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