%% Created by Maple 6 (IBM INTEL NT)
%% Source Worksheet: nulcircst.mws
%% Generated: Tue Feb 17 16:30:47 2004
%\documentclass{article}
%\usepackage{maple2e}
 \def\emptyline{\vspace{12pt}}
\DefineParaStyle{Heading 1}
\DefineParaStyle{Heading 2}
\DefineParaStyle{Heading 3}
\DefineParaStyle{Heading 4}
\DefineParaStyle{Maple Output}
\DefineParaStyle{Warning}
\DefineCharStyle{2D Comment}
\DefineCharStyle{2D Math}
\DefineCharStyle{2D Output}
%\begin{document}
%\pagestyle{empty}
\begin{maplegroup}
\begin{Heading 1}
Reissner-Nordstrom solution
\end{Heading 1}

\begin{Heading 2}
Null geodesics
\end{Heading 2}

\begin{Heading 3}
Double root - stable circular orbit
\end{Heading 3}

\begin{Heading 4}
\mapleinline{inert}{2d}{M^2 < Q_^2;}{%
$M^{2} < \mathit{Q\_}^{2}$%
} \TEXTsymbol{<}
\mapleinline{inert}{2d}{9*M^2/8;}{%
$\frac {9\,M^{2}}{8}$%
}
\end{Heading 4}

\begin{mapleinput}
\mapleinline{active}{1d}{restart:}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{ph := arcsin((2*c*xi+b)/sqrt(4*Q_^2*c+b^2)) /sqrt(-c):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{b := 2 *(M-2*Q_^2*u_c):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{c := u_c *(3*M-4*Q_^2*u_c):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{r := 1/u:}{%
}
\end{mapleinput}

\end{maplegroup}\bigskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{f := (u-u_c)^2
*(-Q_^2*u^2+2*(M-Q_^2*u_c)*u+u_c*(M-Q_^2*u_c)):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{u_c := 1/r_cr:}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{r_cr := 1.5*M*(1-sqrt(1-8*Q_^2/9/M^2)):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{M := 11/14: Q_ := 0.8:}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{u := solve(f, u);}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{u := 1.178725989, 1.178725989, .2941015908, -.1961964261;}{%
\[\color{blue}
u := 1.178725989, \,1.178725989, \,.2941015908, \,-.1961964261
\]
%
}
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{u_2 := u[3]:}{%
}
\end{mapleinput}

\begin{maplegroup}


\begin{flushleft}
\vspace{-5mm}\hspace{18pt} The turning point is \begin{mapleinput}
\mapleinline{active}{1d}{1/u[3];}{%
}
\end{mapleinput}
\end{flushleft}

\mapleresult\vspace{-4mm}
\begin{maplelatex}
\mapleinline{inert}{2d}{3.400185620;}{%
\[\color{blue}
3.400185620
\]
%
}
\end{maplelatex}

\end{maplegroup}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{u := 1/xi +u_c:}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{-1/u_c;}{%
}
\end{mapleinput}

\begin{flushleft}
\vspace{-5mm}\hspace{18pt}\mapleinline{inert}{2d}{xi;}{%
$\xi $%
} must not exceed this
\end{flushleft}

\mapleresult\vspace{-4mm}
\begin{maplelatex}
\mapleinline{inert}{2d}{-.8483735909;}{%
\[\color{blue}
-.8483735909
\]
%
}
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{with(plots):}{%
}
\end{mapleinput}
{\color{blue}
\mapleresult\vspace{-8mm}
\begin{maplettyout}
Warning, the name changecoords has been redefined
\end{maplettyout}}
\smallskip
\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{Fi := polarplot([r, ph, xi = 1/(u_2-u_c)..-.85], -5..30,
-10..10, color = blue):}{%
}
\end{mapleinput}
\smallskip
\end{maplegroup}
\begin{maplegroup}\vspace{-4mm}
\begin{mapleinput}
\mapleinline{active}{1d}{rC := polarplot(r_cr, color = blue):}{%
}
\end{mapleinput}
\smallskip
\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{plotsetup(ps, plotoutput = `nullcst_2.eps`, plotoptions =
`color, portrait, noborder`):}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{display(\{Fi, rC\}, axes = frame, scaling =
constrained, thickness = 3);}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\end{mapleinput}

\end{maplegroup}
%\end{document}
%% End of Maple 6 Output
