[Commits] [svn:einsteintoolkit] EOS_Omni/trunk/doc/ (Rev. 43)

[cott at tapir.caltech.edu]

User: cott
Date: 2011/04/19 07:12 PM

Modified:
 /trunk/doc/
  documentation.tex

Log:
 * describe hybrid EOS

File Changes:

Directory: /trunk/doc/
======================

File [modified]: documentation.tex
Delta lines: +57 -3
===================================================================
--- trunk/doc/documentation.tex	2011-04-19 15:16:15 UTC (rev 42)
+++ trunk/doc/documentation.tex	2011-04-20 00:12:05 UTC (rev 43)
@@ -76,6 +76,18 @@
 %  homepage at www.cactuscode.org)
 \usepackage{../../../../doc/latex/cactus}
 
+\newenvironment{equationarray}
+{\arraycolsep 0.14 em
+\begin{eqnarray}}
+{\end{eqnarray}}
+
+\newenvironment{equationarray*}
+{\arraycolsep 0.14 em
+\begin{eqnarray*}}
+{\end{eqnarray*}}
+
+
+
 \begin{document}
 
 % The author of the documentation
@@ -290,8 +302,46 @@
 
 \subsection{Hybrid}
 
-\textbf{TODO: Not yet documented.}
+The hybrid EOS was introduced by \cite{janka:93} for use in simplified
+simulations of stellar collapse to mimic (1) the stiffening of the
+nuclear EOS at nuclear density and (2) to include thermal pressure in
+the postbounce phase.  It consists of two polytropes characterized by
+($K_1$, $\gamma_1$) and ($K_2$, $\gamma_2$) and a thermal $\gamma-$law
+component described by $\gamma_\mathrm{th}$.  Polytrope 1 is soft and
+describes a gas of relativistic degenerate electrons with $\gamma_1
+\approx 4/3$.  It is used below nuclear density ($\rho_\mathrm{nuc}
+\approx 2\times10^{14}\,\mathrm{g\,cm}^{-3}$) and smoothly matched to
+polytrope 2 which applies above $\rho_\mathrm{nuc}$, is stiff, and
+models the repulsive core of the strong force above nuclear density
+($\gamma_2 \gtrsim 2.5$).  $K_2$ is completely determined by
+$P_1(\rho_\mathrm{nuc}) = P_2(\rho_\mathrm{nuc})$ and $K_1, \gamma_1,$
+and $\gamma_2$. The full functional form of the EOS
+$P=P(\rho,\epsilon)$ with the thermal component (which takes into
+account shock heating) is given by
+\begin{equationarray}
+  P & = & \frac{\gamma - \gamma_{\rm th}}{\gamma - 1}
+  K \rho_{\rm nuc}^{\gamma_1 - \gamma}
+  \rho^{\gamma} - \frac{(\gamma_{\rm th} - 1) (\gamma - \gamma_1)}
+  {(\gamma_1 - 1) (\gamma_2 - 1)}
+  K \rho_{\rm nuc}^{\gamma_1 - 1} \rho 
+  + (\gamma_{\rm th} - 1) \rho \epsilon\,.
+  \label{eq:hybrid_eos}
+\end{equationarray}%
 
+The \texttt{EOS\_Omni} parameters for the hybrid EOS are the following:
+
+\begin{tabular}{ll}
+\texttt{hybrid\_gamma1}     & $\gamma_1$,  $\gamma_1 = 1.325$ is an appropriate choice.\\
+\texttt{hybrid\_gamma2}     & $\gamma_2$,  $\gamma_2 = 2.5$ is an appropriate choice.\\
+\texttt{hybrid\_gamma\_th}   & $\gamma_\mathrm{th}$, perhaps $1.5$.\\
+\texttt{hybrid\_k1}         & $K_1$, $0.4640517$ in solar units for relativistic degenerate e$^{-}$.\\
+\texttt{hybrid\_rho\_nuc}    & nuclear density, standard is $3.238607\times 10^{-4}$ in solar units.
+\end{tabular}
+
+
+
+
+
 \subsection{Finite-Temperature Nuclear}
 
 \textbf{TODO: Not yet documented; not even mentioned yet in some other
@@ -363,10 +413,14 @@
 
 
 
-%\begin{thebibliography}{9}
+\begin{thebibliography}{9}
+\bibitem{janka:93} Janka, H.-T., Zwerger, T., \& Moenchmeyer, R.\ 1993, Astron. Astrophys., 268, 360 
 
-%\end{thebibliography}
 
+
+
+\end{thebibliography}
+
 % Do not delete next line
 % END CACTUS THORNGUIDE