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# Kerr Escape & Recoil Solver

Yukterez Dec 19th, 2019 21 Never
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1. (* yukterez.net - Kerr Escape & Recoil Solver - Syntax: Mathematica *)
2.
3. a=1/2;
4. r=isco;
5. ℧=0;
6. M1=1; M2=10;
7. vt=vφ;
8.
9. Σ[r_]:=r^2;
10. Δ[r_]:=r^2-2 r+a^2+℧^2;
11. Χ[r_]:=(r^2+a^2)^2-a^2 Δ[r];
12. κ[r_]:=a;
13. j[v_]:=Sqrt[1-v^2];
14. Ы[r_]:=Sqrt[Χ[r]/Σ[r]];
15. ω[r_]:=(a (2 r-℧^2))/Χ[r];
16.
17. rA=1+Sqrt[1-a^2-℧^2];
18. rE=1+Sqrt[1-℧^2];
19. rp=rf/.Solve[4 a^2 (rf-℧^2)-(rf^2-3 rf+2 ℧^2)^2==0&&rf>=1,rf]; "rPht"->N[rp]
20.
21. isco=Quiet[Min[RI/.NSolve[0==RI (6 RI-RI^2-9 ℧^2+3 a^2)+4 ℧^2 (℧^2-a^2)-8 a (RI-℧^2)^(3/2)&&RI>=If[Element[rA,Reals],rA,0],RI]]]; "isco"->isco
22.
23. ε[r_]:=Sqrt[Δ[r] Σ[r]/Χ[r]]/j[vt]+Lz[r] ω[r];
24. Lz[r_]:=vφ Ы[r]/j[vt];
25.
26. red[r_]:=Quiet[Reduce[dt==(Lz[r] (-a (a^2+r^2)+Δ[r] κ[r])+ε[r] ((a^2+r^2)^2-Δ[r] κ[r]^2))/(Δ[r] Σ[r])&&0==((a^2+(-2+r) r+℧^2) (16 a dt d\[CapitalPhi] r (r-℧^2)+8 dt^2 r (-r+℧^2)+d\[CapitalPhi]^2 r (8 r (-a^2+r^3)+a^2 (4 a^2+4 ℧^2-4 (a-℧) (a+℧)))))/(8 r^6)&&d\[CapitalPhi]==(Lz[r] (-a^2+Δ[r])+ε[r] (a (a^2+r^2)-Δ[r] κ[r]))/(Δ[r] Σ[r])&&vt>0,{vt,d\[CapitalPhi],dt},Reals]];red[r]
27.
28. vPro=red[r][[1,2]]; "vPro"->vPro
29. vEsc=Quiet@Reduce[ε[r]==1,vt][[2,2]]; "vEsc"->vEsc
30. vDif=Quiet[vd/.Solve[(vPro+vd)/(1+vPro vd)==vEsc,vd][[1]]]; "vDif"->vDif
31.
32. v1=vDif;sol=Quiet[Simplify[Reduce[
33. (M1/Sqrt[1-v1^2]-M1)+(M2/Sqrt[1-v2^2]-M2)==Ek&&((M1 v1)/Sqrt[1-v1^2])+((M2 v2)/Sqrt[1-v2^2])==0&&
34. Ek>0&&M1>0&&M2>M1&&v1>0&&v2<0, {Ek,v2}, Reals]]];
35.
36. "vRec"->sol[[2,2]]
37. "Ek"->sol[[1,2]]
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