%% Created by Maple 6 (IBM INTEL NT)
%% Source Worksheet: tlmgcirc.mws
%% Generated: Thu Mar 30 12:22:35 2006
%\documentclass{article}
%\usepackage{maple2e}
 \def\emptyline{\vspace{12pt}}
\DefineParaStyle{Heading 1}
\DefineParaStyle{Heading 2}
\DefineParaStyle{Heading 4}
\DefineParaStyle{Maple Output}
\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}
Timelike geodesics
\end{Heading 2}

\begin{Heading 4}
Unstable orbits with constant radius
\end{Heading 4}

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

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{eta_s := r_s^2/a^2/(r_s-M)
*(r_s*(4*a^2*M-r_s*(r_s-3*M)^2)/(r_s-M)
+(r_s*(r_s-2*M)^2 -a^2*M)/E^2
-2*r_s*M/(r_s-M)*Delta*(1-sqrt(1-(1-M/r_s)/E^2))):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{Delta := r_s^2 -2*M*r_s +a^2:}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{a := sqrt(.84): r_s := 2.154: }{%
}r\_s = 1.85 gives negative values for 
\mapleinline{inert}{2d}{eta_s;}{%
$\mathit{eta\_s}$%
}
\mapleinline{active}{1d}{  M := 1: E := sqrt(1-.02):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{eta_s; }{%
}should be positive
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{5.551938391;}{%
\[\color{blue}
5.551938391
\]
%
}
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{Q := eta_s*E^2;}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{Q := 5.440899624;}{%
\[\color{blue}
Q := 5.440899624
\]
%
}
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{gama := -a^2*eta_s/r_s^2 +2*M*r_s;}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{gama := 4.208591961;}{%
\[\color{blue}
\mathit{gama} := 4.208591961
\]
%
}
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{alpha := 1-1/E^2:}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{De := -4*((r_s*(1-1/E^2)+M/E^2)^2
+alpha^2*eta_s/r_s^2*(1-1/E^2));}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{De := -3.813805160;}{%
\[\color{blue}
\mathit{De} := -3.813805160
\]
%
}
\end{maplelatex}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{res := EllipticF(sin(psi),k)/alpha/mu_plus =
ln(abs(2*sqrt(gama*F)+2*gama*x+beta))/sqrt(gama):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{F := beta*x +gama*x^2:}{%
}
\end{mapleinput}

\begin{mapleinput}
\mapleinline{active}{1d}{beta := 4*r_s*(1-1/E^2) +2*M/E^2:}{%
}
\end{mapleinput}

\begin{mapleinput}
\mapleinline{active}{1d}{x := 1/(r - r_s):}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{k := mu_minus/mu_plus: psi := arcsin(mu/mu_minus):
mu_plus := (xi_s^2+eta_s+alpha^2
+sqrt((xi_s^2+eta_s+alpha^2)^2-4*alpha^2*eta_s))
/2/alpha:}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{mu_minus := (xi_s^2+eta_s+alpha^2
-sqrt((xi_s^2+eta_s+alpha^2)^2-4*alpha^2*eta_s)) /2/alpha:}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{mu := cos(theta):}{%
}
\end{mapleinput}
\medskip
\begin{mapleinput}
\mapleinline{active}{1d}{xi_s := (M*(r_s^2-a^2)
-r_s*Delta*sqrt(1-(1-M/r_s)/E^2)) /a /(r_s-M);}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{xi_s := 2.805967772;}{%
\[\color{blue}
\mathit{xi\_s} := 2.805967772
\]
%
}
\end{maplelatex}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{lim := solve(eta_s -(xi_s^2+eta_s+alpha^2)*mu^2
+alpha^2*mu^4 = 0, mu); mu_max := lim[4]:}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{lim := -143.6502598, 143.6502598, -.2895717129, .2895717129;}{%
\[\color{blue}
\mathit{lim} := -143.6502598, \,143.6502598, \,-.2895717129, \,
.2895717129
\]
%
}
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{solve(subs(r, res), r);}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{2.154000000, .1030562634;}{%
\[\color{blue}
2.154000000, \,.1030562634
\]
%
}
\end{maplelatex}

\end{maplegroup}

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

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