diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzjzcz" "b/data_all_eng_slimpj/shuffled/split2/finalzzjzcz" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzjzcz" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nThe intersection pairing between two divisors on a projective non-singular surface is the unique bilinear and symmetric pairing with values in $\\mathbb Z$ that satisfies some very natural properties: it counts the number of intersection points (when the divisor are normal crossing) and it is invariant if we ``move'' any of the two divisors in their linear class of equivalence. With the same philosophy, such a definition of intersection pairing between divisors can be extended naturally for projective non-singular varieties of any dimension: we list a number of natural properties and we find a unique multi-linear symmetric pairing satisfying them. It turns out that this unique intersection pairing on algebraic varieties can be expressed explicitly in terms of the Euler-Poincare characteristics of (invertible sheaves associated to the) divisors. For example, for a surface over a field $k$ we have the well known formula:\n\\begin{equation}\\label{eq_intro}\nC.D=\\chi_k(\\mathscr O_X)-\\chi_k(\\mathscr O_X(C)^{-1})-\\chi_k(\\mathscr O_X(D)^{-1})+\\chi_k(\\mathscr O_X(C)^{-1}\\otimes\\mathscr O_X(D)^{-1}) \n\\end{equation}\nwhich is involved in the proof of the Riemann-Roch theorem for surfaces (see for example \\cite{Beau}).\n\n\nFor a relative scheme $X\\to S$, if we don't appeal to any ``compactification arguments'' of $X$ and $S$, there is in general no hope for finding a reasonable intersection pairing for divisors which is invariant up linear equivalence. Let's see an example in the case of an arithmetic surface $X\\to \\spec \\mathbb Z$. Consider a prime $p\\in\\spec\\mathbb Z$, the fibre $X_{p}$ is a principal vertical divisor on $X$. Now let $D$ be an irreducible horizontal\ndivisor on $X$, then certainly $D$ meets $X_p$, which means $D.X_p> 0$ because our phantomatic intersection pairing should count the number of intersection points with multiplicity. Thus, for any other divisor $E\\in \\Div(X)$ we obtain\n$D.E < D.(E + X_p) $\nbut $E \\sim(E + X_p)$ since $X_p$ is principal by construction.\n\nThe closest object to an intersection pairing on a relative scheme $X\\to S$ of relative dimension $n$ is called the Deligne pairing: it is a map\n$$\\left<\\;,\\ldots,\\,\\right>_{X\/S}:\\operatorname{Vec}_1(X)^{n+1}\\to \\operatorname{Vec}_1(S)\\,,$$ \nwhere $\\operatorname{Vec}_1(\\cdot)$ denotes the set of invertible sheaves, which descends to a symmetric, multi-linear map at the level of Picard groups. This pairing is of crucial importance in arithmetic geometry, since it gives ``the schematic contribution'' to the Arakelov intersection number.\n\nThe Deligne pairing was constructed by Deligne in \\cite{Del} for arithmetic surfaces and then generalised to any dimension in \\cite{Elk}, \\cite{Zh} and \\cite{Gar}. Its definition was not built as the unique solution of a universal problem, but it was given a list of properties satisfied by the Deligne pairing. A set of axioms that uniquely identify the Deligne pairing were find recently in the preprint \\cite{Xia}. \n\n\nFor arithmetic surfaces one can show the following isomorphism of invertible sheaves which turns out to be very useful in the proof of Faltings-Riemann-Roch theorem (see for example \\cite{Del} and \\cite{MB} for more details):\n\\begin{equation}\\label{eq_intro1}\n\\left<\\mathscr L,\\mathscr M\\right>_{X\/S}\\cong \\det Rf_\\ast(\\mathscr O_X)\\otimes (\\det Rf_\\ast(\\mathscr L))^{-1}\\otimes(\\det Rf_\\ast(\\mathscr M))^{-1}\\otimes\\det Rf_\\ast(\\mathscr L\\otimes \\mathscr M)\\,.\n\\end{equation}\nOne can notice immediately the similarities between equations (\\ref{eq_intro}) and (\\ref{eq_intro1}). The only substantial difference is that for algebraic surfaces we use the Euler-Poincare characteristic whereas for arithmetic surfaces we use the determinant of the cohomology. Such a distinction makes perfectly sense because the determinant of the cohomology is constructed to be the arithmetic analogue of the Euler-Poincare characteristic.\n\nAt this point the natural question is the following one: is it possible to give an explicit definition of the Deligne pairing (in the most general case) in terms of the determinant of cohomology\\footnote{This was briefly conjectured already in \\cite{Elk}: `` (...) Dans le cas g\\'en\\'eral, c'est-\\`a-dire en dimension quelconque, et sans hypoth\\`ese de lissit\\' e, l'intersection doit aussi s'exprimer en termes de d\\'eterminants d'images directes (...) malheureusement, pour l'instant, des probl\\'e mes de signe obscurcissent s\\'e rieusement la situation.''}? In this paper we give an affirmative answer. By working in complete analogy of the theory of algebraic varieties, we write down a simple explicit formula for the Deligne pairing in terms of the determinant of cohomology. Let $f:X\\to S$ be a proper, flat morphism of reduced Noetherian schemes, and assume that $f$ has pure dimension $n$, then we put\n\n\\begin{equation}\\label{eq_intro2}\n\\left\\{\\begin{array} {lll}\n\\left<\\mathscr L_0,\\ldots, \\mathscr L_n\\right>_{X\/S}:={\\det Rf_\\ast\\left(c_1(\\mathscr L^{-1}_0)c_1(\\mathscr L^{-1}_1)\\ldots c_1(\\mathscr L^{-1}_n)\\mathscr O_X\\right)}\\, & \\textrm{if } n>0\\\\\n&\\\\\n\\left< \\mathscr L_0\\right>_{X\/S}:={\\det Rf_\\ast\\left(c_1(\\mathscr L^{-1}_0)\\mathscr O_X\\right)}^{-1}=N_{X\/S}(\\mathscr L_0)& \\textrm{if } n=0\\\\\n\\end{array}\\right.\n\\end{equation}\n\n\\noindent where $c_1(\\mathscr L)\\mathscr F:=\\mathscr F-\\mathscr L^{-1}\\otimes \\mathscr F\\in K_0(X)$ for any coherent sheaf $\\mathscr F$ and any invertible sheaf $\\mathscr L$ (here only the class of $\\mathscr F$ in the Grothendieck group matters). Moreover $N_{X\/S}$ is the norm, relative to $f$, of an invertible sheaf. We show that definition (\\ref{eq_intro2}) satisfies the axioms of \\cite{Xia}, and this implies that our definition is exactly the Deligne pairing.\n\n\nLet's mention some other papers that previously investigated in our direction: an explicit formula for the Deligne pairing when $X$ and $S$ are integral schemes over $\\mathbb C$ was announced in \\cite{BSW}, although a complete proof is not given. The approach of \\cite{BSW} is essentially different from ours, indeed the authors work on local trivializations of invertible sheaves. A more complicated expression of the Deligne pairing in terms of symmetric difference of the functor $\\det Rf_\\ast$ is proved in \\cite{Ducr} with heavy usage of category theory. An explicit calculation of the Deligne pairing $\\left<\\mathscr L,\\ldots,\\mathscr L\\right>$ (here $\\mathscr L$ is always the same) is given in \\cite{Phon}.\\\\\n\nThis paper is organized in the following way: in section \\ref{endo} we introduce the map $c_1(\\mathscr L):K_0(X)\\to K_0(X)$ with all its properties. Section \\ref{int_var} is a review of intersection theory for algebraic varieties and it gives to the reader the philosophical guidelines for the case of relative schemes. In section \\ref{rel_sch} we give the axioms of the Deligne pairing and we show with all details that if such pairing exists it must be unique (we follow \\cite{Xia}). Afterwards we show that the pairing (\\ref{eq_intro2}) satisfies all the axioms. appendix \\ref{det} is a review of the determinant of cohomology, this part is crucial in order to understand section \\ref{rel_sch}. Finally appendix \\ref{ori_constr} is a ``bonus section'' where we show with all details the original construction of Deligne pairing of \\cite{Del} (very often this construction is just sketched in the literature).\n\n\\paragraph{Acknowledgements.}\nThe author wants to express his gratitude to R.S. De Jong for his time spent in discussing the topic during summer 2019 in Nottingham and for his precious comments. A special thanks goes also to P. Corvaja, I. Fesenko, S. Urbinati and F. Zucconi.\\\\\n\n\\noindent This research was supported by the Italian national grant ``Ing. Giorgio Schirillo\" conferred by INdAM and partially by the EPSRC programme grant EP\/M024830\/1 (Symmetries and correspondences: intra-disciplinary developments and applications).\n\n\\section{An endomomorphism of the group $K_0(X)$}\\label{endo}\n Let's briefly recall the abstract construction of the Groethendieck group $K_0(\\catname C)$. Fix an abelian category $\\catname{C}$ and let $F(\\catname C)$ be the free abelian group over the set $\\Ob(\\catname C)\/\\cong$, where $\\cong$ is the isomorphism relation. If $C\\in \\Ob(\\catname C)$, then $(C)$ denotes isomorphism class in $\\Ob(\\catname C)\/\\cong$. To any short exact sequence in $\\catname C$:\n$$\\mathcal S\\colon \\quad 0\\to C'\\to C\\to C''\\to 0$$\nwe associate an element $ Q(\\mathcal S):=(C)- (C')-(C'')\\in F(\\catname C)$. Now $H(\\catname C)$ is the subgroup of $F(\\catname C)$ generated by all the elements $Q(\\mathcal S)$ for $\\mathcal S$ running over all short exact sequences. Then:\n$$K_0(\\catname C):= F(\\catname C)\/H(\\catname C)\\,,$$\nand $[C]\\in K_0(\\catname C)$ denotes the equivalence class associated to $C\\in \\Ob(\\catname C)$.\n\n\n\nLet's fix a reduced Noetherian scheme $X$, then $K_0(X):=K_0(\\catname {Coh}(X))$, where $\\catname {Coh}(X)$ is the category of coherent sheaves on $X$. From now on, by an abuse of notation we identify any coherent sheaf $\\mathscr F$ with its class in $K_0(X)$. In this paper, with the notation $\\catname {Coh}_r(X)$ we denote the category of coherent sheaves on $X$ whose support has dimension at most $r$, and we define $K_{0,r}(X):= K_0(\\catname {Coh}_r(X))$. Clearly when $0\\le i\\le j$, then $K_{0,i}(X)\\subseteq K_{0,j}(X)$.\n\nFor any invertible sheaf $\\mathscr L$ on $X$ we define a map:\n\n\\begin{equation}\n\\begin{aligned}\n c_1(\\mathscr L):K_0(X)&\\to K_0(X)\\\\\n\\mathscr F &\\mapsto c_1(\\mathscr L)\\mathscr F:= \\mathscr F-\\mathscr L^{-1}\\otimes \\mathscr F\\,.\n\\end{aligned}\n\\end{equation}\nNote that it is well defined because tensoring with an invertible sheaf is an exact functor, moreover it defines and endomorphism of the group $K_0(X)$. Since the notation for the function $c_1(\\mathscr L)$ is multiplicative, the symbol $c_1(\\mathscr L)c_1(\\mathscr L')$ denotes the composition of functions. The properties of the operator $c_1(\\mathscr L)$ are well described in \\cite[Appendix B]{Kl}, so here we just recall them.\n\n\\begin{proposition}\\label{propc_1}\nThe following properties hold for the operator $c_1(\\mathscr L)$:\n\\begin{itemize}\n\\item[$(i)$] $c_1(\\mathscr L)c_1(\\mathscr M)=c_1(\\mathscr L)+ c_1(\\mathscr M)-c_1(\\mathscr L\\otimes \\mathscr M)$, where clearly the sum is taken in $\\operatorname{End}(K_0(X))$.\n\\item[$(ii)$] $c_1(\\mathscr M) c_1(\\mathscr L)=c_1(\\mathscr L) c_1(\\mathscr M).$\n\\item[$(iii)$] If $Z\\subset X$ is a closed subscheme and $\\mathscr L_{|Z}=\\mathscr O_Z(D)$ where $D$ is an effective Cartier divisor on $Z$, then $c_1(\\mathscr L) \\mathscr O_Z=\\mathscr O_D$.\n\\end{itemize}\n\\end{proposition}\n\\proof\nBoth sides of the equality in $(i)$ expand to:\n\n\\begin{equation}\\label{e_comm}\n\\mathscr F-\\mathscr L^{-1}\\otimes \\mathscr F-\\mathscr M^{-1}\\otimes \\mathscr F\\mathscr +\\mathscr L^{-1}\\otimes\\mathscr M^{-1}\\otimes\\mathscr F\\,.\n\\end{equation}\n$(ii)$ Follows easily by looking at equation (\\ref{e_comm}). For $(iii)$ Consider the short exact sequence\n$$0\\to\\mathscr O_Z(-D)\\to\\mathscr O_Z\\to\\mathscr O_D\\to 0\\,.$$\n\\endproof\n\\begin{proposition}\nLet $\\mathscr F\\in K_{0,r}(X)$ and let $Z_1,\\ldots, Z_s$ the $r$-dimensional irreducible components of $\\supp(\\mathscr F)$ whose generic points are denoted respectively by $z_i$. Let $n_i=\\len \\mathscr F_{z_i}$. Then in $K_{0,r}(X)$ we have the equality:\n$$\\mathscr F\\equiv \\sum^s_{i=1} n_i\\mathscr O_{Z_i}\\mod K_{0,r-1}(X)\\,.$$\n\\end{proposition}\n\\proof\nSee \\cite[Lemma B4]{Kl}.\n\\endproof\n\n\\begin{proposition}\\label{decreasing_sup}\nlet $\\mathscr L$ an invertible sheaf on $X$, then $c_1(\\mathscr L)K_{0,r}(X)\\subset K_{0,r-1}(X)$ for any $r\\ge 0$.\n\\end{proposition}\n\\proof\nSee \\cite[Lemma B5]{Kl}.\n\\endproof\n\n\n\\begin{remark}\nThe operator $c_1(\\mathscr L)$ can be ``extended'' to bounded complexes of coherent sheaves on $X$. Let $\\mathscr F^\\bullet$ be a bounded complex of objects in $\\catname{Coh}(X)$ then we can define:\n$$c_1(\\mathscr L)\\mathscr F^\\bullet:=\\sum_i(-1)^ic_1(\\mathscr L)\\mathscr F^i\\in K_0(X)\\,.$$\nSuch a map is clearly zero on short exact sequences. \n\\end{remark}\n\n\\section{Intersection theory for algebraic varieties}\\label{int_var}\n\n\n\\begin{definition}\\label{inters_var}\nLet $X$ be a $n$-dimensional projective, non-singular algebraic variety over a field $k$. An \\emph{intersection pairing} on $X$ is a map:\n\\begin{equation}\n\\begin{aligned}\n\\Div(X)^n&\\to \\mathbb Z\\\\\n(D_1,\\ldots,D_n) &\\mapsto D_1.D_2.\\,\\ldots\\,.D_n\n\\end{aligned}\n\\end{equation}\nsatisfying the following properties:\n\\begin{itemize}\n\\item[$(1)$] It is symmetric and $\\mathbb Z$-multilinear.\n\\item[$(2)$] It descends to a pairing $\\Pic(X)^n\\to \\mathbb Z$.\n\\item[$(3)$] Let $D_i$ be an effective divisor for any $i$ and let $e_{i,x}\\in\\mathscr O_{X,x}$ be a local equation of $D_i$ at the point $x$. Assume that for all $x$ in the support of all divisors $D_i$, the $e_{i,x}$'s form a regular sequence in $\\mathscr O_{X,x}$ (i.e. the divisors are in general position), then: \n$$D_1.D_2.\\,\\ldots\\,.D_n=\\sum_{x\\in\\cap D_i}\\len\\frac{\\mathscr O_{X,x}}{(e_{1,x}, e_{2,x},\\ldots, e_{n,x})}$$ \n\\end{itemize}\n\\end{definition}\nNow we show that if an intersection pairing exists, it is uniquely defined by the three axioms of Definition \\ref{inters_var}.\n\\begin{proposition}\nIf an intersection pairing exists, then it is unique. \n\\end{proposition}\n\\proof\nLet $\\left<\\;\\right>_1$ and $\\left<\\;\\right>_2$ two pairings satisfying the axioms $(1)-(3)$ and fix $D_1,\\ldots,D_n\\in \\Div(X)^n$; by $(1)$ we can assume that all $D_i$ are effective. Thanks to Chow's moving lemma we can find some divisors $D'_i$ such that $D\\sim D_i'$ and $D'_1,\\ldots,D'_n$ are in general position. Therefore by using $(2)$ and $(3)$ we get:\n$$\\left_1=\\left_1=\\left_2=\\left_2\\,.$$\n\\endproof\nThe remaining part of this section is devoted to give the explicit expression of the intersection pairing on $X$ as in \\cite{Sn1}, \\cite{Sn2} and later \\cite{Ca}; then we see that the axioms of definition \\ref{inters_var} are satisfied. Such an intersection pairing uses the endomorphism defined in section \\ref{endo} and the Euler-Poincare characteristic for coherent sheaves. \n\nWe actually give a definition of the intersection pairing in a more general setting, in fact we will assume that $X$ is a relative scheme over a scheme $S$ and we define a ``partial'' intersection number of a particular subclass of divisors: roughly speaking, if we restrict to the case of arithmetic surfaces, we define an intersection number between two divisors provided that one of them is vertical. \n\nFrom now on in this section we assume that $X\\to S$ is a flat and proper morphism of irreducible Noetherian schemes. Let's denote with $\\catname{Coh}(X\/S)$ the category of coherent sheaves on $X$ whose schematic support is a proper over a $0$-dimensional subscheme of $S$. Moreover $\\catname{Coh}_r(X\/S)$ is the subcategory of $\\catname{Coh}(X\/S)$ made of sheaves whose support has dimension at most $r$. The motivation behind the restriction to sheaves with this kind of support is that for any $\\mathscr F\\in \\catname{Coh}(X\/S)$ we have a well defined notion of Euler-Poincare characteristic. In fact if $T$ is the schematic support of $\\mathscr F$ and $S_0=f(T)$, we know that $S_0$ is Noetherian of dimension $0$, so $S_0=\\spec A$ with $A$ artinian; at this point we can put\n\n$$\\chi_S(\\mathscr F):=\\sum_{i\\ge 0} (-1)^i\\len_A H^i(X, \\mathscr F)\\,.$$\nWhen $S=\\spec k$, then $\\chi_S$ is the usual Euler-Poicare characteristic (for coherent sheaves with proper support). Thanks to the ``additivity'' of $\\chi_S$ with respect to short exact sequences, it is immediate to notice that we have a naturally induced group homomorphism $\\chi_S:K_0(\\catname{Coh}_r(X\/S))\\to\\mathbb Z$.\n\\begin{definition}\nLet $X\\to S$ as above and let $\\mathscr F\\in\\catname{Coh}_r(X\/S)$. Then the \\emph{intersection number of the invertible sheaves $\\mathscr L_1,\\ldots\\mathscr L_r$ (with respect to $\\mathscr F$)} is defined as:\n\n$$(\\mathscr L_1.\\mathscr L_2.\\,\\ldots\\,.\\mathscr L_r,\\mathscr F):=\\chi_S\\left(c_1(\\mathscr L_1)c_1(\\mathscr L_2)\\ldots c_1(\\mathscr L_r)\\mathscr F\\right)\\,.$$\nWhen $\\mathscr F=\\mathscr O_X$, which implies $r\\ge\\dim(X)$, we put for simplicity \n\n$$\\mathscr L_1.\\mathscr L_2.\\,\\ldots\\,.\\mathscr L_r:=(\\mathscr L_1.\\mathscr L_2.\\,\\ldots\\,.\\mathscr L_r,\\mathscr O_X)\\,.$$\nMoreover if $\\mathscr L_i=\\mathscr O_X(D_i)$ for a Cartier divisor $D_i$ on $X$, then:\n$$D_1.D_2.\\,\\ldots\\,.D_r:=\\mathscr O_X(D_1).\\mathscr O_X(D_2).\\,\\ldots\\,.\\mathscr O_X(D_r)\\,.$$ \n\n\\end{definition}\n\n\n\n\n\\begin{example}\nIf $X$ is a surface over $k$ and $C,D$ are two divisors, then:\n$$C.D=\\chi_k(c_1(\\mathscr O_X(C)) c_1(\\mathscr O_X(D))\\mathscr O_X)=\\chi_k(c_1(\\mathscr O_X(C))(\\mathscr O_X-\\mathscr O_X(D)^{-1}))=$$\n$$=\\chi_k(\\mathscr O_X)-\\chi_k(\\mathscr O_X(C)^{-1})-\\chi_k(\\mathscr O_X(D)^{-1})+\\chi_k(\\mathscr O_X(C)^{-1}\\otimes\\mathscr O_X(D)^{-1}) $$\n\\end{example}\n\nThe mere definitions tell us that we can intersect a number of divisors which is greater or equal to the dimension on $X$. On the other hand the next lemma shows that the intersection of a number of divisor which is strictly bigger than the dimension of $X$ is always $0$.\n\\begin{lemma}\\label{no_more_dim}\nIf $\\mathscr F\\in\\catname{Coh}_{r}(X\/S)$, then $(\\mathscr L_1.\\mathscr L_2.\\,\\ldots\\,.\\mathscr L_{r+1},\\mathscr F)=0$.\n\\end{lemma}\n\\proof\nIt follows directly from Proposition \\ref{decreasing_sup}.\n\\endproof\n\\begin{proposition}\\label{bilinear}\nThe intersection number of $\\mathscr L_1,\\ldots,\\mathscr L_m$ with respect to $\\mathscr F$ is a $\\mathbb Z$-multilinear map in the $\\mathscr L_j$'s (the operation is the tensor product).\n\\end{proposition}\n\\proof\nFollows by Proposition \\ref{propc_1}(i) and Lemma \\ref{no_more_dim}.\n\\endproof\n\\begin{proposition}\\label{mor_int}\nLet $g:X'\\to X$ be an morphism of $S$-schemes and let $\\mathscr F\\in\\catname{Coh}_r(X'\/S)$, then\n$$(g^\\ast\\mathscr L_1.g^\\ast\\mathscr L_1\\,.\\ldots g^\\ast\\mathscr L_n, \\mathscr F)=(\\mathscr L_1.\\mathscr L_1\\,.\\ldots \\mathscr L_n, g_\\ast\\mathscr F)\\,.$$ \n\\end{proposition}\n\\proof\nSee \\cite[Lemma B.15]{Kl}.\n\\endproof\n\nWe can give the explicit expression of the intersection number on varieties:\n\\begin{proposition}\nLet $X$ be an algebraic variety of dimension $r$ over a field $k$. The pairing $$(D_1,\\ldots,D_r)\\mapsto D_1.D_2.\\,\\ldots\\,.D_r$$ defines the intersection number on $X$.\n\\end{proposition}\n\\proof\nAxiom $(1)$ is satisfied thanks to Proposition \\ref{bilinear}. Axiom $(2)$ is obvious and axiom $(3)$ is \\cite[IV, Theorem 2.8]{Kol}.\n\\endproof\n\nFinally we state a proposition regarding the intersection along fibres:\n\n\\begin{proposition}\\label{gen_fiber}\nLet $s\\in S$ be a closed point and let $X_s$ be the fibre over $b$. Assume that $\\dim(X_s)=d$, then the map:\n$$s\\mapsto (\\mathscr L_1,\\ldots,\\mathscr L_d; \\mathscr O_{X_s})$$\nis locally constant on $S$.\n\\end{proposition}\n\\proof\nSee \\cite[VI, Proposition 2.10]{Kol}.\n\\endproof\n\n\\section{The case of schemes over a general base}\\label{rel_sch}\n\n\n\\subsection{Multi-monoidal and symmetric functors}\nThe Deligne pairing will be expressed as a collection of functors, so in this section we recall what is the functorial equivalent of a multi-linear homomorphism of abelian groups. \n\nWe assume that the reader is familiar with some basic notions of category theory and the concept of \\emph{Picard groupoid}. Roughly speaking a Picard groupoid is a category where morphisms are only isomorphisms and moreover there is a ``group-like'' operation between the object of the category. The simplest example is the \\emph{Picard category} $\\catname{Pic}(X)$ made of all invertible sheaves over a scheme $X$ where the morphisms are just the isomorphisms. The ``operation'' in $\\catname{Pic}(X)$ is clearly the tensor product of invertible sheaves and the identity element is the structure sheaf. The morphisms we want to consider between Picard groupoids are monoidal functors, i.e. functors that preserve the monoidal structure of the categories: \n\n\nFor the remaining part of this subsection we fix two Picard groupoids $(\\catname{C},\\otimes)$ and $(\\catname{D},\\otimes)$.\n\\begin{definition}\nA monoidal functor $\\catname{C}\\to\\catname{D}$ is a collection $(F,\\epsilon, \\mu)$ where $\\mu:=\\{\\mu_{X,Y}\\}_{X,Y\\in\\text{Obj(}\\catname C)}$, satisfying the following properties:\n\\begin{enumerate}\n\\item[\\,\\begin{picture}(-1,1)(-1,-3)\\circle*{3}\\end{picture}\\ ] $F:\\catname{C}\\to\\catname{D}$ is a functor.\n\\item[\\,\\begin{picture}(-1,1)(-1,-3)\\circle*{3}\\end{picture}\\ ] $\\epsilon\\colon F(1_{\\catname C})\\xrightarrow{\\cong} 1_{\\catname D}$ is an isomorphism.\n\\item[\\,\\begin{picture}(-1,1)(-1,-3)\\circle*{3}\\end{picture}\\ ] $\\mu_{X,Y}\\colon F(X)\\otimes F(Y)\\xrightarrow{\\cong} F(X\\otimes Y)$ is an isomorphism functorial in $X$ and $Y$ which satisfies associativity and unitality in the obvious categorical sense.\n\\end{enumerate}\nFor simplicity we often omit $\\epsilon$ and $\\mu$ and we say that $F$ is a monoidal functor between $\\catname C$ and $\\catname{D}$. In symbols we write $F\\in L^1(\\catname C,\\catname D)$.\n\\end{definition}\n\n\\begin{definition}\\label{nat1}\nA \\emph{natural transformation} between monoidal functors $(F, \\epsilon,\\mu)$ and $(F',\\epsilon', \\mu')$ is a monoidal natural transformation $\\alpha:F\\to F'$ which maps $\\epsilon$ to $\\epsilon'$ and $\\mu$ to $\\mu'$.\n\\end{definition}\nIn order to give the next definition we need to introduce some notations. An object of the category $\\catname{C}^n$ (i.e. a $n$-uple of objects of $\\catname{C}$) is denoted by $ X=(X_1,\\ldots, X_n)$. Let $X, Y\\in \\catname{C}^n$ and let $i\\in\\{1,\\ldots,n\\}$ such that for any $j\\in\\{1,\\ldots,n\\}$ with $j\\neq i$ we have $X_j=Y_j$, then we define $ X\\otimes_i Y\\in\\catname{C}^n$ in the following way:\n\n$$\n(X\\otimes_i Y)_j=\\left\\{\\begin{array} {ll}\nX_j\\otimes Y_j & \\text{if } i=j\\\\\nX_j & \\text{if } i\\neq j\\\\\n\\end{array}\\right.\n$$\n\n\n\\begin{definition}\nA multi-monoidal\\footnote{Very often in literature one can find the term multi-additive.} functor $\\catname{C}^n\\to\\catname{D}$ is the datum of\n\\begin{enumerate}\n\\item[\\,\\begin{picture}(-1,1)(-1,-3)\\circle*{3}\\end{picture}\\ ] A functor $F:\\catname{C}^n\\to \\catname{D}$.\n\\item[\\,\\begin{picture}(-1,1)(-1,-3)\\circle*{3}\\end{picture}\\ ] For any functor $F':\\catname{C}\\to\\catname{D}$ obtained by fixing $n-1$ components in $\\catname C$, we have a collection $\\mu'$ such that $(F',\\mu')$ is a monoidal functor $\\catname{C}$ and $\\catname D$.\n\\item[\\,\\begin{picture}(-1,1)(-1,-3)\\circle*{3}\\end{picture}\\ ] For every $i,j\\in\\{1,\\ldots, n\\} $ and $X,Y,Z,W\\in\\catname{C}^n$ such that $X_k=Y_k=Z_k=W_k$ for all $k\\neq i,j$, a commutative diagram:\n$$\n\\adjustbox{scale=0.8,center}{%\n\\begin{tikzcd}\n & F(X)\\otimes F(Y)\\otimes F(Z)\\otimes F(W) \\arrow{dl}\\arrow{dr} &\\\\\n(F(X)\\otimes_j F(Y))\\otimes (F(Z)\\otimes_j F(W))\\arrow{d}& & (F(X)\\otimes_i F(Y))\\otimes (F(Z)\\otimes_i F(W))\\arrow{d}\\\\\nF((X\\otimes_j Y)\\otimes_i (Z\\otimes_j W))\\arrow[\"=\"]{rr}&& (F(X)\\otimes_i F(Y))\\otimes_j (F(Z)\\otimes_i F(W))\n\\end{tikzcd}\n}\n$$\n\\end{enumerate}\n\\end{definition}\nThe notion of symmetry is what one expects:\n\\begin{definition}\nA multi-monoidal functor $\\catname{C}^n\\to\\catname{D}$ is \\emph{symmetric} if for any $c_i\\in\\catname C$ and any permutation $\\sigma\\in\\Sigma_n$ we have $F(c_1,\\ldots c_n)\\cong F(c_{\\sigma(1)},\\ldots c_{\\sigma(n)})$.\n\\end{definition}\n\nThe set of symmetric multi-monoidal morphisms from $\\catname{C}^n$ to $\\catname D$ is denoted by $L^n(\\catname C,\\catname D)$.\n\n\\begin{definition}\n\\emph{A natural transformation between two multi-monoidal functors} $F,F'\\in L^n(\\catname C,\\catname D)$ is a functorial isomorphism $\\alpha: F\\to F'$ which restricts to a natural transformation to each component in the sense of definition \\ref{nat1}.\n\\end{definition}\n\n\\subsection{Axiomatic Deligne pairing}\nThe Deligne pairing was introduced in \\cite{Del} as a bilinear and symmetric map $\\left<\\,,\\,\\right>:\\operatorname{Vec}_1(X)\\times\\operatorname{Vec}_1(X)\\to\\operatorname{Vec}_1(S)$, where $X\\to S$ is an arithmetic surface. Such a definition is quite implicit, in fact requires the choice of rational sections of invertible sheaves and some operations in the \\'etale topology on the base $S$. The Deligne pairing satisfies some compatibility conditions: it behaves well with respect to the base change and the pullback functor; moreover it is strictly related to the norm functor. In \\cite{Elk} Deligne's construction was extended straight away for proper flat morphisms of integral schemes of any dimension.\n\nLet $f:X\\to S$ a proper flat morphism between Noetherian reduced schemes, the guiding idea of this paper is that the Deligne pairing relative to $f$ should be the generalisation the intersection pairing described in section \\ref{int_var}. We want to work in complete analogy with the case of algebraic varieties, so in this section we give a set of ``natural axioms'' that uniquely define the Deligne pairing\\footnote{We follow \\cite{Xia}, but we prefer to give a self contained presentation with all details.}. The explicit construction of the Deligne pairing will be carried out in section \\ref{det_del}.\n\nLet $X$ and $S$ be two Noetherian reduced schemes, with the symbol $\\mathcal F^n(X,S)$ we denote the set of all proper flat morphisms $X\\to S$ of pure dimension $n$.\n\n\\begin{definition}\\label{axiom_deligne}\nA \\emph{Deligne pairing} consists of the following data for any two irreducible Noetherian schemes $X,S$, any $n\\in\\mathbb N$ and any $f\\in\\mathcal F^n(X,S)$: a functor \n$$\\left< \\,\\;,\\;,\\ldots,\\;\\right>_f=\\left< \\,\\;,\\;,\\ldots,\\;\\right>_{X\/S}\\in L^{n+1}(\\catname{Pic}(X), \\catname{Pic}(S))$$\nand a collection of natural transformations $\\alpha,\\beta,\\gamma,\\delta$ described below:\n\\begin{enumerate}\n\\item[(1)] For any commutative square given by a base change $g:S'\\to S$ which is proper, flat and with connected fibres \n\n$$\n\\begin{tikzcd}[row sep=large, column sep = huge]\nX'=X\\times_S S'\\arrow[\"f'\"]{r}\\arrow[\"g'\"]{d} & S'\\arrow[\"g\"]{d}\\\\\nX\\arrow[\"f\"]{r} & S\n\\end{tikzcd}\n$$\na natural transformation between multi-monoidal functors $\\alpha_{f,g}:\\catname{Pic}(X)^{n+1}\\to \\catname{Pic}(S')$ such that\n $$\\alpha_{f,g}:g^\\ast\\left<\\mathscr L_0,\\ldots,\\mathscr L_n\\right>_{X\/S}\\xrightarrow{\\cong}\\left_{X'\/S'}\\,.$$\n \n \\item[(2)] When $n>0$ and $D\\in\\Div(X)$ is an effective relative Cartier divisor, a natural transformation between multi-monoidal functors $\\beta_{f,D}:\\catname{Pic}(X)^{n}\\to \\catname{Pic}(S)$ such that\n $$\\beta_{f,D}:\\left<\\mathscr L_1,\\ldots,\\mathscr L_n, \\mathscr O_X(D)\\right>_{X\/S}\\xrightarrow{\\cong}\\left<\\mathscr L_1|_D,\\ldots,\\mathscr L_n|_D\\right>_{D\/S}\\,.$$\nMoreover $\\beta_{f,D}$ is natural with respect to base change in the following sense: for a base change diagram as in axiom $(1)$ we have a commutative diagram: \n $$\n\\begin{tikzcd}[row sep=large, column sep = huge]\n\\left_{X'\/S'}\\arrow[\"\\beta_{f',g'^\\ast D}\"]{r}\\arrow[\"\\cong\"]{d} & \\left_{{g'^\\ast D}\/S'}\\arrow[\"\\cong\"]{d}\\\\\ng^\\ast\\left<\\mathscr L_1,\\ldots,\\mathscr L_n, \\mathscr O_X(D)\\right>_{X\/S} & g^\\ast \\left<\\mathscr L_1|_D,\\ldots,\\mathscr L_n|_D\\right>_{D\/S}\\arrow[\"g^\\ast \\beta_{f,D}\"]{l} \n\\end{tikzcd}\n$$\nWhere the vertical isomorphisms are given by $\\alpha_{f,g}$ (remember that $g'^\\ast\\mathscr O_X(D)=\\mathscr O_{X'}(g'^\\ast D)$).\n \\item[(3)] When $n>0$, a natural transformation between multi-monoidal functors $\\gamma_{f}:\\catname{Pic}(S)\\times\\catname{Pic}(X)^{n}\\to \\catname{Pic}(S)$\n such that\n $$\\gamma_{f}:\\left_{X\/S}\\xrightarrow{\\cong}\\mathscr L^{(\\mathscr L_1|_{X_s}.\\mathscr L_2|_{X_s}.\\ldots\\mathscr L_n|_{X_s};\\mathscr O_{X_s})}$$\nwhere $X_s$ is a generic fibre of $f$ (see Proposition \\ref{gen_fiber}). Moreover $\\gamma_{f}$ is natural with respect to base change in the following sense: for a base change diagram as in axiom $(1)$ we have a commutative diagram: \n $$\n\\begin{tikzcd}[row sep=large, column sep = huge]\n\\left_{X'\/S'}\\arrow[\"\\gamma_{f'}\"]{r}\\arrow[\"\\cong\"]{d} & g^\\ast \\mathscr L^{(g'^\\ast \\mathscr L_1|_{g'^\\ast X_s}. g'^\\ast \\mathscr L_2|_{g'^\\ast X_s}.\\ldots g'^\\ast\\mathscr L_n|_{g'^\\ast X_s};\\mathscr O_{g'^\\ast X_s})}\\arrow[\"=\"]{d}\\\\\ng^\\ast\\left_{X\/S} & g^\\ast\\mathscr L^{(\\mathscr L_1|_{X_s}.\\mathscr L_2|_{X_s}.\\ldots\\mathscr L_n|_{X_s};\\mathscr O_{X_s})}\\arrow[\"g^\\ast \\gamma_{f}\"]{l} \n\\end{tikzcd}\n$$\nwhere the vertical isomorphism is given $\\alpha_{f,g}$ and the equality follows from proposition \\ref{mor_int} and the properties of $g$.\n\n\\item[(4)] When $n=0$, a natural transformation between monoidal functors $\\delta_{f}:\\catname{Pic}(X)\\to \\catname{Pic}(S)$\nsuch that\n $$\\delta_{f}:\\left<\\mathscr L\\right>_{X\/S}\\xrightarrow{\\cong} N_{X\/S}(\\mathscr L)$$\n where $N_{X\/S}$ is the norm of $f$ (see Definition \\ref{norm}). Moreover $\\delta_f$ is natural with respect to base change in the following sense: for a base change diagram as in axiom $(1)$ we have a commutative diagram:\n$$\n\\begin{tikzcd}[row sep=large, column sep = huge]\n\\left_{X'\/S'}\\arrow[\"\\delta_{f'}\"]{r}\\arrow[\"\\cong\"]{d} & N_{X'\/S'}(g'^\\ast\\mathscr L)\\arrow[\"\\cong\"]{d}\\\\\ng^\\ast\\left<\\mathscr L\\right>_{X\/S} & g^\\ast N_{X\/S}(\\mathscr L)\\arrow[\"g^\\ast \\delta_f\"]{l} \n\\end{tikzcd}\n$$\nwhere the vertical isomorphisms are given respectively by $\\alpha_{f,g}$ and thanks to the properties of the norm.\n\\end{enumerate} \n\\end{definition}\nWe have to show that if a Deligne pairing exists, then it is unique. Roughly speaking we will show that any two pairings $(\\left< \\,\\;,\\;,\\ldots,\\;\\right>^i,\\alpha^i,\\beta^i,\\gamma^i,\\delta^i)$, with $i=1,2$, satisfying the axioms of definition \\ref{axiom_deligne} are related by natural transformation of functors that respects all the data. We will work by induction on the relative dimension of the morphism $f$. Note that we cannot use straight away property $(2)$ to pass from relative dimension $n$ to $n-1$, since the whole construction will depend on the choice of the relative divisor $D$, but we want our constructions to be natural in a functorial way. Let's describe a general well known procedure to reduce the relative dimension of $f$ without any particular choice of divisors. It is called \\emph{universal extension}.\n\nLet $f\\in\\mathcal F^n(X,S)$ and let $\\mathscr L$ be an invertible sheaf on $X$. We assume that $\\mathscr L$ is \\emph{sufficiently ample (with respect to $f$)}, i.e. that the following properties are satisfied: $\\mathscr L$ is very ample (with respect to $f$) and $R^i f_\\ast\\mathscr L=0$ for $i>0$. \n\\begin{remark}\nThe following properties hold for sufficient ampleness:\n\\begin{itemize}\n\\item[\\,\\begin{picture}(-1,1)(-1,-3)\\circle*{3}\\end{picture}\\ ] It is preserved after base change.\n\\item[\\,\\begin{picture}(-1,1)(-1,-3)\\circle*{3}\\end{picture}\\ ] If $f\\in\\mathcal F^n(X,S)$ and $\\mathscr L$ is sufficiently ample on $X$, then $f_\\ast \\mathscr L$ is a locally free sheaf on $S$.\n\\item[\\,\\begin{picture}(-1,1)(-1,-3)\\circle*{3}\\end{picture}\\ ] If $\\mathscr L_0$ is an invertible sheaf on $X$, then there exists a sufficiently ample $\\mathscr L$ such that $\\mathscr L_0\\otimes \\mathscr L$ is sufficiently ample. In particular we can always find on $X$ a sufficiently ample invertible sheaf.\n\\end{itemize}\n\\end{remark}\nPut $\\mathscr M=(f_\\ast\\mathscr L)^{\\vee}$ and let $\\mathbb P:=\\mathbb P_S(\\mathscr M)$ be the projective vector bundle associated to $\\mathscr M$, over $S$. Then we obtain the following base change diagram: \n\\begin{equation}\\label{diag_univ}\n\\begin{tikzcd}[row sep=large, column sep = huge]\n\\mathbb X:=\\mathbb P\\times_S X\\arrow[\"p_2\"]{r}\\arrow[\"p_1\"]{d} & X\\arrow[\"f\"]{d}\\\\\n\\mathbb P\\arrow[\"\\pi\"]{r} & S\n\\end{tikzcd}\n\\end{equation}\nConsider now the invertible sheaf $\\mathscr L_f:=p_1^\\ast\\mathscr O_\\mathbb P(1)\\otimes p_2^\\ast\\mathscr L$ on $\\mathbb X$. We want to construct a \\emph{canonical global section} $\\sigma$ of $\\mathscr L_f$. It is enough to find a canonical non-zero element in $\\mathscr L^{-1}_f$, because if $\\phi\\in\\Hom(\\mathscr O_X,\\mathscr L_f)=\\mathscr L^{-1}_f$ then we put $\\sigma:=\\phi_X(1)$.\nFirst of all we construct a surjective canonical morphism \n$$\\Psi:f^\\ast \\mathscr M\\to\\mathscr L \\,.$$ \nThanks to the properties of the pullback we have a canonical isomorphism $f^\\ast \\mathscr M \\cong (f^\\ast f_\\ast \\mathscr L)^\\vee$. Since $\\mathscr L$ is sufficiently ample, we have a canonical isomorphism $(f^\\ast f_\\ast \\mathscr L)^\\vee\\cong \\mathscr L^\\vee$. Moreover there is a surjective canonical map $\\mathscr L^\\vee\\to\\mathscr L$ given in the following way:\n\\begin{eqnarray*}\n\\Hom(\\mathscr O_X(U), \\mathscr L(U))& \\to& \\mathscr L(U)\\\\\n\\varphi_U &\\mapsto& \\varphi_U(1)\\,.\n\\end{eqnarray*}\nBy taking all compositions, we finally get our surjective $\\Psi$. We have to prove that $\\Psi$ induces a canonical element in $\\mathscr L^{-1}_f$ (in order to get $\\sigma$). Note that $\\mathscr L^{-1}_f$ is canonically isomorphic to $\\Hom(\\mathscr L^{-1}, {p_2}_\\ast p_1^\\ast \\mathscr O_{\\mathbb P}(1))$, but \n$${p_2}_\\ast p_1^\\ast \\mathscr O_{\\mathbb P}(1)=f^\\ast \\pi_\\ast\\mathscr O_{\\mathbb P}(1)=f^\\ast (\\mathscr M^\\vee)=(f^\\ast \\mathscr M)^\\vee\\,.$$\nWe conclude that the dual map of $\\Psi$ induces the non-zero element of $\\mathscr L^{-1}_f$ that we were searching for.\n\nFrom now on we will say that the section $\\sigma$ constructed above is the universal section relative to $\\mathscr L$. The following remark explains why we can use the universal section for our inductive step in the proof of uniqueness:\n\n\\begin{remark}\nIn \\cite[2.2]{Gar} it is shown that $\\sigma$ is a regular section, which is equivalent to say that the zero locus of $\\sigma$ (considered with its reduced scheme structure):\n$$Z(\\sigma):=\\{x\\in\\mathbb X\\colon 0=\\sigma(x)\\in \\mathscr L_f\/\\mathfrak m_x \\mathscr L_f\\}$$\nis a relative Cartier divisor on $\\mathbb X$. In this case we also have that $\\mathscr L_f$ is canonically isomorphic to $\\mathscr O_{\\mathbb X}(Z(\\sigma))$. Now consider the restriction \n$$p:=(p_1)|_{Z(\\sigma)}: Z(\\sigma )\\to \\mathbb P\\,;$$\nLet $U$ be the flat locus of $p$ and put $V:=p(U)$. Then $V$ is open in $\\mathbb P$, and we denote its closed complementar with $W$, then we conclude that \n\\begin{equation}\\label{section_map}\np: Z(\\sigma)-p^{-1}(W)\\to V\n\\end{equation}\nis flat of relative dimension $n-1$.\n\\end{remark}\n\n\n\n\n\n\n\nThe following theorem ensures the unicity of the Deligne pairing:\n\\begin{theorem}\nThe Deligne pairing is unique: given two sets of data $(\\left< \\,\\;,\\;,\\ldots,\\;\\right>^i,\\alpha^i,\\beta^i,\\gamma^i,\\delta^i)$, with $i=1,2$, satisfying the conditions of Definition \\ref{axiom_deligne}, there is a unique multi-additive morphism $\\left< \\,\\;,\\;,\\ldots,\\;\\right>^1\\to \\left< \\,\\;,\\;,\\ldots,\\;\\right>^2$ that transforms $\\alpha^i,\\beta^i,\\gamma^i,\\delta^i$ accordingly.\n\\end{theorem}\n\\proof\nWe proceed by induction on $n$. When $n=0$, the claim follows directly from axiom $(4)$. Let's work now with $n>0$; first of all we want a functorial isomorphism:\n\\begin{equation}\\label{iso_to_show}\n\\Psi(\\mathscr L_0,\\ldots, \\mathscr L_n):\\left<\\mathscr L_0,\\ldots,\\mathscr L_n\\right>^1_{X\/S}\\xrightarrow{\\cong}\\left<\\mathscr L_0,\\ldots,\\mathscr L_n\\right>^2_{X\/S}\\,.\n\\end{equation}\nLet's first construct it by assuming that one invertible sheaf $\\mathscr L=\\mathscr L_0$, is chosen sufficiently ample; we will denote it as $\\Psi'(\\mathscr L_0,\\ldots, \\mathscr L_n)$. Let's construct for $\\mathscr L$ the base change diagram (\\ref{diag_univ}), with the same notations. Then $\\sigma$ is the universal section of $\\mathscr L_f$ and we also have the map $p$ described in equation (\\ref{section_map}). Thanks to \\cite[Lemme 21.13.2]{EGAIV}, in order to give isomorphism (\\ref{iso_to_show}), it is enough to give a functorial isomorphism:\n\\begin{equation}\\label{iso_to_show1}\n(\\pi|_{V})^\\ast\\left<\\mathscr L, \\mathscr L_1,,\\ldots,\\mathscr L_n\\right>^1_{(\\mathbb X-p_1^{-1}(W))\/V}\\xrightarrow{\\cong}(\\pi|_{V})^\\ast\\left<\\mathscr L,\\mathscr L_1,\\ldots,\\mathscr L_n\\right>^2_{(\\mathbb X-p_1^{-1}(W))\/V}\\,.\n\\end{equation}\nwhere $V\\subset \\mathbb P$ is the image of the flat locus of $p$ (remember that $V$ is open) and $W=\\mathbb P-V$. Let's now put $q:=p_2|_{\\mathbb X-p_1^{-1}(W)}$. By applying axiom $(1)$, it is enough to get a functorial isomorphism:\n\\begin{equation}\\label{iso_to_show2}\n\\left^1_{(\\mathbb X-p_1^{-1}(W))\/V}\\xrightarrow{\\cong}\\left^2_{(\\mathbb X-p_1^{-1}(W))\/V}\\,.\n\\end{equation}\nNow remember that by definition of $\\mathscr L_f$ we have:\n$$q^\\ast\\mathscr L=\\mathscr L_f|_{(\\mathbb X-p_1^{-1}(W))}\\otimes (p_1^\\ast\\mathscr O_\\mathbb P(1))^{-1}$$ \nLet's put for simplicity of notations $\\mathscr M:=\\mathscr L_f|_{(\\mathbb X-p_1^{-1}(W))}$; by multi-additivity and axiom $(3)$ we only need to find a functorial isomorphism\n\\begin{equation}\\label{iso_to_show3}\n\\left<\\mathscr M,q^\\ast\\mathscr L_1\\ldots,q^\\ast\\mathscr L_n\\right>^1_{(\\mathbb X-p_1^{-1}(W))\/V}\\xrightarrow{\\cong}\\left<\\mathscr M,q^\\ast\\mathscr L_1\\ldots,q^\\ast\\mathscr L_n\\right>^2_{(\\mathbb X-p_1^{-1}(W))\/V}\\,.\n\\end{equation}\nAt this point put $Z'(\\sigma):=Z(\\sigma)-p_1^{-1}(W)$; thanks to axiom $(2)$, it is enough to find a functorial isomorphism: \n\\begin{equation}\\label{iso_to_show4}\n\\left<(q^\\ast\\mathscr L_1)|_{Z'(\\sigma)}\\ldots,q^\\ast(\\mathscr L_n)|_{Z'(\\sigma)}\\right>^1_{Z'(\\sigma)\/V}\\xrightarrow{\\cong}\\left<(q^\\ast\\mathscr L_1)|_{Z'(\\sigma)}\\ldots,q^\\ast(\\mathscr L_n)|_{Z'(\\sigma)}\\right>^2_{Z'(\\sigma)\/V}\\,.\n\\end{equation}\nThe relative dimension of the map $p:Z'(\\sigma)\\to V$ is now $n-1$ and we can apply the inductive hypothesis.\n\nWe still have to prove the the existence of $\\Psi(\\mathscr L_0,\\ldots, \\mathscr L_n)$ for a general $\\mathscr L_0$. For any invertible sheaf $\\mathscr L_0$ there exists a sufficiently ample one $\\mathscr M$ such that $\\mathscr L_0\\otimes\\mathscr M$ is again sufficiently ample. So we can put:\n$$\\Psi(\\mathscr L_0,\\ldots, \\mathscr L_n):=\\Psi'(\\mathscr L_0\\otimes \\mathscr M,\\ldots, \\mathscr L_n)\\otimes \\Psi'(\\mathscr M,\\ldots, \\mathscr L_n)^{-1}$$\nprovided that the construction doesn't depend on the choice of $\\mathscr M$. Such a claim is equivalent to show that $\\Psi'(\\cdot\\,,\\mathscr L_1,\\ldots, \\mathscr L_n)$ is additive with respect to sufficiently ample invertible sheaves.\n\nNow we consider two sufficiently ample invertible sheaves $\\mathscr L^{(i)}$ for $i=1,2$ and the associated diagrams:\n\\begin{equation}\\label{diag_univ1}\n\\begin{tikzcd}[row sep=large, column sep = huge]\n\\mathbb X^{(i)}:=\\mathbb P^{(i)}\\times_S X\\arrow[\"p^{(i)}_2\"]{r}\\arrow[\"p^{(i)}_1\"]{d} & X\\arrow[\"f\"]{d}\\\\\n\\mathbb P^{(i)}\\arrow[\"\\pi^{(i)}\"]{r} & S\n\\end{tikzcd}\n\\end{equation}\nwhere clearly $\\mathbb P^{(i)}:=\\mathbb P_S(\\mathscr M^{(i)})$ for $\\mathscr M^{(i)}:=(f_\\ast(\\mathscr L^{(i)}))^\\vee$. On the other hand if we put $\\mathscr L:=\\mathscr L^{(1)}\\otimes \\mathscr L^{(2)}$ and $\\mathbb P:=\\mathbb P_S((f_\\ast\\mathscr L^{(1)})^\\vee \\otimes (f_\\ast\\mathscr L^{(2)})^\\vee)$ we end up with the diagram (\\ref{diag_univ1}). There is a natural map \n$$\\iota: \\mathbb X^{(1)}\\times_X \\mathbb X^{(2)}\\to\\mathbb X$$\nLet $q_i$ the projections of $\\mathbb X^{(1)}\\times_X \\mathbb X^{(2)}$ on the factors $\\mathbb X^{(i)}$, then one can show that:\n$$\\iota^\\ast\\mathscr L_f=q_1^\\ast\\mathscr L^{(1)}_f\\otimes q_2^\\ast\\mathscr L^{(2)}_f\\,,$$\n$$\\iota^\\ast \\sigma= q_1^\\ast\\sigma^{(1)}\\otimes q_2^\\ast\\sigma^{(2)}\\,.$$\nFrom the properties of the universal extension discussed in \\cite[I.2]{Elk} the claim follows.\n\nIt remains to show that $\\Psi$ transforms $\\alpha^{1},\\beta^{1},\\gamma^{1},\\delta^{1}$ to $\\alpha^{2},\\beta^{2},\\gamma^{2},\\delta^{2}$. Let's do it for $\\alpha^{i}$, the other cases are similar. In particular we have to prove that, given a base change diagram as in axiom (1), we get a commutative diagram:\n\\begin{equation}\\label{diag_alpha}\n\\begin{tikzcd}[row sep=large, column sep = huge]\ng^\\ast\\left<\\mathscr L_0,\\ldots,\\mathscr L_n\\right>^{1}_{X\/S}\\arrow[\"\\cong\"]{r}\\arrow[\"\\cong\"]{d} & \\left^1_{X'\/S'}\\arrow[\"\\cong\"]{d}\\\\\ng^\\ast\\left<\\mathscr L_0,\\ldots,\\mathscr L_n\\right>^{2}_{X\/S}\\arrow[\"\\cong\"]{r} & \\left^2_{X'\/S'}\\,.\n\\end{tikzcd}\n\\end{equation}\nIn order to construct (\\ref{diag_alpha}) it is enough to proceed similarly as we did above: we work by induction on $n$. If $n=0$ the claim follows from the property of $\\delta^i$ with respect to base change. For the generic $n$ we can use the universal extension procedure described above and the properties of $\\beta^i$ with respect to base change to reduce to $n-1$.\n\\endproof\n\n\\subsection{Deligne pairing in terms of determinant of cohomology}\\label{det_del}\nIn this section we heavily use the proprieties of the determinant functor presented in appendix \\ref{det} in order to give an explicit expression of the Deligne pairing in terms of the determinant of cohomology.\n\n \nLet $f:X\\to S$ be a flat morphism between Noetherian irreducible schemes. If $\\catname{Vec}(X)$ is the category of locally free sheaves of finite rank on $X$, the first think to notice is that $\\det Rf_\\ast $ descends to a map on $K_0(\\catname{Vec}(X))$. Now we put:\n$$\n\\left\\{\\begin{array} {lll}\n\\left<\\mathscr L_0,\\ldots, \\mathscr L_n\\right>_{X\/S}:={\\det Rf_\\ast\\left(c_1(\\mathscr L^{-1}_0)c_1(\\mathscr L^{-1}_1)\\ldots c_1(\\mathscr L^{-1}_n)\\mathscr O_X\\right)}\\, & \\textrm{if } n>0\\\\\n&\\\\\n\\left< \\mathscr L_0\\right>_{X\/S}:={\\det Rf_\\ast\\left(c_1(\\mathscr L^{-1}_0)\\mathscr O_X\\right)}^{-1}=N_{X\/S}(\\mathscr L_0)& \\textrm{if } n=0\\\\\n\\end{array}\\right.\n$$\nWe want to show that this defines the Deligne pairing i.e. that there are some ``canonical'' natural transformations associated to $\\left<\\;,\\ldots, \\;\\right>_{X\/S}$ satisfying all axioms of definition \\ref{axiom_deligne}. \n\\begin{remark}\nWhen $n=1$, after some simple algebraic manipulations we obtain the expected result:\n$$\\left<\\mathscr L,\\mathscr M\\right>=\\det Rf_\\ast(c_1(\\mathscr L^{-1}) c_1(\\mathscr M^{-1})\\mathscr O_X)=\\det Rf_\\ast(c_1(\\mathscr L^{-1})(\\mathscr O_X-\\mathscr M))=$$\n$$=\\det Rf_\\ast(\\mathscr O_X)\\otimes\\det Rf_\\ast(\\mathscr L)^{-1}\\otimes\\det Rf_\\ast(\\mathscr M)^{-1}\\otimes\\det Rf_\\ast(\\mathscr L\\otimes\\mathscr M) $$\n\\end{remark}\n\nLike in the case of algebraic varieties, Proposition \\ref{propc_1} ensures that $\\left<\\;,\\ldots, \\;\\right>_{X\/S}$ is multi-monoidal and symmetric. Moreover axiom (4) of Definition \\ref{axiom_deligne} is trivially satisfied by definition (see Definition \\ref{norm}). So it remains to show that axioms (1)-(3) are satisfied. \n\n\n\\begin{proposition}[Axiom (1) holds]\nFor any commutative square given by a base change $g:S'\\to S$ which is proper, flat and with connected fibres \n\n$$\n\\begin{tikzcd}[row sep=large, column sep = huge]\nX'=X\\times_S S'\\arrow[\"f'\"]{r}\\arrow[\"g'\"]{d} & S'\\arrow[\"g\"]{d}\\\\\nX\\arrow[\"f\"]{r} & S\n\\end{tikzcd}\n$$\nthere is a natural transformation between multi-additive morphisms $\\alpha_{f,g}:\\catname{Pic}(X)^{n+1}\\to \\catname{Pic}(S')$ such that\n $$\\alpha_{f,g}:g^\\ast\\left<\\mathscr L_0,\\ldots,\\mathscr L_n\\right>_{X\/S}\\xrightarrow{\\cong}\\left_{X'\/S'}\\,.$$\n\\end{proposition}\n\\proof\nFirst of all we have that for any invertible sheaf $\\mathscr L$ on $X$ and any coherent sheaf $\\mathscr F'$ on $X'$:\n$$c_1(\\mathscr L)Rg'_\\ast\\mathscr F'=Rg'_\\ast (c_1(g'^\\ast\\mathscr L)\\mathscr F')$$\n(see for example the proof of \\cite[Lemma B.15]{Kl} for a detailed explanation of the above equality). Therefore\n$$c_1(\\mathscr L_0)\\ldots c_1(\\mathscr L_n)Rg'_\\ast(\\mathscr O_{X'})=c_1(\\mathscr L_0)\\ldots c_1(\\mathscr L_{n-1})Rg'_\\ast (c_1(g'^\\ast\\mathscr L_n)\\mathscr O_{X'})=\\ldots$$\n$$\\ldots=Rg'_\\ast(c_1(g'^\\ast\\mathscr L_0)\\ldots c_1(g'^\\ast\\mathscr L_n)\\mathscr O_{X'})\\,.$$\nIt means that:\n\n\\begin{equation}\\label{functor_mess}\ng^\\ast\\det Rf_\\ast\\left(c_1(\\mathscr L_0)\\ldots c_1(\\mathscr L_n)Rg'_\\ast(\\mathscr O_{X'})\\right)=g^\\ast\\det Rf_\\ast \\left(Rg'_\\ast(c_1(g'^\\ast\\mathscr L_0)\\ldots c_1(g'^\\ast\\mathscr L_n)\\mathscr O_{X'})\\right)\\,.\n\\end{equation}\nBut thanks to the properties of the morphism $g:S'\\to S$ we have that $g'_\\ast\\mathscr O_{X'}=\\mathscr O_X$ (see for example \\cite[Exercise 3.11]{Kol}, so it follows that the left hand side of equation (\\ref{functor_mess}) is \n$$g^\\ast\\det Rf_\\ast\\left(c_1(\\mathscr L_0)\\ldots c_1(\\mathscr L_n)\\mathscr O_X\\right)\\,.$$\n On the right hand side of equation (\\ref{functor_mess}) note that we have the composition of the following functors:\n\\begin{equation}\\label{functor_mess1}\ng^\\ast\\circ\\det{\\!}_{S}\\circ Rf_\\ast\\circ Rg'_\\ast\\,.\n\\end{equation}\nBy the properties of the determinant functor equation (\\ref{functor_mess1}) is naturally isomorphic to\n$$\n\\det{\\!}_{S'}\\circ Lg^\\ast \\circ Rf_\\ast\\circ Rg'_\\ast\\cong\\det{\\!}_S'\\circ Rf'_\\ast=\\det Rf'_\\ast\\,.\n$$\nIn other words we obtained that the right hand side of equation (\\ref{functor_mess}) is naturally isomorphic to \n$$\\det Rf'_\\ast\\left(c_1(g'^\\ast\\mathscr L_0)\\ldots c_1(g'^\\ast\\mathscr L_n)\\mathscr O_{X'}\\right)\\,.$$\n\\endproof\n\n\\begin{remark}\nFor axioms (2) and (3), we only have to show that the natural transformations exist, since their ``good behaviour'' with respect to base change is ensured by the properties of the determinant of cohomology with respect to base change i.e. equation (\\ref{det_bchange}).\n\\end{remark}\n\n\\begin{proposition}[Axiom (2) holds]\nWhen $n>0$ and $D\\in\\Div(X)$ is an effective relative Cartier divisor, there is a natural transformation between multi-additive morphisms $\\beta_{f,D}:\\catname{Pic}(X)^{n}\\to \\catname{Pic}(S)$ such that\n $$\\beta_{f,D}:\\left<\\mathscr O_X(D),\\ldots,\\mathscr L_n\\right>_{X\/S}\\xrightarrow{\\cong}\\left<\\mathscr L_1|_D,\\ldots,\\mathscr L_n|_D\\right>_{D\/S}\\,.$$\nMoreover such a transformation is natural with respect to base change.\n\\end{proposition}\n\\proof\nThis follows by the simple fact that $c_1(\\mathscr O_X(D))\\mathscr O_X=\\mathscr O_D$ (see Proposition \\ref{propc_1}(iii)).\n\\endproof\n\n\n\\begin{proposition}[Axiom (3) holds]\nWhen $n>0$ there is a natural transformation between multi-additive morphisms $\\gamma_{f}:\\catname{Pic}(S)\\times\\catname{Pic}(X)^{n}\\to \\catname{Pic}(S)$\n such that\n $$\\gamma_{f}:\\left_{X\/S}\\xrightarrow{\\cong}\\mathscr L^{(\\mathscr L_1|_{X_s}.\\mathscr L_2|_{X_s}.\\ldots\\mathscr L_n|_{X_s};\\mathscr O_{X_s})}$$\nwhere $X_s$ is a generic fibre of $f$. Moreover such a transformation is natural with respect to base change.\n\\end{proposition}\n\\proof\nLet's put $\\mathscr E:= c_1(\\mathscr L_1)\\ldots c_1(\\mathscr L_n)\\mathscr O_X$, then:\n$$\n\\det Rf_\\ast (c_1(f^\\ast \\mathscr L)c_1(\\mathscr L_1)\\ldots c_1(\\mathscr L_n)\\mathscr O_X)=\\det Rf_\\ast(c_1(f^\\ast \\mathscr L)\\mathscr E)=\\det Rf_\\ast(\\mathscr E-(f^\\ast \\mathscr L)^{-1}\\otimes \\mathscr E)=\n$$\n$$\n=\\det Rf_\\ast(\\mathscr E)\\otimes \\det Rf_\\ast((f^\\ast \\mathscr L)^{-1}\\otimes \\mathscr E)^{-1}=(\\ast)\n$$\nNow, thanks to Proposition \\ref{det_chi} the above chain of equalities can be continued in the following way through a canonical isomorphism:\n$$\n(\\ast)\\cong\\det Rf_\\ast(\\mathscr E)\\otimes \\det Rf_\\ast(\\mathscr E)^{-1}\\otimes \\mathscr L^{\\chi_S(\\mathscr E|_{X_s})}=\\mathscr L^{\\chi_S(\\mathscr E|_{X_s})}\n$$\nwhere $X_s$ is a generic fibre. In order to conclude, it is enough to notice that $\\mathscr E|_{X_s}=c_1(\\mathscr L_1|_{X_s})\\ldots c_1(\\mathscr L_n|_{X_s})\\mathscr O_{X_s}$. \n\n\\endproof\n\n\n\\begin{appendices}\n\n\n\n\n\\section{Determinant functor and determinant of cohomology}\\label{det}\n\nIn this section we briefly discuss, without proofs, the determinant functor by following \\cite{KM}. First we define the determinant for locally free sheaves, then we extend it to complexes of locally free sheaves and then we extend it further for perfect complexes. We will use some basic notions from the theory of derived category (see \\cite{Hoche} for a concise introduction.)\n\n\nWe fix a Noetherian irreducible scheme $X$. A \\emph{graded invertible sheaf} on $X$ is a couple $(\\mathscr L, \\alpha)$ where $\\mathscr L$ is an invertible sheaf on $X$ and $\\alpha: X\\to\\mathbb Z$ is a continuous function. A morphism of graded invertible sheaves $\\phi: (\\mathscr L, \\alpha)\\to(\\mathscr M,\\beta)$ is a morphism of invertible sheaves such that the following condition hold: for any $x\\in X$, if $\\alpha(x)\\neq \\beta(x)$, then $\\phi_x=0$. We denote with $\\catname{Gr}(X)$ the category of graded invertible sheaves, and $\\catname{isGr}(X)$ is the category whose objects are graded invertible sheaves and the morphisms are just the isomorphisms; note that $\\catname{isGr}(X)$ is a Picard groupoid. The tensor product (i.e. the group operation) between graded invertible sheaves is defined as $(\\mathscr L, \\alpha)\\otimes(\\mathscr M, \\beta)=(\\mathscr L\\otimes\\mathscr M,\\alpha+\\beta)$. The unit graded invertible sheaf is $(\\mathscr O_X,0)$. Furthermore we can define the isomorphism $\\tau:\\mathscr L\\otimes\\mathscr M\\to\\mathscr M\\otimes\\mathscr L$ such that locally and on pure tensors is given by:\n$$\\tau(l\\otimes m)=(-1)^{\\alpha\\beta}m\\otimes l\\,.$$\nLet $\\catname{Vec}(X)$ be the category of locally free sheaves on $X$ (of finite rank) and let $\\catname{isVec}(X)$ its subcategory where the morphisms are only the isomorphisms. \n\n For a locally free sheaf $\\mathscr E$ of rank $r$, we denote with the symbol $\\bigwedge^r\\mathscr E$ the sheafification of the following presheaf:\n$$U\\mapsto\\bigwedge^r\\mathscr E(U)\\,.$$\nThen we can construct graded invertible sheaf $\\det{\\!}^\\star\\mathscr E$ in the following way:\n$$\\det{\\!}^\\star\\mathscr E:=\\left(\\bigwedge^r\\mathscr E, r\\right)\\,.$$\nThen we have a functor:\n$$\\det{\\!}_X^\\star\\colon \\catname{isVec}(X)\\to\\catname{isGr}(X)\\,.$$\nFor any short exact sequence of locally free sheaves:\n\\begin{equation*}\n\\begin{tikzcd}\n0\\arrow[r]&\\mathscr H\\arrow[r,\"\\alpha\"]&\\mathscr F\\arrow[r,\"\\beta\"]&\\mathscr G\\arrow[r]& 0\\\n\\end{tikzcd}\n\\end{equation*} \nthere is an isomorphism of graded invertible sheaves\n\\begin{equation*}\n\\begin{aligned}\ni_X^\\star(\\alpha,\\beta):\\det{\\!}_X^\\star(\\mathscr H)\\otimes\\det{\\!}_X^\\star(\\mathscr G)\\xrightarrow{\\cong} \\det{\\!}_X^\\star(\\mathscr F)\n\\end{aligned}\n\\end{equation*}\nthat locally is given in the following way: assume that $\\mathscr H$ has rank $r$ and $\\mathscr G$ has rank $s$, then for any local sections $h_i\\in\\mathscr H(U)$ and $\\beta(f_i)\\in\\mathscr G(U)$, for $f_i\\in\\mathscr F(U)$ we have:\n\n$$i_X^\\star(\\alpha,\\beta)((h_1\\wedge\\ldots\\wedge h_r)\\otimes(\\beta(f_1)\\wedge\\ldots\\wedge\\beta(f_s)))=\\alpha(h_1)\\wedge\\ldots\\wedge \\alpha(h_r)\\wedge f_1\\wedge\\ldots\\wedge f_n\\,.$$\nWe are ready to give the definition of determinant for bounded complexes of locally free sheaves.\n\\begin{definition}\\label{def_det}\nLet $\\catname{isVec}_b^\\bullet(X)$ be the category of bounded complexes in $\\catname{Vec}(X)$ where the morphism are just the quasi-isomorphisms between complexes. Then a \\emph{determinant functor on $X$} consists of the following data:\n\\begin{enumerate}\n\\item[$(1)$] A functor $\\mathfrak f_X:\\catname{isVec}_b^\\bullet(X)\\to\\catname{isGr(X)}$\n\\item[$(2)$] For any short exact sequence in $\\catname{isVec}_b^\\bullet(X)$:\n\\begin{equation*}\n\\begin{tikzcd}\n0\\arrow[r]&\\mathscr H^\\bullet\\arrow[r,\"\\alpha\"]&\\mathscr F^\\bullet\\arrow[r,\"\\beta\"]&\\mathscr G^\\bullet\\arrow[r]& 0\\\n\\end{tikzcd}\n\\end{equation*}\nan isomorphism:\n\n\\begin{equation*}\n\\begin{aligned}\ni_X(\\alpha,\\beta):\\mathfrak f_X(\\mathscr H^\\bullet)\\otimes\\mathfrak f_X(\\mathscr G^\\bullet)\\xrightarrow{\\cong} \\mathfrak f_X(\\mathscr F^\\bullet)\n\\end{aligned}\n\\end{equation*}\n\n\\end{enumerate} \n\nMoreover $(\\mathfrak f_X, i_X)$ satisfies the following conditions:\n\\begin{enumerate}\n\\item[$(i)$] Given a commutative diagram in $\\catname{isVec}_b^\\bullet(X)$:\n\\begin{equation*}\n\\begin{tikzcd}\n0\\arrow[r]&\\mathscr H^\\bullet\\arrow[r, \"\\alpha\"]\\arrow[d,\"\\lambda'\"]&\\mathscr F^\\bullet\\arrow[r, \"\\beta\"]\\arrow[d, \"\\lambda\"]&\\mathscr G^\\bullet\\arrow[r]\\arrow[d,\"\\lambda''\"]& 0\\\\\n0\\arrow[r]&\\mathscr H_1^\\bullet\\arrow[r, \"\\alpha_1\"]&\\mathscr F_1^\\bullet\\arrow[r, \"\\beta_1\"]&\\mathscr G_1^\\bullet\\arrow[r]& 0\\\n\\end{tikzcd}\n\\end{equation*}\nsuch that the rows are short exact sequences, then the following diagram commutes\n$$\n\\begin{tikzcd}[row sep=large, column sep = large]\n\\mathfrak f_X({\\mathscr H}^\\bullet)\\otimes \\mathfrak f_X({\\mathscr G}^\\bullet)\\arrow[\"{i_X(\\alpha,\\beta)}\"]{r}\\arrow[\"\\mathfrak f_X(\\lambda')\\otimes \\mathfrak f_X(\\lambda'')\"]{d} & \\mathfrak f_X({\\mathscr F}^\\bullet)\\arrow[\"\\mathfrak f_X(\\lambda)\"]{d}\\\\\n\\mathfrak f_X({\\mathscr H_1}^\\bullet)\\otimes \\mathfrak f_X({\\mathscr G_1}^\\bullet)\\arrow[\"{i_X(\\alpha_1,\\beta_1)}\"]{r} & \\mathfrak f_X({\\mathscr F_1}^\\bullet)\n\\end{tikzcd}\n$$\n\n\\item[$(ii)$] Given a commutative diagram in $\\catname{isVec}_b^\\bullet(X)$:\n$$\n\\begin{tikzcd}\n & 0\\arrow{d} & 0\\arrow{d} & 0\\arrow{d} &\\\\\n 0 \\arrow{r} & \\mathscr H_1^\\bullet \\arrow[r,\"\\alpha_1\"]\\arrow[d,\"\\gamma_1\"] & \\mathscr F_1^\\bullet \\arrow[r,\"\\beta_1\"]\\arrow[d,\"\\gamma\"] & \\mathscr G_1^\\bullet \\arrow{r}\\arrow[d,\"\\gamma_2\"] & 0\\\\\n 0 \\arrow{r} & {\\mathscr H}^\\bullet \\arrow[r,\"\\alpha\"]\\arrow[d,\"\\delta_1\"] & {\\mathscr F}^\\bullet \\arrow[r,\"\\beta\"]\\arrow[d, \"\\delta\"] & {\\mathscr G}^\\bullet \\arrow{r}\\arrow[d,\"\\delta_2\"] & 0\n\\\\\n 0 \\arrow{r} & {\\mathscr H_2}^\\bullet \\arrow[r,\"\\alpha_2\"]\\arrow[d] & {\\mathscr F_2}^\\bullet \\arrow[r,\"\\beta_2\"]\\arrow[d] & {\\mathscr G_2}^\\bullet \\arrow{r}\\arrow[d] & 0\\\\\n & 0 & 0 & 0 &\n\\end{tikzcd}\n$$\nsuch that rows and columns are a short exact sequence, then the following diagram commutes\n$$\n\\begin{tikzcd}[row sep=huge, column sep = 4.5cm]\n\\mathfrak f_X({\\mathscr H_1}^\\bullet)\\otimes \\mathfrak f_X({\\mathscr H_2}^\\bullet)\\otimes\\mathfrak f_X({\\mathscr G_1}^\\bullet)\\otimes \\mathfrak f_X({\\mathscr G_2}^\\bullet)\\arrow[\"{i_X(\\gamma_1,\\gamma_2)\\otimes i_X(\\delta_1,\\delta_2)}\"]{r}\\arrow[\"{i_X(\\alpha_1,\\beta_1)\\otimes i_X(\\alpha_2,\\beta_2)}\"]{d} & \\mathfrak f_X({\\mathscr H}^\\bullet)\\otimes \\mathfrak f_X({\\mathscr G}^\\bullet)\\arrow[\"{i_X(\\alpha,\\beta)}\"]{d}\\\\\n\\mathfrak f_X({\\mathscr F_1}^\\bullet)\\otimes \\mathfrak f_X({\\mathscr F_2}^\\bullet)\\arrow[\"{i_X(\\gamma,\\delta)}\"]{r} & \\mathfrak f_X({\\mathscr F}^\\bullet)\n\\end{tikzcd}\n$$\n\\item[$(iii)$] $\\mathfrak f_X$ and $i_X(\\,,\\,)$ both commute with the base change of $X$. The explicit expression of such a property is the following: fix a morphism of schemes $\\psi:Y\\to X$ and let $L\\psi^\\ast:\\catname D_-(\\catname{ QCoh}(X))\\to\\catname D_-(\\catname {QCoh}(Y))$ be the left derived functor of to the pullback $\\psi^\\ast$; then the following properties hold:\n\\begin{enumerate}\n\\item[\\,\\begin{picture}(-1,1)(-1,-3)\\circle*{3}\\end{picture}\\ ] There is a natural transformation between functors: $\\eta(\\psi)\\colon\\mathfrak f_Y\\circ L\\psi^\\ast\\xrightarrow{\\cong}\\psi^\\ast\\circ\\mathfrak f_X$.\n\\item[\\,\\begin{picture}(-1,1)(-1,-3)\\circle*{3}\\end{picture}\\ ] For any short exact sequence in $\\catname{isVec}_b^\\bullet(Y)$:\n\\begin{equation*}\n\\begin{tikzcd}\n0\\arrow[r]&\\mathscr H^\\bullet\\arrow[r,\"\\alpha\"]&\\mathscr F^\\bullet\\arrow[r,\"\\beta\"]&\\mathscr G^\\bullet\\arrow[r]& 0\\\n\\end{tikzcd}\n\\end{equation*}\nthen the following diagram commutes\n$$\n\\begin{tikzcd}[row sep=large, column sep = huge]\n\\mathfrak f_Y(L\\psi^\\ast{\\mathscr H}^\\bullet)\\otimes\\mathfrak f_Y(L\\psi^\\ast{\\mathscr G}^\\bullet)\\arrow[\"{i_Y(L\\psi^\\ast(\\alpha,\\beta))}\"]{r}\\arrow[\"\\eta(\\psi)(\\mathscr H^\\bullet)\\otimes \\eta(\\psi)(\\mathscr G^\\bullet)\"]{d} & \\mathfrak f_Y(L\\psi^\\ast{\\mathscr F}^\\bullet)\\arrow[\"\\eta(\\psi)(\\mathscr F^\\bullet)\"]{d}\\\\\n\\psi^\\ast\\mathfrak f_X({\\mathscr H}^\\bullet)\\otimes \\psi^\\ast\\mathfrak f_X({\\mathscr G}^\\bullet)\\arrow[\"{\\psi^\\ast i_X(\\alpha,\\beta)}\"]{r} & \\psi^\\ast\\mathfrak f_X({\\mathscr F}^\\bullet)\n\\end{tikzcd}\n$$\n\\end{enumerate}\n\n\\item[$(iv)$] $\\mathfrak f_X(0)=(\\mathscr O_X, 0)$. Moreover for the short exact sequence:\n\\begin{equation*}\n\\begin{tikzcd}\n0\\arrow[r]&\\mathscr F^\\bullet\\arrow[r,\"\\id\"]&\\mathscr F^\\bullet\\arrow[r,\"0\"]&0\\arrow[r]& 0\\\n\\end{tikzcd}\n\\end{equation*}\nwe have that $i_X(\\id,0)\\colon \\mathfrak f_X(\\mathscr F^\\bullet)\\otimes(\\mathscr O_X,0)\\to \\mathfrak f_X(\\mathscr F^\\bullet)$ is the canonical map (i.e. the ``projection on the first component'').\n\\item[$(v)$] If we canonically identify $\\catname{isVec}(X)$ as a subcategory of $\\catname{isVec}_b^\\bullet(X)$, then $\\mathfrak f_X$ restricts to $\\det{\\!}_X^\\star$ and $i_X(\\,,\\,)$ restricts to $i^\\star_X(\\,,\\,)$.\n\\end{enumerate}\n\\end{definition}\n\n\n\n\n\\begin{theorem}\nUp to natural transformation of functors there exists a unique determinant $(\\mathfrak f_X, i_X)$ on $X$. It will be denoted as $(\\det_X, i_X)$.\n\\end{theorem}\n\\proof\nSee \\cite[Theorem 1]{KM} for a complete proof. Here we write down just the explicit expressions for $\\det_X$:\n\\begin{equation}\n\\det{\\!}_X(\\mathscr F^\\bullet)=\\bigotimes_i\\det{\\!}^\\star_X(\\mathscr F^i)^{(-1)^i}\\,.\n\\end{equation}\n\n\\endproof\n\nThe category $\\catname{isVec}_b^\\bullet(\\cdot)$ is quite restrictive, for example it doesn't behave well with respect to the pushforward functor. Therefore, we would like to have a determinant functor for a more general category.\n\\begin{definition}\nA complex $\\mathscr F^\\bullet $ of $\\mathscr O_X$-modules is said \\emph{perfect} if for any $x\\in X$ there exist an open neighbourhood $U\\ni x$, a complex $\\mathscr G^\\bullet $ in $\\catname {Vec}^\\bullet_b(U)$ and a quasi isomorphism of complexes of $\\mathscr O_U$-modules $\\mathscr G^\\bullet\\to\\mathscr F^\\bullet_{|U}$. The category of perfect complexes on $X$ is denoted by $\\catname{Perf}(X)$, whereas $\\catname{isPerf}(X)$ denotes the category of perfect complexes where the morphisms are just the quasi-isomorphisms.\n\\end{definition}\n\\begin{theorem}\nThe determinant $(\\det_X,i_X)$ can be extended uniquely, up to natural transformation, to a determinant on $\\catname{isPerf}(X)$. We will denote this extension again with the symbol $(\\det_X,i_X)$ and formally it is the datum of:\n\\begin{enumerate}\n\\item[$(1)$] A functor $\\det_X\\colon\\catname{isPerf}(X)\\to\\catname{isGr(X)}$\n\\item[$(2)$] For any short exact sequence in $\\catname{isPerf}(X)$:\n\\begin{equation*}\n\\begin{tikzcd}\n0\\arrow[r]&\\mathscr H^\\bullet\\arrow[r,\"\\alpha\"]&\\mathscr F^\\bullet\\arrow[r,\"\\beta\"]&\\mathscr G^\\bullet\\arrow[r]& 0\\\n\\end{tikzcd}\n\\end{equation*}\nan isomorphism:\n\\begin{equation*}\n\\begin{aligned}\ni_X(\\alpha,\\beta):\\det{\\!}_X(\\mathscr H^\\bullet)\\otimes\\det{\\!}_X(\\mathscr G^\\bullet)\\xrightarrow{\\cong} \\det{\\!}_X (\\mathscr F^\\bullet)\n\\end{aligned}\n\\end{equation*}\n\\end{enumerate} \nMoreover the properties $(i)-(v)$ listed in definition \\ref{def_det} are satisfied in $\\catname{isPerf}(X)$. \n\\end{theorem}\n\\proof\nSee \\cite[Theorem 2]{KM}.\n\\endproof\nOne of the most important applications of the determinant functor appears in arithmetic geometry if we consider its interaction with the usual pushfoward functor.\n\n\\begin{definition}\nLet $f:X\\to S$ be a flat morphism between irreducible Noetherian schemes and let $\\mathscr E$ be a locally free sheaf of finite rank on $X$. It is well known (\\cite[Exp. 3, Proposition 4.8]{SGA}) that the complex $Rf_\\ast\\mathscr E$ induced by the right derived functor of $f_\\ast$ is a perfect complex on $S$. Then, just by composing $Rf_\\ast$ with $\\det_S$ it is possible to define the functor: \n$$\\det Rf_\\ast:=\\det{\\!}_S\\circ Rf_\\ast : \\catname{isVec}(X)\\to \\catname{isGr}(S)$$\nwhich is called the \\emph{determinant of cohomology (relative to $f$)}. Very often, for simplicity we want to forget about the graduation on the target of the determinant of cohomology, so it becomes a functor $\\catname{isVec}(X)\\to \\catname{Pic}(S)$.\n\\end{definition}\n\nSince the right derived functor $Rf_\\ast$ is exact in the derived sense (see \\cite{Hoche}) it is not hard to show that for any short exact sequence of locally free sheaves on $X$\n\\begin{equation*}\n\\begin{tikzcd}\n0\\arrow[r]&\\mathscr H\\arrow[r,\"\\alpha\"]&\\mathscr F\\arrow[r,\"\\beta\"]&\\mathscr G\\arrow[r]& 0\\\n\\end{tikzcd}\n\\end{equation*} \nthere is an isomorphism of graded invertible sheaves\n\\begin{equation*}\n\\begin{aligned}\ni_f(\\alpha,\\beta):\\det Rf_\\ast\\mathscr H\\otimes\\det Rf_\\ast\\mathscr G\\xrightarrow{\\cong} \\det Rf_\\ast\\mathscr F\n\\end{aligned}\n\\end{equation*}\nMoreover the whole construction behaves well with respect to flat base change in the following sense: assume that the following commutative square is given by a flat base change from $S$ to $S'$\n\n$$\n\\begin{tikzcd}[row sep=large, column sep = huge]\nX'=X\\times_S S'\\arrow[\"f'\"]{r}\\arrow[\"g'\"]{d} & S'\\arrow[\"g\"]{d}\\\\\nX\\arrow[\"f\"]{r} & S\n\\end{tikzcd}\n$$\nthen for any locally free sheaf $\\mathscr E$ on $X$ we have\n\n\\begin{equation}\\label{det_bchange}\ng^\\ast(\\det Rf_\\ast \\mathscr E)\\cong \\det Rf'_\\ast(g'^\\ast\\mathscr E)\\,.\n\\end{equation}\nWe will also need an important property of the determinant of cohomology:\n\n\\begin{proposition}\\label{det_chi}\nLet $f:X\\to S$ be a flat morphism between irreducible Noetherian schemes and let $\\mathscr E$ be a locally free sheaf of finite rank on $X$. Moreover let $\\mathscr L$ be an invertible sheaf on $S$. Then there is a canonical isomorphism between invertible sheaves on $S$:\n$$\\det Rf_\\ast(f^\\ast \\mathscr L\\otimes \\mathscr E)\\xrightarrow{\\cong} \\mathscr L^{\\otimes \\chi_S(\\mathscr E|_{X_s})}\\otimes \\det Rf_\\ast(\\mathscr E)$$\nwhere $X_s$ is a generic fibre of $f$ .\n\\end{proposition}\n\\proof\nSee \\cite{KM}.\n\\endproof\n\n\\begin{definition}\\label{norm}\nLet $\\varphi:X\\to S$ be a finite morphism between irreducible Noetherian schemes, then \\emph{the norm of $\\varphi$} is defined as the functor:\n\\begin{eqnarray*}\nN_{\\varphi}=N_{X\/S}:\\catname{Pic}(X) &\\to & \\catname{Pic}(S)\\\\\n\\mathscr L &\\mapsto & \\det R\\varphi_\\ast \\mathscr L\\otimes (\\det R\\varphi_\\ast \\mathscr O_X)^{-1}\\,.\n\\end{eqnarray*} \n\\end{definition}\n\n\n\\section{Original construction of Deligne pairing}\\label{ori_constr}\nIn this section we give all details of the construction of the Deligne pairing described in \\cite{Del}.\n\n\n$S$ is an irreducible Dedekind scheme and we put $K:=K(S)$. $\\varphi:X\\to S$ is a $S$-scheme satisfying the following properties:\n\\begin{enumerate}\n\\item[\\,\\begin{picture}(-1,1)(-1,-3)\\circle*{3}\\end{picture}\\ ] $X$ is two dimensional, integral, and regular. The generic point of $X$ is $\\eta$ and the function field of $X$ is denoted by $K(X)$.\n\\item[\\,\\begin{picture}(-1,1)(-1,-3)\\circle*{3}\\end{picture}\\ ] $\\varphi$ is proper and flat.\n\\item[\\,\\begin{picture}(-1,1)(-1,-3)\\circle*{3}\\end{picture}\\ ] The generic fibre, denoted by $X_K$, is a geometrically integral, smooth, projective curve over $K$. \n\\end{enumerate} \nWe say that $X$ is an \\emph{arithmetic surface over $S$}. \n\nIn this section we will also need recall the norm operator $\\mathcal N$ in dimension $1$ and $2$ (it is formally different from the norm of an invertible sheaf defined above). \n\\begin{definition}\nlet $C$ be a projective, non-singular curve over a field $k$, then for a closed point $x\\in C$ and any non-zero rational function $f\\in K(C)^\\times$ we put\n\\begin{equation}\\label{N_oncurves}\n\\mathcal N_x(f):=N_{k(x)|k} \\left(f(x)\\right)\\,,\n\\end{equation}\nwhere $f(x)$ is the obvious element of $k(x)$ associated to $f$. So if $D=\\sum_{x\\in X}v_x(D)[x]\\in \\Div(X)$ and $f\\in k(X)^\\times$ is a non-zero rational function such that $(f)$ and $D$ have no common components, then it is well defined the following element:\n$$ \\mathcal N_D(f):=\\prod_{x\\in X} \\mathcal N_x(f)^{v_x(D)}\\quad\\in k^\\times$$\n\\end{definition}\nThe well known Weil reciprocity law says that:\n$$\\mathcal N_{(g)}(f)=\\mathcal N_{(f)}(g)\\rlap.$$\n\n\\noindent Coming back to our arithmetic surface $\\varphi:X\\to S$, consider \n $$\\Upsilon:=\\Set{(D,E)\\in\\Div(X)\\times\\Div(X)\\colon\\textrm{$D$ and $E$ have no common components}}\\rlap,$$\nand note that if $(D_j,E_j)\\in\\Upsilon$ with $j=1,2$, then $(D_1+D_2,E_1+E_2)\\in\\Upsilon$. \n\\begin{definition}\\label{def1}\nLet $(D,E)\\in \\Upsilon$ such that $D$ and $E$ are both effective, then for any closed point $x\\in X$ we put:\n\n$$i_x(D,E):= \\len_{\\mathscr O_{X,x}}\\mathscr O_{X,x}\/\\left (\\mathscr O_X(-D)_x+\\mathscr O_X(-E)_x\\right)\\,.$$\nThis is called the \\emph{local intersection number} of $D$ and $E$ at $x$. \n\\end{definition}\nThe local intersection number assigns the multiplicity of the intersection at each point of $X$, and the following basic result summarizes its naive properties. \n\\begin{proposition}\\label{prop_loc_int}\nLet $(E,D)\\in \\Upsilon$ and $(E_j,D_j)\\in\\Upsilon$ with $j=1,2$ such that all the divisors are effective, then\n\\begin{enumerate} \n\\item[$(1)$]$i_x(D,E)=i_x(E,D)$.\n\\item[$(2)$]$\\displaystyle i_x(D_1+D_2,E_1+E_2)=\\sum_{j,k=1}^2 i_x(D_j,E_k)$.\n\\item[$(3)$] $i_x(D,E)\\neq 0$ if and only if $x\\in\\supp(D)\\cap\\supp(E)$.\n\\item[$(4)$] If $x\\in E$, $i_x(D,E)=\\mult_x(D|_E)$.\n\\end{enumerate}\n\\end{proposition}\n\\proof\n$(1)$ and $(3)$ are obvious. For $(2)$ and $(4)$ see \\cite[lemma 9.1.4]{Liu}.\n\\endproof\nAny divisor $D\\in\\Div(X)$ can be written in a unique way as $D=D_+-D_-$ where both $D_+$ and $D_-$ are effective and if $(D,E)\\in\\Upsilon$, then $(D_\\pm,E_\\pm)\\in\\Upsilon$. We can use definition \\ref{def1} in order to have the local intersection at $x$ of $D$ and $E$ when $(D,E)$ is any element of $\\Upsilon$ (so not necessarily effective):\n$$i_x(D,E):=i_x(D_+,E_+)-i_x(D_+,E_-)-i_x(D_-,E_+)+i_x(D_-,E_-)\\,.$$\n\\begin{definition}\\label{def2}\nLet $(D,E)$ be an element of $\\Upsilon$, then we define the $0$-cycle on $X$ given by:\n$$i(D,E):=\\sum_{x\\in X^{(0)}}i_x(D,E)[x]\\rlap,$$\nwhere here $[x]$ is a shorthand of $[\\overline{\\{x\\}}].$\n\\end{definition}\n\\begin{remark}\nThe sum in definition \\ref{def2} is finite because if $D$ and $E$ are effective without common components, then $i_x(D,E)=\\mult_x(D|_E)$ (proposition \\ref{prop_loc_int}(4)) and there is only a finite number of points on $E$ at which the divisor $D|_E$ has non-zero multiplicity. \n\\end{remark}\n\\begin{proposition}\nIf $(D,E),(D_j,E_j)\\in\\Upsilon$ with $j=1,2$, then the following properties hold for $i(D,E)$:\n\\begin{enumerate}\n\\item[\\,\\begin{picture}(-1,1)(-1,-3)\\circle*{3}\\end{picture}\\ ] $i(D,E)=i(E,D)$ (symmetry)\\,.\n\\item[\\,\\begin{picture}(-1,1)(-1,-3)\\circle*{3}\\end{picture}\\ ] $\\displaystyle i(D_1+D_2,E_1+E_2)=\\sum_{j,k=1}^2 i(D_j,E_k)$ (bilinearity)\\,.\n\\end{enumerate}\n\\end{proposition}\n\\proof\nIt follows immediately from proposition \\ref{prop_loc_int}.\n\\endproof\n\\begin{definition}\\label{finite_int_numb}\nWe have the symmetric and bilinear pairing on $\\Upsilon$:\n\\begin{eqnarray*}\n\\Upsilon &\\to& \\Div(S)\\\\\n(D,E) &\\mapsto& \\left\n\\end{eqnarray*}\nwhere\n$$\\left:=\\varphi_\\ast i(D,E)=\\sum_{x\\in X} [k(x):k(\\varphi(x))]\\,i_{x}(D,E)\\, [\\varphi(x)]\\,.$$\n\\end{definition}\n\nLet $\\Gamma$ be a prime divisor of $X$ with generic point $\\gamma$ and consider a non-zero rational function $f\\in K(X)^\\times$ such that $(f)$ and $\\Gamma$ have no common components, then define $\\mathcal N_{\\Gamma}(f)\\in K^\\times$ in the following way:\n$$\\mathcal N_{\\Gamma}(f):=\\left\\{\\begin{array} {cc}\nN_{K(\\Gamma)|K}(f|_{\\Gamma}) & \\textrm{if $\\Gamma$ is horizontal}\\\\\n1 & \\textrm{if $\\Gamma$ is vertical}\\\\\n\\end{array}\\right.\n$$\nwhere $N_{K(\\Gamma)|K}$ is the usual field norm and $f|_{\\Gamma}$ is defined as follows: since $(f)$ and $\\Gamma$ have no common components it follows that $v_{\\gamma}(f)=0$, that is $f\\in\\mathscr O_{X,\\gamma}^\\times$. So $f|_{\\Gamma}$, is the natural image of $f$ in $k(\\gamma)=K(\\Gamma)$. At this point for any $D=\\sum_i n_i\\Gamma_i\\in \\Div(X)$ such that $D$ and $(f)$ have no common components we have:\n$$\\mathcal N_{D}(f):=\\prod_i\\mathcal N_{\\Gamma_i}(f)^{n_i}\\quad \\in K^\\times$$\nSince $K(X)$ is the function field of any open subscheme $U\\subseteq X$ and of $X_K$ we can restrict the operator $\\mathcal N_\\ast(\\cdot)$ to $U$ and to $X_K$.\n\n\\begin{proposition}\\label{restr_prop}\nLet $f\\in K(X)^\\times$ and let $D\\in Div(X)$ such that $(f)$ and $D$ have no common components, then the following claims hold:\n\\begin{itemize}\n\\item[$(1)$] Let $U\\subseteq X$ be an open subscheme, then $\\mathcal N_{D|_U}(f)=\\mathcal N_D(f)$.\n\n\\item[$(2)$] $\\mathcal N_{D|_{X_K}}(f)=\\mathcal N_D(f)$, where the left hand side is the one-dimensional operator defined in equation (\\ref{N_oncurves}).\n\n\\end{itemize}\n\\end{proposition}\n\\proof\nIn both items we can restrict to the case when $D=\\Gamma$ is an irreducible horizontal divisor.\\\\\n$(1)$ The function fields and the generic points of $\\Gamma$ and $\\Gamma|_U$ coincide, so the claim follows trivially.\\\\\n$(2)$ Let $\\gamma\\in X_K$ be the generic point of $\\Gamma$, it is a closed point of $X_K$ such that $k(\\gamma)=K(\\Gamma)$. By the bare definitions we can check the required equality. \n\\endproof\n\n\\begin{proposition}\\label{intwithrat}\nLet $f\\in K(X)^\\times$ and let $D\\in\\Div(X)$ a divisor such that $D$ and $(f)$ have no common components, then\n$$\\left=\\left(\\mathcal N_D(f)\\right)\\;\\in \\Princ(S)\\,.$$ \n\\end{proposition}\n\\proof\nSee \\cite[Proposition 4.3]{Mor}.\n\\endproof\n\n\n\n\nNow we will construct the Deligne pairing and see the relation with the pairing $\\left$ for divisors. We divide the construction in two steps:\n\n\\emph{Step 1.} Definition of the $K$-vector space $\\left<\\mathscr L,\\mathscr M\\right>_K$.\\\\\n\nConsider the sets:\n\n\\begin{alignat*}{2}\n \\Upsilon_K &:= \\biggl\\{(D,E)\\in \\Div(X)\\times\\Div(X) &&\\;\\colon\\; \\pctext{2in}{$D|_{X_K}$ and $E|_{X_K}$ have no common components (as divisors on $X_K$)}\\biggr\\}\\rlap, \n\\end{alignat*}\n\\begin{alignat*}{2}\n \\Sigma_K &:= \\biggl\\{(l,m) &&\\;\\colon\\; \\pctext{3.5in}{ $l$ and $m$ are non-zero meromorphic sections of $\\mathscr L$ and $\\mathscr M$ such that $(\\divi(l),\\divi(m))\\in \\Upsilon_K$}\\biggr\\}\\rlap. \n\\end{alignat*}\nNote that $\\Upsilon_K$ is just the set of couple of divisors with no common horizontal components. Now we define some vector spaces over $K$.\n$$V:= K^{(\\Sigma_K)}\\,,$$\nnamely $V$ is the free $K$-vector space over $\\Sigma_K$. \n\\begin{equation}\\label{rel1}\nW':=\\left\\{(fl,m)-\\mathcal N_{\\divi(m)|_{X_K}}(f)\\cdot(l,m)\\colon f\\in K(X)^\\times, (l,m),(fl,m)\\in \\Sigma_K\\right\\}\\,,\n\\end{equation} \n\n\\begin{equation}\\label{rel2}\nT':=\\left\\{(l,gm)-\\mathcal N_{\\divi(l)|_{X_K}}(g)\\cdot(l,m)\\colon g\\in K(X)^\\times, (l,m),(l,gm)\\in \\Sigma_K \\right\\}\\,.\n\\end{equation} \nNote that the above ``$\\mathcal N_{\\ast}(\\cdot)$'' is the one-dimensional operator of definition \\ref{N_oncurves} considered on the curve $X_K$.\n\n\\begin{remark}\n$\\mathcal N_{\\divi(m)|_{X_K}}(f)$ and $\\mathcal N_{\\divi(l)|_{X_K}}(g)$ are well defined since $(l,m)$, $(fl,m)$, $(l,gm) \\in \\Sigma_K$, so $\\divi(m)|_{X_K}$ and $(f)$ have no common components. The same holds for $\\divi(l)|_{X_K}$ and $(g)$.\n\\end{remark}\n\nDefine the free vector spaces $W:= K^{(W')}$ and $T:=K^{(T')} $; moreover put \n$$\\left<\\mathscr L,\\mathscr M\\right>_K:=V\/(W+T)\\,\\rlap,$$\nwhich is considered as a constant sheaf (of $K$-vector spaces) over $X$. The natural image of any element $(l,m)\\in \\Sigma_K\\subset V$ in $\\left <\\mathscr L,\\mathscr M\\right>_K$ is denoted as $\\left _K$.\\\\\n\\begin{proposition}\\label{sstep1}\n $\\left<\\mathscr L,\\mathscr M\\right>_K$ is a one-dimensional vector space over $K$.\n\\end{proposition}\n\\proof\nFix $(l_0,m_0)\\in\\Sigma_K$, then for any $(l,m)\\in\\Sigma_K$ there are two elements $f_0,g_0\\in K(X)^\\times$ such that $l=f_0l_0$, $m=g_0m_0$ and moreover:\n$$((f_0),(g_0)),\\;((f_0),\\;\\divi(m_0)),\\;((g_0),\\divi(l_0))\\in\\Upsilon_K\\,.$$\nBy equations (\\ref{rel1}) and (\\ref{rel2}), in $\\left<\\mathscr L,\\mathscr M\\right>_K$ we can write:\n\\begin{equation}\\label{rel3}\n\\left_K= \\left_K=[f_0,g_0]\\,\\mathcal N_{\\divi(m_0)|_{X_K}}(f_0)\\,\\mathcal N_{\\divi(l_0)|_{X_K}}(g_0)\\left_K\\rlap.\n\\end{equation}\nwhere, in order to simplify the notations, we put $[f_0,g_0]:=\\mathcal N_{(f_0)}(g_0)$ intended as operation on the curve $X_K$. This shows that $\\left<\\mathscr L,\\mathscr M\\right>_K$ has dimension at most $1$ over $K$.\nDefine the homomorphism of $K$-vector spaces:\n$$\\theta:V\\to K$$\nsuch that\n$$\\theta(l,m):=[f_0,g_0] \\mathcal N_{\\divi(m_0)|_{X_K}}(f_0) N_{\\divi(l_0)|_{X_K}}(g_0)\\,.$$\n Note that $\\theta$ is non-trivial, so surjective, since $\\theta(l_0,m_0)=1$. Now by using the Weil reciprocity law we prove that $\\theta$ descends to a non-trivial morphism $\\overline\\theta:\\left<\\mathscr L,\\mathscr M\\right>_K\\to K$, indeed for $f,g\\in K(X)^\\times$:\n\\begin{eqnarray*}\n\\theta(fl,m)&=&[ff_0,g_0]\\, \\mathcal N_{\\divi(m_0)|_{X_K}}(ff_0)\\, \\mathcal N_{\\divi(l_0)|_{X_K}}(g_0)=\\\\\n&=& [f,g_0]\\,[f_0,g_0]\\, \\mathcal N_{\\divi(m_0)|_{X_K}}(f)\\, \\mathcal N_{\\divi(m_0)|_{X_K}}(f_0)\\, \\mathcal N_{\\divi(l_0)|_{X_K}}(g_0)=\\\\\n&=&[g_0,f]\\, \\mathcal N_{\\divi(m_0)|_{X_K}}(f)\\, \\theta(l,m)=\\\\\n&=& \\mathcal N_{\\divi(m)|_{X_K}}(f)\\, \\theta(l,m)\\rlap.\n\\end{eqnarray*}\nSimilarly it holds that\n$$\\theta(l,gm)=\\mathcal N_{\\divi(l)|_{X_K}}(g)\\, \\theta(l,m)\\rlap.$$\nIn other words equation \\ref{rel3} can we written as:\n$$\n\\left_K=\\overline{\\theta}(\\left_K)\\left_K\n$$\nhence, by the non triviality of $\\overline{\\theta}$ we conclude that $\\left<\\mathscr L,\\mathscr M\\right>_K$ has dimension $1$.\n\\endproof\n\\emph{Step 2.} Definition of $\\left<\\mathscr L,\\mathscr M\\right>$.\\\\\n Let $U\\subseteq S$ be a non-empty open subset and denote with $X_U$ the schematic inverse image of $U$ with respect to $\\varphi$. We clearly have a flat map $X_U\\to U$, so we define:\n\\begin{alignat*}{2}\n \\Upsilon_U &:= \\biggl\\{(D,E)\\in \\Div(X)\\times\\Div(X) &&\\;\\colon\\; \\pctext{2in}{$D|_{X_U}$ and $E|_{X_U}$ have no common components (as divisors on $X_U$)}\\biggr\\}\\rlap, \n\\end{alignat*}\n\\begin{alignat*}{2}\n \\Sigma_U &:=\\Bigg\\{(l,m) &&\\;\\colon\\; \\pctext{3in}{$l$ and $m$ are non-zero meromorphic sections of $\\mathscr L$ and $\\mathscr M$ such that $(\\divi(l),\\divi(m))\\in\\Upsilon_U$ and $\\left<\\divi(l)|_{X_U}, \\divi(m)|_{X_U} \\right>$ is effective on $U$}\\Bigg\\}\\rlap. \n\\end{alignat*}\nMoreover notice that $\\Sigma_U\\subset \\Sigma_K$. We define a sheaf of $\\mathscr O_S$-modules $\\mathscr A$ on $X$ given by:\n$$\\mathscr A|_{U}:={\\mathscr O_{S}|_{U}}^{(\\Sigma_{U})}\\,.$$\nFinally consider the morphism of sheaves: $\\Phi:\\mathscr A\\to\\left<\\mathscr L,\\mathscr M\\right>_K$ which sends $(l,m)\\in\\Sigma_U$ to $\\left_K$ and define\n$$\\left<\\mathscr L,\\mathscr M \\right>:=\\faktor{\\mathscr A}{\\ker(\\Phi)}\\rlap.$$\nThe canonical image of $(l,m)\\in\\Sigma_U$ in $\\mathscr A(U)$ is denoted as $\\left_U$.\n\n\\begin{proposition}\\label{sstep2}\nLet $(l,m)\\in \\Sigma_{U}$ such that $\\left<\\divi(l)|_{X_{U}}, \\divi(m)|_{X_{U}}\\right>=0\\in \\Div(U)$. Then for any $(l',m')\\in\\Sigma_{U}$ there exists an element $a\\in\\mathscr O_S(U)$ such that $\\left_{U}=a\\left_{U}$.\n\\end{proposition}\n\\proof\n\nThere are two elements $f,g\\in K(X)^\\times$ such that $l'=fl$, $m'=gm$ and moreover:\n$$((f),(g)),\\;((f),\\;\\divi(m)),\\;((g),\\divi(l))\\in\\Upsilon_K\\,.$$\nHence by using proposition \\ref{intwithrat}:\n$$\\left<\\divi(l')|_{X_{U}}, \\divi(m')|_{X_{U}}\\right>=\\left<(f)|_{X_{U}}\\divi(l)|_{X_{U}}, (g)|_{X_{U}}\\divi(m)|_{X_{U}}\\right>=$$\n$$=\\left< (f)|_{X_{U}},(g)|_{X_{U}}\\right>+\\left< (f)|_{X_{U}},\\divi(m)|_{X_{U}}\\right>+\\left<\\divi(l)|_{X_{U}}, (g)|_{X_{U_i}}\\right>+0=$$\n$$= \\left(\\mathcal N_{(f)|_{X_{U}}}(g)\\right)+\\left(\\mathcal N_{\\divi(m)|_{X_{U}}}(f)\\right)+\\left(\\mathcal N_{\\divi(l)|_{X_{U}}}(g)\\right)=$$\n$$=\\left(\\mathcal N_{(f)|_{X_{U}}}(g)\\, \\mathcal N_{\\divi(m)|_{X_{U}}}(f)\\, \\mathcal N_{\\divi(l)|_{X_{U}}}(g) \\right)\\rlap.$$\nSince $\\left<\\divi(l')|_{X_{U}}, \\divi(m')|_{X_{U}}\\right>$ is effective, then \n$$a:=\\mathcal N_{(f)|_{X_{U}}}(g)\\,\\mathcal N_{\\divi(m)|_{X_{U}}}(f)\\, \\mathcal N_{\\divi(l)|_{X_{U}}}(g)\\in \\mathscr O_S(U)\\,.$$\nOn the other hand\n$$\\left_K=[f,g] \\mathcal N_{\\divi(m)|_{X_K}}(f)\\mathcal N_{\\divi(l)|_{X_K}}(g)\\left_K$$\ntherefore by proposition \\ref{restr_prop} we can conclude that:\n$$\\left_{U}=\\mathcal N_{(f)|_{X_{U}}}(g) \\mathcal N_{\\divi(m)|_{X_{U}}}(f)\\mathcal N_{\\divi(l)|_{X_{U}}}(g)\\left_U=a\\left_{U}\\,.$$\n\\endproof\n\nWe are ready to show that $\\left<\\mathscr L,\\mathscr M\\right>$ is an invertible sheaf on $S$. By proposition \\ref{sstep1} $\\left<\\mathscr L,\\mathscr M\\right>$ is non-zero; now assume $\\mathscr L=\\mathscr O_X(D)$, $\\mathscr M=\\mathscr O_X(E)$ and fix a point $s_0\\in S$. By the moving lemma we can find a divisor $D'$ such that $D'\\sim D$ and $D'$ doesn't have components in $X_{s_0}$. Suppose that $x_1,\\ldots x_m$ are the intersection points of $D'$ and $X_{s_0}$, by applying again the moving lemma we can find a divisor $E'$ such that: $E'\\sim E$, $E'$ and $D'+X_{s_0}$ have no common components, and $E$ doesn't pass by $x_1,\\ldots, x_m$. Consider the finite subset of $S$ \n$$C:=\\{s\\in S'\\colon D'\\cap E'\\cap X_s\\neq \\emptyset\\}\\,$$\nand note that its complement $U:=S\\setminus C$ has the following properties: $s_0\\in U$ and $\\left=0$. At this point any two meromorphic sections of $\\mathscr L$ and $\\mathscr M$ corresponding respectively to the divisors $D'$ and $E'$ will satisfy the hypothesis of proposition \\ref{sstep2} on $U$. This implies that $\\left<\\mathscr L,\\mathscr M\\right>$ is an invertible sheaf.\n\nOne can show that the pairing constructed above satisfies the axioms (1)-(4) of definition \\ref{axiom_deligne} for $n=1$, moreover we have the following additional properties:\n\n\\begin{theorem}\\label{del_thm}\nThe Deligne pairing $(\\mathscr L,\\mathscr M)\\to \\left<\\mathscr L,\\mathscr M\\right>$ satisfies the properties listed below. We assume that $\\mathscr L$ and $\\mathscr M$ are two invertible sheaves on $X$.\n\\begin{itemize}\n\\item[$(1)$] The induced map $\\Pic(X)\\times\\Pic(X)\\to\\Pic(S)$ is bilinear and symmetric. \n\\item[$(2)$] Let $l$ and $m$ be two non-zero meromorphic sections of $\\mathscr L$ and $\\mathscr M$, respectively, such that $\\divi(l)$ and $\\divi(m)$ have no common components. Then, there exists a non-zero meromorphic section $\\left$ with the following properties: \n\\begin{itemize}\n\\item[$(i)$] If $f,g\\in K(X)^\\times$ such that $(\\divi(fl),\\divi(m)),(\\divi(l),\\divi(gm))\\in \\Upsilon$, then:\n$$\\left=\\mathcal N_{\\divi(m)}(f)\\left$$\n$$\\left=\\mathcal N_{\\divi(l)}(g)\\left$$\n\\item[$(ii)$] There is an isomorphism of invertible sheaves \n$$\\left<\\mathscr L,\\mathscr M\\right>\\cong\\mathscr O_S(\\left<\\divi(l),\\divi(m)\\right>)$$\nMoreover, under the above isomorphism $\\left$ corresponds to $1_{\\left<\\divi(l),\\divi(m)\\right>}$. In particular:\n$$\\divi(\\left)=\\left<\\divi(l),\\divi(m)\\right>\\,.$$\n\\end{itemize}\n\\end{itemize}\n\\end{theorem}\n\\proof\nSee \\cite[Theorem 4.7]{Mor}.\n\\endproof\n\\begin{remark}\\label{del_vectspace}\nNote that when $S=\\spec k$ for any field $k$ (in other words $X$ is an algebraic curve), then $\\left<\\mathscr L,\\mathscr M\\right>$ is just a one dimensional $k$-vector space.\n\\end{remark}\n \n\\begin{remark}\nIf $S$ is a non-singular projective curve over a field $k$, and $\\varphi: X\\to S$ is a morphism over $\\spec k$ (i.e. $X$ is a fibred surface over $S$), then it is evident that $(D,E)\\mapsto \\deg\\left<\\mathscr O_X(D),\\mathscr O_X(D)\\right>$ satisfies all the axioms of definition \\ref{inters_var}, so it is the intersection pairing on $X$. The same argument holds also when $X$ has generic dimension $n$.\n\\end{remark}\n\n\n\\end{appendices}\n\n\\bibliographystyle{hplain.bst}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{INTRODUCTION AND MOTIVATION}\nObservation of rare processes demands a careful study and implementation of signal processing and detector design to reach new levels of background suppression. Both the potential backgrounds and detector response to backgrounds must be thoroughly understood. Radon (Rn) gas provides a problematic class of backgrounds due to its terrestrial presence and its long-lived progeny. Exposure to radon at any stage of assembly of an experiment can result in surface contamination by progeny supported by the long half-life (22 yr) of \\nuc{210}{Pb} on sensitive surfaces of a detector \\cite{ams07, leu05,gui11}. The radon progeny surface contaminants can continue to produce unwanted background even after the detector is moved to its final laboratory or configuration with a lower radon level. In the case of neutrinoless double-beta decay experiments, energetic $\\alpha$ and $\\beta$ decays of \\nuc{210}{Po} and \\nuc{210}{Bi}, respectively, produce a background near or above the region of interest. Particularly problematic is the degraded alpha deposit that can fall under the region of interest requiring pulse shape or fiducial cuts to be rejected. In the case of a dark matter detector, the $\\alpha$ decay of \\nuc{210}{Po} produces a nuclear recoil similar to that of a WIMP-nucleon scatter. Because of the importance of this class of backgrounds, all low background rare event searches pay close attention to Rn exposure and are exploring ways to mitigate and control Rn-generated backgrounds. \n\nWhile the Rn progeny deposit directly on a material surface, the nuclear recoils of the alpha decay of \\nuc{218}{Po} and \\nuc{214}{Po} cause the daughters to be implanted deeper into the surface. Modeling and surface alpha measurements \nindicate that the Rn-borne \\nuc{210}{Pb} is implanted down to 0.05 -- 0.1 $\\mu$m in metals due to the deposition of progeny earlier in the chain. Further, exposure to Rn gas invites diffusion of Rn that can carry progeny deeper into the surface layers of material. The diffused Rn progeny and any possible bulk contamination of \\nuc{210}{Pb} creates an additional path of material for the degradation of an emitted alpha background. Though the majority of alpha emission may come from deposited daughters close to the surface, the surface roughness of a material adds an additional layer of material an alpha must traverse \\cite{per13}. \nThe net effect is an emitted alpha with a peak near its full energy, but with a low energy tail extended down to low energies as if the alpha originated deeper than the implantation depth of 0.05 - 0.1 $\\mu$m. For a typical metal, the range of a 5 MeV alpha is 20 $\\mu$m. One method to mitigate surface alpha contamination is to chemically remove the Rn progeny that exist down to a surface depth that they are imbedded.\n\nThe deposition of Rn progeny continues to be thoroughly modeled and studied by low background experiments. The findings are helping establish the infrastructure and facility designs needed for future experiments to control the exposure to Rn gas. Naturally, the deposition studies are complimented by separate studies evaluating the cleaning and surface removal of Rn progeny. Since the start of the Low Radioactivity Techniques workshop series, there have been over 10 papers focusing on the modeling, deposition, or removal of surface Rn progeny \\cite{leu05,woj07,gui11,woj11,jil13,pat13,per13,sch13,bru15,kob15,zuz15}, not to mention the papers where Rn studies are described in a more general experimental program overview paper or studies reported elsewhere. The findings of surface progeny removal studies have shown mixed results. Generally, the Pb and Bi progeny has been found to be easily removed using a variety of standard cleaning methods. However, in some materials, Po has been more difficult to remove and more aggressive chemical cleaning methods are often recommended. \n\nThe mixed results for the removal of surface \\nuc{210}{Po} are worrisome given the needs of next generation low background, rare event searches. With larger experiments, there will be a greater number of parts with a more stringent surface contamination control required. When deciding on a cleaning procedure, next generation experiments must consider: 1) efficiencies of removing Rn progeny, 2) quantities and purities of chemicals needed, 3) chemical waste, 4) generation and mitigation of chemical fumes, 5) underground cleaning limitations and logistics, 6) number of parts and process automation compatibility, and 6) maintaining dimensional and mechanical tolerances.\n\n\\section{Po REMOVAL TECHNIQUES}\n\nSurface cleaning techniques have been evaluated with varying results. In this section, we describe some of the methods used to highlight the generally accepted findings. The work of \\citet{hop07} looked at the cleaning of Cu and evaluated two methods. The Cu samples used in the study were initially loaded with $^{209}$Po by electrodeposition and the removal of the Po atoms was used to evaluate the various surface cleaning methods. The most effective treatment of removing Po surface contamination was a concentrated nitric acid etch. Lowering the concentration of nitric also lowered the effectiveness of Po removal. In the paper, the authors also introduce an alternate acidified peroxide cleaning method containing a dilute piranha solution of 1\\% H$_2$SO$_4$ and 3\\% H$_2$O$_2$ (hereinafter referred to the PNNL solution for cleaning Cu). This PNNL solution can effectively etch away surface layers in a more controlled fashion than concentrated nitric solution. Though it is shown to be effective at removing Cu atoms from the surface, it has limited effect of removing Po surface contamination. The authors stated some concern with Po solubility in the solution given they found improved Po removal when starting with Cu samples of lower initial Po activity. \n\nSimilar tests were conducted by other groups and two studies are worth mentioning here. \\citet{zuz15} looked at the removal of all three long-lived Rn progeny ($^{210}$Pb, $^{210}$Bi, $^{210}$Po) from samples of Cu, stainless steel and Ge. Using the standard PNNL acidified peroxide solution, the authors found that cleaning Cu samples showed better reduction of Bi and Pb than of Po, which showed very little, if any, reduction. Explanations for the poor removal of Po, including redeposition in solution, are given in Ref. \\cite{zuz12}. The removal of Rn progeny from contaminated samples of steel and Ge faired a bit better than Cu indicating the substrate material has a significant effect on contamination removal efficiency. The authors additionally tried a separate electropolishing technique on Cu and stainless steel. The electropolishing treatment significantly improved the removal efficiency of all three progeny, including a significant improvement for Po. The next study given in Ref. \\cite{sch13} also found effective reduction of Po through an electropolishing procedure on stainless steel. Interestingly, this study reduced Po to detection sensitivity levels after a depth of 0.6 $\\mu$m steel was removed, while the implantation depth of Rn progeny is modeled to be 0.05 - 0.1 $\\mu$m in most metals. \n\nThe fact that methods to remove Po have had mixed results indicates the importance of understanding the chemical behavior of Po in solution and the substrate being cleaned. \nMost methods tried using some form of an acidic solution, but given the behavior of Po in the Pourbaix diagram (see Ref. \\cite{ans12}), neutral Po can exist in solution over the entire pH range. In the neutral state, Po will likely redeposit onto a surface. However at low pH, Po can be driven into the Po(IV) oxidized state. When in a ion state, Po should favor staying in solution rather than redepositing based on the aqueous solution chemistry. The Po can be converted into a stable ion state by applying an oxidation potential or an oxidizing agent. The standard electrode potentials for Po in an acidic solution suggests that a potential of 0.73 V separates neutral Po from Po$^{4+}$, which is expected to be the most stable Po ion in solution \\cite{ans12}.\n\nWorking off the scenario that the ion state of Po is a significant factor in its removal from a contaminated sample, the following requirements are needed. First, the Po must be converted into an ion state through oxidation and favorably into the Po$^{4+}$ state. Solubility plays a role so a sufficient amount of surface area should be given within a cleaning solution for the amount of Po present. Finally, the stability of the ion state must be maintained to avoid reduction back to a neutral atom. The substrate atoms (e.g Cu) will likely play a role as they compete for the oxidizer (either an applied potential or an oxidizing agent). Given the separate oxidation potentials, the kinetics of the Po and the substrate atom removal may occur at different rates. That is, the depth of the substrate that is removed may not necessarily mean all other species located within that depth have had time to be oxidized before redepositing back onto the surface. Instead, sufficient exposure to the oxidizer may be the determining factor of effective Po removal, rather than the depth of the substrate removed. Greater exposure can be accomplished several ways: the contaminated sample can be agitated to make use of the volume of solution exposure, greater concentrations of an oxidizing agent, longer exposure times, larger volumes of solution surrounding the contaminated sample, etc. \n\nReturning to the mixed results for Po noted earlier, we can now put the results in the context of the Po ion state scenario. The tests with nitric acid \\cite{hop07} worked very well for all progeny. Nitric acid is an excellent oxidizer and is aggressive to almost all metals. A problem with concentrated nitric is that it will remove a large amount of material in the process in an uncontrolled way due to thermodynamic changes during dissolution. Electropolishing was found to be very efficient in removing Po and the other progeny \\cite{zuz15,sch13}. Electropolishing recipes typically call for concentrated acids to keep the pH low and an applied electric potential provides the necessary oxidation. This method drives off a large portion of the surface layers, which could compromise mechanical properties if used on small parts. The PNNL acidified peroxide solution uses H$_2$0$_2$ as the oxidizing agent. With this method, there is a modest and controlled material removal, which is an important consideration for tight dimensional tolerances on small parts. Though this solution meets some of the the requirements for removal of Po, the previous tests had found this method does not effectively remove Po from Cu. It is possible the method can be effective with increased exposure to the oxidizing agent, which is what we explore and test in the next section. \n\n\\section{Po REMOVAL STUDY}\n\nFor our study, we wish to explore the hypothesis of the role played by oxidation in removing Po from a sample. Given previous studies have shown the poor removal of Po from Cu, we will focus only on quantifying the removal of Po from Cu samples. Cu foil samples 50 mm in diameter and 0.5-mm thick are exposed to a 100 kBq radon source for about 1 month. The samples are left for over 2 years to allow the $^{210}$Po activity to grow-in to equilibrium with the $^{210}$Pb. The net surface alpha rate achieved is around 300 cts\/day (not corrected for detection efficiency). An alpha spectrometer using an ion-implanted silicon detector is used to count the Cu samples before and after chemical treatment. The alpha detector has a background count rate of 6 counts\/day. The goals of this study are to explore certain features of chemical treatment to assess the role played by oxidizers on the removal of Po from the contaminated surface. Based on the ratio of the pre-treatment and post-treatment alpha rates, the fraction of Po removed is determined from the background-subtracted alpha rates, which is not dependent on the detector counting efficiency since the geometry remains unchanged. \n\n\\section{RESULTS AND DISCUSSION}\n\nThe first test was to simply look at the effect of increasing the H$_2$0$_2$ concentration from 3\\% to 9\\% but otherwise following the PNNL acidified peroxide solution process. Each sample was exposed to a fresh solution for a varying amount of time to achieve a range of Cu thickness removed from each face of the sample. The samples were agitated equally during all tests and all but one test was performed with a 100 ml solution. The results in Fig. \\ref{fig:1}(a) show a general trend of improved Po removal with greater Cu removal up to the maximum 100\\% Po removal (within counting uncertainties consistent with the background rate of the alpha detector). However, one should not generally conclude that the depth of Cu removed down to the expected Po implantation depth should determine when all of the Po is removed. Recall, the implanted Po should exist down to 0.05 $\\mu$m. Instead, the amount of Cu removed is likely a proxy for quantifying the exposure to the oxidizing agent converting Po into the Po$^{4+}$ ion state. The conditions present may allow the Cu removal to indicate when sufficient exposure for removal of the Po has occurred. Further, the greater concentration of H$_2$0$_2$ does not have a strong effect on the Po removal; but it does give sufficient exposure in a shorter time as confirmed by the amount of Cu removed. \n\n\\begin{figure}[t]\n\\centering\n\\begin{tabular}[b]{c}\n \\includegraphics[width=0.49\\textwidth]{PoRemovalStd.eps} \\\\\n \\small (a)\n\\end{tabular}\n \\quad\n\\begin{tabular}[b]{c}\n \\includegraphics[width=0.49\\textwidth]{PoRemovalWithVolt.eps}\\\\\n \\small (b) \n \\end{tabular}\n \\caption{(a) The fraction of Po removed from a cleaning solutions plotted against the depth of Cu removed. The standard PNNL acidified peroxide solution is used for a varying amount of time, though the H$_2$0$_2$ is increased for one set of samples. (b) The same data points from (a), but with an additional set where a cell potential is applied.}\n \\label{fig:1}\n\\end{figure}\n\nTo test if a cell potential could aid in the oxidation of Po in the same acidified peroxide solution, three additional samples were cleaned with an applied potential greater than 0.73 V. The results of the three tests are given in Fig. \\ref{fig:1}(b) and plotted with the same non-potential tests from Fig. \\ref{fig:1}(a), where there is no longer a distinction made for the concentration of H$_2$0$_2$. The results show no added benefit of adding a cell potential, which suggests the oxidizing agent present is the dominant oxidizing mechanism under these conditions. Instead, the additional samples follow the same general trend of increasing Po removal with greater exposure, as indicated by the increased Cu removal. With the additional samples present, the variability among samples is more visible when plotted against exposure. This observation indicates another variable may be controlling the effectiveness of Po removal.\n\n\n\nThe data from Fig. \\ref{fig:1}(b) with the most effective Po removal (where $>2\\, \\mu$m of Cu removed) is plotted in Fig. \\ref{fig:2}(a) as a function of initial $^{210}$Po surface activity. The fraction of Po removed appears to generally increase as the initial $^{210}$Po activity decreases, with one notable exception. The sample with a starting activity of 400 cts\/day had an acidified peroxide exposure sufficient to remove 8 $\\mu$m of Cu while the other two samples with starting activity $>$300 cts\/day had exposures removing $<6\\,\\mu$m of Cu. The results suggest the initial Po activity dictates the exposure to the solution required. When the exposure to the oxidizing agent is increased, commensurate with increased initial $^{210}$Po activity, the Po removal can be effective. \n\nIn a final series of tests, some alternative cleaning methods are explored to further investigate the role of oxidizers. \nGiven there was no added benefit to adding a cell potential to the standard acidified peroxide solution, we attempted to increase the acidity to see if the cell potential could be made to play a larger role. The standard PNNL solution is adjusted to 1M H$_2$SO$_4$ and a slight increase of H$_2$0$_2$ to 6\\%. Two separate cleanings were made with this solution while one has an applied cell potential. The results in Fig. \\ref{fig:2}(b) show that both cleanings were effective at removing Cu ($\\sim 8\\,\\mu$m of depth) while the very little Po removal occurred, though the sample with the cell potential present hints at slightly more removal of Po. The chemical conditions of this solution provide a competition between the Po and Cu with an advantage to Cu with its lower oxidation potential. The test further confirms that, for this type of solution, there is no advantage to be gained by applying an oxidizing potential when a strong oxidizing agent is present.\n\nTo seek some benefit of an oxidizing potential, a test is performed without H$_2$0$_2$ but with a 6M H$_2$SO$_4$ solution for an even lower pH solution. Again, two samples are cleaned in separate solutions while one sample has an applied cell potential. From the results in Fig. \\ref{fig:2}(b), the acidic solution alone was not effective at removing either Cu or Po, while the applied cell potential was sufficient to effectively remove both. This observation suggests that a cell potential alone can oxidize Po in the presence of Cu in an acidic environment, which is consistent with the previous finding of the effectiveness of electropolishing.\n\n\\begin{figure}[ht]\n\\centering\n\\begin{tabular}[b]{c}\n \\includegraphics[width=0.49\\textwidth]{PoRemovalByActivity.eps} \\\\\n \\small (a)\n\\end{tabular}\n \\quad\n\\begin{tabular}[b]{c}\n \\includegraphics[width=0.49\\textwidth]{PoRemovalSpecials.eps}\\\\\n \\small (b) \n \\end{tabular}\n \\caption{(a) The fraction of Po removed from a cleaning solutions plotted against the initial $^{210}$Po surface activity. for the samples in Fig. \\ref{fig:1} where $>2 \\mu$m of Cu was removed. See the discussion in the text regarding the sample with an activity of 400 cts\/day. \n (b) The fraction of Po removed from a cleaning solutions plotted against the depth of Cu removed for two alternate cleaning methods. In one set, an acidified peroxide solution of lower pH is used with and without an applied cell potential resulting a small fraction of Po removal. In the second set, only a stronger acidic solution is used with and without an applied cell potential. Removal of Cu and Po is found with the application of a cell potential.\n }\n \\label{fig:2}\n\\end{figure}\n\n\\section{OUTLOOK}\nThere is a rich history of studying Rn progeny deposition and plate-out on surfaces to help guide the design of low background rare event searches. Likewise, successful studies demonstrate methods to remove Rn progeny from metal surfaces, including the problematic $^{210}$Po. Next-generation experiments will have even more stringent demands for the cleaning and removal of Rn progeny surface contamination. Given the need, it is desirable to use methods that provide both efficient progeny removal and ease of process implementation. \n\nSeveral factors are found that determine if the problematic $^{210}$Po progeny will stay in solution during cleaning to be fully removed from a contaminated surface. Oxidizing the Po atoms should keep them in solution and prevent redeposition. As shown here, oxidation of Po can be achieved by an oxidizing agent or an applied potential in the right environment. So the ultimate method to achieve full progeny removal will be dependent on the ideal chemical conditions for the material being cleaned and the chemistry required to achieve and maintain oxidation of the progeny. In the case of Cu samples, the controlled PNNL acidified peroxide solution is capable of fully removing Po atoms rather than more aggressive methods. However, the cleaning must be performed in a way to give sufficient oxidizing agent exposure commensurate wth the initial $^{210}$Po surface activity. Increased exposure can be achieved through agitation of the samples, greater solution volume in contact with the samples, greater concentration of H$_2$0$_2$, or a longer duration time in solution.\n\nFurther studies are needed to fully explore the process variables tested here. The effects of Po solubility, as noted by previous studies \\cite{hop07}, will aid in understanding the correlation between $^{210}$Po activity and Po removal. The role of the kinetics should be explored to understand why a greater exposure is required to oxidize the Po atoms and prevent redeposition than that needed for the substrate atoms. In studies to determine acceptable cleaning solutions, it is desirable to seek out chemical conditions and processes that favor keeping the progeny in solution.\n\n\n\\section{ACKNOWLEDGMENTS}\nWe thank our {\\sc Majorana} collaborators for fruitful discussions. \nThis material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics under Award Number DE-SC0012612. This material is based upon cooperation with S.R. Elliott and Los Alamos National Laboratory.\n\n\\bibliographystyle{aipnum-cp}%\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nSubluminous B stars (subdwarf B stars or sdBs) are stars with thin hydrogen envelopes, currently undergoing helium-core burning, which are found on the extreme horizontal branch (EHB). Their masses were determined to be around $0.47$\\,M$_{\\rm \\odot}$ \\citep{heber09, heber16}. About half of the known single-lined sdB stars are found to be members of short-period binaries \\citep[P $\\lesssim$ 30 d, most even with P $\\lesssim$ 10 d,][]{maxted01,napiwotzki04a,kupfer15}. A large mass loss on the red giant branch (RGB) is required to form these stars, which can be caused by mass transfer to the companion, either via stable Roche lobe overflow or the formation and eventual ejection of a common envelope \\citep{han02,han03}. \nFor the existence of apparently single sdB stars binary evolution might play an important role as well, as such stars could be remnants of helium white dwarf mergers \\citep{webbink84,ibentutukov84} or from engulfing a substellar object, which might get destroyed in the process \\citep{soker98,nelemans98}.\n\nEclipsing sdB+dM binaries (HW\\,Vir systems) having short orbital periods ($0.05-1\\,{\\rm d}$) and low companion masses between $0.06$\\,M$_{\\rm \\odot}$ and $0.2$\\,M$_{\\rm \\odot}$ \\citep[see][for a summary of all known HW Vir systems]{schaffenroth18,schaffenroth19} have been known for decades \\citep{menzies} and illustrate that objects close to the nuclear burning limit of $\\sim 0.070-0.076$\\,M$_{\\rm \\odot}$ for an object of solar metallicity and up to $0.09\\,\\rm M_{\\rm \\odot}$ for metal-poor objects \\citep[see][for a review]{2014AJ....147...94D} can eject a common envelope and lead to the formation of an sdB. The light travel-time technique was used to detect substellar companion candidates to sdB stars \\citep[e.g.][and references therein]{beuermann12, kilkenny12}. However, in these systems the substellar companions\nhave wide orbits and therefore cannot have influenced the evolution of the host star.\n\n\nThe short-period eclipsing HW~Vir type binary SDSS J082053.53+000843.4, hereafter J08205+0008, was discovered as part of the MUCHFUSS project \\citep{geier11a, geier11b}. \\citet{geier11c} derived an orbital solution based on time resolved medium resolution spectra from SDSS \\citep{abazajian09} and ESO-NTT\/EFOSC2. The best fit orbital period was $P_{\\rm orb}=P=0.096\\pm0.001\\,{\\rm d}$ and the radial velocity (RV) semi-amplitude $K=47.4\\pm1.9\\,{\\rm km\\,s^{-1}}$ of the sdB. An analysis of a light curve taken with Merope on the Mercator telescope allowed them to constrain the inclination of the system to $85.8^{\\rm \\circ}\\pm0.16$. \n\nThe analysis resulted in two different possible solutions for the fundamental parameters of the sdB and the companion. As the sdB sits on the EHB the most likely solution is a core-He burning object with a mass close to the canonical mass for the He flash of $0.47 \\,\\rm M_\\odot$. Population synthesis models \\citep{han02,han03} predict a mass range of $M_{\\rm sdB}=0.37-0.48\\,\\rm M_\\odot$, which is confirmed by asteroseismological measurements \\citep{fontaine12}. A more massive ($2-3\\,\\rm M_\\odot$) progenitor star would ignite the He core under non-degenerate conditions and lower masses down to $0.3\\,\\rm M_\\odot$ are possible. Due to the shorter lifetime of the progenitors such lower mass hot subdwarfs would also be younger. Higher masses for the sdB were ruled out as contemporary theory did not predict that. By a combined analysis of the spectrum and the light curve the companion was derived to have a mass of $0.068\\pm0.003\\,\\rm M_\\odot$. However, the derived companion radius for this solution was significantly larger than predicted by theory.\n\nThe second solution that was consistent with the atmospheric parameters was a post-RGB star with an even lower mass of only $0.25\\,\\rm M_\\odot$. Such an object can be formed whenever the evolution of the star on the RGB is interrupted due to the ejection of a common envelope before the stellar core mass reaches the mass, which is required for helium ignition. Those post-RGB stars, also called pre-helium white dwarfs, cross the EHB and evolve directly to white dwarfs.\nIn this case the companion was determined to have a mass of $0.045\\pm0.003\\,\\rm M_\\odot$ and the radius was perfectly consistent with theoretical predictions.\n\n\nThe discovery of J08205+0008 was followed by the discovery of two more eclipsing systems with brown dwarf (BD) companions, J162256+473051 \\citep{schaffenroth14} and V2008-1753 \\citep{schaffenroth15}, both with periods of less than 2 hours. Two non-eclipsing systems were also discovered by \\citet{schaffenroth14a}, and a subsequent analysis of a larger population of 26 candidate binary systems by \\citet{schaffenroth18} suggests that the fraction of sdB stars with close substellar companions is as high as 3 per cent, much higher than the $0.5\\pm0.3$ per cent that is estimated for brown dwarf companions to white dwarfs (e.g. \\citealt{steele11}). Seven of the nine known white dwarf-brown dwarf systems have primary masses within the mass range for a He-core burning hot subdwarf and might therefore have evolved through this phase before. \n\nIn this paper, we present new phase-resolved spectra of J08205+0008 obtained with ESO-VLT\/UVES and XSHOOTER and high cadence light curves with ESO-NTT\/ULTRACAM. Combining these datasets, we have refined the radial velocity solution and light curve fit. We performed an in-depth analysis of the sdB atmosphere and a fit of the spectral energy distribution using the ULTRACAM secondary eclipse measurements to better constrain the radius and mass of the sdB primary and the companion. We also present our photometric campaign using the SAAO\/1m-telescope and BUSCA mounted at the Calar Alto\/2.2m telescope which has been underway for more than 10 years now, and which has allowed us to derive variations of the orbital period.\n\n\\section{Spectroscopic and photometric data}\n\n\\subsection{UVES spectroscopy}\nWe obtained time-resolved, high resolution ($R\\simeq40\\,000$) spectroscopy of J08205+0008 with ESO-VLT\/UVES \\citep{dekker04} on the night of 2011-04-05 as part of program 087.D-0185(A). In total 33 single spectra with exposure times of $300\\,{\\rm s}$ were taken consecutively to cover the whole orbit of the binary. We used the 1\" slit in seeing of $\\sim$ 1\" and airmass ranging from 1.1. to 1.5. The spectra were taken using cross dispersers CD\\#2 and CD\\#3 on the blue and red chips respectively to cover a wavelength range from \\SI{3300}{\\angstrom} to \\SI{6600}{\\angstrom} with two small gaps ($\\simeq$ \\SI{100}{\\angstrom}) at \\SI{4600}{\\angstrom} and \\SI{5600}{\\angstrom}. \n\nThe data reduction was done with the UVES reduction pipeline in the \\textsc{midas} package \\citep{midas}. In order to ensure an accurate normalisation of the spectra, two spectra of the DQ type white dwarf WD\\,0806$-$661 were also taken \\citep{subasavage09}. Since the optical spectrum of this carbon-rich white dwarf is featureless, we divided our data by the co-added and smoothed spectrum of this star.\n\nThe individual spectra of J08205+0008 were then radial velocity corrected using the derived radial velocity of the individual spectra as described in Sect. \\ref{rvs} and co-added for the atmospheric analysis.\nIn this way, we increased the signal-to-noise ratio to S\/N\\,$\\sim$\\,90, which was essential for the subsequent quantitative analysis.\n\n \n\n\\subsection{XSHOOTER spectroscopy}\nWe obtained time resolved spectra of J08205+0008 with ESO-VLT\/XSHOOTER \\citep{vernet11} as part of programme 098.C-0754(A). The data were observed on the night of 2017-02-17 with 300~s exposure times in nod mode and in seeing of $0.5-0.8$\". We obtained 24 spectra covering the whole orbital phase (see Fig. \\ref{change of atmospheric parameters vs. orbital phase}) in each of the UVB ($R\\sim$\\,5400), VIS ($R\\sim$\\,8900) and NIR ($R\\sim$\\,5600) arms with the $0.9-1.0$\" slits. The spectra were reduced using the ESO \\textsc{reflex} package \\citep{reflex} and the specific XSHOOTER routines in nod mode for the NIR arm, and in stare mode for the UVB and VIS arms.\n\nTo correct the astronomical observations for atmospheric absorption features in the VIS and NIR arms, we did not require any observations of telluric standard stars, as we used the \\texttt{molecfit} software, which is based on fitting synthetic transmission spectra calculated by a radiative transfer code to the astronomical data \\citep{2015A&A...576A..77S, 2015A&A...576A..78K}. The parameter set-up (fitted molecules, relative molecular column densities, degree of polynomial for the continuum fit, etc.) for the telluric absorption correction evaluation of the NIR-arm spectra were used according to Table 3 of \\citet{2015A&A...576A..78K}. Unfortunately, the NIR arm spectra could not be used after the telluric corrections since the signal-to-noise (S\/N) ratio and the fluxes are too low. Figure \\ref{VIS X-Shooter arm before and after telluric correction with molecfit} shows an example comparison between the original and the telluric absorption corrected XSHOOTER VIS arm spectra. The quality of the telluric correction is sufficient to allow us to make use of the hydrogen Paschen series for the quantitative spectral analysis.\n\nAccurate radial velocity measurements for the single XSHOOTER spectra were performed within the analysis program SPAS \\citep{2009PhDT.......273H}, whereby selected sharp metal lines listed in Table \\ref{list of lines detected} were used. We used a combination of Lorentzian, Gaussian and straight line (in order to model the slope of the continuum) function to fit the line profiles of the selected absorption lines.\nAfter having corrected all single spectra by the averaged radial velocities, a co-added spectrum was created in order to achieve S\/N\\,$\\sim 460\/260$ in the UVB and VIS channels, respectively.\n\nThe co-added spectrum then was normalized also within SPAS. Numerous anchor points were set where the stellar continuum to be normalized was assumed. In this way, the continuum was approximated by a spline function. To obtain the normalized spectrum, the original spectrum was divided by the spline. \n\n\\subsection{ULTRACAM photometry}\\label{ULTRACAM}\nLight curves in the SDSS $u'g'r'$ filters were obtained simultaneously using the ULTRACAM instrument \\citep{dhillon} on the 3.5m-ESO-NTT at La Silla. The photometry was taken on the night of 2017-03-19 with airmass $1.15-1.28$ as part of programme 098.D-679 (PI; Schaffenroth). The data were taken in full frame mode with 1$\\times$1 binning and the slow readout speed with exposure times of 5.75~s resulting in 1755 frames obtained over the full orbit of the system. The dead-time between each exposure was only 25 msec. We reduced the data using the HiperCam pipeline (\\url{http:\/\/deneb.astro.warwick.ac.uk\/phsaap\/hipercam\/docs\/html}). The flux of the sources was determined using aperture photometry with an aperture scaled variably according to the full width at half-maximum. The flux relative to a comparison star within the field of view (08:20:51.941 +00:08:21.64) was determined to account for any variations in observing conditions. This reference star has SDSS magnitudes of $u'$=15.014$\\pm$0.004, $g'$=13.868$\\pm$0.003, $r'$=13.552$\\pm$0.003 which were used to provide an absolute calibration for the light curve. \n\n\n\\subsection{SAAO photometry}\\label{saao}\n\n\nAll the photometry was obtained on the 1m (Elizabeth) telescope at the Sutherland\nsite of the South African Astronomical Observatory (SAAO). Nearly all observations\nwere made with the STE3 CCD, except for the last two (Table H1), which were made\nwith the STE4 camera. The two cameras are very similar with the only difference being the pixel size as the STE3 is $512\\times512$ pixels in size and the STE4 is $1024\\times1024$. We used a $2\\times2$ pre-binned mode for each CCD resulting in a read-out time of around 5 and 20s, respectively, so that\nwith typical exposure times around 10-12s, the time resolution of STE4 is only about\nhalf as good as STE3. Data reduction and eclipse analysis were carried out as outlined\nin \\citep{kilkenny11}; in the case of J08205+008, there are several useful comparison \nstars, even in the STE3 field, and - given that efforts were made to observe eclipses near the meridian - usually there were no obvious \"drifts\" caused by differential\nextinction effects. In the few cases where such trends were seen, these were removed\nwith a linear fit to the data from just before ingress and just after egress. The stability of the\nprocedures (and the SAAO time system over a long time base) is demonstrated by the\nconstant-period system AA Dor \\citep[Fig.1 of ][]{kilkenny14} and by the intercomparisons in Fig. 8 of \\citet{baran18}, for example. \n\n\n\\subsection{BUSCA photometry}\\label{calaralto}\nPhotometric follow-up data were also taken with the Bonn University\nSimultaneous CAmera \\cite[BUSCA; see][]{busca}, which is mounted to the\n2.2 m-telescope located at the Calar Alto Observatory in Spain.\nThis instrument observes in four bands simultaneously giving a very accurate eclipse measurement and good estimate of the errors. The four different bands we used in our observation are given solely by the intrinsic transmission curve given by the\nbeam splitters (UB, BB, RB, IB, \\url{http:\/\/www.caha.es\/CAHA\/Instruments\/BUSCA\/bands.txt}) and the efficiency of the CCDs, as no filters where used to ensure that all the visible light is used most efficiently. \n\nThe data were taken during one run on 25 Feb 2011 and 1 Mar 2011. We used an exposure time of 30~s. Small windows were defined around the target and four comparison stars to decrease the read-out time from 2 min to 15 s. As comparison stars we used stars with similar magnitudes ($\\Delta m< 2$mag) in all SDSS bands from $u$ to $z$, which have been pre-selected using the SDSS DR 9 skyserver (\\url{http:\/\/skyserver.sdss.org\/dr9\/en\/}). The data were reduced using IRAF\\footnote{http:\/\/iraf.noao.edu\/}; a standard CCD reduction was performed using the IRAF tools for bias- and flatfield-correction. Then the light curves of the target and the comparison stars were extracted using the aperture photometry\npackage of DAOPHOT. The final light was constructed by dividing the light curve of the target by the light curves of the comparison stars.\n\n\n\n\n\\section{Analysis}\n\n\\subsection{The hybrid LTE\/NLTE approach and spectroscopic analysis}\\label{The spectroscopic analysis technique}\nBoth the co-added UVES and XSHOOTER (UVB and VIS arm) spectra were analyzed using the same hybrid local thermodynamic equilibrium (LTE)\/non-LTE (NLTE) model atmospheric approach. This approach has been successfully used to analyze B-type stars (see, for instance, \\citealt{2006BaltA..15..107P, 2006A&A...445.1099P, 2011JPhCS.328a2015P}; \\citealt{2007A&A...467..295N, Nieva_2008}) and is based on the three generic codes \\textsc{atlas12} \\citep{1996ASPC..108..160K}, \\textsc{detail}, and \\textsc{surface} (\\citealt{1981PhDT.......113G}; \\citealt{Butler_1985}, extended and updated). \n\nBased on the mean metallicity for hot subdwarf B stars according to \\citet{naslim13}, metal-rich and line-blanketed, plane-parallel and chemically homogeneous model atmospheres in hydrostatic and radiative equilibrium were computed in LTE within \\textsc{atlas12}. \nOccupation number densities in NLTE for hydrogen, helium, and for selected metals (see Table \\ref{summary of model atoms used for the hybrid LTE\/NLTE approach}) were computed with \\textsc{detail} by solving the coupled radiative transfer and statistical equilibrium equations. The emergent flux spectrum was synthesized afterwards within \\textsc{surface}, making use of realistic line-broadening data.\nRecent improvements to all three codes \\citep[see][for details]{2018A&A...615L...5I} with regard to NLTE effects on the atmospheric structure as well as the implementation of the occupation probability formalism \\citep{1994A&A...282..151H} for H\\,{\\sc i} and He\\,{\\sc ii} and new Stark broadening tables for H \\citep{2009ApJ...696.1755T} and He\\,{\\sc i} \\citep{1997ApJS..108..559B} are considered as well. For applications of these models to sdB stars see \\citet{schneider18}. \n\n\n\nWe included spectral lines of H and \\ion{He}{i}, and in addition, various metals in order to precisely measure the projected rotational velocity ($v\\sin i$), radial velocity ($v_{\\rm rad}$), and chemical abundances of J08205+0008. The calculation of the individual model spectra is presented in detail in \\citet{2014A&A...565A..63I}.\nIn Table \\ref{Hybrid LTE\/NLTE model grid used for the quantitative spectral analysis of SDSS J08205+0008}, the covered effective temperatures, surface gravities, helium and metal abundances for the hybrid LTE\/NLTE model grid used are listed.\n\n\n\n\nThe quantitative spectral analysis followed the methodology outlined in detail in \\citet{2014A&A...565A..63I}, that is, the entire useful spectrum and all 15 free parameters ($T_{\\rm eff}$, $\\log g$, $v_{\\rm rad}$, $v\\sin i$, $\\log{n(\\text{He})}:=\\log{\\left[\\frac{\\text{N(He)}}{\\text{N(all elements)}}\\right]}$, plus abundances of all metals listed in Table \\ref{summary of model atoms used for the hybrid LTE\/NLTE approach}) were simultaneously fitted using standard $\\chi^2$ minimization techniques. Macroturbulence $\\zeta$ and microturbulence $\\xi$ were fixed to zero because there is no indication for additional line-broadening due to these effects in sdB stars \\citep[see, for instance,][]{geier:2012,schneider18}.\n\n\n\\begin{table}\n\\caption{Metal abundances of J08205+0008 derived from XSHOOTER and UVES.$^\\dagger$}\\label{abundance table}\n\\centering\n\\begin{tabular}{lll}\n\\hline\\hline\nParameter & XSHOOTER & UVES\\\\\n\\hline\n$\\log{n(\\text{C})}$ & $-4.38\\pm0.05$ & $-4.39^{+0.04}_{-0.03}$\\\\\n$\\log{n(\\text{N})}$ & $-4.00^{+0.03}_{-0.02}$ & $-3.98\\pm0.03$\\\\\n$\\log{n(\\text{O})}$ & $-4.01^{+0.05}_{-0.06}$ & $-3.86^{+0.07}_{-0.06}$\\\\\n$\\log{n(\\text{Ne})}$ & $\\leq -6.00$ & $\\leq -6.00$\\\\\n$\\log{n(\\text{Mg})}$ & $-4.98^{+0.05}_{-0.04}$ & $-5.03\\pm0.05$\\\\\n$\\log{n(\\text{Al})}$ & $-6.20\\pm0.03$ & $\\leq -6.00$\\\\\n$\\log{n(\\text{Si})}$ & $-5.13\\pm0.04$ & $-5.17^{+0.07}_{-0.08}$\\\\\n$\\log{n(\\text{S})}$ & $-5.31^{+0.11}_{-0.10}$ & $-5.12^{+0.06}_{-0.08}$\\\\\n$\\log{n(\\text{Ar})}$ & $-5.54^{+0.15}_{-0.27}$ & $-5.32^{+0.19}_{-0.23}$\\\\\n$\\log{n(\\text{Fe})}$ & $-4.39\\pm0.04$ & $-4.41^{+0.04}_{-0.05}$\\\\\n\\hline\n\\multicolumn{3}{l}{$\\dagger$: Including 1$\\sigma$ statistical and systematic errors.}\\\\\n\\multicolumn{3}{l}{$\\log{n(\\text{X})}:=\\log{\\left[\\frac{\\text{N(X)}}{\\text{N(all elements)}}\\right]}$}\n\\end{tabular}\n\\end{table}\\noindent\n\n\\subsection{Effective temperature, surface gravity, helium content and metal abundances} \\label{Effective temperature, surface gravity, and helium content}\nThe excellent match of the global best fit model spectrum to the observed one is demonstrated in Fig. \\ref{XSHOOTER hydrogen and helium lines 1} for selected spectral ranges in the co-added XSHOOTER spectrum of J08205+0008 (UVB + VIS arm).\n\nThe wide spectral range covered by the XSHOOTER spectra allowed, besides the typical hydrogen Balmer series and prominent $\\ion{He}{i}$ lines in the optical, Paschen lines to be included in the fit, which provides additional information that previously could not be used in sdB spectral analysis, but provides important consistency checks.\\\\\n\nIn the framework of our spectral analysis, we also tested for variations of the atmospheric parameters over the orbital phase as seen in other reflection effect systems \\citep[e.g.][]{heber04,schaffenroth13}. As expected, due to the relatively weak reflection effect of less than 5\\%, the variations were within the total uncertainties given in the following and can therefore be neglected (see also Fig. \\ref{change of atmospheric parameters vs. orbital phase} for details).\n\n \n \\begin{figure*}\n \\begin{center}\n \\includegraphics[trim = 0cm 0cm 0cm 0cm, clip, scale=0.49]{SDSS0820_hydrogen_and_helium_line_profiles_1_v2.pdf}\n \\includegraphics[trim = 0cm 0cm 0cm 0cm, clip, scale=0.49]{SDSS0820_hydrogen_and_helium_line_profiles_2_v2.pdf}\n \\includegraphics[trim = 0cm 0cm 0cm 0cm, clip, scale=0.49]{SDSS0820_hydrogen_and_helium_line_profiles_3_v2.pdf}\n \\caption{Comparison between observation (solid black line) and global best fit (solid red line) for selected spectral ranges in the co-added XSHOOTER spectrum of J08205+0008. Prominent hydrogen and $\\ion{He}{i}$ lines are marked by blue labels and the residuals for each spectral range are shown in the bottom panels, whereby the dashed horizontal lines mark mark deviations in terms of $\\pm1\\sigma$, i.e., values of $\\chi=\\pm1$ (0.2\\% in UVB and 0.4\\% in VIS, respectively).} Additional absorption lines are caused by metals (see Fig. \\ref{SDSS0820_metal_line_profiles}). Spectral regions, which have been excluded from the fit, are marked in grey (observation) and dark red (model), respectively. Since the range between $\\ion{H}{i}$ \\SI{9230}{\\angstrom} and $\\ion{H}{i}$ \\SI{9546}{\\angstrom} strongly suffers from telluric lines (even after the telluric correction with \\texttt{molecfit}), it is excluded from the figure.\n \\label{XSHOOTER hydrogen and helium lines 1}\n \\end{center}\n \\end{figure*}\n\n\n\n\n \n \n \n \n\n \n \\begin{figure}\n \\begin{center}\n \\includegraphics[width=\\linewidth]{J0820_kiel.pdf}\n \\caption\n $T_{\\text{eff}}-\\log{(g)}$ diagram for J08205+0008. While the blue square represents the UVES solution, the red square results from XSHOOTER. The grey square marks the LTE solution of \\citet{geier11c}. The zero-age (ZAEHB) and terminal-age horizontal branch (TAEHB) for a canonical mass sdB are shown in grey as well as evolutionary tracks for a canonical mass sdB with different envelope masses from \\citet{1993ApJ...419..596D} with black dotted lines. Additionally we show evolutionary tracks with solar metallicity for different sdB masses with hydrogen layers of $0.005\\rm\\,M_\\odot$, according to \\citet{han02} to show the mass dependence of the EHB. The error bars include 1$\\sigma$ statistical and systematic uncertainties as presented in the text (see Sect. \\ref{Effective temperature, surface gravity, and helium content} for details).}\n \\label{Kiel diagram}\n \\end{center}\n \\end{figure}\n \n \\begin{figure}\n \\begin{center}\n \\includegraphics[trim = 0cm 0cm 0cm 0cm, clip, scale=0.45]{NLTE_abundances_plot_v2.pdf}\n \\caption{The chemical abundance pattern of J08205+0008 (red: XSHOOTER, blue: UVES) relative to solar abundances of \\citet{asplund09}, represented by the black horizontal line. The orange solid line represents the mean abundances for hot subdwarf B stars according to \\citet{naslim13} used as the metallicity for our quantitative spectral analysis. Upper limits are marked with downward arrows and $\\left[\\frac{\\text{N}(\\text{X})}{\\text{N}(\\text{total})}\\right]:=\\log_{10}{\n \\left\\{\\frac{\\text{N}(\\text{X})}{\\text{N}(\\text{total})}\\right\\}}-\\log_{10}{\\left\\{\\frac{\\text{N}(\\text{X(solar)})}{\\text{N}(\\text{total})}\\right\\}}$.}\n \n \\label{abundance}\n \\end{center}\n \\end{figure}\n \n \n \n \\begin{figure*}\n \\begin{center}\n \\includegraphics[trim = 0cm 0cm 0cm 0cm, clip, scale=0.5]{SDSS0820_metal_line_profiles_1.pdf}\n \\includegraphics[trim = 0cm 0cm 0cm 0cm, clip, scale=0.5]{SDSS0820_metal_line_profiles_2.pdf}\n \\caption{Selected metal lines in the co-added XSHOOTER spectrum of J08205+0008. The observed spectrum (solid black line) and the best fit (solid red line) are shown. Solid blue vertical lines mark the central wavelength positions and the ionization stages of the individual metal lines according to Table \\ref{list of lines detected}.}\n \\label{SDSS0820_metal_line_profiles}\n \\end{center}\n \\end{figure*}\n \nThe resulting effective temperatures, surface gravities, and helium abundances derived from XSHOOTER and UVES are listed in Table \\ref{tab:par}. The results include 1$\\sigma$ statistical errors and systematic uncertainties according to the detailed study of \\citet{2005A&A...430..223L}, which has been conducted in the framework of the ESO Supernova Ia Progenitor Survey. For stars with two exposures or more, \\citet{2005A&A...430..223L} determined a systematic uncertainty of $\\pm$374\\,K for $T_{\\text{eff}}$, $\\pm$\\,0.049\\,dex for $\\log{(g)}$, and $\\pm$\\,0.044\\,dex for $\\log{n(\\text{He})}$ (see Table 2 in \\citealt{2005A&A...430..223L} for details).\n\nFigure \\ref{Kiel diagram} shows the $T_{\\text{eff}}-\\log{(g)}$ diagram, where we compare the UVES and XSHOOTER results to predictions of evolutionary models for the horizontal branch for a canonical mass sdB with different envelope masses from \\citet{1993ApJ...419..596D}, as well as evolutionary tracks\nassuming solar metallicity and masses of $0.50$\\,M$_{\\rm \\odot}$ and $0.55$\\,M$_{\\rm \\odot}$ \\citep{han02}. With $T_{\\text{eff}}=26\\,000\\pm400$\\,\\si{\\kelvin} and $\\log{(g)}=5.54\\pm0.05$ (XSHOOTER, statistical and systematic errors) and $T_{\\text{eff}}=25\\,600\\pm400$\\,\\si{\\kelvin} and $\\log{(g)}=5.51\\pm0.05$ (UVES, statistical and systematic errors), J08205+0008 lies within the EHB, as expected. Our final result ($T_{\\text{eff}}=25\\,800\\pm290$\\,\\si{\\kelvin}, $\\log{(g)}=5.52\\pm0.04$), the weighted average of the XSHOOTER and UVES parameters, is also in good agreement with the LTE results of \\citet{geier11c}, which are $T_{\\text{eff}}=26\\,700\\pm1000$\\,\\si{\\kelvin} and $\\log{(g)}=5.48\\pm0.10$, respectively.\n\nThe determined helium content of J08205+0008 is $\\log{n(\\text{He})}=-2.06\\pm0.05$ (XSHOOTER, statistical and systematic errors) and $\\log{n(\\text{He})}=-2.07\\pm0.05$ (UVES, statistical and systematic errors), hence clearly subsolar (see \\citealt{asplund09} for details).\nThe final helium abundance ($\\log{n(\\text{He})}=-2.07\\pm0.04$), the weighted average of XSHOOTER and UVES, therefore is comparable with \\citet{geier11c}, who measured $\\log{n(\\text{He})}=-2.00\\pm0.07$, and with the mean helium abundance for sdB stars from \\citet{naslim13}, which is $\\log{n(\\text{He})}=-2.34$ (see also Fig. \\ref{abundance}).\n\n \n\nMoreover, it was possible to identify metals of various different ionization stages within the spectra (see Table \\ref{list of lines detected} and Fig. \\ref{SDSS0820_metal_line_profiles}) and to measure their abundances. Elements found in more than one ionization stage are oxygen ($\\ion{O}{i\/ii}$), silicon ($\\ion{Si}{ii\/iii}$), and sulfur ($\\ion{S}{ii\/iii}$), whereas carbon ($\\ion{C}{ii}$), nitrogen ($\\ion{N}{ii}$), magnesium ($\\ion{Mg}{ii}$), aluminum ($\\ion{Al}{iii}$), argon ($\\ion{Ar}{ii}$), and iron ($\\ion{Fe}{iii}$) are only detected in a single stage. We used the model grid in Table \\ref{Hybrid LTE\/NLTE model grid used for the quantitative spectral analysis of SDSS J08205+0008} to measure the individual metal abundances in both the co-added XSHOOTER and the UVES spectrum. We were able to fit the metal lines belonging to different ionization stages of the same elements similarly well (see Fig. \\ref{SDSS0820_metal_line_profiles}). The corresponding ionization equilibria additionally constrained the effective temperature.\\\\\nAll metal abundances together with their total uncertainties are listed in Table 1. Systematic uncertainties were derived according to the methodology presented in detail in \\citet{2014A&A...565A..63I} and cover the systematic uncertainties in effective temperature and surface gravity as described earlier.\n\nThe results of XSHOOTER and UVES are in good agreement, except for the abundances of oxygen, sulfur, and argon, where differences of $0.15$\\,dex, $0.19$\\,dex, and $0.22$\\,dex, respectively, are measured. However, on average these metals also have the largest uncertainties, in particular argon, such that the abundances nearly overlap if the corresponding uncertainties are taken into account. According to Fig. \\ref{abundance}, J08205+0008 is underabundant in carbon and oxygen, but overabundant in nitrogen compared to solar (\\citealt{asplund09}), showing the prominent CNO signature as a remnant of the star's hydrogen core-burning through the CNO cycle. Aluminum and the alpha elements (neon, magnesium, silicon, and sulfur) are underabundant compared to solar. With the exception of neon, which is not present, the chemical abundance pattern of J08205+0008 generally follows the metallicity trend of hot subdwarf B stars (\\citealt{naslim13}), even leading to a slight enrichment in argon and iron compared to solar. The latter may be explained by radiative levitation, which occurs in the context of atomic transport, that is, diffusion processes in the stellar atmosphere of hot subdwarf stars (\\citealt{Greenstein_1967}; see \\citealt{Michaud_Atomic_Diffusion_in_Stars_2015} for a detailed review). \n\n \n\nDue to the high resolution of the UVES (and XSHOOTER) spectra, we were also able to measure the projected rotational velocity of J08205+0008 from the broadening of the spectral lines, in particular from the sharp metal lines, to $v\\sin{i}=66.0\\pm0.1\\,{\\rm km\\,s^{-1}}$ (UVES, 1$\\sigma$ statistical errors only) and $v\\sin{i}=65.8\\pm0.1\\,{\\rm km\\,s^{-1}}$ (XSHOOTER, 1$\\sigma$ statistical errors only)\n\n\n\n \n\n\\subsection{Search for chemical signatures of the companion} \nAlthough HW\\,Vir type systems are known to be single-lined, traces of the irradiated and heated hemisphere of the cool companion have been found in some cases. \\citet{wood99} discovered the H$\\alpha$ absorption component of the companion in the prototype system HW\\,Vir \\citep[see also][]{edelmann08}. \n\nMetal lines in emission were found in the spectra of the hot sdOB star AA\\,Dor by \\citet{vuckovic16} moving in antiphase to the spectrum of the hot sdOB star indicating an origin near the surface of the companion. After the removal of the contribution of the hot subdwarf primary, which is dominating the spectrum, the residual\nspectra showed more than 100 shallow emission lines originating from the heated side of the secondary, which show their maximum intensity close to the phases\naround the secondary eclipse. They analysed the residual spectrum in order to model the irradiation of the low-mass companion by the hot subdwarf star.\nThe emission lines of the heated side of the secondary star allowed them to determine the radial velocity semi-amplitude of the centre-of-light. After the correction to the centre-of-mass of the secondary they could derive accurate masses of both components of the AA Dor system, which is consistent with a canonical sdB mass of $0.46\\,\\rm M_\\odot$ and a companion of $0.079\\pm0.002\\rm\\,M_\\odot$ very close to the hydrogen burning limit. They also computed a first generation atmosphere model\nof the low mass secondary including irradiation effects. \n\nJ08205+0008 is significantly fainter and cooler than AA Dor but with a much shorter period.\nWe searched the XSHOOTER spectra for signs of the low-mass companion of J08205+0008.\nThis was done by subtracting the spectrum in the secondary minimum where the companion is eclipsed from the spectra before and after the secondary eclipse where most of the heated atmosphere of the companion is visible. \nHowever, no emission or absorption lines from the companion were detected (see Fig. \\ref{balmer_companion_uvb} and \\ref{balmer_companion_vis}). Also, in the XSHOOTER NIR arm spectra, no emission lines could be found.\n\n\n\n\n\n\n\n\n\n\n\n \n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n \n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n \n\\subsection{Photometry: Angular diameter and interstellar reddening}\\label{Photometric analysis}\\label{SED fitting}\n\nThe angular diameter of a star is an important quantity, because it\nallows the stellar radius to be determined, if the distance is known e.g. from\ntrigonometric parallax. The angular diameter can be determined by\ncomparing observed photometric magnitudes to those calculated from\nmodel atmospheres for the stellar surface. \nBecause of contamination by the reflection effect the apparent magnitudes of the hot subdwarf can be measured only during \nthe secondary eclipse, where the companion is completely eclipsed by the larger \nsubdwarf. We performed a least squares fit to the flat bottom of the secondary eclipse in the ULTRACAM light curves to \ndetermine the apparent magnitudes and \nderived $u'=14.926\\pm$ 0.009mag, $g'=15.025\\pm$ 0.004mag, and $r'=15.450\\pm$ 0.011mag (1$\\sigma$ statistical errors).\n\nBecause the star lies at low Galactic latitude (b=19$^\\circ$) interstellar reddening is expected to be significant.\nTherefore, both the angular diameter and the interstellar colour\nexcess have to be determined simultaneously. \nWe used the reddening law of \\citet{2019ApJ...886..108F} and matched a synthetic flux distribution calculated from the same grid of model \natmospheres that where also used in the quantitative spectral analysis (see Sect. \\ref{The spectroscopic analysis technique}) to the observed magnitudes as \ndescribed in \\citet{2018OAst...27...35H}. The $\\chi^2$ based fitting routine uses two free parameters: the angular diameter $\\theta$, \nwhich shifts the fluxes up and down according to $f(\\lambda)=\\theta^2 F(\\lambda)\/4$, where f($\\lambda$) is the observed flux at the detector \nposition and $F(\\lambda)$ is the synthetic model flux at the stellar surface, and the color excess \n\\footnote{\\citet{2019ApJ...886..108F} use $E(44-55)$, the monochromatic equivalent of the usual $E(B-V)$ in the Johnson system, \n using the wavelengths $\\lambda$ =4400\\,\\AA and 5000\\,\\AA, respectively. In fact, $E(44-55)$ is identical to $E(B-V)$ \n for high effective temperatures as determined for J08205+0008.}.\nThe final atmospheric parameters and their respective uncertainties derived from the quantitative spectral analysis (see Sect. \\ref{Effective temperature, surface gravity, and helium content}) \nresult in an angular diameter of $\\theta= 6.22\\, (\\pm 0.15) \\cdot 10^{-12}\\,\\rm rad$ and an interstellar reddening of \n$E(B-V)=0.041 \\pm 0.013$ mag.\nThe latter is consistent with values from reddening maps of \\citet{1998ApJ...500..525S} and \\citet{2011ApJ...737..103S}: 0.039 mag and 0.034 mag, \nrespectively. \n\nIn addition, ample photometric measurements of J08205+0008 are available in different filter systems, covering the spectral range all the way \nfrom the ultraviolet (GALEX) through the optical (e.g. SDSS) to the infrared (2MASS; UKIDDS; WISE). However, those measurements are mostly averages of \nobservations taken at multiple epochs or single epoch measurements at unknown orbital phase. \nTherefore, those measurements do not allow us to determine the angular diameter of the sdB because of the contamination by light from the heated hemisphere of the companion. However, an average spectral energy distribution of the system\ncan be derived. This allows us to redetermine the interstellar reddening and to search for an infrared excess caused by light from the \ncool companion.\n\nThe same fitting technique is used in the analysis of the SED as\ndescribed above for the analysis of the ULTRACAM magnitudes. Besides\nthe sdB grid, a grid of synthetic spectra of cool stars \\citep[$2300\\,{\\rm K} \\le T_{\\rm eff} \\le 15 000\\,\\rm K$,][]{2013A&A...553A...6H} is used.\nIn addition to the angular diameter and reddening parameter, the temperature of the cool companion as well as the surface ratio are \nfree parameters in the fit. The fit results in $E(B-V) = 0.040 \\pm\n0.010$ mag, which is fully consistent with the one derived from the ULTRACAM photometry as well as with the reddening map. \nThe apparent angular diameter is larger than that from ULTRACAM photometry by 2.8\\%, which is caused by the contamination by \nlight from the companion's heated hemisphere. The effective temperature of the companion is unconstrained and the best match is achieved for the surface ratio of zero, which means there is no signature from the \ncool companion.\nIn a final step we allow the effective temperature of the sdB to vary and determine it along with the angular diameter and the interstellar reddening, which results in $T_{\\rm eff}$= 26900$^{+1400}_{-1500}$\\,K in agreement with the spectroscopic result.\n\n\n \n\n\\begin{figure}\n \\centering\n \\includegraphics[width=1.05\\columnwidth]{photometry_SED.pdf}\n \\caption{\\label{fig:photometry_sed}Comparison of synthetic and\n observed photometry: \\textit{Top panel:} Spectral\n energy distribution: Filter-averaged fluxes converted from\n observed magnitudes are shown in different colours. \n The respective full width at tenth maximum are shown as dashed\n horizontal lines. The best-fitting model, degraded to a\n spectral resolution of 6\\,{\\tiny\\AA} is plotted in gray. \n In order to reduce the steep SED slope the\n flux is multiplied by the wavelength cubed.\n \\textit{Bottom panel:} Difference between synthetic and observed magnitudes\n divided by the corresponding uncertainties (residual $\\chi$).\n The following color code is used for the different photometric\n systems: GALEX\n \\citep[violet,][]{2017yCat.2335....0B}; SDSS\n \\citep[golden,][]{2015ApJS..219...12A}; Pan-STARRS1\n \\citep[dark red,][]{2017yCat.2349....0C}; Johnson\n \\citep[blue,][]{2015AAS...22533616H};\n {\\it Gaia}\n \\citep[cyan,][with corrections and\n calibrations from \\citet{2018A&A...619A.180M}]{2018A&A...616A...4E}; 2MASS\n \\citep[red,][]{2003yCat.2246....0C}; UKIDSS \n \\citep[pink,][]{2007MNRAS.379.1599L}; \n WISE \\citep[magenta,][]{2014yCat.2328....0C,2019ApJS..240...30S}.}\n \\end{figure}\n \n\\subsection{Stellar radius, mass and luminosity}\\label{Stellar radius, mass and luminosity}\n \nSince Gaia data release 2 \\citep[DR2;][]{2018A&A...616A...1G}, trigonometric parallaxes are available for a\nlarge sample of hot subdwarf stars, including J08205+0008\nfor which 10\\% precision has been reached. We corrected for the Gaia DR2 parallax zero point offset of $-$0.029 mas \\citep{2018A&A...616A...2L}.\n\n Combining the parallax measurement\nwith the results from our quantitative spectral analysis ($\\log{g}$ and $T_{\\rm eff}$) and with the angular diameter $\\theta$ derived\nfrom ULTRACAM photometry, allows for the determination of the mass of the sdB primary in J08205+0008 via:\n\\begin{equation}\n M = \\frac{g \\theta^2}{4 G \\varpi^2}\\label{mass}\n\\end{equation}\n\nThe respective uncertainties of the stellar parameters are derived by Monte Carlo error propagation. The uncertainties are dominated by the error of the parallax measurement. Results are summarized in Table \\ref{tab:par}. \n Using the gravity and effective temperature derived by the spectroscopic analysis, the mass for the\nsdB is $M= 0.48^{+0.12}_{-0.09}$ M$_\\odot$ and its luminosity is $L=16^{+3.6}_{-2.8}$ L$_\\odot$ in agreement with canonical models for EHB stars \\citep[see Fig. 13][]{1993ApJ...419..596D}.\nThe radius of the sdB is calculated by the angular diameter and the parallax to\n$R = 0.200^{+0.021}_{-0.018}$ R$_\\odot$.\n \n \n \\subsection{Radial velocity curve and orbital parameters}\\label{rvs}\n\n \\begin{figure}\n \\begin{center}\n \\includegraphics[width=\\linewidth]{rv_j0820.pdf}\n \\caption{Radial velocity of J08205+0008 folded on the orbital period.\n The residuals are shown together with a prediction of the Rossiter-McLaughlin effect using the parameters derived in this paper in blue and a model with a higher rotational velocity assuming bound rotation in green. The radial velocities were determined from spectra obtained with XSHOOTER (red circles), UVES (black triangles), EFOSC2 (black circles), and SDSS (black rectangles). The EFOSC2 and SDSS RVs have been corrected by a systematic shift (see text for details).}\n \\label{rv}\n \\end{center}\n \\end{figure}\n \n\nThe radial velocities of the individual XSHOOTER spectra were measured by fitting all spectral features simultaneously to synthetic models as described in Sect.~\\ref{The spectroscopic analysis technique}. \n\nDue to lower S\/N of the individual UVES spectra, which were observed in poor conditions, only the most prominent features in the spectra are suitable for measuring the Doppler shifts. After excluding very poor quality spectra, radial velocities of the remaining 28 spectra were measured using the {\\sc fitsb2} routine \\citep{napiwotzki04b} by fitting a set of different mathematical functions to the hydrogen Balmer lines as well as He\\,{\\sc i} lines. The continuum is fitted by a polynomial, and the line wings and line core by a Lorentzian and a Gaussian function, respectively. The barycentrically corrected RVs together with formal $1\\sigma$-errors are summarized in Table~\\ref{RVs}. \n\nThe orbital parameters $T_{\\rm 0}$, period $P$, system velocity $\\gamma$, and RV-semiamplitude $K$ as well as their uncertainties were derived with the same method described in \\citet{geier11b}. To estimate\nthe contribution of systematic effects to the total error budget additional to the statistic errors determined by the {\\sc fitsb2} routine, we normalised the $\\chi^{2}$ of the most probable solution by adding systematic errors to each data point $e_{\\rm norm}$ until the reduced $\\chi^{2}$ reached $\\simeq1.0$.\n\nCombining the UVES and XSHOOTER RVs we derived $T_{0}=57801.54954\\pm0.00024\\,{\\rm d}$, $P=0.096241\\pm 0.000003\\,{\\rm d}$, $K=47.9\\pm0.4$\\,\\si{\\kilo\\metre\\per\\second} and the system velocity $\\gamma=26.5\\pm0.4$\\,\\si{\\kilo\\metre\\per\\second}. No significant systematic shift was detected between the two datasets and the systematic error added in quadrature was therefore very small $e_{\\rm norm}=2.0$\\,\\si{\\kilo\\metre\\per\\second}. The gravitational redshift is significant at $1.6_{+0.05}^{-0.02}$\\,\\si{\\kilo\\metre\\per\\second} and might be important if the orbit of the companion could be measured by future high resolution measurements \\citep[see, e.g.,][]{vos13}.\n\nTo improve the accuracy of the orbital parameters even more we then tried to combine them with the RV dataset from \\citet{geier11c}, medium-resolution spectra taken with ESO-NTT\/EFOSC2 and SDSS. A significant, but constant systematic shift of $+17.4$\\,\\si{\\kilo\\metre\\per\\second} was detected between the UVES+XSHOOTER and the SDSS+EFOSC2 datasets. Such zero-point shifts are common between low- or medium-resolution spectrographs. It is quite remarkable that both medium-resolution datasets behave in the same way. However, since the shift is of the same order as the statistical uncertainties of the EFOSC2 and SDSS individual RVs we refrain from interpreting it as real. \n\nAdopting a systematic correction of $+17.4$\\,\\si{\\kilo\\metre\\per\\second} to the SDSS+EFOSC2 dataset, we combined it with the UVES+XSHOOTER dataset and derived $T_{0}\\,(\\rm BJD_{TDB})=2457801.59769\\pm0.00023\\,{\\rm d}$, $P=0.09624077\\pm 0.00000001\\,{\\rm d}$, which is in perfect agreement with the photometric ephemeris, $K=47.8\\pm0.4$\\,\\si{\\kilo\\metre\\per\\second} and $\\gamma=26.6\\pm0.4$\\,\\si{\\kilo\\metre\\per\\second}. This orbital solution is consistent with the solution from the XSHOOTER+UVES datasets alone. Due to the larger uncertainties of the SDSS+EFOSC2 RVs, the uncertainties of $\\gamma$ and $K$ did not become smaller. The uncertainty of the orbital period on the other hand improved by two orders of magnitude due to the long timebase of 11 years between the individual epochs. Although this is still two orders of magnitude larger than the uncertainty derived from the light curve (see Sect.~\\ref{timing}), the consistency with the light curve solution is remarkable. The RV curve for the combined solution phased to the orbital period is given in Fig.~\\ref{rv}. Around phase 0 the Rossiter-McLaughin effect \\citep{rossiter,mclaughlin} is visible. This effect is a RV deviation that occurs as parts of a rotating star are blocked out during the transit of the companion. The effect depends on the radius ratio and the rotational velocity of the primary. We can derive both parameters much more precisely with the spectroscopic and photometric analysis, but we plotted a model of this effect using our system parameters on the residuals of the radial velocity curve to show that is consistent. \n\nExcept for the corrected system velocity, the revised orbital parameters of J08205+0008 are consistent with those determined by \\citet{geier11c} ($P=0.096\\pm 0.001\\,{\\rm d}$, $K=47.4\\pm1.9$\\,\\si{\\kilo\\metre\\per\\second}), but much more precise.\n\n \n \\subsection{Eclipse timing}\\label{timing}\n \\begin{figure}\n \\centering\n \\includegraphics[width=\\linewidth]{o-c_J0820_res}\n \\caption{(O--C) diagram for J08205+0008 using eclipse times observed with Merope (red squares), BUSCA (blue diamonds), ULTRACAM (green triangles) and the SAAO-1m\/1.9m telescope (black circles). The solid line represents a fit of a parabola to account for the period change of the orbital period. The derived quadratic term is given in the legend. The parameters of the fit are provided in the legend. In the lower panel the residuals between the observations and the best fit are shown.}\n \\label{o-c}\n\\end{figure}\n \n Since the discovery that J08205+00008 is an eclipsing binary in November 2009, we have monitored the system regularly using BUSCA mounted at the 2.2m-telescope in Calar Alto, Spain, ULTRACAM and the 1m in Sutherland Observatory (SAAO), South Africa. Such studies have been performed for several post-common envelope systems with sdB or white dwarf (WD) primaries and M dwarf companions \\citep[see][for a summary]{lohr14}. In many of those systems period changes have been found\n \nThe most convenient way to reveal period changes is to construct an observed minus calculated (O--C) diagram. Thereby we compare the observed mid-eclipse times (O) with the expected mid-eclipse times (C) assuming a fixed orbital period $P_0$ and using the mid-eclipse time for the first epoch $T_0$. Following \\citet{kepler91}, if we expand the observed mid-eclipse of the Eth eclipse ($T_E$ with $E=t\/P$) in a Taylor serie\n, we get the (O--C) equation:\n\\begin{equation}\n \\mathrm{O-C}=\\Delta T_0+\\frac{\\Delta P_0}{P_0}t+\\frac{1}{2}\\frac{\\dot{P}}{P_0}t^2+...\n\\end{equation}\nThis means that with a quadratic fit to the O--C data we can derive the ephemeris $T_0$, $P$, and $\\dot{P}$ in $BJD_{TDB}$.\n\nTogether with the discovery data observed with Merope at the Mercator telescope on La Palma \\citep{geier11c} it was possible to determine timings of the primary eclipse over more than 10 years, as described in Sect. \\ref{saao} and \\ref{calaralto}. \nAll measured mid-eclipse times can be found in Table \\ref{ecl_time}.\n\n\n\n We used all eclipse timings to construct an O--C diagram, which is shown in Fig. \\ref{o-c}. We used the ephemeris given in \\citet{geier11c} as a starting value to find the eclipse numbers of each measured eclipse time and detrended the O-C diagram by varying the orbital period until no linear trend was visible to improve the determination of the orbital period. During the first $7-8$ years of observations, the ephemeris appeared to be linear. This was also found by \\citet{pulley:18}. As their data show a large scatter, we do not use it in our analysis. However, in the last two years a strong quadratic effect was revealed. The most plausible explanation is a decrease in the orbital period of the system.\n This enabled us to derive an improved ephemeris for J08205+0008:\n \\begin{eqnarray*}\n T_0&=&2455165.709211(1)\\\\\n P&=&0.09624073885(5)\\rm\\,d\\\\\n \\dot{P}&=&-3.2(8)\\cdot 10^{-12}\\,\\rm dd^{-1}\n \\end{eqnarray*}\n \n \n \n \\subsection{Light curve modeling}\n \n \n\n With the new very high quality ULTRACAM $u'g'r'$ light curves we repeated the light curve analysis of \\citep{geier11c} obtaining a solution with much smaller errors.\n\n For the modeling of the light curve we used {\\sc lcurve}, a code written to model detached and accreting binaries containing a white dwarf \\citep[for details, see][]{copperwheat10}. It has been used to analyse several detached white dwarf-M dwarf binaries \\citep[e.g.,][]{parsons_nnser}. Those systems show very similar light curves with very deep, narrow eclipses and a prominent reflection effect, if the primary is a hot white dwarf. Therefore, {\\sc lcurve} is ideally suited for our purpose.\n \n The code calculates monochromatic light curves by subdividing each star into small elements with a geometry fixed by its radius as measured along the line from the center of one star towards the center of the companion. The flux of the visible elements is always summed up to get the flux at a certain phase. A number of different effects that are observed in compact and normal stars are considered, e.g. Roche distortions observed when a star is distorted from the tidal influence of a massive, close companion, as well as limb-darkening and gravitational darkening. Moreover, lensing and Doppler beaming, which are important for very compact objects with close companions, can be included. The Roemer delay, which is a light travel-time effect leading to a shift between primary and secondary eclipse times due to stars of different mass orbiting each other and changing their distance to us, and asynchronous orbits can be considered. The latter effects are not visible in our light curves and can hence be neglected in our case. \n \n \\begin{figure}\n \\centering\n \\includegraphics[width=1.05\\linewidth]{lc.pdf}\n \\caption{ULTRACAM $u'g'r'$ light curves of J08205+0008 together with the best fit of the most consistent solution. The light curves in the different filters have been shifted for better visualisation. The lower panel shows the residuals. The deviation of the light curves from the best fit is probably due to the fact that the comparison stars cannot completely correct for atmospheric effects due to the different colour and the crude reflection effect model used in the analysis is insufficient to correctly describe the shape of the reflection effect.}\n \\label{lc_0820}\n \\end{figure}\n \n \\begin{figure*}\n \\centering\n \\includegraphics[width=1.0\\linewidth]{mcmc_g_newer.pdf}\n \\caption{MCMC calculations showing the distributions of the parameter of the analysis of the ULTRACAM g'-band light curve.}\n \\label{mcmc_r}\n\\end{figure*}\n\n\n \n As we have a prominent reflection effect it is very important to model this effect as accurately as possible. The reflection effect, better called the irradiation effect, results from the huge difference in temperature between the two stars, together with their small separation.\nThe (most likely) tidally locked companion is heated up on the side facing the hot subdwarf because of the strong irradiation by the hot primary. Therefore, the contribution of the companion to the total flux of the system varies with phase and increases as more of the heated side is visible to the observer. We use a quite simple model, which calculates the fluxes from the temperatures of both companions using a black body approximation. The irradiation is approximated by assigning a new temperature to the heated side of the companion \n \\begin{equation}\n \\sigma T'^4_{\\rm sec}=\\sigma T^4_{\\rm sec}+F_{\\rm irr}=\\sigma T^4_{\\rm sec}\\left[1+\\alpha\\left(\\frac{T_{\\rm prim}}{T^{\\rm sec}}\\right)^4\\left(\\frac{R_{\\rm prim}}{a}\\right)^2\\right],\n \\end{equation}\n with $\\alpha$ being the albedo of the companion and $F_{\\rm irr}$ the irradiating flux, accounting for the angle of incidence and distance from the hot subdwarf. The irradiated side is heated up to a temperature of $13\\,000-15\\,000$ K similar to HW Vir \\citep{kiss00}, which is slightly hotter but has a longer period. Hence, the amplitude of the effect is increasing from blue to red as can be seen in Fig. \\ref{lc_0820}, as the sdB is getting fainter compared to the companion in the red. If the irradiation effect is very strong, the description given above might not be sufficient, as the back of the irradiated star is completely unaffected in this description, but heat transport could heat it up, increasing the luminosity of unirradiated parts as well. This is not considered in our simple model. \n\nAs the light curve model contains many parameters, not all of them independent, we fixed as many parameters as possible (see Table \\ref{param}). The temperature of the sdB was fixed to the temperature determined from the spectroscopic fit. We used the values determined by the coadded XSHOOTER spectra, as they have higher signal-to-noise.\nThe gravitational limb darkening coefficients were fixed to the values expected for a radiative atmosphere for the primary \\citep{von_zeipel} and a convective atmosphere for the secondary \\citep{lucy} using a blackbody approximation to calculate the resulting intensities. For the limb darkening of the primary we adopted a quadratic limb darkening law using the tables by \\citet{claret}. As the tables include only surface gravities up to $\\log{g}=5$ we used the values closest to the parameters derived by the spectroscopic analysis. \n\nAs it is a well-separated binary, the two stars are approximately spherical, which means the light curve is not sensitive to the mass ratio. Therefore, we computed solutions with different, fixed mass ratios. To localize the best set of parameters we used a \\textsc{simplex} algorithm \\citep{press92} varying the inclination, the radii, the temperature of the companion, the albedo of the companion (absorb), the limb darkening of the companion, and the time of the primary eclipse to derive additional mid-eclipse times. Moreover, we also allowed for corrections of a linear trend, which is often seen in the observations of hot stars, as the comparison stars are often redder and so the correction for the air mass is often insufficient. This is given by the parameter \"slope\". The model of the best fit is shown in Fig. \\ref{lc_0820} together with the observations and the residuals.\n\nTo get an idea about the degeneracy of parameters used in the light curve solutions, as well as an estimation of the errors of the parameters we performed Markov-Chain Monte-\nCarlo (MCMC) computations with \\textsc{emcee} \\citep{emcee} using the best solution we obtained with the \\textsc{simplex} algorithm as a starting value varying the radii, the inclination, the temperature of the companion as well as the albedo of the companion. As a prior we constrained the temperature of the cool side of the companion to $3000\\pm500$ K. Due to the large luminosity difference between the stars the temperature of the companion is not significantly constrained by the light curve. The computations were done for all three light curves separately.\n\nFor the visualisation we used the python package \\textsc{corner} \\citep[see Fig. \\ref{mcmc_r}]{corner}. The results of the MCMC computations of the light curves of all three filters agree within the error (see Table \\ref{param}). A clear correlation between both radii and the inclination is visible as well as a weak correlation of the albedo of the companion (absorb) and the inclination. This results from the fact that the companion is only visible in the combined flux due to the reflection effect and the eclipses and the amplitude of the reflection effect depends on the inclination, the radii, the separation, the albedo and the temperatures. Looking at the $\\chi^2$ of the temperature of the companion we see that all temperatures give equally good solutions showing that the temperature can indeed not be derived from the light curve fit. The albedo we derived has, moreover, a value > 1, which has been found in other HW Vir systems as well and is due to the simplistic modeling of the reflection effect. The reason for the different distribution in the inclination is not clear to us. However, it is not seen in the other bands. It might be related to the insufficient correction of atmospheric effects by the comparison stars. \n\n\\begin{table}\\caption{Parameters of the light curve fit of the ULTRACAM u'g'r' band light curves}\\label{param}\n\\begin{tabular}{llll}\n\t\\hline\\hline\n\tband & u'&g'&r'\\\\\\hline\n\t\\multicolumn{4}{c}{Fixed Parameters}\\\\\\hline\n\tq&\\multicolumn{3}{c}{0.147}\\\\\n\t$P$&\\multicolumn{3}{c}{0.09624073885}\\\\\n\t$T_{\\rm eff,sdB}$&\\multicolumn{3}{c}{25800}\\\\\n\t$x_{1,1}$&0.1305&0.1004&0.0788\\\\\n\t$x_{1,2}$&0.2608&0.2734&0.2281\\\\\n\t$g_1$&\\multicolumn{3}{c}{0.25}\\\\\n\t$g_2$&\\multicolumn{3}{c}{0.08}\\\\\n\t\\hline\n\t\\multicolumn{4}{c}{Fitted parameters}\\\\\\hline\n\t$i$&$85.3\\pm0.6$&$85.6\\pm0.2$&$85.4\\pm0.3$\\\\\n\t$r_1\/a$&$0.2772\\pm0.0029$&$0.2734\\pm0.0010$&$0.2748\\pm0.0014$\\\\\n\t$r_2\/a$&$0.1322\\pm0.0018$&$0.1297\\pm0.0006$&$0.1304\\pm0.0008$\\\\\n\t$T_{\\rm eff,comp}$&$3000\\pm500$&$2900\\pm500$&$3200\\pm560$\\\\\n\tabsorb&$1.54\\pm0.08$&$1.58\\pm0.03$&$2.08\\pm0.05$\\\\\n\t$x_2$&0.70&0.78&0.84\\\\\n\t$T_0$ [MJD]&57832.0355&57832.0354&57832.0354\\\\\n\tslope&-0.000968&-0.002377&0.00013417\\\\\n\t$\\frac{L_1}{L_1+L_2}$&0.992578&0.98735&0.97592\\\\\n\t\\hline\n\\end{tabular}\n\\end{table}\n\n\n\\subsection{Absolute parameters of J08205+00008}\\label{Absolute parameters of J08205+00008}\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\linewidth]{sdB_python.pdf}\n \\caption{Mass of the sdB versus the photometric $\\log g$ for J08205+0008 for different mass ratios from $0.11-0.20$ in steps of 0.01 (red solid line). The parameters were derived by combining the results from the analysis of the light curves and radial velocity curve. The grey area marks the spectroscopic $\\log{g}$ that was derived from the spectroscopic analysis.\n The blue dashed lines indicate the $\\log g$ derived by the radius from the SED fitting and the {\\it Gaia} distance for different sdB masses.\n The red area marks the mass range for the sdB for which we get a consistent solution by combining all different methods. The red vertical line represents the solution for a canonical mass sdB.} \n \\label{sdb}\n\\end{figure}\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\linewidth]{brown_dwarf_python.pdf}\n \\caption{Comparison of theoretical mass-radius relations of low-mass stars \\citep{baraffe,baraffe_2} to results from the light curve analysis of J08205+0008. We used tracks for different ages of 1 Gyr (dashed), 5 Gyr (dotted-dashed) and 10 Gyr (dotted). Each red square together with the errors represents a solution from the light curve analysis for a different mass ratio ($q = 0.11-0.20$ in steps of 0.01). The red vertical line represents the solution for a canonical mass sdB. The red area marks the mass range of the companion corresponding to the mass range we derived for the sdB.}\n \\label{bd}\n\\end{figure}\nAs explained before, we calculated solutions for different mass ratios ($q=0.11-0.20$). We obtain equally good $\\chi^2$ for all solutions, showing that the mass ratio cannot be constrained by the light curve fit as expected. Hence, the mass ratio needs to be constrained differently.\nHowever, the separation, which can be calculated from the mass ratio, period, semi-amplitude of the radial-velocity curve and the inclination, is different for each mass ratio.\nThe masses of both companions can then be calculated from the mass function. From the relative radii derived from the light curve fit together with the separation, the absolute radii can be calculated. This results in different radii and masses for each mass ratio. \n\nAs stated before, the previous analysis of \\citet{geier11c} resulted in two possible solutions: A post-RGB star with a mass of 0.25 $\\rm M_\\odot$ and a core helium-burning star on the extreme horizontal branch with a mass of $\\rm 0.47\\,M_\\odot$.\nFrom the analysis of the photometry together with the Gaia magnitudes (see Sect. \\ref{Stellar radius, mass and luminosity}) we get an additional good constraint on the radius of the sdB. Moreover, the surface gravity was derived from the fit to the spectrum. This can be compared to the mass and radius of the sdB (and a photometric $\\log g$: $g=GM\/R^2$) derived in the combined analysis of radial velocity curve and light curve. This is shown in Fig. \\ref{sdb}. We obtain a good agreement for of all three methods (spectroscopic, photometric, parallax-based) for an sdB mass between 0.39-0.60 $\\rm M_\\odot$. This means that we can exclude the post-RGB solution. The position of J0820 in the $T_{\\rm eff}-\\log{g}$ diagram, which is shown in Fig. \\ref{Kiel diagram}, gives us another constraint on the sdB mass. By comparing the atmospheric parameters of J08205+0008 to theoretical evolutionary tracks calculated by \\citet{han02} it is evident that the position is not consistent with sdB masses larger than $\\sim 0.50\\,\\rm M_\\odot$, which we, therefore, assume as the maximum possible mass for the sdB.\n\nAccordingly, we conclude that the solution that is most consistent with all different analysis methods is an sdB mass close to the canonical mass ($0.39-0.50\\,\\rm M_\\odot$). For this solution we have an excellent agreement of the parallax radius with the photometric radius only, if the parallax offset of $-0.029$ mas suggested by \\citet{2018A&A...616A...2L} is used. Otherwise the parallax-based radius is too large. The companion has a mass of $0.061-0.71\\,\\rm M_\\odot$, which is just below the limit for hydrogen-burning. Our final results can be found in Table \\ref{tab:par}. The mass of the companion is below the hydrogen burning limit and the companion is hence most likely a massive brown dwarf.\n\nWe also investigated the mass and radius of the companion and compared it to theoretical calculations by \\citet{baraffe} and \\citet{baraffe_2} as shown in Fig.~\\ref{bd}. It is usually assumed that the progenitor of the sdB was a star with about $1-2\\,\\rm M_\\odot$ \\citep{heber09,heber16}. Therefore, we expect that the system is already quite old (5-10 Gyrs). For the solutions in our allowed mass range the measured radius of the companion is about 20\\% larger than expected from theoretical calculations. Such an effect, called inflation, has been observed in different binaries and also planetary systems with very close Jupiter-like planets. A detailed discussion will be given later. This effect has already been observed in other hot subdwarf close binary systems \\citep[e.g.][]{schaffenroth15}. \n\nHowever, if the system would still be quite young with an age of about 1 Gyr, the companion would not be inflated. We performed a kinematic analysis to determine the Galactic population of J08205+0008. As seen in Fig. \\ref{toomre} the sdB binary belongs to the thin disk where star formation is still ongoing and could therefore indeed be as young as 1 Gyr, if the progenitor was a $2\\,\\rm M_\\odot$ star. About half of the sdO\/Bs at larger distances from the Galactic plane (0.5 kpc) are found in the thin disk \\citep{martin17}. However, it is unclear whether a brown dwarf companion can eject the evelope from such a massive 2 $\\rm M_{\\odot}$ star. Hydrodynamical simulations performed by \\citet{kramer20} indicate that a BD companion of $\\sim0.05-0.08\\,\\rm M_\\odot$ might just be able to eject the CE of a lower mass ($1\\,\\rm M_\\odot$) red giant.\n\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\linewidth]{{SDSSJ082053.53+000843.4_Toomre_AS}.pdf}\n \\caption{Toomre diagram of J08205+0008: the quantity $V$ is the velocity in direction of Galactic rotation, $U$ towards the Galactic center, and $W$ perpendicular to the Galactic plane. The two dashed ellipses mark boundaries for the thin (85\\,km\\,s$^{-1}$) and thick disk (180\\,km\\,s$^{-1}$) following Fuhrmann (2004). The red cross marks J08205+0008, the yellow circled dot the Sun, and the black plus the local standard of rest. The location of J08205+0008 in this diagram clearly hints at a thin disk membership. }\n \\label{toomre}\n\\end{figure}\n\n\\begin{table}\n\\caption{Parameters of J08205+0008.\n}\\label{tab:par}\n\\vspace{0.5cm}\n\\begin{tabular}{lll}\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\n\\multicolumn{3}{l}{SPECTROSCOPIC PARAMETERS}\\\\\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\n$\\gamma$& [${\\rm km\\,s^{-1}}$] & {$26.5\\pm0.4$} \\\\\n$K_1$ & [${\\rm km\\,s^{-1}}$] & {$47.8\\pm0.4$} \\\\\n$f(M)$& [$M_{\\rm \\odot}$] & {$0.0011\\pm0.0001$} \\\\\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\n$T_{\\rm eff,sdB}$ & [K] & {$25800\\pm290^\\ast$} \\\\ \n$\\log{g,sdB}$ & & {$5.52\\pm0.04^\\ast$} \\\\\n$\\log{n(\\text{He})}$ & & {$-2.07\\pm0.04^\\ast$} \\\\\n$v\\sin{i}$ & [${\\rm km\\,s^{-1}}$] & $65.9\\pm0.1^\\dagger$\\\\\n\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\n$a$ &[$\\rm R_\\odot$]&$0.71\\pm0.02$\\\\\n$M_{\\rm 1}$&[$M_{\\rm \\odot}$]& $0.39 - 0.50$ \\\\\n$M_{\\rm 2}$ & [$M_{\\rm \\odot}$] & $0.061-0.071$ \\\\\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\n\\multicolumn{3}{l}{PHOTOMETRIC PARAMETERS}\\\\\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\n$T_0$ & [BJD$_{TDB}$]&2455165.709211(1)\\\\\n$P$&[d]&$0.09624073885(5)$\\\\\n$\\dot{P}$&dd$^{-1}$&$-3.2(8)\\cdot 10^{-12}$\\\\\n$i$&[$^\\circ$]&$85.6\\pm0.3$\\\\\n$R_{\\rm 1}$ & [$R_{\\rm \\odot}$] & $0.194\\pm0.008$ \\\\\n$R_{\\rm 2}$ & [$R_{\\rm \\odot}$] & $0.092\\pm 0.005$ \\\\\n$\\log g$ & &$5.52\\pm0.03$\\\\\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\n\\multicolumn{3}{l}{SED FITTING}\\\\\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\n$\\varpi_{\\text{Gaia}}$ & [mas] & $0.6899\\pm0.0632^\\dagger$ \\\\\n$E(B-V)$ & [mag] & $0.040\\pm0.010^\\dagger$ \\\\\n$\\theta$ & [$10^{-12}$\\,rad] & $6.22\\pm0.15^\\ast$ \\\\\n$R_{\\text{Gaia}}$ & [$R_{\\odot}$] & 0.200$^{+0.021\\ast}_{-0.018}$ \\\\\n$M_{\\text{Gaia}}$ & [$M_{\\odot}$] & 0.48$^{+0.12\\ast}_{-0.09}$ \\\\\n$\\log{(L_{\\text{Gaia}}\/L_{\\odot})}$ & & 16$^{+3.6\\ast}_{-2.8}$ \\\\\n\\noalign{\\smallskip}\n\\hline\n\\multicolumn{3}{l}{Gaia: Based on measured \\textit{Gaia} parallax, but applying a zero}\\\\ \\multicolumn{3}{l}{point offset of $-0.029$\\,mas (see Sect. \\ref{Stellar radius, mass and luminosity} for details).}\\\\\n\\multicolumn{3}{l}{$\\dagger$: 1$\\sigma$ statistical errors only.}\\\\\n\\multicolumn{3}{l}{$\\ast$: Listed uncertainties result from statistical and systematic}\\\\ \\multicolumn{3}{l}{errors (see Sects. \\ref{Effective temperature, surface gravity, and helium content} and \\ref{SED fitting} for details).}\\\\\n\\end{tabular}\n\\end{table}\n\n\n\\section{Discussion}\n\n\\subsection{Tidal synchronisation of sdB+dM binaries} \\label{syncro}\n\n\nIn close binaries, the rotation of the components is often assumed to be synchronised to their orbital motion. In this case the projected rotational velocity can be used to put tighter constraints on the companion mass. \\citet{geier10b} found that assuming tidal synchronisation of the subdwarf primaries in sdB binaries with orbital periods of less than $\\simeq1.2\\,{\\rm d}$ leads to consistent results in most cases. In particular, all the HW\\,Vir type systems analysed in the \\citet{geier10b} study turned out to be synchronised. \n\nIn contrast to this, the projected rotational velocity of J08205+0008 is much smaller than is required for tidal synchronisation. We can calculate the expected rotational velocity ($v_{\\rm rot}$) using the inclination ($i$), rotational period ($P_{\\rm rot}$) and the radius of the primary ($R_1$) from the light curve analysis if we assume the system is synchronised:\n\t\\begin{equation}\n\tP_{\\rm rot, 1}=\\frac{2\\pi R_1}{v_{\\rm rot}}\\equiv P_{\\rm orb} \\rightarrow v_{\\rm synchro} \\sin i=\\frac{2\\pi R_1\\sin i}{P_{\\rm orb}}.\n\t\\end{equation}\n\nDue to the short period of this binary, the sdB should spin with $v_{\\rm syncro}\\simeq102\\,{\\rm km\\,s^{-1}}$ similar to the other known systems \\citep[see][and references therein]{geier10b}. \n\n\nOther observational results in recent years also indicate that tidal synchronisation of the sdB primary in close sdB+dM binaries is not always established in contrast to the assumption made by \\citet{geier10b}. New theoretical models for tidal synchronisation \\citep{preece:18,preece:19} even predict that none of the hot subdwarfs in close binaries should rotate synchronously with the orbital period. \n\nFrom the observational point of view, the situation appears to be rather complicated.\n\\citet{geier10b} found the projected rotational velocities of the two short-period ($P=0.1-0.12\\,{\\rm d}$) HW\\,Vir systems HS\\,0705+6700 and the prototype HW\\,Vir to be consistent with synchronisation. \\citet{charpinet08} used the splitting of the pulsation modes to derive the rotation period of the pulsating sdB in the HW\\,Vir-type binary PG\\,1336$-$018 and found it to be consistent with synchronised rotation. This was later confirmed by the measurement of the rotational broadening \\citep{geier10b}. \n\nHowever, the other two sdBs with brown dwarf companions J162256+473051 and V2008-1753 \\citep{schaffenroth14,schaffenroth15} have even shorter periods of only 0.07 d and both show sub-synchronous rotation with 0.6 and 0.75 of the orbital period, respectively, just like J0820+0008. AA Dor on the other hand, which has a companion very close to the hydrogen burning limit and a longer period of 0.25 d, seems to be synchronised \\citep[and references therein]{vuckovic16}, but it has already evolved beyond the EHB and is therefore older and has had more time to synchronise.\n\n\\citet{pablo11} and \\citet{pablo12} studied three pulsating sdBs in reflection effect sdB+dM binaries with longer periods and again used the splitting of the pulsation modes to derive their rotation periods ($P\\simeq0.39-0.44\\,{\\rm d}$). All three sdBs rotate much slower than synchronised. But also in this period range the situation is not clear, since a full asteroseismic analysis of the sdB+dM binary Feige\\,48 ($P\\simeq0.38\\,{\\rm d}$) is consistent with synchronised rotation. \n\nSince synchronisation timescales of any kind \\citep{geier10b} scale dominantely with the orbital period of the close binary, these results seem puzzling. Especially since the other relevant parameters such as mass and structure of the primary or companion mass are all very similar in sdB+dM binaries. They all consist of core-helium burning stars with masses of $\\sim0.5\\,M_{\\odot}$ and low-mass companions with masses of $\\sim0.1\\,M_{\\odot}$. And yet 5 of the analysed systems appear to be synchronised, while 6 rotate slower than synchronised without any significant dependence on companion mass or orbital period. \nThis fraction, which is of course biased by complicated selection effects, might be an observational indication that the synchronisation timescales of such binaries are of the same order as the evolutionary timescales\n\nIt has to be pointed out that although evolutionary tracks of EHB stars exist, the accuracy of the derived observational parameters (usually $T_{\\rm eff}$ and $\\log{g}$) is not high enough to determine their evolutionary age on the EHB by comparison with those tracks as accurate as it can be done for other types of stars (see Fig.~\\ref{Kiel diagram}). As shown in Fig. \\ref{Kiel diagram}, the position of the EHB is also dependent on the core and envelope mass and so it is not possible to find a unique track to a certain position in the $T_{\\rm eff}-\\log{g}$ diagram and in most sdB systems the mass of the sdB is not constrained accurately enough.\n\n\\citet{2005A&A...430..223L} showed that sdB stars move at linear speed over the EHB and so the distance from the zero-age extreme horizontal branch (ZAEHB) represents how much time the star already spent on the EHB.\nIf we look at the position of the non-synchronised against the position of the synchronised systems in the $T_{\\rm eff}-\\log{g}$ diagram (Fig. \\ref{Kiel diagram synchro}), it is obvious that all the systems, which are known to be synchronised, appear to be older. There also seems to be a trend that systems with a higher ratio of rotational to orbital velocity are further away from the ZAEHB. This means that the fraction of rotational to orbital period might even allow an age estimate of the sdB. \n\nThe fact that the only post-EHB HW\\,Vir system with a candidate substellar companion in our small sample (AA\\,Dor) appears to be synchronised, while all the other HW\\,Vir stars with very low-mass companions and shorter periods are not, fits quite well in this scenario. \nThis could be a hint to the fact that for sdB+dM systems the synchronisation timescales are comparable to or even smaller than the lifetime on the EHB.\nHot subdwarfs spend $\\sim100\\,{\\rm Myrs}$ on the EHB before they evolve to the post-EHB stage lasting $\\sim10\\,{\\rm Myrs}$. So we would expect typical synchronisation timescales to be of the order of a few tens of millions of years, as we see both synchronised and unsynchronised systems.\n\n\n\n\n\n \\begin{figure}\n \\begin{center}\n \\includegraphics[width=\\linewidth]{J0820_synchro.pdf}\n \\caption{$T_{\\text{eff}}-\\log{(g)}$ diagram for the sdB+dM systems with known rotational periods mentioned in Sect. \\ref{syncro}. The filled symbols represent synchronized systems, the open symbols, systems which are known to be non-synchronised. The square marks the position of J08205+0008. The sizes of the symbols scale with the orbital period, with longer periods having larger symbols. Plotted error bars are the estimated parameter variations due to the reflection effect, as found e.g. in \\citet{schaffenroth13}. The zero-age (ZAEHB) and terminal-age extreme horizontal branch (TAEHB) for a canonical mass sdB as well as evolutionary tracks for a canonical mass sdB with different envelope masses from \\citet{1993ApJ...419..596D} are also shown.}\n \\label{Kiel diagram synchro}\n \\end{center}\n \\end{figure}\n\n \\subsection{A new explanation for the period decrease}\n\n There are different mechanisms of angular momentum loss in close binaries leading to a period decrease: gravitational waves, mass transfer (which can be excluded in a detached binary), or magnetic braking \\citep[see][]{quian:08}.\n Here, we propose that tidal synchronisation can also be an additional mechanism to decrease the orbital period of a binary.\n \n From the rotational broadening of the stellar lines (see Sect. \\ref{syncro}) we derived the rotational velocity of the subdwarf to be about half of what would be expected from the sdB being synchronised to the orbital period of the system. This means that the sdB is currently spun up by tidal forces until synchronisation is reached causing an increase in the rotational velocity.\n As the mass of the companion is much smaller than the mass of the sdB, we assume synchronisation for the companion. \n \n The total angular momentum of the binary system is given by the orbital angular momentum $J_{\\rm orb}$ and the sum of the rotational angular momentum of the primary and secondary star $I_{\\rm spin,1\/2}$, with $\\omega$ being the orbital angular velocity and $\\Omega_i$ the rotational, angular velocity\n \\begin{align}\n\tJ_{\\rm tot}&=J_{\\rm orb}+\\sum_{i=1}^{2}I_{\\rm spin,i} \\\\\n\tJ_{\\rm orb}&=(m_1a_1^2+m_2a_2^2)\\,\\omega = \\frac{m_1m_2}{m_1+m_2}a^2\\omega\\\\%\\frac{2\\pi}{P_{\\rm orb}} \\\\\n\ta^2&=\\left(\\frac{G(m1+m2)}{\\omega^2}\\right)^{2\/3}\\\\%\\left(\\frac{G(m_1+m_2)}{4\\pi^2}\\right)^{2\/3}P_{\\rm orb}^{4\/3} \\\\\n I_{\\rm spin,i}&=k_r^2M_iR_i^2\\Omega_\n \\end{align}\n with $k_r^2$ the radius of gyration of the star. It refers to the distribution of the components of an object around its rotational axis. It is defined as $k_r^2=I\/MR^2$, where $I$ is the moment of inertia of the star. \\citet{geier10b} used a value of 0.04 derived from sdB models, which we adopt.\n\nFor now we neglect angular momentum loss due to gravitational waves and magnetic braking.\nIf we assume that the companion is already synchronised and its rotational velocity stays constant ($\\frac{\\rm d\\Omega_{2}}{\\mathrm{d}t}=0$) and that the masses and radii do not change, as we do not expect any mass transfer after the common envelope phase, we obtain \n\t\\begin{equation}\n\t\\frac{dJ_{\\rm tot}}{dt}=p_1\\frac{\\mathrm{d}\\omega^{-1\/3}}{\\mathrm{d}t}+p_2\\frac{\\rm d\\Omega_1}{\\mathrm{d}t}=-p_1\\frac{\\dot{\\omega}}{3\\omega^{4\/3}}+p_2\\dot{\\Omega}=0\n\t\\end{equation}\n\twith \n\t\\begin{equation}\n\tp_1=\\frac{m_1m_2G^{2\/3}}{(m_1+m_2)^{1\/3}}\n\t\\end{equation}\n\tand\n\t\t\\begin{equation}\n\tp_2=k_r^2m_1R_1^2\n\t\\end{equation}\nThis shows that from an increase in the rotational velocity of the primary, which is expected from tidal synchronisation, we expect an increase of the orbital velocity, which we observe in the case of J08205+0008.\nWe can now calculate the current change of orbital velocity:\n\\begin{equation}\n\t\\dot{\\Omega}_1=\\frac{p_1}{3p_2}\\frac{\\dot{\\omega}}{\\omega^{4\/3}}=\\frac{m_2G^{2\/3}}{3k_r^2R_1^2(m_1+m_2)^{1\/3}}\\frac{\\dot{\\omega}}{\\omega^{4\/3}}\n\t\\end{equation}\n\nFrom this equation we can clearly see that rotational velocity change depends on the masses of both stars, the radius of the primary, the orbital velocity change and the current orbital velocity. An increasing rotational velocity causes an increasing orbital velocity and hence a period decrease.\n\n\\subsection{Synchronisation timescale}\n\nIf we assume that the observed period decrease is only due to the rotational velocity change, we can calculate the rate of the rotational velocity change and the timescale until synchronisation is reached.\n According to \\citet{preece:18}, the change of rotational angular velocity is given by\n\\begin{equation}\n \\frac{\\mathrm{d}\\Omega}{\\mathrm{d}t}=\\frac{\\omega}{\\tau_{\\rm tide}}\\left(1-\\frac{\\Omega}{\\omega}\\right)\\frac{M_2}{M_1+M_2}\\frac{a^2}{R^2k_r^2}\\propto \\left(1-\\frac{\\Omega}{\\omega}\\right)\\label{eq}\n\\end{equation}\nwhere $\\tau_{\\rm tide}$ is the tidal time-scale depending on the density, radius and mass of the star and the viscous time-scale of the convective region. The current position of J08205+0008 on the $T_{\\rm eff}-\\log g$ diagram and the mass we derived from our analysis suggest that the sdB is currently in the evolutionary phase of helium-burning. The lifetime of this phase is approximately 100 Myrs. So we do not expect the structure of the star to change significantly in the next few Myr. Because the moment of inertia of an sdB star is small compared to that of the binary orbit, the change in separation and angular velocity can be neglected.\n\nTherefore, we can calculate the timescale until synchronisation is reached using the equation given in \\citet{zahn:89}:\n\\begin{equation}\n\\frac{1}{T_{\\rm sync}}=-\\frac{1}{\\Omega_1-\\omega} \\frac{\\mathrm{d}\\Omega_1}{\\mathrm{d}t}\n\\end{equation}\nUsing our equation (\\ref{eq}) and calculating and substituting the angular velocities by the periods we derive an expression for the synchronisation time scale:\n\\begin{equation}\n T_{\\rm sync}=\\left(1-\\frac{2\\pi R_1\\sin i}{P_{\\rm orb}v\\sin i}\\right)\\frac{P_{\\rm orb}^{2\/3}v\\sin i}{\\dot{P}_{\\rm orb}\\sin i}\\frac{3(2\\pi)^{1\/3}k_r^2R_1(m_1+m_2)^{1\/3}}{m_2G^{2\/3}}\n\\end{equation}\nUsing the orbital period, the masses, radii and inclination from our analysis, we calculate a synchronisation time $T_{\\rm sync}$ of $2.1\\pm0.1$ Myrs, well within the lifetime of a helium burning object on the extreme horizontal branch. The orbital period will change by about 200 s (3.5\\%) in this 2 Myrs, which means a change in the separation of only 0.01 $\\rm R_\\odot$, which shows that our assumption of a negligible change in separation is valid. If we assume that the rotation after the common envelope phase was close to zero, the total timescale until the system reaches synchronisation is about 4 Myrs. This assumption is plausible as most red giant progenitors rotate slowly and the common envelope phase is very short-lived and so no change of the rotation is expected.\n\nThis means that this effect could significantly add to the observed period decrease. The fact that the synchronised systems appear to be older than the non-synchronised ones confirms that the synchronisation timescale is of the expected order of magnitude and it is possible that we might indeed measure the synchronisation timescale.\n\nAs mentioned before \\citet{preece:18} predict that the synchronisation timescales are much longer than the lifetime on the EHB and that none of the HW Vir systems should be synchronised.\n\\citet{preece:19} investigated also the special case of NY Vir, which was determined to be synchronised from spectroscopy and asteroseismolgy, and came to the conclusion that they cannot explain, why it is synchronised. They proposed that maybe the outer layers of the sdB were synchronised during the common envelope phase. However, observations show that synchronised sdB+dM systems are not rare, but that synchronisation occurs most likely during the phase of helium-burning, \nwhich shows that synchronisation theory is not yet able to predict accurate synchronization time scales\n\n\n\\subsection{Orbital period variations in HW Vir systems}\n\nAs mentioned before, there are several mechanisms that can explain period changes in HW Vir systems. \nThe period change due to gravitational waves is usually very small in HW Vir systems and would only be observable after observations for many decades \\citep[e.g.][]{kilkenny14}. Using the equation given in \\citet{kupfer20} with the system parameters derived in this paper, we predict an orbital period decay due to gravitational waves of $\\dot{P}=4.5\\cdot10^{-14}\\,\\rm ss^{-1}$.\nThe observed change in orbital period is hence about 100 times higher than expected by an orbital decay due to gravitational waves\n \n\nHW Vir and NY Vir have also been observed to show a period decrease of the same order of magnitude \\citep{quian:08,kilkenny14} but have been found to rotate (nearly) synchronously.\nBoth also show additionally to the period decrease a long-period sinusoidal signal \\citep{Lee:09,Lee:14}. These additional variations in the O--C diagram have been interpreted as caused by circumbinary planets in both cases, however the solutions were not confirmed with observations of longer baselines. Observations of more than one orbital period of the planet would be necessary to confirm it. The period decrease was explained to be caused by angular momentum loss due to magnetic stellar wind braking.\n\nFollowing the approach of \\citet{qian:07} we calculated the relation between the mass-loss rate and the Alfv\\'{e}n radius that would be required to account for the period decrease in J08205+0008 due to magnetic braking. This is shown in Fig. \\ref{mag_braking}. Using the tidally enhanced mass-loss rate of \\citet{tout:88} we derive that an Alfv\\'{e}n radius of $75\\,\\rm R_\\odot$ would be required to cause the period decrease we measure, much larger than the Alfv\\'{e}n radius of the Sun. This shows that, as expected, the effect of magnetic braking in a late M dwarf or massive brown dwarf is very small at best and cannot explain the period decrease we derive.\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\linewidth]{mag_braking.pdf}\n \\caption{Correlation between the Alfv\\'{e}n radius and the mass loss rate\nfor the companion of J08205+0008. The red dashed line marks the Alfv\\'{e}n radius for the Sun, the blue dotted line indicates the tidally enhanced mass-loss rate determined using the parameters of the sdB using the formula of \\citet{tout:88}.}\n \\label{mag_braking}\n\\end{figure}\n\n\\citet{bours:16} made a study of close white dwarf binaries and observed that the amplitude of eclipse arrival time variations in K dwarf and early M dwarf companions is much larger than in late M dwarf, brown dwarf or white dwarf companions, which do not show significant orbital period variations. They concluded that these findings are in agreement with the so-called Applegate mechanism, which proposes that variability in the binary orbits can be driven by magnetic cycles in the secondary stars. In all published HW Vir systems with a longer observational baseline of several years quite large period variations on the order of minutes have been detected \\citep[see][for an overview]{zorotovic:13, pulley:18}, with the exception of AA Dor \\citep{kilkenny14}, which still shows no sign of period variations after a baseline of about 40 years. Also the orbital period decrease in J08205+0008 is on the order of seconds and has only been found after 10 years of observation and no additional sinusoidal signals have been found as seen in many of the other systems. This confirms that the findings of \\citet{bours:16} apply to close hot subdwarf binaries with cool companions. The fact that the synchronised HW Vir system AA Dor does not show any period variations also confirms our theory that the period variations in HW Vir systems with companions close to the hydrogen-burning limit might be caused by tidal synchronisation. In higher-mass M dwarf companions the larger period variations are likely caused by the Applegate mechanism and the period decrease can be caused dominantly by magnetic braking and additionally tidal synchronisation.\n\nIt seems that orbital period changes in HW Vir systems are still poorly understood and have also not been studied observationally to the full extent. More observations over long time spans of synchronised and non-synchronised short-period sdB binaries with companions of different masses will be necessary to understand synchronisation and orbital period changes of hot subdwarf binaries. Most likely it cannot be explained with just one effect and is likely an interplay of different effects.\n\n\\subsection{Inflation of brown dwarfs and low-mass M dwarfs in eclipsing WD or sdB binaries}\nClose brown dwarf companions that eclipse main sequence stars are rare, with only 23 known to date \\citep{carmichael20}. Consequently, brown dwarf companions to the evolved form of these systems are much rarer with only three (including J08205+0008) known to eclipse hot subdwarfs, and three known to eclipse white dwarfs. These evolved systems are old ($>$ 1 Gyr), and the brown dwarfs are massive, and hence not expected to be inflated \\citep{thorngren18}.\n\nSurprisingly, of the three hot subdwarfs with brown dwarf companions, J08205+0008 is the one that receives the least irradiation - almost half that received by V2008-1753 and SDSSJ162256.66+473051.1, both of which have hotter primaries (32000~K, 29000~K) and shorter periods ($\\sim$1.6~hr) than J08205+0008. This suggests that more irradiation, and more irradiation at shorter wavelengths does not equate to a higher level of inflation of a brown dwarf. Indeed this finding is consistent with that for brown dwarfs irradiated by white dwarfs, where the most irradiated object with a measured radius is SDSS J1205-0242B, in a 71.2 min orbit around a 23681~K white dwarf and yet the brown dwarf is not inflated \\citep{parsons17}. The brown dwarf in this system only receives a hundredth of the irradiation that J08205+0008 does. However, WD1032+011, an old white dwarf ($T_{\\rm eff} \\sim$ 10000~K) with a high mass brown dwarf companion (0.0665 M$_{\\odot}$) does appear to be inflated \\citep{casewell:2020}. As can be seen from Figure \\ref{MR}, the majority of the low mass brown dwarfs (M$<$35 M$_{\\rm Jup}$) are inflated, irrelevant of how much irradiation they receive. For the few old (5-10 Gyr), higher mass inflated brown dwarfs, the mechanism leading to the observed inflation is not yet understood.\n\n\n\\begin{figure*}\n \\centering\n \\includegraphics[width=1.05\\linewidth]{sds_mr_jul31.pdf}\n \\caption{All known eclipsing binary white dwarfs with detached brown dwarf (triangles: \\citealt{parsons17, littlefair14}) and late M dwarf companions (triangles) from \\citet{parsons18}, hot subdwarfs with eclipsing brown dwarf companions (circles: \\citealt{schaffenroth14a, schaffenroth15}) and all known eclipsing brown dwarf companions to main sequence stars (+: \\citealt{carmichael20}). J08205+0008 is plotted as the filled square. The colour is proportional to the effective temperature of the primary in each system and the coloured circle size is proportional to the amount of total incident radiation the secondary receives. Also shown are the Sonora Bobcat brown dwarf evolutionary models of \\citet{bobcat} for solar and sub-solar metallicity and the NextGen models \\citep{baraffe97}.}\n \\label{MR}\n\\end{figure*}\n\n\n\\subsection{Previous and future evolution of the system}\nAs stated before, stars with a cool, low-mass companion sitting on the EHB are thought to have formed by a common-envelope phase from a progenitor of up to two solar mass on the RGB. Due to the large mass ratio only unstable mass transfer is possible. If the mass transfer happened at the tip of the RGB, a core-helium burning object with about $0.5\\,\\rm M_\\odot$ will be formed. If the mass transfer happened earlier then the core of the progenitor has not enough mass to start He-core burning and the pre-He WD will move to the WD cooling track crossing the EHB. Our analysis of J08205+0008 showed that a low-mass solution ($0.25\\,M_\\odot$, as discussed previously) can be excluded and that the primary star is indeed currently a core He-burning object. \n\n\\citet{kupfer15} calculated the evolution of J08205+0008 and considering only angular momentum loss due to gravitational waves and found that the companion will fill its Roche lobe in about 2.2 Gyrs and mass transfer is expected to start forming a cataclysmic variable.\nWe detected a significantly higher orbital period decrease in this system than expected from gravitational waves. Up to now, we could not detect any change in the rate of this period decrease. If we assume that the orbital period change is due to rotational period change\nuntil synchronisation is reached and afterwards the period decrease will be solely due to gravitational waves, \nwe can calculate when the companion will fill its Roche lobe and accretion to the primary will start. To calculate the Roche radius the equation derived in \\citet{eggleton83} was used:\n\\begin{equation}\n R_L=\\frac{0.49q^{2\/3}}{0.6q^{2\/3}+\\ln(1+q^{1\/3})}a\n\\end{equation}\nUsing the values derived in our analysis we calculate that the Roche lobe of the companion will be filled at a system separation of 0.410 $\\rm R_\\odot$, 56\\% of the current separation, which is reached at a period of 3525 s.\nFrom this we calculate a time scale of 1.8 Gyrs until the Roche lobe will be filled.\n\nSystems with a mass ratio $q=M_2\/M_1<2\/3$, with $M_1$ being the mass of the accretor, are assumed to be able to undergo stable mass transfer. Our system has a mass ratio of $0.147\\ll2\/3$.\nThe subdwarf will already have evolved to a white dwarf and a cataclysmic variable will be formed. It is expected that the period of an accreting binary with a hydrogen-rich donor star will decrease until a minimum period of $\\simeq70$ min is reached at a companion mass around $0.06\\,M_\\odot$ and the period will increase again afterwards \\citep{nelson18}. Such systems are called period bouncers. Our system comes into contact already close to the minimum period and should hence increase the period when the mass transfer starts.\n\nThe future of the system depends completely on the period evolution. A longer baseline of observations of this system is necessary to confirm that the period decrease is indeed stable and caused by the tidal synchronisation. \n\n\n\\section{Conclusion and summary}\n\nThe analysis of J08205+0008 with higher quality data from ESO-VLT\/XSHOOTER, ESO-VLT\/UVES and ESO-NTT\/ULTRACAM allowed us to constrain the masses of the sdB and the companion much better by combining the analysis of the radial velocity curve and the light curve. We determine an sdB mass of $0.39-0.50\\,\\rm M_\\odot$ consistent with the canonical mass and a companion mass of $0.061-0.071\\,\\rm M_\\odot$ close to the hydrogen burning limit. Therefore, we confirm that the companion is likely be a massive brown dwarf. \n\nThe atmospheric parameters and abundances show that J08205+0008 is a typical sdB and comparison with stellar evolution tracks suggest that the mass has to be less than $0.50\\,M_\\odot$ consistent with our solution and also the mass derived by a spectrophotometric method using Gaia parallaxes and the SED derived in the secondary eclipse, where the companion is not visible. \n\nIf the sdB evolved from a $1\\,\\rm M_\\odot$ star, the age of the system is expected to be around 10\\, Gyrs. In this case the radius of the brown dwarf companion is about 20\\% inflated compared to theoretical calculations. Such an inflation is observed in several sdB\/WD+dM\/BD systems but not understood yet. However, the inflation seems not to be caused by the strong irradiation.\nThe sdB binary belongs to the thin disk, as do about half of the sdB at this distance from the Galacic plane. This means that they also could be young, if they have evolved from a more massive progenitor. Then we get a consistent solution without requiring inflation of the companion. However, a brown dwarf companion might not be able to remove the envelope of a more massive progenitor.\n\nWe detected a significant period decrease in J0820+0008. This can be explained by the spin-up of the sdB due to tidal sychronisation. We calculated the synchronisation timescale to 4 Myrs well within the lifetime on the EHB. The investigation of the parameters of all known Vir systems with rotational periods (see Sect. \\ref{syncro}) shows that the synchronised systems tend to be older, showing that the synchronisation timescale seems to be comparable but smaller than the lifetime on the EHB in contrast to current synchronisation theories.\n\nBy investigating the known orbital period variations in HW Vir systems, we can confirm the findings by \\citet{bours:16} that period variations in systems with higher mass M dwarf companions seem to be larger. Hence, we conclude that the large period variations in those systems are likely caused by the Applegate mechanism and the observed period decreases dominantly by magnetic braking. In lower-mass companions close to the hydrogen-burning limit, on the other hand, tidal synchronisation spinning up the sdB could be responsible for the period decrease, allowing us to derive a synchronisation timescale. \n\n\nThe results of our analysis are limited by the precision of the available trigonometric parallax. As the Gaia mission proceeds, the precision and accuracy of the trigonometric parallax will improve, which will narrow down the uncertainties of the stellar parameters.\nA very important goal is to detect spectral signatures from the companion and to measure the radial velocity curve of the companion. We failed to do so, because the infrared spectra at hand are of insufficient quality. The future IR instrumentation on larger telescopes, such as the ESO-ELT, will be needed. A high precision measurement of the radial velocity curves of both components will then allow us to derive an additional constraint on mass and radius from the difference of the stars' gravitational redshifts \\citep{vos13}. \nSuch measurements will give an independent determination of the nature of the companion and will help to test evolutionary models for low mass star near the hydrogen burning limit via the mass-radius relation.\n\nThe combination of many different methods allowed us to constrain the masses of both components much better without having to assume a canonical mass for the sdB. This is only the fourth HW Vir system for which this is possible.\n\n\n\n\n\n\n\n\\section*{Acknowledgements}\nD.S. is supported by the Deutsche Forschungsgemeinschaft (DFG) under grant HE 1356\/70-1 and IR190\/1-1.\nV.S. is supported by the Deutsche Forschungsgemeinschaft, DFG through grant GE 2506\/9-1. S.L.C. is supported by an STFC Ernest Rutherford Fellowship ST\/R003726\/1.\nDK thanks the SAAO for generous allocations of telescope time and the National Research Foundation of South Africa and the University of the Western Cape for financial support. VSD, SPL and ULTRACAM are supported by the STFC.\n We thank J. E. Davis for the development of the \\texttt{slxfig} module, which has been used to prepare figures in this work. \\texttt{matplotlib} \\citep{2007CSE.....9...90H} and \\texttt{NumPy} \\citep{2011CSE....13b..22V} were used in order to prepare figures in this work. This work has made use of data from the European Space Agency (ESA) mission\n{\\it Gaia} (\\url{https:\/\/www.cosmos.esa.int\/gaia}), processed by the {\\it Gaia}\nData Processing and Analysis Consortium (DPAC, \\url{https:\/\/www.cosmos.esa.int\/web\/gaia\/dpac\/consortium}). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the {\\it Gaia} Multilateral Agreement. Based on observations at the Cerro Paranal Observatory of the European Southern Observatory (ESO) in Chile under the program IDs 087.D-0185(A), and 098.C-0754(A). Based on observations at the La Silla Observatory of the European Southern Observatory (ESO) in Chile under the program IDs 082.D-0649(A), 084.D-0348(A), and 098.D-679. \nThis paper uses observations made at the South African Astronomical Observatory.\nWe made extensive use of NASAs Astrophysics Data System Abstract Service (ADS) and the SIMBAD and VizieR database, operated at CDS, Strasbourg, France.\n\n\\section*{Data availability statement}\nMost data are incorporated into the article and its online supplementary material. All other data are available on request.\n\n\n\n\n\\bibliographystyle{mnras}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction} \\label{sec:intro}\n\nThe Apache Point Observatory Galactic Evolution Experiment 2 (APOGEE-2; \\inprep{S.~Majewski et al.~in prep}) is one of the programs in the Sloan Digital Sky Survey IV \\citep[SDSS-IV;][]{blanton_2017} and the successor survey to APOGEE \\citep[referred to as APOGEE-1, hereafter, for clarity;][]{majewski_2017} in SDSS-III \\citep{eisenstein_2011}. \nAPOGEE-2 operates in both the Northern and Southern hemispheres, with the original APOGEE spectrograph on the Sloan Foundation Telescope at Apache Point Observatory \\citep[APO;][]{gunn_2006} and an almost clone spectrograph on the Ir\\'en\\'ee DuPont telescope at Las Campanas Observatory \\citep[LCO;][]{Bowen_1973}; both instruments are described in \\citet{wilson_2019}.\nWhen referring to the SDSS-III program, we will use ``APOGEE-1'' and, for the combination of APOGEE-1 and APOGEE-2, we will use ``APOGEE'' to encompass the joint dataset.\\footnote{Because SDSS data release are cumulative, the user will find the distinction between APOGEE-1 and APOGEE-2 to be academic. Yet, in terms of targeting, targeting-flags, and some elements of operations, there are many notable distinctions.}\nWhen necessary to specify a specific APOGEE-2 survey component, ``APOGEE-2N'' (for APOGEE-2 North) refers to the survey, data, or instrument associated with APO, and ``APOGEE-2S'' (for APOGEE-2 South) to those at LCO. \nIf we need to refer to the spectrograph specifically, APOGEE-N is the spectrograph at APO and APOGEE-S is the spectrograph at LCO.\n\n\n\\defcitealias{zasowski_2017}{Z17}\n\\defcitealias{zasowski_2013}{Z13}\n\nThe scientific goals of the APOGEE-1 survey \\revise{in SDSS-III \\citep{eisenstein_2011}}, and how those scientific goals were mapped into hardware, software, and data processing requirements, are given in \\citet{majewski_2017}. \nThe overall targeting strategy for APOGEE-1, including a description of APOGEE-1's ancillary programs, was given in \\citet[][hereafter Z13]{zasowski_2013}. \nThe cornerstone targeting strategy for APOGEE-1 was the use of a simple set of color and magnitude criteria in de-reddened color magnitude diagrams (CMDs) that permits precise modelling of the survey selection function; this targeting strategy forms the ``main red star sample'' that aims to target late-type giants based on their intrinsic colors in the near-infrared (NIR) and mid-infrared (MIR). \nThere were, however, deviations from this strategy when deemed necessary to best achieve specific scientific goals---for example, the use of Washington$+DDO51$ photometry to pre-select likely red giant stars in the Milky Way halo among the dominant disk foreground of dwarf type stars \\citep[the technique is described in][]{majewski_2000}. \nWhile some of these deviations require only a modification to the selection function, other programs demanded the selection of individual stars for explicit inclusion in the survey, such as confirmed member-stars in open clusters \\citep[a detailed description is given in][]{frinchaboy_2013}, or the targets required for ancillary programs \\citep[such as the M\\,31 star clusters presented in][]{sakari_2016}, reflecting the specificity of their focused scientific goals.\nThus, \\citetalias{zasowski_2013} not only included the methodology for target selection, but also how to identify the methodology for a given source using a set of targeting flags. \n\\revise{APOGEE-1 observations occurred from September 2011 until July 2014, with Data Releases in 2013 \\citep[DR10;][]{dr10_sdss,dr10_apogee}, 2015 \\citep[DR12][]{dr12_sdss,holtzman_2015}, and the final final data release from SDSS-III in 2016 \\citep[DR13][]{dr13_sdss,holtzman_2015}.}\n\n\\revise{In SDSS-IV \\citep{blanton_2017}}, APOGEE-2 continues the large-scale goal of chemo-dynamical mapping of the key structural components of the Milky Way and its environment in both the Northern Hemisphere and the Southern Hemisphere, where it expands to new scientific areas due to its access to the full sky \\revise{and a six-year operational timescale for APOGEE-2N} (\\inprep{S.~Majewski et al.~in prep}).\nAPOGEE-2 largely adopted the same underlying targeting strategies described above for APOGEE, including its foundation ``main red star sample.''\n\\revise{However,} APOGEE-2 also elevated several ancillary projects from APOGEE-1 to key ``core'' programs, such that the science goals of these programs became part of APOGEE-2's primary \\revise{scientific goals}, while, at the same time, granted time to new ancillary programs that further expand the impact and legacy of APOGEE \\revise{as a scientific project}.\n\\revise{APOGEE-2 also included new targeting classes, spanning from RR Lyrae in the inner Galaxy to red giants in the dwarf Spheroidal companions to the Milky Way. \nWith its broader goals, longer timeline, and dual-instruments, APOGEE-2 presented a significant change from the overall targeting strategies of APOGEE-1.}\n\nThe initial targeting plans for APOGEE-2 were presented in \\citet[][hereafter Z17]{zasowski_2017} and focused on the overall strategy for the \\revise{joint APOGEE-2N and APOGEE-2S observing programs.}\nThe publication of \\citetalias{zasowski_2017} was timed to accompany the first \\revise{APOGEE-2} data release from SDSS-IV \\citep[Data Release 14;][DR14]{dr14} \\revise{that contained new observations with the APOGEE-N spectrograph from July 2014 to July 2016 \\citep{holtzman_2018}.}\n\\revise{A subsequent APOGEE-2 Data Release occurred in 2019 with Data Release 16 \\citep[DR16;][]{dr16,Jonsson_2020} that included observations from 2016 to 2018, including $\\sim$1 year of observations from APOGEE-2S.\nThe final Data Release will occur in December 2021 (DR17) and will include the complete observational program from APOGEE-2 alongside new processing of APOGEE-1 observations.} \n\nWhen \\citetalias{zasowski_2017} was written, modifications to this \\revise{base plan} were anticipated, \\revise{at a minimum} due to the then-incomplete commissioning of the APOGEE-2S instrument \\citep[see][]{wilson_2019} and the on-going APOGEE-2N ancillary program allocation process \\revise{(there were application cycles in 2015 and 2017, with implementation of programs often taking months, and observations taken over several years)}.\n\\revise{Since \\citetalias{zasowski_2017}, additional changes in the targeting plan have also occurred through the Contributed Programs in APOGEE-2S and the Bright Time Extension (BTX) for APOGEE-2N, the latter served as an effective 1.5 year extension due to unanticipated gains in operational efficiency at APO.}\n\nThe objective of this paper is to present the summation of modifications to the field plan and targeting strategy for APOGEE-2N that was presented in \\citepalias[][]{zasowski_2017} as it applies to the \\revise{now complete} APOGEE-2N survey from APO (with observations from 2014 to 2020). \nA companion paper, F.~Santana et al. (submitted), provides a complementary presentation of the changes made in the APOGEE-2S observing program at LCO \\revise{that began in February 2017 and completed in January 2021 as were required by on-sky performance and time allocations.} \n\\revise{These papers are intended to be a formal presentation of survey strategy and motivations.}\n\\revise{The material is presented separately for APOGEE-2N and APOGEE-2S to make apparent the different contexts through which the survey plans from \\citetalias{zasowski_2017} were modified in terms of nights available, target visibility, instrumental throughput, and operational procedures that required distinct planning and implementation strategies.}\n\\revise{The web documentation accompanying Data Release 17 contains a streamlined and, perhaps more practical, presentation of the total observational programs spanning APOGEE-1 and APOGEE-2.\\footnote{The DR16 Documentation for Targeting is here: \\url{https:\/\/www.sdss.org\/dr16\/irspec\/targets\/}, for Special Programs here: \\url{https:\/\/www.sdss.org\/dr16\/irspec\/targets\/special-programs\/}, and a discussion of Selection Biases and the Survey Selection Function is here: \\url{https:\/\/www.sdss.org\/dr16\/irspec\/targets\/selection-biases\/}. The URLs can be updated for ``dr17'' to point to the DR17 versions when DR17 becomes public in December 2021.} }\n\n\n\\revise{This paper for APOGEE-2N and its companion for APOGEE-2S (F.~Santana et al. submitted)\\ are} intended to serve in supplement to the existing APOGEE-2 targeting paper, \\citetalias{zasowski_2017}, \\revise{and are intended to accompany Data Release 17 (DR17) planned for December 2021 that will release the cumulative APOGEE-1 and APOGEE-2 observations (\\inprep{J.~Holtzman et al.~in prep.}).}\nThus, readers new to APOGEE-2 are strongly advised to review \\citetalias{zasowski_2017} as well as \\revise{the APOGEE-1 Targeting Paper \\citetalias{zasowski_2013} that provides the base-line strategies}.\nOnly those programs \\revise{in APOGEE-2N} that were modified relative to the descriptions in \\citetalias{zasowski_2017} or programs that are completely new are discussed in this paper. \nHowever, certain keystone-information regarding targeting is repeated from \\citetalias{zasowski_2017} to provide enough context that elements of this paper can stand alone \\revise{(predominantly in \\autoref{sec:prelims})}.\nBecause of the broad scientific scope of the programs in APOGEE-2N, descriptions of prior work from APOGEE-1 along with other scientific background is required to explain how that prior work has influenced the targeting and implementation of programs within APOGEE-2N. \n\nThis paper is organized as follows. \nA summary of general information regarding APOGEE-2N targeting and the general motivation for the modifications of its strategy are given in \\autoref{sec:prelims}.\nThe paper is organized by the \\revise{significance of the} modification from \\revise{the original survey strategies and plans for APOGEE-2N described in} \\citetalias{zasowski_2017} from the most significant to the least significant.\n\\autoref{sec:strategy} details \\revise{changes to the targeting strategies or selection algorithms described in \\citetalias{zasowski_2017}.}\n\\autoref{sec:new_programs} describes the newly added programs that were not described in \\citetalias{zasowski_2017} \\revise{because they were added after that publication because survey observations beyond the initial six year plan became feasible}.\n\\autoref{sec:expansion} describes the extensions of some programs in \\citetalias{zasowski_2017} during the BTX \\revise{to include additional fields or span longer time baselines}.\nFinally, \\autoref{sec:wrapup} provides a summary that reflects on the overall process of modifying the combined APOGEE-1 and APOGEE-2 survey over its decade of operations \\revise{at APO}, with a focus on highlighting intentional choices in our survey-planning \\revise{specific to its targeting operations} that enabled the project to grow and adapt. \n\\revise{As discussed in F.~Santana et al. (submitted), these strategies were pivotal in the successful completion of the APOGEE-2S scientific program.}\n\n\n\\section{Preliminaries} \\label{sec:prelims}\n\nTo aid the reader, in the subsections that follow, we briefly review observing and targeting concepts from \\citetalias{zasowski_2013} that continue to be used in APOGEE-2 (\\autoref{sec:obsoverview} and \\autoref{sec:targoverview}).\nA number of SDSS- and APOGEE-specific terms and acronyms will be introduced and, as an additional aid, a glossary of these terms is given in \\autoref{sec:glossary} (following that of \\citetalias{zasowski_2013} and \\citetalias{zasowski_2017}, but also including new terms used for this present paper). \nAfter these summaries, we provide a broad overview of the updates to the APOGEE-2N survey that motivated modifications to its targeting (\\autoref{ssec:ancprograms_overview} and \\autoref{ssec:btx_overview}).\nWe close by reviewing the final field plan for the APOGEE-2 Survey (\\autoref{ssec:field_plan}) and summarizing the datasets used in this paper (\\autoref{ssec:datasets}).\n\n\\subsection{Observational Framework} \\label{sec:obsoverview}\n\nAs described in depth by \\citet{wilson_2019}, both APOGEE spectrographs are fed by fibers that are held to a target position using plug-plates; thus, the specifics of the plate and fiber positioning places fundamental constraints on the targeting for the survey.\nPlates designed for the APOGEE-N spectrograph have a 3$^{\\circ}$ diameter field-of-view; each fiber is approximately 3\\arcsec\\ in diameter on the sky, and two adjacent fibers cannot be placed closer than $\\sim$72\\arcsec\\ apart, the ``fiber collision radius''. \n\\revise{In addition, no targets can be placed within 96$''$ of the field center due to the central post that supports the plate \\citep{Owen_1994}.}\nEach unique APOGEE pointing on the sky is referred to as a field, and each unique set of targets selected for that field is called a design; even a difference of a single target will create a unique design that will be indicated by a unique design ID (an integer assigned to each design).\nDesigns are then drilled onto plates, but the exact locations of target holes on given plate is set by the intended hour angle at observation, to minimize the impacts of differential refraction across the field-of-view and during an integration \\citep[see discussion in][]{majewski_2017,wilson_2019}. \nThus, a given field can have multiple designs, and any given design can have multiple associated plates. \nThe final APOGEE-2 field plan is given in \\autoref{fig:fieldmap}, where each circle represents a single field, but the number of targets per field depends on the number of designs and the number of stars common to those designs. \n\n\nEach observation of a given plate is known as a ``visit''; a typical visit consists of about one hour of integration that is broken into eight exposures separated into two sets of spectral dithering sequences (each an ``ABBA'' sequence).\\footnote{Half pixel dithering in the spectral dimension is employed by APOGEE to recover Nyquist sampling of the intrinsic spectrograph resolution \\citep[see][]{majewski_2017,wilson_2019}.}\nDeviations from this procedure occur when a plate has been designed specifically to focus on faint targets, and the changes are in two forms, intended to improve the overall signal-to-noise ($S\/N$) in the visit spectra: \n\\begin{enumerate} \\itemsep -2pt\n \\item beginning in late 2017, when a given plate has more than five faint targets (defined as $H$\\textgreater~13.5), only a single dither sequence is used over the same time frame, with each individual exposure being doubled in length \n (such a visit is referred to as ``DAB''); and \n \\item beginning in late 2019, when a given plate is dominated by faint stars, the plate receives an extra dither pair in a single DAB-style visit, such that its visit would sum to an hour and a half of exposure time (referred to as ``TDAB''). \n\\end{enumerate}\nA given field is planned to have a specific number of visits determined by its target magnitude depth, and these visits are implemented according to specific temporal spacing requirements, referred to as the ``cadence rule''.\n\\revise{Cadence rules vary by the scientific program. \nIn the main red star sample, the cadence rules were designed with a spacing optimized to the anticipated radial velocity variation from close binaries on the red giant branch with the goal of, at a minimum, removing such stars as a potential source of uncertainty in studies of detailed Galactic dynamics and, more optimally, to determine the true systematic velocity of the binary system for use in such studies.\nWe note that programs specifically aiming to characterize and not just detect such variations have more complicated cadence rules.}\n\nGenerally, APOGEE aims for spectra with a minimum $S\/N$ of 100 per pixel to ensure the highest quality stellar parameters and chemical abundances. \nThis $S\/N$ target is a fundamental constraint on the targeting and, along with the intended magnitude limit, determines the number of visits a plate will receive. \nBecause the APOGEE reduction pipeline performance has proven to be similarly reliable from $S\/N$ = 70 to 100 \\citep[for a detailed performance assessment, see][]{Jonsson_2020}, for some programs the targeting-imposed magnitude limits have been altered; when this is the case, it will is noted.\\footnote{For the purposes of this paper, ``reliable'' from $S\/N$ = 70 to 100 refers to the observation that the stellar parameter and chemical abundance uncertainties, generally, over this $S\/N$-range are similar. In contrast, for $S\/N<70$, the uncertainties increase rapidly. \nBy this reasoning, the \\texttt{STARFLAG} bit for \\texttt{SN\\_WARN} is automatically set for all stars with $S\/N<70$ \\citep{holtzman_2015,holtzman_2018,Jonsson_2020}.} \n \n \n \n\\autoref{tab:maglims} summarizes the approximate $S\/N$ attained for a typical star at the intended faint $H$ magnitude limit on some common design configurations (we note that the brightest target allowed is $H\\sim7$ due to instrument detector saturation concerns). \nThe visit number needed to reach $S\/N = 100$ for a given plate magnitude limit is shown in \\autoref{tab:maglims} and provides the guideline for survey planning and scheduling. \nPlates are generally categorized as ``3-visit plates'', ``6-visit plates'', etc., according to the prescriptions shown. \nObviously, the actual on-sky $S\/N$ performance achieved during visits varies due to observing conditions.\n\nHowever, as can be seen, any $H=11$ star that might happen to be in a 24-visit plate, would obtain an estimated $S\/N\\sim490$, greatly exceeding what is required to achieve our scientific goals. \nFor this reason, stars in targeted fields are often grouped by magnitude bins to form ``cohorts''; cohorts are normally only observed for the number of visits required to obtain $S\/N\\sim$100 for the faintest star in the cohort. \nMany fields have more than one cohort per design and this strategy combines each faint cohort with several brighter cohorts, so that the same faint stars are included in multiple designs while bright targets are switched out; this strategy increases the overall number of stars in a field that are included in the survey.\nAs a rule, a design will have no more than three cohorts, generally referred to as the `short,' `medium', and `long' cohorts, where the short cohorts require the fewest visits, and the long cohorts are included on the full complement of field designs to achieve the maximum exposure time.\nThe visits per cohort and relative number of stars in each cohorts are specific to a scientific program \\citepalias{zasowski_2013,zasowski_2017}. \n\n\\input{Tables\/table1_signaltonoise}\n\n\\subsection{General Targeting Overview} \\label{sec:targoverview}\n\n`Targeting' is our term for the implementation of the observational strategies of APOGEE to achieve its scientific goals through the assignment of fibers to targets. \nBoth APOGEE spectrographs have 300 total fibers. \nFor all plates designed in APOGEE-2, 15 fibers are assigned to hot (more ``featureless'') stars for the derivation of telluric absorption corrections, 35 fibers are assigned to ``blank sky'' positions distributed across the plate \\citepalias[see][]{zasowski_2013}, and 250 fibers are available for science programs.\n\nBecause our data are only as good as the calibrations, the selection of suitable calibration fibers, both tellurics and blank sky, occur at high priority.\nTelluric stars are selected to be the bluest stars in a given field that can also achieve $S\/N>100$ in a single visit; because of their importance, telluric stars are selected and assigned first.\nCandidate ``blank sky'' positions are selected as regions with no 2MASS point source \\citep{Skrutskie_06_2mass} within 6$\\arcsec$ of the position.\nThough the selection of a large number of candidate sky regions occurs early in the design process, their fibers are not assigned until the science targets have been selected and \\revise{then the 35 blank sky fibers are selected from the candidate positions and distributed uniformly across the plate. \nThe design will not be drilled unless the 15 tellurics and 35 blank sky fibers are successfully allocated.}\n\nAllocation of science fibers occurs in a two phase process. \nIn the first phase, `special targets' are assigned following star-by-star priorities.\nSpecial targets are generally stars that are unlikely to be picked from our standard algorithms; specific cases for the main survey include \n (i) extremely rare or sparse targets (for example, a specific type of photometric variable star), or \n (ii) known member stars of a substructure (for example, a star cluster or dwarf galaxy).\nAll targets for scientifically-focused Survey programs (for example, programs specifically targeting stars with {\\it Kepler}\\ observations) and targets from Ancillary Science Programs (\\autoref{ssec:ancprograms_overview}) are considered special targets. \nTypically, special target lists are prepared and prioritized field-by-field by the relevant science working group or Ancillary Science Program principal investigator (PI) and submitted to the targeting team for plate design. \nIn all cases, special targets are identified by special targeting flags (\\autoref{tab:targeting_bits}).\nIf multiple sets of special targets exist for a given field, the special programs themselves are given a priority schema such that the special program with the smallest number of special targets is given the highest priority; the one exception to this rule is a field assigned to a specific program, like an open cluster, in which case the targets for that program come at highest priority. \n\nOnce special targets are assigned, the remaining fibers are allocated following specific selection algorithms.\nThe core of APOGEE is the ``main red star sample'' \\citepalias{zasowski_2013,zasowski_2017}, which is selected by fixed and simple color-magnitude criteria.\nThe color-limit and color-selection criteria adopted are set by the primary Galactic component targeted by the plate (effectively, disk, bulge, halo) and the magnitude range is set in accordance with the number of visits and cohorting scheme for a given field.\nThe main red star sample is well described in \\citetalias{zasowski_2013} and \\citetalias{zasowski_2017}, with modifications for the APOGEE-2S survey given in F.~Santana et al. (submitted). \nThe definitions of the plates designed to target specific Galactic components and their associated color cuts are given in \\autoref{tab:colorcuts}. \n\nFor fields that are devoted to specific science cases, the majority of fibers are assigned to that science case as special targets and the remaining targets are drawn from the appropriate ``main red star sample'' selection function for that part of the sky. \nFor example, fibers not assigned to dwarf spheroidal galaxy members or candidate members would be assigned following the halo selection function and cohort scheme for a 24-visit field \\citepalias[see \\autoref{sec:dsph} in this work and also][]{zasowski_2017}.\n\n\\input{Tables\/table2_colorcuts}\n\n\\subsubsection{Targeting Bits} \\label{sec:targbits}\nAPOGEE-2 uses bit flags to convey the targeting schema. \nThe flags are not a comprehensive way of identifying stars that meet particular scientific criteria. \nRather the flags serve to identify why a particular set of stars was targeted to enable study of (and correction for) the selection function of the survey. \n\\autoref{tab:targeting_bits} provides a summary and description of the APOGEE-2 targeting bits that span four bit flags; three are used currently, \\texttt{APOGEE2\\_TARGET1}, \\texttt{APOGEE2\\_TARGET2}, and \\texttt{APOGEE2\\_TARGET3}, and the fourth, \\texttt{APOGEE2\\_TARGET4}, was added to the DR17 data model but is not currently in use. \nThe specific bits that have been put into use since \\citetalias{zasowski_2017} are shown in \\autoref{tab:targeting_bits} in bold for ease of identification. \nThe newly allocated bits largely indicate those targets selected under a specific schema, which will be described in the sections that follow. \nF.~Santana et al. (submitted)\\ provides a more comprehensive discussion of modifications to the targeting bits relative to that described in \\citetalias{zasowski_2017}. \n\n\\input{.\/Tables\/table3_bitflags}\n\n\\subsection{Ancillary Science Programs} \\label{ssec:ancprograms_overview}\n\nIn APOGEE-1 and in APOGEE-2N, a fiber reserve was intentionally budgeted into the survey plan for Ancillary Science Programs. \nThis aspect of the survey design has been exceptionally beneficial, as Ancillary Science Programs from APOGEE-1 (e.g., the APOKASC and KOI programs, see \\citetalias{zasowski_2013}) eventually became core components of APOGEE-2N \\citepalias{zasowski_2017}.\n\nIn APOGEE-2N, approximately 5\\% of the fiber hours \\revise{for a six years of bright time operations} were reserved for Ancillary Science Programs. \nA fiber hour is defined as one visit for a single fiber, such that a single plate represents 265 fiber hours allocated for stars; \\revise{allocation by fiber hour} allows for more flexibility in the implementation of Ancillary Science Programs. \nThese were awarded by way of a competitive, internal review process that resulted in the selection of 23 programs \\revise{over two application cycles}. \nThe corresponding allocations could be through sparse fibers across many plates, a concentration of targets in dedicated (often new) fields or APOGEE-N observations via the fiber link to the NMSU 1-meter telescope \\citep{holtzman_2010,holtzman_2015}.\n\\citetalias{zasowski_2017} described the general process of selecting and implementing Ancillary Science Programs, but, because the full implementation and even allocation of some of these programs was then still underway, could not include detailed descriptions of the programs, as had been done for APOGEE-1 Ancillary Science Programs in \\citetalias{zasowski_2013}.\n\nA \\revise{description} of each program is given in \\autoref{sec:ancillary2015} for the 2015 programs and \\autoref{sec:ancillary2017} for the 2017 programs. \n\\revise{These descriptions include the scientific motivations for the observations, the observations undertaken, and the specific goals of the program. \nThe scope of the programs vary a great deal. \nAll Ancillary Targets are input into the targeting procedure as ``special targets'' and are then drilled onto plates at high priority. \nOnly in rare cases would the targets be modified (e.g., if the star was too bright and could compromise other observations). \n}\n\n\nFor ease, \\autoref{tab:anc_sum} summarizes the 23 programs with their title and subsection reference, contact scientists, and appropriate targeting flag.\nDedicated fields for ancillary programs have \\texttt{PROGRAMNAME} given as ``ancillary,'' \\revise{such that whole fields could be identified and all} individual targets are flagged as summarized in \\autoref{tab:anc_sum}.\nWe note that the timing of the 2017 call for Ancillary Science Programs relative to the Bright Time Extension (BTX; \\autoref{ssec:btx_overview}) resulted in some programs from the former being absorbed into the latter; if this occurred for a particular program, it is noted both in the main text and in the appendices. \n\n\\input{.\/Tables\/table4_ancprograms.tex} \n\n\\subsection{The Bright Time Extension} \\label{ssec:btx_overview}\nThe \\revise{Bright Time Extension} (BTX) was an expansion of APOGEE-2N programs to fill an excess of bright time anticipated toward the end of the SDSS-IV survey as a result of improved observational efficiencies and better than average weather at APO during the first few years of the survey. \nPlanning for the BTX began in mid-2017 and fiber-hours were awarded through an open call across SDSS-IV that anticipated approximately $\\sim$1200 hours would be available (corresponding to the equivalent of $\\sim$1.5 years, \\revise{or 20\\%}, of APOGEE-2 bright time observations). \nThis time was divided between a mix of programs initiated by the APOGEE-2 team and jointly with the ``After SDSS-IV (AS4)'' scientific collaboration. \nThe latter has since been formally established as SDSS-V\\footnote{\\url{https:\/\/www.sdss5.org\/}}, but the implementation of these programs used the acronym ``as4'' and throughout this paper we will use both AS4 and SDSS-V when referring to these programs. \n\n\nThe general APOGEE-2N BTX strategy had three primary components:\n\\begin{enumerate}\n \\item expanding ``core'' programs and modifying their target selection to better meet the strategic science objectives of the APOGEE-2 survey (\\autoref{sec:strategy}), \n \\item the construction of new programs as a reaction to developments in the scientific community, to secure a more comprehensive legacy of the APOGEE survey (\\autoref{sec:new_programs}), or to build synergy with the After Sloan-IV collaboration, and\n \\item the expansion of some programs to better meet their overall scientific goals (\\autoref{sec:expansion}).\n\\end{enumerate}\nAll observations for the BTX have `\\_btx' appended to their \\texttt{FIELD} and \\texttt{PROGRAMNAME} tag; one exception to this policy was the ``odisk'' program.\\footnote{We note that the Outer Disk program was a completely new program and did not need `\\_btx' appended to differentiate it and its targeting strategy from a similar non-BTX program.}\n\n\n\\subsection{The Final Field Plan} \\label{ssec:field_plan}\n\n\\autoref{fig:fieldmap} shows the final field plan for the APOGEE-2 survey overlaid on the \\citet{sfd98} all sky infrared dust map in Galactic coordinates, with colored circles representing APOGEE-2N (this paper and \\citetalias{zasowski_2017}) and grey circles representing APOGEE-1 \\citepalias{zasowski_2013} and APOGEE-2S F.~Santana et al. (submitted). \nThe color-coding in \\autoref{fig:fieldmap} is used to indicate the primary scientific program for a given field, which, in some cases, combine programs core to APOGEE-2 (e.g., disk, halo) and BTX expansions. \nThe labels roughly correspond to terms used in the \\texttt{programname} tag in the summary files produced for the SDSS-IV data releases \\citep[e.g.,][]{Jonsson_2020}\\footnote{See also the DR16 documentation: \\url{https:\/\/www.sdss.org\/dr16\/irspec\/dr_synopsis\/}}; the tag(s) specific to a program will be given with its description. \n\n\\begin{figure*}\n \\begin{mdframed}\n \\centering\n \\includegraphics[width=\\textwidth]{f01_fieldplan.pdf}\n \\caption{The complete APOGEE-1 and APOGEE-2 field plan overlaid on the \\citet{sfd98} all sky infrared dust map and shown in Galactic coordinates. \n Fields designed and observed for APOGEE-2N are color-coded by program and fields designed and those observed in APOGEE-1 or APOGEE-2S are shown in gray.\n The FOV for APOGEE-S is smaller than APOGEE-N, such that APOGEE-S pointings can be distinguished from APOGEE-N pointings by the point size. \n The general motivations for this field plan are given in \\citetalias{zasowski_2013} and \\citetalias{zasowski_2017}, whereas this paper describes the Ancillary Science Programs and the BTX. \n }\n \\label{fig:fieldmap}\n \\end{mdframed}\n\\end{figure*}\n\n\\subsection{APOGEE Datasets Used in this Paper} \\label{ssec:datasets}\n\n\\revise{Throughout this paper, we will show results from APOGEE-1 and APOGEE-2 using the final sample and pipeline to be released as Data Release 17 (DR17) planned for December 2021 (J.~Holtzman et al., in prep.). \nEach of the image reduction, radial velocity measurements, spectral combination, and the derivation of stellar parameters and chemical abundances have been modified from that of Data Release 16 described in \\citet{Jonsson_2020}. \nFor the purposes of this paper, we typically show targets using targeting-related planes (typically \\emph{Gaia} color-magnitude diagrams), but for the evaluation of some targeting strategies, we will use the stellar parameters ($\\log{g}$\\ and $T_{\\rm eff}$).\nFrom the perspective of what is used in this paper, however, the impacts are overall small and the reader may use intuition from \\citet{Jonsson_2020} to understand these limited ASPCAP-results when presented.\n}\n\nWe will often refer to ``tags'' or ``fields'' that occur in various APOGEE-2 data products using their official names.\nSuch ``tags'' are found in multiple APOGEE data products, as in the headers affiliated with APOGEE spectra, as well as in the more commonly used summary files, \\texttt{allStar} and \\texttt{allVisit}. \nGenerally the ``tags'' described here will refer to those in the summary files, unless otherwise noted and tags will be referred to in true-type fonts, e.g., {\\tt APOGEE\\_ID}. \nA full description of the data products and their affiliated data models are given in the online documentation for the Data Release.\\footnote{For DR16: \\url{https:\/\/www.sdss.org\/dr16\/irspec\/spectro_data\/}}\n\nTo estimate how successful our targeting methods were, throughout the paper we will classify stars using their DR17 ASPCAP $\\log{g}$\\ into dwarfs, sub-giants, and giants. \nTo determine the numbers of stars observed in each of these stellar classifications, we will only compare the numbers of those stars with calibrated measurements \\citep[see][(J.~Holtzman et al., in prep.)]{Jonsson_2020}, e.g., those with the \\texttt{LOGG} tag populated. \nWe will use the following definitions:\n a dwarf is a star with $\\log{g}$ \\textgreater\\ 4.1,\n a giant has -1 \\textless $\\log{g}$ \\textless 3.5, and \n subgiants have 3.5 \\textless $\\log{g}$ \\textless 4.1. \n\n\\revise{Because spectro-photometric can be determined for stars that are well beyond the current reach of trigonometric parallaxes from {\\it Gaia}, will also use spectro-photometric distances} following the methods of \\citet{Rojas-Arriagada_2017,Rojas-Arriagada_2019,Rojas-Arriagada_2020}, but applied to the \\sout{internal, incremental data release mentioned above that uses the DR16 pipeline}\n\\revise{final DR17 dataset described above (e.g. not the specific datasets described in those works, but using the same spectro-photometric distance method)}. \n\\revise{As described in \\citet[][their Section 2.2]{Rojas-Arriagada_2020}, the APOGEE spectroscopic parameters ($T_{\\rm eff}$, $\\log{g}$, and [M\/H]) are used to match an observed star to potential absolute magnitudes on PARSEC stellar evolution tracks \\citep{Bressan_2012,Marigo_2017}.\nUsing the observed 2MASS photometry ($JHK_s$), the distance and extinction can be determined.}\nA detailed comparison of these distances to those derived by other studies \\revise{including {\\it Gaia}~trigonometric parallaxes} is given in \\citet[][their appendix A]{Rojas-Arriagada_2020}.\n\n\\section{Targeting Strategy Changes in the Bright Time Extension} \\label{sec:strategy}\n\nThis section discusses changes to the general targeting strategy established in \\citetalias{zasowski_2017} as they apply to observations planned in the BTX. Fields that are subject to these changes in strategy have `\\_btx' appended to their field name (and given in the \\texttt{FIELD} tag) and also have `btx' appearing in the \\texttt{PROGRAMNAME} tag.\\footnote{There is one exception to this rule in the case of the new Outer Disk BTX program which can be identified according to it's `odisk' \\texttt{PROGRAMNAME} tag (\\autoref{sec:odisk}).} \n\\revise{Two strategy modifications are described: (1) a change in priority star selection (\\autoref{sec:mastar}) and (2) the remaining sections discuss a major targeting change aimed at bolstering the sample of distant halo stars.}\n\n\\subsection{Telluric Selection and MaStar Co-Targeting} \\label{sec:mastar}\n\nThe SDSS-IV MaNGA Stellar program \\citep[MaStar;][]{yan_2019} uses the MaNGA fiber bundles \\revise{\\citep{bundy_2015}} to collect stellar spectra; this is distinct from APOGEE-2N co-targeting with MaNGA galaxy observations that is described in \\citetalias{zasowski_2017}. \n\\revise{The aim of these observations is to build a \\emph{fully empirical} stellar library with a broad span in stellar type and abundance for use in the MaNGA project \\citep[][]{yan_2016,yan_2019}.}\nBecause the observations for MaStar occur in tandem with the APOGEE-2N observations using the same plates, the plate design process takes into account the locations of MaStar targets, and this can influence APOGEE-2N targeting.\n\n\nPrior to the BTX, the MaStar targets were included on the plate at the lowest priority, with the intent of having the smallest impact on the APOGEE-2 targeting. \nHowever, this resulted in the under-sampling of some of some of MaStar's target classes due to targeting collisions or conflicts with APOGEE-2N targets. \nOne example deficiency in the MaStar sampling occurred for very luminous B and A type stars with low foreground extinction; this is a natural consequence of APOGEE's reliance on these stars as telluric calibrators, which are the highest priority targets.\n\nTo remedy the lack of such stars in the MaStar sample, the priority for plate design within the BTX fields was altered so that the highest priority MaStar targets were selected first. \nThe revised default priority scheme for the BTX is as follows: \n\\begin{enumerate} \\itemsep -2pt\n \\item MaStar high priority targets (but see description of reconciliation process below),\n \\item APOGEE telluric calibration stars,\n \\item APOGEE targets (but see description of reconciliation process below),\n \\item MaStar low priority targets,\n \\item MaStar standard stars,\n \\item MaStar sky fibers,\n \\item APOGEE sky fibers.\n\\end{enumerate}\nThough this scheme solves the MaStar under-sampling issue, it poses a potential problem for APOGEE-2N science goals.\n\nThere are two impacts: the selection of the same target (a conflict) and the selection of a neighboring target the precludes another target (a collision).\nFor the latter, the MaStar fiber-bundle collision radius (102\\arcsec) is larger than that for the APOGEE-N fibers (72\\arcsec) and, as a result, MaStar targets are not a one-to-one replacement of an APOGEE-N target in a given plate design; this is particularly challenging for plates with spatially clustered targets. \nSome of the more complicated cases include targeting for star clusters, photometric objects of interest (e.g., {\\it Kepler}, {K2}, and {\\it TESS}), and confirmed distant halo stars; in most of these cases, the scientific motivation to collect data for a specific target is similar for MaStar and APOGEE-2N. \nThis competition for targets created a logistical and managerial challenge.\n\nThus, a target reconciliation process was constructed that compared the highest priority MaStar and APOGEE-2N targets for a given plate to determine conflicts (same target desired by both surveys) and collisions (where the MaStar fiber bundle precludes an APOGEE-2N target).\nA team of MaStar and APOGEE-2N scientists carefully evaluated these issues (at a typical rate of only a 1-2 incidents per plate, but potentially dozens for the ensemble of plates being designed at a given time) with the aim that both programs were ensured success of their science goals.\nSolutions to these conflicts and collisions included: \n (i) one survey ceding the target, \n (ii) for a multi-visit field, splitting the visits between MaStar and APOGEE-2N data collection, \n (iii) inclusion of an additional design or visit to satisfy the needs of both surveys. \nThis process was time-consuming, but assured mutual success of the respective goals for each survey. \n\nWhile we anticipate the net impact of these priority changes to be small, those science investigations requiring a detailed selection function analysis may want to exclude all plates designed with MaStar modifications --- i.e., all plates that have ``\\_btx'' appended to their field name.\n\n\\begin{figure*}[h]\n \\begin{mdframed}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{f02_halo_mainsurvey}\n \\caption{Evaluation of the Washington+$DDO51$ giant pre-selection in the APOGEE-2N Main Survey halo program for distant halo stars (e.g., \\texttt{PROGRAMNAME}=``halo'' and \\texttt{TELESCOPE}=``apo25m''). \n \n (a) Spectroscopic $\\log{g}$\\ versus $T_{\\rm eff}$\\ for APOGEE-observed giant candidates (grey) and dwarf candidates (black) demonstrating the overall effectiveness of the Washginton$+DDO51$ pre-selection technique. \n Stars with distances between 15 and 25 kpc are shown as green circles and those with distances greater than 25 kpc as yellow diamonds.\n \n (b) Spectro-photometric distance against apparent $H$ magnitude for giant candidates (grey) and the distant star samples (green and yellow). \n Horizontal dashed lines indicate the magnitude limits for targeting at the specified number of visits (labeled to the right).\n \n (c) After separating the giant candidate sample into distance bins ($d$~\\textless~15~kpc, 15 \\textless~$d$~\\textless~25~kpc, 25~\\textless~$d$), the fraction of the sample identified in the long (24-visit), medium (12-visit) and short (3- or 6-visit) ``cohorts'' on the plate designs. \n \n (d) The yield of distant stars within each ``cohort'' defined as $N_{\\rm distant}\/N_{\\rm total}$, with $N_{\\rm total}$ being the number of giant candidates.\n }\n \\label{fig:halo_bad}\n \\end{mdframed}\n\\end{figure*}\n\n\n\\subsection{Sampling Stars in the Distant Halo} \\label{sec:halo}\n\nA high-level scientific goal of APOGEE is to define the chemo-dynamical fingerprint for stars in all of the structural components of the Milky Way. \nOne particularly difficult component to sample is the distant halo, both because it is sparsely populated and because the stars are fainter due to their distance. \nTo sample this distant and diffuse component of the Milky Way, the APOGEE-2 Science Requirements Document (SRD) set a benchmark APOGEE-2N goal of 1000 stars at distances beyond 15 kpc, such that at least 100 were ``distant'' stars beyond 25~kpc and the remaining $\\sim$900 were at ``intermediate'' distances between 15 and 25 kpc.\nThe SRD was specific that these stars were to be identified from ``halo'' targeting (e.g., not counting stars in dwarf satellite galaxies, but including deliberately targeted streams or serendipitous targets in the background or foreground of dwarf satellite galaxies). \n \n \n \n \n \n \n \n \n \n \nBecause only giants can be seen at these distances (given APOGEE's magnitude limits), giants are the targets of interest to probe the Milky Way halo, however, local dwarf stars in the disk of the Milky Way provide significant levels of contamination, providing a challenge to identify giants of interest. \nTo mitigate this foreground contamination, APOGEE-1 used the Washington+$DDO51$ dwarf-giant separation technique \\citep[as defined by][]{majewski_2000} to pre-select likely giant stars.\nThis technique combines the $DDO51$ intermediate-band filter \\citep{McClure_1973}, which is centered on the surface gravity sensitive Mgb triplet at 5051~\\AA, with the Washington $M$ and $T_2$ filters \\citep{Canterna_1976}, which are optimized for temperature separation in late-spectral types.\nAs demonstrated in \\citetalias{zasowski_2013} for the APOGEE-specific use of Washington+$DDO51$, a star is classified as a likely dwarf or giant based on its location in the $M-DDO51$ versus $M-T_2$ color-color diagram.\n\nAs discussed in detail in \\citetalias{zasowski_2017}, the APOGEE-1 Washington+$DDO51$ dwarf-giant separation methodology was adopted for APOGEE-2N. \nThose stars meeting our NIR color-magnitude criteria {\\it and} classified as giants using the Washington$+DDO51$ criterion were targeted at the highest priority for halo fields, with the likely dwarf stars targeted at lowest priority \\citepalias[see Section~7.1 of][]{zasowski_2013}.\nThe APOGEE-1 strategy was continued in APOGEE-2 \\citepalias{zasowski_2017} in new 3-visit ``short'' fields and additional visits to APOGEE-1 fields (so-called ``deep-drill fields'') for a total of 24 visits per field (to reach $H\\sim13.8$).\nThe ``short'' fields had a single short cohort of 3-visits, while the ``deep-drill fields'' had four short cohorts observed for 6 visits each, two medium cohorts of 12 visits, and a single long cohort of 24 visits. \n\n\nAs discussed in \\autoref{sec:badhalo}, at early targeting reviews, it was clear that the halo program was deficient in its number of observed giants in both the intermediate and distant samples defined in the SRD. \nThe subsections that follow first explain the halo-targeting problem in more detail (\\autoref{sec:badhalo}), then describe our BTX targeting scheme where we attempted to remedy this sampling problem (\\autoref{sec:known_giants} and \\autoref{sec:new_halo_targeting}), evaluate the new targeting scheme (\\autoref{sec:how_did_we_do}), and summarize our investigation into halo targeting (\\autoref{sec:halo_sum}). \n\n\\subsection{Detailed Evaluation of Halo Targeting} \\label{sec:badhalo}\n\nThe spectroscopic $T_{\\rm eff}$--$\\log{g}$\\ diagram for the stars in the APOGEE-2N halo program (selected using the \\texttt{PROGRAMNAME} `halo' and \\texttt{TELESCOPE} `apo25m'), for which target selection relied on Washington$+DDO51$ photometry, is shown in \\autoref{fig:halo_bad}a.\nThe data points in \\autoref{fig:halo_bad}a are color-coded by their Washington$+DDO51$ photometric classification as a dwarf (black filled circle; \\texttt{APOGEE2\\_TARGET1} bit 8) or a giant (grey filled circle; \\texttt{APOGEE2\\_TARGET1} bit 7).\n\nFor our initial, targeting review evaluations of the efficacy of our halo targeting and whether we were meeting our SRD goal of probing the more distant halo, we computed spectro-photometric distances of our observed halo stars. \nHere we reproduce that initial assessment from our targeting reviews, but using updated distances \\citep[using the methods of][]{Rojas-Arriagada_2020} and sub-divide the stars into the three SRD-relevant distance bins as shown in \\autoref{fig:halo_bad}a:\n (i) stars with $d$~\\textless~15 kpc (black for dwarfs and grey for giants), \n (ii) stars with 15~\\textless~$d$~\\textless~25 kpc (green circles), and \n (iii) stars with 25~\\textless~$d$ (orange diamonds). \nAs visible in \\autoref{fig:halo_bad}a, the distant stars (green and orange) are among the most intrinsically luminous in this sample.\n\nOverall, the Washington$+DDO51$ pre-selection has proven very effective at identifying giant stars, since, of the stars with spectroscopically measured parameters, only 12\\% of those pre-selected to be giant candidates turned out to be dwarf stars. \nMany of the pre-selected giants do not fall into these intermediate and distant halo bins, and instead appear to be at distances associated with the thin or thick disk. \n\\autoref{fig:halo_bad}b compares spectro-photometric distances, $d$, of stars to their apparent $H$ magnitude (note that extinction is negligible in these fields) for the giant candidate sample having reliable ASPCAP results and distances \\revise{(1214 stars)} with the same color-coding as \\autoref{fig:halo_bad}a.\nThe dashed horizontal lines in \\autoref{fig:halo_bad}b indicate the magnitude limits from \\citetalias{zasowski_2013} and \\citetalias{zasowski_2017} to obtain $S\/N\\sim$100 in the labeled number of visits. \nThe majority of stars with $d > $1~kpc are fainter than $H$=12.2 (the 3-visit depth), but the $d$\\textgreater~25~kpc stars are largely fainter than $H$=12.8 (the 6-visit depth). \n\n\n\\autoref{fig:halo_bad}c dissects the sample into the three distance bins by the targeting cohort; specifically, there were \\revise{195, 395, and 624 stars} in the short (3- or 6-visit), medium (12-visit) and long (24-visit) cohorts, respectively.\n\\revise{The fractions of the stars in the nearest distance-bin were targeted using the magnitude selection for the the short, medium, and long cohorts were 17\\%, 32\\%, 50\\%, respectively}. \nFor the intermediate and distant distance-bins, \\revise{just over} half of the sample was targeted using the long cohort strategy \\revise{(57\\% and 61\\% )} with the other half coming from the sum of the medium and short cohorts \\revise{(43\\% and 39\\%)}.\nDespite the greater 0.7 mag greater $H$ limit and, especially, the commitment of $\\sim$12 hours more observing time, \\autoref{fig:halo_bad}c demonstrates that the long, 24-visit cohort is not more effective at yielding distant halo stars than the combination of medium and short cohorts.\n\n\n\\autoref{fig:halo_bad}d, which shows the intermediate and distant star targeting efficiency for each targeting cohort, further clarifies the optimal observing strategy to yield such stars. \nWhile the efficiency in the medium cohort is double (triple) that of the short for intermediate (distant) stars, there is no commensurate gain in efficiency from the long cohort. \nThus, despite requiring two to four times more fiber hours per target, the long cohort did not provide a commensurate gain in the harvest of intermediate and distant star samples. \nFurthermore, while the medium cohorts yield a higher percentage of distant stars than the short cohorts, these also required more visits to reach their requisite $S\/N$, meaning that for the same amount of time, the short cohorts were nearly as efficient at accumulating distant halo stars.\nThe ultimate reason for this is difficult to diagnose, but it could be reflective of a number of operational reasons (specific fields targeted, the depth of the Washington$+DDO51$ photometry, etc.) or astrophysical reasons (halo giant luminosity function, global density law, the presence of halo substructure, variations of chemistry and luminosity functions in accreted systems, etc.).\n\n\n\nOf the $\\sim$900 intermediate distance stars and the 100$+$ distant stars required by the SRD, there are only \\revise{91 stars in the intermediate bin} and \\revise{59 in the distant bin} for the data shown in \\autoref{fig:halo_bad} (\\revise{this is only for the Washington-selected sample and} most of this data was in hand by the time of BTX planning). \nThat means that around the time of the BTX planning, only \\revise{$\\sim$15\\% of the goal} was met for stars beyond 15~kpc and \\revise{59\\% of the goal} for beyond 25~kpc.\nFortunately, the BTX program offered us an opportunity not only to add to, but also to course correct, our halo targeting to help meet our SRD goal. \n\nBecause of the diminishing returns of the medium and long cohorts (corresponding to the deep-drill fields in the original APOGEE-2N halo plan), we opted to take a ``broad and shallow'' approach to the BTX halo targeting program (corresponding roughly 1\/3 of the APOGEE-2N BTX allocation). \nWe used 6-visit, single cohort halo fields rather than deeper 12- or 24-visit fields, which, despite their comparable effective yields, are more challenging to schedule and complete. \nCombined with this alteration in how observing hours were distributed by field, we also modified our target selection strategy, with the goal of a higher yield of more distant stars collected with a higher efficiency.\nThis modified target selection strategy employed a two-tiered prioritization scheme, with one tier focusing on fields containing confirmed distant halo stars previously identified by the SEGUE survey (\\autoref{sec:known_giants}), and the second tier relying on new giant star candidate selection criteria that exploited \\revise{the (then) newly available {\\it Gaia}~DR1 astrometry} (\\autoref{sec:new_halo_targeting}). \n\n\\subsection{BTX Halo Targeting Method 1: Known Halo Giants in SEGUE} \\label{sec:known_giants}\nCoincident with the BTX planning was the submission of an Ancillary Science Program on the distant halo, described in \\autoref{anc:distanthalo}, that intended to use spectro-photometric distances derived from SEGUE and SEGUE-II observations in SDSS-II and SDSS-III, respectively, to expand upon the APOGEE-2 halo targeting at large heliocentric distances \\citep[][\\inprep{C.~Rockosi et al.~in prep}]{yanny_2009, eisenstein_2011}. \nWe opted to fold the Ancillary Science Program into the BTX halo program, making the former much larger in scope and implemented at a high priority.\n\n\\citet{xue_2014} identified over 6000 giants in the SEGUE sample, many of which were both metal poor and distant; exploiting this dataset is an ideal way to ensure that APOGEE-2 adequately samples the chemical fingerprint of the distant halo.\nAPOGEE-2N fields were selected to contain 2-3 of the \\citeauthor{xue_2014} K-giants. \nWe did not place constraints on the distance or metallicity of the \\citeauthor{xue_2014} stars and, instead, opted to target any stars from the \\citeauthor{xue_2014}, as any overlap with SEGUE expands upon our inter-survey cross-targeting (\\autoref{sec:cross}). \n\nIn the figures developed in the discussion to follow, the targets from \\citeauthor{xue_2014} are always shown as filled black symbols.\nBecause we consider these targets part of our ``BTX Halo'' strategy, they will be included in efficiency metrics, in part because all of the \\citet{xue_2014} targets would have been selected from our algorithmic targeting strategy (\\autoref{sec:new_halo_targeting}) and, in effect, we have just prioritized their selection by including them as special targets. \n\nStars targeted from \\citet{xue_2014} have \\texttt{APOGEE2\\_TARGET2} bit 20 set and, because a different scheme was used for the APOGEE-2S Halo Program, will have a \\texttt{TELESCOPE} tag of ``apo25m''. \n\n\\begin{figure*}[h]\n \\begin{mdframed}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{f03_halo_newstrategy}\n \\caption{Giant star pre-selection using the total proper motion ($\\mu_{\\rm tot}$) and fractional proper motion uncertainty ($\\sigma_{\\mu_{tot}}\/ \\mu_{tot}$) from HSOY \\citep{altmann_2017,altmann_2017_catalog}. \n The original halo targeting, (a) and (b), was evaluated in all ``deep'' fields to demonstrate that distant giants almost always have $\\mu_{\\rm tot}$ \\textless~5~mas yr$^{-1}$ and $\\sigma_{\\mu_{tot}}\/ \\mu_{tot}$ \\textgreater~0.4 (indicated by dotted lines in these panels). \n Similar to \\autoref{fig:halo_bad}, confirmed giant candidates are in grey, confirmed dwarf candidates in black, and the distant samples are in green for 15~\\textless~$d$~\\textless~25~kpc and yellow for 25~\\textless~$d$~kpc. \n Open symbols are colored by distance bin and represent stars from the ``main red star sample'' that were targeted without Washington$+DDO51$ classification. \n The result of using this proper motion-based strategy in the BTX is shown in panels (c) and (d), with spectroscopically-confirmed giants in grey, dwarfs in black, and the distant samples as in (a) and (b).\n }\n \\label{fig:halo_pm}\n \\end{mdframed}\n\\end{figure*}\n\n\\subsection{BTX Halo Targeting Method 2: Exploiting Astrometry} \\label{sec:new_halo_targeting}\nDue to the short time between the BTX planning and the beginning of its observations, it was impossible to acquire and process Washington$+DDO51$ pre-imaging for plate design. \nThus, we used the existing APOGEE observations, spectro-photometric distances, and, at that time, the recently released Hot Stuff for One Year \\citep[HSOY;][]{altmann_2017,altmann_2017_catalog} catalog of proper motions, combining URAT1 \\citep{urat1} and {\\it Gaia}~DR1 \\citep{gaia_dr1} astrometry, to see if an alternative strategy for increasing our yield of distant halo stars could be constructed.\nOur original investigations of how to identify distant giants effectively explored any fields in APOGEE reaching sufficient depth (e.g., 6, 12, or 24 visits) and included tests with a number of different spectro-photometric distance codes; but for simplicity, we will demonstrate some of the test results using the original APOGEE-2 halo sample and the spectro-photometric distances from \\autoref{fig:halo_bad}. \n\nFor these stars, \\autoref{fig:halo_pm}a compares the spectro-photometric distance to the total proper motion from HSOY, where $\\mu_{\\rm tot}$ is taken to be the quadrature sum of $\\mu_{\\alpha}$ and $\\mu_{\\delta}$.\n\\autoref{fig:halo_pm}a demonstrates that the vast majority of the Washington$+DDO51$-selected distant stars (green circles and orange diamonds as in \\autoref{fig:halo_bad}) have $\\mu_{\\rm tot}$ \\textless 10 mas year$^{-1}$. \n\\autoref{fig:halo_pm}b compares the derived distance to the fractional proper motion uncertainty from the HSOY measurements. \nWe found that all of our distant stars not only had small $\\mu_{\\rm tot}$ but also had large uncertainties relative to those motions. \nThese studies on data in-hand led to an algorithmic strategy that had three criteria to select halo candidates:\n\\begin{enumerate} \\itemsep -2pt\n \\item $J-K_{\\rm s}$ \\textgreater\\ 0.5 \n \\item $\\mu_{\\rm tot}$ \\textless\\ 10 mas year$^{-1}$ \n \\item $\\sigma_{\\mu_{tot}}\/ \\mu_{tot}$ \\textgreater\\ 0.4\n\\end{enumerate}\nThe latter two, proper motion criteria are illustrated in \\autoref{fig:halo_pm}c and \\ref{fig:halo_pm}d, along with the targets observed in BTX halo fields, again color-coding distant halo stars (green circles and orange diamonds as in \\autoref{fig:halo_bad}).\n \nTo further increase our yield and acknowledging that ASPCAP produces reliable stellar parameters and chemical abundances for spectra with $S\/N$ as low as 70 \\citep[][]{holtzman_2015,holtzman_2018,Jonsson_2020}, we also increased the $H$-magnitude limits to $H$=13.5 in 6-visit fields (i.e., that corresponding to $S\/N\\sim$70 per pixel for the faintest stars; \\autoref{tab:maglims}) for any halo star candidates; stars drawn from the main red star sample were still restricted to $H$=12.8 for a 6-visit field (\\autoref{tab:maglims}).\nTargets selected following this scheme have \\texttt{APOGEE2\\_TARGET2} bit 21 set and, because a different method was used for APOGEE-2S, will have \\texttt{TELESCOPE} of ``apo25m''.\n\n\\begin{figure*}[ht]\n \\begin{mdframed}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{f04_halo_fulleval} \n \\caption{\n Evaluation of the new halo targeting using HSOY proper motions described in \\autoref{sec:new_halo_targeting} \\citep{altmann_2017}.\n (a) Kiel diagram for BTX halo targeting. \n Giant candidates are shown in grey, SEGUE targets are shown in black, and the distant samples are in green for the intermediate bin and yellow for the distant bin.\n (b) Spectro-photometric distance versus apparent $H$ magnitude with color-coding as in (a). \n The depth of various targeting strategies is labeled.\n These panels can be compared with their counterparts for the original halo targeting \\autoref{fig:halo_bad}a and \\ref{fig:halo_bad}b.\n (c) Total number of stars in distance bins for three targeting methods: the original halo selection scheme in blue circles, the BTX scheme in purple triangles, and stars collected via MaNGA co-targeting in red down-triangles. \n As discussed in the text, the strategies are quite different, but produce similar numbers of distant stars.\n (d) The number of stars in a given bin normalized by the total fiber hours applied to a given targeting scheme, following the color-coding in (c). \n Here the increased yield from the BTX halo targeting is evident; indeed, the BTX targeting improved over the original halo targeting \\revise{by a factor of $\\sim$3$\\times$ to $\\sim$5$\\times$ for all distance bins and by 5$\\times$ to 18$\\times$ over the MaNGA Co-Targeting strategy in the more distant bins.}\n Broadly we interpret the efficiency panels to indicate that dampening the dwarf-star-foreground has led to our gains.\n The data used for panels c and d is given in \\autoref{tab:halo_distant_stars}.\n }\n \\label{fig:halo_newtarg}\n \\end{mdframed}\n\\end{figure*}\n\n\\input{Tables\/table5_halonumbers}\n\n\\subsection{Effectiveness of the BTX Halo Targeting Strategies} \\label{sec:how_did_we_do}\n\nAfter several years of observing, it is now possible to evaluate the effectiveness of the strategies employed for the BTX observing.\n\\autoref{fig:halo_newtarg} illustrates the current status of the BTX program by way of the metrics used to understand the original halo targeting (e.g., \\autoref{fig:halo_bad}a and \\autoref{fig:halo_bad}b).\n\\autoref{fig:halo_newtarg}a shows the Kiel diagram for all targets in halo fields (grey), SEGUE distant halo stars (black), and distant stars identified based on our altered BTX targeting scheme described in Section 3.5 (green and orange). \nAs may be seen in \\autoref{fig:halo_newtarg}a and \\autoref{fig:halo_bad}a, the distant stars in the BTX targeting span a larger range in $\\log{g}$, meaning that they provide a more broadly representative sample of giant stars, not just the most intrinsically luminous ones, as in the original scheme. \n\\autoref{fig:halo_newtarg}b and \\autoref{fig:halo_bad}b also show that the BTX targeting has identified stars over a wide range of apparent magnitudes than in the original halo targeting scheme.\nBased on \\autoref{fig:halo_newtarg}, we believe our targeting criteria are actually building an overall less biased sample of stars at large distances.\n\nThe remaining panels of \\autoref{fig:halo_newtarg} show metrics to quantify the relative success of our BTX halo targeting strategy against those of the other targeting scenarios used in APOGEE-2. \nHere we focus on three in particular:\n\\begin{itemize} \\itemsep -2pt\n \\item {\\it MaNGA Co-Targeting:} As described in \\citetalias{zasowski_2017}, APOGEE-2 observations were obtained in tandem with MaNGA observations of their galaxy sample. \n These fields are in the North Galactic Cap, which is in our halo ($\\ell$,$b$) range. \n The APOGEE-2N targeting strategy for these ``free fibers'' adopted the high-latitude halo color-cut (\\autoref{tab:colorcuts}), but a faint limit of $H\\sim$11.5.\\footnote{The pointings for MaNGA co-targeting are not shown in \\autoref{fig:fieldmap} because their positions are not determined by APOGEE-2; these fields and their data are shown in the data release papers, see e.g., \\citet[][see their Figure 1]{Jonsson_2020} and \\inprep{S.~Majewski et al. (in prep.)}.} These pointings can be identified using the \\texttt{PROGRAMNAME} of ``manga'' and the program is comprised of 3-visit fields. \n While not formally part of the APOGEE-2N Halo Program, the MaNGA Co-Targeting sample is a good comparison set as it is a true \\revise{near-infrared-based} magnitude-color selection without any other interventions or modifications aiming to isolate giants. \\revise{There were 606 individual, but sometimes overlapping, fields in the MaNGA Co-Targeting program all to equal depth and using identical selection.}\n %\n \\item {\\it Original Halo:} As described both in \\citetalias{zasowski_2017} and in this paper, the original halo strategy is a combination of the Washington+$DDO51$ strategy and the color-magnitude criteria (\\autoref{tab:colorcuts}). The results of this strategy are summarized in \\autoref{fig:halo_bad}. This program consists of 52 pointings with a range of depths. \n %\n \\item {\\it BTX Halo:} Halo giant candidates selected according to the criteria given in \\autoref{sec:known_giants} and \\autoref{sec:new_halo_targeting} or as filler according to typical color-magnitude criteria (\\autoref{tab:colorcuts}). This program is comprised of 32 pointings of 6-visits each.\n\\end{itemize}\n\\autoref{tab:halo_distant_stars} summarizes statistics for these three strategies, including the total number of targets and total number of fiber hours. \nThe row ``All Targets'' is the number of targets with calibrated stellar parameters (e.g., {\\tt LOGG} and {\\tt TEFF} tags in the summary file), but excluding duplicates ({\\tt EXTRATARG} bit 4) and telluric observations ({\\tt EXTRATARG} bit 2) \\citep{Jonsson_2020}. \nWe use the same spectro-photometric distances to separate targets into four distance bins: \n (1) 5 \\textless~$d$~\\textless 10 kpc, \n (2) 10 \\textless~$d$~\\textless 15 kpc,\n (3) 15 \\textless~$d$~\\textless 25 kpc, and \n (4) $d$ \\textgreater 25kpc. \nIn \\autoref{fig:halo_newtarg}c, the total number of targets in each of these distance bins is plotted for each of the three strategies; here we see that the MaNGA Co-Targeting strategy actually produces a large sample of distant stars in all distance bins, larger even than the focused halo plates, with the exception of the most distant bin.\nHowever, the raw counts of stars is not necessarily the best metric of success and we should take into account the ``resource cost'' for different means of targeting.\n\nIn \\autoref{fig:halo_newtarg}d, the number of targets in a given distance bin is normalized by the total fiber hours in the program; this normalization takes into account the different spatial areas (e.g., number of distinct fields) and magnitude-depths of the strategies. \nTo compute fiber hours, we sum the number of visits contributing to the final spectrum ({\\tt NVISITS}) for all targets meeting the criteria of the sample; the total fiber hours is the total {\\tt NVISITS} for all targets (not just those in a given distance bin). \nWe note that the MaNGA Co-targeting and the Original Halo programs used approximately the same number of total fiber hours (see \\autoref{tab:halo_distant_stars}), but used them differently, with MaNGA Co-targeting accumulating \\revise{606 unique, but sometimes overlapping, fields} to a depth of $H\\sim$11.5 and the Original Halo only targeting 52 fields, but employing the wedding-cake targeting strategy of deeper cohorts going as faint as $H\\sim$13.8 (see \\autoref{fig:halo_bad}b, and the star counts enumerated in \\autoref{tab:halo_distant_stars}).\nHere, the overall inefficiency of the MaNGA Co-Targeting for distances beyond 10~kpc is evident, however, these fields were not designed specifically for reaching distant halo stars, and, remarkably, the Original Halo strategy does not perform distinctly better. \nIn the end, these performance metrics indicate that distant stars are sufficiently sparse on the sky that that a shallow-depth, but wide-area sampling strategy has benefits in terms of overall sample size.\n\nIn contrast, the BTX Halo program used \\revise{about $\\sim$30\\% of the fiber hours} as employed in the original halo targeting (\\revise{in contrast, MaNGa Co-Targing used 2x the fiber hours}), and employed them to target only 32 fields (61\\% of fields in original halo \\revise{and 5\\% of MaNGA Co-Targeting}). \nThe efficiency of the BTX halo program, as shown in the \\autoref{fig:halo_newtarg}d, is \\revise{$\\sim$3$\\times$ to $\\sim$5$\\times$} more efficient than the original halo targeting and \\revise{from $\\sim$2$\\times$ to $\\sim$18$\\times$} more efficient than the MaNGA co-targeting strategy (the exact numbers are given in \\autoref{tab:halo_distant_stars}). \nThe greatest BTX gains are at the largest distances and, using \\autoref{fig:halo_newtarg}b, we can see that the bulk of the distant stars have apparent magnitudes fainter than the depth probed by the MaNGA co-targeting.\n\n\\subsection{Summary of the Efficacy of APOGEE-2N Halo Targeting} \\label{sec:halo_sum}\n\nThanks to routine evaluations of its targeting strategies, the APOGEE-2 targeting team was able to identify the shortcomings of the original halo targeting strategy, and then determine a set of new strategies to reduce the effect of the dominant foreground contamination of disk stars, and increase both the total number of distant halo stars in the sample as well as the efficiency at which they were accrued. \nAs a result, the BTX program netted a comparable number of distant stars as the original strategy, \\revise{but with only $\\sim$30\\% of the total fiber hours.}\nIt is worth noting that the strategy was successful not so much through more efficient, deliberate targeting of distant halo stars, but rather by more effectively limiting the amount of foreground disk contamination.\nWe also compromised between depth and area, choosing to increase to an $H\\sim$13.5 limit using 6-visits, rather than using past strategies that prioritized depth with $H\\sim$13.8 limit in 24-visit fields.\nIn the end, the BTX Strategy of dampening the foreground improved our yield per fiber hour \\revise{by factors of $\\sim$3 to $\\sim$5} over that of the original halo targeting strategy for stars with distances greater than 5~kpc.\n\nAPOGEE-2 also sampled the halo using a simple magnitude-color criterion with the MaNGA co-targeting. \nThis sample represents a wide-area, but shall depth ($H\\sim$11.5) targeting strategy that differs from the original APOGEE-2 strategy and that adopted in the BTX using proper motions. \nWe find that the MaNGA co-targeting produced comparable, or even substantially larger, samples of stars at heliocentric distances greater than 5~kpc. \nHowever, this was due to the sheer number of fibers allocated in this way, and the halo stars accumulated were collected very inefficiently; \\revise{the BTX strategy is more efficient by factors of $\\sim$2 to $\\sim$18} over the MaNGA co-targeting program in terms of yield per fiber hour (\\autoref{tab:halo_distant_stars}). \n\nThe counts of the MaNGA sample suggest that even relatively shallow but wide-area strategies can accumulate large numbers of stars at halo-relevant distances; however, the stars are accumulated highly inefficiently and adding in a proper-motion based criteria would improve the efficiency at reaching distant halo stars.\nWe note that in many cases the highest-latitude fields are so sparse in stars of any type that cohorted targeting strategies like the Original Halo will ``run-out'' of suitable bright targets.\nThis too motivates the sample building potential of a wide-and-shallow approach, like those taken in the MaNGA co-targeting or BTX halo.\n\nFuture surveys aiming to target the halo will need to weigh their observing strategies carefully, as we did here, but also take into account the impacts on the selection function.\nFor APOGEE-2, we deliberately altered our halo targeting strategy during the course of the survey to explicitly build a sufficiently large sample (e.g., hundreds of stars) at large distances so that we might better probe the chemical distributions of stars in the outer halo.\nWe made this choice while fully recognizing the impact it would have in complicating our selection function and other scientific investigations.\nFor example, the proper motion-prior imposed in the BTX halo targeting could impact dynamics-focused studies of the halo using APOGEE-2N data. On the other hand, the halo star sample that resulted is far less-biased in $\\log{g}$\\ and $T_{\\rm eff}$, and therefore better samples the luminosity function. \nAnd for all of the subsamples of halo stars within the APOGEE dataset, there are complex distance-luminosity biases imposed by the varying magnitude limits employed.\nSampling the halo is difficult and compromises have to be made that are driven by the scientific aims. \n\n\n\\section{New Programs in the Bright Time Extension} \\label{sec:new_programs}\nThis section discusses programs undertaken in the BTX that add new scientific objectives to the APOGEE-2N program.\nFor each, a scientific motivation is provided to place the targeting needs and constraints in context.\n\\revise{There are four new programs:\n (1) mapping of the California Giant Molecular Cloud (\\autoref{sec:calicloud}), \n (2) coverage in the TESS Continous Viewing Zone (\\autoref{fig:tess}),\n (3) probing the outer Galacitic disk (\\autoref{sec:odisk}), \n (4) main sequence calibrations (\\autoref{sec:mainsequence}).\n }\n\n\\subsection{Mapping the Interstellar Medium in the California Giant Molecular Cloud}\\label{sec:calicloud}\n\nThis program is a pathfinder to understand the observational limits for a larger program to map Giant Molecular Clouds (GMCs) planned for SDSS-V \\citep{sdssv}.\nTo understand GMC evolution, one needs to understand better the relative importance of colliding flows, gravitational contraction, magnetic support, and turbulence over the range of physical scales and internal conditions spanned by GMCs \\citep[e.g.,][]{VazquezSemadeni_2007_GMCformation,Clark_2012_GMCformation,Heitsch_2013_GMCformation,Fujimoto_2014_GMCformation}.\nThe velocity field of the GMC's environment may hold the key: for a GMC to contract and form stars, material must flow together.\nA promising way to probe these flows around the GMC formation regime is to use the 1.5272 \\micron\\ Diffuse Interstellar Band (DIB) present in the APOGEE spectral range \\citep{Zasowski_2015a}, in combination with line-of-sight dust column measurements \\citep{Zasowski_2019_highDdust}.\n\nThe California GMC\\footnote{As noted in \\citet{Harvey_2013}, this cloud is called ``Auriga'' in the {Spitzer}\\ Legacy survey. \\citet{Harvey_2013} use the name ``Auriga-California Cloud.'' We stay consistent with the naming convention within APOGEE-2N of ``California Cloud'' following \\citet{lada_2009}.} is a nearby ($d_{\\rm los} = 400$~pc), massive (10$^{5}$ $M_{\\odot}$), isolated, quiescent GMC with a large reservoir of surrounding dust and gas \\citep{lada_2009,Harvey_2013}. \n{\\it Gaia}~DR2 \\citep{gaia_dr2} provides distances accurate to 10~pc for a large number of stars within about 50~pc of the California GMC. \nThus, APOGEE spectra can be used to measure the mean line-of-sight velocity of the 1.5272~$\\mu$m DIB, map the flow of material onto or away from the molecular cloud, and thus probe the dynamical environment of the California GMC. \nMany of the stars in the vicinity of the California GMC are either F-type stars or red giants with $H$ \\textless\\ 11.5 that can be targeted with single, one-hour visits to provide $S\/N\\sim$100 spectra suitable for measuring the DIB in $A_{\\rm V}\\sim1$ material \\citep[a discussion of observational requirements for DIB measurements can be found in][]{Zasowski_2015a}. \nA target density of at least 1 star per 10 pc$^{3}$ is estimated to meet these scientific aims, which requires $\\sim$1200 stars within 50 pc of the California GMC.\nLater catalogs from {\\it Gaia}\\ will be released during the operation of SDSS-V and are likely to provide comparable stellar distance accuracy to GMCs that are $\\sim$1 kpc away, enabling an expansion of this program to other spiral arms and star formation environments \\citep[SDSS-V Science Drivers are explained in][]{sdssv}.\nThis pilot program was designed to determine what quantities can be recovered from different observing strategies and thus optimize the upcoming SDSS-V observations. \n\nThis program was implemented with seven dedicated 1-visit fields with field names of the form `CA\\_$lll$-$bb$\\_btx', where $lll$-$bb$ are the Galactic coordinates of the field center. \nThe \\texttt{PROGRAMNAME} is `as4\\_btx,' where `as4' stands for After Sloan-IV, the project now established as SDSS-V. \nIndividual stars observed as part of this program have \\texttt{APOGEE2\\_TARGET2} bit 1 set. \n\n\n\\subsection{{\\it TESS}\\ Northern Continuous Viewing Zone} \\label{sec:tessncvz}\n\n\n\\begin{figure*}\n \\begin{mdframed}\n \\centering\n \\includegraphics[width=0.95\\textwidth]{f05_tess}\n \\caption{Overview of the {\\it TESS}~N-CVZ program in APOGEE-2N. \n (a) CMD from {\\it Gaia}~DR2 photometry and using {\\it Gaia}~DR2 parallaxes to compute a distance modulus ($\\mu_{\\rm {\\it Gaia}~DR2}$). \n The color-coding indicates the four targeting categories of \n (1) OBAF stars (red), \n (2) stars from the {\\it TESS}~Guest Investigator programs (orange), \n (3) stars identified as dwarf-type from the TESS Candidate Target List (CTL; green),and \n (4) stars identified as giants in the CTL (blue). \n (b) Sky distribution of targets. \n (c) Total $m_{\\rm H}$ distribution of targets (black), as well as histograms for each of the the four targeting categories separately (colored lines).\n }\n \\label{fig:tess}\n \\end{mdframed}\n\\end{figure*}\n\nAs described in \\citet{ricker2015}, the {\\it TESS}\\ mission was approved to enter Phase B implementation in 2013 with a launch no earlier than March 2018.\\footnote{ \\url{https:\/\/tess.mit.edu\/science\/}}.\nThis timeline precluded large-scale consideration of the {\\it TESS}\\ mission for APOGEE-2 planning given that APOGEE-2N operations began in 2014 (APOGEE-2S began in 2017), and with the APOGEE-2 science requirements, field plan, and targeting schema largely in place prior to even the {\\it TESS}\\ Phase B approval.\nThus, no specific effort to coordinate with {\\it TESS}\\ observations was included in the original APOGEE-2 targeting plan \\citepalias{zasowski_2017}.\n\nThe BTX thus provided a key opportunity to capitalize on the scientific opportunities feasible from a joint-analysis of {\\it TESS}\\ and APOGEE data products.\nSuch opportunities include, but are not limited to: \n characterization of planet-hosting stars \\citep[e.g.,][]{canas_2019}, \n radial velocity monitoring \\citep[e.g.,][]{troup_2016}, \n binary star identification \\citep[e.g.,][]{El-Badry_2018}, \n and stellar astrophysics \\citep[e.g.,][]{apokasc,apokasc2}. \nMany of these science goals are fundamental rationales for SDSS-V \\citep{sdssv}, and indeed SDSS-V planned to synergize with the {\\it TESS}\\ mission to help address them. \nThe BTX represented a timely opportunity to test several strategies for SDSS-V, and this motivated a joint effort from APOGEE-2 and the AS4 teams. \nBecause SDSS-V will operate with robotic fiber positioners that lend it a greater ability to survey sparsely-spaced {\\it TESS}\\ targets across the full sky, the APOGEE-2 observations of {\\it TESS}\\ targets have focused on the {\\it TESS}\\ Continuous Viewing Zones (CVZ, hereafter) in the Northern and Southern Hemispheres, which correspond to a circular area 15$^{\\circ}$ in diameter around the ecliptic poles. \nThe latter regions of sky have received 365 day coverage at 30-minute cadence by {\\it TESS}\\ in the original two year mission, with additional observations occurring in the on-going {\\it TESS}\\ extended mission.\nWhile similar in concept and yielding complementary data, the Southern (F.~Santana et al. submitted)\\ and Northern {\\it TESS}\\ CVZ programs differ in their logistical implementation.\n\nA {\\it TESS}-focused APOGEE-2N program is challenging because, despite spanning a large area on the sky, the CVZ is only accessible for observations over a limited range of local sidereal time (LST) for ground-based observations (with a bulk of the N-CVZ accessible to the APOGEE-N spectrograph from roughly at LSTs of 17 to 19 hours).\nMoreover, this LST range is already over-subscribed in APOGEE-2N owing to the location of the {\\it Kepler}\\ field (R.A.,Dec = 19:22:40,+44:30:00) and the cadence requirements for observing programs in the {\\it Kepler}\\ field (see \\autoref{sec:koi} and discussion in \\citetalias{zasowski_2017}). \nThus, only a limited number (75) of hours were available and allocated to this program over the 1.5 years of the BTX, and these observations had to be spread out over several years because of these LST constraints.\n\n\nPlanning how to implement the 75 1-visit fields was a joint effort between the scientific teams within SDSS-IV and SDSS-V, who identified four classes of targets to observe, and which are, in priority order (see Fig. \\ref{fig:tess}a): \n\\begin{enumerate} \\itemsep -2pt\n \\item hot stars of OBAF spectral type (\\texttt{APOGEE2\\_TARGET2} bit 27), \n \\item stars on {\\it TESS}\\ 2-min cadence, largely those either from the {\\it TESS}\\ Guest Investigator programs or candidates for such programs (\\texttt{APOGEE2\\_TARGET2} bit 28), \n \\item dwarf stars in the Asterosiesmic Target List \\citep[ATL;][]{tess_atl} or Candidate Target List \\citep[CTL;][]{stassun_2019} (\\texttt{APOGEE2\\_TARGET2} bit 29),\n \\item giant type stars generally meeting the specifications of the APOGEE `main red star sample' \\revise{and drawn from the TESS Input Catalog (TIC;} \\texttt{APOGEE2\\_TARGET2} bit 30). \n\\end{enumerate}\nAn effort was made to have roughly 50\\% dwarfs and 50\\% giants on a given plate to ensure broad coverage in the Hertzsprung-Russell diagram.\nThese four classes of science targets are briefly described below, before discussion of the detailed implementation of the {\\it TESS}\\ N-CVZ program (\\autoref{sec:tessncvz_implement}). \n\n\\subsubsection{OBAF Stars} \\label{sec:tessncvz_obaf}\nHigh-resolution spectra for oscillating OBAF stars are the foundation for an goal of ``dynamical asteroseismology'' for stars with convective cores; this program was designed in synergy with the program planned for SDSS-V. \nThese stars contribute to the dynamical and chemical evolution of galaxies, but the models of their stellar structure and evolution are known to be inadequate. \nBy comparing seismically determined parameters with spectroscopic parameters and dynamical masses from modelling multi-epoch radial velocities, we will infer precise constraints, for example on the size of the convective core.\nSuch constraints are necessary for the calibration and improvement of the present-day models of stellar structure and evolution for these stellar types \\citep[see e.g.,][]{Pedersen_2018}. \nThe SDSS-IV observations for the OBAF stars provide a first epoch radial velocity measurement to aid in the orbital determinations as well as allow for a first pass on their stellar parameters.\nThese stars have \\texttt{APOGEE2\\_TARGET2} bit 27 set. \n\n\\subsubsection{Stars with 2-min Cadence Observations} \\label{sec:tessncvz_gi}\n\n{\\it TESS}\\ Cycle 1 and Cycle 2 observations produced two data products: full-frame images (FFIs) at 30 minute cadence and ``postage stamp'' images at 2 minute cadence. \nThe former category ensures that every bright star has some data, whereas the latter category was reserved for Guaranteed Time Observations (GTO) from the {\\it TESS}\\ team and Guest Investigator (GI) observations from the broader astronomical community, with the latter awarded through competitive proposal cycles. \nThus, a portion of our program was reserved for stars with 2-minute cadence observations from Guest Investigator samples and given high priority to ensure these rare-target classes would be sufficiently sampled. \nSuch targets include known planet hosts, cool dwarfs, and subgiants. \nThese targets are all identified by \\texttt{APOGEE2\\_TARGET2} bit 28. \n\n\\subsubsection{Dwarf Stars} \\label{sec:tessncvz_ctl}\n\nAfter the rare target classes were selected, roughly half of the fibers ($\\sim$125) were reserved for dwarf-type stars drawn from two {\\it TESS}\\ scientific target lists: \n (i) first, stars from the \\revise{Asterosiesmic Target List \\citep[ATL;][]{tess_atl}} and then \n (ii) stars from the \\revise{Candidate Target List \\citep[CTL;][]{stassun_2019}}.\n\nStars with solar-like oscillations were selected from the {\\it TESS}\\ ATL produced by the {\\it TESS}\\ Asteroseismic Science Consortium (TASC)\\footnote{\\url{http:\/\/tasoc.dk}} as of Version 4.\\footnote{This is Version 4 of the ATL produced in $\\sim$October 2017 (see TASOC website \\url{https:\/\/tasoc.dk\/wg1\/Targetselection}) that implemented discussion from the TASC3\/KASC10 workshop; \\url{https:\/\/www.tasc3kasc10.com\/}} \nThe final ATL sample and its detailed derivation is described in \\cite{tess_atl}, but we provide a brief summary of the intermediate catalog and priority scheme that was available for our plate design.\nStars were selected based on their assigned priority in the ATL. \nThe ATL priorities that we used were assessed based on a term known as $P_{\\rm mix}$ that is a linear combination of the likelihood of the detection of seismic modes and the likelihood that such modes are falling-off beyond detectability based on the stellar parameters relative to the known space in the Hertzsprung-Russell diagram with seismic modes.\\footnote{$P_{\\rm mix} = (1-\\alpha)~P_{vary} + \\alpha~P_{\\rm fix}$, where $\\alpha$=0.5 in the version of the catalog used for APOGEE-2 targeting.}\nThese parameters were selected to balance competing scientific objectives in the ATL.\nThis is similar to what is described in \\citet[][their Subsubsection 4.3.3]{tess_atl} and the APOGEE-2 sample should mimic the overall distribution of the larger ATL. \nAt the time of plate design, $\\sim$900 ATL stars with solar-like oscillations were in the {\\it TESS}~N-CVZ footprint. \nThe entire list was folded into the priority schema and the ATL stars will have \\texttt{APOGEE2\\_TARGET2} bit 28 and \\texttt{APOGEE2\\_TARGET2} bit 29 set because they were targeted as part of a GI program {\\it and} counted toward the dwarf-stars quota for each plate.\n\nAfter selecting candidates from the ATL, the remaining fibers for dwarf-type targets were assigned from the CTL version 8 (CTL8) \\citep[][]{stassun_2019}.\nAs described in \\citet{stassun_2018, stassun_2019}, the CTL is a set of stars selected from the \\revise{{\\it TESS}~Input Catalog (TIC)} that are ideal candidates for the {\\it TESS}\\ planet-finding mission;\nCTL stars are high-likelihood main-sequence or sub-giant type stars with their likelihood determined from the broad range of photometric and astrometric measurements collated into the TIC.\n\nA major component of the classification of stars from the TIC and into the CTL was the likelihood that a star was a dwarf. \nAccordingly, a key component of this determination was the use of the reduced proper motion diagram (RPM), more specifically the NIR version of RPM developed by \\citet{colliercameron_2007} known as RPMJ. \nThis means that astrometric information (proper motions) are important for the CTL and the classification of sources may have some dependence on the astrometric catalog being adopted \\citep[for evaluations pre- and post- adoption of {\\it Gaia}\\ astrometry see][]{stassun_2019}. \nWe will return to this in \\autoref{sec:tessncvz_implement} for the {\\it TESS}~N-CVZ program and, because the RPMJ technique is used in the Outer Disk Program, more discussion is included in \\autoref{sec:outerdist_assess}.\nAPOGEE-2N targets selected from the CTL will have \\texttt{APOGEE2\\_TARGET2} bit 29 set. \n\nTwo sets of fiber assignment were performed from the CTL: \n a ``bright'' set limited to targets with $H<$ 12 and \n a ``faint'' set of targets than with $12 < H < 14$; \nthe latter ``faint'' selection was performed after selecting ``bright'' giants but before selecting ``faint'' giants as described in the next subsection.\nIn either selection round, the stars in the CTL were ranked by their CTL-priority \\citep[see][their Section 3.4]{stassun_2018}, which, briefly, is the probability of detecting a transit signal from a small, rocky planet from a typical {\\it TESS}\\ 2-min postage stamp observation.\n\n\\subsubsection{Giant Stars} \\label{sec:tessncvz_giants}\n\nAfter the rare target classes were selected, roughly half of the fibers ($\\sim$125) were reserved for candidate giant-type stars drawn from the TIC \\citep{stassun_2018,stassun_2019}. \nTwo rounds of fiber assignment were performed, a ``bright'' with $H<$13 and a ``faint'' set with $13 < H < 14$. \nThese two rounds occurred subsequent to the``bright'' and ``faint'' dwarf samples, respectively, to both prioritize dwarfs over giants while prioritizing bright stars over faint stars. \nIn both cases, the giant-candidates were selected following the dwarf-giant classification criteria given in \\citet{stassun_2018} using reduced-proper motion diagrams \\citep[the criterion of][]{colliercameron_2007}. \nGiants selected from the TIC have \\texttt{APOGEE2\\_TARGET2} bit 30 set. \n\nAfter all of the above {\\it TESS}\\ target selections were completed, any remaining fibers were assigned following the standard ``main red star sample'' criteria in APOGEE-2 \\citepalias[][]{zasowski_2013,zasowski_2017} and are flagged accordingly.\n\n\\subsubsection{Implementation of the N-CVZ Program} \\label{sec:tessncvz_implement}\nThe first year of APOGEE-2N observations of the {\\it TESS}~N-CVZ was planned in advance of the {\\it TESS}~Guest Investigator cycles and the release of {\\it Gaia}~DR2 \\citep{gaia_dr2}, both of which were deemed to likely have a significant impact on how to optimize our limited number of fiber hours to apply to the {\\it TESS}\\ science cases.\nThus, the first set of APOGEE-2N observations of the {\\it TESS}~N-CVZ were drilled around the locations of rare, O and B type stars (priority 1), with the remaining fibers assigned following our general schema but using the Tycho-{\\it Gaia}\\ Astrometric Solution from {\\it Gaia}~DR1 \\citep[TGAS][]{gaia_dr1} for the RPMJ computations;\n21 such plates were designed around these OB stars (\\texttt{FIELD} names of CVZ\\_OB\\#\\#\\_btx) plus three additional plates (\\texttt{FIELD} names of CVZ\\_FILL\\#\\#\\_btx).\n\nA second set of plates were designed to attain more-or-less uniform spatial coverage of the N-CVZ; \n51 plates were required and these have \\texttt{FIELD} names CVZTILE\\_$lll\\pm bb$\\_btx, with $lll\\pm bb$ representing the Galactic coordinates of the field center. \nThese plates were designed simultaneously from a consistent list of input targets and relied on {\\it Gaia}~DR2 \\citep{gaia_dr2} for the RPMJ computation; the plates, themselves, were drilled over time to optimize our use of this LST range (e.g., selecting Hour Angles of observation that optimized our observing schedule).\nAll plates from this program have \\texttt{PROGRAMNAME} `cvz\\_btx' with individual targets flagged as described above.\n\n\n\\autoref{fig:tess} gives a summary of the targeting for the {\\it TESS}~N-CVZ program.\n\\autoref{fig:tess}a shows a {\\it Gaia}~DR2 color-absolute magnitude diagram with the distinct targeting classes identified using colors (OBAF stars are red, GI are orange, dwarfs are green, and giants are blue). \n\\autoref{fig:tess}b provides the sky distribution of the targets.\nDuring the first year of observations, an error in target lists led to spatial distributions for science targets that did not fill the full plate footprint. These plates are visible as oblong footprints in \\autoref{fig:tess}b, surrounded by telluric standards that did utilize the full circular footprint of the plate. \n\\autoref{fig:tess}c provides histograms of the magnitude range spanned by each of the four targeting classes; these plates were only intended to have single visits, but span a larger range of magnitudes than typical 1-visit plates in APOGEE-2, as a test of strategies planned for SDSS-V.\n\n\\subsection{Probing the Outer Disk} \\label{sec:odisk}\n\nThe advent of photometric and spectroscopic surveys over large areas of the sky have led to a revolution in our understanding of the structure of the outer disk.\nFar from there being a slow ramp down of the inner disk properties, the outer disk is abundant with star clusters \\citep[e.g.,][]{Zasowski_2013clusters}, apparent substructure \\citep[e.g.,][]{Slater_2014}, and perhaps additional, yet undiscovered features. \nWith the advent of large-area imaging surveys, such as SDSS and 2MASS, star-count maps revealed apparent stellar overdensities in the outer disk; these include the \n Monoceros Ring \\citep[sometimes included into the ``Galactic Anticenter Stellar Structure,'' or GASS;][]{Newberg_2002,crane_2003,yanny2003}, \n Canis Major \\citep{martin_2004}, \n Triangulum-Andromeda \\citep[TriAnd;][]{majewski_2004,rocha-pinto_2004}, \n and A13 \\citep{Sharma_2010,li_2017}, \n among other, smaller features \\citep[for a more exhaustive list see the overview from][]{grillmair_carlin_2016}. \n\nDue to their discovery at a time when clearly identifiable stellar streams from dwarf galaxies were also being uncovered, such as the Sagittarius stream \\citep{ivezic_2000,Newberg_2002,majewski_2003}, these overdensities were broadly interpreted to be debris from dwarf satellite mergers \\citep{rocha-pinto_2003,penarrubia_2005,chou_2010,chou_2011,sollima_2011,sheffield_2014}. \nSpectroscopic follow-up of such features largely seemed to justify these interpretations \\citep{crane_2003,chou_2011,sheffield_2014, deason_2014}, but typically limited themselves to only sampling the most likely member stars and not broadly examining the larger scale behavior of the stars in these areas of the Milky Way.\n\nWhile the dwarf galaxy debris origin was originally favored, evidence grew that these Galactic Anticenter overdensities could also be related to the Milky Way disk, in particular as perturbations to the disk excited by orbiting dwarf galaxies \\citep{kazantzidis_2008,purcell_2011,gomez_2013,gomez_2016,price-whelan_2015,xu_2015,li_2017,newberg_2017,laporte_2018}. \nIn this picture, density waves in the Milky Way disk would be expressed as vertical oscillations of the disk midplane and and the crests and troughs of these waves would appear as apparent overdensities \\citep[although in reality, because of the low density in the outer disk, these overdensities may instead appear to be more feathery, spiral arm-like features rather than a strictly rippled disk; for more realistic examples see][]{laporte_2018}. \n\nVarious studies argued for this revised picture of the Galactic Anticenter overdensities, given that\n (1) the ratio of RR Lyrae to M-giant type stars implied a lack of old stars, which are normally seen in dwarf spheroidal galaxies \\citep{price-whelan_2015}, \n (2) star counts suggest a connection between the apparent substructures that could be interpreted as one continuous feature originating from the Galactic disk \\citep{xu_2015}, and \n (3) we may expect to see these kinds of corrugations across the outer disk excited by known dwarf galaxy satellites like the Sagittarius dSph \\citep[e.g.,]{laporte_2018}. \nHowever, the few existing spectroscopic analyses of the Galactic Anticenter overdensities largely focused on small numbers of stars, were of moderate spectral resolution, and lacked strong sampling of the chemo-dynamics of the outer disk, where these overdensities are located, so that chemistry could not decisively be brought to bear on the origin of these overdensities.\n\nThe original targeting plan for APOGEE-2N contained five pointings on one of these overdensities, Triangulum-Andromeda \\citepalias[referred to as TriAnd in][]{zasowski_2017}, each of which contains only a handful of confirmed members of the TriAnd feature \\citep[drawn from][]{chou_2011,sheffield_2014}, but designed with the hope that the strategy for the main star red sample would naturally identify additional members \\revise{over the extent of TriAnd on the sky (from 100$^{\\circ}$\\textless$\\ell$\\textless150$^{\\circ}$ and -50$^{\\circ}$\\textless$b$\\textless-15$^{\\circ}$).}\nDespite the relatively small sample of TriAnd stars as well as the limited sample of APOGEE targets in the outer disk available at that time, \\citet{hayes_2018} used samples of TriAnd and available outer disk stars to show convincingly that the TriAnd chemistry in [$\\alpha$\/Fe]-[Fe\/H] space is a natural extension of disk chemical patterns to lower metallicity, a finding in agreement with the concurrent study by \\citet{bergemann_2018}.\nThese high resolution spectroscopic studies revealed the potential for chemistry to be used to help clarify the origin of these other Galactic Anticenter overdensities and better understand their evolution, and motivated a more thorough probe of the Milky Way's outer disk with available time in the BTX.\n\n\\begin{figure*}\n \\begin{mdframed}\n \\centering\n \\includegraphics[width=0.95\\textwidth]{f06_odisk_skymap.pdf}\n \\caption{\n PanStarrs DR1 star-count map \\citep[for stars at distances between 7.6 $-$ 11.0 kpc, adapted from][]{Slater_2014} showing also the main survey APOGEE-2 fields (grey) and the outer disk fields for the BTX (red). \n Previously identified member stars in the outer disk substructures are indicated as well; more specifically, \n GASS \\citep{crane_2003} in yellow diamonds, \n A13 \\citep{li_2017} as blue triangles, and\n TriAnd \\citep{rocha-pinto_2004,sheffield_2014} as green circles.\n }\n \\label{fig:outerdisk_map}\n \\end{mdframed}\n\\end{figure*} \n\n\\subsubsection{Outer Disk Program Implementation} \\label{sec:outerdist_implement}\n\nThe \\citet{hayes_2018} study motivated a larger-scale effort to use the APOGEE-2N BTX program to trace the outer disk using a more deliberate targeting strategy; this was essentially accomplished by systematically extending the APOGEE disk field grid out to larger Galactic latitudes ($|b|$~\\textgreater~20$^{\\circ}$), while also pursuing a focused targeting of previously studied stars in GASS, TriAnd, and A13 to define robustly the multi-abundance ``chemical fingerprint'' of these systems relative to the disk \\citep[with targets drawn from][]{crane_2003,chou_2011,sheffield_2014,li_2017}.\nStars targeted from these prior surveys have bit 7 set in \\texttt{APOGEE2\\_TARGET2}.\n\nThe BTX Galactic Anticenter plan encompasses 50 fields, with 26 having $|b|$~\\textless~$24^{\\circ}$ but that expand the disk grid out of the Galactic plane to probe lower latitude features like the Monoceros Ring \\citep{Newberg_2002,yanny2003}, another 19 fields with $|b|$ $\\gtrsim 24^{\\circ}$ that target known members of these features (described below), and two fields, 162$+$34\\_btx and 186$+$31\\_btx, designed to fall on regions of the sky along the AntiCenter Stream \\citep[ACS;][]{grillmair_2006}.\nThe three remaining fields, 108$-$31\\_btx, 114$-$25\\_btx, and 129$-$21\\_btx, were intended to target known TriAnd members but due to an error in the design of these fields, instead probe the lower latitude disk closer to the disk midplane. \n\nEach of these pointings is comprised of two short (3-visit) cohorts (7 \\textless\\ $H$ \\textless\\ 12.2) and 1 medium (6-visit) cohort (12.2 \\textless\\ $H$ \\textless\\ 13.3). \nKnown members of anticenter structures were always given the highest priority.\nNext, likely dwarf stars were identified and removed using a reduced proper motion (RPM) diagram in the NIR; we followed the the example of \\citet{colliercameron_2007}, who adapted a \\emph{Tycho}-based algorithm established by \\citet[][]{gould_2003} to the 2MASS $J$ filter \\citep{Skrutskie_06_2mass}. \nWe used the best fit from \\citet{colliercameron_2007} to separate dwarfs from giants as follows.\nFirst, the RPMJ for a star is defined as follows, \n \\begin{equation}\n \\label{eq:rpmj_def}\n RPMJ = m_{\\rm J} + 5\\log(\\mu) \\\\\n \\end{equation}\n\\noindent and the empirical division between giant- and dwarf-like RPMJ, is \n \\begin{equation}\n \\label{eq:rpmj_division}\n RPMJ \\ {\\rm Division} = -141.25 (J-H)^{4} + 473.18 (J-H)^{3} - 583.6 (J-H)^{2} + 313.42 (J-H) - 58.\n \\end{equation}\nThus, for source $i$, if RPMJ$_{i}$ is brighter than \\autoref{eq:rpmj_division}, then the star is a RPMJ giant candidate and, conversely, if its RPMJ$_{i}$ is fainter, then the star is a RPMJ dwarf candidate; we will refer to \\autoref{eq:rpmj_division} this as the ``RPMJ Division'' in the text that follows.\nAny source defined as an RPMJ dwarf candidate using the URAT1 proper motions \\citep{urat1} was removed from initial consideration and only the remaining, giant candidates were available for the first phase of targeting; however, the dwarf candidates would still be eligible for selection in``main red star sample.'' \nStars targeted as RPMJ giant candidates have bit 8 set in \\texttt{APOGEE2\\_TARGET2}. \n\nFor any remaining fibers, targets were selected largely following the normal red star sample color-cuts (without consideration its RPMJ; \\autoref{eq:rpmj_def}). \nFields with $|b|$\\textless $24^{\\circ}$ followed a disk-like cut with $(J-K)_{0}$ \\textgreater\\ 0.5 and a requirement that 50\\% of the stars were assigned to each the short and medium samples per design.\nFields with $|b|$\\textgreater $24^{\\circ}$ followed a halo-like cut with $(J-K)_{0}$ \\textgreater\\ 0.3, but also had a 50:50 short\/medium cohort requirement.\nThe targeting flags for these selections are as given in \\autoref{tab:colorcuts}. \n\n\n\\subsubsection{Outer Disk Program Assessment} \\label{sec:outerdist_assess}\n\nThe implementation of the foreground removal using the RPMJ division (\\autoref{eq:rpmj_division}) represents a different targeting strategy than that typically employed in APOGEE \\citepalias{zasowski_2013,zasowski_2017}. \nThus, we provide evaluation of the targeting strategy in three ways:\n (i) by the fraction of spectroscopic giants and dwarfs obtained using the RPMJ strategy compared to that obtained in the Main Survey,\n (ii) by the reliability of the RPMJ giant selection using newer, and more precise, proper motions, and \n (iii) by the fraction of targets at the distances of the overdensities of interest.\n\\autoref{fig:pm_comp} is used to illustrate these evaluations, with the columns in this figure representing the BTX targeting (\\autoref{fig:pm_comp}a and \\ref{fig:pm_comp}c) and a selection of stars from the ``main red star sample'' at a similar location in the sky, i.e., in original APOGEE-1 and -2 fields, which we refer to as the ``Main Survey'' sample below (\\autoref{fig:pm_comp}b and \\ref{fig:pm_comp}d). \n\nThe BTX Sample is selected using \\texttt{PROGRAMNAME} of ``odisk''.\nThe ``Main Survey'' sample covers the same area on the sky (see \\autoref{fig:outerdisk_map}; $90^{\\circ} < \\ell < 220^{\\circ}$ and $5^{\\circ}<|b|<40^{\\circ}$), but we have eliminated fields dominated by special targeting (e.g., using \\texttt{PROGRAMNAME} to remove stars in the young stellar clusters, radial velocity monitoring, and contributed programs, among others) and removed the mid-plane to avoid extremely dusty sightlines ($|b| > 5^{\\circ}$). \nThe Main Survey and BTX Samples used to compare the Outer Disk targeting each have $\\sim$18,000 stars.\nBecause the BTX targeting (\\autoref{fig:outerdisk_map}) was designed to target high-latitude fields, the samples are not perfectly matched in terms of the underlying stellar density, but the samples are comparable enough for present purposes.\n\nThe rows in \\autoref{fig:pm_comp} show the differences in the RPMJ (\\autoref{eq:rpmj_def}) distribution obtained using the URAT1 proper motions adopted for the BTX design \\citep{urat1} (a and b) and what would be obtained using the \\revise{{\\it Gaia}~eDR3 proper motions \\citep{gaia_edr3}} (c and d).\nThe \\citet{colliercameron_2007} RPMJ division between dwarf- and giant- candidates (\\autoref{eq:rpmj_division}) is shown in each panel as the dashed line.\nThe color-coding is based on spectroscopic luminosity class using the calibrated $\\log{g}$, such that dwarfs are blue, giants are red, and subgiants are green (see \\autoref{ssec:datasets} for the $\\log{g}$\\ limits). \nGiant-type stars in the spectro-photometric distance catalog with heliocentric distances larger than 12~kpc, placing them at the distance of the TriAnd overdensity (or beyond), are shown as the larger, yellow symbols. \n\n\n\\subsubsection{RPMJ Efficacy for Identifying of Giants}\n\n\\autoref{fig:pm_comp}a shows the BTX Sample with the RPMJ determined from the URAT1 proper motions; as designed, 98\\% of the targets are giant-candidates using the RPMJ division (above the dashed line; \\autoref{eq:rpmj_division}) and the 2\\% of targets that are classified as RPMJ dwarf candidates are all from the ``main red star sample'' (using the targeting flags).\nIn \\autoref{fig:pm_comp}b, the Main Survey sample (that only followed the ``main red star sample'' color-magnitude criteria) RPMJ-color distribution using URAT1 is shown. \nFor the Main Survey sample, only \\revise{81\\% of the targets} would have been classified as giants and \\revise{19\\% being classified} as dwarfs following the RPMJ division (\\autoref{eq:rpmj_division}).\nThus, \\revise{19\\%} of the targets in the Main Survey sample could have been excluded were the RPMJ criterion employed, and opened up fibers for more giant candidates to be observed.\n\nIn the end, the efficacy of the RPMJ criterion can be evaluated using the spectroscopic parameters. \nOf the BTX Sample, \\revise{69\\% of the stars} are spectroscopic giants comprise whereas 72\\% of the targets from the Main Survey sample were giants; thus, the end yield of giant stars were similar from either program.\nHowever, it is important to note that the bulk of the BTX Sample were targeting regions of dramatically lower stellar density than the Main Survey. \nThe initial targeting simulations of APOGEE-1 illustrated that the main red star sample would achieve a $\\sim 50-75$\\% giant fraction in traditional disk fields, but at higher latitudes such as the BTX outer disk fields, the respective giant fraction was significantly lower, around $\\sim 25-50$\\% \\citep[see Appendix D. of][]{majewski_2017}. \nTherefore, achieving the same fraction of giants in the BTX and Main Survey samples is impressive.\nPerhaps most importantly, 75\\% of the spectroscopic dwarfs from the Main Survey sample would have been identified as dwarf-candidates from using the RPMJ division.\n\n\\begin{figure}[h]\n \\begin{mdframed}\n \\centering\n \\includegraphics[width=0.8\\textwidth]{f07_odisk_rpmj}\n \\caption{ \n RPMJ selection criteria applied in the outer disk (left column) and the same visualized for the main survey sample in the same region of the sky (right column) for different proper motion catalogs, at top URAT1 and middle \\revise{{\\it Gaia}~eDR3 \\citep{gaia_edr3}}. \n The dashed line shows the RPMJ division given in \\autoref{eq:rpmj_division}.\n The points are color-coded by spectroscopic giants (red), spectroscopic sub-giants (green), and spectroscopic dwarfs (blue) with the larger symbols indicating spectroscopic giants at heliocentric distances greater than 12 kpc. \n The BTX and Main Survey samples both have $\\sim$18,000 stars. \n The BTX sample has systematically bluer colors than the main survey, though the main survey sample does have higher-extinction sight-lines. \n (e) Fraction of targets at a given heliocentric distance for the main survey (thin blue) and the BTX (thick purple). \n }\n \\label{fig:pm_comp}\n \\end{mdframed}\n\\end{figure} \n\n\\subsubsection{Reliability of URAT1 RPMJ}\nWhile scientifically interesting, the BTX Outer Disk program was designed to fill an imminent need for observations in specific regions of the sky and it was one of the first programs planned in the BTX.\nAt that time, the URAT1 proper motions were the best available and, for the purposes of screening out nearby stars, should have been more than sufficient given our goals.\nSince that time, {\\it Gaia}~eDR3 proper motions were released that attain significantly higher precision \\citep{gaia_edr3}.\n\\autoref{fig:pm_comp}c and \\ref{fig:pm_comp}d investigate \\revise{if the more precise, {\\it Gaia}~eDR3 proper motions could have impacted our selection}.\\footnote{\\revise{We also performed this exercise with {\\it Gaia}~DR2 \\citep{gaia_dr2} and drew identical conclusions.}}\n\nIn the BTX Sample, the number of targets classified as RPMJ giant candidates changes from 98\\% using URAT1 to 92\\% using {\\it Gaia}~eDR3 (\\revise{note $\\sim$1\\%} of the targets did not have proper motions in {\\it Gaia}~eDR3, so if we only consider the sample of stars with measured proper motions, the number of giant candidates \\revise{increases to 93\\%}).\nThus, using the same RPMJ division and the more precise {\\it Gaia}~DR2 proper motions would reclassify $\\sim$6\\% of the targets. \nHowever, or the BTX Sample, 99.8\\% of the spectroscopic giants were RPMJ giants \\revise{in {\\it Gaia}~eDR3} (with a similar fraction from the Main Survey), so therefore, this $\\sim$6\\% reclassification is predominantly changing spectroscopic dwarfs from giant-candidates to dwarf-candidates and all coming from the ``main red star sample''. \n\nBecause the majority of the fields in the BTX Sample still had open fibers after selecting all of the available RPMJ giant candidates and the {\\it Gaia}~eDR3 proper motions tended to only reclassify ``main red star sample'' dwarfs, it is unclear that {\\it Gaia}~eDR3 proper motions would have had a strong impact on the BTX targeting.\nThus, we see no significant impact to our targeting by having used the less precise URAT1 motions over those of {\\it Gaia}~eDR3, largely because we use these not to select the giants, but to suppress the dwarf foreground and the dwarf foreground has sufficiently large proper motions that the precision of the underlying astrometric catalog has less of an impact.\n\nWe further note that the RPMJ division defined by \\citet{colliercameron_2007} was designed to construct a ``pure'' sample of dwarf stars and our comparisons reinforce the reliability of this tool for that purpose. \nHowever, inspection of \\autoref{fig:pm_comp}c and \\ref{fig:pm_comp}d suggests that the {\\it Gaia}~eDR3 proper motions are sufficiently precise that the RPMJ division could be refined.\nThe refinements could act to build a more complete dwarf sample by including the ``cloud'' of spectroscopic dwarfs just above the RPMJ division and, in turn, to make a more pure giant sample with their exclusion. \n\n\n\\subsubsection{Using RPMJ to Access Distant Stars}\n\nLastly, \\autoref{fig:pm_comp}e is a histogram of fraction of targets at a given heliocentric distance for the Main Survey (thin, light blue) and BTX outer disk sampling (thick, purple). \nOverall, the BTX targeting scheme has a higher fraction of stars at larger distances ($d>15$~kpc) and a similar fraction of stars from $10$ 4-5 $M_{\\odot}$) AGB stars form very different isotopes (such as $^{87}$Rb, $^{7}$Li, $^{14}$N) from the isotopes formed by lower mass AGB stars and supernova explosions, as a consequence of different dominant nuclear reaction mechanisms \\citep{abia_2001,garcia-hernandez_2007}. \nStars evolving from the AGB phase to the PNe stage also form complex organic molecules (such as polycyciclic aromatic hydrocarbons or PAHs, fullerenes, and graphene) and inorganic solid-state compounds. \nThus, the $\\sim$10$^{2}$ to $\\sim$10$^{4}$ years of evolution following the end of the AGB phase represent a most fascinating laboratory for astrochemistry. \n\nIn AGB stars, the CNO elemental and isotopic abundances, aluminum as well as the abundances of several $\\alpha$-elements together with s-process abundances (Rb, Sr, Y, Cs, Nd, Ce) can be measured with APOGEE spectra.\nComplementary, circumstellar effects on the chemical abundances in AGB stars will be also investigated and we will attempt the identification of high excitation s-process lines for bright PNe (e.g., [Kr {\\sc III}], [Se {\\sc IV}], [Rb {\\sc IV}]).\n\nThe targets were selected using four different sources:\nThe General Catalogue of Galactic Carbon stars \\citep{alksnis_2001}, \nThe Strasbourg-ESO Catalogue of Galactic Planetary Nebulae \\citep{acker_1992},\nThe Torun catalogue of Galactic post-AGB and related objects \\citep{szczerba_2007} and the PhD thesis of Pedro Garcia-Lario (1992).\nTargets were selected with 7.0 \\textless\\ $H$ \\textless\\ 13.5.\n\\section{2017 Ancillary Programs} \\label{sec:ancillary2017}\nAncillary programs were solicited via an open call across the SDSS-IV collaboration (i.e., not limited to just APOGEE) and were awarded based on a review that focused on technical and scientific feasibility.\nThe 2017 proposals were due in February 2017, selected by peer review during March\/April 2017, and implemented starting in August\/September 2017. \nBecause the bulk of the Main Survey observations had already been drilled when implementation began, the 2017 Ancillary Science Programs had more limited access to certain parts of the sky.\n\nThe timing of the 2017 Ancillary Call and the formulation phases of the Bright Time Extension, while distinct processes, meant that programs were merged during implementation for those cases where the scientific or targeting goals were aligned. \nSuch instances have been noted in the main text, but given that these projects were approved based on their scientific merits independent of the Bright Time Extension, we include them here.\n\\subsection{M33 Globular Clusters} \\label{anc:m33}\nM33 has both young and old globular clusters (GCs) that span from 1 to 12 Gyr in age \\citep{chandar_2006,beasley_2015}. We might expect both old and young GCs to show the CNO and Na\/O anomalies -- extragalactic GCs have been inferred to host Na\/O anti-correlations since many have high [Na\/Fe] as measured from integrated light \\citep{colucci_2014, sakari_2015}.\nSurprisingly, abundance variations have not been observed in young to intermediate-aged LMC GCs \\citep[e.g.,][]{sakari_2017}. \nThis motivated a program to collect integrated-light spectra on young and old GCs in the Triangulum Galaxy (M\\,33) to test this hypothesis. \n\nM\\,33 is the third most massive galaxy in the Local Group and it is is much less studied than the Milky Way (MW), Andromeda (M31) and the Magellanic Clouds (MCs).\nThe Pan-Andromeda Archaeological Survey (PANDAS) revealed large-scale substructures of low surface brightness, including arcs, stream and globular clusters, connecting the M\\,31 and M\\,33 galaxies \\citep{huxor_2011}. Substructure in our own Galactic halo reveals its merger history, and certain globular clusters (GCs) appear to be associated with specific accreted satellites (e.g., the Sagittarius dwarf spheroidal galaxy).\nHence, Globular Cluster Systems (GCSs) can be used as tracers of the formation processes and the assembly history of a galaxy. \n\nTo date, integrated-light spectroscopy in the optical has been obtained for only twelve M33 GCs \\citep{beasley_2015}; no such data exist in the NIR.\nYet the APOGEE $H$-band wavelength coverage confers some significant advantages for integrated-light spectroscopy: insensitivity to hot stars, but high sensitivity to red giant branch (RGB) and asymptotic giant branch (AGB) stars, a feature that simplifies integrated-light analysis \\citep{schiavon_2004,sakari_2014}. \nThe $H$ also offers access to some chemical features not easily available to optical spectroscopy, in particular, the presence of strong molecular lines of CN, CO and OH, which enable determinations of C, N and O abundances \\citep{smith_2013,sakari_2016}Other useful lines for this work are those for Mg, Al, Si, Ca and Ti. \nAPOGEE spectra also bring the opportunity to probe multiple populations in GCs, the ability to detect [O\/Fe], and to probe directly the Na\/O anti-correlation \\citep{sakari_2016}. \n\nWe selected our targets from the catalogue with homogeneous $UBVRI$ photometry of 708 M\\,33 star clusters and cluster candidates based on archival images from the Local Group Galaxies Survey \\citep{fan_2014}.\nWe use the photometry for the M\\,31 GCs \\citep[][their table 1]{sakari_2016} to convert from the $V$ to $H$ in the \\citet{fan_2014} catalogue. \nWe select all the clusters with 12.5 \\textless\\ $H$ \\textless\\ 15 for a total of 132~clusters; these also have initial estimates of metallicity, age and mass from \\citet{fan_2014}.\n\\subsection{Cepheids Calibrators} \\label{anc:cephcalib}\nThe goal of this program is to provide homogeneous chemical characterization of Galactic Cepheids that have multi-wavelength photometric characterization and will have sub-percent precision trigonometric parallaxes from {\\it Gaia}. \nMultiple epochs of APOGEE spectra were obtained for each star using the NMSU 1-meter fiber feed. Chemical abundances will be used to calibrate the metallicity effects in the period-luminosity relationship for ten photometric bands and will provide high signal-to-noise templates over multiple phase points for other Cepheid-based programs in APOGEE-2.\n\nFor nearly a century, Cepheid variables have been the de facto standard candle buttressing the extra-galactic distance scale. While nearby dwarf galaxies provide excellent probes of the Leavitt law at low metallicity, the overall metallicity sensitivity of the Leavitt Law remains relatively poorly unconstrained due to the lack of high metallicity calibrators (the Galactic field Cepheids) with independent distances. \nThe trigonometric parallaxes delivered by {\\it Gaia}\\ are the first to provide these independent distances for a large sample of Cepheids in the Galaxy. \nThis NMSU 1-meter fiber extension ancillary project targets a set of the most well characterized Cepheids in the Galaxy that are poised to be the best calibrators in {\\it Gaia}.\nThese stars serve not only an important scientific role in themselves, but will support other Cepheid projects in APOGEE-2 and SDSS-V.\n\nThe first goal of the program is to calibrate the metallicity dependency of the Leavitt Law and utilize the \\citet{fouque_2007} sample of Cepheids with magnitudes in eight optical and near-infrared bandpasses, to which we add newer observations in the near-infrared and mid-infrared for a total of ten bands. \nThis sample includes the most nearby Cepheids that will, in turn, have the best {\\it Gaia}\\ parallax measurements. \nThe stars will be sampled over a number of epochs to study the stability of the metallicity measurements over phase, which helps to place potential single phase measurements in context. \nA second goal of the project is to calibrate the \\citet{scowcroft_2016b} mid-infrared color-metallicity relationship using a homogeneous metallicity characterization spanning 2 dex in [Fe\/H] (in combination with other APOGEE-2 Cepheid programs).\nThis Galactic sample will have the best mid-infrared light curves for Cepheids \\citep[][]{monson_2012}. \n\nWe selected the sample from \\citet{fouque_2007}, which has homogeneously derived magnitudes in eight bands, to which we add 3.6\\micron\\ and 4.5\\micron\\ measurements from the Carnegie Hubble Program \\citep{monson_2012}, better NIR sampling \\citep{monson_2011}, and revised line-of-sight extinction measurements \\citep{madore_2017}. \n\nThe \\citet{fouque_2007} multi-wavelength sample contains 60 stars with 2MASS apparent magnitudes in the range 1.8 \\textless\\ $H$ \\textless\\ 8.2 (note that for bright stars the 2MASS magnitude uncertainties are 50\\% of the pulsation amplitude of the star). Restricting to those objects with $\\delta$ \\textgreater\\ $-30^{\\circ}$, there are 34 remaining sources that range in magnitude over 1.8 \\textless\\ $H$ \\textless\\ 7.7. We prioritized the sources by putting those sources with TGAS parallaxes as top priority and those without TGAS parallaxes as lower priority \\citep{gaia_dr1}. \nDue to saturation limits with {\\it Gaia}, this distinction amounts to down-weighting the brightest sources (in the range 1.8 \\textless\\ $H$ \\textless\\ 5). \n\n\\subsection{Brown Dwarfs} \\label{anc:browndwarf}\nThis program aims to build a library of late-M and L dwarf APOGEE spectra for the purpose of (1) measuring spatial and rotational kinematics for the nearby population and (2) extending spectral modeling for abundance analysis to $T_{\\rm eff}$\\ \\textless\\ 2700~K. Sources with spectral types later than M7 and 11 \\textless\\ $H$ \\textless\\ 14.5 were selected. The desired scientific outcome is improved spectral models across the hydrogen-burning limit, with an eye toward improving characterization of potential low-mass terrestrial exoplanet host systems.\n\nThe transition between the M dwarf and L dwarf spectral classes for very low mass, $M_{*}$ $\\leq$ 0.1 $M_{\\odot}$ \\citep[VLM;][]{kirkpatrick_2005}, stars and brown dwarfs is a critical benchmark in studies of Galactic populations, substellar evolution, stellar magnetic field generation, angular momentum evolution, star formation processes and history, and exoplanet atmospheric chemistry and habitability. \nThis transition spans the temperature range for atmospheric condensate formation, a focus of current star and exoplanet spectral modeling work \\citep{helling_2008,marley_2010}; and the decoupling of atmospheres from internally-generated magnetic fields \\citep{mohanty_2002}, which results in sharp declines in the incidence and strength of H$\\alpha$ and X-ray non-thermal emission \\citep{west_2011} -- albeit with dramatic exceptions \\citep[e.g.,][]{schmidt_2014}, and reduced angular momentum transport resulting in rotation periods as short as 1-3 hr \\citep{konopacky_2010,irwin_2011}. \nLate-M and L dwarfs also span the hydrogen-burning mass limit ($M_{*}=0.07~M_{\\odot}$) and are the densest hydrogen-rich bodies known, probing a minimum in the mass-radius relationship and potentially exotic states of matter \\citep{burrows_2001}. \nThe long lifetimes of these sources (trillions of years) and their limited fusion and full convective mixing make them ideal time capsules for Galactic star formation and chemical evolution history.\nFinally, their small radii ($R_{*} \\sim$0.1 $R_{\\odot}$), close-in habitable zones ($d R_{*}^{-1} \\sim$ 10-30), and apparent preference for forming Earth-sized planets \\citep{dressing_2013} make VLM dwarfs ideal targets for probing Galactic habitability through the transit method \\citep[e.g., Trappist-1,][]{gillon_2017}.\n\nCharacterizing the physical and populative properties of local VLM dwarfs is optimally accomplished with high-resolution infrared spectroscopy. \nAPOGEE's resolution and sensitivity are well matched to the infrared magnitudes and rapid rotation of these objects (up to 80 km s$^{-1}$), while its broad spectral coverage is critical for measuring atmospheric abundances, probing both bulk composition and atmospheric chemical dynamics. \nHowever, late-M and L dwarfs fall below current APOGEE stellar modeling temperature limits \\citep[][]{garciaperez_2016,schmidt_2016,souto_2017}. \nDR14 included $\\sim$25 VLM dwarfs \\citep{holtzman_2018} and this provided only a narrow scope for examining population properties. \nThis program aims to increase the observe sample 20-fold by targeting up to 448 M6-L6 dwarfs. \nThese data will provide radial velocities, which, combined with PanSTARRS\/{\\it Gaia}\\ proper motions, will: yield precise kinematics (\\textless 1-2 km s$^{-1}$); measure rotational velocities down to 5 km s$^{-1}$, to examine angular momentum evolution and magnetic activity trends in conjunction with ancillary optical spectral observations; enable searches for close-separation companions down to planetary masses through RV variability measurements; improve spectral modeling in the low-temperature regime; and characterize planet-host candidates targeted by MEarth \\citep{nutzman_2008}, SPECULOOS \\citep{gillon_2013}, and {\\it TESS}\\ \\citep{ricker2015}, among others.\n\nTargets were selected from compilations of known VLM dwarfs, including \n Dwarf Archives\\footnote{\\url{DwarfArchives.org}}; \n Pan-STARRS DR1 \\citep{best_2017}; \n BOSS Ultracool Dwarfs \\citep{schmidt_2015}, \n BASS \\citep{gagne_2015}, \n LaTE-MoVeRS \\citep{theissen_2017}, \n MEarth \\citep{newton_2017}, \n SPECULOOS \\citep{gillon_2013}, and \n the {\\it TESS}\\ Input Catalog \\citep{stassun_2018}. \nTargets were selected to have reported spectral types later than M7 ($T_{\\rm eff}$\\ \\textless 2700~K) and 11 \\textless $H$ \\textless 14.5. \n\\subsection{Distant Halo Giants} \\label{anc:distanthalo}\nAs discussed in Section \\autoref{sec:halo}, the initial targeting strategy of APOGEE unfortunately yielded relatively few spectra for distant halo stars ($D$\\textgreater10~kpc).\nOn the other hand, the SEGUE\ncatalog consists of 6000 confirmed K~giants \\citep[][]{xue_2014}, many of which are at large heliocentric distances.\nThe 0.5 \\textless\\ ($g$-$r$) \\textless\\ 1.3 color range is within the temperature range of the APOGEE pipeline. \nObservations that deliberately target known SEGUE giants will dramatically increase the number of distant stars observed by APOGEE and, in particular, those in the distant halo. \nBy comparison there were approximately 60 halo giants observed by APOGEE in DR14 \\citep[based on metallicity, velocity and distance;][]{holtzman_2018,dr14} from approximately 100 halo or stream plates observed by APOGEE to that point in the survey \\citepalias[][]{zasowski_2013,zasowski_2017}.\nThe halo is lumpy and so the number of giants within the APOGEE magnitude varies greatly, but by targeting known stars we will confidently boost this sample of stars and ensure that APOGEE obtains a chemical fingerprint of this Galactic component.\nThis program was subsumed, fully, into the BTX Halo targeting described in \\autoref{sec:new_halo_targeting}. \n\\subsection{The Young Galaxy} \\label{anc:younggal}\n{\\it Gaia}\\ DR2 has increased our understanding of the Galaxy. \nHowever, a complete description of the Galactic thin disk remains challenging, because even {\\it Gaia}\\ has severe limitations in the Galactic plane, where dust extinction dims stars below the detection limit at distances greater than 5~kpc. \nWe target faint\/distant young Cepheids that were identified in the PanSTARRs \\citep[PS1;][]{panstars_dr1} multi-epoch catalog to sample the farthest obscured reaches of the Galactic disk.\n\nCepheids are powerful probes of both the structure and the recent history of the Milky Way: \nthey are luminous and can be seen to great distances, even through substantial dust extinction;\ntheir individual ages and distances can be precisely determined from their periods; and, with ages of 20-150 Myrs, they are young stars but they are relatively cool and hence their spectra show a rich metal-line absorption spectrum, from which many element abundances can be determined. \nThe new PanSTARRs (PS1) catalog of variable stars is currently still under construction.\n\nWe selected a list of targets determined on the basis of the following steps:\n[1] identification of Cepheids-like variables in the entire PS1 database, by using the variability parameters by \\citet{Hernitschek_2016} and colors defined on the basis of magnitudes in the $grizy$ (PS1) and near-infrared bands from 2MASS \\citep{Skrutskie_06_2mass} and ALLWISE \\citep{cutri_2014}; \n[2] multi-band template fitting of the sparsely sampled PS1 light curves for all the 200,000 sources selected and determination of pulsation period and other parameters (including distance modulus and extinction);\n[3] use of machine-learning techniques to extract the variables classified as Cepheids with highly significant probability (i.e., higher classification score).\nThe targets existing in APOGEE fields will be targeted for multi-epoch observations as is feasible.\n\\subsection{Kapteyn Selected Area 57} \\label{anc:sa57}\n\nSelected Area 57 (SA57) is the nearest of Kapteyn's Selected Areas (an ambitious program to systmatically study the Milky Way first organized by Jacobus Kapteyn in the year 1906) to the North Galactic Pole. \nSA57, therefore, lies in a direction relatively free of reddening and where ``in situ'' halo stars can most easily be accessed. \nSA57 has traditionally played a significant role for a variety of astronomical studies, from probing the vertical density laws of the Galactic stellar populations to deep studies of galaxies and quasars, the latter because of the minimal stellar foreground. \nAs a result of more than a century of interest in this direction of the sky, SA57 has received extensive attention by surveys of photometry, astrometry and spectroscopy --- but never, to this point, high-resolution spectroscopy. \n\nThe APOGEE observations here are intended to tie state-of-the-art chemistry of giant stars in SA57 to the rich legacy of previous observations in this field. \nStars were selected following the criteria of main red star sample \\citepalias{zasowski_2013,zasowski_2017}.\nFour overlapping plate centers (N, S, E, W) were targeted yielding three cohorts in a ``wedding-cake'' arrangement that follows the magnitude limits of the main red star sample, but also takes into account the overlap between the plates to reach the full $S\/N$.\n\\subsection{Tidally Synchronized Binaries} \\label{anc:tlockbinary}\nIt is now common knowledge that more stars have binary companions than are truly isolated stars \\citep{Raghavan_2010}. \nWhile the formation of single-star systems is still under active study, the formation and evolution of binaries demonstrates rich behavior and, yet, is even more poorly understood. \nBinary formation is governed by the interplay between fragmentation and the circumprotobinary disk\n\\citep{Artymowicz_1991,Artymowicz_1994,Bonnell1994,Bate_2000,gunther_2002}, while the subsequent evolution occurs due to dynamical interactions in the cluster \\citep{Bodenheimer_2001,clarke_2001}.\nUnfortunately, obtaining data on the field binary population to compare to evolutionary models is notoriously difficult.\nThe selection function for field binaries is so heterogeneous and biased that statistical analyses do not provide strong constraints. \nBinaries in clusters are more easily interpretable; however, binaries currently in clusters may not be representative of the escaped field population.\n\nThe aim of this project is to collect a verified sample of tidally-synchronized binaries selected from the rapid rotators in the {\\it Kepler}\\ field. \nThe modulation amplitude in tidally-synchronized binaries should be easily measured because rapid rotation enhances starspot activity \\citep{Basri_1987}.\nApplying solar neighborhood binary properties \\citep{Raghavan_2010} to the {\\it Kepler}\\ field predicts around 300 non-eclipsing tidally-synchronized binaries showing detectable rotational modulation, compatible with the actual number of rapid-rotators found.\n\\subsection{SubStellar Companions} \\label{anc:substellar}\nWith the ever-growing sample of companions (both stellar and substellar) being discovered around stars from ever more diverse environments, population studies comparing host environments are now possible.\nFor example, recent work suggests the hot Jupiter occurrence rate in M67 is higher than in the field \\citep{Brucalassi_2016}, although it is not clear whether this is the norm for open clusters generally.\nWe aim to expand upon the substellar companion science program in APOGEE-2 \\citepalias{zasowski_2017} to include stars in a variety of cluster environments with a wide range of ages, metallicities, and densities.\nThese observations will help constrain the role environment has to play in the formation and evolution of stellar systems containing companions of various masses. \nCritically, this program will both extend the time baseline for targets observed in APOGEE-1 and increase the number of RV measurements to enable robust orbital fits in the final dataset. \nThe \\texttt{FIELD} names from APOGEE-1 are: IC348\\_RV, M67\\_RV, and M3\\_RV. \n\n\\subsection{M dwarfs in {K2}} \\label{anc:mdwarfk2}\nM-dwarf stars are increasingly important objects in both the fields of exoplanet searches, due to the emphasis on cool dwarfs in the target lists of {K2}\\ and {\\it TESS}, as well as studies of Milky Way stellar populations, as these are the most numerous stars in the Galaxy. \nM-dwarfs are notoriously difficult to analyze via optical spectroscopy, due to intense molecular line absorption; this difficulty is alleviated greatly at infrared wavelengths. \nOur team is using APOGEE spectra of M dwarfs to pioneer the derivation of fundamental stellar parameters and detailed chemical compositions for these types of stars \\citep{souto_2017,Souto_2020}. \nM dwarfs are a particularly important component in both transit and radial-velocity searches for exoplanets, thanks to the enhanced detectability of small planets due to low stellar masses, low luminosities, and small stellar radii. \nThis work opens a new window into a class of planet-hosting stars that will play an increasingly important role in ongoing and future planet surveys and missions, such as {K2}\\ and {\\it TESS}. \nTargets from this program are shown in \\autoref{fig:k2}a.\n\\subsection{Local Group Stellar Populations in Integrated Light} \\label{anc:m31}\n\nThe MW provides us with a unique opportunity to constrain galaxy evolution using individual interstellar clouds and single stars, both of which serve as the fundamental building blocks of galaxies. \nThe results from these stellar measurements in the Galaxy have guided our understanding of the chemo-dynamical processes that impact galaxies on all spatial scales.\nThis understanding, then, relies on the assumption that the processes required to produce these spatial and chemical distributions are universally available and commonplace for galaxies generally. M\\,31 provides one opportunity for testing this assumption.\n\nWith 3\\arcsec\\ fibers at APO, APOGEE-2 cannot resolve individual stars in M\\,31, so integrated light observations are required to identify multiple stellar populations with distinct chemistry and dynamics.\nAPOGEE collected integrated light spectra of the inner disk of M\\,31, spaced in a grid of points within $R_{\\rm M31} < 5$~kpc, and of the centers of M\\,32 and M\\,110. \nAll fiber positions were visually checked against optical and near-IR imaging and shifted by up to 10\\arcsec\\ to ensure that no bright stars, clusters, or emission line regions fell within the aperture. \nFibers that could not be fit onto these targets due to fiber packing and plugging limitations were placed on positions observed by the MaNGA survey and on Luminous Blue Variable stars in the field of view.\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction \\label{section_intro} }\n\n\nNGC~2419 is a globular cluster (GC) in the outer halo of the Galaxy at\na distance of 84~kpc\\footnote{The distance and other parameters for NGC~2419\n are from the 2010 version of the on-line GC database of\n \\cite{harris96}, based on the photometry of \\cite{harris97}.}.\n\\cite{sigmav09} determined a velocity dispersion for this cluster\nbased on 40 stars which leads to $M\/L = 2.05\\pm0.50~M_{\\odot}\/L_{\\odot}$, a normal\nvalue for an old stellar system, with no evidence for dark\nmatter at the present time. \\citet*{cls10} further asserted that this cluster could not\nhave formed in a now-defunct dark matter halo. \\cite{harris97}\nobtained a deep CMD of NGC~2419 with HST\/WFPC2 and demonstrated that\nthere is no detectable difference in age between it and M92, an\nancient, well-studied inner halo GC of comparable metallicity.\nFurthermore, there is no evidence from CMDs of multiple stellar\npopulations in NGC~2419.\n\n\nHowever, some of the characteristics of NGC~2419 are anomalous for a\nGC. It has the highest luminosity ($M_V \\sim -9.6$~mag) of any GC\nwith Galactocentric radius $R > 20$~kpc, higher than all other GCs\nwith $R > 15$~kpc with the exception of M54, which is believed to represent\nthe nuclear region of the Sgr dSph galaxy now being accreted by the Milky Way\n\\citep{ibata95,sarajedini95}. NGC~2419 also has an unusually large\nhalf-light radius ($r_h = 19$~pc) and core radius ($r_c = 7$~pc)\nfor a massive Galactic GC and a relaxation time which exceeds the Hubble\ntime, also unusual for a GC. Every other luminous GC in the outer\nhalo is considerably more compact. The 2010 version of the on-line\ndatabase of \\cite{harris96} lists only three Galactic GCs that lie\nanomalously above the bulk of the Galactic GCs in the plane $r_h$ vs\n$M_V$, one of which is NGC~2419. The other two are M54 and $\\omega$\nCen, the most luminous Galactic GC, and one with a large internal\nspread in metallicity, also believed to be the stripped core of a\ndwarf galaxy accreted by the Milky Way. For these reasons,\n\\cite{vdb04} and \\cite{mackey05}, among others, suggested that\nNGC~2419 is also the remnant of an accreted dwarf galaxy. In fact,\n\\citet{irwin99} and \\citet{newberg03} suggested that NGC~2419 is one\nof many GCs associated with the disrupted Sgr dSph, but \\citet{law10}\nfirmly concluded that it is not. Therefore, the cluster is potentially\na piece of an unidentified, tidally disrupted dSph.\n\n\n\nTo follow up on these earlier suggestions, \\cite{deimos} obtained\nmoderate resolution spectra of a large sample of luminous giants in\nNGC~2419 and analyzed line strengths in the region of the\nnear-infrared Ca triplet. We found that there is a small but measurable\nspread in Ca abundance of $\\sim$0.3~dex within this massive metal-poor\nglobular cluster. We present here an analysis of high dispersion\nspectra of seven luminous giant members of NGC~2419, whose chemical\ninventory we compare\nto that inferred from a similar sized sample of luminous RGB members of the nearby\ninner halo globular cluster M30 (NGC~7099), observed and analyzed\nin the same way as the NGC~2419 stars. Note that\nthese two globular clusters\nhave the same metallicity in the 2003 version of the\non-line database of \\cite{harris96}.\nWe explore two issues:\n\n\\begin{enumerate}\n\n\\item Is the chemical inventory in such a distant cluster identical to\n that of a globular cluster of similar [Fe\/H] in the inner halo?\n\n\\item What star-to-star variations are revealed by our high-dispersion\nspectra?\n\n\n\\end{enumerate}\n\n\n\\section{Observations and Analysis\\label{section_obs} }\n\n\nWe obtained high-spectral-resolution and reasonably high-SNR spectra\nof 7 luminous giants in NGC~2419 using HIRESr \\citep{vogt94} on\nthe Keck~I 10~m Telescope. \nThe stars were selected from both the early spectroscopic study of\n\\cite{suntzeff88} and the current version of the\n on-line photometric database described by \\cite{stetson05}.\nThe sample stars range in $V$ from 17.2 to\n17.6~mag. The spectral range is 3900 to 8350~\\AA\\ with gaps\nbetween the reddest echelle orders. We used a 1.1~arcsec slit,\nequivalent to 6.3 pixels at 15$\\mu$\/pixel, to achieve a spectral\nresolution of 34,000. The maximum total exposure time for one star\nwas 2.5 hours, split into shorter segments to improve cosmic ray\nremoval. The code MAKEE\\footnote{MAKEE was developed\nby T.A. Barlow specifically for reduction of Keck HIRES data. It is\nfreely available on the world wide web at the\nKeck Observatory home page, \nhttp:\\\\www2.keck.hawaii.edu\/inst\/hires\/data\\_reduction.html.}\nwas used to reduce the HIRES spectra, which\nwere obtained in various runs over the past 5\nyears. The majority of the seven spectra have a SNR exceeding 90 per\nspectral resolution element at 5500~\\AA\\ in the continuum. The SNR\ndegrades towards the blue, and we often eliminated lines bluer than\n5000~\\AA\\ for species with many detected features. The radial\nvelocities for these seven stars have a mean of $-20.6$~km\/sec with\n$\\sigma$ of 3.7~km\/sec. \\cite{deimos}\ndiscussed membership issues for NGC~2419 in detail. \nOur sample of 5 RGB\nmembers in the sparse nearby GC M30 (NGC~7099) were observed\nwith the same HIRES configuration; all the resulting spectra\nare of high SNR. Table~\\ref{table_sample} gives details of the samples and\nspectra. Figs.~\\ref{figure_7099_cmd} and \\ref{figure_2419_cmd}\nshow CMDs for the full Stetson on-line database \\citep{stetson05}\nas well as our HIRES sample for each of these two GCs\nwith 12~Gyr isochrones from the Y2 grid \\citep{yale03} superposed.\n\n\nThe determination of stellar parameters followed our earlier papers,\n\\cite[see, e.g.][~and references therein]{cohen05a}\nand is based on $V-I$, $V-J$, and $V-K_s$ where the optical colors are from\n\\cite{stetson05} and the infrared colors are from 2MASS\n\\citep{2mass1,2mass2}. The uncertainties in 2MASS $K_s$ are\nrather large for the faint NGC~2419 giants, and only two of the\nNGC~7099 giants have an $I$~mag in the \\cite{stetson05} database. \nWe use the predicted color grid as a function of $T_{\\rm eff}$,\nlog($g$), and [Fe\/H]\\footnote{The \nstandard nomenclature is adopted; the abundance of\nelement $X$ is given by $\\epsilon(X) = N(X)\/N(H)$ on a scale where\n$N(H) = 10^{12}$ H atoms. Then\n[X\/H] = log[N(X)\/N(H)] $-$ log[N(X)\/N(H)]$_{\\odot}$, and similarly\nfor [X\/Fe].} \nfrom \\cite{houdashelt00}. If we use the recent $T_{\\rm eff}$ --\ncolor relatoins of \\cite{irfm_teff}, which are not calibrated\nfor such luminous giants, we obtain $T_{\\rm eff}$ $\\sim$35~K\nlower for the cooler luminosity stars in our sample for\nNGC~2419, ranging up to 105~K lower for the most luminous\nstar in this GC.\nThe surface gravities\nwere calculated assuming a mass of 0.8~$M_{\\odot}$, the known\ndistance, and the (low) interstellar extinction to each GC. The isochrone of\nthe upper RGB for a very metal-poor old stellar system\nis such that an error of 100~K in $T_{\\rm eff}$ produces\nan error of 0.2~dex in the inferred surface gravity using\nthis procedure.\n\nThe red giants in our sample in NGC~2419 span 0.36~mag\nin $V$ and a range of less than 200~K in $T_{\\rm eff}$.\nThe RGB stars in our sample for NGC~7099 are somewhat hotter\nin the mean\ndue both to the somewhat lower metallicity of this GC \nand to the scarcity of upper RGB stars in this rather sparse\ncluster.\n\n\nThe detailed abundance analysis for each of the cluster giants follows\nJ.~Cohen's previous work \\citep[see, e.g.,][]{cohen05a} using\n\\cite{kurucz93} LTE plane-parallel \nmodel stellar atmospheres and the analysis code MOOG\n\\citep{moog}. The adopted transition probabilities are largely\nfrom NIST version 3. Note that the $gf$ values adopted here for\nthe observed Mg~I and Ca~I lines have\nbeen updated from those we consistently used\nup to the present \\citep[see, e.g.][]{cohen05a}\nto match the current on-line values in\nthe NIST version 4.0 \\citep{nist} database. \nThe changes \nin abundance resulting from these updates are small, about\n$-0.05$~dex for Mg for a typical set of observed lines,\n and $+0.03$~dex for Ca~I\nwhere more lines are usually observed,\nonly some of which have $gf$ values that have been updated.\n\n\nHyper-fine energy level splitting was used where\nappropriate, with many such patterns taken from\n\\cite{prochaska}. The adopted damping constants are those\ndescribed in \\cite{cohen05a}.\n\n\nAlthough it appears that small non-LTE corrections\nshould be applied for several species, for example Na~I \\citep{takeda03}\nand Ba~II \\citep{nonlte_ba2}, for consistency with\nour earlier work,\nthe only non-LTE correction we have\nimplemented is a fixed value of $+0.60$~dex for\nthe resonance Al~I doublet at 3944 and 3961~\\AA\\ following\n\\cite{al1_nonlte}, \\citep[see also][]{andrievsky08}.\nNon-LTE for K~I is discussed in \\S\\ref{section_s1131}.\n\n\nEquivalent widths were measured in some cases by the script\ndescribed in {\\S}3 of \\cite{cohen05a},\nwhich automatically\nsearches for absorption features, fits a Gaussian,\nthen matches those found\nto a master line list given the radial velocity of the star.\nFor other sample stars,\nW.~Huang used an IDL script to determine $W_{\\lambda}$\nfor the set of lines in the master list.\nThere was extensive hand checking\nof weak features, of most rare earth lines, and of many\nof the strongest accepted lines. Lines with $W_{\\lambda} > 175$~m\\AA\\\nwere rejected unless the species had only a few detected features.\nThe resulting values, together with the\natomic parameters for each line used, are given\nin Tables~\\ref{table_2419_eqw} and \\ref{table_7099_eqw}.\n\nThe $W_{\\lambda}$ for the 3961~\\AA\\ Al~I line, when given, \nare particularly uncertain for the NGC~2419 giants; the\n3944~\\AA\\ line was never used to due contamination by CH.\nBoth lines are uncomfortably far in the blue,\nwhere the SNR of the HIRES spectra for such faint stars\nis degraded, but the 3961~\\AA\\ line is a key feature for this element as \nthe 6690~\\AA\\ Al~I doublet is very weak and difficult to\ndetect at the low metallicity of this GC.\n\nBecause of the uncertainty of the photometry for the NGC~2419 giants, \nwe felt free to make\nsmall adjustments to the photometrically derived stellar parameters,\nprimarily changing $T_{\\rm eff}$. These adjustments,\ndetailed in Table~\\ref{table_temp}, improved the\nexcitation equilibrium for \\ion{Fe}{1} lines and the ionization balance\nbetween neutral and singly ionized lines of Fe and Ti. No such\nadjustment exceeded 100~K, and was less than\n50~K for four of the NGC~2419 giants. In the case of M30,\nonly one star had $T_{\\rm eff}$ modified by more than 20~K.\nFor each star in NGC~2419, we measured 67 to 108 \\ion{Fe}{1}\nlines, which allowed us to determine the microturbulent velocity\ndirectly from the spectra; every star in our sample in NGC~7099\nhad 99 or more detected Fe~I lines.\n\nTable~\\ref{table_fe_slopes} gives the slope of a linear\nfit to the Fe abundance deduced from each of the Fe~I lines\ndetected in a specific sample star\nas a function of excitation potential, reduced equivalent\nwidth ($W_{\\lambda}\/{\\lambda}$), and $\\lambda$, where the\nfirst is most sensitive to the adopted $T_{\\rm eff}$, the\nsecond to $v_t$, and the third indicates any systematic\nproblems in continuum opacity as a function of wavelength.\nValues are given for both GCs. In the ideal case, these slopes\nare zero. For NGC~2419 the typical\nrange in $\\chi$ for the Fe~I lines is 0.9 to 4.2~eV;\nthe stars with the best spectra span a larger\nrange from 0.0 to 4.8~eV. The wavelength range for Fe~I lines\nin most of the NGC~2419 (NGC~7099) spectra is $\\sim$2400~\\AA\\\n($\\sim$3000~\\AA),\nand a typical range in reduced equivalent width for\nsuch spectra is $-5.4$ to $-4.5$~dex for NGC~2419\nand $-6.3$ to $-4.4$~dex for NGC~7099. \nThe crucial slope with $\\chi$ ranged from $-0.083$\nto $-0.038$~dex\/eV for the NGC~2419 sample, and is\n$-0.048\\pm{0.007}$~dex\/eV for the M30 RGB sample.\n \n \nThe resulting abundances for the 7 RGB stars in NGC~2419\nand the 5 in M30 are given in\nTables~\\ref{table_ngc2419_abund} and \\ref{table_7099_abund},\nwhere all abundances are relative to Fe~I.\nThe adopted Solar absolute abundances for each element can be\ninferred from the entries in Table~\\ref{table_7099_abund}.\nWith regard to the ionization equilibrium, that for Fe~I\nvs Fe~II is generally quite good, within 0.1~dex, for all\nstars in both NGC~2419 and in M30. \nThe ionization equilibrium of Ti~I vs Ti~II is somewhat worse, typically\nwithin $\\sim$0.2~dex, with the Ti~II lines giving a somewhat\nhigher Ti abundance.\n\nTables~\\ref{table_abund_sens_abs} and \\ref{table_abund_sens_rel}\ngive the sensitivity of the derived\nabsolute abundances [X\/H] and those relative to Fe [X\/Fe]\nfor small variations in the various relevant stellar and\nanalysis parameters.\n\n\n\n\n\n\\section{Alternative Choices of Stellar \n Parameters \\label{section_caspread} }\n\n\nThe choice of stellar parameters is crucial for determining\nabsolute stellar abundances, although less critical for abundance\nratios [X\/Fe] as many of the systematic effects cancel,\nas is shown by comparing the entries for each species in\nTable~\\ref{table_abund_sens_abs} with those in \nTable~\\ref{table_abund_sens_rel}. We have\ntherefore explored several methods of determining $T_{\\rm eff}$.\nWe call the $T_{\\rm eff}$ determined by the mean of that\ninferred from each available\ncolor ($V-I,~V-J,$ and $V-K_s$), the photometric $T_{\\rm eff}$. \nThe value of $T_{\\rm eff}$ adopted for the abundance analysis,\n $T_{\\rm eff}$(adopt), is initially\nset to $T_{\\rm eff}$(phot), with small adjustments permitted to improve\nthe Fe~I slopes and the ionization equilibrium as described above.\n\nAnother\n$T_{\\rm eff}$ can be determined by assuming that all the stars are\nRGB members of their cluster, and that all the cluster stars\nhave the same initial chemical inventory and age. Hence they must \nlie along a single old very metal-poor isochrone.\nHere we are using a single magnitude,\nchosen on the basis of its high measurement accuracy\nand discrimination along the isochrone,\nto determine $T_{\\rm eff}$(iso).\nThis removes the issue of random errors in colors,\nwhich is of particular concern for NGC~2419, with its large distance and\nhence faint RGB stars near the limit for 2MASS photometry.\nWith a large enough sample, the ``isochrone''\nitself can be defined from the mean locus of the stars in the cluster CMD,\nas was done in \\cite{cohen05a}, see also the large VLT\nstudy of \\cite{carretta10} and references therein.\nIn the present case, even if the HIRES sample in NGC~2419\nwere much larger, \nthe concern\nthat this particular GC may not be chemically uniform would \nlimit the applicability of such an approach.\nIf the isochrone chosen is appropriate\nfor the cluster,\nthen the mean ${\\Delta}[T_{\\rm eff}(phot) - T_{\\rm eff}(iso)]$ will be zero.\nIf the isochrone is close, but not exactly that of the cluster, then\nthe mean difference will be a constant.\n\nTable~\\ref{table_temp} gives the values of the adopted, photometric,\nand isochrone $T_{\\rm eff}$ for the sample of RGB stars in NGC~2419.\nThe rms $\\sigma$ about the mean for the set of available colors\nfrom the three we consider is\ngiven for the $T_{\\rm eff}$(phot) in parentheses. \nThe photometric $T_{\\rm eff}$ was adopted for the detailed\nabundance analysis for 3 of the 5\nstars in NGC~7099. For the other two, the differences are 18~K (0.8 $\\sigma$)\nfor S31 and 79~K (1.2 $\\sigma$) for S34. The isochrone and photometric\nvalues of $T_{\\rm eff}$ for this nearby GC are essentially identical.\nFor NGC~2419 that is not the case. The brightest star (S223)\nat the RGB tip is somewhat redder than would be expected from the isochrone.\n\n\n\\section{Inner vs Outer Halo }\n\nA comparison of the mean abundance ratios in the two globular\nclusters discussed here\nM30 (NGC~7099), at a distance of only 8~kpc, with the very distant\nouter halo\nglobular cluster NGC~2419 is given in Table~\\ref{table_abund_mean}\nand illustrated in detail in Fig.~\\ref{figure_abund}.\nThere is no substantive difference between the elemental abundance\nratios for luminous red giants in the outer halo cluster NGC~2419 and\nthe ratios of inner halo clusters. The inner halo abundance pattern \nis also\nobserved in other distant clusters: NGC~7492\n\\citep[$R_G \\sim 25$~kpc,][]{cohen05b}, the low luminosity GC Pal~3\n\\citep[$R_G \\sim 90$~kpc,][]{pal3}, and in coadditions of\nhigh-resolution, low-SNR spectra of 19 luminous giants in the very\ndistant, low-luminosity GC Pal~4 \\citep[$R_G \\sim 109$~kpc,][]{pal4}.\nThe low [C\/Fe] ratios seen among the NGC~2419 and M30 \nluminous giants are common\namong such stars in inner halo GCs \\citep[see, e.g.][]{m13c}, \npresumably due to depletion via deep mixing.\n\nWhen comparing Galactic GC abundances to those of dSph\nsatellites of the Galaxy, it has become apparent\nvia extensive surveys in recent years \n\\citep{cohen_draco, cohen_umi} and at moderate resolution\nthe extensive work of \\cite{kirby4} that\nthe metal-poor end of all known systems seems to\nhave, at least to first order, an identical chemical \ninventory. That should not be a surprise, since differences\nproduced through different star formation rates\nor gas flow history (accretion and\/or outflows) between the dSph satellites and\nthe Galactic GCs will only show up at late times. Initially\nonly the metals produced in and ejected by SNII\nenrich the system's gas, and hence,\nexcept for small metallicity dependent effects\non nucleosynthetic yields, all\nvery metal-poor stellar systems should have identical abundance\nratios. Those GCs with peculiar abundance ratios,\nincluding no enhancement of the $\\alpha$-elements, have\nintermediate [Fe\/H] and most are\nknown to be associated with the ongoing accretion of the Sgr dSph;\nexamples include\nPal~12 \\citep{cohen_pal12} with [Fe\/H] $\\sim -0.7$~dex, \nTerzan~7 \\citep{terzan7} with\n[Fe\/H] $-0.6$~dex, as well as\nRup~106 \\citep{rup106} with [Fe\/H] $-1.45$~dex, whose age is several\nGyr less than the bulk of the halo, suggesting an accretion origin.\n\n\n\n\\section{Comparison with Previously Published Abundance Analyses}\n\nThe only previously published high-dispersion analysis of any star in\nNGC~2419 is for the star S1305, which was observed by\n\\cite{shetrone01}\\footnote{\\cite{shetrone01} call this star RH~10.\nThe coordinates of this star, privately communicated from\n M.~Shetrone, match those of S1305.}. Our measurement of [Fe\/H] is\n0.13~dex higher than the value they obtained, which we ascribe\nlargely to our adopted $T_{\\rm eff}$ being 125~K\nhigher than theirs. The two sets of [X\/Fe] for all elements in \ncommon through atomic number 30 (Zn) are in good agreement\nexcept for Na. Na is a difficult case as the NaD lines are somewhat\ncorrupted by interstellar features at the cluster's $v_r$,\nwhile the 5682, 5688~\\AA\\ doublet used by \\cite{shetrone01}\nis very weak. Among the heavier elements detected in\nboth studies, only Y (in the form of Y~II)\nhas a large discrepancy. There are 3 detected lines from\nour spectra, and a claim of 4 from theirs, but the line \nat 4900.1~\\AA\\ is quite crowded, and perhaps should be\nignored. For the two Y~II lines in common, our $W_{\\lambda}$\nare $\\sim$20\\% smaller. Considering how faint this star is \n($V = 17.61$~mag) we regard the overall agreement\nin abundances between the two analyses to be very good.\n\n\\cite{shetrone03} observed one star in NGC~7099. Once\nthe difference in adopted solar Fe abundance is taken into\naccount, their derived [Fe\/H] for this star differs from\nthe mean for our sample of five stars by only 0.05~dex.\nThe abundance ratios agree satisfactorily in most cases,\nwith 7 elements differing in [X\/Fe] by less than 0.10~dex.\nTheir UVES observations cover only 4800 to 6800~\\AA,\nand so are missing many key blue lines for the rare earths,\nwhere the differences in relative abundances between\nthe two studies tend to be the largest. The choice of\nadopted transition probabilities, particularly for the rare earths, also contributes;\nwe adopted them from the recent studies of Lawler\nand collaborators, \n\\citep[see, e.g.][for Eu as an example]{lawler01} \nwhen available, while \\cite{shetrone03} tended to use older values.\n\nThe Padua group (see, e.g. Carretta et al 2009abc and\nCarretta et al 2010) has completed a large VLT study\nof the Na\/O anti-correlation in GCs. Although their survey does not\ninclude NGC~2419, they have a observed a\nsample of 10 RGB stars in M30 with UVES. Thus far they have published\nabundances for only the light elements. \nTable~\\ref{table_7099_comp} presents a comparison of our\nresults from HIRES at Keck with their UVES\/VLT study\nand highlights the good agreement between\nthese two completely independent detailed abundance\nanalyses for RGB stars in M30.\n\n\n\n\n\n\n\n\n\n\\section{Consistency with Our Previous Deimos Range in Ca Abundance \n\\label{section_ca_2419} }\n\n\n\n\nOur previous medium-resolution study of RGB stars in NGC~2419\n\\citep{deimos} uncovered a range in [Ca\/H] of a factor of \ntwo or three among the sample of 43 stars found to be definite\nmembers of this GC. The range in $V$ of these stars is from 17.4 to 19.3~mag.\nFig.~\\ref{figure_cah_hist} shows a histogram of [Ca\/H] for these\nstars, on which is superposed the distribution of the\ngiants in NGC~2419 with high quality HIRES spectra analyzed \nhere.\nThe two additional probable members discussed in \n\\cite{deimos}, each of which has\nsome remaining concern about membership, are indicated in this figure\nwith cross hatching. \n\n\nThe first point to note is that the histogram of [Ca\/H] for an unbiased\nsample of members of NGC~2419, the moderate resolution Deimos survey\nof \\cite{deimos}, is sharply peaked at [Ca\/H] $-1.95$~dex.\nWhile there is a rapidly declining tail extending towards higher metallicities,\nthe low metallicity peak dominates, and the fraction of stars\nin the extended tail is small. The minimum [Ca\/H] is $-2.0$~dex.\nOnly 13 (30\\%) of the 43 definite members lie outside the range\n$-2.0$ to $-1.85$~dex.\n\nOur sample of RGB stars in NGC~2419 with HIRES\nspectra good enough for a detailed abundance analysis \ncontains only 7 stars.\nThis sample was, to a large extent, selected by brightness before\nthe full analysis of the moderate resolution spectra was completed.\nOne of these 7 stars (S1131) has\n[Ca\/H] $-1.72$~dex from its HIRES spectrum ($-1.76$~dex from\nits Deimos Ca triplet region analysis) and a second star (S973) is\nmarginally above the $-1.85$~dex [Ca\/H] cutoff at\n$-1.81$~dex from our detailed abundance analysis\n($-1.93$~dex from its moderate resolution Deimos spectrum).\nThe others all are more metal-poor than [Ca\/H] $= -1.85$~dex as deduced\nfrom both their HIRES and Deimos spectra.\n\nThe total range in Ca and Fe abundance within the NGC~2419 sample\nis small (see Table~\\ref{table_ngc2419_abund}.\nWe used the three different choices of $T_{\\rm eff}$\ndiscussed in \\S\\ref{section_obs}\nand given in Table~\\ref{table_temp} as an indication of the maximum\nrange in this parameter that might be appropriate for each of the NGC~2419 \nstars\\footnote{We adjusted log($g$) appropriately in each case to maintain the star\non the RGB.}.\nIn effect, we carried out the abundance analysis three times for each star\nusing different sets of adopted stellar parameters.\nWe find that for each of these three choices, S1131 {\\it{always}} has\nthe highest value of [Ca\/H] of the 7 stars in the HIRES sample for NGC~2419,\n and it is ${\\sim}0.2$~dex (2 to 3 $\\sigma$) higher than\nthe mean value for the other 6 stars in this cluster.\nThe Fe abundance for S1131, as derived from either Fe~I or Fe~II,\nis always the highest or second highest as well, but is\nwithin 1~$\\sigma$ of the mean of the other cluster stars. \n\nThe uncertainties\nin absolute abundances (see Table~\\ref{table_abund_sens_abs})\nare daunting for such a small range in Ca abundance in such\na distant cluster with spectra and near-IR photometry\nthat are less than ideal. Lowering \nthe $T_{\\rm eff}$ of S1131 in NGC~2419 by $\\sim$100~K\nrelative to the other RGB stars in this cluster in our HIRES\nsample would eliminate its high Ca and Fe abundance.\nBut this requires a change \nof this one star relative to all the others which\nis about half of the\nentire range in $T_{\\rm eff}$ spanned by the 7 cluster giants,\nfor which there is no justification.\nGiven the small size of our HIRES sample,\nall we can say is that the Ca\/H distribution of the HIRES\nsample in NGC~2419 is consistent with that of the Deimos sample, which\ndoes appear to show a small but measurable range in Ca\/H among\nthe cluster members \\citep{deimos}.\n\n\nBoth of two probable additional members of NGC~2419 discussed\nin \\cite{deimos} from the Deimos sample, shown by cross hatching in\nFig.~\\ref{figure_cah_hist},\nhave been observed with HIRES. However observations were\nterminated after the first 30~min exposure revealed\nconcerns about cluster membership. Hence\nthe resulting HIRES spectra for these two stars are not good enough for a detailed\nabundance analysis.\nOne of these (S951) has a spectrum which appears to be that of a RGB\nstar in this very metal-poor GC, but with $v_r$ off from the cluster mean\nby $\\sim$15~km s$^{-1}$. This star has a value from an analysis of\nthe Ca triplet lines in its Deimos spectrum of\n[Ca\/H] of $-1.91$~dex, consistent with the bulk of the cluster\npopulation. The concern with the second star, S1673, is twofold; it lies\nslightly bluer than the main RGB locus of NGC~2419 in $V-I$\n(see Fig.~\\ref{figure_2419_cmd}) and its HIRES spectrum\nappears metal-rich. The [Ca\/H] inferred from its Deimos \nspectrum ( $-1.68$~dex)\nputs this star, if it is a member, firmly in the metal-rich\ntail of the histogram shown in Fig.~\\ref{figure_cah_hist}. \nSee \\cite{deimos} for further discussion\nregarding membership in NGC~2419 of these two stars.\n\nThere are 13 stars in the Deimos sample that may\nbe members of NGC~2419 with $V < 17.7$, which is slightly fainter\nthan the faintest star in the present HIRES sample analyzed here.\nIn addition to all 7 RGB stars in our present HIRES sample,\nboth\nof the two probable cluster members discussed above\nare brighter than that cutoff.\nThis leaves 4 stars brighter than $V = 17.7$~mag without\nHIRES observations at present,\n3 of which have [Ca\/H] $\\sim -1.9$~dex. Only one is in the high\nmetallicity tail for NGC~2419. \n\nOur present\nHIRES sample, which was selected primarily on the\nbasis of brightness given the 84~kpc distance to NGC~2419, \nis not ideal for exploring the star-to-star variation.\nThe small variations of [Ca\/H] or [Fe\/H] from the 7 stars with\nhigh dispersion spectra are only marginally larger than the\nuncertainties, but are broadly consistent with our\nDeimos results \\citep{deimos} which found a range\nin [Ca\/H] within NGC~2419. In the future we hope to provide a sample that better\nprobes the metal-rich tail of NGC~2419 as\ndefined by the larger Deimos sample of \\cite{deimos}.\n\n\n\n\\section{One Star with a Peculiar Abundance Pattern \\label{section_s1131} }\n\nOne star, NGC~2419 S1131,deviates from the\narchetypal GC abundance pattern. \n S1131 is\na definite member of NGC~2419 based on its \nHIRES radial velocity (its heliocentric $v_r$ is $-17.2$~km s$^{-1}$)\nand spectrum; see also \\cite{deimos}. As discussed above\nin \\S\\ref{section_ca_2419}\nour detailed abundance analysis\nsuggests that this star is slightly more metal-rich \nthan the other NGC~2419 giants\nin Fe and Ca as well as \nseveral other elements with many detected lines.\nIt has an\nenhancement of $\\sim 0.2$~dex in [Ca\/H] \n(with slight variation depending on the choice of\n$T_{\\rm eff}$ discussed in \\S\\ref{section_caspread}), corresponding\nto Ca enhanced by a factor of 1.6, compared to the mean \nof the other 6 NGC~2419 giants.\n\nIt is interesting to note that S1131 is the only one of the six\nNGC~2419 sample giants included in the moderate resolution study of\n\\cite{deimos} found to be Ca-rich. The remaining 6 NGC~2419 giants in\nthe HIRES sample all lie at the low Ca\nabundance end of the distribution inferred by \\cite{deimos}. \nHowever, given the small abundance\nrange involved, observations of additional giants believed\nto be more metal-rich than the bulk of the cluster stars\nare required for a definitive confirmation for star-to-star\nvariation of Ca, Fe, and other heavy elements within\nNGC~2419.\n\n\nFurthermore, NGC~2419 S1131 has some anomalous abundance ratios,\nirrespective of whether its $T_{\\rm eff}$ has been overestimated. It\nhas [Mg\/Fe] = $-0.47$~dex from five \\ion{Mg}{1} lines, in contrast\nto the mean of the other six stars,\n$+0.44\\pm0.06$~dex. In deriving the Mg abundance for S1131, the upper limit\nto the strength of the 5711~\\AA\\ line was considered as a detection,\nand the two unblended Mg triplet lines were retained. These two lines\ngive Mg abundances slightly higher than the mean, but the highest\nMg abundance is from the 5711~\\AA\\ line, 0.18~dex above the mean\nof the 5 lines used. Note that the first ionization potential of\nMg is only 0.25~eV lower than that of Fe. The [Mg\/Fe]\nmeasurement would increase by only 0.08~dex if $T_{\\rm eff}$ were\nreduced by 100~K (see Table~\\ref{table_abund_sens_rel}). \nThe regions of the spectra of two \\ion{Mg}{1} lines\nin this star and in S1209, also known as Suntzeff~16\n\\citep{suntzeff88}, with $T_{\\rm eff}$ lower than that of S1131 by\nonly 85~K, are shown in Fig.~\\ref{figure_spectra}. Although most GCs\nshow a small spread in [Mg\/Fe] \\citep{gratton_araa}, \ngenerally interpreted as the result of proton-burning\nof Mg into Al in\nintermediate mass AGB stars,the expected enhancement of Al does not \nappear to be present in S1131. The abundance pattern of this\nspecific star cannot be reproduced by proton-burning.\nFurthermore $-0.47$~dex is an\nunprecedentedly low [Mg\/Fe] abundance for a globular cluster star; the\nonly stars that have such low [Mg\/Fe] ratios are the halo star\nCS22966-043 \\citep{ivans03}, a hot SX~Phe variable, \nand stars in dSph satellite galaxies, such\nas COS~171 in Ursa Minor \\citep{cohen_umi} and S58 in Sextans\n\\citep{shetrone01}. NGC~2419\nS1131 is unique in that no other elements appear depleted, including\nSi. A very low value of Mg and super-solar values of other alpha\nelements are inconsistent with any Type~II SN yields\n\\citep[e.g.,][]{woosleyweaver,nomoto06,heger10}.\n\n\nIn addition, NGC~2419 S1131 also has an unusually high [K\/Fe] of\n$+1.13$~dex, also\nillustrated in Fig.~\\ref{figure_spectra}. This figure includes\nthe spectrum of a rapidly rotating B star to demonstrate that\nany telluric absorption at the wavelength of the K~I line,\ntaking into account the $v_r$ of this GC, is negligible.\nFurthermore, two other NGC~2419 sample stars were observed\non the same run (see Table~\\ref{table_sample}) and do not\nshow this abnormality.\nThis abundance ratio would\ndecrease by less than 0.10~dex if $T_{\\rm eff}$ were reduced by 200~K. \nThis is an internal comparison within our NGC~2419\n sample, which has a total range in $T_{\\rm eff}$ of less than 200~K.\nStudies of non-LTE corrections for the 7699~\\AA\\ \nresonance line of K~I\n(the only line used here) by \\cite{andrievsky10} and by \\cite{takeda_k_nonlte}\n suggest that the appropriate non-LTE corrections\nchange by only 0.15~dex over such a small range in $T_{\\rm eff}$,\nranging from about $-0.35$ to $-0.2$~dex.\nThus differential non-LTE corrections for K~I within our HIRES \nsample of luminous RGB stars are much too small to explain the\ndeviant potassium abundance of S1131.\n\n\\cite{takeda_k_nonlte} very recently found a red giant in M13 and another\nin M4 with similarly high [K\/Fe]. They ascribe the unexpectedly\nstrong K~I resonance lines to an increase in activity or turbulent\nvelocity fields high in the stellar atmosphere where \nthe core of such a strong line is formed.\nHowever, while S1131 does show emission on both the red and blue wings\nof H$\\alpha$, the emission wings of H$\\alpha$ \nin the spectrum of NGC~2419 S223 are considerably stronger,\nyet this star does not show an anomalously strong K~I line.\nFurthermore if this explanation is correct, \nother strong resonance lines arising from neutral species\nof elements with (low) first ionization potentials comparable \nto that of K should be similarly affected. Unfortunately\nthe NaD lines, the most likely candidates to check, are\nbadly corrupted\nby interstellar absorption due to the radial velocity of NGC~2419.\n\n\nA careful inspection of Fig.~\\ref{figure_spectra} \nstrongly suggests, by comparison with the spectrum of NGC~2419\nS1209, which has a $V$ magnitude only 0.2~mag brighter than S1131, \nthat S1131 is indeed\nslightly more metal-rich in other species as well as K. \n\nThe Li~I blend at 6707~\\AA\\ was too weak to be detected\nin any of the NGC~2419 stars. The upper limit for the equivalent\nwidth of this feature of 7~m\\AA\\ in\nS1131 corresponds to log[$\\epsilon$(Li)]\nof $-0.01$~dex, indicates extensive depletion from the Li production \nin the Big Bang, typical of red giants, which have extensive\nsurface convection zones.\n\n\nThe stellar parameters, spectra, and\nanalyses have been carefully checked and these specific differences in\nabundance ratios discussed here (i.e. for Mg and for K) \nwithin the NGC~2419 sample are real. \n\n\n\n\\section{Summary}\n\nOur key result from the analysis of 7 luminous giant members of the\ndistant outer halo globular cluster NGC~2419 is that one of these\nstars is extremely peculiar, having a very low [Mg\/Fe] ratio, but\nbeing normal in all other element ratios except for a high [K\/Fe].\nSimilarly low [Mg\/Fe] ratios are seen in stars in dSph satellites of the\nGalaxy, specifically in Ursa Minor \\citep{cohen_umi} and in Sextans\n\\citep{shetrone01}. But in both cases these stars have low values of\nother $\\alpha$ elements, which is not the case for NGC~2419 S1131.\nJ.~Cohen has examined and analyzed spectra of $\\gtrsim$100 stars in\nGalactic GCs, and has never seen one similar to S1131 in NGC~2419.\n\nFurthermore, the HIRES spectra support the small range in\nmetallicity within the giants in NGC~2419 found by \\cite{deimos},\nwhich might suggest that this distant outer halo GC is\nthe remnant of an accreted dwarf galaxy. A definitive result\nwill require further observations of the faint upper RGB\nstars in this GC, concentrating on those whose moderate\nresolution Deimos spectra from \\cite{deimos}\nsuggest they are more metal-rich than\nthe bulk of the stars in this cluster, which will be carried out\nin a future campaign.\n \n\nIgnoring the limited peculiarities of S1131, we find that there is\nno substantive difference between the mean behavior of the abundance\nratios for six other red giants in NGC~2419 and those ratios characteristic of\nthe inner halo despite the structural peculiarities of NGC~2419. \nThe influence of star formation rates\nand other aspects of the detailed chemical evolution of a stellar\nsystem only affect such ratios once star formation\nhas been underway long enough for sources other than SNII to\nbecome important contributors, which does not seem to be the case\nin very metal-poor systems such as NGC~2419, nor in the\nmost metal-poor stars in the dSph satellites of the Milky Way\n(Cohen \\& Huang 2009, 2010; Kirby et al 2011).\n\nThus, like the previous studies based on detailed\nabundance analyses of individual or co-added stellar spectra of stars\nin distant outer halo clusters, the chemical inventory of the globular\ncluster system appears to be independent of location. With the\nexception of those clusters known to be associated with the\ncurrently disrupting Sgr dSph galaxy, every globular cluster\nexhibits the same trends as a function of overall cluster\nmetallicity---parameterized by [Fe\/H]---irrespective of kinematics or\nlocation within the halo. \n\nThe existence of a metallicity gradient in the Galactic halo is\ncontroversial. \\citet{carollo07} found evidence for a break in the\nmetallicity distribution of halo field stars in the sense that the\nmore distant stars are more metal-poor; this conclusion is\nsupported by \\cite{rix10}. Furthermore, \\cite{carollo07} showed that\nthe metallicity within the outer halo population displays a gradient\ntoward even lower metallicities for the most distant stars or stars\nshowing the highest magnitude of rotation retrograde to the Milky Way\ndisk. However, both \\citet{sesar11} and \\citet{schoenrich10} improved\nupon \\citeauthor{carollo07}'s study. These new works,\nwhich are disputed by \\cite{beers11}, found no\nevidence for a dichotomous halo, particularly in regard to the\nkinematics and density profile. Our measurements for NGC~2419 \nadditionally provide\nevidence against a dichotomy in detailed abundances. The outer halo\nglobular clusters show the same bulk abundance\nproperties---and, by extension, (short) star formation timescales---as the\ninner halo globular clusters.\n\n\n\n\\acknowledgements\n\n\nWe are grateful to the many people who have worked to make the Keck\nTelescope and its instruments a reality and to operate and maintain\nthe Keck Observatory. The authors wish to extend special thanks to\nthose of Hawaiian ancestry on whose sacred mountain we are privileged\nto be guests. Without their generous hospitality, none of the\nobservations presented herein would have been possible. We thank \nAlex Heger and Ken Nomoto for helpful conversations. J.G.C.\nand W.H. thank NSF grants AST-0507219 and\nAST-0908139 for partial support. Work by E.N.K. was supported by NASA\nthrough Hubble Fellowship grand HST-HF-01233.01 awarded to ENK by the\nSpace Telescope Science institute, which is operated by the\nAssociation of Universities for Research in Astronomy, Inc., for NASA,\nunder contract NAS 5-26555. \n\n\n\n\n\\clearpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}