%% Created by Maple 6 (IBM INTEL NT)
%% Source Worksheet: nullcirc.mws
%% Generated: Thu Jul 01 05:24:18 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{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}
Null geodesics
\end{Heading 2}

\begin{Heading 3}
\mapleinline{inert}{2d}{eta;}{%
$\eta $%
} positive
\end{Heading 3}

\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}{res := EllipticF(sin(ps),k)/a/sqrt(mu_plus^2 + mu_minus^2)
= ln(abs(2*sqrt(c*(1 + 4*r_s*x + c*x^2)) + 2*c*x + 4*r_s))
/ sqrt(c):}{%
}
\end{mapleinput}

\end{maplegroup}
\medskip
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{ph := E *EllipticF(sin(ps), k) /sqrt(mu_plus^2+mu_minus^2)
-Lz/a /sqrt(mu_plus^2+mu_minus^2)/(1-mu^2)
*EllipticPi(sin(ps), alpha^2, k)
+a*(((r_p^2+a^2)*E-a*Lz)/(r_p-r_m)/(r_p-r_s)*f_p
-((r_m^2+a^2)*E-a*Lz)/(r_p-r_m)/(r_m-r_s)*f_m
-((r_s^2+a^2)*E-a*Lz) /(r_p-r_s) /(r_m-r_s) /sqrt(c)
*ln(abs(2*sqrt(c)*sqrt(y_s^2+4*r_s*y_s+c)/y_s +2*c/y_s
+4*r_s*y_s))):}{%
}
\end{mapleinput}

\end{maplegroup}
\vspace{-8pt}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{c := 3*r_s^2 - a^2*eta_s/r_s^2:}{%
}
\end{mapleinput}
\vspace{-8pt}
\begin{mapleinput}
\mapleinline{active}{1d}{x := 1/(r - r_s):}{%
}
\end{mapleinput}

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

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{alpha := mu_plus/sqrt(mu_plus^2-1):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{r_p := M +sqrt(M^2-a^2): r_m := M -sqrt(M^2-a^2):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{k := mu_plus/sqrt(mu_plus^2 + mu_minus^2):}{%
}
\end{mapleinput}

\begin{mapleinput}
\mapleinline{active}{1d}{ps := arccos(mu/mu_plus):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{mu := cos(theta):}{%
}
\end{mapleinput}
\bigskip
\begin{mapleinput}
\mapleinline{active}{1d}{mu_plus := sqrt((sqrt((xi^2+eta_s-a^2)^2+4*a^2*eta_s)
-(xi^2+eta_s-a^2)) /a^2/2):}{%
}
\end{mapleinput}
\bigskip
\begin{mapleinput}
\mapleinline{active}{1d}{mu_minus :=sqrt((sqrt((xi^2+eta_s-a^2)^2+4*a^2*eta_s)
+(xi^2+eta_s-a^2)) /a^2/2):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{y_p := r_p -r: y_m := r_m -r: y_s := r -r_s:}{%
}
\end{mapleinput}

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

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{a := .8: r_s := 1.85: E := 1:}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{subs(M = 1, r_p^2 -a^2 *eta_s/r_s^2 +2*r_p*r_s);}{%
}
\end{mapleinput}

\mapleresult\color{blue}
\begin{maplelatex}
\mapleinline{inert}{2d}{8.189697232;}{%
\[
8.189697232
\]
%
}
\end{maplelatex}

\end{maplegroup}
\vspace{-8pt}
\begin{maplegroup}
\begin{flushleft}
\hspace{21pt}above is positive so
\end{flushleft}
\vspace{0mm}
\begin{mapleinput}
\mapleinline{active}{1d}{f_p := -1/sqrt(r_p^2 -a^2 *eta_s/r_s^2 +2*r_p*r_s)
*ln(abs(2 *sqrt(r_p^2 -a^2 *eta_s/r_s^2 +2*r_p*r_s)
*sqrt(y_p^2+(-2*r_s-2*r_p)*y_p +r_p^2 -a^2 *eta_s/r_s^2
+2*r_p*r_s) /y_p +2 *sqrt(r_p^2 -a^2 *eta_s/r_s^2 +2*r_p*r_s)
/y_p -2(r_p+r_s))):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{subs(M = 1, r_m^2 -a^2 *eta_s/r_s^2 +2*r_m*r_s);}{%
}
\end{mapleinput}

\mapleresult\color{blue}
\begin{maplelatex}
\mapleinline{inert}{2d}{1.349697232;}{%
\[
1.349697232
\]
%
}
\end{maplelatex}

\end{maplegroup}
\vspace{-8pt}
\begin{maplegroup}
\begin{flushleft}
\hspace{21pt}above is positive so
\end{flushleft}

\begin{mapleinput}
\mapleinline{active}{1d}{f_m := -1/sqrt(r_m^2 -a^2 *eta_s/r_s^2 +2*r_m*r_s)
*ln(abs(2 *sqrt(r_m^2 -a^2 *eta_s/r_s^2 +2*r_m*r_s)
*sqrt(y_m^2+(-2*r_s-2*r_m)*y_m +r_m^2 -a^2 *eta_s/r_s^2
+2*r_m*r_s) /y_m +2 *sqrt(r_m^2 -a^2 *eta_s/r_s^2
+2*r_m*r_s) /y_m -2(r_m+r_s))):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{r := 3.2:}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{res1 := subs(M = 1, res):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{teta := solve(res1,theta);}{%
}
\end{mapleinput}

\mapleresult\color{blue}
\begin{maplelatex}
\mapleinline{inert}{2d}{teta := 1.833106054, 1.308486600;}{%
\[
\mathit{teta} := 1.833106054, \,1.308486600
\]
%
}
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{evalf(subs(\{M = 1, theta = teta[1]\}, ph));}{%
}
\end{mapleinput}

\mapleresult\color{blue}
\begin{maplelatex}
\mapleinline{inert}{2d}{-1.507997117;}{%
\[
-1.507997117
\]
%
}
\end{maplelatex}

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

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