%% Created by Maple 6 (IBM INTEL NT)
%% Source Worksheet: tmbdcirc.mws
%% Generated: Thu Mar 30 21:31:27 2006
%\documentclass{article}
%\usepackage{maple2e}
 \def\emptyline{\vspace{12pt}}
\DefineParaStyle{Heading 1}
\DefineParaStyle{Heading 2}
\DefineParaStyle{Heading 3}
\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 3}
\mapleinline{inert}{2d}{eta;}{%
$\eta $%
} positive
\end{Heading 3}

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

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

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{rov1 := M*r^2 -Q*r +3*M*Q = 0: }{%
}
\end{mapleinput}

\begin{mapleinput}
\mapleinline{active}{1d}{rov2 := e*r^2 -M*(3*e-1)*r -4*M^2 =0:}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{M :=1: a := sqrt(.84): e := -.08: E := sqrt(e+1);}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{E := .9591663047;}{%
\[\color{blue}
E := .9591663047
\]
%
}
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{aa := solve(\{rov1, rov2\}, \{r, Q\});}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{aa := \{r = 4.577855615, Q = 13.28179957\}, \{Q = 15.05820043, r =
10.92214439\};}{%
\[\color{blue}
\mathit{aa} := \{r=4.577855615, \,Q=13.28179957\}, \,\{Q=
15.05820043, \,r=10.92214439\}
\]
%
}
\end{maplelatex}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{r_s := 10.92214439: Q_minus := 15.05820043: Lz :=
2*M*a*E/(2*M-r_s): eta_s := Q_minus/E^2;}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{eta_s := 16.36760916;}{%
\[\color{blue}
\mathit{eta\_s} := 16.36760916
\]
%
}
\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 := 21.72903672;}{%
\[\color{blue}
\mathit{gama} := 21.72903672
\]
%
}
\end{maplelatex}

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

\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 := -.7493980729e-1;}{%
\[\color{blue}
\mathit{De} := -.07493980729
\]
%
}
\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 := 1-1/E^2+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}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{xi_s := Lz/E:}{%
}
\end{mapleinput}

\end{maplegroup}
\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}

\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{mu_plus := -188.7131272;}{%
\[\color{blue}
\mathit{mu\_plus} := -188.7131272
\]
%
}
\end{maplelatex}

\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}

\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{mu_minus := -.8673276975e-1;}{%
\[\color{blue}
\mathit{mu\_minus} := -.08673276975
\]
%
}
\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);}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{lim := -46.58541571, 46.58541571, -.9987125027, .9987125027;}{%
\[\color{blue}
\mathit{lim} := -46.58541571, \,46.58541571, \,-.9987125027, \,
.9987125027
\]
%
}
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{solve(subs(r = 8+1/3, res), mu);}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{-.8465346478e-1;}{%
\[\color{blue}
-.08465346478
\]
%
}
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{2*M/3/abs(e);}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{8.333333333;}{%
\[\color{blue}
8.333333333
\]
%
}
\end{maplelatex}

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

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