the linear increase and strong decrease occur are identical within Models 1 and 2. A physical interpretation of the thermal gluon mass can thus be given independently of the considered Model. \n\\begin{figure}[ht]\n\\includegraphics*[width=\\columnwidth]{fig3.eps}\n\\caption{(Color online) Interaction measure of the gluon plasma computed with Model 1 (solid line) and Model 2 (dashed line), and compared to the lattice data of Ref.~\\cite{boyd95} (circles). Model 1 is defined by Eqs.~(\\ref{model1}) with the gluon mass~(\\ref{mfit1}) and Model 2 is defined by Eqs.~(\\ref{model2}) with the gluon mass~(\\ref{mfit2}). The dispersion relation~(\\ref{disper}) and the values~(\\ref{betas}) for the integration constants are used.}\n\\label{fig3}\n\\end{figure}\n\\begin{figure}[ht]\n\\includegraphics*[width=\\columnwidth]{fig4.eps}\n\\caption{(Color online) Same as Fig.~\\ref{fig01}, but the results obtained with Model 1 are also plotted for comparison. Model 1 is defined by Eqs.~(\\ref{model1}) with the gluon mass~(\\ref{mfit1}), the dispersion relation~(\\ref{disper}), and the value~(\\ref{betas}) for the integration constant.}\n\\label{fig4}\n\\end{figure}\n\nThe thermal gluon mass has been fitted on one of the three thermodynamical quantities available in lattice QCD: energy density for Model 1, and entropy density for Model 2. The remaining quantities can now be numerically computed within both Models by using Eqs.~(\\ref{model1}) and (\\ref{model2}) with the dispersion relation~(\\ref{disper}), provided that the integration constants ensuring an optimal agreement with lattice QCD are known. The following fitted values\n\\begin{equation}\\label{betas}\nT_{\\text{c}}\\, \\beta^{(1)}_\\star=0.435,\\quad {\\rm and}\\quad T_{\\text{c}}\\, \\beta^{(2)}_\\star=0.445,\n\\end{equation}\nlead to an excellent agreement with the available lattice data, as it can be seen in Figs.~\\ref{fig3} and \\ref{fig4}. \n\nThe interaction measures computed with Model 1 and Model 2 are quasi indistinguishable from each other and from the lattice data. Again, both formalisms lead to nearly identical results. That is why, for clarity, we have only plotted the results of Model 1 in Fig.~\\ref{fig4}: The curves computed with Model 2 would have been indistinguishable from those of Model 1. Notice that in our approach, the thermal gluon mass is fitted so that the asymptotic behavior of the interaction measure corresponds to lattice QCD, that is $e-3p\\propto T^2$ (see Fig.~\\ref{fig02}). Such a quadratic increase is compatible with previous theoretical results~\\cite{pisa06} and with the more recent unquenched lattice study~\\cite{chen07}. It is worth mentioning that other approaches rather favor $e-3p\\propto T$~\\cite{zwan04,mill06,giac07}. The interaction measure is thus a quantity that would deserve further studies since there is not yet a general agreement concerning its asymptotic growing.\n\nThe values we find for the integration constants are almost equal: Their average value is $T_{\\text{c}}\\, \\bar\\beta=0.44$, corresponding to the temperature $\\bar T\/T_{\\text{c}}\\simeq 2.27$. This is the typical temperature at which $m^{(1)}(\\bar T)=m^{(2)}(\\bar T)$, as it can be seen in Fig.~\\ref{fig2}. It is not a coincidence: Models 1 and 2 are designed to reproduce the same data. Then if $m^{(1)}(\\bar T)=m^{(2)}(\\bar T)$, the only way for both Models to give identical results is to have $B^{(i)}(\\bar\\beta)=0$, thus $\\beta^{(1)}_\\star=\\beta^{(2)}_\\star=\\bar \\beta$, in rough agreement with the fitted values~(\\ref{betas}). The integration constant is thus not really a free parameter since its value can constrained by the thermal gluon mass once Model 1 and Model 2 are compared.\n\nFinally, it is important to stress that the terms $B^{(i)}$, given by Eqs.~(\\ref{b1}) and (\\ref{b2}), are not small corrections as one could have thought. Without these terms, even the qualitative behavior of the various thermodynamical quantities is wrong. \n\n\\subsection{Color interactions above $T_{\\text{c}}$}\n\\label{ssec:scenar}\n\nWe have considered up to now that the gluon plasma is an ideal boson gas, where gluons are transverse and free but have a temperature-dependent mass $m(T)$. From Sec.~\\ref{ssec:genem} we can conclude that reproducing the lattice QCD results demands first that $m(T\\gg T_{\\text{c}})\\sim T$ and second that $m(T\\gtrsim T_{\\text{c}})$ decreases fast enough (see Fig.~\\ref{fig2}). Such a nontrivial behavior can be intuitively explained by invoking color interactions above $T_{\\text{c}}$. It is indeed widely accepted that, at the critical temperature $T_{\\text{c}}$, the medium undergoes a phase transition and becomes deconfined. It does not mean however that the color interactions vanish: They are actually screened because of the great amount of color charges in the medium, and the residual potential is no longer confining as for $T=0$. These residual color interactions can be rather important for $T\/ T_{\\text{c}}\\simeq1-2$, as suggested by several lattice QCD studies~\\cite{pet05}. \n\nOne can think about the mean field approximation to have a first guess about the influence of screened color interactions. In this picture, the gluon dispersion relation should be modified as follows:\n\\begin{equation}\n\\label{disper2}\n\t\\sqrt{k^2+m^2(T)}=\\sqrt{k^2+\\bar m^2(T)}+\\bar V(T),\n\\end{equation}\nwhere $\\bar V(T)$ is the effective mean potential energy felt by a gluon, and where $\\bar m(T)$ is \\textit{a priori} different of $m(T)$. When $T$ becomes very large, it can reasonably be assumed that $\\bar V(T)$ vanishes. Consequently, one has $\\bar m(T\\gg T_{\\text{c}})=m(T\\gg T_{\\text{c}})=\\bar m\\, T$. Our main physical assumption is then the following: Since $\\bar m(T)$ is the thermal gluon mass in a temperature range when the gluons are free, it can be seen as the rest mass of a free gluon for any $T\\geq T_{\\text{c}}$. Let us now consider that $\\bar V(T\\gtrsim T_{\\text{c}})\\gg \\bar m(T\\gtrsim T_{\\text{c}})$, \\textit{i.e.} that the color interactions become dominant near $T_{\\text{c}}$. Then, by squaring Eq.~(\\ref{disper2}), one gets $m^2(T)=\\bar m^2(T)+\\bar V^2(T)+2\\sqrt{k^2+\\bar m^2(T)}\\bar V(T)$ and consequently $m^2(T\\gtrsim T_{\\text{c}})\\approx \\bar V^2(T\\gtrsim T_{\\text{c}})$ since the potential term dominates the right hand side.\n\nWe are thus led to the following interpretation for $m(T)$: At large $T$ (region III in Fig.~\\ref{fig2}), $m(T)$ tends to the rest mass of a free gluon in the gluon plasma according to the perturbative QCD result. But near $T_{\\text{c}}$ (region I in Fig.~\\ref{fig2}), the behavior of $m(T)$ is dominated by the existence of non negligible screened color interactions. Region II in Fig.~\\ref{fig2} is finally a transition regime in which these interactions progressively vanish. According to this scenario, screened color interactions play an important role for $T\/T_{\\text{c}}\\simeq1-2$, and one can wonder whether glueballs can form or not at these temperatures. The next section is devoted to answer to this question. Notice that the possible glueball formation above $T_{\\text{c}}$ has already been suggested in Ref.~\\cite{castor07} as a mechanism explaining the sudden increase of the effective gluonic degrees of freedom near $T_{\\text{c}}$ that is observed in this last work.\n\n\\section{Existence of glueballs above $T_{\\text{c}}$}\n\\label{sec:glue}\n\n\\subsection{Effective Hamiltonian for glueballs}\n\\label{ssec:hamglu}\n\nIn a constituent gluon (or quasiparticle) picture such as the one we develop here, a glueball is a bound state of at least two gluons. Let us focus on two-gluon glueballs. Being the lightest and presumably the most strongly bound ones, they should be the easiest glueballs to be produced in the gluon plasma. In a deconfined medium, a binary gluon state may exist in several colored configurations following the decomposition $\\bm 8\\otimes\\bm 8=\\bm 1\\oplus\\bm 8\\oplus\\bm 8\\oplus\\bm{10}\\oplus\\overline{\\bm{10} }\\oplus\\bm{27}$. As the strength of color interactions is proportional to the color Casimir operator of the gluon pair, the last three configurations are irrelevant as far as glueball formation is concerned since they lead to interactions which are either vanishing ($\\bm{10},\\ \\overline{\\bm{10}}$) or repulsive $\\bm{27}$~\\cite{shur04,boul08}. However, both the singlet and octet configurations lead to attractive interactions, these interactions in the singlet channel being twice as large as in the octet one. The most favored glueball from an energetic point of view is thus a two-gluon bound state with the gluon pair in a color singlet. The dynamics of the gluon pair also comes into play at this stage: The most strongly bound gluon pairs will be those with a minimal value of the radial quantum number ($n=0$) and of the orbital angular momentum. If the gluons were longitudinal the minimal value of the square orbital angular momentum would be $\\left\\langle \\bm L^2\\right\\rangle=0$ for the $0^{++}$ state. However, we have seen that the large-$T$ behavior of the gluon plasma is compatible with transverse gluons. In this case, $\\left\\langle \\bm L^2\\right\\rangle=2$ is the minimal allowed value, corresponding to the $0^{\\pm+}$ glueballs as shown in Ref.~\\cite{math08}. \n\nDenoting the static potential between a color-singlet quark-antiquark pair by $V(r,T)$, a relativistic Hamiltonian describing the aforementioned lightest glueballs is the following spinless Salpeter one\n\\begin{equation}\\label{hamglu}\n\tH_G=2\\left.\\sqrt{\\bm p^2+\\bar m^2(T)}\\right|_{\\left\\langle \\bm L^2\\right\\rangle=2}+\\frac{9}{4}\\, V(r,T),\n\\end{equation}\nwhere $\\bm p^2=p^2_r+\\left\\langle \\bm L^2\\right\\rangle\/r^2$ and where\n\\begin{equation}\n\\frac{\\bar m(T)}{T_{\\text c}}=\\bar m\\, \\frac{T}{T_{\\text c}}\t=0.973\\, \\frac{T}{T_{\\text c}}\n\\end{equation}\nis the free gluon mass introduced in the previous section. The $9\/4$ factor comes from the color Casimir operator. Such a Casimir scaling for the static energy between sources in various color representation has been confirmed by the lattice study~\\cite{colo}. We choose for $\\bar m$ the value that reproduces that saturation value of the thermodynamical quantities following the analysis of Sec.~\\ref{ssec:genem}. Notice that, since $T_{\\text c}$ is estimated to be around $270$~MeV by pure glue lattice calculations~\\cite{boyd}, one has\n\\begin{equation}\n\\bar m(T)=0.263\\, \\frac{T}{T_{\\text c}}~\\ {\\rm GeV}.\n\\end{equation}\n\nWe point out that building an effective glueball Hamiltonian by starting from a best-known quark-antiquark one has already led to a successful description of glueballs at $T=0$~\\cite{math08}. That is why we find relevant to apply it in this case also. A last remark has to be done: Our framework leads by construction to the same mass for the scalar and pseudoscalar glueballs. At $T=0$ this degeneracy can be lifted by the introduction of instanton-induced forces~\\cite{math08}. Such forces are not taken into account by the present model and one can expect that the ground state of Hamiltonian~(\\ref{hamglu}), whose mass is denoted $M_G(T)$, is rather an average mass of the scalar and pseudoscalar glueballs. Although the current understanding of this topic is far from being complete, we can nevertheless mention that instantons effects might be less important at high temperatures following Ref.~\\cite{shur96}. Actually, we have checked that the results that we obtain in the following do not demand an accurate knowledge of $M_G(T)$.\n\nA key ingredient in Hamiltonian~(\\ref{hamglu}) is the potential energy $V(r,T)$ between a static quark-antiquark pair. It is well known from lattice QCD that this potential is compatible with a funnel shape $ar-b\/r$ at $T=0$~\\cite{bali00}, but the situation is less clear when $T>0$. The potential energy that is the most readily obtained in lattice QCD is the quark-antiquark free energy $F(r,T)$~\\cite{mcler,boyd}. We recall that, thermodynamically speaking, the free energy of a system is the energy that is available in the system to produce a work once the energy losses due to the increase of the entropy have been subtracted. As also noticed in Ref.~\\cite{shur04}, in a potential approach however, the potential energy of the system should be the total energy that it contains, no matter it will be lost or not in heat transfers. Such a potential energy corresponds to the internal energy of the system, usually denoted by $U=F+TS$, where $S$ is the entropy. The internal energy of a quark-antiquark pair is thus the quantity we choose as potential term. It has been computed in lattice QCD in Refs.~\\cite{pet05}; we give a plot of these results in Fig.~\\ref{fig5}. Notice that those $N_f=0$ computations are the most relevant for our purpose since we consider a genuine gluon plasma. \n\n\\begin{figure}[ht]\n\\includegraphics*[width=\\columnwidth]{fig5.eps}\n\\caption{(Color online) Internal energy of a static quark-antiquark pair computed in lattice QCD for different values of $T\/T_{\\text{c}}$ and for $N_f=0$ (dots). Lattice data are taken from Refs.~\\cite{pet05} and compared to the fitted form~(\\ref{Vfit}) for some values of $T\/T_{\\text{c}}$ (solid lines).}\n\\label{fig5}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\includegraphics*[width=\\columnwidth]{fig6.eps}\n\\caption{(Color online) Values of $a(T)$, $b(T)$, and $c(T)$ obtained by a fit of the lattice QCD data to the form (\\ref{vfit2}) (dots), compared to the analytical curves (\\ref{fitcoeff}) (solid lines).}\n\\label{fig6}\n\\end{figure}\n\nIt can be checked in Figs.~\\ref{fig5} and \\ref{fig6} that the lattice data are accurately fitted by the following form\n\\begin{subequations}\\label{Vfit}\n\\begin{equation}\\label{vfit2}\n\tV(r,T)=-a(T)\\ {\\rm e}^{-b(T)\\, r}+c(T),\n\\end{equation}\nwhere \n\\begin{eqnarray}\n\\label{fitcoeff}\n\ta(T)&=&\\frac{a_0}{\\ln\\left(\\frac{T}{a_1T_{\\text{c}}}\\right)} + a_2 \\ln\\left(\\frac{T}{a_1T_{\\text{c}}}\\right), b(T)= b_0+ b_1\\, \\frac{T}{T_{\\text{c}}},\\nonumber \\\\ \n\tc(T)&=&\\frac{c_0}{\\ln\\left(\\frac{T}{c_1T_{\\text{c}}}\\right)},\\quad {\\rm and}\n\\end{eqnarray}\n\\begin{eqnarray}\n a_0&=&0.459~{\\rm GeV},\\quad a_1=0.915,\\quad a_2=1.159~{\\rm GeV}\t,\\nonumber\\\\\n b_0&=&0.111~{\\rm GeV},\\quad b_1=0.489~{\\rm GeV}, \\nonumber \\\\\n c_0&=&0.341~{\\rm GeV},\\quad c_1=0.808.\n\\end{eqnarray}\n\\end{subequations}\nIt has to be stressed that the form~(\\ref{Vfit}) is the one that gives the best fit of the lattice data but is not motivated by any physical theory predicting such a form. We just use it as a convenient parameterization of the lattice QCD results. It clearly appears from Eq.~(\\ref{vfit2}) that the potential energy is no longer confining. Exponential potentials indeed only admit a finite number of bound states. \n\n\\subsection{Numerical results}\n\\label{ssec:numres} \n\nAll the terms appearing in Hamiltonian~(\\ref{hamglu}) are now explicitly known, and its ground state mass can be numerically computed. To this aim we use the Lagrange mesh method, which is a numerical procedure allowing in particular to accurately solve eigenequations associated to relativistic Hamiltonians~\\cite{sem01}. The evolution of the lowest-lying glueball mass with the temperature is given in Fig.~\\ref{fig7}. The numerically computed evolution of the glueball mass with $T$ is accurately fitted by the form\n\\begin{subequations}\\label{mfit}\n\\begin{equation}\\label{mgit2}\n\tM_G(T)=\\frac{9}{4}c(T)+2 \\bar m(T)+2b(T)\\, \\varepsilon(T)\\quad {\\rm for}\\quad T\\leq 1.13\\, T_{\\text{c}} ,\n\\end{equation}\nwhere \n\\begin{equation}\\label{mfit3}\n\\varepsilon(T)=\\frac{\\varepsilon_0 T\/T_{\\text{c}}-\\varepsilon_1}{T\/T_{\\text{c}}-\\varepsilon_2},\n\\end{equation}\n\\begin{equation}\\label{mfit4}\n\t\\varepsilon_0=0.818,\\quad \\varepsilon_1=0.921,\\quad \\varepsilon_2=0.958.\n\\end{equation}\n\\end{subequations}\nThe glueball mass we find is around $1.8$~GeV at $T\\approx \\, T_{\\text{c}}$, then increases to reach a maximal value of about 2.8~GeV. Notice that the mass near the critical temperature is similar to the one obtained at zero temperature~\\cite{glub}. To our knowledge, the behavior of the scalar glueball mass versus the temperature has not been studied a lot in the literature. We can nevertheless quote the lattice study of Ref.~\\cite{glub2} that finds a reduction of 20\\% of the scalar glueball mass when one goes from $T=0$ to $T=T_{\\text{c}}$, and the more recent work~\\cite{meng} finding an almost constant glueball mass from $T=0$ to $T=T_{\\text{c}}$. Beyond the qualitative behavior of $M_G(T)$, an important result we find is that the ground state is bound up to $T=1.13\\, T_{\\text{c}}$ and then dissociates in the medium above this temperature. Numerically, the dissociation temperature is reached when the binding energy of the system vanishes. Our model thus predicts the existence of glueballs in the temperature range $T\/T_{\\text{c}}=1-1.13$, but the existence of bound states is a very stringent criterion: Glueball resonances can indeed appear in the continuum even if the gluons are not bound. Following the lattice results of Ref.~\\cite{meng}, glueball resonances can even be expected up to $1.9\\, T_{\\text{c}}$. \n\n\\begin{figure}[t]\n\\includegraphics*[width=\\columnwidth]{fig7b.eps}\n\\caption{(Color online) Numerically computed lowest-lying glueball mass, that is the ground state mass of Hamiltonian~(\\ref{hamglu}), versus $T\/T_{\\text{c}}$ (solid line). The fitted form~(\\ref{mfit}) is also plotted for comparison (dashed line). The curve stops at the glueball dissociation temperature, namely 1.13\\, $T_{\\text{c}}$.}\n\\label{fig7}\n\\end{figure}\n\nSince glueballs can be present in the deconfined medium, we propose to recompute the thermodynamical properties of the gluon plasma by assuming that it is a mixing between an ideal gas of transverse gluons and an ideal gas of glueballs; the glueball abundance $n(T)$ depending on the temperature. This last approach will be referred to as Model 3; it shares with Model 1 the property that $\\beta=1\/T$ and that the form of the energy is preserved. Its spirit is a bit similar to the one of the hadronic resonance gas model, assuming that the hot hadronic medium can be described as an ideal gas made of all\npossible resonance species (see Refs.~\\cite{hgm,hgm2} for more informations). We actually consider that the screened color interactions ``generate\" color singlet scalar and pseudoscalar glueballs in a first stage, and that these glueballs behave as free particles in the gluon plasma in a second stage. Consequently, if \n\\begin{equation}\ne_0(d,m,\\beta)=\\frac{d}{2\\pi^2} \\int^\\infty_{\\beta m} \\frac{k^2\\sqrt{k^2-(\\beta m)^2}}{{\\rm e}^k-1}\\, dk \n\\end{equation}\nis the energy density of an ideal gas of bosons with mass $m$ and with $d$ degrees of freedom, then the total energy density of the mixed gluon-glueball gas is\n\\begin{subequations}\\label{model3}\n\\begin{eqnarray}\n\\label{emodel3}\n\te^{(3)}&=&[1-n(T)]\\, e_0(16,\\bar m T,1\/T)\\nonumber\\\\\n\t&&+n(T)\\, e_0(2,M_G(T),1\/T),\n\\end{eqnarray}\nwhere two degrees of freedom are associated to the glueball gas, accounting for the lowest-lying $0^{\\pm+}$ states. \\textit{A priori}, $n(T)$ should vanish above $1.13\\, T_{\\text{c}}$ because glueballs are then not bound anymore. However, two-gluon resonances can in principle appear in the continuum above the dissociation temperature. The simplest way to take this phenomenon into account is to allow $n(T)$ to be nonzero above the dissociation temperature. In this sector, formula~(\\ref{mgit2}) remains well-defined, roughly simulating a gluon pair in the continuum. \n \\begin{figure}[t]\n\\includegraphics*[width=\\columnwidth]{fig8b.eps}\n\\caption{(Color online) Glueball abundance computed by fitting Eq.~(\\ref{emodel3}) to lattice QCD versus $T\/T_{\\text{c}}$ (solid line). The fitted form~(\\ref{nt}) is plotted for comparison (dashed line).}\n\\label{fig8}\n\\end{figure} \n\nThe unknown function $n(T)$ can be computed by fitting Eq.~(\\ref{emodel3}) to the lattice energy. The result is given in Fig.~\\ref{fig8}; it appears that the numerically computed curve is accurately described by the following form\n\\begin{equation}\n\\label{nt}\n\tn(T)={\\rm e}^{-n_0\\, (T\/T_{\\text{c}} -1)^{n_1}},\n\\end{equation}\nwith $n_0=3.358$ and $n_1=0.541$. The glueball abundance is nearly $100$\\% at $T=T_{\\text{c}}$, then decreases to reach 33\\% at 1.13 $T_{\\text{c}}$, the dissociation temperature of the two-gluon glueballs. But as we said previously, resonances are then still expected to form in the continuum, justifying a nonzero glueball abundance at higher temperatures. Finally, $n(T)$ is less than 5\\% at 1.9 $\\ T_{\\text{c}}$. Such a negligible value is coherent with the fact that glueball resonances are expected to disappear above that temperature~\\cite{meng}. We have checked that the quantitative behavior of $n(T)$ is not very sensitive to the glueball mass, $M_G(T)$. The key result of Model 3 is rather that $e_0(16,\\bar m T,1\/T)$ alone is unable to fit the available data and consequently that an additional term accounting for glueballs is needed. \n\nThe entropy density can be computed from Eqs.~(\\ref{emodel3}) and (\\ref{link-energy-entropy}). It reads\n\\begin{equation}\\label{smodel3}\n\ts^{(3)}=\\int^{1\/T_{\\text{c}}}_{1\/T}\\beta\\, \\partial_\\beta e^{(3)}(1\/\\beta)\\, d\\beta.\n\\end{equation}\nThe upper bound of this last integral ensures that $s^{(3)}(T_{\\text{c}})=0$, in qualitative agreement with lattice QCD. Finally, the pressure can be computed thanks to the definition~(\\ref{pression-def}), that is \n\\begin{equation}\\label{pmodel3}\np^{(3)}=T\\, s^{(3)}-e^{(3)}\t.\n\\end{equation}\n\\end{subequations}\nThe results are plotted in Figs.~\\ref{fig9} and \\ref{fig10} and compared to lattice QCD. As it was the case for Models 1 and 2, Model 3 leads to an excellent agreement with the lattice data, although relying on a different physical picture of the gluon plasma.\n\n\\begin{figure}[ht]\n\\includegraphics*[width=\\columnwidth]{fig9.eps}\n\\caption{Same as Fig.~\\ref{fig3}, but lattice data are this time compared to Model 3 (solid gray line) defined by Eqs.~(\\ref{model3}).}\n\\label{fig9}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\includegraphics*[width=8.0cm]{fig10.eps}\n\\caption{(Color online) Same as Fig.~\\ref{fig4}, but lattice data are this time compared to Model 3 (solid gray line) defined by Eqs.~(\\ref{model3}).}\n\\label{fig10}\n\\end{figure}\n\n\\section{Conclusions and outlook}\n\\label{sec:conc}\n\nIt is now widely accepted that the equation of state of the gluon plasma, coming from pure gauge lattice QCD computations, can be accurately reproduced by modeling the gluon plasma as a gas of transverse gluons with a temperature-dependent mass. As we have outlined in the beginning of this paper indeed, such a quasiparticle model is indeed in disagreement with lattice QCD if a constant gluon mass is used. One is thus led to deal with temperature-dependent Hamiltonians. In that case, standard formulas in statistical mechanics have to be modified in order to enforce the thermodynamical consistency, but the procedure to achieve such a task varies from one work to another. In the frameworks that can be found in the literature so far, the standard expression of only one thermodynamical quantity can be preserved in order to enforce the thermodynamic consistency, \\textit{i.e.} to satisfy the laws of thermodynamics. The expressions of the other quantities have to be modified: Either the pressure~\\cite{golo93}, the entropy~\\cite{gore95}, or the energy~\\cite{bann07} is kept invariant. \n\nIn this work, we have clarified the situation by showing that all the existing formulations can be derived in a simple unified way. In the process, we have uncovered a new possible formulation for which the standard form of each thermodynamical quantity is preserved but for which $\\beta$ is no longer equal to $1\/T$. The function $\\beta(T)$ has to be extracted from a first order nonlinear differential equation expressing the fulfillment of the laws of thermodynamics. We think that this last formalism is the most fundamental one, since it only demands a change in the definition of the Lagrangian multiplier $\\beta$, which has no physical meaning \\textit{a priori}. Moreover, the corrections to standard statistical mechanics implied by this new formulation are only local in $T$ -- {\\it i.e.} they vanish in regions where the Hamiltonian does not depend on $T$ -- while corrections found in other formulations are non-local in $T$. However, this new formulation is far more complicated to deal with in numerical applications when the dependence of the Hamiltonian on temperature is not known. That is why the other approaches are also useful to study the quark-gluon plasma. \n\nConsequently, we focused on two formulations: The ones that preserve the form of the energy and of the entropy. It can be analytically shown that, independently of the considered formulation, reproducing the lattice data leads to constraints on the thermal gluon mass, $m(T)$. It must be strongly decreasing just after the critical temperature and grow linearly asymptotically. A numerical fit of the thermal mass on the available data confirms this behavior and eventually leads to an excellent agreement with lattice QCD. Both frameworks lead to nearly indistinguishable results as expected, and to very similar thermal gluon mass. \n\nMean-field-inspired arguments show that the singular behavior of the thermal gluon mass near $T_{\\text{c}}$ accounts for residual color interactions, which are still strong in the early stages after deconfinement. The potential energy coming from such screened color interactions has already been computed in lattice QCD, allowing us to build a consistent Hamiltonian describing the interactions between two transverse gluons in a color singlet, that is the channel in which the color interactions are the strongest. It appears that the two-gluon ground state, corresponding to the scalar and pseudoscalar glueballs, remains bound up to $T=1.13\\, T_{\\text{c}}$. We have then proposed a last description of the gluon plasma, in which this medium is seen as a ideal mixture of free gluons and colorless glueballs. The agreement with lattice QCD is as good as with the previous approaches, with a glueball abundance that is very large near the critical temperature, takes the lower value of 33\\% at the dissociation temperature of the lightest glueballs, and becomes negligible after $1.9\\, T_{\\text{c}}$, where even continuum glueball resonances are expected to disappear~\\cite{meng}. This interpretation of the gluon plasma draws a bridge between the quasiparticle approach and other models focusing on the existence of bound states after deconfinement~\\cite{shur04}. \n\nFrom an experimental point of view, the main result of the present study is the prediction that the gluon plasma, and thus presumably the quark-gluon plasma, might be a glueball-rich medium in the early stages after deconfinement. This brings support to previous studies arguing that an important amount of glueballs can be formed in relativistic heavy ion collisions~\\cite{gluprod,gluprod2}. The experimental detection of the scalar glueball in the quark-gluon plasma could be achieved through the scenario developed in Refs.~\\cite{vento06} which roughly suggests that, although the bare scalar glueball would be nearly stable in the quark-gluon plasma, it should mix with scalar mesons. Then, such a ``physical\" glueball, denoted as $G$, could decay mostly in the channels $G\\rightarrow\\pi\\pi$ and $G\\rightarrow\\gamma\\gamma$ through its mesonic component, leading to an enhancement of the number of events versus the two-photon (or two-pion) invariant mass. In our model, the bare glueball mass is mostly located around $2.8$~GeV; a peak in the $\\gamma\\gamma$ or $\\pi\\pi$ channels can thus be expected not too far of $2.8$~GeV, depending on the strength of the meson-glueball coupling.\n\nWe finally stress that, if quarks were included in our model, the number of bound states above the critical temperature would increase since mesons, diquarks, quark-gluon states, \\textit{etc.} can also form. We leave the extension of our approach to the full quark-gluon plasma for future works. The effects of a nonzero chemical potential will also be leaved for subsequent studies.\n\n\\begin{acknowledgments}\nThe authors thank Vincent Mathieu, Francesco Giacosa and O. Kaczmarek for useful discussions and suggestions about the present work. F. Buisseret thanks the F.R.S.-FNRS Belgium for financial support. F. Brau acknowledges financial support from a return grant delivered by the Federal Scientific Politics.\n\\end{acknowledgments}\n\n\\begin{appendix}\n\\section{Determination of $\\beta(T)$ for temperature-dependent Hamiltonians: an example}\n\\label{app:Tdepend}\nIn order to illustrate the general procedure given in Sec.~\\ref{ssec:genef}, we study the particular case of a classical ideal gas with temperature-dependent mass $m(T)$, or equivalently $m(f(\\beta))$ because of the definition~(\\ref{fbeta-def}). The Hamiltonian reads\n\\begin{equation}\n\tH=\\frac{k^2}{2m(f(\\beta))},\n\\end{equation}\nand one finds that, for system of $N$ particles, \n\\begin{equation}\n\t{\\cal Z}=V^N\\left[\\frac{2\\pi m(f(\\beta))}{\\beta}\\right]^{3N\/2},\n\\end{equation}\nwhere $V$ is the volume of the system. The normalized probability density is thus known and it can be computed that \n\\begin{equation}\n E=\\frac{3N}{2\\beta},\\quad {\\rm and} \\quad\t\\overline{\\partial_\\beta H}=-\\frac{3N}{2\\beta}\\frac{m'(f(\\beta)) f'(\\beta)}{m(f(\\beta))},\n\\end{equation}\nwhere the prime denotes a partial derivation with respect to the argument of the considered function. These last two equalities allow to rewrite Eq.~(\\ref{fbeta}) as \n\\begin{equation}\\label{deq1}\n\t\\beta\\frac{m'(f(\\beta)) }{m(f(\\beta))}f'(\\beta)+f(\\beta)-\\beta=0.\n\\end{equation}\n\nLet us consider the following form for $m(T)$ to illustrate the procedure\n\\begin{equation}\n\tm(T)=m_0 e^{(1-\\sqrt{1+4\\delta\/T})\/2},\n\\end{equation}\nwith $\\delta\\ge 0$. The equation for $f(\\beta)$ then reads\n\\begin{equation}\n\t\\label{tempf}\n\t\\delta \\beta^2 f'(\\beta)=(f(\\beta)-\\beta) \\sqrt{1+4 \\delta f(\\beta)}.\n\\end{equation}\nTo determine uniquely the function $f$, we need a boundary condition. For $T\\to \\infty$, the mass tends to $m_0$ which is constant. In this limit, one recover the standard statistical mechanics and $f(\\beta)=\\beta=1\/T$. Consequently the boundary condition is $f(0)=0$. The unique solution of the nonlinear differential equation (\\ref{tempf}) is then\n\\begin{equation}\n\tf(\\beta)=\\beta + \\delta \\beta^2.\n\\end{equation}\nThe relation between $\\beta$ and $T$ is then given by\n\\begin{equation}\n\t\\beta=\\frac{-1+\\sqrt{1+4\\delta\/T}}{2\\delta}.\n\\end{equation}\nFor $T\\gg 4\\delta$ (or $\\delta \\to 0$), we just recover the standard relation $\\beta=1\/T$. In this formalism, we also find that\n\\begin{equation}\n\t\\frac{E}{N}=\\frac{3\\delta}{-1+\\sqrt{1+4\\delta\/T}}.\n\\end{equation}\n\n\\begin{figure}[t]\n\\includegraphics*[width=8.0cm]{fig11.eps}\n\\caption{(Color online) Comparison between the corrections to the energy for the new formalism and Model 2 as a function of $T\/\\delta$. We used $T^*\/\\delta=1$. The evolution of $m(T)\/m_0$ is also presented.}\n\\label{fig11}\n\\end{figure}\n\nWe can now compare this last energy formula with the energy obtained within Models 1 and 2. For the Model 1, where the expression for the energy is preserved, we simply have the standard expression $E\/N=3T\/2$ while for the Model 2, where the expression for the entropy is preserved, the energy takes the form (remember that in this formalism $\\beta=1\/T$)\n\\begin{eqnarray}\n\t\\frac{E}{N}&=&\\frac{3T}{2}-\\int_{\\beta^*}^{\\beta}\\overline{\\partial_{\\beta} H}|_{\\beta=\\nu} d\\nu, \\nonumber \\\\\n\t&=& \\frac{3T}{2}+\\frac{3}{2}\\int_{\\beta^*}^{\\beta} \\frac{\\partial_{\\nu} m(\\nu)}{m(\\nu)} \\frac{1}{\\nu} d\\nu, \\\\\n\t&=& \\frac{3T}{2}+\\frac{3\\delta}{2}\\left[\\ln\\left(\\frac{1+\\sqrt{1+4\\delta \\nu}}{-1+\\sqrt{1+4\\delta \\nu}} \\right) \\right]_{\\nu=1\/T^*}^{\\nu=1\/T}.\\nonumber \n\\end{eqnarray}\nThe correction to the energy ($E\/N-3T\/2$) can be compared for each formalism. Of course, for Model 1, this correction is vanishing; in this case corrections would be associated to the entropy. Consequently, in Fig.~\\ref{fig11}, we compare only corrections to the energy obtained with Model 2 and with the new formalism proposed in this paper together with the evolution of the mass as functions of the temperature $T\/\\delta$.\n\nWe notice that the corrections to the energy from Model 2 (and corrections to the entropy from Model 1) are non-local since they involve integrals over some range in temperature, see Eqs.~(\\ref{new-entropy}) and ~(\\ref{new-energy}). This means that those corrections are still significative in regions where the mass is essentially constant (in this example the corrections are logarithmic in $T$) while the corrections to the energy from the new formalism are essentially localized around the region where the mass depends significantly on the temperature. This is indeed what we expect: If the Hamiltonian does not essentially depend on $T$ over some large interval of temperature, the statistical mechanics in this interval of $T$ should be essentially the same than the standard statistical mechanics.\n\\end{appendix}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\nRecent years have seen a surge of interest in the study of active matter and active particles. \nThe term `active particle' refers to a class of self-propelled particles which can generate dissipative directed motion by consuming energy directly from their environment \\cite{Romanczuk,soft,BechingerRev,Ramaswamy2017,Marchetti2017,Schweitzer}. Examples of active matter can be found in nature at all length scales, ranging from micro-organisms like bacteria \\cite{Berg2004,Cates2012} to granular matter \\cite{gran1,gran2}, flock of birds \\cite{flocking1, flocking2} and fish-schools \\cite{Vicsek,fish}. Apart from a diverse set of novel collective behaviours like clustering \\cite{cluster1,cluster2,evans}, motility induced phase separation \\cite{separation1, separation2, separation3}, and absence of well defined pressure \\cite{Kardar2015}, active particles show many intriguing features even at the single particle level. One such interesting feature is that, in the presence of external potentials and confining boundaries, active particles show very different behaviour than their passive counterparts, including non-Boltzmann stationary state, clustering near the boundaries of the confining region \\cite{Solon2015,Potosky2012,ABP2019,RTP_trap,Malakar2019} and unusual relaxation and persistence properties \\cite{RTP_free,ABP2018, Singh2019}. There have been numerous recent studies focusing on the behaviour of active particles in the presence of external potentials and confinements, both theoretical \\cite{Franosch2016,Das2018,Caprini2019,Sevilla2019} and experimental \\cite{Hagen2014,Takatori,Deblais2018,Dauchot2019}. \n\n\n\n\nThe theoretical attempts to characterise the behaviour of active particles focus on studying simple models of such systems. Run-and-tumble particle (RTP) is one of the most studied models of an active particle. An RTP is an overdamped particle which moves with a constant speed $v_0$, or `runs,' along the direction of an internal `spin' degree of freedom. The orientation of the spin can change randomly resulting in a sudden change, or `tumble,' in the direction of motion of the particle. The simplest example is an RTP moving in one spatial dimension with two possible values of the spin $\\sigma = \\pm 1.$ In this case, the particle moves with velocity $v_0$ or $-v_0;$ the reversal of direction occurs stochastically with rate $\\gamma,$ with the flipping of the spin $\\sigma \\to - \\sigma.$ In the presence of an external potential $U(x),$ the position $x(t)$ of this two-state RTP evolves according to the Langevin equation,\n\\begin{eqnarray}\n\\dot x = f(x) + v_0 \\sigma(t) \\label{eq:RTP_2st}\n\\end{eqnarray}\nwhere $f(x)= - U'(x)$ is the deterministic force acting on the particle. The spin variable $\\sigma$ plays the role of the noise, its dichotomous nature giving rise to the `activity'. In fact, it is clear from the auto-correlation $\\langle \\sigma(t) \\sigma(t') \\rangle = e^{-2 \\gamma |t-t'|}$ that $\\sigma(t)$ is a coloured noise with a finite memory, characterised by the persistence time $\\tau = (2\\gamma)^{-1}.$ Despite the apparent simplicity of the model, the two-state RTP shows a lot of intriguing features typical to active particles including non-Boltzmann stationary distribution\\cite{RTP_trap, RTP_free}. \n\nFor any confining potential, the stationary position distribution of a two-state RTP is known exactly, and is given by, \n\\begin{eqnarray}\nP_\\textrm{st}(x) \\propto \\frac {1}{v_0^2-f^2(x)}\\exp{\\bigg[2 \\gamma \\int_0^x \\textrm{d} y \\frac{f(y)}{v_0^2-f^2(y)}\\bigg]} \\label{eq:Pst_2st}\n\\end{eqnarray}\nup to a normalization constant.\nThe above result was first obtained long ago in the context of quantum optics \\cite{q-optics1,q-optics2,q-optics3,q-optics4}, and later to study the role of coloured noise in dynamical systems \\cite{colored}. More recently, it has been re-derived in the context of active particles \\cite{Kardar2015, RTP_trap}. In particular, the stationary distribution \\eref{eq:Pst_2st} has been analysed for specific confining potentials of the type $U(x) \\propto |x|^p$ with $p>0$ in Ref. \\cite{RTP_trap}. The case $p=2$ corresponds to a harmonic potential which is of particular interest, not only from theoretical but also from an experimental point of view \\cite{Takatori,Dauchot2019}. For a harmonic potential $U(x) = \\mu x^2\/2,$ the stationary distribution \\eref{eq:Pst_2st} simplifies to,\n\\begin{eqnarray}\nP_\\textrm{st}(x)= \\frac{2 \\mu}{4^{\\beta}B(\\beta, \\beta)v_0} \\left [1- \\left(\\frac{\\mu x}{v_0} \\right)^2 \\right]^{\\beta -1} \\label{eq:2st_harmonic}\n\\end{eqnarray}\nwhere $\\beta = \\gamma\/\\mu$ and $B(u,v)$ is the beta-function. This distribution is symmetric in $x$ and has a finite support in the region $-\\frac{v_0}{\\mu} \\le x \\le \\frac{v_0}{\\mu}.$ Consequently, the particle is confined within this region in the stationary state. This stationary position distribution shows an interesting shape-transition as a function of $\\beta.$ For $\\beta > 1$ the distribution is convex shaped, with a peak at the origin $x=0$ and $P_\\textrm{st}(x)$ vanishing at the boundaries $x = \\pm \\frac{v_0}{\\mu}.$ On the other hand, for $\\beta < 1$ $P_\\textrm{st}(x)$ has a concave shape with divergences at the boundaries and a minimum at the origin. For $\\beta=1,$ the distribution is uniform. Thus by varying $\\beta,$ one can observe a transition from a double-peaked (at the boundaries) to a single-peaked distribution. The double-peaked nature of the distribution for $\\gamma < \\mu$ signifies an `active phase', where the persistence time of the spin-orientation is larger than $\\mu^{-1},$ the relaxation time-scale of the potential. On the other hand, $\\gamma > \\mu,$ \\textit{i.e.}, when the persistence time is smaller compared to $\\mu^{-1},$ corresponds to a passive phase, where the stationary distribution resembles that of a passive particle in a trap, with a single peak at the centre of the trap. Indeed, in the diffusive limit when $v_0 \\to \\infty$, $\\gamma \\to \\infty$ but keeping the ratio $v_0^2\/2\\gamma = D$ fixed, the dynamics of the RTP in the harmonic trap converges to the Ornstein-Uhlenbeck process. This is also exhibited in the stationary state where the distribution in Eq. (\\ref{eq:2st_harmonic}) converges to a Boltzmann distribution, which in this case is a simple Gaussian $P_{\\textrm{st}}(x) \\propto e^{- \\frac{\\mu}{D}x^2}$. \n \n\n\nIt is then natural to ask how the stationary distribution changes if the RTP has more than two internal states. In fact, an RTP with many internal degrees have been studied where the internal degrees can take a set of discrete values and evolve following some discrete jump processes \\cite{Seifert2016,Maes2018}. \nHowever, most of these studies are numerical and to the best of our knowledge no analytical results are available for the stationary state of a multi-state RTP in the presence of an external potential.\n\n\n\nIn this article, we study a run-and-tumble active particle in one spatial dimension with three discrete internal states, with positive, negative and zero velocities, respectively. We show that such a multi-state dynamics naturally arises when one considers an RTP in higher spatial dimensions and project it to one-dimension. We calculate exactly the stationary position probability distribution in the presence of a harmonic potential of strength $\\mu$ for arbitrary flip-rate $\\gamma$ among the internal states. It turns out that the presence of the zero-velocity internal state leads to a rich behaviour of the position distribution $P(x)$. As in the two-state case, it turns out that the shape of the stationary state distribution is governed by one single parameter\n\\begin{eqnarray}\\label{def_beta}\n\\beta=\\frac{\\gamma}{\\mu} \\;.\n\\end{eqnarray}\nWe show that $P(x)$ has a finite support on the real line and undergoes a transition in shape as $\\beta=\\gamma\/\\mu$ is varied : For $\\beta <1,$ $P(x)$ diverges both at the origin and the boundaries with the same exponent $\\beta -1.$ Thus, in this case, the position distribution has a double-concave shape, with three peaks, namely at the boundaries and the origin. For $\\beta=1,$ $P(x)$ shows a logarithmic divergence near the origin. On the other hand, for $\\beta >1,$ the distribution converges to a finite value at the origin while it vanishes at the boundaries, implying a convex shape with a single peak at the origin (see Fig. \\ref{fig:px}). \n\n\n\n\\section{Model}\n\nOur model of a three-state RTP in one-dimension is motivated by a natural ``clock-like'' model for a two-dimensional RTP. \nLet us indeed consider an overdamped particle moving on a two dimensional $(xy)$ plane with an internal orientational degree of freedom or `spin' $\\sigma$ associated with it. In the absence of any external potential the particle moves with a constant speed $v_0$ along the direction of $ \\sigma,$ which is a unit vector with four possible discrete orientations, denoted by $E,W,N,S$ (along $\\pm x$ and $\\pm y$ axes respectively). The spin $\\sigma$ evolves in time following a Markov jump process -- its orientation can change via a rotation of $\\frac\\pi 2$ either clockwise or anti-clockwise, both with rate $\\frac \\gamma 2.$ This jump process is schematically represented in Fig. \\ref{fig:4st_3st}(a). Additionally, we consider an external harmonic potential $U(x,y) = \\frac \\mu 2(x^2+y^2)$ which exerts a force $f(x,y) = -\\nabla U(x,y)$ on the RTP. \n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=8 cm]{.\/spin_4st2.pdf} \\hspace{1.5 cm}\\includegraphics[width=3.5 cm]{.\/reduced_3st.pdf} \n\n \\caption{(a) Schematic representation of the jump process through which the orientation $\\sigma$ evolves. (b) The equivalent 3-state jump process for $\\sigma_x$.}\n \\label{fig:4st_3st}\n\\end{figure}\nThe time-evolution of the position $(x(t),y(t))$ of the RTP can be conveniently expressed in terms of the Langevin equations,\n\\numparts\n\\begin{eqnarray}\n\\dot x(t) &=& -\\mu x(t) + v_0 \\sigma_x(t) \\label{eq:2dmodel_x} \\\\\n\\dot y(t) &=& - \\mu y(t) + v_0 \\sigma_y(t) \\label{eq:2dmodel_y}\n\\end{eqnarray}\n\\endnumparts\nwhere $\\sigma_{x,y}(t)$ are components of the spin vector $\\sigma(t)$ at any time $t,$ along the $x$ and $y$ axes respectively (see Fig. \\ref{fig:4st_3st}(a)). \n\nThe position probability distribution $\\cal P(x,y,t)$ is given by the sum $\\cal P(x,y,t) = \\sum_{\\sigma} \\cal P_\\sigma(x,y,t)$ where $\\cal P_\\sigma(x,y,t)$ denotes the probability that the particle has the position $(x,y)$ and orientation $\\sigma = E, N, W, S$ at time $t.$ These probabilities evolve according to the Fokker-Planck (FP) equations,\n\\numparts\n\\begin{eqnarray}\n\\fl \\qquad \\frac{\\partial}{\\partial t} \\cal P_E(x,y,t) = \\frac{\\partial }{\\partial x} \\bigg[(\\mu x - v_0)\\cal P_E \\bigg] + \\frac{\\partial }{\\partial y} \\bigg[\\mu y \\cal P_E \\bigg] + \\frac \\gamma 2 (\\cal P_N+\\cal P_S) - \\gamma \\cal P_E \\label{eq:FP_2d1}\\\\\n\\fl \\qquad \\frac{\\partial}{\\partial t} \\cal P_N(x,y,t) = \\frac{\\partial }{\\partial x} \\bigg[ \\mu x \\cal P_N \\bigg] + \\frac{\\partial }{\\partial y} \\bigg[(\\mu y - v_0) \\cal P_N \\bigg] + \\frac \\gamma 2 (\\cal P_E+\\cal P_W) - \\gamma \\cal P_N \\\\\n\\fl \\qquad \\frac{\\partial}{\\partial t} \\cal P_W(x,y,t) = \\frac{\\partial }{\\partial x} \\bigg[ (\\mu x + v_0)\\cal P_W \\bigg] + \\frac{\\partial }{\\partial y} \\bigg[\\mu y \\cal P_W \\bigg] + \\frac \\gamma 2 (\\cal P_N+\\cal P_S) - \\gamma \\cal P_W \\\\\n\\fl \\qquad \\frac{\\partial}{\\partial t} \\cal P_S(x,y,t) = \\frac{\\partial }{\\partial x} \\bigg[ \\mu x\\cal P_S \\bigg] + \\frac{\\partial }{\\partial y} \\bigg[(\\mu y + v_0) \\cal P_S \\bigg] + \\frac \\gamma 2 (\\cal P_E+\\cal P_W) - \\gamma \\cal P_S \\label{eq:FP_2d}\n\\end{eqnarray}\n\\endnumparts\nwhere we have suppressed the argument of $P_\\sigma$ on the right hand side for the sake of brevity. \nIt is hard to find an analytical form of $\\cal P(x,y,t)$ as these equations are difficult to solve, even in the stationary state.\n\nHowever, it is also interesting to look at the $x$-process only, governed by Eq.~\\eref{eq:2dmodel_x}. This describes an effective one-dimensional RTP where the internal spin $\\sigma_x$ has three possible discrete values, $1,0,-1.$ \nAs illustrated in Fig. \\ref{fig:4st_3st}(a), both $\\sigma=N$ and $\\sigma=S$ correspond to $\\sigma_x=0$ while $\\sigma=E$ and $\\sigma=W$ corresponds to $\\sigma_x=1$ and $\\sigma_x=-1,$ respectively. The jump from $\\sigma_x=1$ to $0$ can, thus, occur through two different channels ($E \\to N$ and $E \\to S$), resulting in a jump rate $\\gamma$ for $\\sigma_x=1 \\to \\sigma_x=0.$ Similarly, $\\sigma_x=-1 \\to \\sigma_x =0$ occurs with rate $\\gamma,$ while $0 \\to \\pm 1$ occurs with rate $\\frac \\gamma2$ (since there is only one way to make this transition). This effective 3-state jump process in one-dimension is schematically shown in Fig.~\\ref{fig:4st_3st}(b). Let $P_{i}(x,t)$ denote the probability that the RTP is at a position $x$ at time $t$ with $\\sigma_x=i.$ The corresponding FP equations read,\n\\numparts\n \\begin{eqnarray}\n \\frac{\\partial}{\\partial t} P_{1}(x,t) &=& \\frac{\\partial }{\\partial x} [(\\mu x - v_0 ) P_{1}] + \\frac\\gamma 2 P_0 - \\gamma P_1 \\label{eq:FP_3st1}\\\\\n \\frac{\\partial}{\\partial t} P_{-1}(x,t) &=& \\frac{\\partial }{\\partial x} [(\\mu x + v_0 ) P_{-1}] + \\frac\\gamma 2 P_0 - \\gamma P_{-1} \\label{eq:FP_3stm1}\\\\\n \\frac{\\partial}{\\partial t} P_{0}(x,t) &=& \\frac{\\partial }{\\partial x} [\\mu x P_{0}] + \\gamma (P_1 + P_{-1}) - \\gamma P_0. \\label{eq:FP_3st0}\n \\end{eqnarray} \n \\endnumparts\nWe note that this set of FP equations can also be obtained from Eqs.~\\eref{eq:FP_2d1}-\\eref{eq:FP_2d} by integrating both sides over $y$ and then identifying $P_1(x,t) = \\int \\textrm{d} y ~\\cal P_E(x,y,t),$ $P_{-1}(x,t) = \\int \\textrm{d} y ~\\cal P_W(x,y,t),$ and $P_0(x,t)= \\int \\textrm{d} y ~[\\cal P_N(x,y,t) + \\cal P_S(x,y,t)].$ \n\n \n \nIn the presence of the confining harmonic potential, in the long time limit the RTP is expected to reach a stationary state where the left hand side (l. h. s.) of the Eqs.~\\eref{eq:FP_3st1} - \\eref{eq:FP_3st0} would vanish. The corresponding stationary distributions $P_i(x) = \\lim_{t \\to \\infty} P_i(x,t)$ then satisfy a set of coupled linear differential equations (obtained by putting $\\frac{\\partial P_i}{\\partial t}=0$),\n\\numparts\n\\begin{eqnarray}\n\\frac{\\textrm{d} }{\\textrm{d} x} [(\\mu x - v_0 ) P_{1}] + \\frac\\gamma 2 P_0 - \\gamma P_1 &=& 0 \\label{eq:st_P1} \\\\\n\\frac{\\textrm{d} }{\\textrm{d} x} [(\\mu x + v_0 ) P_{-1}] + \\frac\\gamma 2 P_0 - \\gamma P_{-1}&=& 0 \\label{eq:st_Pm1}\\\\\n \\frac{\\textrm{d}}{\\textrm{d} x} [\\mu x P_{0}] + \\gamma (P_1 + P_{-1}) - \\gamma P_0&=&0. \\label{eq:st_P0} \n\\end{eqnarray}\n\\endnumparts\nOur objective is to solve this set of equations to find $P_i(x)$ in the stationary state. \\\\\n\n\\noindent {\\bf Boundary Conditions:} To proceed with the solution we first need to specify the boundary conditions for $P_i(x).$ To determine these boundary conditions, we first note that, in the stationary state, the RTP is confined within a finite region bounded by $x_\\pm = \\pm v_0\/\\mu.$ This can be understood easily from the following argument: from the Langevin equation \\eref{eq:2dmodel_x} it is clear that\nif the particle is outside the region $[x_-,x_+],$ it always feels a drift towards the origin, irrespective of the value of $\\sigma_x.$ As a result, if the particle starts from some initial position $x_0 > x_+,$ or $x_0 < x_-,$ it will eventually reach the region $[x_-,x_+].$ Consequently, the stationary distribution has a finite support in the region $[x_-,x_+]$ \nand it is zero outside. To solve Eqs.~\\eref{eq:st_P1} - \\eref{eq:st_P0} then, we need to specify the boundary conditions at these two points. Let us first look at the behaviour of $P_1(x)$ near $x=x_-.$ During an infinitesimal time increment $\\Delta t,$ $P_1(x_-,t)$ evolves as,\n\\begin{eqnarray}\nP_1(x_-,t+ \\Delta t) = (1- \\gamma \\Delta t) P_1(x_- -\\Delta x,t) + \\frac \\gamma 2 \\Delta t P_0(x_-,t) \\label{eq:P1_dt}\n\\end{eqnarray}\nwhere the first term on the right hand side (r.h.s.) represents the transition when the position of the particle changes by an amount $\\Delta x$ during interval $\\Delta t,$ and the second term corresponds to the case when $\\sigma_x$ changes from $0$ to $1;$ the pre-factors $(1- \\gamma \\Delta t)$ and $\\gamma \\Delta t\/2$ denotes the probabilities for these two occurrences, respectively. Now, in the stationary state, the probabilities $P_i(x)$ are independent of time, hence, we have from \\eref{eq:P1_dt},\n\\begin{eqnarray} \\label{Eq:P1}\nP_1(x_-) = (1- \\gamma \\Delta t) P_1(x_- -\\Delta x) + \\frac \\gamma 2 \\Delta t P_0(x_-)\n\\end{eqnarray}\nMoreover, from Eq.~\\eref{eq:2dmodel_x} we have, for $\\sigma_x=1$ and near $x_-,$ $\\Delta x \\simeq (- \\mu x_- + v_0) \\Delta t = 2 v_0 \\Delta t >0,$ thus $P_1(x_- - \\Delta x) = P_1(x_--2 v_0 \\Delta t)$ which vanishes in the stationary state, as the argument $x_- - 2 v_0 \\Delta t$ is outside the region $[x_-,x_+]$. Then, taking $\\Delta t \\to 0$ limit in Eq. (\\ref{Eq:P1}), we get $P_1(x_-)=0.$ Using similar arguments for $P_{-1}$ and $P_0$, one finds the full set of boundary conditions to be satisfied by the set of equations \\eref{eq:st_P1} - \\eref{eq:st_P0},\n\\begin{eqnarray}\nP_1(x_-)=0, \\; P_{-1}(x_+) =0, \\; P_0(x_-) =0, \\; P_0(x_+) =0. \\label{eq:bc2}\n\\end{eqnarray}\nNote that the behaviour of $P_1(x_+)$ and $P_{-1}(x_-)$ remain unspecified. The set of boundary conditions for $P_1$ and $P_{-1}$ is very similar to the case of 2-state RTP \\cite{RTP_trap}. However, as we will see below, the presence of the third state $\\sigma_x=0$ leads to a richer behaviour in the present case. \\\\\n\n\n\\section{Exact Solution} \n\nThe straightforward strategy to solve a set of coupled first order equations like Eqs.~\\eref{eq:st_P1} - \\eref{eq:st_P0} is to decouple them and find separate equations for $P_i(x).$ However, our primary goal is to find the marginal position distribution of the particle, \\textit{i.e.}, the probability that the effective one-dimensional RTP has a position $x,$ irrespective of the spin-orientation $\\sigma_x.$ This is given by\n\\begin{eqnarray}\nP(x) &=& P_0(x) + P_{1}(x) + P_{-1}(x). \\label{eq:P-def}\n\\end{eqnarray}\nIn the following we attempt to derive an equation for $P(x)$ using Eqs.~\\eref{eq:st_P1} - \\eref{eq:st_P0}. To this end, we first define,\n\\begin{eqnarray}\nQ(x) = P_{1}(x) + P_{-1}(x),\\quad \\textrm{and} \\quad R(x) = P_{1}(x) - P_{-1}(x). \\label{eq:QR-def}\n\\end{eqnarray}\nIt is straightforward to see that in terms of these functions $P$ and $Q$, the four boundary conditions given by Eq.~\\eref{eq:bc2} translate to,\n\\begin{eqnarray}\nP(x_+) = Q(x_+), \\quad \\textrm{and} \\; P(x_-)=Q(x_-) \\label{eq:bc_PQ}\n\\end{eqnarray}\nNote that the boundary conditions of $R(x)$ remain unspecified. \nWe proceed by expressing Eqs.~\\eref{eq:st_P1} - \\eref{eq:st_P0} in terms of these functions $P$ and $Q.$ For this purpose, we first add equations \\eref{eq:st_P1}, \\eref{eq:st_Pm1} and \\eref{eq:st_P0} to get, \n\\begin{eqnarray}\n\\frac{\\textrm{d}}{\\textrm{d} x} \\bigg[\\mu x P(x) - v_0 R(x)\\bigg] =0 \\; \\Rightarrow \\mu x P(x) - v_0 R(x) = C \\label{eq:C}\n\\end{eqnarray}\nwhere $C$ is a constant independent of $x.$ To determine $C,$ we substitute $x=x_+$ in the above equation. Using the definitions of $P$ and $R,$ along with the boundary condition \\eref{eq:bc2}, we get, $C = (\\mu x_+ - v_0) P_1(x_+) = 0.$ Hence, from Eq.~\\eref{eq:C} we have,\n\\begin{eqnarray}\nR(x) = \\frac{\\mu x}{v_0} P(x) \\label{eq:PR}\n\\end{eqnarray}\nfor all values of $x.$ Now, adding Eqs.~\\eref{eq:st_P1} and \\eref{eq:st_Pm1} and using Eq.~\\eref{eq:PR}, we get, \n\\begin{eqnarray}\n\\mu x P'(x) + (\\mu - \\gamma) P(x) = \\mu x Q'(x) + (\\mu - 2 \\gamma) Q(x) \\label{eq:PQ_add}\n\\end{eqnarray}\nwhere $'$ denotes the derivative with respect to (w.r.t.) the argument of the functions. Next, we subtract Eq.~\\eref{eq:st_Pm1} from Eq.~\\eref{eq:st_P1} to get, \n\\begin{eqnarray}\n(\\mu x)^2 P'(x) + \\mu (2 \\mu - \\gamma) x P(x) = v_0^2 Q'(x). \\label{eq:PQ_subtract}\n\\end{eqnarray} \nEqs.~\\eref{eq:PQ_add} and \\eref{eq:PQ_subtract} are two coupled linear differential equations involving $P(x)$ and $Q(x).$ In the following, we use them to get two separate differential equations for $P(x)$ and $Q(x).$ But, first, it is convenient to use a change of variable $z= (\\frac{\\mu x}{v_0})^2$ with $0 \\le z \\le 1.$ Let us denote $\\tilde P(z)=P(x=v_0\\sqrt{z}\/\\mu)$ and $\\tilde Q(z)= Q(x=v_0\\sqrt{z}\/\\mu).$ Eqs.~\\eref{eq:PQ_add} and \\eref{eq:PQ_subtract} then become,\n\\begin{eqnarray}\n2z \\tilde P'(z) +(1-\\beta) \\tilde P(z) &= & 2z \\tilde Q'(z) +(1- 2 \\beta)\\tilde Q(z)\\label{eq:PQz1} \\\\\nz \\tilde P'(z) + \\left(1- \\frac \\beta 2 \\right) \\tilde P(z) &=& \\tilde Q'(z) \\label{eq:PQz2}\n\\end{eqnarray}\nwhere $\\beta= \\gamma \/\\mu.$ The two boundary conditions in Eq.~\\eref{eq:bc_PQ} reduce to a single condition for $\\tilde P$ and $\\tilde Q,$\n\\begin{eqnarray}\n\\tilde P(z=1) = \\tilde Q(z=1) \\;. \\label{eq:bc_z}\n\\end{eqnarray}\nAs we will see below, this boundary condition is enough to solve the differential equations uniquely. \n\n\nTo get an equation involving $\\tilde P(z)$ only, we take derivative of Eq.~\\eref{eq:PQz1} w.r.t. $z.$ Then, using Eq.~\\eref{eq:PQz2}, we immediately arrive at a second order differential equation,\n\\begin{eqnarray}\n\\fl \\qquad z(1-z) \\tilde P''(z) + \\left[\\frac{3-\\beta}2 - \\frac 12(7-3\\beta)z \\right]\\tilde P'(z) -\\left(1- \\frac \\beta 2 \\right)\\left(\\frac 32 - \\beta\\right)\\tilde P(z)=0 \\label{eq:Pz}\n\\end{eqnarray}\nIt is straightforward to check that the above equation is in the form of a hypergeometric differential equation,\n\\begin{eqnarray}\nz(1-z) \\tilde P''(z) +[c_1-(a_1+b_1+1)z] \\tilde P'(z) - a_1b_1\\tilde P(z) =0 \\label{eq:Hyp}\n\\end{eqnarray}\nwith the parameters,\n\\begin{eqnarray}\na_1= 1-\\frac \\beta2; \\quad b_1 = \\frac 32-\\beta; \\quad c_1 = \\frac {3 - \\beta}2.\n\\end{eqnarray}\nOne can also get a similar second order equation for $\\tilde Q(z).$\nTo this end, we first express $P'(z)$ in terms of $\\tilde Q(z)$ and $\\tilde Q'(z)$, \\textit{i.e.}, in a form similar to Eq.~\\eref{eq:PQz2}. Multiplying Eq.~\\eref{eq:PQz1} by $(1-\\frac \\beta 2)$ and Eq.~\\eref{eq:PQz2} by $(1-\\beta),$ and subtracting the latter resulting equation from the former, we get,\n\\begin{eqnarray}\nz \\tilde P'(z) = (1- 2 \\beta)\\left(1- \\frac \\beta 2 \\right)\\tilde Q - [1- \\beta -(2- \\beta)z] \\tilde Q'(z) \\label{eq:dPz}\n\\end{eqnarray}\nTaking a derivative of Eq.~\\eref{eq:PQz2} and using Eq.~\\eref{eq:dPz}, we get,\n\\begin{eqnarray}\n\\fl \\qquad z(1-z) \\tilde Q''(z) + \\left[\\frac{1- \\beta}{2} - \\frac 12(5 - 3\\beta)z \\right] \\tilde Q'(z) -\\left(1- \\frac \\beta2 \\right)\\left(\\frac 12 - \\beta\\right) \\tilde Q(z) =0\\label{eq:Qz}\n\\end{eqnarray}\nClearly, this is also a hypergeometric differential equation of the form \\eref{eq:Hyp}, but with a different parameter set,\n\\begin{eqnarray}\na_2 = 1- \\frac \\beta 2 = a_1, \\quad b_2 = \\frac 12 -\\beta = b_1 -1, \\quad c_2 = \\frac{1-\\beta}2 = c_1 -1.\n\\end{eqnarray}\n\n\\subsection{Position distribution for $\\beta \\ne 1$}\n\nThe general solutions for Eqs.~\\eref{eq:Pz} and \\eref{eq:Qz} can be written in terms of the hypergeometric function $_2 F_1(a,b,c;z)$ \\cite{dlmf}. For $c_1 \\ne 1,$ \\textit{i.e.}, for $\\beta \\ne 1,$ these general solutions read,\n\\begin{eqnarray}\n\\fl \\quad P(z) &=& A_1~\\left[ _2F_1(a_1,b_1,c_1;z)\\right] + B_1 z^{1-c_1}~ \\left[_2F_1(a_1-c_1+1,b_1-c_1+1,2-c_1;z)\\right] \\label{eq:Pz_sol} \\\\\n\\fl \\quad Q(z) &=& A_2~ \\left[_2F_1(a_2,b_2,c_2;z)\\right] + B_2 z^{1-c_2}~ \\left[_2F_1(a_2-c_2+1,b_2-c_2+1,2-c_2;z)\\right ] \\label{eq:Qz_sol} \n\\end{eqnarray}\nwhere $A_1, A_2,B_1,B_2$ are arbitrary constants. The case $\\beta=1$ is special, which we discuss later. To determine the constants $A_1, A_2,B_1,B_2$, we first use the original first order equations \\eref{eq:PQz1} and \\eref{eq:PQz2} which must be satisfied by the solution. Substituting Eqs.~\\eref{eq:Pz_sol} and \\eref{eq:Qz_sol} in Eq.~\\eref{eq:PQz2} and using well known identities involving the hypergeometric function, we get, $B_2 = \\frac{B_1}{1 + \\beta}$ and $A_2 = \\frac{A_1(1-\\beta)}{1- 2 \\beta}.$\nNext, we impose the boundary condition \\eref{eq:bc_z}. Once again, using properties of hypergeometric functions, we get\n\\begin{eqnarray}\nB_1 = \\frac{2 A_1}{\\sqrt{\\pi}} \\frac{\\Gamma(\\frac{3-\\beta}2)\\Gamma(\\frac 12 +\\beta)}{(1- 2 \\beta)\\Gamma(\\frac {1+\\beta}2)} \\label{eq:B1}\n\\end{eqnarray}\nTo completely specify $\\tilde P(z)$ we still need $A_1$ which can be determined using the normalization condition,\n\\begin{eqnarray}\n\\int_{x_-}^{x_+}\\textrm{d} x ~ P (x) =1 \\Rightarrow \\int_{0}^{v_0\/\\mu}\\textrm{d} x ~ \\tilde P \\left[\\left(\\frac{\\mu x}{v_0}\\right)^2 \\right] =\\frac 12. \\label{eq:norm}\n\\end{eqnarray}\nFortunately, this integral can be performed analytically and yields,\n\\begin{eqnarray}\n\\fl A_1 = \\frac{\\mu}{2 v_0}\\left[~_3F_2\\left({\\frac 12 \\;\\; \\frac 32 - \\beta \\;\\; 1-\\frac \\beta 2 \\atop \\frac 32 ~~ \\frac{3-\\beta}2}; 1 \\right) - \\frac{1}{\\beta \\sqrt{\\pi}} \\frac{\\Gamma(\\frac{3-\\beta}{2})\\Gamma(\\beta -\\frac 12)}{\\Gamma(\\frac{1+\\beta}2)}~ _3F_2\\left({\\frac 12 \\;\\; 1 - \\frac \\beta 2 \\;\\; \\frac \\beta 2 \\atop \\frac {1+\\beta}2 ~~ \\frac{\\beta}2+1}; 1 \\right) \\right]^{-1}\\;\\; \\label{eq:A1}\n\\end{eqnarray}\nwhere $_pF_q({a_1, a_2, \\dots a_p \\atop b_1, b_2, \\dots b_q};z )$ denotes the generalized hypergeometric function \\cite{dlmf}. Finally, we can write an explicit expression for the stationary position probability distribution,\n\\begin{eqnarray}\n\\fl P(x) &=& A_1~\\left [ _2F_1\\left(1- \\frac \\beta 2,\\frac 32 - \\beta,\\frac{3-\\beta}2;\\left(\\frac{\\mu x}{v_0}\\right)^2 \\right) \\right. \\cr\n\\fl &&\\left. +\\frac{2}{\\sqrt{\\pi}} \\frac{\\Gamma(\\frac{3-\\beta}2)\\Gamma(\\beta+\\frac 12)}{(1- 2 \\beta)\\Gamma(\\frac {\\beta+1} 2)}\\left(\\frac{\\mu x}{v_0}\\right) ^{\\beta -1}~ _2F_1 \\left(\\frac 12,1-\\frac \\beta 2,\\frac{\\beta +1}2;\\left(\\frac{\\mu x}{v_0}\\right)^2\\right) \\right] \\label{eq:Px_sol}\n\\end{eqnarray}\nwhere the normalization constant $A_1$ is given by Eq.~\\eref{eq:A1}. Note that, $P(x)$ is an even function of $x$ and it depends on the flip rate $\\gamma$ comes through the ratio $\\beta=\\gamma\/\\mu$ only. $P(x)$ takes particularly simple form for certain specific values of $\\beta,$\n\\begin{eqnarray}\nP(x) = \\left \\{\\begin{array}{ccc}\n \\frac{\\Gamma(\\frac 34)}{\\sqrt{\\pi} \\Gamma(\\frac 14)} \\frac{\\sqrt{\\mu v_0}}{\\sqrt{|x|(v_0^2-\\mu^2x^2)}} \\qquad \\qquad\\qquad \\qquad & \\textrm{for}& \\beta= \\frac 12 \\cr\n \\frac{\\mu}{v_0}(1-\\frac{\\mu |x|}{v_0}) \\qquad \\qquad \\qquad \\qquad \\quad \\qquad & \\textrm{for}& \\beta= 2 \\cr\n \\frac {6\\mu}{5v_0} \\left[1- 5 (\\frac{\\mu x}{v_0})^2 - \\left(\\frac{\\mu |x|}{v_0}\\right)^3\\left((\\frac{\\mu x}{v_0})^2-5 \\right) \\right] & \\textrm{for}& \\beta= 4.\n \\end{array}\n \\right.\\label{eq:Px_simple}\n\\end{eqnarray}\nOne can also write an explicit expression for $Q(x)$ using Eqs.\\eref{eq:Qz_sol} and \\eref{eq:B1},\n\\begin{eqnarray}\n\\fl Q(x) &=& \\frac{A_1}{(1-2\\beta)}\\left [(1-\\beta)~_2F_1\\left(1- \\frac \\beta 2,\\frac 12 - \\beta,\\frac{1-\\beta}2;\\left(\\frac{\\mu x}{v_0}\\right)^2 \\right) \\right. \\cr\n\\fl &&\\left. + \\frac{2}{\\sqrt{\\pi}} \\frac{\\Gamma(\\frac{3-\\beta}2)\\Gamma(\\beta+ \\frac 12)}{(1+\\beta)\\Gamma(\\frac {\\beta+1} 2)}\\left(\\frac{\\mu x}{v_0}\\right)^{\\beta +1}~ _2F_1 \\left(\\frac 12,1-\\frac \\beta 2,\\frac{\\beta +3}2;\\left(\\frac{\\mu x}{v_0}\\right)^2\\right) \\right]. \\label{eq:Qx_sol}\n\\end{eqnarray}\nFrom Eqs.~\\eref{eq:Px_sol} and \\eref{eq:Qx_sol} and using the relation \\eref{eq:PR} between $P(x)$ and $R(x)$ we can also calculate $P_i(x)$ individually in a straightforward manner. However, we do not give explicit expressions for them here. \nFigure \\ref{fig:px}(a) and (c) show plots of $P(x)$ as a function of $x$ for different values of $\\beta$ calculated from Eq.~\\eref{eq:Px_sol} along with the data obtained from numerical simulations. It appears that, similar to the 2-state RTP, the distribution shows two different behaviours near the boundary $x=x_\\pm$ depending on the value of $\\beta.$ Moreover, it appears from the plots that for $\\beta<1,$ $P(x)$ also diverges near the origin $x=0$ while it shows a cusp-like behaviour for large $\\beta.$ In the following we investigate the behaviour of $P(x)$ in more details and characterise this change in shape. \\\\\n\n\n\n\\noindent{\\bf Behaviour near $x=0$:} To understand the behaviour of $P(x)$ near the origin we use the series expansion of the hypergeometric function $_2F_1(a,b,c;z)$ near $z=0,$\n\\begin{eqnarray}\n_2F_1(a,b,c;z) = 1+ \\frac{ab}{c}z+\\frac{ab(1+a)(1+b)}{2c(1+c)} z^2+ \\cal O(z^3)\n\\end{eqnarray}\nUsing this expansion in Eq.~\\eref{eq:Px_sol}, we have, near $x=0,$\n\\begin{eqnarray} \\label{asympt_x0}\nP(x) \\sim \\left \\{ \\begin{array}{l}\n B_1 \\left( \\frac{\\mu}{v_0} x\\right)^{\\beta-1} \\quad \\qquad \\quad \\textrm{for} \\;\\; \\beta < 1 \\cr\n A_1(1 - C_1 x^{\\beta -1}) \\quad \\textrm{for} \\;\\; 1 < \\beta < 3 \\cr\n A_1(1 - C_2 x^2) \\qquad \\textrm{for} \\;\\; \\beta > 3\n \\end{array}\n\\right.\n\\end{eqnarray}\nwhere $B_1$ and $A_1$ are given respectively in Eqs. \\eref{eq:B1} and~\\eref{eq:A1} while $C_1$ and $C_2$ are given~by\n\\begin{eqnarray}\n&&C_1 = \\frac{2}{\\sqrt{\\pi}} \\frac{\\Gamma(\\frac{3-\\beta}2)\\Gamma(\\beta+\\frac 12)}{(2 \\beta-1)\\Gamma(\\frac {\\beta+1} 2)} \\left( \\frac{\\mu}{v_0}\\right)^{\\beta-1}\\label{C1} \\;, \\\\\n&&C_2 = \\frac{2 \\beta^2 + 6 - 7\\beta}{2(\\beta-3)} \\left( \\frac{\\mu}{v_0}\\right)^{2} \\label{C2} \\;.\n\\end{eqnarray}\nClearly, for $\\beta <1,$ $P(x)$ diverges near the origin whereas for $\\beta > 1$ it approaches a finite value. The approach also depends on the value of $\\beta:$ for $1 < \\beta < 3,$ $P(x)$ has a cusp-like behaviour near the origin while for $\\beta \\ge 3$ it shows a quadratic behaviour, resembling a Gaussian around the origin. Indeed, in the diffusive limit, when $\\gamma \\to \\infty$ and $v_0 \\to \\infty$ keeping $v_0^2\/(2 \\gamma) = D$ fixed (as a consequence $\\beta \\to \\infty$ in this limit), we find from Eq. \\eref{C2} that $C_2 \\to \\mu\/(2D)$. As a result, from the third line of \\eref{asympt_x0}, we recover the Boltzmann distribution $P(x) \\sim e^{-\\mu\/(2D) x^2}$ which actually holds for all $x$. \\\\\n\n\\noindent{\\bf Behaviour near $x=x_\\pm$:} The position distribution $P(x)$ also shows an interesting behaviour near the boundaries $x= x_{\\pm}.$ As $P(x)$ is symmetric in $x,$ it suffices to explore its nature near one boundary, say $x_+.$ To characterise the same we use the series expansion of $_2F_1(a,b,c;z)$ near $z=1.$ From Eq.~\\eref{eq:Pz_sol}, we have, for $z\\to 1^-,$\n\\begin{eqnarray}\n\\tilde P(z) \\sim \\left\\{ \\begin{array}{l}\n (1-z)^{\\beta -1} \\quad \\textrm{for} \\;\\; \\beta < 3 \\cr\n (1-z)^2 \\quad \\textrm{for} \\;\\; \\beta > 3 \\cr \\;.\n \\end{array}\n \\right.\n \\end{eqnarray}\nHence, near $x=x_+,$ we have the following behaviour of $P(x):$\n\\begin{eqnarray}\nP(x) \\sim \\left\\{ \\begin{array}{l}\n(x_+ - x)^{\\beta -1} \\quad \\textrm{diverges for} \\;\\; \\beta < 1 \\cr\n (x_+ - x)^{\\beta -1} \\quad \\textrm{vanishes for} \\;\\; 1< \\beta \\le 3 \\cr\n (x_+ - x)^2 \\quad \\textrm{vanishes for} \\;\\; \\beta > 3 \\;.\n \\end{array}\n \\right.\n \\end{eqnarray}\nA similar behaviour is seen also near $x=x_-$. Note that this ``freezing'' for the leading behaviour for $\\beta > 3$ occurs only for the three-state\nmodel, but not for the two-state model \\cite{RTP_trap}. \n\n\n \\begin{figure}[t]\n\\centering \\includegraphics[width=10.3 cm]{.\/Px1.pdf} \\includegraphics[width=5.1 cm]{.\/Px2.pdf}\n \\caption{Stationary position distribution $P(x)$ as a function of $x$ for the 3-state model for (a) $\\beta <1$, (b) $\\beta=1,$ and (c) $\\beta>1.$ Here $v_0=1$ and $\\mu=1.$ The symbols correspond to the data obtained from numerical simulations while solid lines are obtained from the exact result [see Eq.~\\eref{eq:Px_sol} and Eq.~\\eref{eq:Px_a1}].}\\label{fig:px}\n \\end{figure}\n\n \n\n\\subsection{Position distribution for $\\beta=1$} \n\nAs mentioned before, the case $\\beta=1$ is special. In this case,\nthe differential equations~\\eref{eq:Pz} and \\eref{eq:Qz} reduce to,\n\\begin{eqnarray}\nz(1-z) \\tilde P''(z) + (1-2z) \\tilde P'(z) - \\frac 14 \\tilde P(z) =0 \\label{eq:Pz_al1}\\\\\nz(1-z) \\tilde Q''(z) -z \\tilde Q'(z) + \\frac 14 \\tilde Q(z) =0. \\label{eq:Qz_al1}\n\\end{eqnarray}\nwhich correspond to two hypergeometric equations with $c_1=1$ and $c_2=0,$ along with $a_1=a_2=b_1=1\/2,b_2=-1\/2.$ Eq.~\\eref{eq:Pz_sol} is not a general solution anymore as the two hypergeometric functions therein become identical. We use Mathematica to solve Eqs.~\\eref{eq:Pz_al1} and \\eref{eq:Qz_al1} and it turns out that the general solutions can be expressed in the form,\n\\begin{eqnarray}\n\\tilde P(z) &=& \\frac {2 A_1} \\pi K(1-z) + B_1 \\cal Q_{-\\frac 12}(2z-1) \\label{eq:Pz_al1_sol}\\\\\n\\tilde Q(z) &=& A_2 z ~_2F_1\\left(\\frac 12, \\frac 32, 2;z \\right) + B_2 ~G^{20}_{22}\\left({\\frac 12~ \\frac 32 \\atop 0 ~1};z \\right). \\label{eq:Qz_al1_sol}\n\\end{eqnarray}\nHere $K(u)$ is the Legendre's complete elliptic integral of the first kind (see Ref.~\\cite{Gradshteyn} and Eq.~19.2.8 in Ref.~\\cite{dlmf}), $G^{mn}_{pq}({a_1,\\dots a_p \\atop b_1 \\dots b_q};z)$ is the Meijer's G-function (see Ref.~\\cite{Gradshteyn} and Eq.~16.17.1 in Ref.~\\cite{dlmf}) and $\\cal Q_\\nu(u)$ is the Legendre function of the second kind (see Eq.~14.3.7 in Ref.~\\cite{dlmf}). \n\nTo determine the arbitrary constants $A_1,A_2,B_1$ and $B_2$ we use the same strategy as in the previous section. First, we note that the solutions in Eqs.~\\eref{eq:Pz_al1_sol} and \\eref{eq:Qz_al1_sol} must satisfy the original first order equations \\eref{eq:PQz1} and \\eref{eq:PQz2} with $\\beta=1$ for all values of $z.$ We then look at the behaviour of $\\tilde P(z)$ and $\\tilde Q(z)$ in Eqs.~\\eref{eq:Pz_al1_sol} and \\eref{eq:Qz_al1_sol} near $z=0$. In this limit both $K(1-z)$ and $G^{20}_{22}\\left({\\frac 12~ \\frac 32 \\atop 0 ~1};z \\right)$ diverge logarithmically whereas the Legendre and hypergeometric functions approach a constant value. \nSubstituting the series expansions of these functions back into Eq.~\\eref{eq:PQz2} and comparing coefficients of $\\ln z$ and different powers of $z,$ we get, $B_2 = A_1,$ and $\\quad A_2 = -\\frac \\pi 4 B_1.$ It is also straightforward to check that Eq.~\\eref{eq:PQz1} gives the same relation. We still have two independent constants $A_1$ and $B_1.$ To determine these we use the boundary condition \\eref{eq:bc_z}. Using the limiting behaviours of the special functions we have, for $z \\to 1^{-},$ $\\tilde P(z) - \\tilde Q(z) = B_1 + \\cal O(1-z)$ which immediately implies $B_1=0$ [see Eq.~\\eref{eq:bc_z}].\nThe last remaining constant $A_1$ can be determined from the normalization condition \\eref{eq:norm} and yields $A_1=\\frac{\\mu}{\\pi v_0}.$\nFinally, we have, for $\\beta=1,$\n\\begin{eqnarray}\nP(x) = \\frac {2 \\mu}{\\pi^2 v_0} K\\left (1- \\frac{\\mu^2 x^2}{v_0^2} \\right),\\quad \\textrm{and} \\quad Q(x) = \\frac{\\mu}{\\pi v_0} G^{20}_{22}\\left({\\frac 12~ \\frac 32 \\atop 0 ~1}; \\frac{\\mu^2 x^2}{v_0^2} \\right). \\label{eq:Px_a1}\n\\end{eqnarray}\nFigure \\ref{fig:px}(b) shows a plot of $P(x)$ for $\\beta=1$ together with the same obtained from numerical simulations. \nTo understand the behaviour near the origin $x=0$ and the boundaries $x=x_\\pm$ we look at the series expansion of $P(x).$ Near $x=0,$ a logarithmic divergence is seen, $P(x) \\sim - \\ln x.$\nOn the other hand, near the boundaries $x=x_\\pm,$ $P(x)$ approaches a constant value, $\\lim_{x \\to x_\\pm} P(x) = \\frac{\\mu}{\\pi v_0}.$\n\n\n\n\n \n\n\\section{Conclusion}\n\nIn this paper, we have solved exactly the stationary position distribution of a one-dimensional run-and-tumble (RTP) particle with three discrete internal states and subjected to an external harmonic potential. To our knowledge, this is the first exact solution with three states that generalizes the well-known result for the standard two-state RTP. We showed that the stationary state exhibits a rich behavior as a function of the single parameter $\\beta = \\gamma\/\\mu$ (where $\\gamma$ represents the rate at which the internal state changes and $\\mu$ is the stiffness of the trap). One of the interesting outcomes is that the stationary distribution undergoes a shape-transition at $\\beta=1$.\n\nWhile we were able to characterise the stationary state of a three-state RTP in a harmonic trap exactly, it would be interesting\nto study the relaxational dynamics towards this stationary state, as was recently done for the two-state RTP \\cite{RTP_trap}. \nIt would also be natural to extend our studies to non-harmonic potentials, such as $U(x) \\sim |x|^p$, with $p>0$. Another natural extension \nwould be to consider an RTP particle with more than $3$ internal states. Finding even the stationary state of a general $n$-state RTP with $n > 3$ remains a challenging open problem. \n\n\n\n \n\\ack\nWe acknowledge support from the project 5604-2 of the Indo-French Centre for the Promotion of Advanced Research (IFCPAR). \nS. N. M. acknowledges the support from the Science and Engineering Research Board (SERB, government of India) under the VAJRA faculty scheme (Ref.VJR\/2017\/000110) during a visit to the Raman Research Institute in 2019, where part of this work was carried out. U. B. acknowledges support from Science and Engineering Research Board (SERB), India under Ramanujan Fellow-ship (Grant No. SB\/S2\/RJN-077\/2018) and CNRS for a one month visit to LPTMS, Univ. Paris-Sud. \n \n \n\\vspace*{0.5 cm} \n \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nThis paper presents polarimetric observations of the \nearthshine on the Moon's dark side for a characterization \nof the integrated polarimetric properties of the planet Earth \nfor the future investigation of Earth-like extra-solar planets.\n\nWith the rapid progress in observational techniques the detection \nof reflected light from terrestrial or even Earth-like extra-solar\nplanets may become possible in the near future with high-contrast imaging of\nvery nearby $(d \\mathrel{\\hbox{\\rlap{\\lower.55ex \\hbox{$\\sim$}} \\kern-.3em \\raise.4ex \\hbox{$<$}}} 5pc)$ stars. Statistical studies based on the\nradial velocity survey of stellar reflex motions due to \nlow mass planets \\citep{Mayor2011}\nor the planetary transit frequency \nof small planets by the KEPLER satellite \n\\citep{Howard2012} \nindicate both \nthat terrestrial planets could be present with high probability \naround every nearby star. The detection of a periodic RV-signal \nin $\\alpha$ Cen B by\n\\citet{Dumusque2012}, \nwhich was \nattributed to a planet with a mass of $\\approx 1 M_{\\rm Earth}$, \ndemonstrates that the very nearest stars are really excellent \ntargets for the search of extra-solar planets.\n\nThe intensity contrast between a reflecting \nplanet and the parent star is\n\\begin{equation}\n C_I(\\alpha,\\lambda)= f(\\alpha,\\lambda)(R_p\/d_p)^2\\,, \n\\end{equation}\nwhere $\\alpha$ is the phase angle, $R_p$ the radius of the planet, \n$d_p$ its separation to the star, and $f(\\alpha,\\lambda)$ the phase\ndependent reflectivity. Thus the contrast is high for small\nseparations $d_p$ and therefore the prospect for direct detection \nis particularly favorable for close-in planets $d_p \\mathrel{\\hbox{\\rlap{\\lower.55ex \\hbox{$\\sim$}} \\kern-.3em \\raise.4ex \\hbox{$<$}}} 0.3$~AU \naround nearby stars for which such a small separation planet can still be\nspatially resolved. However, detecting a faint signal from a \nreflecting planet at an angular separation of about 0.1 arcsec \nfrom a bright star is challenging and requires an instrument\nwith high spatial resolution and very high contrast capabilities\nbased on coronagraphy and some kind of differential imaging.\nThe upcoming planet finder instruments SPHERE \n\\citep{Beuzit2008} and \nGPI \\citep{Macintosh2012}\nwill provide both much improved performance \nfor substantial progress in this direction. A particularly \npromising technique for the search of reflected light from \nplanets around the nearest stars is differential polarimetric\nimaging available with the SPHERE instrument. With sensitive \npolarimetry one can search for a polarized signal due to the \nscattered and therefore polarized light from the planet in the halo \nof the unpolarized light from the star\n\\citep[e.g.][]{Schmid2006}. \nThe measureable\npolarization contrast can be described similar to the intensity\ncontrast by \n\\begin{equation}\nC_p(\\alpha,\\lambda)= p(\\alpha,\\lambda) f(\\alpha,\\lambda)(R_p\/d_p)^2\\,,\n\\label{cpol}\n\\end{equation} \nwhere $p(\\alpha,\\lambda)$ is the\nintegrated fractional polarization. Therefore, the investigation of\n$p(\\alpha,\\lambda)$ and the polarization flux \n\\mbox{$p(\\alpha,\\lambda) \\times f(\\alpha,\\lambda)$} \nof planet Earth is important for the planing of future observing projects on\nextra-solar planetary systems and the interpretation of observational \ndata. Up to now only very limited data are availabe for the\nintegrated polarization of the planet Earth. The oldest and still best\npolarization phase curve of Earth originates from the \nobservations of the earthshine by\n\\citet{Dollfus1957}.\n\nHe determined the earthshine polarization phase curve \n$p_{\\rm es}(\\alpha)$ with visual observations (i.e. in the V band) \nfor Earth phase angles from $\\alpha =22^\\circ$, about 1.5 days after\nnew moon, to about $\\alpha = 140^\\circ$, between half and full moon. \n\\citet{Dollfus1957} finds for dark regions (maria) on the Moon\na steady increase of the fractional polarization of the\nearthshine from about $p_{\\rm es}\\approx 2$~\\% around $\\alpha=30^\\circ$, \nto a maximum polarization of about $p_{\\rm es}\\approx 10$~\\% \nfor $\\alpha\\approx 100^\\circ$, and a decline for larger\n$\\alpha$'s down to $p_{\\rm es}\\approx 4$~\\% at $\\alpha\\approx 140^\\circ$. \n\\citet{Dollfus1957}\nalso finds a higher fractional polarization for\nthe back-scattered light for dark regions with surface albedo of\nabout $a=0.1$ than for bright regions with $a=0.2$. In addition\nhe notes a wavelength dependence in the fractional polarization\nof the earthshine with higher values at shorter wavelength.\nOne should note that the back-scattering by the lunar surface \nintroduces a depolarization of the earthshine. Thus, the \nfractional polarization of the light scattered by Earth is\nhigher by a factor of about 2 to 3 than the measured value from the\nback-scattered earthshine. \n\nSpace experiments did not provide much progress because full Earth\npolarimetry was to our knowledge not taken or at least not published. \nEarth observing satellites with polarimetric capabilities took usually \nmeasurements of only small fractions of the Earth surface from which it is \ndifficult to determine the net polarization for the entire planet. \nA result from the POLDER satellite was reported by \n\\citet{Wolstencroft2005} \nwho obtained fractional polarization values for $\\alpha=90^{\\circ}$ for three wavelengths and different\ncloud coverages. For a typical value for the \naverage cloud coverage of 55 \\% \nthey derive for the polarization of the planet Earth:\n$p(443~{\\rm nm})=22.6$~\\%, $p(670~{\\rm nm})=8.6$~\\%, $p(865~{\\rm nm})=7.3$~\\%. \n\n\nAn interesting\nnew result on the Earth polarization from earthshine measurements is\nthe VLT spectro-polarimetry from \n\\citet{Sterzik2012}\nwhich show\nnarrow spectral features due to water, O$_2$, and O$_3$ \nabsorptions in the Earth atmosphere and a rise of the fractional \npolarization towards the blue due to Rayleigh scattering.\n\nModel calculations have been made for the fractional\npolarization of the reflected light from Earth-like planets \n\\citep{Stam2008} \nas well as the polarization produced by reflecting clouds\n\\citep[e.g.][]{Karalidi2011, Karalidi2012, Bailey2007}\nor glint from ocean water surfaces \n\\citep{Williams2008}. \nThe models\nprovide an adequate description of the dominating scattering processes\nand the signatures of different surface types. However, the overall\nnet polarization of Earth depends strongly on the not so well\nknown contributions of the different areas to the total signal. \nTherefore it is very desirable to have better observational\ndata which constrain between the various model options. \n \nLunar earthshine observations are very attractive for the investigation\nof the intensity and polarization of the reflected light of the\nEarth because they provide the integrated scattered light signal \nfrom the whole planet Earth from the ground. However, for the\nretrieval of the real level of scattered intensity $f_{\\rm E}$ and\nfractional polarization $p_{\\rm E}$ of the Earth from the measured \nearthshine signals $f_{\\rm es}$ and $p_{\\rm es}$ \none needs also to consider the back-scattering properties of \nthe absorbing and depolarizing lunar surface. \n\nIn addition, earthshine observations are very special because \nthe Moon is a bright and large target for modern astronomical\ninstrumentation and because the contrast between the bright crescent \nand the dark side of the Moon is very high.\nIt is therefore not straightforward to disentangle\nthe earthshine from the disturbing contributions of the variable \natmospheric (and instrumental) stray light from the bright moonshine \nand of the twilight. \n\nIn this paper we describe new earthshine polarization measurements\ntaken with an imaging polarimeter specially designed for earthshine\nobservations. Our data cover Earth phase angles from $30^{\\circ}$ to $110^{\\circ}$\nproviding calibrated\n$p_{\\rm es}(\\alpha)$ curves in the four broad-band filters B, V, R and\nI for lunar maria and highlands which are corrected for the stray light\nfrom the moonshine and the sky background. Section \\ref{s:instrument} describes\nour instrument and Sect. \\ref{s:observations} our measurements while Sect. \\ref{s:datareduction} discusses\nthe data reduction. The observational results are given in\nSect. \\ref{s:results} and then we discuss in Sect. \\ref{s:retro-reflection} our correction for the depolarizing\neffect of the lunar surface. The final polarization phase curves $p_{\\rm E}(\\alpha,\\lambda)$\nfor Earth and the derivation of the polarization flux $p_{\\rm E} \\times f_{\\rm E}$\nand the Earth-Sun polarization contrast $C_{\\rm p}$ are given in Sect. \\ref{s: earth polarization}.\nThe last Section \\ref{s:conclusions} gives a summary and discusses the potential of\nearthshine polarization measurements.\n\n\n\n \n\\section{Instrumentation}\\label{s:instrument}\n\n\\subsection{Instrument requirements}\n \n \\begin{figure}\n \\centering\n \\includegraphics[width=9cm]{evsm.eps}\n \\caption{Reflectivity of the Earth $f_{\\rm E}(\\alpha_{\\rm E},\\lambda)$\n and the Moon $f_{\\rm M}(\\alpha_{\\rm E},\\lambda)$ for \n B (solid), V (dotted), R (dashed) and I (dash-dot). The thick dashed line is the difference of the surface brightness between \n earthshine $B_{\\rm es}(\\alpha_{\\rm E},\\lambda)$ and moonshine $B_{\\rm M}(\\alpha_{\\rm E},\\lambda)$ for the \\mbox{400-700 nm} pass band. \n }\n \\label{evsm} \n \\end{figure}\n\n\n \\begin{figure*}[!htb]\n \\centering\n \\includegraphics[width=18cm]{ESPOL_paper_setup_V3.eps}\n \\caption{Schematic overview of ESPOL with optical\n design and components drawn to scale. The focal mask is\n located in the focal plane of the 21cm telescope.\n }\n \\label{setup} \n \\end{figure*}\n\nThe (surface) brightness \nof the lunar earthshine, the moonshine, and the contrast between\nthe bright and dark side of the Moon are described in the literature and summarized\nin Fig. \\ref{evsm}.\nThe reflectivity of the Moon \n$f_{\\rm M}(\\alpha_{M},\\lambda)$ \nis from \\citet{Kieffer2005} and plotted as function of the phase angle of the Earth\n$\\alpha_{\\rm E}$ using $\\alpha_{\\rm E}=180^\\circ - \\alpha_{\\rm M}$.\nThe reflectivity of the Earth $f_{\\rm E}(\\alpha_{E},\\lambda)$ \nis from earthshine observations and model calculations \nby \\citet{Palle2003} for the pass band 400-700 nm.\nDue to the lack of multi-color earthshine observations the same shape \nfor the reflectivity phase curve given by \n\\citet{Palle2003}\nis adopted for all colors and the curves for the different bands are just scaled\nwith the factors derived from earthshine spectra from \\citet{Arnold2002} as described in Sect. \\ref{flux contrast}.\nMeasurements of the surface brightness of the earthshine $B_{\\rm es} (\\alpha,\\lambda)$ and the moonshine $B_{\\rm M} (\\alpha,\\lambda)$ \nwere derived from observations of fiducial patches of highland regions \\citep[see also][]{Qiu2003} \nby \n\\citet{Montanes2007} for the pass band \\mbox{400-700 nm} and Earth phase angles between $\\alpha_{\\rm E} = 30^{\\circ}-140^{\\circ}$.\nIn Figure \\ref{evsm} we adopt their waxing Moon results\nand derive the mean daily surface\nbrightness contrast $B_{\\rm es}(\\lambda) - B_{\\rm M}(\\lambda)$ (thick dashed line) between the dark and the bright side of the Moon.\nFor $\\alpha=90^\\circ$ (half moon) the\nsurface brightness of the earthshine is 14.9 mag\/arcsec$^{2}$. The difference between earthshine\nand moonshine ranges from 8 - 12 mag\/arcsec$^{2}$ between $\\alpha_{\\rm E} = 30^{\\circ}-140^{\\circ}$.\n\nThe surface brightness of the earthshine is highest for the\nnew moon phase and decreases with the Sun-Earth-Moon phase angle $\\alpha$\n(Fig. \\ref{evsm}). Observations for small $\\alpha$ near new moon require\ndaytime or twilight observations for which a correction\nfor the sky light is impossible or difficult. For $\\alpha\\approx\n45^\\circ$ an observational window of roughly 30 minutes with reasonably\ndark sky conditions becomes available after sunset or before sunrise\nfor useful earthshine polarization measurements. \n\n\nObservations during the night with \nmuch reduced sky background levels are possible for larger $\\alpha$,\nbut the brightness of the moonshine due to the solar illumination increases rapidly \n(\\mbox{Fig. \\ref{evsm}}). At around $\\alpha\\approx 90^\\circ$ the contrast between moonshine and earthshine\nbecomes greater than about 10$^{4}$ and the light scattering in the Earth atmosphere and the instrument becomes\nmore and more a problem for earthshine observations particularly in the red where the\nmoonshine is strong and the earthshine weak.\nThe light from the twilight sky and \nthe moonshine are both strongly polarized $p>3~\\%$ and this needs\nto be considered for an accurate measurement of the earthshine \npolarization. The polarization of the moonshine is discussed in detail in Sect. \\ref{s: moonshine}. \n\n\\subsection{The earthshine polarimeter}\\label{s:ESPOL}\n\nThe EarthShine POLarimeter (ESPOL) measuring concept takes the\nbackground and stray light conditions for earthshine\nobservations into account. The instrument allows imaging polarimetry\nof the entire Moon and the surrounding sky regions in order to\nmeasure the polarization signal of the weak earthshine on top of \nthe strong stray light from the moonshine \nand\/or the light contribution from the sky. ESPOL includes in\naddition exchangeable focal plane masks to block the light from\nthe bright moonshine. The blocking of the bright crescent is required to allow for \nintegrations of a few seconds without heavy detector saturation.\n \nESPOL is a dual-beam imaging linear polarimeter based on the\nrotating half-wave retarder plate and Wollaston polarization\nbeam splitter concept. A schematic overview of the instrument is given \nin Fig. \\ref{setup}. The instrument includes a holder for exchangeable\nfocal masks with different Moon phase shapes to block the light \nfrom the bright lunar crescent in order to avoid \nheavy detector saturation. ESPOL uses a super-achromatic \n$\\lambda \/ 2$ retarder plate on a motorized rotational stage for \npolarization beam switching and the selection of the $Q$ and $U$\npolarization direction. The following Wollaston prism splits\nthe light into the ordinary $i_\\parallel$ and extraordinary $i_\\perp$ beams with polarization\nperpendicular to each other. Both beams, each with a field of view\nof $50'\\times 40'$ \nare imaged on the same 3072 x 2048 pixel CCD \ndetector with a pixel scale of 1.5 arcsec\/pixel. For our measurements we used\na pixel binning of $3\\times3$ pixels which reduced the spatial resolution\nto roughly 10 arcsec. \n\nColor or neutral \ndensity filters with a diameter of 5 cm or \\mbox{2 inches} \ncan be inserted into the five-position filter wheel\nlocated in the collimated beam or into the camera \nfilter wheel respectively. In order to optimally align the focal\nmasks to the orientation of the bright lunar crescent the \nmask holder can be rotated around the optical axis. In addition\nthe whole instrument can be rotated around the optical axis to\nfix the zero-point of the polarization direction to any desired\norientation. \n\nESPOL was built in-house for low costs using equipment for amateur\nastronomers and standard polarimetric and optical components for\nthe wavelength range of 360-860 nm. The instrument is attached to \nan equatorially mounted 21 cm Dall-Kirkham Cassegrain telescope. \nAs CCD system a thermo-electrically cooled SBIG-STL 6303E camera system \nwith integrated filter wheel and shutter is used. \n \nFigure \\ref{raw} \nillustrates the data format delivered by the CCD. \nDue to the large field of view and the use of simple optical\ncomponents the system shows quite some image distortion in\nnorth-south direction where the Moon diameter between ordinary\nand extraordinary beam differs by about 5\\%. \nThese distortions can be tolerated because we are not interested\nin high spatial resolution but in the fractional polarization\nof extended surface regions.\n\n\n \\begin{SCfigure}\n \\centering\n \\includegraphics[width=5.4cm]{espol_raw.eps}\n \\caption{ESPOL raw frame with the two polarization\n images $i_\\parallel$ and $i_\\perp$ of the\n earthshine on the Moon and the dark focal mask which blocks\n the light from the bright crescent.}\n \\label{raw} \n \\end{SCfigure}\n\n\n \n \n\n\\section{Observations}\\label{s:observations}\n\nWith ESPOL we measured the polarization of the earthshine \nfor different phase angles and different wavelengths using a Bessell\nB, V, R, I filter set \\citep{Bessell2012}. \n\nTo minimize read-out overheads and detector noise \nthe CCD was operated with $3\\times3$ pixel binning providing a\nspatial resolution of about 10 arcsec, which is sufficiently high\nfor distinguishing mare and highland regions on the Moon. \n\nTo minimize differential instrumental effects in the polarimetric\nsignal the measurements were performed in beam-exchange mode \n\\citep{Tinbergen1996}. \nOnly the linear Stokes components \n$Q\/I$ and $U\/I$ are measured. \n\nOne polarimetric cycle consists of\ntwo measurements with half-wave plate position $0^\\circ$ and\n$45^\\circ$ for $Q\/I$ and two measurements with position $22.5^\\circ$ and\n$-22.5^\\circ$ for $U\/I$. Typical integration times per exposure\nwere about 2-10 s per half-wave plate position so that one full cycle \ncould be recorded in about \none minute. This is sufficiently fast to avoid problems with\nguiding drifts and strongly changing atmospheric conditions. \nTo improve the S\/N ratio typically a series of about 5 to 10 such datasets \nwere recorded for each wavelength band during one observing night. \n\nESPOL was rotated for all our measurements, including standard stars, \ninto the orientation of the plane Sun-Earth-Moon so that the \nStokes $+Q$ direction is a polarization perpendicular to this plane\nand $-Q$ a polarization in this plane. The Stokes $\\pm U$ directions\nare $\\pm 45^\\circ$ with respect to this plane. The alignment was done by eye\nby rotating the complete instrument until the crescent shaped focal mask completely blocked \nthe moonlight which lead to an\nalignment accuracy of about $\\Delta\\theta=\\pm 2^\\circ$.\n\nFor instrument monitoring and calibration additional polarimetric \nmeasurements of the moonshine and polarized\/unpolarized standard stars \nas well as darks and twilight flatfield calibrations were recorded \nduring each observing night. \n \nOur data were collected during two observing runs in March and \nOctober 2011. For the run in March the instrument was \ninstalled at the former Swiss Federal Observatory in Zurich \nat an altitude of 470 m above sea level. This \nrun served mainly for first instrument testing and data with \nlimited wavelength and phase coverage were obtained. Despite the \nnon-optimal observing location in the heart of the city \nthe data quality was good enough to be included in this study. \nFor the second run we moved the instrument to the former Arosa \nAstrophysical Observatory at an altitude of 2050 m a.s.l. located \nin the Alps of eastern Switzerland. This site provides a much darker\nsky and a much reduced level of light scattering in the Earth\natmosphere allowing measurements of the earthshine polarization for\nlarger phase angles. \n \nBoth observing runs cover phase angles for the waxing Moon only. For\nthe March measurements the earthshine originates mainly from \nthe Atlantic Ocean, the Pacific Ocean, and the american continent while\nin October the earthshine was due to reflected light from South\nAmerica and the Atlantic Ocean. \\mbox{Table \\ref{observations}} gives \nan overview of the observed phase angles of our measurements, the used filters \nand the number of polarization cycles for each filter. \n\n \\begin{table}\n \\caption{Observing log. \n The number of polarization cycles refer to the different\n filters taken for this date.} \n \\label{observations} \n \\centering \n \\begin{tabular}{C{19mm} C{10mm} C{13mm} C{18mm} } \n \\hline \n observing date & phase & filters & \\# pol. cycles \\\\\n \\hline \n 07.03.2011 & $31.5^{\\,\\circ}$ & V & 11 \\\\\n 08.03.2011 & $42.5^{\\,\\circ}$ & B, V, R & 3, 9, 2 \\\\\n 11.03.2011 & $75.5^{\\,\\circ}$ & B, V, R & 8, 19, 4 \\\\\n 02.10.2011 & $73.0^{\\,\\circ}$ & B, V, R, I & 1, 3, 4, 4 \\\\\n 03.10.2011 & $85.5^{\\,\\circ}$ & B, V, R, I & 5, 5, 4, 5 \\\\\n 04.10.2011 & $98.0^{\\,\\circ}$ & B, V, R, I & 16, 15, 21, 6 \\\\\n 05.10.2011 & $109.5^{\\,\\circ}$ & B, V, R, I & 8, 8, 10, 8 \\\\\n \\hline \n \\end{tabular}\n \\end{table}\n \n\n\n\\section{Data reduction}\\label{s:datareduction}\n\n\\subsection{Polarimetric reduction}\n\nFigure \\ref{raw} shows a typical ESPOL raw\nframe with the ordinary $i_\\parallel$ and extraordinary $i_\\perp$\nbeams from the Wollaston showing the earthshine on the Moon in \ntwo opposite polarization directions. The \nbright crescent is blocked by the focal mask in order to suppress\nstray light in the instrument and to avoid disturbing detector saturation. \n \nIn the first data reduction step the raw images were dark subtracted\nbefore the two opposite polarization images $i_\\parallel$ and\n$i_\\perp$ for all half-wave plate orientations\n($0^\\circ,45^\\circ,22.5^\\circ$ and $-22.5^\\circ$) were cut out and aligned.\nThen the fractional Stokes parameter $Q\/I$ images were calculated\naccording to the beam-exchange method described in\n\\citet{Tinbergen1996}:\n\\begin{equation} \\label{beam-exchange calculation}\nq = \\frac{Q}{I} = \\frac{R - 1}{R + 1} \\hspace{0.5 cm} \n with \\hspace{0.5 cm} R^{2} =\n\\frac{i_{0,\\parallel}\/i_{0,\\perp}}{i_{45,\\parallel}\/i_{45,\\perp}},\n\\end{equation}\n where the first index of the image {\\em i} refers to the $\\lambda\/2$\n retarder orientation and $\\parallel$ and $\\perp$ indicate the \n two opposite polarization states from the ordinary and extraordinary\n beams of the Wollaston prism. \n\nThe corresponding intensity images are calculated by\n \\begin{equation}\\label{intensity} \n I_{Q} = 0.5 \\cdot (i_{0,\\parallel}+i_{0,\\perp}+i_{45,\\parallel}+i_{45,\\perp})\\,.\n \\end{equation}\nThe polarization and intensity images for the \\mbox{Stokes U}\n measurements are determined in the same way but using the frames\n taken with $+22.5^\\circ$ and $-22.5^\\circ$ retarder positions.\n \nIn the differential polarization measurements effects like the \nspatial variations of the system throughput, detector pixel-to-pixel\nsensitivity differences and temporal changes between individual measurements\nare compensated to first order with the used double ratio, without\nany application of a flatfielding correction. Therefore, flatfielding was only applied on the intensity image\n$I_{Q}$ described in Eq. (\\ref{intensity}) using an intensity flatfield image produced in the same way. \n\nAs described above there are some image distortions due to the\nlarge field of view and the relatively simple optical setup.\nThese differential distortions between the\nordinary $i_\\parallel$ and extraordinary $i_\\perp$ beams\ndisappear almost entirely in the double ratio method because\nimages from both beams are in the nominator and denominator of\nthat ratio. This first order cancellation effect is not\npresent in the summed intensity images and leads to some spatial\nsmearing. Therefore the limb of the Moon is not sharp in the\nintensity image but on the more relevant larger scales, \ni.e. for the identification of extended mare or highland regions, \nthe image distortions are negligible. Nonetheless \nwe have considered in our data analysis that small scale features\nmay be affected by image distortion effects and the \nassociated alignment inaccuracies. \n\nThe polarimetric properties of ESPOL were tested with observations\nof zero polarization standard stars\n$\\beta$ Tau, $\\beta$ UMa, $\\gamma$ Boo, \n\\citep{Turnshek1990} and Vega\n\\citep{Bhatt2000} which\nshow that the instrumental polarization is $\\le 0.5\\%$ in all \nfilters. From the polarized standard stars\nHD 21291, 9 Gem, $\\phi$ Cas, \\mbox{55 Cyg}\n\\citep{Hsu1982} \nwe deduced a polarimetric efficiency above 98~\\%\nand checked the zero point of the polarization direction.\n \n\\begin{figure}\n \\centering\n \\includegraphics[width=8.5cm]{espol_cuts.eps}\n \\caption{Lunar west-east intensity (top) and \n polarization (bottom) profiles in the B band for phase 42.5$^{\\circ}$ and 98.0$^{\\circ}$ with low (left) \n and high (right) stray light contribution from the moonshine, respectively. \n The panel in the middle shows the corresponding stripe of the\n intensity images. The profiles were extracted from 10 pixel wide\n regions as indicated by the dashed lines in the middle panel.}\n \\label{cuts} \n \\end{figure}\n\n\n \\begin{figure}\n \\centering\n \\includegraphics[width=8.5cm]{extract_es_signal.eps}\n \\caption{Measuring the earthshine signals ${\\Delta\n I},\\,{\\Delta Q}$ for relatively low (phase 73$^{\\circ}$, region \\#1, filter B)\n and high \n (phase 98$^{\\circ}$, region \\#1, filter R) moonshine \n levels with the\n background $x'$ and measuring regions $x$ indicated. The \n dashed lines illustrate the guessed level of\n the background (mainly stray light) and \n background plus constant earthshine regions (reflected from maria).\n The full line is the linear extrapolation of the\n measured background from the $x'$ to the $x$ region.}\n \\label{extract es} \n \\end{figure}\n\n\n \n\\subsection{Extracting the earthshine polarization}\\label{s: extract earthshine}\n\nWe are interested in the measurement of the fractional polarization\nof the earthshine $(Q\/I)_{\\rm es}$, \nwhich needs to be extracted from our data. Our observations show\nthe contributions of three intensity \ncomponents \n\\begin{displaymath}\nI_{\\rm tot}=I_{\\rm es}+I_{\\rm M}+I_{\\rm sky}\n\\end{displaymath}\nfrom the earthshine (es), the scattered light from the moonshine (M) \nand the sky (see Fig. \\ref{cuts}). In our images it is rather\neasy to distinguish these components, assuming that the sky is essentially\nconstant over the whole field of view. The location of the earthshine is well defined and \nits intensity lies in a restricted range between the intensities of dark\nmaria and bright highlands. The scattered light intensity from the moonshine \nhas a more complex geometry. It is increasing rapidly towards the \nbright crescent which is covered in our data by the occulting mask.\n\n\nThe signatures of the three components can also be recognized in the\nfractional polarization W-E cuts extracted from the $Q\/I$ images \nshown in Figure \\ref{cuts}. \nThe B band observation for phase 42.5$^{\\circ}$ \nshows a significant sky contribution from the twilight. The sky \npolarization is about\n1.5 \\% on the west side of the Moon and the earthshine plus sky\npolarization is about 3.5 \\%. The polarization of the moonshine\nis slightly higher ($\\sim$4.5 \\%) as can be seen near the east side of the occulting \nmask where the scattered light of the moonshine dominates. For phase \n98$^{\\circ}$ the\nscattered moonshine with a polarization of about 8 \\% dominates strongly.\nThe fractional polarization is just slightly enhanced at the position \nof the earthshine. The (Q\/I)-images consist of the following\ncontributions\n\\begin{equation}\n\\left({Q\\over I}\\right)_{\\rm tot} = \n {Q_{\\rm es}+Q_{\\rm M}+Q_{\\rm sky}\\over I_{\\rm tot}}\\,.\n\\end{equation} \nBecause of our definition of the $\\pm Q$-directions \nperpendicular and parallel to the scattering plane the $U\/I$ polarization is\nessentially zero ($\\approx \\pm 0.5$ \\%) and dominated by noise.\n\nAfter some investigation we defined a procedure for the extraction of the \nfractional earthshine polarization $(Q\/I)_{\\rm es}$ which provides\nalso good results for large phase angles and the I filter \nfor which the signal is weak and\/or the stray light from the moonshine is very \nstrong. For small phase angles the earthshine signal is strong and \nthe measurement is easy. The basic idea is to measure the signal\nof the earthshine on top of the \"background signal\" in the\nI$_{\\rm tot}$ frame and the Q frame where $Q=(Q\/I)_{\\rm tot} \\cdot I_{\\rm tot}$. \nThe \"background signals\" (bg) are just the sum of the contributions\nof the sky and the moonshine $I_{\\rm bg} = I_{\\rm sky} + I_{\\rm M}$\nand $Q_{\\rm bg} = Q_{\\rm sky} + Q_{\\rm M}$. For this we \nextract radial cuts and extrapolate\nthe background signal from the region $x'$ outside \nto a location $x$ inside the lunar disk \n($I_{\\rm bg}(x')$, $Q_{\\rm bg}(x')\\rightarrow \nI_{\\rm bg}(x)$, $Q_{\\rm bg}(x)$) where we measure the earthshine + \nbackground level. The final signal is then:\n\\begin{equation} \n \\left(\\frac{Q}{I}\\right)_{\\rm es} = \n\\frac{\\Delta Q}{\\Delta I} = \\frac{Q_{\\rm bg+es}(x) -\n Q_{\\rm bg}(x)}{I_{\\rm bg+es}(x) - \n I_{\\rm bg}(x)} \\,.\n\\end{equation} \n\nThis procedure is illustrated in Figure \\ref{extract es}, for\ntwo cases. The first is a strong and clear earthshine signal typical for\nphase angles $\\alpha < 109^{\\circ}$ in the B, V filters and phase angles $\\alpha < 98^\\circ$ in the R filter respectively.\nThe large majority of our data are of this kind. \nThe other case is typical for phase angles $\\alpha \\ge 98^{\\circ}$ \nin the R and I band filter\nfor which the stray light from the moonshine dominates strongly. \nThe Q signal from the earthshine is still\nabove but close to the measuring limit. \nAlso given are fits \nto the background, which consists in these cases mainly of\nthe moonshine plus a constant earthshine level fitted to the \nmare regions. The use of \na linear extrapolation of the background in the $x'$ region \nfor the background correction of the total earthshine plus background signal\nmeasured at $x$ seems reasonable \\citep[see also][]{Qiu2003, Hamdani2006}.\n \n\n \\begin{figure}\n \\centering\n \\includegraphics[width=5.7cm,angle=90]{obs_areas.eps}\n \\caption{Selected mare (\\#1) and highland (\\#2) fields used for\n the earthshine measurements together with their \n background regions (white areas). }\n \\label{obs_areas} \n\t\\end{figure}\n\n\n\n\n \\begin{figure*}\n \\centering\n \\includegraphics[width=17.6cm]{results_all.eps}\n \\caption{Fractional polarization Q\/I and U\/I of the\n earthshine measured for highland (top) and mare regions\n (bottom) for the four different filters B, V, R and I (left to right).\n The solid curves are $q_{\\rm max} \\sin^{2}$ fits to the data. \n The error bars give the statistical\n 1$\\sigma$ noise $\\Delta_{\\rm noise}$ of the data whereas the mare I band data at phase angle $109.5^{\\circ}$ are\n additionally affected by a substantial systematic offset $\\Delta_{\\rm syst} > 0.5~\\%$. \n The dots in the V panel for the mare region indicate the measurements of \n \\citet{Dollfus1957} and a corresponding\n $q_{\\rm max} \\sin^{2}$ fit (dashed line) is also given.\n }\n \\label{results} \n \\end{figure*}\n\n\n\n\n\n \\begin{table*}\n \\caption{Fractional polarization values $(Q\/I)_{\\rm es}$ \n \t\t for the earthshine from the mare and highland regions of all our measurements and corresponding typical statistical 1$\\sigma$ uncertainties $\\Delta_{\\rm noise}$. Also given is the fit parameter \n for $q_{\\rm max} \\sin^{2}(\\alpha)$ derived in Sect. \\ref{phase dependence} and the standard deviation of the data points from the fit $\\sigma_{\\rm d-f}$. \n The polarization efficiencies \n $\\epsilon_{\\#1}$ and $\\epsilon_{\\#2}$ are derived in Sect. \\ref{s:retro-reflection} and the depolarization corrected values $q_{\\rm max, corr}$ are given. \n } \n \\label{tbl: results} \n \\centering \n \\begin{tabular}{C{16mm} | C{24mm} | C{11mm} C{11mm} C{11mm} C{11mm} | C{11mm} C{11mm} C{11mm} C{11mm}} \n \\hline \n date & phase & \\multicolumn{4}{c | }{$Q\/I$ (highland) [\\%]} & \\multicolumn{4}{c}{$Q\/I$ (mare) [\\%]} \\\\\n (2011) & (Sun-Earth-Moon) & B & V & R & I & B & V & R & I \\\\\n \\hline \n 07.03 & 31.5$^{\\circ}$ & - & 1.5 & - & - & - & 2.3 & - & - \\\\\n 08.03 & 42.5$^{\\circ}$ & 4.0 & 3.4 & 2.3 & - & 5.4 & 4.6 & 3.5 & - \\\\\n 02.10 & 73.0$^{\\circ}$ & 8.7 & 6.1 & 4.0 & 2.5 & 11.9 & 8.1 & 5.3 & 2.9 \\\\\n 11.03 & 75.5$^{\\circ}$ & 9.7 & 6.9 & 4.5 & - & 12.9 & 8.7 & 6.4 & - \\\\\n 03.10 & 85.5$^{\\circ}$ & 9.4 & 6.9 & 4.8 & 2.9 & 12.7 & 8.6 & 5.7 & 2.7 \\\\\n 04.10 & 98.0$^{\\circ}$ & 9.1 & 6.9 & 5.0 & 3.4 & 11.9 & 8.6 & 6.1 & 3.7 \\\\\n 05.10 & 109.5$^{\\circ}$ & 7.7 & 5.7 & 4.2 & 3.1 & 11.1 & 8.4 & 5.9 & 8.6$^a$ \\\\\n \\hline\n \\multicolumn{2}{l | }{statistical 1$\\sigma$ uncertainty $\\Delta_{\\rm noise}$ [\\%]} & 0.2 & 0.2 & 0.2 & 0.2 & 0.4 & 0.3 & 0.3 & 0.3 \\\\ \n \\multicolumn{2}{l | }{fit parameter $q_{\\rm max}$ [\\%]} & 9.4 & 7.0 & 4.7 & 3.0 & 12.7 & 9.0 & 6.1 & 3.2 \\\\ \n \\multicolumn{2}{l | }{stddev of data from fit $\\sigma_{\\rm d-f}$ [\\%]} & 0.22 & 0.13 & 0.10 & 0.21 & 0.24 & 0.13 & 0.22 & - \\\\ \n \\multicolumn{2}{l | }{polarization efficiency $\\epsilon$ [\\%]$^b$ } & 36.6 & 34.3 & 32.7 & 30.7 & 54.3 & 50.8 & 48.5 & 45.6 \\\\ \n \\multicolumn{2}{l | }{$\\epsilon$ corrected $q_{\\rm max, corr}$ [\\%]} & 25.8 & 20.3 & 14.4 & 9.6 & 23.4 & 17.8 & 12.6 & 6.9 \\\\\n \\hline\n \\multicolumn{10}{l}{\\vspace{-0.2cm}}\\\\\n \\multicolumn{10}{l}{$^a$value affected by systematic errors due to high stray light level (see text)}\\\\\n \\multicolumn{10}{l}{$^b$the uncertainty of $\\epsilon$ is mainly a systematic offset estimated to be $\\Delta\\epsilon = \\pm 3~\\%$ (see text)} \n \\end{tabular}\n \\end{table*}\n \n\n\n\n\n\n \nWe have investigated more complex background\/straylight correction\nprocedures, e.g. using 3-parameter exponential fits, but they didn't agree better than\nthe linear extrapolation. Important for the accuracy of \nthe earthshine measuring process is the selection of areas close\nto the western limb but not exactly at the limb because image \nalignment uncertainties of the polarimetric data reduction can create\ndisturbing spurious features at the limb. The limb is also not a\ngood measuring region because of the extreme incidence and\nreflection angles (near $90^\\circ$) with respect to the large scale \nsurface normal which represents a situation which is not well explored\nfor its back-scattering properties. \n \nAll our data show that the differences between the lunar dark mare\nand bright highland regions are significant when determining the intensity\nand polarization of the back-scattered earthshine. \nTherefore it is important to carry out separate measurements\nfor these two main lunar surface types. \nBecause of the strong albedo dependence (see Sect. \\ref{s:retro-reflection}) \nit is important that a chosen measurement field on the Moon does not have strong albedo variations. \nUnder these terms we selected one mare field \\#1 in the Oceanus Procellarum area\nand one highland field \\#2 between Mare Humorum and \nthe Moon's limb as indicated in Figure \\ref{obs_areas}. Both fields are close to the western limb far away\nfrom the bright lunar side. They are available for measurements at all phase angles $\\alpha$ taking into account\nincreasing stray light and the lunar libration. \nTherefore, for both fields a consistent data reduction could be carried out.\n \nFor both fields $10$ radial\nI and Q profiles separated by one degree were extracted and $\\Delta I$\nand $\\Delta Q$ was determined as described above. \nTable \\ref{tbl: results} gives the obtained $(Q\/I)_{\\rm es}$ polarization values \nfrom both fields\nwhich are also plotted in Figure \\ref{results} as phase curves together with the\nstatistical 1$\\sigma$ error bars $\\Delta_{\\rm noise}$. \n \nAs long as the S\/N is sufficiently high \nthe linear extrapolation method is robust. The total uncertainty \nfor the obtained fractional polarization of the earthshine for\na highland or mare region at a particular date can be described by the statistical noise\nplus a predominantly positive systematic offset\n $\\Delta(Q\/I)_{\\rm es}= \\Delta_{\\rm syst}\\pm \\Delta_{\\rm noise}$.\n \n\n\nThe statistical $1\\sigma$ uncertainty $\\Delta_{\\rm noise}$ is small ($< 0.3~\\%$).\nThis follows from\nthe scatter of the obtained values from different extraction\ncuts of the same day and includes random noise, but also hard to\nquantify systematic effects due to small image drifts on the detector,\nchanging stray light levels of the moonshine related to non-stable\natmospheric conditions, and perhaps other unidentified effects. \n\nFor observations\nwith very small earthshine signals (i.e. at large phase angles and\/or strong stray light of the moonshine) \nthe linear extrapolation of the background \nintroduces a systematic overestimate $\\Delta_{\\rm syst}$ of \nthe result. This is because the moonshine dominated stray light background\nincreases with an upward curvature towards the illuminated \ncrescent. For the B, V, and R measurements at phase angles $< 100^{\\circ}$ this offset is negligible or small ($< 0.5~\\%$).\nHowever in the I band filter at phase angle $109.5^{\\circ}$ the systematic offset is \nclearly dominating and the mare I band result for $109.5^{\\circ}$ is no longer useful\n(see Fig. \\ref{results}). For this reason we disregard the mare I band result at $109.5^\\circ$.\n\n\n\n\n \\begin{figure}\n \\centering\n \\includegraphics[width=8cm]{results_m_vs_h.eps}\n \\caption{Correlation between the fractional polarization \n for mare and highland regions measured simultaneously. The different symbols\n indicate the colors \n B($+$), V($\\ast$), R($\\diamond$), and I($\\square$). The line \n shows the derived proportionality factor $1.30\\,\\pm 0.01$.}\n \\label{ratio mh} \n \\end{figure}\n\n\n \n\n\n\n\\section{Earthshine polarization results}\\label{s:results}\n\n\\subsection{Data}\n\nThe results for the fractional earthshine polarization $Q\/I$\nmeasured in the Bessell B, V, R and I bands are presented\nin \\mbox{Table \\ref{tbl: results}} and Figure \\ref{results}. \nThe plots in Fig. \\ref{results} also include the $U\/I$ data points and the estimated\nstatistical 1$\\sigma$ uncertainties of the individual data points $\\Delta_{\\rm noise}$.\nThe mare V band panel shows also the measurements by \\citet{Dollfus1957}\nwhich are in good agreement with our data.\n\nOur earthshine data show a very good correlation between the \npolarization taken simultaneously \nfor the highland and mare regions. Independent of color filter and phase\nangle the polarization for the mare region is a factor of $1.30\\pm0.01$ higher\nthan for the highland region as illustrated in\n\\mbox{Figure \\ref{ratio mh}}. \n\n\nGood correlations are also found between different colors taken for the same observing date. \nWhen we plot the polarization $(Q\/I)_{\\rm es}$ in the V, R and I band versus the \npolarization in the B band (Fig. \\ref{ratio BF}) \nwe find that the ratios are independent of $\\alpha_{\\rm E}$. We get the ratios \n$0.72\\pm0.02$, $0.49\\pm0.02$ and $0.28\\pm0.05$ for the ratios of the polarization between V and B band, \nR and B band, and I and B band respectively. Therefore, we conclude that to \nfirst order we can assume the same shape for the polarization phase curve for\nall wavelengths. \n\n\n \\begin{figure*}\n \\centering\n \\includegraphics[width=18cm]{results_ratio_BF.eps}\n \\caption{\n Fractional polarization of the earthshine reflected at highland ($+$) and mare ($\\ast$)\n regions in V, R and I band (left to right) with respect to the B band. The lines indicate linear fits\n both for highland (solid) and mare (dashed) data separately. \n }\n \\label{ratio BF} \n \\end{figure*}\n\n\n\n \n\n\\subsection{Fits for the phase dependence}\\label{phase dependence}\n\nThe phase dependence of the earthshine polarization looks symmetric\nand can be well fitted with a simple $q_{\\rm max}\\sin^{2}(\\alpha)$ curve. \nThe model simulations by \\citet{Stam2008} for Earth-like planets also support phase curves $q_{\\rm max}\\sin^{p}(\\alpha)$.\nShe calculates polarization phase curves assuming a range of \nsurface types (e.g. forest-covered areas with Lambertian reflection, dark ocean with \nspecular reflection) and cloud coverages.\nWe find that the broad shape of her model phase curves can be well fitted by\ncurves $\\sim q_{\\rm max} sin^{p}(\\alpha + \\alpha_{0})$ with $p \\approx 1.5-3$ and $\\alpha_0 \\approx 0^{\\circ}-10^{\\circ}$. \n\nFurthermore, she finds characteristic features at low \nphase angles due to the rainbow effect and negative polarization at large phase angles due to second\norder scattering. We cannot assess the presence of such features because of \nthe coarse phase sampling of our data.\n\nBesides the $q_{\\rm max} \\sin^{2}(\\alpha)$ curve we also tried functions with more free parameters to fit the data,\ne.g. using a curve like $q_{\\rm max} \\sin^{p}(\\alpha+\\alpha_0)$\nand varying the exponent $p$ between\nvalues of 1.5-3 and by introducing a phase shift $\\alpha_0$. However, such fits provide\nnot a significantly better match to the data. \nBecause our data cover predominantly phase angles around quadrature the\nshape of the phase curve is not very well constrained. \n\nThe derived $q_{\\rm max}$ fit parameters for the different phase curves are given in Table \\ref{tbl: results}\ntogether with the standard deviation of the data points from the fit $\\sigma_{\\rm d-f}$. For $Q\/I$ the typical $\\sigma_{\\rm d-f}$ is $\\approx 0.2~\\%$ in good\nagreement with the typical 1$\\sigma$ uncertainty of the individual data points $\\Delta_{\\rm noise}$. The standard deviation of the derived\n$U\/I$ values from the expected zero-value is only slightly higher and typically \\mbox{$\\approx 0.3~\\%$} indicating that the instrument alignment with respect to the\nSun-Earth-Moon plane was excellent (see Sect. \\ref{s:observations}). Note that the $U$ signal is at the level of the measurement noise $|U|\\approx\\Delta_{\\rm noise}(U)$. \nTherefore one should not use the normalized \ntotal polarization ${\\rm p}=\\sqrt{(Q\/I)^2+(U\/I)^2}$ because the square in this formula introduces systematic errors. However, we estimate that the\nimpact of $U\/I$ to the total polarization ${\\rm p}$ is less than $\\pm 0.05~\\%$.\nTherefore we use ${\\rm p}\\cong Q\/I$ and neglect the $U$ component in the subsequent discussion.\n\n \n\n \n\\subsection{The moonshine polarization}\\label{s: moonshine}\n\n\nAs a check of our polarimetry we can compare the polarization\nof the stray light from the moonshine with literature values from\n\\citet{Coyne1970} and \\citet{Dollfus1971}. \n\nForward scattering in the Earth atmosphere with scattering angles less than a few degrees does not introduce a\nsignificant polarization effect. \nTherefore, we can assume that the polarization of the lunar stray light \n$(Q\/I)_{\\rm M}$ represents well the polarization of\nthe bright lunar crescent.\nFor areas just east of the Moon close to the focal mask (see Fig. \\ref{cuts}) the scattered moonshine dominates\nstrongly. There we can neglect the contribution of the sky background $(Q\/I)_{\\rm sky}$ and assume that\n\\mbox{$(Q\/I)_{\\rm bg} \\approx (Q\/I)_{\\rm M}$}. \n\nFigure \\ref{moonshine} compares our results with\nthe waxing Moon values given by \\citet{Coyne1970}. They used different\nfilters but their $B'$ and $G_{\\rm m}$ bands \n($\\lambda_{\\rm eff}[\\mu m]=0.45,\\, 0.53$) are close to our B\nand V band respectively and the good agreement with our data\nunderlines the consistency of our polarimetric data reduction.\n\nUnfortunately \\citet{Coyne1970} used no red filters\nbut the scaled $G_{\\rm m}$ phase curve fits also well our \nR and I band data if scaling parameters of 0.80 and 0.65 are used respectively.\nThis is in good agreement with the wavelength dependency of the maximum degree\nof polarization $P_{\\rm max}$ of the whole Moon presented in\n\\citet{Dollfus1971}:\n\\begin{equation} \nP_{\\rm max,\\lambda_1}\/P_{\\rm max,\\lambda_0}=(\\lambda_1\/\\lambda_0)^{- 1.137} \\,.\n\\end{equation} \nWith this formula we obtain color ratios of\n$q(R)\/q(G_{\\rm m})=0.81$ and $q(I)\/q(G_{\\rm m})=0.64$ for the Moon polarization in good agreement\nwith the above scaling parameters derived from our stray light data.\n \n \n \n\n \\begin{figure}\n \\centering\n \\includegraphics[width=8cm]{moonshine.eps}\n \\caption{Measured polarization of the lunar stray light near the focal mask\n in the B, V, R, and I filters (from top to bottom by filled dots and dashed lines). \n Also indicated are the\n polarization values given by \\citet{Coyne1970} for the disk \n integrated moonshine\n of the waxing Moon in their $B'$ ($\\diamond$) and $G_m$ band ($+$).\n }\\label{moonshine} \n \\end{figure}\n\n \n \n\n \n \n\n\\section{A correction for the depolarization due to the \nback-scattering from the lunar surface }\\label{s:retro-reflection}\n\nThe polarized light from the Earth is depolarized by the\nback-scattering from the particulate surface of the Moon. \nWe express this effect as polarization efficiency $\\epsilon$,\ndescribing the fraction of linear polarization preserved. \nWe simplify the treatment by considering only the $Q$ linear \npolarization direction perpendicular and parallel to the\nscattering plane Sun-Earth-Moon. Then the polarization\nefficiency is\n\\begin{displaymath}\n \\epsilon = {(Q\/I)_{\\rm es} \\over (Q\/I)_{\\rm E}}\\,,\n\\end{displaymath}\nwhere $(Q\/I)_{\\rm E}$ is the Earth polarization. \nThe polarization efficiency $\\epsilon(\\lambda,a_\\lambda)$ depends on the wavelength and the\nsurface albedo. We neglect the phase dependence\nin the back-scattering because the scattering angle is always\n$179^\\circ\\pm 0.5^\\circ$. \n \nThe depolarization of the lunar surface \nwas already investigated by \\citet{Lyot1929} and \\citet{Dollfus1957}. \nThey measured the depolarization of the back-scattering \nof volcanic ashes and fines, which were used \nas proxy for the lunar soil. They found a well defined \nanticorrelation between albedo and polarimetric efficiency. \n\nMost important for the determination of $\\epsilon(\\lambda,a)$\nare the albedo and polarization measurements for \nthe reflection from several Apollo lunar soil\nsamples by \\citet{Hapke1993, Hapke1998}. They illuminated \neight samples under an inclination of 5~degrees (to avoid specular\nreflection) with 100~\\% polarized blue and red light and \nmeasured the ratio of $I_\\perp\/I_\\parallel$ for phase angles $1^\\circ$ \n($\\sim$ back-scattering) to $19^\\circ$ or scattering angles of \n$179^\\circ$ to $161^\\circ$. \nThe results for the linear polarization ratio are presented in \\citet{Hapke1993} in graphical form \nand we extracted the data for phase angle $1^\\circ$ and derived \nthe polarization efficiency $\\epsilon$. The normal albedos are only\ngiven for a phase angle of $5^{\\circ}$ \\citep[][Table~1]{Hapke1993} and we converted them into \nearthshine back-scattering\nalbedos corresponding to $1^{\\circ}$ phase angle by applying a conversion factor of $1.25\\pm 0.05$.\nWe derived this factor from albedo phase curves presented in \\citet{Velikodsky2011} where they give a \ncomprehensive summary of the results of\nvarious independent photometric observations of the Moon including their own, Clementine data, and ROLO data.\n\n\n \\begin{figure}\n \\centering\n \\includegraphics[width=8cm]{poleff_vs_albedo.eps}\n \\caption{Linear polarization efficiency as function of the normal albedo at $1^{\\circ}$ (Eq. \\ref{eq:pvsa}) for \n the B (dashed), V (dash-dot), R (solid), and I (dash-dotdot) band.\n More information about the\n fit procedure to the \\citet{Hapke1993} samples in the red ($\\diamond$) and the blue ($+$) is given in the text.\n Measurements of the same sample are connected by dotted lines.\n The thick black lines show the derived wavelength\n and albedo dependent polarization efficiencies for our two measurement areas \\#1 and \\#2 (Fig.~\\ref{obs_areas}).\n }\n \\label{pvsa} \n \\end{figure}\n\n\n\n \n \n\nThe samples from \\citet{Hapke1993} include 5\nlow albedo samples $a_{\\rm red}\\approx 0.09-0.13$ representative \nfor maria, 2 higher albedo samples $a_{\\rm red}\\approx 0.15 - 0.19$\nrepresentative for highlands and one non-typical, extremly high albedo \nsample with normal albedo $a_{\\rm red}>0.35$. This sample with \nNASA number 61221 was taken from white material at the bottom of\na trench \n(see The Lunar Sample Compendium\\footnote{http:\/\/curator.jsc.nasa.gov\/lunar\/compedium.cfm}) \nand therefore we treat this sample as special case in \nour analysis. \n\n\n \\begin{figure*}\n \\centering\n \\includegraphics[width=18cm]{results_all_corrected.eps}\n \\caption{Depolarization corrected polarization phase curves of Earth in the B, V, R, and I bands. \n The solid line indicates the mean of the mare (dashed) and highland (dash-dot) results\n based on the polarization efficiency correction derived in this work. \n }\n \\label{results corrected} \n \\end{figure*}\n\n\n\nFigure \\ref{pvsa} shows the polarization efficiencies for the measurements\nof \\citet{Hapke1993} as function of the $1^\\circ$-albedo for \na blue wavelength ($\\lambda=442$~nm) and a red wavelength \n($\\lambda=633$~nm). \nThe figure illustrates nicely the clear anticorrelation between\nalbedo $a$ and polarization efficiency $\\epsilon$. \n\nWe consider now in more detail the back-scattering properties of\nthe lunar samples. All samples, except 61221, show a very similar color \ndependence in their albedo with $a_{\\rm red}\/a_{\\rm blue}= 1.35$ ($\\sigma= 0.05$).\nThis is in agreement with the spectral variation of the\nmean lunar albedo $\\bar{a}$ from \n\\citet{Dollfus1971} (see also \\citealt{Gehrels1964}, Table XIII; and \\citealt{Velikodsky2011}, Table 2)\ndescribed by \n\\begin{equation}\\label{albedo dependence}\n{\\rm log}\\, \\bar{a} = 0.83 \\cdot {\\rm log}\\, \\lambda [\\mu{\\rm m}] - 0.80\\,.\n\\end{equation}\nInserting the wavelengths of the \\citet{Hapke1993} measurements\ninto this formula yields \n$\\bar{a}_{\\rm red}\/\\bar{a}_{\\rm blue} = 0.108\/0.0805 = 1.34$\nin very good agreement with the above derived albedo ratio for the lunar samples. \n \nFor most samples the polarization efficiency is slightly higher \n(in one case equal) in the blue than in the red. \nThere is one notable exception which is sample number 79221.\nAlthough the albedo of sample 79221 is lower in the blue\nthan in the red its polarization efficiency is not higher\nas in all other samples. Also when looking into reflectivity\nstudies for this sample \\citep[e.g.][]{Noble2001}\nit is not clear why this sample\ncould behave different in its depolarization properties \nthan other maria soils. Therefore, we treat sample 79221 as\nan exception. \n \nIf we disregard sample 79221, then the remaining 6 samples \nhave an average color dependence for their polarization \nefficiency ratio of $\\epsilon_{\\rm red}\/\\epsilon_{\\rm blue}= 0.91$ \n($\\sigma = 0.05$). Including sample 79221 gives a mean ratio of\n0.96 but a standard deviation which is with 0.14 significantly\nhigher. \n\nBased on these back-scattering measurements of lunar samples we\nderive a two dimensional linear fit for the polarization\nefficiency ${\\rm log}\\,\\epsilon$ as function of ${\\rm log}\\,a_{603}$, the albedo at 603 nm, and ${\\rm log}\\,\\lambda$ for\nthe wavelength\n\n\\begin{equation}\\label{eq:pvsa}\n{\\rm log}\\, \\epsilon(\\lambda, a_{603}) = -0.61 \\,{\\rm log}\\, a_{603}\n- 0.291 \\, {\\rm log}\\, \\lambda [\\mu{\\rm m}] - 0.955 \\,.\n\\end{equation}\nFor this fit the wavelength dependence of the albedo has been assumed to be according to Eq. \\ref{albedo dependence} and it was normalized to 603 nm. \nBy fitting the red data points we find the logarithmic slope $-0.61\\pm 0.04$ between the polarization efficiency\n$\\epsilon$ and the \\mbox{albedo $a_{\\rm red}$} which we also adopt\nfor the blue data points. Finally, by fitting over the red and blue points separately we determine the other two parameters\n$-0.291$ and $-0.955$. The resulting relation for the linear polarization efficiency as function of the normal albedo at $1^\\circ$ is shown in\nFig. \\ref{pvsa} for the B, V, R, and I band.\n\n\nFor the derivation of the albedos of our measurement regions we used the results of\n\\citet{Velikodsky2011} who present \nmaps of lunar apparent and equigonal albedos at \nphase angles $1.7^{\\circ}-73^{\\circ}$ at wavelength \\mbox{603 nm}. We extrapolated their results to a \nphase angle of $1^{\\circ}$ and we get albedos \n$a_{\\#1}(603~\\rm nm)=0.11 \\pm 0.01$ and $a_{\\#2}(603~\\rm nm)=0.21 \\pm 0.01$.\nThe resulting polarization efficiencies\n$\\epsilon_{\\#1}(\\lambda,a_{603})$ and $\\epsilon_{\\#2}(\\lambda,a_{603})$ are\nlisted in Table \\ref{tbl: results} for the B, V, R, and I band and shown in \\mbox{Figure \\ref{pvsa}} giving the $\\epsilon(\\lambda, a_{603})$ \nfits.\nOverall we estimate the uncertainty of the derived polarization efficiency to be\nin the order of $\\Delta\\epsilon\\approx \\pm 3~\\%$.\n\n\n\nThe main uncertainty of this derivation stems from the uncertainty in the above mentioned albedo conversion from $\\alpha=5^\\circ$ into albedos corresponding\nto $1^\\circ$ phase angle where the conversion factor $1.25\\pm 0.05$ leads to an uncertainty of the polarization efficiency of about $\\Delta\\epsilon=\\pm 1.5~\\%$. \nTo significantly improve the determination of $\\epsilon$ accurate lunar albedo maps\nfor back-scattering geometry are required. This is because\nfor back-scattering at $\\approx 1^\\circ$\nreflection is\nstrongly influenced by the opposition effect of the lunar surface, i.e. a steep brightness surge due to coherent backscattering and shadow-hiding\n(e.g. \\citealt{Shkuratov2011} and references therein). In addition to that the \\citet{Hapke1993, Hapke1998} sample might not be representative for the surface properties of the Moon.\n\nThe logarithmic slope $-0.61\\pm 0.04$ is better constraint for the low albedo samples and it introduces an uncertainty $\\Delta\\epsilon = \\pm 1~\\%$ towards the higher albedo samples.\nMoreover, the logarithmic relation might not be valid\nover the complete albedo range between $a_{\\lambda}=0.09-0.19$ and two slopes, one for the maria and one for the highlands, might be necessary. However, based on the\navailable samples this is not obvious and one log fit may not be the best representation of the data. \nMore direct measurements of the polarization efficiency of the lunar back-scattering are required to reduce this source of uncertainty.\n\n\n\n\n\n\n\\section{Polarization of planet Earth} \\label{s: earth polarization}\n\n\n\\subsection{Fractional polarization derived from the earthshine}\n\nIn Figure \\ref{results corrected} we present the depolarization-corrected polarization\nphase curves of planet Earth\nin the B, V, R, and I bands and the corresponding corrected fit parameters $q_{\\rm max,corr}$ are listed in Table \\ref{tbl: results}. \nFor the B band we obtain a maximum polarization of about $25~\\%$ which decreases with wavelength to about $8~\\%$\nin the I band. For perfect measurements and perfect polarization efficiency corrections the same Earth polarization $q_{\\rm max,corr}$\nvalues should be obtained for the mare and highland regions. We note that the corrected highland results are systematically\nhigher than the mare results by a factor of about $1.1$ for the B, V, R bands and 1.4 for the I band. This reflects also the uncertainty in our determination of the\npolarization efficiency of the back-scattering $\\Delta \\epsilon \\approx \\pm 3~\\%$ derived in Section \\ref{s:retro-reflection}.\n\n\n\\subsection{Comparison with previous measurements}\n\n\n\n \\begin{figure}\n \\centering\n \\includegraphics[width=8cm]{figcolordep.eps}\n \\caption{Top: Earthshine polarization results at quadrature for maria ($\\ast$) and highlands ($+$).\n The thin lines give the \\citet{Sterzik2012} spectro-polarimetry for waning (dashed) and waxing (dotted)\n moon at Earth phases $87^\\circ$ and $102^\\circ$ respectively and the \\citet{Takahashi2013} spectro-polarimetry (dash-dot) at $96^\\circ$. \n The circles are the \\citet{Dollfus1957}\n values $q_{\\rm max}$ from Fig. \\ref{results} (filled) and two additional observations at\n Earth phase \\mbox{$\\approx100^\\circ$} (open). $2^{nd}$ panel: Earth polarization $p_{\\rm E}$ from Table \\ref{tbl: quadrature results} \n ($\\diamond$) compared to the \n POLDER\/ADEOS results of \\citet{Wolstencroft2005} ($\\times$) and two \\citet{Stam2008} models with\n 40~\\% (dash-dot) and 60~\\% (dash-dotdot) cloud coverage.\n Bottom two panels: spectral reflectivity of Earth $f_{\\rm E}$ and\n polarized reflectivity of Earth $p_{\\rm E}\\times f_{\\rm E}$.}\n \\label{ColorDep} \n \\end{figure}\n\n\n\n\nIn Fig.~\\ref{results} we compare our earthshine \nmeasurements with the pioneering\nstudy of \\citet{Dollfus1957}, who obtained his data with visual\nobservations using a ``fringed-field polariscope''. The agreement with\nour V band phase curve for the mare region is excellent. If we apply\n$q_{\\rm max} sin^2$ fits (see Sect. \\ref{phase dependence}) to both data sets \nthe quadrature signals only differ by\n0.8~\\%. \nThe small deviations between \\citet{Dollfus1957} and us can be explained\nby different mare regions observed and the expectedly non-equal effective wavelengths of the two\ncompletely different types of measurements.\nFor one night at $\\alpha\\approx 100^\\circ$ \\citet{Dollfus1957} reports also \nthe earthshine polarization in two filters, namely $p=5.4~\\%$ for 0.55 $\\mu$m (V' band)\nand 3.5~\\% for 0.63 $\\mu$m (R' band). The ratio $p_{V'}\/p_{R'}= 1.54$ is again in\nexcellent agreement with our\npolarization ratio $q_{\\rm max,V}\/q_{\\rm max,R}=1.47$.\nThis indicates that the filters used by \\citet{Dollfus1957} must match quite well\nour filter pass bands. \n \nThe spectral dependence of the earthshine polarization observed with a spectral resolution\nof 3~nm was recently\npublished by \\citet{Sterzik2012}. These sensitive spectro-polarimetric data reveal weak,\nnarrow features of the planet Earth due to O$_2$ and H$_2$O on \na smooth polarization spectrum decreasing steadily from the blue towards longer wavelengths. \nThey present measurements for two epochs\nwith phase angles $\\alpha=87^\\circ$ for a waning moon phase \nand $\\alpha=102^\\circ$ for the waxing moon phase. \nFor the waning\nmoon case they obtained an earthshine polarization of about $p_{\\rm B}=12.1$~\\% in \nthe B band, $p_{\\rm V}=7.7$~\\% in V, $p_{\\rm R}=5.6$~\\% in R, and $p_{\\rm I}=3.9$~\\% in I, and \na significantly higher polarization for the waxing moon phase with $p_{\\rm B}=16.6$~\\%, \n$p_{\\rm V}=9.7$~\\%, $p_{\\rm R}=8.0$~\\%, $p_{\\rm I}=6.7$~\\% as plotted in Fig. \\ref{ColorDep}.\nUnfortunately it is not clear whether they measured \nthe back-scattering from maria or highlands.\n\\citet{Sterzik2012} attribute\nthe polarization differences between the two epochs mainly to intrinsic differences \nof the polarization of Earth because the earthshine stems from \ndifferent surface areas and were taken for days with different cloud coverage. \nConsidering our polarization values for highlands and\nmaria then it could be possible that the differences\nmeasured by \\citet{Sterzik2012} are at least partly due to the mare\/highland depolarization difference \n(or surface albedo difference).\n\n \n\nAnother spectra-polarimetric observation of the earthshine was published by \\citet{Takahashi2013}. \nThey also find a rise of the fractional polarization of the earthshine towards the blue but with a much flatter slope.\nUnfortunately they do not report whether their results were obtained from maria or highlands either. Therefore, only a qualitative\ncomparison with our data can be made. The observations of \\citet{Takahashi2013} are conducted at 5 consecutive nights\nand cover phase angles $\\alpha=49^\\circ - 96^\\circ$. In the blue they find that the maximum polarization\nis reached at $\\alpha \\approx 90^\\circ$. However, for wavelengths $> 600~{\\rm nm}$ \nthe polarization keeps increasing up to and including their last measurement at $\\alpha=96^\\circ$.\nThey conclude that the phase with the highest fractional polarization $\\alpha_{\\rm max}$\nis shifted towards larger phase angles which could be explained by an increasing contribution of the\nEarth surface reflection.\nIn our data we do not see this shift but neither can we exclude it because \nwe were not able to derive meaningful data due to the very strong stray light from the moonshine and the weak signal from\nthe earthshine.\nIn this regime our linear extrapolation method to subtract\nthe background stray light from the earthshine signal introduces a strong systematic overestimate $\\Delta_{\\rm syst}$ of the result (see Sect. \\ref{s: extract earthshine}).\n\\citet{Takahashi2013} also use a linear extrapolation method to determine the earthshine polarization but unfortunately they do not describe their data reduction in detail.\nTherefore, considering the limitations of our linear extrapolation, it could be possible that the shift of $\\alpha_{\\rm max}$ reported by \\citet{Takahashi2013}\nis due to the strong stray light at phase angles $> 90^\\circ$.\n\nOverall, the spectral dependence of the polarization of \\citet{Sterzik2012} and \\citet{Takahashi2013} is qualitatively\nsimilar to our measurements but the level and slope of the fractional polarization differ quantitatively. \nBecause \\citet{Sterzik2012} and \\citet{Takahashi2013} provide no information about the lunar surface albedo for their\nmeasuring area and do not assess the stray light effects from the bright moonshine their results cannot be used for a\nquantitative test of our results.\nThe spectral slope of \\citet{Sterzik2012} is slightly steeper than ours while the slope of \\citet{Takahashi2013} is slightly flatter. \n\n\nFor an assessment of the polarization efficiency for the lunar back-scattering we used \nliterature data for polarimetric\nmeasurements of lunar samples by \\citet{Hapke1993, Hapke1998} and we derive\na wavelength and surface albedo dependent polarization \nefficiency relation $\\epsilon(\\lambda,a_{603})$ \nwhich gives for mare in the V band\n $\\epsilon(V,0.11)=50.8~\\%$. This value is\nsignificantly higher than the 33~\\% derived by \n\\citet{Dollfus1957} which he based on the analysis of volcanic\nsamples from Earth used as a proxy for the lunar maria. \nBecause of this, the Earth polarization derived in this\nwork is much lower than the value given in \\citet{Dollfus1957}. \nWe are not aware of other studies on the polarization efficiency $\\epsilon$\nfor the lunar back-scattering. Relying the determination of $\\epsilon$ on\nreal lunar soil is certainly an important step in the right direction for a more accurate\ndetermination of the polarization of Earth.\n\nVery valuable are the reported Earth polarization values\nfrom \\citet{Wolstencroft2005} based on direct\npolarization measurements with the POLDER instrument on the\nADEOS satellite. They derived the fractional polarization for the\nwavelengths 443 nm (B'), 670 nm (R') and 865 nm (I') for different surface types and\ncloud coverage. \nWeighted mean values representative for an integrated planet \nEarth observation (55~\\% cloud coverage) \nof 22.6~\\%, 8.6~\\% and 7.3~\\% in the \nB', R' and I' band are obtained which are also indicated in the second panel of\nFig.~\\ref{ColorDep}. The good agreement between our \nderivation based on the earthshine and the values from \n\\citet{Wolstencroft2005}\nconfirms our determination of the polarization efficiency. \nUnfortunately, it is not possible to assess whether the R band point of this study\ndiffers significantly from the value of \\citet{Wolstencroft2005}\nbecause they give no description of their data and uncertainties.\n\n\\subsection{Comparison with the models from Stam (2008)}\n\nThe study of \\citet{Stam2008} is unique for the modeling of\nthe spectral dependence of the fractional polarization of\nEarth-like planets. In her work she explored also dependencies on a\nrange of physical properties different from Earth. For our\ncomparison we pick the model for an\ninhomogeneous Earth-like planet with 70~\\% of the surface\ncovered by a specular reflecting ocean and 30~\\% by deciduous \nforrest (lambertian reflector with an albedo for forest), and cloud coverages 40~\\% and 60~\\%. When compared to our\nEarth polarization determinations (Fig. \\ref{ColorDep}, second panel), these models\nagree with our measurements at short wavelengths but\nshow a clear deficit in the fractional polarization at long wavelengths in the I band.\nThis is not surprising\nsince the models were not tuned to the case of Earth. In the models only very thick, liquid water clouds were included but no thin liquid water clouds and no ice clouds.\n\\citet{Karalidi2012} showed that with more realistic cloud properties for Earth the degree of polarization can vary strongly depending on cloud optical thickness.\nHence, our data could now be used to test and to improve model calculations for the Earth polarization.\n\n\n\\subsection{Polarization flux contrast for the Earth - Sun system} \\label{flux contrast}\n\n\n \\begin{table}\n \\caption{Geometric albedo $A_g$, phase integral $A_s\/A_g$ and quadrature results for the planet Earth. \n \t\t Given are the fractional polarization $p_{\\rm E}$, the Earth reflectivity $f_{\\rm E}$,\n \t\t the polarized reflectivity \\mbox{$p_{\\rm E}\\times f_{\\rm E}$}, and the polarization contrast \n\t\t $C_{\\rm p}=p_{\\rm E} \\times f_{\\rm E} \\times \\left(R_{\\rm E}\/d_{\\rm S-E}\\right)^2$ for an Earth-Sun system.\n\t\t The systematic offset uncertainty $\\Delta p_{\\rm E}$ is due to the uncertainty of the \n\t\t depolarization and albedo of the lunar surface.\n } \n \\label{tbl: quadrature results} \n \\centering \n \\begin{tabular}{L{25mm} | C{10mm} C{10mm} C{10mm} C{10mm}} \n \\hline \n & B & V & R & I \\\\\n \\hline \n $A_g$ & 0.504 & $0.367^1$ & 0.320 & 0.301 \\\\\n $A_s\/A_g$ & 0.86 & 0.86 & 0.86 & 0.86 \\\\\n \\hline\n values for $\\alpha = 90^\\circ$ & & & & \\\\\n \\hline \n $p_{\\rm E}$ [\\%] & 24.6 & 19.1 & 13.5 & 8.3 \\\\\n $\\Delta p_{\\rm E}$ [\\%] & $\\pm 1.2$ & $\\pm 1.3$ & $\\pm 0.9$ & $\\pm 1.4$ \\\\\n $f_{\\rm E}$ & 0.136 & 0.099 & 0.086 & 0.081 \\\\\n $p_{\\rm E} \\times f_{\\rm E}$ [$10^{-3}$] & 33.41 & 18.90 & 11.66 & 6.73 \\\\\n $C_{\\rm p}$ [$10^{-11}$] & 6.07 & 3.44 & 2.12 & 1.22 \\\\\n \\hline\n\n \\multicolumn{5}{l}{\\vspace{-0.2cm}}\\\\\n \\multicolumn{5}{l}{$^1$\\citet{Cox2000}}\n \\end{tabular}\n \\end{table}\n \n\n\n\n\n\n \\begin{figure}\n \\centering\n \\includegraphics[width=8cm]{phasefunction.eps}\n \\caption{Polarized reflectivity phase curve $p_{\\rm E}(\\alpha)\\times f_{\\rm E}(\\alpha)$ for Earth in the B (solid), V (dotted), R (dashed), and I band (dash-dot).}\n \\label{phasefunction} \n \\end{figure}\n\n\n \nA key parameter for the polarimetric search and characterization of Earth-like \nextra-solar planets is the polarization flux $p \\times f$ of a planet or the\npolarization flux contrast $C_p$ as described in \\mbox{Eq. \\ref{cpol}}. \nThe polarization flux of a highly polarized planet is easier to \nmeasure than the fractional polarization $p$, because the reflected intensity\ncannot be distinguished easily from the scattered light halo of the central\nstar in high contrast observations. But because stars are essentially unpolarized, it should be possible\nto detect a differential signal of polarized light from extra-solar planets\nwith high contrast polarimetric imagers as foreseen for the upcoming instrument \nSPHERE\/VLT and planed for future facilities like the \\mbox{E-ELT}\n\\citep[e.g.][]{Schmid2006, Beuzit2008, Kasper2010}.\nThe polarimetric detection of an Earth-like planet is difficult but, nonetheless, \nit is useful to have accurate values for the expected signal for the planning\nof such observations.\n\nThe prediction of the polarization flux of an exo-Earth requires\nbesides the fractional polarization $p(\\alpha,\\lambda)$ determined in this \nwork also the reflected intensity $f(\\alpha,\\lambda)$. \n\nThe reflected intensity of Earth can be split into a wavelength\ndependent geometric albedo term $A_g(\\lambda)=f(0^\\circ,\\lambda)$\nand a normalized phase dependence $\\Phi(\\alpha)\/\\Phi_0$ \nwhere $\\Phi_0=\\Phi(\\alpha=0)$ according to:\n\\begin{displaymath}\nf(\\alpha,\\lambda) = A_g(\\lambda) {\\Phi(\\alpha)\\over \\Phi_0}\\,.\n\\end{displaymath}\nWith this approach we neglect the color dependence of the phase curve, which \nis not known but certainly small when compared to the measuring\nuncertainties for the spectral albedo $A_g (\\lambda)$ and the uncertainties in the fractional\npolarization $p_{\\rm E}(\\lambda, \\alpha)$. \n \nThe visual geometric albedo of Earth is $A_g(V)=0.367$ \\citep{Cox2000}.\nWith the relative spectral reflectance \nmeasured by \\citet{Arnold2002}, \\citet{Woolf2002}\nand \n\\citet{Montanes2005}, we deduce the geometric albedo for the\nindividual B, V, R, and I filters as given in \\mbox{Table \\ref{tbl: quadrature results}}.\n\nFor the phase dependence $\\Phi(\\alpha)\/\\Phi_0$ we use \nthe phase curve determined by \\citet{Palle2003} for the 400-700~nm filter\nnormalized to the B, V, R, and I band geometric albedos derived above. The derived\nphase curves are given in Fig. \\ref{evsm} and \ntheir value for $\\Phi(90^\\circ)\/\\Phi_0 = 0.27$ yields \nthe polarized reflectivity for quadrature phase $p_{\\rm E}(90)\\times f_{\\rm E}(90)$\nfor the B, V, R, and I filters as plotted in Fig. \\ref{ColorDep}\nand given in Table \\ref{tbl: quadrature results}. \n\nThe phase curve $\\bar{f}_{\\rm E}$ of \\citet{Palle2003} is \nbased on earthshine measurements\nat phases between $\\alpha = 30^\\circ - 145^\\circ$ extrapolated to $\\alpha = 0^\\circ - 180^\\circ$. \nThis broad phase angle coverage is unique and remains, to our knowledge, the only available observation of the \nphase dependence $\\Phi(\\alpha)$ of $f_{\\rm E}$. For future reference we also give in Table \\ref{tbl: quadrature results} the \nphase integral parameter $A_s\/A_g$, the ratio between spherical and geometric albedo, derived from the \n\\citet{Palle2003} data. \n\n\nThe spectral dependence of the polarization flux $p_{\\rm E}(\\lambda, 90^\\circ) \\times f_{\\rm E}(\\lambda, 90^\\circ)$ \nof Earth decreases steeply towards longer wavelength, because\nboth, the fractional polarization $p_{\\rm E}$ and the reflectivity $f_{\\rm E}$ are\nhigher for the blue than the red. The $p_{\\rm E} \\times f_{\\rm E}$ signal in the B band is about a \nfactor 5 times stronger than in the I band. \n\nThe polarization contrast $C_p(\\lambda,90^\\circ)$ according to Eq. \\ref{cpol}\nis determined from $p_{\\rm E} \\times f_{\\rm E}$ and using $R_E^2\/{\\rm AU}^2$. This yields values\nat the level of a few times $10^{-11}$ only \n(Table \\ref{tbl: quadrature results}). One should note that\nan Earth-like planet in the habitable zone of a M star with\n$L=0.02~L_\\odot$ is at a much smaller separation of 0.14 AU. In this\ncase the expected polarization contrast is about a factor of 50 higher\nand within reach for a high contrast imaging polarimeter at\nan ELT (Kasper et al. 2010). \n\n\nThe B, V, R, and I phase curves for the polarized reflectivity \n\\mbox{$p_{\\rm E} \\times f_{\\rm E}$} are plotted in Fig. \\ref{phasefunction}. \nThe maximum signal occurs near $\\alpha\\approx 65^\\circ$, which is thus the \nbest phase for a detection. \n\n\n\n\n\\section{Summary and Discussion}\\label{s:conclusions}\n\nThis work presents measurements of the earthshine polarization in the\nB, V, R and I bands. The data were acquired with a specially designed\nwide field imaging polarimeter using a focal plane mask for the suppression\nof the light from the bright lunar crescent. Thanks to this measuring\nmethod we can accurately correct for contributions from the (twilight) sky and \nthe scattered light from the bright lunar crescent and derive\nvalues with well understood uncertainties. We derive phase curves\nfor the fractional polarization for the earthshine reflected from maria\nand highlands for the different filter bands. The phase curves can be\nfit with the sine-square function $q_{\\rm max} \\sin^2(\\alpha)$. The amplitude\n$q_{\\rm max}$ decreases strongly with wavelength from about \n13~\\% in the B band to about 3~\\% in the I band (see Table \\ref{tbl: results}). The fractional \npolarization of the earthshine is about a factor 1.3 higher for \nthe dark mare region when compared to the bright highland. \nOur phase curve for the mare region in the V band is in very good agreement with \nthe historic visual polarization phase curve from \\citet{Dollfus1957}. \n\n\nWe study the depolarization introduced by the back-scattering from\nthe lunar surface based on published polarimetric\nmeasurements of lunar samples \\citep{Hapke1993, Hapke1998}. \nWe derive a 2-dimensional fit function for the polarization efficiency \n$\\epsilon(\\lambda,a_{603})$ for the back-scattering \nwhich depends on wavelength and surface albedo. \nEarthshine measurements plus $\\epsilon$ correction yield\nas main result of this paper the fractional polarization of the \nreflected light from the planet Earth as function of phase in four bands. \nThe polarization of Earth at quadrature phase is as high as \n25~\\% in the B band and decreases steadily with wavelength to 8~\\% in the\nI band (see Table \\ref{tbl: quadrature results}). \nSimilar values were reported from direct satellite measurements\nof the Earth polarization \\citep{Wolstencroft2005}. \n\nThis work provides the most comprehensive measurements\nof the polarization of the integrated light of the planet Earth up to now.\nThe determined values can be used as benchmark values\nfor tests of polarization models and for predictions for future polarimetric \nobservations of Earth-like extra-solar planets. \nIn particular we describe accurately our measurements and assess the uncertainties.\nIn addition we apply for the first time a polarization efficiency $\\epsilon$ correction\nwhich is based on lunar soil measurements, and which is significantly different from \npreviously used volcanic rock measurements.\n\nSimilar to our data of Earth the models of \\citet{Stam2008} for horizontally \ninhomogeneous Earth-like planets with thick liquid water cloud coverage show \nalso a decrease in fractional\npolarization with wavelength but with a significantly steeper slope. \nThis may indicate that other\nscattering components, for example aerosols, thin liquid water clouds, and ice clouds contribute\nsignificantly to the Earth polarization in the I band \\citep[see][]{Karalidi2012}.\n\n\nAre our polarization values for the planet Earth representative\nor should we expect large temporal variations? Our data were\ntaken during two observing runs lasting each a few days.\nTwo data sets are from similar phase angles,\n$73.0^\\circ$ and $75.5^\\circ$, but they were taken 7 months apart. \nThe measured fractional polarization differs by about $\\Delta q\/q\\approx 0.1$. \nAlso the deviation of the data points from the fit $q_{\\rm fit}=q_{\\rm max}\\sin^2\\alpha$ is \nat the same level $(|q-q_{\\rm fit}|)\/q\\approx 0.1$. This scatter is at the level \nof our calibration errors. Therefore, variation of the intrinsic polarization signal of\nEarth on the 10~\\% level could be present in our data without\nbeing recognized. Our measurements show certainly no changes at\nthe $\\Delta q\/q\\approx 0.3$ level as suggested by \\citet{Sterzik2012}.\nVariations in the fractional polarization are of interest because they\ncould be used as diagnostic tool for investigations of surface structures\nor temporal changes in the cloud coverage of extra-solar planets. \n\nBecause our study includes a detailed assessment of the \nuncertainties for each step in our determination, we can now discuss \nhow the Earth polarization measurement could be improved. \n\n\nPolarization variability studies could be carried out\nwith enhanced sensitivity selecting observing periods and filters\nwith strong earthshine polarization signals in order to minimize\nstatistical noise and systematic effects in the data extraction. \nObservations in the B and V filter, and for phase\nangles in the range from $\\alpha=40^\\circ$ to $100^\\circ$ would be\nideal for such studies. Measurements taken for several\nconsecutive nights would allow a sensitive search for day to day\nvariation at a level of \n$\\Delta q\/q \\approx 0.03$ due to variable cloud \ncoverage. Also multiple epoch data could be collected for an investigation \nof long term and seasonal polarization changes. \n\nThe determination of a more accurate wavelength dependence of the \nearthshine polarization could be established with long integrations\nfor phase angles between $50^\\circ - 80^\\circ$ when the earthshine\npolarization signal is strong, the level of scattered light from the\nmoonshine still low, and the time for observations after twilight long\nenough for observations in multiple filters. \n\nMore accurate phase curves require a careful analysis of the data\nfrom different phases because the earthshine observing conditions\nand the associated measuring and calibration procedures change\nstrongly with lunar phase. If these problems can be \nsolved then one could determine accurately the peak in the fractional \npolarization curve near $\\alpha = 90^\\circ$ as function of wavelength\nand investigate the presence of a rainbow feature in the polarization data\naround $\\alpha=40^\\circ$ \\citep[see][]{Stam2008}. \n\nA more accurate absolute value for the polarization of the planet Earth\nrequires first more data in order to average out intrinsic \nvariations. Equally important is a more accurate determination of\nthe surface albedo for the measuring region and the associated\npolarization efficiency $\\epsilon(\\lambda,a_\\lambda)$ for the correction\nof the lunar back-scattering. \n\nThe imaging polarimetry of the earthshine presented in this study\nand the spectro-polarimetric results from \\citet{Sterzik2012} and \\citet{Takahashi2013} demonstrate\nthat the investigation of the Earth polarization via earthshine measurements \nis very useful and attractive. Detailed and versatile investigations \nare possible with existing polarimetric instruments as used by \\citet{Sterzik2012} and \\citet{Takahashi2013}\nor with small, specific experiments as demonstrated in this work. \nThe obtained results can be compared with model calculations like\nthose described in \\citet{Stam2008} and teach us about light scattering\nprocesses of planets. Because we know so well our Earth we can also\ninvestigate subtle effects, which are potentially important in other\nplanets. Building up our knowledge on scattering polarization from Earth\ncould therefore become also important for the future polarimetric\ninvestigation of extra-solar planets. \n\n\n\n\n \n\\begin{acknowledgements}\n Part of this work was supported by the FINES research fund by a grant through the Swiss National Science Foundation (SNF).\n\\end{acknowledgements}\n\n\n \n\n\\bibliographystyle{aa} \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction} \\label{sec:intro}\n\nFor decades, star clusters have been recognized as useful tracers of star formation. Rather than representing uniform tracers of star formation, we now understand that non-embedded, long-lived, gravitationally-bound star clusters emerge from natal regions with relatively high gas surface density and star formation efficiency \\citep{Elmegreen97, Kruijssen12, Grudic21}. Star clusters contain $\\sim$5--30\\% of the stellar mass formed \\citep{Johnson16_gamma, Adamo20} and are long-lasting remnants of peaks in hierarchically structured star-forming regions that survived the stellar feedback and gas removal dissolution that unbind most stellar groupings and natal structures.\n\nWe now have broad samples of clusters spanning a wide variety of galactic environments, where long-lived bound star clusters are tell-tale tracers of past episodes of intense, efficient star formation. Large imaging surveys using the \\textit{Hubble Space Telescope} (HST) have made significant progress in cataloging and characterizing star cluster populations in nearby (3--30 Mpc) galaxies \\citep[e.g., LEGUS, PHANGS-HST;][]{Adamo17, Lee22}. While the diversity of galactic environments included in these samples is very useful for purposes of galaxy-to-galaxy comparisons, individual star clusters at these distances are only marginally resolved, limiting observational measurements to integrated properties.\n\nIn contrast, studies of neighboring galaxies in the Local Group provide a unique opportunity for detailed studies of external galaxies and their star clusters, yielding a rich picture of star formation observed at and below molecular cloud spatial scales --- a level of detail not possible in more distant extragalactic targets. Due to their proximity and the spatial resolving power of HST, Local Group galaxies provide an unmatched opportunity to construct high quality cluster catalogs and make detailed observations of these systems and their environments \\citep[e.g.,][]{Johnson12}. Local Group cluster catalogs reach low cluster mass completeness limits leading to increased sample sizes and diversity. Observations at these distances resolve individual cluster member stars leading to marked improvements in age-dating precision and usefulness to stellar evolution studies \\citep{Johnson16_gamma, Girardi20}. \n\nThis paper studies the star cluster population of the Triangulum Galaxy (M33), whose intermediate galaxy mass and relatively active star formation providing a point of comparison to studies of the stellar cluster populations of the Andromeda Galaxy (M31) and the Magellanic Clouds. Notably, M33 hosts a larger star formation rate surface density ($\\Sigma_{\\mathrm{SFR}}$) than the bulge-dominated, relatively quiescent M31 \\citep{Williams21}. Therefore, we expect observations of M33's young cluster population to unlock valuable new insight into star cluster formation and evolution. Triangulum's relatively face-on orientation \\citep[inclination angle of 55$^{\\circ}$;][]{Koch18} also presents an advantage over Andromeda in terms of line-of-sight dust attenuation and projection effects.\n\nStudies of the star cluster population in M33 include results from both ground-based \\citep[e.g.,][]{Christian82, SanRoman10} and space-based \\citep[e.g.,][]{Chandar99, Chandar01, Park07, SanRoman09} observations, as summarized by \\citet{Sarajedini07}. Previous work demonstrated HST's utility for identifying star clusters, but spatial coverage of M33's star-forming disk was sparse and largely non-contiguous, preventing systematic studies of the cluster population. As a result, much of the past work to characterize M33's star clusters makes use of ground-based imaging and photometry \\citep[e.g.,][]{deMeulenaer15, Fan14}, especially from the Local Group Galaxy Survey \\citep[LGGS;][]{Massey06}.\n\nWe note that an alternative catalog of M33 young star cluster candidates was published by \\citet{Sharma11} based on mid-infrared Spitzer 24$\\mu$m\\ source identification. This catalog should be sensitive to embedded clusters that are not detected by an optical search, and has been used for analysis of M33 clusters by a number of groups \\citep[e.g.,][]{GonzalezLopezlira12, Pflamm13, Corbelli17}. However, significant concerns about this catalog's contamination by non-cluster objects and its suitability for star cluster studies have been well articulated by \\citet{Sun16}. HST observations will have comparatively limited sensitivity to the earliest embedded stages of star cluster formation (1--3 Myr), but its high spatial resolution (0.1 arcsec v.\\ $\\sim$6 arcsec for Spitzer 24$\\mu$m images) remains the best avenue for identification and analysis of star clusters at nearly every other age.\n\nThe Panchromatic Hubble Andromeda Treasury: Triangulum Extended Region survey \\citep[PHATTER;][]{Williams21} of M33 delivers contiguous, multi-band imaging of a majority of the galaxy's star-forming disk, extending the same quality of data obtained by the Panchromatic Hubble Andromeda Treasury survey \\citep[PHAT;][]{Dalcanton12} in M31 to M33. Similarly, this work moves cluster studies in M33 into a new era using techniques and analysis that were employed to construct the PHAT cluster catalog \\citep[][hereafter J15]{Johnson15_AP} for M31.\n\nFacing the absence of a robust algorithmic method for identifying clusters in Local Group galaxy images, we launched an online citizen science project, the Local Group Cluster Search (LGCS), to perform a visual search of the PHATTER data. We employ the crowdsourced methodology developed for PHAT and the Andromeda Project \\citepalias{Johnson15_AP} to construct a star cluster catalog. This approach improves on the subjectivity of expert-led searches conducted in the past in M33 \\citep[e.g.,][]{Christian82, Chandar99} using a ``wisdom of the crowds'' consensus classification technique, where an unbiased, repeatable result is obtained by averaging over tens of independent image classifications. In addition to producing a robust cluster catalog, we characterize catalog completeness using synthetic clusters inserted into the search images. Not only does this methodology produce useful results, but it facilitates meaningful engagement with project volunteers regarding astronomy and star cluster science.\n\nIn this paper, we present the survey-wide cluster catalog for PHATTER. We describe the LGCS project, its input data and preparation, and data collection results in \\S\\ref{sec:data}. We analyze image classifications and outline the steps required to produce a cluster catalog in \\S\\ref{sec:catcon}. We present the final catalog in \\S\\ref{sec:catalog}, followed by a characterization of the catalog's completeness in \\S\\ref{sec:catcompleteness}. We derive integrated light ages and masses in \\S\\ref{sec:slugfitting} and place the new catalog in context with previous work in M33 and similar work in M31 in \\S\\ref{sec:discussion}.\n\nThis catalog serves as the foundation for PHATTER survey cluster science. Future work includes the measurement of the cluster mass function \\citep{Wainer22}, measurement of the high-mass stellar initial mass function, calibration of stellar evolution models, and more. These studies will build upon and benefit from comparisons to the PHAT star cluster studies of M31, including measurements of star cluster formation efficiency \\citep{Johnson16_gamma}, the cluster mass function \\citep{Johnson17}, and the high mass stellar initial mass function \\citep{Weisz15}.\n\nThroughout this work, we assume a distance to M33 of 859 kpc \\citep[distance modulus: 24.67][]{deGrijs14} where 1 arcsec is equivalent to $\\sim$4.2 pc.\n\n\\section{Data} \\label{sec:data}\n\nIn this section, we describe the Local Group Cluster Search citizen science project and the underlying HST data that enables this study. We begin by describing the PHATTER imaging used for the project (\\S\\ref{sec:phatdata}) and the LGCS website interface (\\S\\ref{sec:interface}). Next, we discuss data collection and statistics regarding image classifications and project volunteers (\\S\\ref{sec:datacollection}). Finally, we discuss the creation of synthetic clusters used to characterize catalog completeness (\\S\\ref{sec:synclst}).\n\n\\subsection{PHATTER Images and Resolved Star Photometry} \\label{sec:phatdata}\n\nThe HST images analyzed by the LGCS project were obtained as part of the PHATTER survey. Full details of the survey are presented in \\citet{Williams21}, but here we highlight the features of this survey that are relevant to star cluster catalog work.\n\nThe PHATTER survey uses the same imaging strategy as the PHAT survey, where parallel observations are efficiently obtained with the Advanced Camera for Surveys (ACS) and Wide Field Camera 3 (WFC3). These observations are organized into three contiguous ``bricks'', a 3$\\times$6 mosaic of WFC3 footprints formed from pairs of parallel ACS and WFC3 images that combine to create a rectangular region of fully-overlapped spatial coverage in all observed passbands. This observing strategy yields images in six filters: F475W and F814W in the optical obtained with ACS\/WFC; F275W and F336W in the near-UV obtained with WFC3\/UVIS; F110W and F160W in the NIR obtained with WFC3\/IR. The PHATTER survey's three bricks (54 individual fields of view) span the inner disk of M33, extending out to a galactocentric radius of $\\sim$4~kpc. We use three types of image products from the survey. First, drizzled single-pointing ACS images were used to create optical images with synthetic clusters inserted to minimize computational effort (see \\S\\ref{sec:synclst}). Second, brick-wide mosaic images for each of the six filters were used for aperture photometry of the clusters, which provide the best overlapping spatial coverage and artifact removal (i.e., chip gaps and cosmic rays). Third, LGCS search images (see \\S\\ref{sec:interface}) were extracted from survey-wide optical mosaic images. All images have an image scale of 0.05~arcsec pixel$^{-1}$ are astrometrically aligned to Gaia DR2 with 3~mas (7~mas) residuals for ACS\/WFC and WFC3\/UVIS (WFC3\/IR), and are combined and distortion corrected using \\texttt{AstroDrizzle} from the \\texttt{DrizzlePac} package \\citep{DrizzlePac12,Hack13,Avila15}.\n\nIn addition to the images, we also use PHATTER resolved star photometry catalogs presented in \\citep{Williams21}. This PSF photometry was measured simultaneously in all six filters using DOLPHOT\\footnote{\\url{http:\/\/americano.dolphinsim.com\/dolphot\/}}, an updated version of the HSTphot photometry package \\citep{Dolphin00}.\n\nWe use the PHATTER photometry catalogs to quantify and map stellar density across the survey footprint. Specifically, we define and use two quantities: $N_{\\mathrm{MS}}$, the number of upper main sequence stars selected using a color magnitude cut of F475W $<$ 24 and F475W$-$F814W $<$ 1; $N_{\\mathrm{RGB}}$, the number of bright RGB stars defined using a polygon region in the optical color magnitude diagram (CMD) that mimics a NIR-based selection used in M31 by \\citetalias{Johnson15_AP}, where F475W$-$F814W $> $1.5 and F814W brighter than $\\sim$22.5.\n\n\n\\subsection{LGCS Interface} \\label{sec:interface}\n\nThe Local Group Cluster Search (LGCS)\\footnote{\\url{https:\/\/www.clustersearch.org\/}} is a citizen science project built and hosted on the Zooniverse\\footnote{\\url{https:\/\/www.zooniverse.org\/}} platform. The project is a direct follow-on of the Andromeda Project \\citepalias{Johnson15_AP}, but was built using the Zooniverse's Project Builder\\footnote{\\url{https:\/\/www.zooniverse.org\/lab}} platform tools rather than being built as a custom project-specific website. The Project Builder platform allowed the research team to build and configure the project without the effort or assistance of the Zooniverse web development team, though LGCS has fewer custom features than the Andromeda Project (e.g., no interactive walk-through of interface tools during project tutorial). The main capabilities provided by the Zooniverse platform, however, remain the same: an interactive user interface that enables image annotation, web hosting for the project page and image data, subject image selection and queuing, feedback confirming the correct identification of synthetic clusters in the images (see \\ref{sec:synclst}), and storage of classification responses.\n\nThe scope of the LGCS project extends beyond the search of PHATTER imaging presented here; the project hosts a visual cluster search of SMASH \\citep{Nidever17} imaging of the Large and Small Magellanic Clouds, and will expand to additional datasets in the future. This study focuses solely on the results of the PHATTER M33 search, while results from other LGCS searches will be published separately in future work.\n\n\\begin{figure*}\n \\centering\n \\includegraphics[width=0.75\\textwidth]{LGCS_website.pdf}\n \\caption{Screenshot of the Local Group Cluster Search annotation interface.}\n \\label{fig:website}\n\\end{figure*}\n\nVolunteers who participate in the LGCS search of the PHATTER data use a simple annotation interface to mark objects of interest, as shown in Figure~\\ref{fig:website}. Specifically, participants are asked to mark star clusters, background galaxies, and nebulous emission regions using a circular tool with adjustable size. Users click the center of an object in the image and drag outward to set a circular marker's radial size. Two images are shown to participants: a color image constructed from F475W and F814W bands, and an inverted grayscale F475W image. The two-band composite image provides important color information, while the inverted single-band image provides high contrast to improve the detection of faint clusters. Individual $\\sim$36$\\times$25 arcsec ($\\sim$150$\\times$100 pc) subimages are extracted from survey-wide drizzled mosaic images. These subimages spatially overlap by 100 pixels (5 arcsec) to minimize edge effects on search results.\n\nIn addition to the image annotation interface, the LGCS project also hosts helpful resources for volunteers. Participants are presented a tutorial for instruction about the task during their first visit to the classification page, which includes a short demonstration video and text instructions. Volunteers can also access a number of ``About'' pages that summarize the project's research goals and background information, a field guide that provides detailed examples of target objects, and a forum called ``Talk'' to facilitate interactions between with the research team and project participants.\n\n\\subsection{Data Collection and Classification Statistics} \\label{sec:datacollection}\n\nThe LGCS project collected image classifications from 8 January 2019 to 28 February 2019. During this time, LGCS volunteers submitted 269,645 classifications, where one classification denotes a participant's response specifying the location and size of any objects they identify in an image. Each search image was classified by at least 60 unique volunteers.\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=8.5cm]{Cumulative_Fraction.pdf}\n \\caption{Classification statistics of individual LGCS users, sorted in decreasing order of total submitted classifications. Top: The cumulative fraction of the LGCS's $\\sim$269 thousand classifications. Bottom: The number of classifications submitted by the Nth ranked volunteer. The red dashed-dotted lines illustrate that 50\\% of all classifications were performed by the 50 most active volunteers who each contributed at least 1267 classifications. The blue dashed-dotted lines illustrate that 90\\% of all classifications were performed by 463 volunteers who each contributed at least 62 classifications.}\n \\label{fig:userstats}\n\\end{figure}\n\nA total of 2757 users participated in the LGCS project, 1517 of whom participated as registered users using a Zooniverse account. Contributions from individuals who were not logged in are identified and grouped via an anonymized hash of the participant's IP address. We note that while non-registered users represent 45\\% of volunteers by number, these users only contributed 8\\% of the total classifications. Beyond this difference between registered and non-registered users, the distribution of effort across the pool of participants varies significantly, as shown in Figure~\\ref{fig:userstats}. The median number of classifications per person is nine (nineteen for registered users), but half of all classification effort was provided by the 50 most active volunteers --- each of whom contributed at least 1267 classifications. This behavior is consistent with trends seen for the Andromeda Project \\citepalias{Johnson15_AP} and other Zooniverse projects \\citep{Spiers19}, where a relatively small group of active volunteers contribute a bulk of the total effort.\n\n\\pagebreak\n\\subsection{Synthetic Cluster Generation} \\label{sec:synclst}\n\nA key part of our analysis is incorporating synthetic clusters with known ages, masses, and radii into the visual search. Synthetic clusters were inserted into the same LGCS images used for the cluster search, but were analyzed separately. The synthetic clusters were created following a procedure similar to the one used by \\citetalias{Johnson15_AP}. In short, a synthetic cluster's individual member stars were drawn with masses following a \\citet{Kroupa01} IMF and stellar properties for a specified age using PARSEC 1.2S + COLIBRI PR16 isochrones \\citep{Bressan12,Marigo17}. Spatial positions were drawn from a \\citet{King62} profile with a specified effective radius, $R_{\\mathrm{eff}}$, and a concentration ($R_{\\mathrm{tidal}}\/R_{\\mathrm{core}}$) of 30. The sample of synthetic clusters were created with the following properties:\n\\begin{enumerate}\n \\item Ages were drawn randomly from a grid of log(Age\/yr) values ranging from 6.6 to 10.1 incremented every 0.05 dex.\n \\item Masses were drawn randomly from a continuous uniform distribution of log(Mass\/$M_{\\sun}$) values ranging from 2.0 to 5.0.\n \\item A fixed Solar metallicity ($Z$=0.0152) was assumed for ages younger than 5 Gyr. For older ages, the metallicity was randomly drawn from a set of five discrete values ($Z$=[0.0152,0.005,0.0015,0.0005,0.00015]) so that the sample of older synthetic clusters would span a metallicity range resembling that of Galactic and extragalactic globular clusters.\n \\item Extinctions were drawn from an exponential $A_V$ distribution which ranges from the foreground Milky Way extinction value of 0.11 to 3.0 mags, following the expression: $P(A_V) \\propto e^{-(A_V \/ 1.34)}$; this is the same distribution used for M31 synthetic clusters by \\citetalias{Johnson15_AP}. \n \\item Effective radii ($R_{\\mathrm{eff}})$ were drawn from the distribution of measured values obtained for M31 clusters by \\citetalias{Johnson15_AP}, but biased to larger $R_{\\mathrm{eff}}$ values to ensure sufficient number statistics of diffuse clusters in the high-$R_{\\mathrm{eff}}$ tail for completeness determination purposes.\n\\end{enumerate}\n\nAfter creating a parent population of artificial clusters, we chose a subset to insert into LGCS images. This selection was based on cluster magnitude and age, and it was designed to produce a sample of synthetic clusters that spans the full range of detectability, from easily detected to undetectable. Specifically, we adopt the following magnitude limits: $18.5 < m_{\\mathrm{F475W}} < 22$ for $6.6 < \\log(\\mathrm{Age\/yr}) < 8.0$; $19.5 < m_{\\mathrm{F475W}} < 22.5$ for $8.0 < \\log(\\mathrm{Age\/yr}) < 9.0$; $20 < m_{\\mathrm{F475W}} < 22.5$ for for $9.0 < \\log(\\mathrm{Age\/yr}) < 10.0$.\n\nWe inserted the magnitude-selected sample of synthetic clusters into F475W and F814W images using DOLPHOT. One synthetic cluster was added per LGCS search image, positioned pseudo-randomly within the image, avoiding positions within 120 pixels of the edge. Because DOLPHOT places the synthetic clusters into each individual frame before drizzling, we use single-field images as opposed to multi-image mosaics for insertion to minimize computational complexity. Insertion locations were chosen to avoid chip edges and gaps to ensure that the synthetic images were essentially identical to the original search images. \n\nWe created two batches of synthetic clusters, each with 848 objects for a total of 1696 synthetic clusters. The first batch was randomly assigned to LGCS images spanning the entire PHATTER survey footprint, resulting in a diverse set of cluster-image pairs across the full range of galactic environments. The second batch was assigned spatial locations in a targeted manner, such that young clusters ($<$100 Myr) were placed in regions of the footprint with a high density of bright, blue stars, as defined by their high $N_{\\mathrm{MS}}$ values ($N_{\\mathrm{MS}} > 1200$ stars per search image). The remaining older synthetic clusters were distributed across the remaining fields with lower $N_{\\mathrm{MS}}$ values. The targeted placement of this second batch ensures sufficient numbers of young synthetic clusters fall within young star-forming regions, safeguarding our ability to derive catalog completeness for the key population of young star clusters.\n\n\\section{Catalog Construction} \\label{sec:catcon}\n\nThe process of converting LGCS image classifications into a star cluster catalog involves combining 60 independent classifications from each image into a consensus result regarding the presence, location, and size of candidate clusters (and other objects). This target number of classifications per image is selected as a balance of keeping statistical errors on classification results low and maintaining a reasonable total runtime for the project.\n\nThe first step in this process is to compile and combine the candidate identifications made by LGCS participants. We merge identifications following the procedure described in detail in Appendix A of \\citetalias{Johnson15_AP}: we aggregate markings for each individual search image by clustering marker centers and merging overlapping candidates, then we combine the per-image lists of identifications into a survey-wide data product by running a spatial match to merge duplicate candidates in regions of overlapping image coverage.\n\nWe use the fraction of classifications where the object is detected as the principal indicator of significance. We compute four fractional metrics to characterize each candidate: \n\\begin{enumerate}\n \\item $f_{\\mathrm{view}}$ is the fraction of total classifications where a candidate is identified as any class of object.\n \\item $f_{\\mathrm{cluster}}$ is the fraction of total classifications where a candidate is identified as a star cluster\n \\item $f_{\\mathrm{galaxy}}$ is the fraction of total classifications where a candidate is identified as a background galaxy\n \\item $f_{\\mathrm{emission}}$ is the fraction of total classifications where a candidate is identified as an emission region.\n\\end{enumerate}\nThese quantities are related by:\n\\begin{equation}\n f_{\\mathrm{view}} = f_{\\mathrm{cluster}} + f_{\\mathrm{galaxy}} + f_{\\mathrm{emission}}\n\\end{equation}\nWe note that the definitions of these metrics differ slightly from those used by \\citetalias{Johnson15_AP}, such that all four metrics are normalized by the total number of available image classifications.\n\nThe aggregation process produced a set of 10926 unique identifications. This total number includes many low significance objects, with only 4780 candidates having $f_{\\mathrm{view}} \\ge 0.1$. For this paper, we focus primarily on the cluster candidates; please see Appendix \\ref{sec:app_othercat} for discussion of background galaxy and emission region results. \n\n\\begin{figure}\n \\centering\n \\includegraphics[width=8.5cm]{f_cluster_hist.pdf}\n \\caption{Distribution of $f_{\\mathrm{cluster}}$ values for our full sample of cluster candidates.}\n \\label{fig:f_clst_hist}\n\\end{figure}\n\nWe present a histogram of $f_{\\mathrm{cluster}}$ values in Figure~\\ref{fig:f_clst_hist}. Our visual inspection of cluster candidates confirmed that as $f_{\\mathrm{cluster}}$ decreases, the quality of the cluster candidates is lower. We find that for $f_{\\mathrm{cluster}} > 0.6$, there are very few contaminants; among 841 candidates, only one has $f_{\\mathrm{galaxy}} > 0.1$, and that object is eliminated by subsequent weighted cuts (see \\S\\ref{sec:userweighting}).\n\n\n\\subsection{User Weighting}\n\\label{sec:userweighting}\n\n\\begin{figure*}[ht]\n \\centering\n \\includegraphics[width=0.75\\textwidth]{Metrics_One_im.pdf}\n \\caption{\n {\\em Left:} $f_{\\mathrm{consensus}}$ and $\\overline{f}_{\\mathrm{cluster}}$ user metric scores used for weighting. Circular points represent metrics for individuals with $>$20 classifications; point shading denotes volunteer classification count, where darker colors highlight individuals with more classifications. Star markers represent the average metrics for a group of users, binned in groups according to classification count. The blue dashed line represents the maximum $\\overline{f}_{\\mathrm{cluster}}$ a volunteer can achieve based on the fraction of high-quality consensus clusters identified.\n {\\em Right:} Star markers correspond to the same binned user groups presented in the left panel, but now showing the average number of classifications for each group on the x-axis. The red number next to each star is the number of users represented in each respective bin. For volunteers with $<$20 classifications (gray shaded region) users were weighted according to aggregate user metric values calculated for the binned group rather than using individual values.\n }\n \\label{fig:user_metrics}\n\\end{figure*}\n\nWhile the $f_{\\mathrm{cluster}}$ metric assumes that each volunteer is equally skilled at identifying star clusters, multiple citizen science projects \\citep[e.g.,][]{Willett13, Jayasinghe19, Eisner21} have found that weighting volunteer responses according to their task performance can increase sample quality and decrease catalog contamination as a function of completeness. To maximize the usefulness of LGCS volunteer contributions, we follow the methodology of \\citetalias{Johnson15_AP} and weight classifications based on the volunteer's performance identifying star clusters. In this section, we demonstrate that employing user weighting significantly improves the resulting cluster catalog.\n\nWe calculate user weights based on the agreement between the classifications of the user and the entire set of LGCS participants. We use two separate weights: one for detections, objects the user identified; one for non-detections, objects the user did not identify. A user's detections are weighted according to the average $f_{\\mathrm{cluster}}$ of all cluster identifications made by the individual, such that classifications by those who tend to identify good candidates with higher $f_{\\mathrm{cluster}}$ are assigned greater weights than those who identify worse candidates with lower $f_{\\mathrm{cluster}}$. A user's cluster non-detections are weighted according to the fraction of high-quality clusters ($f_{\\mathrm{cluster}} > 0.6$) that the user sees and detects, such that classifications by those who rarely miss good clusters carry greater weight than those who are more selective and identify fewer clusters.\n\nWe note that volunteer classifications of synthetic cluster images are omitted from user metric calculations. This omission ensures that user classification metrics are based only on real data, and that catalog completeness is not biased due to user weighting.\n\nWe calculate two user metrics for all volunteers to quantify the detection and non-detection behaviors described above: $\\overline{f}_{\\mathrm{cluster}}$, the average $f_{\\mathrm{cluster}}$ of all clusters a user identifies, and $f_{\\mathrm{consensus}}$, the fraction of high-quality ($f_{\\mathrm{cluster}} > 0.6$) clusters a user saw that they identified. Figure~\\ref{fig:user_metrics} shows these user metrics for all volunteers who contributed more than 20 classifications. Many users lie toward the upper right corner of the plot, representing users who excel in both user metrics. In contrast, users in the top left are conservative classifiers who identify clusters with high $f_{\\mathrm{cluster}}$, but miss a significant fraction of commonly-identified clusters. Those in the bottom right are liberal classifiers who include all good clusters in their identifications, but at the expense of also including lower quality clusters as well.\n\nWe examine trends in user behavior by grouping users into bins according to their total classification count, with the average user metrics of each group plotted as colored stars in Figure~\\ref{fig:user_metrics}. Volunteers with higher classification counts (redder points) tend to have higher user metrics scores, which may indicate that volunteers become more skilled on average as they classify an increasing number of images.\n\nBecause an individual's user metrics become noisy at small numbers of classifications, we replace the metrics of users with $\\leq$20 classifications with the aggregate values of $\\overline{f}_{\\mathrm{cluster}}$ and $f_{\\mathrm{consensus}}$ obtained for their binned groups. The use of aggregate metrics and weights for users with low classification counts has little impact on the catalog results due to the small percentage of total classifications contributed by these individuals (see Figure~\\ref{fig:userstats}).\n\nTo convert a volunteer's user metric results into a classification weight, we adopt a generalized logistic function:\n\\begin{equation}\n W(x) = B \\times \\left( A + \\frac{1}{1 + e^{-m_{\\mathrm{logistic}}(x-b_{\\mathrm{logistic}})}} \\right), \n\\end{equation}\nwhere $x$ represents $\\overline{f}_{\\mathrm{cluster}}$ for detection weights and $f_{\\mathrm{consensus}}$ for non-detection weights, while $m_{\\mathrm{logistic}}$ and $b_{\\mathrm{logistic}}$ are the slope and position of maximum growth of the logistic curve. The coefficients $A$ and $B$ are normalization constants set such that $W$ varies between 0 and 1 over the $x$ interval [0, 1]. We seek to identify values of $m_{\\mathrm{logistic}}$ and $b_{\\mathrm{logistic}}$ for detection and non-detection weights that maximize cluster catalog completeness and minimize contamination.\n\nTo fit for an optimal weighting scheme, we first compile a set of ``expert'' ratings to use as a reference when computing completeness and contamination metrics for a given set of weighting parameters. A group of 4 co-authors visually inspected clusters and scored them on a scale of 1--3: 1 is a definite cluster, 2 is a possible cluster, and 3 is a non-cluster. Four co-authors ranked all marginal cluster candidates ($0.35 < f_{\\mathrm{cluster}} < 0.5$) where we expect the greatest variety in quality. Additionally, one co-author ranked a broader range of candidates ($f_{\\mathrm{cluster}} \\gtrsim 0.25$) to confirm that objects with $f_{\\mathrm{cluster}} < 0.35$ were low quality identifications. The average of these ranks, $S_{\\mathrm{expert}}$, is then used to categorize clusters and contaminants. Clusters with $S_{\\mathrm{expert}} < 1.5$ were declared good clusters, and $S_{\\mathrm{expert}} > 2.5$ were considered contaminants.\n\nUsing the expert ratings, we construct a completeness versus contamination curve for our unweighted sample by varying the $f_{\\mathrm{cluster}}$ threshold from 0 to 1, as shown in Figure~\\ref{fig:comp_vs_cont}. We define the minimum distance from the curve to the lower left corner of this plot (i.e., an optimal sample with 100\\% completeness and no contaminants) as $d_{\\mathrm{optimal}}$, and use this metric to evaluate, rank, and optimize the adjustable weighting parameters.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=8.5cm]{Completeness_vs_Contamination.pdf}\n \\caption{Top: catalog completeness versus contamination for the original, unweighted sample (dashed black line), and the optimal user weighting system (solid black line). The orange dashed line represents the $d_{\\mathrm{optimal}}$ threshold, where $d_{\\mathrm{optimal}}$ is the distance from this point to the bottom right corner of the plot. The green line shows the adopted threshold chosen to reduce catalog contamination. Bottom: catalog completeness versus $f_{\\mathrm{cluster, W}}$, which provides reference to where each threshold is drawn.}\n \\label{fig:comp_vs_cont}\n\\end{figure}\n\nWe conduct an iterative, grid-based search for an optimal set of detection and non-detection weighting function parameters that minimize the $d_{\\mathrm{optimal}}$ metric for the completeness versus contamination curve. For each grid point, we calculate detection and non-detection user weights from $\\overline{f}_{\\mathrm{cluster}}$ and $f_{\\mathrm{consensus}}$ user metrics, respectively, using logistic functions with specified values of $m_{\\mathrm{logistic}}$ and $b_{\\mathrm{logistic}}$ parameter values to make the metric-to-weight transformation. We then compute user-weighted $f_{\\mathrm{cluster}}$ values, $f_{\\mathrm{cluster, W}}$, for each cluster candidate, construct a completeness vs.\\ contamination curve, and calculate an associated $d_{\\mathrm{optimal}}$.\n\nWe identify the set of logistic function parameters that minimize $d_{\\mathrm{optimal}}$, and thus produce an optimal cluster catalog that maximizes completeness and minimizes contamination. We find that the following weighting parameters produce the best weighted catalog: detection weight parameters of $(m_{\\mathrm{logistic}}, b_{\\mathrm{logistic}}) = (30.0, 0.45)$; non-detection weight parameters of $(m_{\\mathrm{logistic}}, b_{\\mathrm{logistic}}) = (30.0, 1.1)$. The optimally weighted catalog has significantly lower contamination as a function of completeness than the unweighted catalog, as shown in Figure~\\ref{fig:comp_vs_cont}, demonstrating that the application of user weights improved the quality of the cluster catalog we produced.\n\n\\subsubsection{Catalog Threshold Selection}\n\\label{sec:threshold}\n\nWith the user weighting parameters fixed, we move on to selecting a catalog threshold. \\citetalias{Johnson15_AP} choose a $f_{\\mathrm{cluster, W}}$ cutoff that corresponds to the point on the completeness versus contamination curve where $d_{\\mathrm{optimal}}$ is minimized. The minimum $d_{\\mathrm{optimal}}$ point corresponds to a $f_{\\mathrm{cluster, W}}$ threshold of 0.568, 94.8\\% completeness, and 7.1\\% contamination, indicated by the orange lines in Figure~\\ref{fig:comp_vs_cont}.\n\nWe note two key differences between the \\citetalias{Johnson15_AP} and LGCS completeness versus contamination curves: $f_{\\mathrm{contamination}}$ values for LGCS are smaller by approximately a factor of 2; the original unweighted curve (and the weighted curve to a lesser degree) shows a distinct change in slope behavior at $f_{\\mathrm{completeness}} \\sim 0.9$ in Figure~\\ref{fig:comp_vs_cont}. In addition, a qualitative evaluation of the $d_{\\mathrm{optimal}}$-based threshold concluded that the resulting cluster sample includes a higher number of contaminants than desired, leading us to reevaluate our choice of $f_{\\mathrm{cluster, W}}$ threshold.\n\nBased on the poor assessment of the initial $f_{\\mathrm{cluster, W}}$ threshold, we seek an alternative, more conservative catalog limit. We target a greater $f_{\\mathrm{cluster, W}}$ value that corresponds to a point on the weighted completeness versus contamination curve near the transition in slope at $f_{\\mathrm{completeness}} \\sim 0.9$. We find that by applying a factor of 2 scaling to the $f_{\\mathrm{contamination}}$ component of the $d_{\\mathrm{optimal}}$ distance calculation, motivated by the $\\sim$2x scaling difference between the LGCS and \\citetalias{Johnson15_AP} curves, we identify a viable threshold that meets all the above criteria. The resulting $f_{\\mathrm{cluster, W}}$ catalog threshold is 0.674, which corresponds to 90.5\\% completeness and 4.4\\% contamination, indicated by the green lines in Figure~\\ref{fig:comp_vs_cont}.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=8.5cm]{Expert_gvb.pdf}\n \\caption{Weighted and unweighted $f_{\\mathrm{cluster}}$ values are plotted for the subset of expert ranked clusters. The blue dots are expert identified good clusters, and the red dots are expert identified contaminants. The dashed green line represents the $f_{\\mathrm{cluster, W}}$ catalog threshold of 0.674, while the orange dashed line represents the $f_{\\mathrm{cluster, W}}$ catalog threshold of 0.568. We prefer the higher adopted $f_{\\mathrm{cluster, W}}$ threshold as it rejects a larger number of expert identified bad cluster candidates, as seen by the larger number of red points that that fall below the green horizontal line.}\n \\label{fig:weights_work}\n\\end{figure}\n\nWe demonstrate the impacts of our weighting system in Figure~\\ref{fig:weights_work} for the expert-classified subsample of cluster candidates with $0.35 \\leq f_{\\mathrm{cluster}} \\leq 0.5$ where weighting and threshold selection has the greatest impact. Expert identified good clusters ($S_{\\mathrm{expert}} < 1.5$) are plotted in blue, and expert identified bad clusters ($S_{\\mathrm{expert}}$ > 2.5) are shown in red. This plot shows how the weighted $f_{\\mathrm{cluster}}$ system is more effective at separating good candidates from bad candidates than the unweighted system, due to the improved separation of blue and red points by horizontal lines of constant $f_{\\mathrm{cluster, W}}$ over vertical lines of constant $f_{\\mathrm{cluster}}$. We can also see that the number of bad candidates that are rejected by the higher, adopted $f_{\\mathrm{cluster, W}}$ threshold is larger than the number rejected by the original, unscaled $d_{\\mathrm{optimal}}$-based $f_{\\mathrm{cluster, W}}$ threshold, justifying our choice of the more conservative catalog threshold.\n\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.44\\textwidth]{LGCS_map.pdf}\n \\caption{Spatial distribution of PHATTER star clusters (blue) overlaid on a F475W survey-wide mosaic image.}\n \\label{fig:map}\n\\end{figure}\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.34\\textwidth]{Final_example_vertical.pdf}\n \\caption{An example cluster: PHATTER 22. Top: Two $12\\times12$ arcsec F475W+F814W color images, where the pink circles represent user cluster identifications, and the green circle represents the final cluster aperture ($R_{\\mathrm{ap}}$ of 2.04 arcsec) derived from the median radius of the individual user apertures. Middle: The PHATTER 22 SED (blue) created from six-band integrated photometry. The results of SLUG SED fitting for PHATTER 22 are printed in the lower right (50th percentile value with 16th-84th percentile confidence interval), and gray lines show the 100 best-fit SLUG models. Bottom: The PHATTER 22 cluster CMD, where stellar photometry for sources within the cluster aperture (blue) are accompanied by surrounding photometry of field stars (red). CMD fitting results from \\citet{Wainer22} are listed in the lower right, and a stellar isochrone (black) depicting the cluster's best fit properties is overplotted.}\n \\label{fig:examplecluster}\n\\end{figure}\n\n\\section{PHATTER Star Cluster Catalog} \\label{sec:catalog}\n\nWe apply the catalog construction techniques and user weighting described in Section \\ref{sec:catcon} to derive a final cluster catalog. We present a sample of 1214 clusters that were selected using an $f_{\\mathrm{cluster, W}}$ threshold of 0.674, which reflects a conservative selection of clusters that minimizes contamination by non-cluster candidates. Table \\ref{tab:cat} reports cluster position, radius, and classification metrics (including weighted $f_{\\mathrm{cluster, W}}$) for each of our cataloged clusters. We show the spatial distribution of the clusters in Figure~\\ref{fig:map}.\n\nWe also present information for an additional 3566 candidate identifications with $f_{\\mathrm{cluster, W}} < 0.674$ and $f_{\\mathrm{view}} \\ge 0.1$ in Appendix \\ref{sec:app_othercat}, allowing catalog users to make alternative choices of catalog $f_{\\mathrm{cluster, W}}$ thresholds based on the specific needs and requirements of a particular science use case.\n\nWe present an example cluster, PHATTER 22, in Figure~\\ref{fig:examplecluster} to illustrate the data available for the PHATTER cluster sample. Optical cutout images display individual cluster identifications made by LGCS volunteers as well as the final cluster aperture for the cluster, and the images show that star clusters appear as collections of individually-resolved members stars in PHATTER imagery. The survey's panchromatic images produce six-band SEDs from integrated light photometry (see \\S\\ref{sec:photometry}) and cluster CMDs from resolved star photometry catalogs, both of which are used to fit for cluster properties (see \\S\\ref{sec:slugfitting} and \\citealt{Wainer22}).\n\n\\movetabledown=2.3in\n\\begin{rotatetable*}\n\\begin{deluxetable*}{cccccccccccccccccccccccc}\n\\tabletypesize{\\footnotesize}\n\\setlength{\\tabcolsep}{0.05in}\n\\tablewidth{0pt}\n\\tablecaption{PHATTER Cluster Catalog \\label{tab:cat}}\n\n\\tablehead{\n\\colhead{ID} & \\colhead{RA (J2000)} & \\colhead{DEC (J2000)} & \\colhead{$R_{\\mathrm{ap}}$ ($\\arcsec$)} & \\colhead{$R_{\\mathrm{eff}}$ ($\\arcsec$)} & \\colhead{$f_{\\mathrm{view}}$} & \\colhead{$f_{\\mathrm{cluster}}$} & \\colhead{$f_{\\mathrm{galaxy}}$} & \\colhead{$f_{\\mathrm{emission}}$} & \\colhead{$f_{\\mathrm{cluster, W}}$} & \\colhead{Flags} & \\colhead{$m_{\\mathrm{apcor}}$} & \\colhead{$m_{275}$} & \\colhead{$\\sigma_{275}$} & \\colhead{$m_{336}$} & \\colhead{$\\sigma_{336}$} & \\colhead{$m_{475}$ }& \\colhead{$\\sigma_{475}$} & \\colhead{$m_{814}$} & \\colhead{$\\sigma_{814}$} & \\colhead{$m_{110}$} & \\colhead{$\\sigma_{110}$} & \\colhead{$m_{160}$} & \\colhead{$\\sigma_{160}$}\n}\n\n\\startdata\n1 & 23.553754 & 30.479462 & 1.90 & 0.23 & 1.0000 & 0.9667 & 0.0000 & 0.0333 & 1.0000 & \\nodata & -0.00 & \\nodata & \\nodata & \\nodata & \\nodata & 18.39 & 0.02 & 17.70 & 0.04 & \\nodata & \\nodata & \\nodata & \\nodata \\\\\n2 & 23.611876 & 30.696172 & 1.66 & 0.48 & 1.0000 & 1.0000 & 0.0000 & 0.0000 & 1.0000 & \\nodata & -0.11 & $>$21.01 & \\nodata & $>$21.10 & \\nodata & 19.63 & 0.06 & 18.73 & 0.08 & \\nodata & \\nodata & \\nodata & \\nodata \\\\\n3 & 23.606064 & 30.699199 & 1.54 & 0.36 & 1.0000 & 1.0000 & 0.0000 & 0.0000 & 1.0000 & \\nodata & -0.05 & 20.09 & 0.18 & 20.42 & 0.01 & 20.21 & 0.10 & 19.41 & 0.25 & \\nodata & \\nodata & \\nodata & \\nodata \\\\\n4 & 23.434333 & 30.514082 & 1.90 & 0.49 & 1.0000 & 0.9833 & 0.0000 & 0.0167 & 0.9999 & \\nodata & -0.07 & $>$21.12 & \\nodata & $>$22.39 & \\nodata & 19.84 & 0.11 & 18.25 & 0.05 & 17.41 & 0.05 & 16.61 & 0.05 \\\\\n5 & 23.583846 & 30.659212 & 1.86 & 0.37 & 0.9836 & 0.9016 & 0.0000 & 0.0820 & 0.9772 & \\nodata & -0.02 & 19.29 & 0.10 & 19.00 & 0.06 & 18.76 & 0.02 & 18.00 & 0.06 & 17.64 & 0.12 & 17.21 & 0.16\n\\enddata\n\\tablecomments{Table \\ref{tab:cat} is published in its entirety in the electronic edition of the {\\it Astrophysical Journal}. A portion is shown here for guidance regarding its form and content. Note that the $R_{\\mathrm{ap}}$ parameter gives the median of the user-clicked radii, which we use as the aperture for the photometry measurements (see Section~\\ref{sec:photometry}).}\n\\end{deluxetable*}\n\\end{rotatetable*}\n\n\\subsection{Cluster Photometry} \\label{sec:photometry}\n\nAperture photometry for candidate clusters and other objects are measured using the same techniques and code employed by \\citetalias{Johnson15_AP}, which we summarize here. Photometric apertures for an object are centered at the mean position and extend to the median radius ($R_{\\mathrm{ap}}$) drawn by LGCS participants. The sky background is measured in 10 annuli, each with an area equal to that of the photometric aperture, extending from 1.2--3.4 $R_{\\mathrm{ap}}$. These sky annuli facilitate robust determinations of sky flux levels and associated uncertainties, which are important given that background variations often dominate the photometry error budget. Photometry is reported in the VEGAMAG system for the native HST bandpasses and calibrated using zeropoints obtained from relevant instrument websites for ACS\\footnote{\\url{https:\/\/acszeropoints.stsci.edu\/}} \\citep{Bohlin16}, WFC3\/UVIS\\footnote{\\url{https:\/\/www.stsci.edu\/hst\/instrumentation\/wfc3\/data-analysis\/photometric-calibration\/uvis-photometric-calibration}} \\citep[2017 values;][]{Deustua17} and WFC3\/IR\\footnote{\\url{https:\/\/www.stsci.edu\/hst\/instrumentation\/wfc3\/data-analysis\/photometric-calibration\/ir-photometric-calibration}} (2012 values). We note that these adopted zeropoints are consistent to within 1\\% (2\\%) to alternative ``2020 values'' for ACS and WFC3\/UVIS (WFC3\/IR).\n\nSix-band integrated photometry for the final cluster sample is presented in Table~\\ref{tab:cat}, and equivalent photometry for ancillary cluster candidates is presented in Appendix \\ref{sec:app_othercat}. Measured magnitudes are reported for detections with S\/N $\\ge$ 3 and 3$\\sigma$ upper limits are reported for non-detections. Blank entries denote cases of incomplete image coverage in that photometric passband.\n\nThe radial light profile of each cluster is measured from the F475W image, and the half-light radius, $R_{\\mathrm{eff}}$, is derived through interpolation of the radial profile. Aperture corrections are computed assuming a \\citet{King62} light profile with fixed concentration ($c = R_{\\mathrm{tidal}}\/R_{\\mathrm{core}} = 7$), scaled to match the measured $R_{\\mathrm{eff}}$. When applied to the aperture magnitudes\\footnote{$m_{\\mathrm{total}} = m_{\\mathrm{ap}}+m_{\\mathrm{apcor}}$}, these corrections yield an estimate of total cluster light that accounts for flux that falls outside the photometric aperture, $R_{\\mathrm{ap}}$. The median correction is $-0.04$ mag and the 25th-to-75th interquartile range spans from $-0.09$ to $-0.01$. We report the F475W $R_{\\mathrm{eff}}$ measurements and aperture corrections in Table~\\ref{tab:cat}.\n\nWe flag objects with large $R_{\\mathrm{eff}}$ ($\\ge$0.8 arcsec or $\\sim$3pc) that are also bright ($m_{\\mathrm{F475W}} < 19.0$) and blue ($m_{\\mathrm{F336W}}-m_{\\mathrm{F475W}} < -0.5$) as possible associations. Eight such objects are identified in Table~\\ref{tab:cat}.\n\nWe plot a UV-optical color-color diagram for the PHATTER cluster sample in Figure~\\ref{fig:colorcolor}. As expected, many of the clusters are quite blue (F336W$-$F475W $<$ 0.5), indicating young ages. When compared to the color-color distribution of PHAT clusters from M31 \\citepalias{Johnson15_AP}, we see the difference between the younger and bluer cluster population from M33 that hosts on-going star formation versus the quiescent cluster population from M31 that hosts relatively larger numbers of globular clusters and other older clusters ($>$1--3 Gyr).\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.35\\textwidth]{LGCS_colcol.pdf}\n \\includegraphics[width=0.35\\textwidth]{LGCS_colcol_compare.pdf}\n \\caption{Top: Color-color diagram of 729 PHATTER clusters with three-band photometric detections. Bottom: Comparison of color-color distribution for PHATTER clusters in M33 and PHAT clusters in M31 \\citepalias{Johnson15_AP} that shows the PHATTER clusters are significantly bluer, denoting younger cluster age.}\n \\label{fig:colorcolor}\n\\end{figure}\n\n\\section{Catalog Completeness} \\label{sec:catcompleteness}\n\nWe determine the completeness of our cluster sample by analyzing synthetic clusters inserted into LGCS images. We analyze the classification metrics and catalog inclusion status of the synthetic cluster sample as a function of cluster luminosity, age, and mass. We also examine the impact of cluster size and environment on cluster detection and catalog completeness.\n\n\\begin{figure*}\n \\centering\n\\includegraphics[width=\\textwidth]{Completeness.pdf}\n \\caption{Completeness results based on analysis of synthetic clusters. The top panels present results as a function of F475W magnitude, the bottom panels as a function of mass. The leftmost panels present results for individual synthetic clusters: detections in black, non-detections in red. The left-middle panels show a binned version of the same results, where shading denotes the fraction of clusters detected in each bin. The middle-right panels show the completeness model fit, and the rightmost panels showing the data-model residuals. The vertical gray dashed lines in each panel highlight the age bins whose data are presented in Figure~\\ref{fig:Logistic_plot}. The blue dot-dashed line in each panel represents the fitted exponential 50\\% completeness model fit to the full synthetic data.}\n \\label{fig:completeness}\n\\end{figure*}\n\nWe process classification results for the synthetic clusters through the same catalog creation pipeline used for the real LGCS cluster candidates, applying the same classification weighting and $f_{\\mathrm{cluster,W}}$ detection threshold. We present the synthetic cluster sample and detection results as an ancillary table in Appendix \\ref{sec:app_othercat}. The synthetic cluster results are shown in the left panels of Figure~\\ref{fig:completeness}, where black points indicate detected clusters and red points indicate non-detections. The middle-left panel shows a 2D binned representation of the results. At higher luminosities and masses, a larger fraction of clusters are detected, and at the highest masses, nearly all synthetic clusters are detected.\n\nThe behavior of completeness with age is somewhat more complicated. Clusters at a fixed mass are more frequently detected at young ages due to their brighter luminosities. At fixed luminosity, clusters are more frequently detected at older ages due to their broader distribution of light across member stars, in contrast to young clusters that typically have a small number of very bright massive stars.\n\nTo characterize these observed trends in completeness on the age-magnitude and age-mass planes, we derive analytic formulae that can be easily applied in future modeling work. The goal here is to characterize the average completeness properties for the entire catalog, reasoning that detailed studies of specific cluster subpopulations may require a more complex completeness model than the one we present here. \n\nIn addition to age, luminosity, and mass, dust extinction and effective radius (or, surface brightness) may also play a role. We find no need to truncate the cluster effective radius distribution, but choose to omit synthetic clusters in the high $A_V$ tail ($>$1.5 mag) when fitting for average model behavior. \n\nWe note that the cluster environment also has an impact on completeness \\citepalias[][]{Johnson15_AP}. However, because the spatial distribution of half the synthetic clusters was designed to replicate the correlation between young clusters and high $N_{\\mathrm{MS}}$ regions, we expect our overall distribution of synthetic cluster environments to be comparable to the true cluster sample. We revisit the impact of cluster environment later in this section. \n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.4\\textwidth]{F475W_Completeness_LF.pdf}\n \\includegraphics[width=0.4\\textwidth]{Mass_Completeness_LF.pdf}\n \\caption{Logistic function fits to catalog completeness as a function of F475W magnitude (top) and mass (bottom) for synthetic clusters with ages $7.3 < {\\rm log(Age\/yr)} < 7.77$. Dashed lines represent the Agresti-Coull binomial proportion confidence interval for each bin.}\n \\label{fig:Logistic_plot}\n\\end{figure}\n\nWe begin our modeling by characterizing completeness as a function of mass and F475W magnitude for bins of log(Age\/yr). We use a bin width of 0.23 dex and a sliding two-bin window to improve number statistics. We use a logistic function to analytically model the completeness curve, as shown in Figure~\\ref{fig:Logistic_plot}, where the black lines show the completeness data and the green line shows the best-fit logistic function. The functional form of the logistic function, $C$, is given by:\n\\begin{equation}\n C = ( 1 + e^{ -k ( x - M_{50}) } )^{-1}\n\\end{equation} \nwhere $k$ is the slope of the logistic function, $M_{50}$ is the 50\\% completeness limit, and $x$ is the physical input parameter --- either the F475W magnitude or log($M\/M_\\odot$).\n\nBased on initial fits, we find that the slope of the logistic function, $k$, is quite similar at all ages. Therefore, we fix $k$ to the median fitted value across all age bins ($-2.52$ for F475W magnitude, $5.84$ for mass) and re-fit. \n\nWe find the age dependence of the 50\\% completeness parameter, $M_{50}$, is well fit by an exponential function over the full synthetic cluster age range ($6.6 < {\\rm log(Age\/yr)} < 10.1$) for both mass and F475W magnitude, following the form:\n\\begin{equation}\n M_{50}(\\tau) = a \\times e^{b (\\tau -\\tau_{\\mathrm{min}})} + c,\n\\end{equation}\nwhere $\\tau \\equiv {\\rm log(Age\/yr)}$, and $\\tau_{\\mathrm{min}}$ is the median $\\tau$ for the youngest bin, which is 6.71 for the full LGCS synthetic cluster sample and binning used here. We note that in this formulation, the 50\\% completeness in mass at $\\tau_{\\mathrm{min}}$ is given by $a+c$. We fit for constants $a$, $b$, and $c$ by minimizing $\\chi^2$ for the full 2D binned histogram shown in the middle-left panels of Figure~\\ref{fig:completeness}. We find the best fit exponential parameter values for mass are $(a, b, c) = (0.1455, 0.7870, 2.7810)$. For F475W magnitude the best fit values are $(a, b, c) = (-2.6767, -0.6154, 21.6834)$. The reduced $\\chi^2$ for the exponential fit to the mass and F475W magnitude planes is 1.06 and 1.18, respectively. The fitted 50\\% completeness exponential is presented as a dashed blue line in Figure~\\ref{fig:completeness}, while the full completeness model (assuming fixed $k$ values) and its residuals relative to the synthetic cluster results are shown in the middle-right and rightmost panels of Figure~\\ref{fig:completeness}.\n\nThe mass completeness behavior in the fiducial age range of $7.0 < {\\rm log(Age\/yr)} < 8.5$ has useful application in follow-up science analyses of young clusters, so we derive an alternate completeness model specific to this fiducial age range. Adopting a fixed median $k$ of 6.02 and $\\tau_{\\mathrm{min}}$ of 7.09, we find the best fit $M_{50}$ exponential parameters are $(a, b, c) = (0.0303, 1.9899, 2.9770)$, with a reduced $\\chi^2$ of 0.82. We recommend this alternative completeness model for any applications that exclude old clusters. \n\nWe note that careful inspection of the binned F475W magnitude data in Figure~\\ref{fig:completeness} (top middle-left panel) and its model residuals (top right panel) show that the youngest age bins do not neatly follow the large scale trends of the age-magnitude plane. The fact that the age-mass plane is more well behaved at these same ages suggests this difference is likely due to stochasticity impacting the F475W magnitudes due to small number statistics of the brightest cluster members \\citep[e.g.,][]{Fouesneau10,Beerman12}. As a result, we recommend use of the mass-based completeness model whenever feasible, as we find it more reliable than our luminosity model in describing these youngest clusters. \n\n\n\\subsection{Additional Completeness Dependencies}\n\nIn addition to age, mass, and F475W magnitude, the spatial distribution of the cluster's member stars is another physical property that effects completeness. Here, we use the synthetic cluster's effective radius, $R_{\\mathrm{eff}}$, to parameterize cluster size and central density. For synthetic clusters with $R_{\\mathrm{eff}} > 3$~pc (33\\% of sample), we find a 50\\% completeness limit mass that is 0.2 dex higher than for smaller clusters. This shows that larger, diffuse clusters are systematically more challenging to detect due to the reduced central density of cluster members, which tends to reduce the cluster's contrast against the field background. As discussed in Section~\\ref{sec:synclst}, we sampled the synthetic cluster radii from the observed cluster radii of the PHAT M31 cluster sample, but biased the distribution toward larger objects. If we remove synthetic clusters with $R_{\\mathrm{eff}} > 3$~pc, the mass completeness presented here improves by a median of 0.05 dex. We note that for our final cluster sample, only 14\\% of clusters have $R_{\\mathrm{eff}} > 3$~pc. \n\nAdditionally, local cluster environment plays a role in detection and catalog completeness. At a basic level, when the contrast between cluster and field is reduced due to an increasing density of nearby field stars of similar color, cluster detection becomes more difficult. For PHATTER images, we can characterize the local stellar density according to the number of main sequence stars per LGCS search image, $N_{\\mathrm{MS}}$, calculated from counts of PHATTER photometric sources selected according to their optical CMD location. $N_{\\mathrm{MS}}$ ranges between $2.0 < log(N_{\\mathrm{MS}}) < 3.75$ across the PHATTER footprint. As $N_{\\mathrm{MS}}$ increases, we see a systematic 0.7 dex increase in the 50\\% mass completeness limit for the young blue star clusters, whose members are also predominately main sequence stars. As such, we recommend that future population analyses take care to account for spatial variation in catalog completeness due to stellar density, especially for trends with respect to galactocentric radius. In particular, we point to the investigation of environmental influence on catalog completeness performed by \\citet{Wainer22} as an example.\n\n\\subsection{Completeness Comparison: PHATTER vs.\\ PHAT}\n\nWe find that the 50\\% mass and luminosity completeness limits for the PHATTER M33 cluster catalog are worse relative to the similar PHAT M31 cluster catalog presented in \\citetalias{Johnson15_AP}; fractional completeness is lower in M33 at a fixed mass or F475W magnitude. At younger ages, the 50\\% mass completeness limit for the M33 catalog is $\\sim$0.3 dex higher than what was found for M31, and correspondingly $\\sim$0.5 mags brighter in F475W magnitude.\n\nWhile M33's larger distance can account for a 0.2 mag difference in luminosity, we believe the completeness differences are primarily due to the PHATTER footprint's central disk location and M33's higher star formation surface density, which together lead to a higher average density of young field stars stars (i.e., high $N_{\\mathrm{MS}}$) and worse catalog completeness. Thanks to the use of an analogous $N_{\\mathrm{MS}}$ definition by \\citetalias{Johnson15_AP} for PHAT cluster work, we can confirm that the PHATTER median $N_{\\mathrm{MS}}$ is a factor of 5 (0.7 dex) larger than the PHAT median $N_{\\mathrm{MS}}$. We also note that unlike for PHAT, using an alternative F475W$_{-3}$ metric, which subtracts the contribution of the three brightest stars from the integrated F475W magnitude, does not remove the age-dependent trend in luminosity completeness.\n\n\n\\section{Cluster SED Fitting} \\label{sec:slugfitting}\n\nIn this section we discuss our method for deriving the cluster properties (age, mass, and extinction) from their integrated light photometry. We use the public source code Stochastically Lighting Up Galaxies (SLUG) \\citep{Krumholz15_SLUG} to build a set of $10^7$ model star clusters that we use to estimate the M33 cluster properties in \\S\\ref{sec:slug_mod}. We discuss fitting of various filter combinations and discuss the reliability of this integrated light fitting in \\S\\ref{sec:slug_sel+fit} based on comparison to CMD-based results for a similar sample of M31 clusters presented in Appendix \\ref{sec:app_slugM31}. We derive SLUG-based estimates for the cluster sample and present the results in \\S\\ref{sec:slug_result_m33}. \n\n\\subsection{Building the SLUG Cluster Library} \\label{sec:slug_mod}\n\nSLUG is a stellar population synthesis code that incorporates stochastic modeling of stellar mass and luminosity distributions. More information and details about SLUG can be found in \\citet{Krumholz15_SLUG}, and examples of its use include \\citet{Krumholz15_LEGUS,Krumholz19_SLUG}. Using SLUG, we build a grid of $10^7$ model star clusters assuming Padova stellar evolution models that include thermally pulsing AGB stars \\citep{Girardi00}, which are distributed with Starburst99 \\citep{Vazquez05}. We use a \\citet{Kroupa01} stellar initial mass function (IMF) which spans from 0.01 $M_\\odot$ to 120 $M_\\odot$ with the ``stop after'' sampling method, which allows for some of the more massive stars to be included in our simulated sample \\citep{Krumholz15_SLUG}. We generate models with SLUG by drawing ages from a $t^{-1}$ distribution over a range of 10$^{6}$ to 10$^{10}$ years, encompassing the majority of clusters in the PHATTER catalog. We draw cluster masses from an $M^{-2}$ distribution and draw dust extinction values from a lognormal centered at an $A_V$ of 1 mag, width of 0.33 mag, and min\/max breakpoints of 10$^{-6}$ and 5 mag. We apply extinction according to a Milky Way extinction curve \\citep{Fitzpatrick99}, compute photometry in Vega magnitudes for six HST filters (ACS: F475W and F814W; WFC3: F275W, F336W, F110W, F160W), and convert from model magnitudes to observed magnitudes using an adopted distance modulus of 24.67 \\citep{deGrijs14}.\n\n\\subsection{Filter Selection and Fitting} \\label{sec:slug_sel+fit}\n\nThe reliability of SLUG cluster property determinations depends on the number of filters we are able to include in the fitted SEDs. Therefore, we begin by analyzing the fraction of clusters with good photometry in various combinations and numbers of filters for the PHATTER sample. Specifically, we choose filter combinations in increasing order of photometric detectability, and present the results for the M33 sample in Table~\\ref{tab:M33_passband_percent}.\n\n\\vspace{-3.5em}\n\n\\begin{deluxetable}{cc}\n\\tabletypesize{\\footnotesize}\n\\tablecaption{Detection Statistics for Passband Combinations \\label{tab:M33_passband_percent}}\n\\tablehead{\\colhead{Passband} & \\colhead{N(Detections)}} \n\\startdata\nF275W+F336W+F475W+F814W+F110W+F160W & 349 (28.7\\%) \\\\\nF336W+F475W+F814W+F110W+F160W & 414 (34.1\\%) \\\\\nF275W+F336W+F475W+F814W & 612 (50.4\\%) \\\\\nF336W+F475W+F814W & 729 (60.0\\%)\n\\enddata\n\\tablecomments{The number and percentage of PHATTER clusters with photometric detections in each of the listed combinations of 3, 4, 5, and 6 filter passbands.}\n\\end{deluxetable}\n\n\\begin{deluxetable*}{ccc|ccc|ccc|ccc}\n\\tabletypesize{\\footnotesize}\n\\tablecaption{SLUG Results \\label{tab:slug_m33}}\n\\tablehead{\n\\colhead{ID} & \\colhead{Error Cut Flag} & \\colhead{Filters Available} & \\multicolumn{3}{|c|}{log($Mass\/M_{\\odot}$)} & \\multicolumn{3}{c|}{log($Age\/yr$)} & \\multicolumn{3}{c}{$A_{V}$} \\\\\n\\colhead{} & \\colhead{} & \\colhead{} & \n\\multicolumn{1}{|c}{P16} & \\colhead{P50} & \\colhead{P84} & \\multicolumn{1}{|c}{P16} & \\colhead{P50} & \\colhead{P84} & \\multicolumn{1}{|c}{P16} & \\colhead{P50} & \\colhead{P84}\n}\n\\startdata\n3 & F & 4 & 2.47 & 3.23 & 3.43 & 7.72 & 8.25 & 8.41 & 0.24 & 0.43 & 0.69 \\\\\n5 & F & 6 & 3.99 & 4.07 & 4.16 & 8.41 & 8.49 & 8.57 & 0.23 & 0.41 & 0.57 \\\\\n7 & F & 6 & 3.97 & 4.05 & 4.12 & 8.40 & 8.49 & 8.58 & 0.38 & 0.58 & 0.74 \\\\\n8 & F & 6 & 4.53 & 4.59 & 4.65 & 8.17 & 8.23 & 8.28 & 0.20 & 0.26 & 0.32 \\\\\n11 & F & 6 & 2.45 & 3.46 & 3.56 & 7.71 & 8.31 & 8.38 & 0.10 & 0.23 & 0.32\n\\enddata\n\\tablecomments{Table \\ref{tab:slug_m33} is published in its entirety in the electronic edition of the {\\it Astrophysical Journal}. A portion is shown here for guidance regarding its form and content. The error cut flag identifies cases where fits have large uncertainties (16th to 84th percentile range $>$1.2~dex in age or $>$1.3~dex in mass) and should be excluded from uses where uncertainties are not factored in explicitly.}\n\\end{deluxetable*}\n\nWe compute SLUG fits for each filter combination listed in Table \\ref{tab:M33_passband_percent} and the corresponding sample of clusters that are detected in all of the combination's selected passbands. The SLUG model grid is trimmed for each passband combination to omit the magnitudes of any filters that are not selected. Once the cluster fits from each filter combination are compiled, we adopt the fit for each cluster that results from the filter combination with the largest number of filters. This ensures that we are not fitting incomplete SEDs and that we obtain fits for a maximum number of total clusters.\n\nWe execute each iteration of SLUG fitting, and process each set of results, using a fixed set of parameters and assumptions. We adopt the following settings that relate to specifics of the fitting process: a photometric bandwidth of 0.02, a physical properties bandwidth of 0.05, and a Gaussian kernel for PDF estimation. And same as for the underlying set of models, we assume a $t^{-1}$ age prior, $M^{-2}$ mass prior, and lognormal $A_V$ prior. The code returns marginalized PDFs for age, mass, and dust extinction of each cluster, from which we can derive 16th, 50th, and 84th percentile values. These percentiles yield median estimates for each physical property accompanied by an associated 1$\\sigma$ uncertainty. We also flag and exclude highly uncertain fits, such that any cluster with a 16th to 84th percentile range greater than 1.2 dex in age or 1.3 dex in mass is identified by an error flag in the fitting results.\n\nWe present an example of a fitted cluster SED in the middle panel of Figure~\\ref{fig:examplecluster} for PHATTER 22. The 100 best-fit SEDs from the SLUG library are plotted along with the observed SED, which show good agreement between models and observations. The median, 16th, and 84th percentiles of the marginalized posterior PDF for cluster age, mass, and dust extinction, computed over the full library of model SEDs, are derived using functions from the \\texttt{cluster\\_slug} package that is included as part of the SLUG code. We find good agreement between SED and CMD fitting results for PHATTER 22.\n\n\\subsection{SLUG Results} \\label{sec:slug_result_m33}\n\nWe derive cluster properties using SLUG for 729 objects with detections in at least one of the filter combinations listed in Table~\\ref{tab:M33_passband_percent}; we report the fitting results in Table \\ref{tab:slug_m33}. We note that the limited number of fitted clusters (729 out of 1214; 60\\%) is due to a minimum three-filter (F336W, F475W, and F814W) detection criteria for SED fitting. As a result, the completeness of this fitted sample of clusters is worse than the overall catalog completeness, and is biased toward younger and brighter clusters.\n\nExcluding flagged cases with broad PDFs, the median ages from the cluster PDFs range from $6.08 < \\mathrm{log(Age\/yr)} < 8.91$ with a median value of 7.96 for our cluster sample. The median cluster PDF masses range from $2.14 < \\log(M\/M_{\\odot}) < 4.59$ with a median value of 3.29. The median 16th to 84th percentile range in mass is 0.46~dex, and the median 16th to 84th percentile range in age is 0.41~dex.\n\nTo gauge the reliability of the SLUG fits for the PHATTER cluster sample, we compare newly-derived SLUG results to high-quality CMD-based cluster fits for a similar sample of clusters in M31. We find that masses are reliably determined via SLUG integrated light fitting, but that SLUG age and dust results suffer from large uncertainties and artifacts. In particular, SLUG fits tend not to reliably recover ages for clusters younger than $\\sim$100~Myr, which instead are often fit with older ages; see Appendix~\\ref{sec:app_slugM31} for full details of the comparison analysis and results. Due to these results, we recommend that CMD-based fits for the younger clusters in the PHATTER cluster sample presented in \\citet{Wainer22} should be preferred over the SLUG fits reported here. At older ages, there are fewer resolved stars, making the SLUG age estimates the better (and sometimes the only) option.\n\n\n\\section{Discussion} \\label{sec:discussion}\n\n\\subsection{Comparison to Existing M33 Cluster Catalogs} \\label{sec:litcat}\n\nWe cross-match the full PHATTER candidate list (clusters, galaxies, emission regions, and remaining ancillary objects; see Appendix~\\ref{sec:app_othercat}) to five different catalogs from the literature: \\citet{Sarajedini07}, \\citet{SanRoman09}, \\citet{SanRoman10}, \\citet{Sharma11}, and \\citet{Corbelli17}. These references are chosen to facilitate three types of comparisons: to comprehensive catalogs (\\S\\ref{sec:litcat_primary}), to HST-based catalogs (\\S\\ref{sec:litcat_hst}), and to infrared catalogs (\\S\\ref{sec:litcat_ir}). We compile the cross-matching results in Table~\\ref{tab:compare} where we list identifiers (and object classes, where relevant) for each catalog, as well as additional alternate names and accompanying references. These matches are based on a one arcsec match radius between cataloged positions, after applying a mean astrometric offset of 0$\\farcs$609 to the \\citet{Sarajedini07} cluster sample before cross-matching.\n\n\\begin{deluxetable*}{cccccccccc}\n\\centering\n\\tabletypesize{\\footnotesize}\n\\setlength{\\tabcolsep}{0.05in}\n\\tablecaption{Literature Cross-matching Results \\label{tab:compare}}\n\\tablewidth{0pt}\n\n\\tablehead{\n\\colhead{ID} & \\colhead{SM07 ID} & \\colhead{SM07 Class\\tablenotemark{a}} & \\colhead{SR10 ID} & \\colhead{SR10 Class\\tablenotemark{b}} & \\colhead{SR09 ID} & \\colhead{S11 ID} & \\colhead{C17 ID} & \\colhead{C17 Class\\tablenotemark{c}} & \\colhead{Alternate Names \\& References\\tablenotemark{d}}}\n\\startdata\n1 & \\nodata & \\nodata & 1849 & 3 & 157 & \\nodata & \\nodata & \\nodata & \\nodata \\\\\n2 & 391 & Unknown & 2039 & 0 & \\nodata & \\nodata & \\nodata & \\nodata & CS U80 \\\\\n3 & \\nodata & \\nodata & 2025 & 0 & \\nodata & \\nodata & \\nodata & \\nodata & \\nodata \\\\\n4 & \\nodata & \\nodata & 1441 & -1 & \\nodata & \\nodata & \\nodata &\\nodata & \\nodata \\\\\n5 & 372 & Cluster & 1959 & 3 & \\nodata & \\nodata & \\nodata & \\nodata & CBF 58; MKKSS 50; CS U91\n\\enddata\n\n\\tablecomments{Table \\ref{tab:compare} is published in its entirety in the electronic edition of the {\\it Astrophysical Journal}. A portion is shown here for guidance regarding its form and content. Literature references: SM07 \\citep{Sarajedini07}, SR10 \\citep{SanRoman10}, SR09 \\citep{SanRoman09}, S11 \\citep{Sharma11}, C17 \\citep{Corbelli17}.}\n\n\\tablenotetext{a}{\\citet{Sarajedini07} Classes: cluster, stellar, galaxy, unknown}\n\\tablenotetext{b}{\\citet{SanRoman10} Classes: $-1$ = galaxy, 0 = unknown extended object, 1 = candidate cluster, 2 = highly probable cluster, and 3 = confirmed cluster}\n\\tablenotetext{c}{\\citet{Corbelli17} Classes: b = associated with clouds, no optical counterpart, c1 = associated with clouds: coincident $H_{\\alpha}$ and mid-infrared peaks, c2 = associated with clouds: coincident $H_{\\alpha}$, mid-infrared and UV peaks, c3 = not associated with clouds but optically detected, d = ambiguous, e = not associated with clouds, mostly mid-infrared peaks only}\n\\tablenotetext{d}{Reference Abbreviations for Alternate Names \\citep[from][]{Sarajedini07}: Hilt \\citep{Hiltner60}, MD \\citep{Melnick78}, CS \\citep{Christian82}, MKKSS \\citep{Mochejska98}, CBF \\citep{Chandar99, Chandar01}, BEA \\citep{Bedin05}, SBGHS \\citep{SarajediniBarker07}}\n\\end{deluxetable*}\n\\vspace{-2em}\n\n\n\\begin{figure*}\n \\centering\n \\includegraphics[width=\\textwidth]{Previous_gals.pdf}\n \\caption{A comparison of LGCS classifications for objects in the \\citet{SanRoman10} and \\citet{Sarajedini07} catalogs that fall within the PHATTER footprint. Top: Left and center panels show $f_{\\mathrm{cluster,W}}$ distributions for \\citet{SanRoman10} catalog, where object classes include ``confirmed star cluster'' (class 3), ``highly probable star cluster'' (class 2), ``candidate star cluster'' (class 1), ``unknown'' (class 0), and ``background galaxy'' (class $-1$). The dashed line represents the $f_{\\mathrm{cluster,W}}$ threshold we use for cluster catalog inclusion. The right panel shows the \\citet{SanRoman10} classes for objects located in the PHATTER footprint that did not match to a LGCS identification. Bottom: same as top, but for matched (left and center) and unmatched (right) objects from the \\citet{Sarajedini07} catalog and its different set of object classes.}\n \\label{fig:catalogcompare}\n\\end{figure*}\n\n\n\\subsubsection{Primary Catalog Comparisons: Sarajedini \\& Mancone (2007) and San Roman et al.\\ (2010)} \\label{sec:litcat_primary}\n\nWe compare the PHATTER cluster catalog to two key M33 cluster catalogs in the literature: \\citet{Sarajedini07} and \\citet{SanRoman10}. We focus on these catalogs for our primary literature comparison due to their comprehensive compilation of published M33 cluster catalogs and their complete, uniform spatial coverage, respectively.\n\nWe summarize the recovery of clusters from the optically-selected catalogs of \\citet{Sarajedini07} and \\citet{SanRoman10} in Figure~\\ref{fig:catalogcompare}. We recover a large fraction of the cataloged cluster candidates that fall within the spatial coverage of the PHATTER survey. Specifically, for objects that \\citet{Sarajedini07} classify as ``cluster'' and lie within the PHATTER footprint, 89\\% are present in the PHATTER cluster catalog. We also classify 85\\% of their ``unknown'' objects as clusters. \n\nFor the portion of the \\citet{SanRoman10} catalog that falls within the PHATTER footprint, 92\\% of their ``confirmed star cluster'' (class 3) objects and 67\\% of their ``highly probable star cluster'' (class 2) objects are present in the PHATTER cluster catalog. Most of the remaining class 2 and 3 \\citet{SanRoman10} objects are recovered by the LGCS search, but lie in a long tail at low $f_{\\mathrm{cluster,W}}$ values, as shown in the top left panel of Figure~\\ref{fig:catalogcompare}. Additionally, 41\\% of their less certain cluster identifications (``candidate star cluster'' \/ class 1 and ``unknown'' \/ class 0) are also identified as clusters in this work. More surprisingly, we find that half of the ``background galaxy'' (class $-1$) objects that lie within the PHATTER footprint are classified as clusters by the LGCS search. Visual inspection of HST images for these objects confirm they are in fact clusters, and thus were misclassified by the ground-based \\citet{SanRoman10} effort.\n\nFinally, we note that a small fraction of \\citet{Sarajedini07} and \\citet{SanRoman10} sources that fall within the PHATTER footprint were not recovered in the LGCS search. We show the class distribution of these objects in the right panels of Figure~\\ref{fig:catalogcompare}. \n\n\\subsubsection{Space-based Comparisons: HST Catalogs} \\label{sec:litcat_hst}\n\nWe compare the PHATTER cluster catalog to three HST-based cluster catalogs from the literature: \\citet[][hereafter, collectively CBF]{Chandar99, Chandar01}, \\citet{Bedin05}, and \\citet{SanRoman09}. We note that CBF and \\citet{Bedin05} were cross-matched as members of the \\citet{Sarajedini07} compilation, and \\citet{SanRoman09} was matched individually. While these literature catalogs were derived from imaging datasets with relatively small spatial footprints, they serve as useful points of comparison for analyzing catalog-specific differences in visual cluster identification of HST imagery.\n\nA comparison between the PHATTER catalog and the CBF catalog shows that the PHATTER catalog recovers nearly all previously identified objects (103 out of 110 that lie within PHATTER footprint). However, the PHATTER catalog includes an additional 445 clusters within the spatial footprint searched by CBF, resulting in a total catalog that is a factor of 5 larger. This discrepancy is expected due to CBF's use of WFPC2 images whose wide field cameras have significantly lower spatial resolution (pixel scale of 0.1 arcsec) than the ACS and WFC3 instruments used by the PHATTER survey (pixel scales of 0.05 and 0.04 arcsec, respectively). Lower spatial resolution images make the identification of faint, low mass clusters much more difficult, leading to worse luminosity and mass completeness limits for the CBF catalog and significantly fewer cluster identifications as a result.\n \nNext we compare the PHATTER catalog to the work of \\citet{Bedin05}, who used a single ACS pointing located within the PHATTER survey footprint to identify 33 clusters. Within this spatial region, the PHATTER catalog includes 33 clusters, where 22 entries are shared between the two works. The 11 unmatched clusters from the \\citet{Bedin05} catalog break down into two categories: (1) 6 objects are explained by object definition differences, where PHATTER categorized these objects as emission regions or loose non-cluster associations; (2) 5 objects are identified by the PHATTER search, but are excluded from the cluster catalog due to low $f_{\\mathrm{cluster, W}}$. For the 11 PHATTER clusters not recovered by \\citet{Bedin05}, we believe the mismatch is due to their use of a single-band F775W ACS image. These clusters have lower $f_{\\mathrm{cluster, W}}$ and fainter $m_{\\mathrm{F475W}}$ than most of the 22 matched clusters. Faint, low mass clusters are identified in PHATTER via a small clustering of blue main sequence stars, and therefore it is expected that these objects would be missed in a search of only red wavelength imagery.\n \nFinally, we compare the PHATTER catalog to the work of \\citet{SanRoman09}, who searched multi-band ACS imagery that partially overlaps with the PHATTER survey footprint and identified 86 clusters in the overlapping region. The PHATTER cluster catalog includes 75 of these previously identified objects, leaving 11 objects that were excluded by the PHATTER catalog selection threshold. Importantly, the PHATTER catalog includes 119 clusters not identified by \\citet{SanRoman09}, resulting in a total catalog that is a factor of 2.3 larger. Upon examination, this significant difference is due to a more conservative selection threshold, where the PHATTER catalog tends to probe to lower $f_{\\mathrm{cluster, W}}$ and fainter $m_{\\mathrm{F475W}}$. Imagery wavelength may also play a role here, as more than half of the PHATTER overlapping fields were only imaged in two redder bands (F606W and F814W) without bluer F475W coverage.\n\n\\begin{figure*}\n \\centering\n \\includegraphics[width=0.65\\textwidth]{Corbeli_Objects.pdf}\n \\caption{Spatially-matched Spitzer 24$\\mu$m (top) and HST F475W$-$F814W (bottom) color cutouts for three \\citet{Corbelli17} and \\citet{Sharma11} objects that represent three scenarios for the objects: an object which shows evidence of being a young embedded star forming region (left; Corbelli YSCC 225), an object where there is sufficient background visible to rule out an embedded cluster (middle; Corbelli YSCC 215), and a rare object which we classify as a star cluster in the PHATTER catalog (right; Corbelli YSCC 297, PHATTER 529).}\n \\label{fig:c_o}\n\\end{figure*}\n\n\n\\subsubsection{Infrared Comparisons: Alternative Catalogs} \\label{sec:litcat_ir}\n\nStudies in the literature have also made use of infrared images to assess M33's cluster population. \\citet{Sharma11} uses Spitzer 24 $\\mu$m images to construct a catalog of young stellar clusters, where it is assumed these objects are still embedded in their natal molecular clouds. Of the 240 \\citet{Sharma11} objects that fall within the PHATTER footprint, we only identify 41 cross-matches, of which we only classify 5 as star clusters while the remainder are mostly low-$f_{\\mathrm{view}}$ candidates. The poor correspondence between these two catalogs corroborates the conclusions of \\citet{Sun16}, who advocate against the use of the \\citet{Sharma11} catalog for star cluster population studies due to contamination by non-cluster objects, as we noted in \\S\\ref{sec:intro}.\n\nThe catalog of \\citet{Sharma11} was used as a starting point for compiling a sample of young cluster candidates for use in a cross-comparison with a CO molecular cloud catalog by \\citet{Corbelli17}. Of the 291 objects from \\citet{Corbelli17} within the PHATTER footprint, we match 91 objects and find just 12 to be star clusters identified in our catalog. As with the \\citet{Sharma11} sample, these cluster candidates are generally not associated with optical star clusters, and caution should be taken when using this catalog for the purpose of star cluster population work.\n\nTo provide a visual example of optical vs.\\ infrared cluster candidates, we present Spitzer 24$\\mu$m and HST F475W+F814W color cutout pairs for three \\citet{Corbelli17} and \\citet{Sharma11} objects in Figure~\\ref{fig:c_o}. The first is an example of a region where there is a bright 24$\\mu$m source, and in the color image there is visible extinction, indicative of molecular gas, and an emission region. These indicators confirm the presence of a young embedded star forming region and potentially (but not certainly) a young star cluster. In the absence of certainty on the presence of a bound star cluster, this object is not identified as a cluster in the PHATTER catalog. The second example is an object where, even though there is a 24$\\mu$m source, the background field in the optical image is fully visible with no sign of dust obscuration. Therefore, we conclude the probability of an embedded cluster is low. The third example is an object that is a classified as a cluster in our catalog. Examination of PHATTER imaging around these positions suggests that for at least 30\\% of \\citet{Corbelli17} objects, we can rule out the presence of an embedded star cluster based on the uniformity of the background at the location of the \\citet{Corbelli17} cluster candidates. Thus it appears that a significant fraction of these candidates are neither young embedded star forming regions nor optically visible star clusters.\n\n\\subsubsection{Catalog Comparison Summary} \\label{sec:litcat_summary}\n\nOverall, the PHATTER cluster catalog significantly enhances the population of known clusters in the inner disk region of M33. Out of 1214 total clusters, 810 (67\\% of the catalog) are identified here for the first time. We compare the luminosity distributions of PHATTER clusters and previously identified objects from the literature in Figure \\ref{fig:lum_comp}, showing that PHATTER's sample probes fainter, lower mass clusters than previous works. The superior spatial resolution of HST images facilitate this marked increase, as tight groups of stars can be differentiated from a single, barely or unresolved source. In addition, the ability to identify small groupings of faint resolved stars in PHATTER images leads to our ability to probe further down the cluster mass function in M33 than ever before.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.47\\textwidth]{Luminosity_Comp.pdf}\n \\caption{Shown here are the F475W magnitude distributions of PHATTER clusters (black), and previously identified literature objects (blue). Shown in black are 1166 PHATTER clusters with F475W detections. In blue are 391 literature objects included in the PHATTER cluster catalog with F475W detections. Literature objects not represented above: 13 objects lacking F475W detections and 178 objects with low $f_{cluster,W}$ omitted from the PHATTER cluster catalog.}\n \\label{fig:lum_comp}\n\\end{figure}\n\n\\subsection{M33 Catalog in Context: Expectations from \\texorpdfstring{$\\Gamma$}{Gamma} and \\texorpdfstring{$M_c$}{Mc} Correlations} \\label{sec:m33_sample_predict}\n\nWe examine whether the PHATTER sample size of 1214 clusters matches expectations based on our knowledge of M33 and the relation between galaxies and their cluster populations. An easy comparison we can make is to the PHAT M31 catalog \\citepalias{Johnson15_AP} and its 2753 clusters. We might naively expect M31's sample to outnumber M33's based on the number of bricks in the survey footprint (23 vs.\\ 3) or the total galaxy stellar mass \\citep[6$\\times$10$^{10}$ $M_{\\odot}$ vs.\\ 3$\\times$10$^{9}$ $M_{\\odot}$;][]{Patel17}.\nIn this context, it seems there should be a much larger than factor of $\\sim$2 difference in cluster count. However, differences in on-going star formation and spatial coverage fraction of HST imaging could plausibly provide an explanation, especially given that both cluster samples are dominated by young star clusters. \nTherefore, in this section we endeavor to make an SFR-based accounting of the M33 cluster sample to test whether we can explain the sample size differences we see between M33 and M31. We will use observed relations with respect to galaxy $\\Sigma_{\\mathrm{SFR}}$\\ to predict cluster formation efficiency ($\\Gamma$) and Schecter trunctation mass ($M_c$), and use these inputs to infer a prediction for cluster sample size.\n\nTo begin, we derive a SFR and $\\Sigma_{\\mathrm{SFR}}$\\ measurement for the PHATTER survey region in M33. Following the methodology laid out in Appendix A of \\citet{Johnson17}, we use \\textit{GALEX} FUV and \\textit{Spitzer} 24 \\textit{$\\mu$m} images and a $\\Sigma_{\\mathrm{SFR}}$\\ prescription from \\citet{Leroy08}. We find log($\\langle \\Sigma_{\\mathrm{SFR}} \\rangle$\\ \/ $M_{\\sun}$ yr$^{-1}$\\ kpc$^{-2}$) of $-2.04^{+0.16}_{-0.18}$ for M33 within the PHATTER survey footprint. We also extract the total star formation rate within the survey region, which we find is 0.12 $M_{\\sun}$ yr$^{-1}$, approximately 46\\% of M33's total SFR.\n\nNext, we can use the PHATTER $\\Sigma_{\\mathrm{SFR}}$\\ value to derive an estimate of the cluster formation efficiency, $\\Gamma$, based on the $\\Gamma$-$\\Sigma_{\\mathrm{SFR}}$\\ relation from \\citet{Johnson16_gamma}. This relation predicts a cluster formation efficiency of 15\\%. We can also derive an estimate for the cluster mass function by assuming a Schechter function form with a $-2$ power law slope and trunctation mass, $M_c$ estimate based on the $M_c$-$\\Sigma_{\\mathrm{SFR}}$\\ relation from \\citet{Johnson17}. This relation predicts a Schechter $M_c$ of 4.3$\\times$10$^4$ $M_{\\sun}$. \n\nWhen we combine the determinations of M33 SFR and $\\Gamma$, we derive a cluster formation rate (CFR) of 0.0180 $M_{\\sun}$ yr$^{-1}$. Interestingly, this is very similar to the PHAT M31 CFR of 0.0186 ($\\Gamma$=6.4\\% via \\citealt{Johnson16_gamma} and SFR=0.29 via \\citealt{Lewis15}). Although the PHATTER M33 SFR is 2.4 times smaller then the PHAT M31 survey SFR (0.12 $M_{\\sun}$ yr$^{-1}$ vs.\\ 0.29 $M_{\\sun}$ yr$^{-1}$), M33's $\\Sigma_{\\mathrm{SFR}}$\\ is almost 4 times higher, which leads to a 2.4x higher $\\Gamma$ (15\\% vs.\\ 6.3\\%) that cancels out the SFR difference.\n\nBased on the well-matched M31 and M33 CFRs, the relative difference in size between the two cluster catalogs (2753 vs.\\ 1214; 2.3x) seems unexpected. Note that the variation in $M_c$ (8.5$\\times$10$^3$ $M_{\\sun}$\\ vs.\\ 4.3$\\times$10$^4$ $M_{\\sun}$) is only expected to make a few percent difference in the number statistics due to the small number of clusters at the high mass end, so that does not explain the difference. However, catalog completeness differences likely play a role. M33's high stellar density within the PHATTER footprint in the central region of the disk leads to a higher mass for the 50\\% completeness limit than in M31: log($M$\/$M_{\\sun}$) of 3.2 vs.\\ 3.0 for a nominal 100--300 Myr age range. However, this 0.2 dex offset only affords a 1.3x--1.7x correction to M33 $N_{\\mathrm{cluster}}$, leaving another factor of 1.3x--1.7x still unexplained. This remaining discrepancy could be due to our assumption of a constant SFH, or perhaps we will find that $\\Gamma$ is not as high in M33 as predicted.\n\nOverall, this exercise shows that the star formation differences between M31 and M33 are likely significant enough to impact cluster populations. We look forward to using robust star formation history fitting \\citep{Lazzarini22} to inform recent SFR and $\\Sigma_{\\mathrm{SFR}}$\\ determinations for M33, and aid future $\\Gamma$ determinations for the PHATTER cluster sample.\n\n\\subsection{Cluster Affiliated Phenomena: X-ray Sources \\& Planetary Nebulae} \\label{sec:xraypn}\n\nOne of the immediate uses of the PHATTER cluster catalog is to cross-match it with objects of interest in M33, such as planetary nebulae (PNe), X-ray emitting sources, and other stellar populations. Identifying associations between clusters and these source populations can provide useful information about the source, such as a cluster-based age or additional information that assists in source classification.\n\nWe begin by searching for cross-matches between the PHATTER cluster catalog and two M33 X-ray source catalogs created from Chandra \\citep[ChASeM33 survey;][]{Tullmann11} and XMM-Newton \\citep{Williams15_xray} observations. We use an initial matching radius of 5 arcsec, but require the X-ray source to fall within the aperture radius of the cluster center (typically $\\sim$1.5 arcsec), resulting in two matches --- one from each X-ray catalog. The matched Chandra source (ChASeM33 393) is paired with PHATTER 675, a known globular cluster \\citep[ID: 275;][]{Sarajedini07} whose association with an X-ray source was identified by \\citet{Tullmann11}. The matched XMM-Newton catalog entry (Source 716) is associated with PHATTER 29, a previously identified young cluster \\citep[ID: 260;][]{Sarajedini07} with a CMD-estimated age of 10 Myr \\citep{Wainer22}. Given the ages of the clusters, the Chandra source in the globular cluster is likely to be a bright low-mass X-ray binary, and the XMM-Newton source in the young cluster is likely to be a bright high-mass X-ray binary.\n\nWe also report the presence of a possible planetary nebula (PN) associated with a PHATTER cluster. The candidate PN associated with the PHATTER 4 cluster was discovered serendipitously while reviewing sources with outlier optical colors in cluster CMDs. PNe are known to appear as anomalous blue sources in F475W$-$F814W color due to strong line emission in the F475W bandpass \\citep{Veyette14}. PHATTER 4 has a very uncertain SLUG integrated light age determination, most likely due to the unmodeled contribution of nebular line emission from the PN, however its CMD-fitted age estimate of $\\sim$1 Gyr \\citep{Wainer22} is not unexpected for a PN-hosting star cluster.\n\nGiven the serendipitous identification of the PN, we conducted a search for additional candidates. We performed a cross-match of the PHATTER clusters with the PN catalog of \\citet{Ciardullo04}, but we found no candidates that lie within the aperture radius from a cluster center. This lack of matches is not unexpected, however, given that a cluster PN would likely have been rejected by the \\citet{Ciardullo04} search of groundbased narrowband imaging due to the presence of coincident continuum emission from the cluster. We also perform a search for other cluster members with anomalous optical colors (F475W$-$F814W $<$ $-1$), but find no other reliable sources among cluster members.\n\nWe will continue to expand PHATTER cluster catalog cross-matching and the analyses of cluster membership to additional source populations in future work. Following on from work conducted for PHAT, cluster membership of AGB stars \\citep{Girardi20}, Cepheid variables \\citep{Senchyna15}, and other populations are ripe for study in the PHATTER data.\n\n\n\\section{Summary} \\label{sec:summary}\n\nWe present the results of a crowdsourced visual star cluster search of M33 conducted as part of the LGCS citizen science project using imaging from the PHATTER survey. The resulting catalog of 1214 star clusters has well-characterized completeness properties and a 50\\% completeness limit of approximately 1500 $M_{\\sun}$\\ at an age of 100 Myr. We derive ages and masses from SED fitting of the subset of clusters with multi-band detections in the catalog's integrated aperture photometry. We find the sample is composed primarily of young, low-mass star clusters, although the SLUG-fitted clusters are a biased subsample of the full PHATTER catalog.\n\nThis cluster catalog builds upon similar Local Group cluster work in M31 \\citepalias{Johnson15_AP} and significantly increases the number of known Local Group star clusters observed with HST. The PHATTER cluster catalog samples higher $\\Sigma_{\\mathrm{SFR}}$\\ galactic properties than M31, which provides leverage for studying how cluster properties like the cluster mass function, cluster formation efficiency, and more depend on star formation intensity. In accompanying work, we use CMDs of individually resolved stars to fit high-precision ages and masses, and to constrain the mass function of young clusters \\citep{Wainer22}. We also expect the sample will also be useful for calibrating models of stellar evolution \\citep[e.g.,][]{Girardi20} and other future M33 cluster studies.\n \n\n\n\\begin{acknowledgements}\nWe enthusiastically thank the $\\sim$2,800 LGCS volunteers who made this work possible. Their contributions are acknowledged individually at \\url{https:\/\/authors.clustersearch.org}. This publication uses data generated via the Zooniverse.org platform, development of which is funded by generous support, including a Global Impact Award from Google, and by a grant from the Alfred P. Sloan Foundation. Support for this work was provided by NASA through grant number HST-GO-14610 from the Space Telescope Science Institute, which is operated by AURA, Inc., under NASA contract NAS5-26555. L.C.J. acknowledges support through a CIERA Postdoctoral Fellowship at Northwestern University. This material is partially based on work by T.M.W. as a CIERA REU student at Northwestern University, supported by the National Science Foundation under grant No. AST-1757792. Some of the data presented in this paper were obtained from the Mikulski Archive for Space Telescopes (MAST). This research made use of NASA's Astrophysics Data System (ADS) bibliographic services.\n\\end{acknowledgements}\n\n\\facilities{HST(ACS, WFC3)}\n\\software{astropy \\citep{Astropy13}, DOLPHOT \\citep{Dolphin00}, DrizzlePac \\citep{DrizzlePac12,Hack13,Avila15}, SLUG \\citep{Krumholz15_SLUG}, TOPCAT \\citep{TOPCAT05}}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}