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SpecialOrbits.m
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SpecialOrbits.m
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(* ::Package:: *)
(* ::Title:: *)
(*SpecialOrbits subpackage of KerrGeodesics*)
(* ::Chapter:: *)
(*Define usage for public functions*)
BeginPackage["KerrGeodesics`SpecialOrbits`",
{"KerrGeodesics`ConstantsOfMotion`"}];
KerrGeoPhotonSphereRadius::usage = "KerrGeoPhotonSphereRadius[a,x] returns the radius of the photon sphere."
KerrGeoISCO::usage = "KerrGeoISCO[a,x] returns the location of the innermost stable circular orbit (ISCO) for pro- and retrograde orbits."
KerrGeoISSO::usage = "KerrGeoISSO[a,x] returns the location of the innermost stable spherical orbit (ISSO)."
KerrGeoIBSO::usage = "KerrGeoISBO[a,x] returns the location of the innermost bound spherical orbit (IBSO)."
KerrGeoSeparatrix::usage = "KerrGeoSeparatrix[a,e,x] returns the value of p at the separatrix."
KerrGeoBoundOrbitQ::usage = "KerrGeoBoundOrbitQ[a,p,e,x] tests if the orbital parameters correspond to a bound orbit."
Begin["`Private`"];
(* ::Chapter:: *)
(*Special orbits (separatrix, ISCO, ISSO etc...) *)
(* ::Section::Closed:: *)
(*Innermost stable circular orbit (ISCO)*)
(* ::Text:: *)
(*Schwarzschild ISCO is at r=6M*)
KerrGeoISCO[(0|0.),x_]:=6
(* ::Text:: *)
(*Kerr inner-most circular orbit ISCO from Bardeen, Press, Teukolsky ApJ, 178, p347 (1972), Eq. 2.21*)
KerrGeoISCO[a_,x_/;x^2==1]:=Module[{M=1,Z1,Z2},
Z1=1+(1-a^2/M^2)^(1/3) ((1+a/M)^(1/3)+(1-a/M)^(1/3));
Z2=(3a^2/M^2 + Z1^2)^(1/2);
M(3+Z2-x ((3-Z1)(3+Z1+2Z2)/x^2)^(1/2))
];
(* ::Section::Closed:: *)
(*Photon Sphere*)
(* ::Text:: *)
(*The photon sphere is at 3M for all radii in Schwarzschild*)
KerrGeoPhotonSphereRadius[(0|0.),x_]:=3
(* ::Text:: *)
(*Radius of photon sphere for equatorial orbits from Bardeen, Press, Teukolsky ApJ, 178, p347 (1972), Eq. 2.18*)
KerrGeoPhotonSphereRadius[a_,1]:=2(1+Cos[2/3 ArcCos[-a]])
KerrGeoPhotonSphereRadius[a_,-1]:=2(1+Cos[2/3 ArcCos[a]])
(* ::Text:: *)
(*For polar orbits the radius was given by E. Teo, General Relativity and Gravitation, v. 35, Issue 11, p. 1909-1926 (2003), Eq. (14)*)
KerrGeoPhotonSphereRadius[a_,(0|0.)]:=1+2Sqrt[1-1/3 a^2]Cos[1/3 ArcCos[(1-a^2)/(1-1/3 a^2)^(3/2)]]
(* ::Text:: *)
(*In the extremal limit we can find the photon sphere radius exactly*)
KerrGeoPhotonSphereRadius[1,x_]:=If[x < Sqrt[3]-1, 1+Sqrt[2] Sqrt[1-x]-x, 1];
(* ::Text:: *)
(*For all other inclinations we have to numerically find the photon sphere radius*)
KerrGeoPhotonSphereRadius[a1_?NumericQ,x0_?NumericQ/;Abs[x0]<=1]/;Precision[{a1,x0}]!=\[Infinity]:=Module[{M=1,a=a1,req,rpolar,\[CapitalPhi],Q,r,u0Sq,prec},
prec=Precision[{a1,x0}];
req=KerrGeoPhotonSphereRadius[a,Sign[x0]];
rpolar=KerrGeoPhotonSphereRadius[a,0];
\[CapitalPhi]=-((r^3-3M r^2+a^2 r+a^2 M)/(a(r-M)));
Q=-((r^3 (r^3-6M r^2+9M^2 r-4a^2 M))/(a^2 (r-M)^2));
u0Sq=((a^2-Q-\[CapitalPhi]^2)+Sqrt[(a^2-Q-\[CapitalPhi]^2)^2+4a^2 Q])/(2a^2);
r/.FindRoot[1-u0Sq-x0^2,Flatten[{r,(req+rpolar)/2,Sort[{req,rpolar}]}],WorkingPrecision->Max[MachinePrecision,prec-1]]//Quiet
(*The final Quiet[] is there to stop FindRoot complaining about the precision of the argument.
This seems to be fine near the equatorial plane but might not be ideal for inclincation near the polar orbit*)
]
(* ::Section::Closed:: *)
(*Innermost bound spherical orbits (IBSO)*)
KerrGeoIBSO[0,x_]:= 4
(* ::Text:: *)
(*Equatorial IBSO results from Bardeen, Press, Teukolsky 1972*)
KerrGeoIBSO[a_,1]:= 2-a+2(1-a)^(1/2)
KerrGeoIBSO[a_,-1]:= 2+a+2(1+a)^(1/2)
(* ::Text:: *)
(*At the IBSO E=1. Solve[KerrGeo[a,p,0,0]==1,p] to get the formula for the IBSO for polar orbits*)
KerrGeoIBSO[a_,0]:=Module[{\[Delta]},
\[Delta]=27 a^4-8 a^6+3 Sqrt[3] Sqrt[27 a^8-16 a^10];
1+Sqrt[12-4 a^2-(6 Sqrt[6] (-2+a^2))/Sqrt[6-2 a^2+(4 a^4)/\[Delta]^(1/3)+\[Delta]^(1/3)]-(4 a^4)/\[Delta]^(1/3)-\[Delta]^(1/3)]/Sqrt[6]+Sqrt[6-2 a^2+(4 a^4)/\[Delta]^(1/3)+\[Delta]^(1/3)]/Sqrt[6]
]
KerrGeoIBSO[1,(0|0.)]:=1/3 (3+(54-6 Sqrt[33])^(1/3)+(6 (9+Sqrt[33]))^(1/3))
KerrGeoIBSO[a1_?NumericQ,x1_?NumericQ/;Abs[x1]<=1]/;Precision[{a1,x1}]!=\[Infinity]:=Block[{a=a1,x=x1,rph,prec,n=1,ru,E0},
prec=Precision[{a1,x}];
rph=KerrGeoPhotonSphereRadius[a,x];
E0=KerrGeoEnergy[a,ru,0,x];
While[(E0/.ru->rph+10^-n)<1,n++];
ru/.FindRoot[E0==1,{ru,rph+10^-n,10},WorkingPrecision->Max[MachinePrecision,prec-1]]
]
(* ::Section::Closed:: *)
(*Separatrix*)
(* ::Text:: *)
(*Schwarzschild*)
KerrGeoSeparatrix[0,e_,x_]:= 6+2e;
(* ::Text:: *)
(*From Glampedakis and Kennefick arXiv:gr-qc/0203086, for a=M we have Subscript[p, s]=1+e*)
KerrGeoSeparatrix[1,e_,1]:= 1+e
(* ::Text:: *)
(*Separatrix for equatorial Kerr from Levin and Periz-Giz arXiv:1108.1819*)
KerrGeoSeparatrix[a1_,e_,x_/;Abs[x]==1]:= Module[{ru,a=a1},
If[x==-1, a = -a];
ru=ru/.Solve[e==(-ru^2+6 ru-8a ru^(1/2)+3a^2)/(ru^2-2ru+a^2),ru][[-1]];
(4 ru (ru^(1/2)-a)^2)/(ru^2-2ru+a^2)
]
(* ::Text:: *)
(*Polar ISSO in extremal case found from playing around with the equations*)
KerrGeoSeparatrix[1,0,0]:=1+Sqrt[3]+Sqrt[3+2 Sqrt[3]]
(* ::Text:: *)
(*For e=1 the Subscript[p, s] is at 2 Subscript[r, ibso]*)
KerrGeoSeparatrix[a_,1,x_]:=2KerrGeoIBSO[a,x]
(* ::Text:: *)
(*This method is an extension of the method in arXiv:1108.1819. See N. Warburton's notes for details.*)
(*The results of the KerrGeoSeparatrix function have also been tested against the recent analytic results in arXiv:1901.02730 (which also extends the method in arXiv:1108.1819) -- see Eqs. (26a-d) in that paper.*)
KerrGeoSeparatrix[a1_?NumericQ,e1_?NumericQ,x1_?NumericQ/;Abs[x1]<=1]/;Precision[{a1,e1,x1}]!=\[Infinity]:=Block[{a=a1,ra2,\[Beta],E0,L0,Q0,e2,ru,x=x1,prec,r1,\[Beta]2,p,ru0},
{E0,L0,Q0}=Values[KerrGeoConstantsOfMotion[a,ru,0,x]];
\[Beta]=(-1+E0^2);
ra2=2 (-a E0+L0)^2+2 Q0+ru^2 (-1-ru \[Beta]+Sqrt[1+\[Beta] (L0^2+Q0-a^2 \[Beta]-2 ru (1+ru \[Beta]))]);
\[Beta]2=ru (2+ru \[Beta]-2 Sqrt[1+L0^2 \[Beta]+Q0 \[Beta]-2 ru \[Beta]-a^2 \[Beta]^2-2 ru^2 \[Beta]^2]);
e2=(ra2-ru \[Beta]2)/(ra2+ru \[Beta]2);
prec=Precision[{a,e1,x}];
r1=KerrGeoIBSO[a,x];
ru0=ru/.FindRoot[e2==e1,{ru,(r1+10)/2,r1,10},WorkingPrecision->Max[MachinePrecision,prec-1]];
p=(2ra2 ru)/(ra2+ru \[Beta]2)/.ru->ru0
]
KerrGeoBoundOrbitQ[a_?NumericQ,p_?NumericQ,e_?NumericQ,x_?NumericQ]:=Module[{ps},
ps = KerrGeoSeparatrix[a,e,x];
If[p >= ps, True, False]
]
(* ::Section::Closed:: *)
(*Innermost stable spherical orbit (ISSO)*)
KerrGeoISSO[a_,x_/;Abs[x]==1]:=KerrGeoISCO[a,x]
KerrGeoISSO[a_,x_]:=KerrGeoSeparatrix[a,0,x]
(* ::Section::Closed:: *)
(*Close the package*)
End[];
EndPackage[];