%% Created by Maple 6 (IBM INTEL NT)
%% Source Worksheet: eqnuplot.mws
%% Generated: Mon Jun 21 21:45:29 2004
%\documentclass{article}
%\usepackage{maple2e}
 \def\emptyline{\vspace{12pt}}
\DefineParaStyle{Heading 1}
\DefineParaStyle{Heading 2}
\DefineParaStyle{Heading 3}
\DefineParaStyle{Maple Output}
\DefineCharStyle{2D Comment}
\DefineCharStyle{2D Math}
\DefineCharStyle{2D Output}
%\begin{document}
%\pagestyle{empty}
\begin{maplegroup}
\begin{Heading 1}
Kerr black hole
\end{Heading 1}

\begin{Heading 2}
Equatorial plane
\end{Heading 2}

\begin{Heading 3}
Null geodesics
\end{Heading 3}

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

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

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{f := 2*M*(a-De)^2*u^3 +(a^2-De^2)*u^2+1:}{%
}
\end{mapleinput}

\end{maplegroup}

\begin{maplegroup}
\begin{flushleft}
\hspace{18pt}\mapleinline{inert}{2d}{M;}{%
$M$%
} should be \TEXTsymbol{>} 
\mapleinline{inert}{2d}{a;}{%
$a$%
}
\end{flushleft}

\begin{mapleinput}
\mapleinline{active}{1d}{a := .96: M := 1: De := 4:}{%
}
\end{mapleinput}

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

\mapleresult\color{blue}
\begin{maplelatex}
\mapleinline{inert}{2d}{u := -.2277020572, .3356987941, .7077927367;}{%
\[
u := -.2277020572, \,.3356987941, \,.7077927367
\]
%
}
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{g := 2/sqrt(u[3] -u[1]):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{flushleft}
\hspace{18pt}\mapleinline{inert}{2d}{k;}{%
$k$%
} should be \TEXTsymbol{<} or = 
\mapleinline{inert}{2d}{1;}{%
$1$%
}
\end{flushleft}

\begin{mapleinput}
\mapleinline{active}{1d}{k := sqrt((u[2]-u[1]) /(u[3]-u[1])); is(k <= 1);}{%
}
\end{mapleinput}

\mapleresult\color{blue}
\begin{maplelatex}
\mapleinline{inert}{2d}{k := .7760470671;}{%
\[
k := .7760470671
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{true;}{%
\[
\mathit{true}
\]
%
}
\end{maplelatex}

\end{maplegroup}
\bigskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{p := 2*M*(a -De)*(M^2 -a^2) +(a^2*De -2*M^2*(a 
-De))*sqrt(M^2 -a^2):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{h := solve(a^2*x^2 -2*M*x +1, x);}{%
}
\end{mapleinput}

\mapleresult\color{blue}
\begin{maplelatex}
\mapleinline{inert}{2d}{h := 1.388888889, .7812500000;}{%
\[
h := 1.388888889, \,.7812500000
\]
%
}
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{u_plus := h[1]: u_minus := h[2]:}{%
}
\end{mapleinput}

\end{maplegroup}
\smallskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{ph1 := g*p/2/(M^2 -a^2)* 
((EllipticPi(sqrt((0.325-u[1])/(u[2]-u[1])),
(u[2]-u[1])/(u_plus-u[1]), k)
+EllipticPi((u[2]-u[1])/(u_plus-u[1]),
k))/(u_plus-u[1])
-EllipticPi(sqrt((0.325-u[1])/(u[2]-u[1])),
(u[2]-u[1])/(u_minus-u[1]), k)
+EllipticPi((u[2]-u[1])/(u_minus-u[1]),
k)/(u_minus-u[1]));}{%
}
\end{mapleinput}

\mapleresult\color{blue}
\begin{maplelatex}
\mapleinline{inert}{2d}{ph1 := 98.93177565;}{%
\[
\mathit{ph1} := 98.93177565
\]
%
}
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{y := 1.19:}{%
}
\end{mapleinput}

\end{maplegroup}
\smallskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{ph2 := g*p/2/(M^2-a^2) 
*((EllipticPi((sqrt(u[3]-u[1])/(y-u[1])),
(u_minus-u[1])/(u[3]-u[1]), k)
+EllipticF((sqrt(u[3]-u[1])/(y-u[1])), k)) /(u_minus-u[1]) 
-(EllipticPi((sqrt(u[3]-u[1])/(y-u[1])),
(u_plus-u[1])/(u[3]-u[1]), k)
+EllipticF((sqrt(u[3]-u[1])/(y-u[1])), k))
/(u_plus-u[1]));}{%
}
\end{mapleinput}

\mapleresult\color{blue}
\begin{maplelatex}
\mapleinline{inert}{2d}{ph2 := 13.49258854;}{%
\[
\mathit{ph2} := 13.49258854
\]
%
}
\end{maplelatex}

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

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