## LTE, Karp

\documentclass[12pt]{article} \usepackage{amsmath} \hbadness 10000 \topmargin -27pt \oddsidemargin 0in \evensidemargin 0in \textheight 23cm \textwidth 16.3cm \def\ave#1{\langle #1 \rangle} \def\pd{\partial} %\def\sp(#1){\noalign{\vskip #1pt}} \title{A Globally Convergent Method for computing the electron density of a partially ionized plasma} \author{Alan H. Karp\\ IBM Palo Alto\\ CA 94304, USA} \date{Received 30 April 1979} \begin{document} \begin{titlepage} \maketitle \thispagestyle{empty} \vspace{0.2cm} \begin{abstract} \end{abstract} A stable Newton Raphson method for Saha and NLTE.\\ {\em JQSRT vol.23 (1980), No.3-D, p. 285-289} \end{titlepage} \section{Introduction} \label{intr} In LTE Saha: \label{saha}  \frac{N_{i,j+1}}{N_{ij}}=\frac{D_{ij}(T)}{N_e}  where N_{ij} is the number density of atoms of atomic number i ionized j times and \begin{equation*} \label{DT}  D_{ij}(T) = \left(\frac{2\pi m_e kT}{h^2} \right)^{3/2} 2 \frac{u_{i,j+1}(T)}{u_{ij}(T)}\mathrm{e}^{-\chi_{ij}/kT} \; . \end{equation*} Since the partition functions u_{ij} depend on the electron density N_e through the depression of the continuum, D_{ij} is also a function of N_e. See Sec.2 why we can ignore this. The fraction of atoms of atomic number i ionized j times is just \label{Nij}  \frac{N_{ij}}{N_{i}}=\frac{N_{ij}}{N_{i0}+N_{i1}+\dots+N_{ii}}= \frac{N_{ij}/N_{i0}}{1+N_{i1}/N_{i0}+\ldots+N_{ii}/N_{i0}} \; ,  where  \frac{N_{ij}}{N_{i0}}=\prod_{k=0}^{j-1}\frac{N_{i,k+1}}{N_{ik}}= N_e^{-j}\prod_{k=0}^{j-1}D_{ik}(T)=A_{ij}(T)N_e^{-j} \; .  If we define  \alpha_i=1+\sum_{j=1}^i A_{ij}(T) N_e^{-j}  and  x_i=\sum_{j=1}^i jA_{ij}(T) N_e^{-j} \; ,  it is clear that each atom of element i gives up an average of x_i/\alpha_i free electrons to the plasma. The number density of atoms is N_A=\rho/\mu_0 m_H, where \mu_0 is mean molecular weight (in hydrogen scale) of the completely recombined gas. If \nu_i is the fractional number abundance of element i (\sum_i \nu_i = 1), then \label{Ne}  N_e = N_A \sum_i \nu_i x_i/\alpha_i \; .  Since Eq.(3) is nonlinear in N_e, an iterative procedure is needed. One approach is to solve simultaneously for all the N_{ij} and N_e using Eqs. (2), (3). This method involves solving a system of equations for each iteration and may become unstable. Since simpler and more stable methods are available, the continued use of this approach is surprising. Another approach, which avoids these problems [1,2]: (1) Guess an N_e ; (2) compute x_i and \alpha_i for all i; (3) use Eq.(3) to compute N_e; repeat. *** It seems Ron Eastman uses this, so we have to change this in {\tt vladsf/ } (Elka?). In nether method is the rate of convergence known. Even though the change from one iteration to the next is small the result may be far from the solution. The radius of convergence is unknown. There is no {\em a priori} way of knowing wether the first guess is close enough to guarantee convergence. The procedure derived in Sec.2 avoids these problems. \section{Iteration for LTE} \label{LTE} We wish to find the root of the nonlinear equation \label{gNe}  g(N_e) = N_A \sum_i \nu_i x_i/\alpha_i -N_e  using the Newton-Raphson method, namely,  N_e^{(n+1)}=N_e^{(n)}-g[N_e^{(n)}]/g'[N_e^{(n)}] ,  where n is the iteration index. Diff. Eq.(4) gives \label{gprim}  g'[N_e^{(n)}]=-\frac{N_A}{N_e} \sum_i \frac{\nu_i}{\alpha_i^2}( \alpha_i y_i -x_i^2)-1 \equiv -\frac{N_A}{N_e}V -1,  where  y_i=\sum_{j=0}^i j^2A_{ij}(T) N_e^{-j} \; .  (Actually, summation starts from j=1, since j^2=0 for j=0, {\it P.Baklanov}.) \noindent Letting  U=\sum_i \nu_i x_i/\alpha_i  gives  N_e^{(n+1)}=N_e^{(n)} + \frac{N_AU-N_e^{(n)}}{N_AV/N_e^{(n)}+1},  which can be rearranged to give \label{nenp1}  N_e^{(n+1)}=N_e^{(n)}\left\{\frac{U+V}{N_e^{(n)}/N_A+V}\right\}.  as n\to\infty, N_AU \to N_e^{(n)} and the quantity in brackets goes to unity, as shown in Sec.3. Frequently one knows the total gas pressure P_g instead of the mass density used to derive Eq.(3). Since the electrons contribute to P_g, the situation is more complicated. Eq.(5) (*** probably must be 4) becomes  g(N_e) = (N-N_e) \sum_i \nu_i x_i/\alpha_i -N_e ,  where N=P_g/kT. Using the same procedure as before gives  N_e^{(n+1)}=N_e^{(n)}\left\{\frac{NU+[N-N_e^{(n)}]V}{N_e^{(n)}(1+U)+[N-N_e^{(n)}]V}\right\}.  Again the quantity in brackets goes to unity as n increases. It is now possible to justify ignoring the dependence of the partition function on N_e. The depression of the continuum by nearby charged particles only affects the highest bound levels [3], i.e.  u_{ij}=\sum_{k=1}^\infty W_{ijk}(T,N_e)g_{ijk}\mathrm{e}^{-E_{ijk}/kT},  where k refers to the atomic levels, g_{ijk} is the statistical weight, and E_{ijk} is the excitation potential. W_{ijk} is the probability that level k exists and deviates from unity only when E_{ijk} is near the ionization potential. Therefore, u_{ij} has an appreciable dependence on N_e only when kT\simeq \chi_{ij}. However, in this situation, the relative abundance of ion j is low and its contribution to the sums in Eq.(3) is small. While in principle the depression of the continuum [3], \Delta\chi=1.8\times10^{-8}N_e^{1/3} eV can be significant, the dependence is weak and is never important in the temperature-density range of interest. *** Then a table is given from [4] showing that convergence goes as  |\epsilon^{(n+1)}| \leq C [\epsilon^{(n)}]^2  with  \epsilon^{(n)} a relative error in N_e on n-th iteration and C of order unity [5], i.e. the convergence is quadratic. \section{Convergence proofs} \label{proofs} Clearly, D_{ij}(T) can be treated as independent of N_e as expected. This assumption has been made implicitly in the proofs given below. Some insight to the convergence properties can be gained by examining x_i/\alpha_i as a function of N_e. When N_e is low (perhaps because the total density is low), x_i \to iA_{ii}N_e^{-1} and \alpha_i \to A_{ii}N_e^{-1} and, therefore, the ratio approaches i. In other words, every atom gives up all its electrons. On the other hand, when N_e gets large, x_i \to A_{i1}/N_e and \alpha_i \to 1+A_{i1}/N_e and so the ratio goes to zero. We expect, therefore, that the first derivative of g(N_e) with respect to N_e will be negative. If, in addition, g''(N_e)>0 for all N_e, i.e. g(N_e) is convex, then the convergence is guaranteed for any first guess [5]. Convergence is proven in the following two theorems. {\em Theorem 1:} g'(N_e)<0 for all N_e>0. {\em Proof:} Since all quantities in Eq.(5) are positive for N_e>0, it suffices to show that D_i=N_{i0}^2( \alpha_i y_i -x_i^2)>0 for all N_e>0. Using  \alpha_i=1+\sum_{j=1}^i N_{ij}/N_{i0},   x_i=\sum_{j=1}^i j N_{ij}/N_{i0} \; ,  and  y_i=\sum_{j=1}^i j^2 N_{ij}/N_{i0} \; ,  gives  D_i=\left[\sum_{j=0}^i N_{ij} \right] \left[\sum_{j=0}^i j^2N_{ij}\right] -\left[\sum_{j=0}^i j N_{ij}\right]^2.  But  \left[\sum_{j=0}^i j N_{ij}\right]^2 = \left[\sum_{j=0}^i j N_{ij}^{1/2} N_{ij}^{1/2} \right]^2 \leq \left[\sum_{j=0}^i N_{ij} \right] \left[\sum_{j=0}^i j^2N_{ij}\right]  by the Schwartz (Cauchy) inequality [6]. An alternate proof proceeds by induction. We note that  D_1=N_{11}(N_{10}+N_{11})-N_{11}^2=N_{11}N_{10}>0  and assume D_{k-1}>0. Then  D_k=\left[N_{kk}+\sum_{j=0}^{k-1} N_{kj} \right] \left[k^2N_{kk}+\sum_{j=0}^{k-1} j^2N_{kj}\right] -\left[kN_{kk}+\sum_{j=0}^{k-1} j N_{kj}\right]^2.  and  D_k=D_{k-1}+N_{kk}\sum_{j=0}^{k-1}(k-j)^2N_{j}>0 \mathrm{***check if} N_{kj}  since both term on RHS are positive. {\em Theorem 2:} g''(N_e)>0 for all N_e>0. {\em Proof:} Differentiating Eq.(5) with respect to N_e gives  g''[N_e^{(n)}]=\frac{N_A}{N_e^2} \sum_i \frac{\nu_i}{\alpha_i^3} [ \alpha_i(\alpha_i y_i-x_i^2) + \alpha_i^2 z_i + \alpha_i x_i y_i+2x_i^3],  where  z_i=\sum_{j=1}^i j^3 N_{ij}/N_{i0} \; .  From theorem 1, \alpha_i y_i > x_i^2 so that  g''(N_e)>0. These theorems prove that the iteration always converges as long as N_e^{(n)}>0 and that the convergence is quadratic. Since  g'(N_e)<0 and  g''(N_e)>0 for  N_e>0, the iteration approaches the solution from below. Therefore, if  N_e^{(1)}<0, it must be replaced by a small positive number. This, and the obvious need to avoid overflows and underflows, are the only precautions needed to ensure convergence. \section{Iteration procedure for NLTE} \label{NLTE} The extension to non-LTE is straightforward. Let a starred quantity represent the LTE value and unstarred the NLTE. Then,  \frac{N_{i,j+1}}{N_{ij}}=b_{ij}\left(\frac{N_{i,j+1}}{N_{ij}}\right)^* ,  where b_{ij} is the departure coefficient. Defining \alpha_i, x_i and y_i by using N_{ij} in place of N_{ij}^* gives, for example,  \alpha_i=1+\sum_{j=1}^i\left(\frac{N_{ij}}{N_{i0}}\right)^* \prod_{k=0}^{j-1} b_{ik},  If the departure coefficient is not a function of N_e the iteration converges as before. Since, in general, b_{ij} depends on N_e the derivative g'(N_e) has not been computed exactly. In spite of this problem the iteration may still converge but more slowly than before. Some plausibility arguments can be made. We consider an atom with one bound level and a continuum for which [7]  b_{i\kappa}=\frac{R_{i\kappa}+N_e C_{i\kappa}}{R_{\kappa i}+N_e C_{\kappa i}}  There are three cases of interest. (1) Collisions dominate, LTE holds, and b_{i\kappa}=1. (2) The radiation is Planckian at the local temperature which implies b_{i\kappa}=1. (3) Radiation processes dominate and the radiation is non-Planckian. Only in this case does b_{ij} depend on N_e, but only through the radiation field. If an estimate of the derivative of b_{ij} with respect to N_e is available, then the iteration of Eq.(6) can be used with the definitions  U=N_A \sum_i \nu_i x_i/\alpha_i   V=N_A \sum_i \nu_i [\alpha_i y_i - x_i^2 + q_i\alpha_i-p_ix_i]/\alpha_i^2,  where  p_i=N_e\sum_{j=1}^i A_{ij}N_e^{-j} \frac{\mathrm{d}}{\mathrm{d}N_e}\prod_{k=0}^{j-1}b_{ik},   q_i=N_e\sum_{j=1}^i jA_{ij}N_e^{-j} \frac{\mathrm{d}}{\mathrm{d}N_e}\prod_{k=0}^{j-1}b_{ik}.  The more accurately the derivative of b_{ij} with respect to N_e is known, the more nearly quadratic the convergence. \bigskip {\em Acknowledgement} ... David Fischel provided the proof for theorem 1. \begin{thebibliography}{} \bibitem{1} C.A.Rouse, ApJ 135, 599 (1962). \bibitem{2} R.L.Kurucz, ATLAS, a computer program for calculating model stellar atmospheres, SAO Special Rep. No. 309 (1970). \bibitem{3} D.Fischel, W.M. Sparks, ApJ 164, 359 (1971). \bibitem{4} J.W.Fowler, Line-Blank model stellar atm. Sirius, unpub. PhD thesis, Univ. maryland, 1972; also NASA-GSFC doc. No. X-670-72-303 \bibitem{5} J.M.Ortega, Numerical Analysis, Academic Press, NY (1972). \bibitem{6} R.R.Goldberg, Methods of real analysis, p.93, Blaisdell, NY (1964). \bibitem{7} D.Mihalas, Stellar atmospheres. Freeman, SF (1970). \end{thebibliography} \end{document}