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Updated eccentric equatorial separatrix to analytic result #58

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Updated eccentric equatorial separatrix to analytic result #58

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25 changes: 20 additions & 5 deletions Kernel/SpecialOrbits.m
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Original file line number Diff line number Diff line change
Expand Up @@ -79,7 +79,7 @@
];


(* ::Section:: *)
(* ::Section::Closed:: *)
(*Photon Sphere*)


Expand Down Expand Up @@ -134,7 +134,7 @@
]


(* ::Section:: *)
(* ::Section::Closed:: *)
(*Innermost bound spherical orbits (IBSO)*)


Expand Down Expand Up @@ -176,7 +176,7 @@
p/.FindRoot[IBSOPoly/.{a->a1,x->x1},{p,KerrGeoIBSO[a1,0],KerrGeoIBSO[a1,-1]},WorkingPrecision->Max[MachinePrecision,prec-1]]];


(* ::Section:: *)
(* ::Section::Closed:: *)
(*Separatrix*)


Expand Down Expand Up @@ -216,6 +216,21 @@
KerrGeoSeparatrix[a_,e_/;e==1,x_]:=2KerrGeoIBSO[a,x]


(* ::Text:: *)
(*General equatorial separatrix analytically from Josh. You'll just have to take my word for it.*)


KerrGeoSeparatrix[a_,e_,x_/;Abs[x]==1]:=Module[{\[CapitalDelta]13, TermOneSq,ats,TermTwoSq,TermOne,TermTwo},
ats=a^2 (1+e);
\[CapitalDelta]13=(-ats^3 (-3+e)^3+(3+e)^6-3 ats (3+e)^3 (15+e^2)+3 ats^2 (153+6 e^2+e^4)+24 Sqrt[3] Sqrt[-ats^3 (-1+ats-e) (ats (-3+e)^3-(-1+e) (3+e)^3)])^(1/3);
TermOneSq=ats (1-e/3)+1/3 (3+e)^2+(1/3 ats^2 (-3+e)^2+1/3 (3+e)^4-2/3 ats (3+e) (15+e^2))/\[CapitalDelta]13+\[CapitalDelta]13/3;
TermOne=Sqrt[TermOneSq];
TermTwoSq=(3+e)^2+ats (3-e+16/TermOne)-TermOneSq;
TermTwo=Sqrt[TermTwoSq];
3+e+TermOne-x*TermTwo
]


(* ::Text:: *)
(*The below methods come from L. Stein and N. Warburton arXiv:1912.07609*)

Expand All @@ -233,9 +248,9 @@
p/.FindRoot[SepPolar/.{a->a1,e->e1},{p,1+Sqrt[3]+Sqrt[3+2Sqrt[3]],8},WorkingPrecision->Max[MachinePrecision,prec-2]]]


KerrGeoSeparatrix[a1_?NumericQ,e1_?NumericQ,1]/;(Precision[{a1,e1}]!=\[Infinity]):=pEquatPro[a1,e1]
(*KerrGeoSeparatrix[a1_?NumericQ,e1_?NumericQ,1]/;(Precision[{a1,e1}]!=\[Infinity]):=pEquatPro[a1,e1]*)
KerrGeoSeparatrix[a1_?NumericQ,e1_?NumericQ,0]/;(Precision[{a1,e1}]!=\[Infinity]):=pPolar[a1,e1]
KerrGeoSeparatrix[a1_?NumericQ,e1_?NumericQ,-1]/;(Precision[{a1,e1}]!=\[Infinity]):=pEquatRet[a1,e1]
(*KerrGeoSeparatrix[a1_?NumericQ,e1_?NumericQ,-1]/;(Precision[{a1,e1}]!=\[Infinity]):=pEquatRet[a1,e1]*)
KerrGeoSeparatrix[a1_?NumericQ,e1_?NumericQ,x1_?NumericQ]/;((Precision[{a1,e1,x1}]!=\[Infinity])&&(1>x1>0)):=With[{prec=Precision[{a1,e1,x1}]},
p/.FindRoot[SepPoly/.{a->a1,x->x1,e->e1},{p,pEquatPro[a1,e1],pPolar[a1,e1]},WorkingPrecision->Max[MachinePrecision,prec-2]]]
KerrGeoSeparatrix[a1_?NumericQ,e1_?NumericQ,x1_?NumericQ]/;((Precision[{a1,e1,x1}]!=\[Infinity])&&(-1<x1<0)):=With[{prec=Precision[{a1,e1,x1}]},
Expand Down