% $Id: notes.tex,v 1.1 2019/05/17 14:59:20 brandenb Exp $
%\documentclass{article}
\documentclass[twocolumn]{article}
\setlength{\textwidth}{160mm}
\setlength{\oddsidemargin}{-0mm}
\setlength{\textheight}{220mm}
\setlength{\topmargin}{-8mm}
%\usepackage{graphicx,natbib,bm,url,color,slashed,amsmath,mathtools}
\usepackage{graphicx,natbib,bm,url,color,mathtools}
\newcommand{\fsl}[1]{\ensuremath{\mathrlap{\!\not{\phantom{#1}}}#1}}
\graphicspath{{./fig/}{./png/}}
\thispagestyle{empty}
\input macros
\def\red{\textcolor{red}}
\def\blue{\textcolor{blue}}
\title{}
\author{}
\date{\today,~ $ $Revision: 1.1 $ $}
\begin{document}
%\maketitle

\section{Newtonian reference case}

1000 mass bins, uniformly stacked, from 0 to 1,
starting at m=0.001, so then we have m=0.002, 0.003,
until m=1.

Initial condition:
\begin{equation}
r=\left({3m\over4\pi\rho_0}\right)^{1/3},\quad
\beta=0.
\end{equation}

\begin{equation}
{\dd r\over\dd t}=\beta
\end{equation}
\begin{equation}
{\dd\beta\over\dd t}=-{m\over r^2}
\end{equation}

\begin{figure}[h!]\begin{center}
\includegraphics[width=\columnwidth]{pnewtonian}
\end{center}\caption[]{
%AB: 1.5 for pde.pro
Newtonian reference case for $\rho_0=1$.
Contours of constant radius, $r=0$, 0.02, etc.
The blue (red) contour corresponds to $r=0.1$ ($r=0.2$).
Fixed time step $\delta t=5\times10^{-5}$.
}\label{pnewtonian}\end{figure}

\section{TOV case}

\begin{equation}
{\dd\beta\over\dd t}=-{m\over|r(r-r_{\rm G})|}.
\end{equation}

\begin{figure}[h!]\begin{center}
\includegraphics[width=\columnwidth]{pTOV}
\end{center}\caption[]{
%AB: 1.6 for pde.pro
Same as \Fig{pnewtonian}, but for TOV potential with $r_{\rm G}=0.2$.
}\label{pTOV}\end{figure}

\section{TOV case with pseudo-relativistic equation of motion}

\begin{equation}
{\dd r\over\dd t}={\beta\over\sqrt{1+\beta^2}}
\end{equation}

\begin{figure}[h!]\begin{center}
\includegraphics[width=\columnwidth]{pTOVpseudo}
\end{center}\caption[]{
%AB: 1.7 for pde.pro
Same as \Fig{pTOV}, but for the pseudo-relativistic equation of motion.
}\label{pTOVpseudo}\end{figure}

%\begin{table}[htb]\caption{
%}\vspace{12pt}\centerline{\begin{tabular}{lccccccc}
%\hline
%\label{Ttimescale}\end{tabular}}\end{table}

\begin{verbatim}
\end{verbatim}

%r e f
%\begin{thebibliography}{}

%\end{thebibliography}

\vfill\bigskip\noindent\tiny\begin{verbatim}
$Header: /var/cvs/brandenb/tex/upasana/collapse/notes.tex,v 1.1 2019/05/17 14:59:20 brandenb Exp $
\end{verbatim}

\end{document}
