(* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *) (* |||| Mathematica Syntax | kerr.yukterez.net | 22.06.2016 - 04.04.2019, Version 17 |||| *) (* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *) ClearAll["Local`*"]; Needs["DifferentialEquations`NDSolveProblems`"]; Needs["DifferentialEquations`NDSolveUtilities`"]; mt1 = {"StiffnessSwitching", Method-> {"ExplicitRungeKutta", Automatic}}; mt2 = {"EventLocator", "Event"-> (r[τ]-1001/1000 rA)}; mt3 = {"ImplicitRungeKutta", "DifferenceOrder"-> 20}; mt4 = {"EquationSimplification"-> "Residual"}; mt0 = Automatic; mta = mt0; (* mt0: Speed, mt3: Accuracy *) wp = MachinePrecision; (* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *) (* |||||||| 1) STARTBEDINGUNGEN EINGEBEN |||||||||||||||||||||||||||||||||||||||||||||||| *) (* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *) A = a; (* pseudosphärisch: A=0, kartesisch: A=a *) crd = 1; (* Boyer-Lindquist: crd=1, Kerr-Schild: crd=2 *) dsp = 1; (* Display Modus *) tmax = 120; (* Eigenzeit *) Tmax = 120; (* Koordinatenzeit *) TMax = Min[Tmax, т[plunge-1*^-3]]; tMax = Min[tmax, plunge]; (* Integrationsende *) r0 = 3; (* Startradius *) r1 = 5; (* Endradius wenn v0=vr0=vr1 *) θ0 = π/2; (* Breitengrad *) φ0 = 0; (* Längengrad *) a = 1; (* Spinparameter *) v0 = 1; (* Anfangsgeschwindigkeit *) α0 = 0; (* vertikaler Abschusswinkel *) ψ0 = δp[r0, a]; (* Bahninklinationswinkel *) (* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *) (* |||||||| 2) GESCHWINDIGKEITS-, ENERGIE UND DREHIMPULSKOMPONENTEN ||||||||||||||||||||| *) (* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *) vr0 = v0 Sin[α0]; (* radiale Geschwindigkeitskomponente *) vθ0 = v0 Cos[α0] Sin[ψ0]; (* longitudinale Geschwindigkeitskomponente *) vφ0 = v0 Cos[α0] Cos[ψ0]; (* latitudinale Geschwindigkeitskomponente *) vrj[τ_]:=R'[τ]/Sqrt[Δi[τ]] Sqrt[Σi[τ] (1+μ v[τ]^2)]; vθj[τ_]:=Θ'[τ] Sqrt[Σi[τ] (1+μ v[τ]^2)]; vφBL[τ_]:=-((Sin[Θ[τ]] Sqrt[1+μ v[τ]^2] (-a^5 ε Cos[Θ[τ]]^2-a ε R[τ]^4+ a^2 Δi[τ] (xJ[τ] ε Cot[Θ[τ]]^2+Φ'[τ] Cos[Θ[τ]]^2 Σi[τ])+ R[τ]^2 (-a^3 ε (1+Cos[Θ[τ]]^2)+Δi[τ] (xJ[τ] ε Csc[Θ[τ]]^2+Φ'[τ] Σi[τ]))))/((a^2 Cos[Θ[τ]]^2+ R[τ]^2) (a^2 Sin[Θ[τ]]^2-Δi[τ]) Sqrt[Χi[τ]/Σi[τ]])); vφKS[τ_]:=(j[v[τ]] Sin[Θ[τ]]^2 (2 a ε R[τ]-Φ'[τ] Δi[τ] Σi[τ]+ a Σi[τ] R'[τ]))/(Ыi[τ] (2 R[τ]-Σi[τ])) vφj[τ_]:=If[crd==2, vφKS[τ], vφBL[τ]]; vtj[τ_]:=Sqrt[vrj[τ]^2+vθj[τ]^2+vφj[τ]^2]; vr[τ_]:=vrj[τ]/vtj[τ]*v[τ]; vθ[τ_]:=vθj[τ]/vtj[τ]*v[τ]; vφ[τ_]:=vφj[τ]/vtj[τ]*v[τ]; x0BL[A_] := Sqrt[r0^2+A^2] Sin[θ0] Cos[φ0]; (* Anfangskoordinaten *) y0BL[A_] := Sqrt[r0^2+A^2] Sin[θ0] Sin[φ0]; z0[A_] := r0 Cos[θ0]; x0KS[A_] := ((r0 Cos[φ0]+A Sin[φ0]) Sin[θ0]); y0KS[A_] := ((r0 Sin[φ0]-A Cos[φ0]) Sin[θ0]); x0[A_] := If[crd==1, x0BL[A], x0KS[A]]; y0[A_] := If[crd==1, y0BL[A], y0KS[A]]; ε = Sqrt[δ Ξ/χ]/j[v0]+Lz ω0; (* Energie und Drehimpulskomponenten *) Lz = vφ0 Ы/j[v0]; pθ0 = vθ0 Sqrt[Ξ]/j[v0]; pr0 = vr0 Sqrt[(Ξ/δ)/j[v0]^2]; Q = Simplify[Limit[pθ0^2+(Lz^2 Csc[ϑ]^2-a^2 (ε^2+μ)) Cos[ϑ]^2, ϑ->θ0]]; (* Carter Q *) k = Q+Lz^2+a^2 (ε^2+μ); (* Carter k *) μ = If[Abs[v0]==1, 0, -1]; (* Baryon: μ=-1, Photon: μ=0 *) (* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *) (* |||||||| 3) RADIUS NACH GESCHWINDIGKEIT UND VICE VERSA ||||||||||||||||||||||||||||||| *) (* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *) rPro = 2 (1+Cos[2/3 ArcCos[-a]]); (* prograder Photonenorbitradius *) rRet = 2 (1+Cos[2/3 ArcCos[+a]]); (* retrograder Photonenorbitradius *) rTeo = 1+2 Sqrt[1-a^3/3] Cos[ArcCos[(1-a^2)/(1-a^2/3)^(3/2)]/3]; (* ISCO *) isco = rISCO/.Solve[0 == rISCO (6 rISCO-rISCO^2+3 a^2)-8 a rISCO^(3/2) && rISCO>=rA, rISCO][[1]]; δp[r_,a_] := Quiet[δi/.NSolve[(a^4(-1+r)+2(-3+r)r^4+a^2r(6+r(-5+3 r))+ (* Eq. Ink. Winkel *) 4a Sqrt[a^2+(-2+r)r](a^2+3 r^2)Cos[δi]-a^2(3+r)(a^2+(-2+r)r)Cos[2δi])/(2r(a^2+ (-2+r)r)(r^3+a^2(2+r)))==0&&δi<=π&&δi>=0,δi][[1]]]; vPro = (a^2-2a Sqrt[r0]+r0^2)/(Sqrt[a^2+(-2+r0)r0](a+r0^(3/2)));(* Kreisgeschwindigkeit + *) vRet = (a^2+2a Sqrt[r0]+r0^2)/(Sqrt[a^2+(-2+r0)r0](a-r0^(3/2)));(* Kreisgeschwindigkeit - *) vr1 = \[Sqrt](((a^2+(-2+r0) r0) (r0^2+a^2 Cos[θ0]^2) ((a^2+r1^2)^2-a^2 (a^2+(-2+ r1) r1) Sin[θ0]^2) (-1+((a^2+(-2+r1) r1) (r1^2+a^2 Cos[θ0]^2) (-(a^2+r0^2)^2+ a^2 (a^2+(-2+r0) r0) Sin[θ0]^2))/((a^2+(-2+r0) r0) (r0^2+a^2 Cos[θ0]^2) (-(a^2+ r1^2)^2+a^2 (a^2+(-2+r1) r1) Sin[θ0]^2))))/((a^2+(-2+r1) r1) (r1^2+ a^2 Cos[θ0]^2) ((a^2+r0^2)^2-a^2 (a^2+(-2+r0) r0) Sin[θ0]^2))); (* v Flucht von r0 bis r1 *) (* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *) (* |||||||| 4) HORIZONTE UND ERGOSPHÄREN RADIEN ||||||||||||||||||||||||||||||||||||||||| *) (* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *) rE = 1+Sqrt[1-a^2 Cos[θ]^2]; (* äußere Ergosphäre *) rG = 1-Sqrt[1-a^2 Cos[θ]^2]; (* innere Ergosphäre *) rA = 1+Sqrt[1-a^2]; (* äußerer Horizont *) rI = 1-Sqrt[1-a^2]; (* innerer Horizont *) RE[A_, w1_, w2_] := Xyz[xyZ[ {Sqrt[rE^2+A^2] Sin[θ] Cos[φ], Sqrt[rE^2+A^2] Sin[θ] Sin[φ], rE Cos[θ]}, w1], w2]; RG[A_, w1_, w2_] := Xyz[xyZ[ {Sqrt[rG^2+A^2] Sin[θ] Cos[φ], Sqrt[rG^2+A^2] Sin[θ] Sin[φ], rG Cos[θ]}, w1], w2]; RA[A_, w1_, w2_] := Xyz[xyZ[ {Sqrt[rA^2+A^2] Sin[θ] Cos[φ], Sqrt[rA^2+A^2] Sin[θ] Sin[φ], rA Cos[θ]}, w1], w2]; RI[A_, w1_, w2_] := Xyz[xyZ[ {Sqrt[rI^2+A^2] Sin[θ] Cos[φ], Sqrt[rI^2+A^2] Sin[θ] Sin[φ], rI Cos[θ]}, w1], w2]; (* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *) (* |||||||| 5) HORIZONTE UND ERGOSPHÄREN PLOT ||||||||||||||||||||||||||||||||||||||||||| *) (* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *) horizons[A_, mesh_, w1_, w2_] := Show[ ParametricPlot3D[RE[A, w1, w2], {φ, 0, 2 π}, {θ, 0, π}, Mesh -> mesh, PlotPoints -> plp, PlotStyle -> Directive[Blue, Opacity[0.10]]], ParametricPlot3D[RA[A, w1, w2], {φ, 0, 2 π}, {θ, 0, π}, Mesh -> None, PlotPoints -> plp, PlotStyle -> Directive[Cyan, Opacity[0.15]]], ParametricPlot3D[RI[A, w1, w2], {φ, 0, 2 π}, {θ, 0, π}, Mesh -> None, PlotPoints -> plp, PlotStyle -> Directive[Red, Opacity[0.25]]], ParametricPlot3D[RG[A, w1, w2], {φ, 0, 2 π}, {θ, 0, π}, Mesh -> None, PlotPoints -> plp, PlotStyle -> Directive[Red, Opacity[0.35]]]]; BLKS := Grid[{{horizons[a, 35, 0, 0], horizons[0, 35, 0, 0]}}]; (* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *) (* |||||||| 6) FUNKTIONEN ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *) (* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *) j[v_] := Sqrt[1+μ v^2]; (* Lorentzfaktor *) mirr = Sqrt[(Sqrt[1-a^2]+1)/2]; (* irreduzible Masse *) я = Sqrt[Χ/Σ]Sin[θ[τ]]; (* axialer Umfangsradius *) яi[τ_] := Sqrt[Χi[τ]/Σi[τ]]Sin[Θ[τ]]; Ы = Sqrt[χ/Ξ]Sin[θ0]; Ыi[τ_] := Sqrt[Χi[τ]/Σi[τ]]Sin[Θ[τ]]; ц = 2r[τ]/Σ; ц0=2r0/Ξ; Σ = r[τ]^2+a^2 Cos[θ[τ]]^2; (* poloidialer Umfangsradius *) Σi[τ_] := R[τ]^2+a^2 Cos[Θ[τ]]^2; Ξ = r0^2+a^2 Cos[θ0]^2; Δ = r[τ]^2-2r[τ]+a^2; Δi[τ_] := R[τ]^2-2R[τ]+a^2; δ = r0^2-2r0+a^2; Χ = (r[τ]^2+a^2)^2-a^2 Sin[θ[τ]]^2 Δ; Χi[τ_] := (R[τ]^2+a^2)^2-a^2 Sin[Θ[τ]]^2 Δi[τ]; χ = (r0^2+a^2)^2-a^2 Sin[θ0]^2 δ; XJ = a Sin[θ[τ]]^2; xJ[τ_] := a Sin[Θ[τ]]^2; Xj = a Sin[θ0]^2; т[τ_] := Evaluate[t[τ]/.sol][[1]]; (* Koordinatenzeit nach Eigenzeit *) д[ξ_] := Quiet[Ξ /.FindRoot[т[Ξ]-ξ, {Ξ, 0}]]; (* Eigenzeit nach Koordinatenzeit *) T := Quiet[д[tk]]; ю[τ_] := Evaluate[t'[τ]/.sol][[1]]; γ[τ_] := If[μ==0, "Infinity", ю[τ]]; (* totale ZD *) R[τ_] := Evaluate[r[τ]/.sol][[1]]; (* Boyer-Lindquist Radius *) Φ[τ_] := Evaluate[φ[τ]/.sol][[1]]; Θ[τ_] := Evaluate[θ[τ]/.sol][[1]]; ß[τ_] := Re[Sqrt[X'[τ]^2+Y'[τ]^2+Z'[τ]^2 ]/ю[τ]]; gs[τ_] := (2 (R[τ]^2-a^2 Cos[Θ[τ]]^2) Sqrt[((a^2+R[τ]^2)^2-a^2 (a^2+(R[τ]- (* Gravitation *) 2) R[τ]) Sin[Θ[τ]]^2)/(a^2+2 R[τ]^2+a^2 Cos[2 Θ[τ]])^2])/(R[τ]^2+a^2 Cos[Θ[τ]]^2)^2; ς[τ_] := Sqrt[Χi[τ]/Δi[τ]/Σi[τ]]; ς0 = Sqrt[χ/δ/Ξ]; (* gravitative ZD *) ω[τ_] := 2R[τ] a/Χi[τ]; ω0 = 2r0 a/χ; ωd=2r[τ] a/Χ; (* Frame Dragging *) Ω[τ_] := ω[τ] Sqrt[X[τ]^2+Y[τ]^2]; (* Frame Dragging beobachtete Geschwindigkeit *) й[τ_] := ω[τ] яi[τ] ς[τ]; й0 = ω0 Ы ς0; (* Frame Dragging lokale Geschwindigkeit *) ж[τ_] := Sqrt[ς[τ]^2-1]/ς[τ]; ж0 = Sqrt[ς0^2-1]/ς0; (* Fluchtgeschwindigkeit *) vEsc = ж0; vd[τ_] := Abs[-((\[Sqrt](-a^4(ε-Lz ωd)^2-2 a^2r[τ]^2 (ε-Lz ωd)^2- r[τ]^4(ε-Lz ωd)^2+Δ(Σ+a^2 Sin[θ[τ]]^2 (ε- Lz ωd)^2)))/(Sqrt[-(a^2+r[τ]^2)^2+ a^2 Sin[θ[τ]]^2 Δ](ε - Lz ωd)))]; v[τ_] := If[μ==0, 1, Evaluate[vlt'[τ]/.sol][[1]]]; (* lokale Dreiergeschwindigkeit *) dst[τ_] := Evaluate[str[τ]/.sol][[1]]; (* Strecke *) pΘ[τ_] := Evaluate[pθ[τ] /. sol][[1]]; pΘks[τ_] := Σi[τ] Θ'[τ]; (* Impuls *) pR[τ_] := Evaluate[pr[τ] /. sol][[1]]; pRks[τ_] := Σi[τ]/Δi[τ] R'[τ]; sh[τ_] := Re[Sqrt[ß[τ]^2-Ω[τ]^2]]; epot[τ_] := ε+μ-ekin[τ]; (* potentielle Energie *) ekin[τ_] := If[μ==0, ς[τ], 1/Sqrt[1-v[τ]^2]-1]; (* kinetische Energie *) (* beobachtete Inklination *) ink0 := б/. Solve[Z'[0]/ю[0] Cos[б]==-Y'[0]/ю[0] Sin[б]&&б>0&&б<2π&&б<δp[r0, a], б][[1]]; (* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *) (* |||||||| 7) DIFFERENTIALGLEICHUNG |||||||||||||||||||||||||||||||||||||||||||||||||||| *) (* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *) dp= Style[\!\(\*SuperscriptBox[\(Y\),\(Y\)]\), White]; n0[z_] := Chop[Re[N[Simplify[z]]]]; (* Boyer-Lindquist-Koordinaten *) pr2τ[τ_] := 1/(Σ Δ) (((r[τ]^2+a^2)μ-k)(r[τ]-1)+r[τ] Δ μ+2r[τ](r[τ]^2+a^2) ε^2- 2 a ε Lz)-(2pr[τ]^2 (r[τ]-1))/Σ; pθ2τ[τ_] := (Sin[θ[τ]]Cos[θ[τ]])/Σ (Lz^2/Sin[θ[τ]]^4-a^2 (ε^2+μ)); DG1={ t'[τ] == ε+(2r[τ](r[τ]^2+a^2)ε-2 a r[τ] Lz)/(Σ Δ), t[0] == 0, r'[τ] == (pr[τ] Δ)/Σ, r[0] == r0, θ'[τ] == pθ[τ]/Σ, θ[0] == θ0, φ'[τ] == (2 a r[τ] ε+(Σ-2r[τ])Lz Csc[θ[τ]]^2)/(Σ Δ), φ[0] == φ0, pr'[τ] == pr2τ[τ], pr[0] == pr0, pθ'[τ] == pθ2τ[τ], pθ[0] == pθ0, str'[τ]== vd[τ]/Max[1*^-16, Abs[Sqrt[1-vd[τ]^2]]], str[0] == 0, vlt'[τ]== vd[τ], vlt[0] == 0 }; (* Kerr-Schild-Koordinaten *) dr = (pr0 δ)/Ξ; dθ=pθ0/Ξ; dφ = (2a r0 ε+(Ξ-2r0)Lz Csc[θ0]^2)/(Ξ δ)+dr a/δ; dΦ = If[θ0==0, 0, If[θ0==π, 0, dφ]]; φk = φ0+cns/.FindRoot[Sqrt[r0^2+a^2] Cos[φ0+cns]-((r0 Cos[φ0]+a Sin[φ0])),{cns,1}]; DG2={ t''[τ] == (2 ((a^4 Cos[θ[τ]]^4+a^2 Cos[θ[τ]]^2 r[τ]-r[τ]^3-r[τ]^4) r'[τ]^2+r[τ] ((a^2 Cos[θ[τ]]^2-r[τ]^2) t'[τ]^2+r[τ] (a^2 Cos[θ[τ]]^2+r[τ]^2)^2 θ'[τ]^2-2 a^3 Cos[θ[τ]] (a^2 Cos[θ[τ]]^2+r[τ]^2) Sin[θ[τ]]^3 θ'[τ] φ'[τ]+Sin[θ[τ]]^2 (r[τ] (a^2 Cos[θ[τ]]^2+r[τ]^2)^2+a^2 (a^2 Cos[θ[τ]]^2-r[τ]^2) Sin[θ[τ]]^2) φ'[τ]^2+a t'[τ] (a (2 a^2 Cos[θ[τ]]^3 Sin[θ[τ]]+r[τ]^2 Sin[2 θ[τ]]) θ'[τ]+2 (-a^2 Cos[θ[τ]]^2+r[τ]^2) Sin[θ[τ]]^2 φ'[τ]))+r'[τ] ((a^4 Cos[θ[τ]]^4+2 a^2 Cos[θ[τ]]^2 r[τ]-2 r[τ]^3-r[τ]^4) t'[τ]+a (a r[τ] (2 a^2 Cos[θ[τ]]^3 Sin[θ[τ]]+r[τ]^2 Sin[2 θ[τ]]) θ'[τ]+(-a^4 Cos[θ[τ]]^4-2 a^2 Cos[θ[τ]]^2 r[τ]+2 r[τ]^3+r[τ]^4) Sin[θ[τ]]^2 φ'[τ]))))/(a^2 Cos[θ[τ]]^2+r[τ]^2)^3, t'[0] == Limit[(ц0 (-dr+a Sin[θ1]^2 dΦ))/(-1+ц0)+\[Sqrt]((1/((-1+ц0)^2 Ξ))(Ξ (dr^2+(-1+ц0) (μ-Ξ dθ^2))+Sin[θ1]^2 dΦ (-2a Ξ dr-(-1+ц0) χ dΦ+ц0^2 a^2 Ξ Sin[θ1]^2 dΦ))), θ1->θ0], t[0] == 0, r''[τ] == (-8 (a^2 Cos[θ[τ]]^2-r[τ]^2) (a^2 Cos[2 θ[τ]]+r[τ] (2+r[τ])) r'[τ]^2+4 r'[τ] (4 (a^2 Cos[θ[τ]]^2-r[τ]^2) (-2 r[τ]+a^2 Sin[θ[τ]]^2) t'[τ]+2 a^2 (a^2 Cos[θ[τ]]^2+r[τ]^2)^2 Sin[2 θ[τ]] θ'[τ]-a Sin[θ[τ]]^2 (2 r[τ] (a^2 Cos[θ[τ]]^2 (-4+a^2+a^2 Cos[2 θ[τ]])+2 r[τ] ((2+a^2+a^2 Cos[2 θ[τ]]) r[τ]+r[τ]^3-a^2 Sin[θ[τ]]^2))+a^4 Sin[2 θ[τ]]^2) φ'[τ])+2 (a^2+(-2+r[τ]) r[τ]) (4 (a^2 Cos[θ[τ]]^2-r[τ]^2) t'[τ]^2+4 r[τ] (a^2 Cos[θ[τ]]^2+r[τ]^2)^2 θ'[τ]^2+8 a (-a^2 Cos[θ[τ]]^2+r[τ]^2) Sin[θ[τ]]^2 t'[τ] φ'[τ]+Sin[θ[τ]]^2 (4 r[τ] ((a^2 Cos[θ[τ]]^2+r[τ]^2)^2-a^2 r[τ] Sin[θ[τ]]^2)+a^4 Sin[2 θ[τ]]^2) φ'[τ]^2))/(8 (a^2 Cos[θ[τ]]^2+r[τ]^2)^3), r'[0] == dr, r[0] == r0, θ''[τ] == (4 a^2 r[τ] Sin[2 θ[τ]] (r'[τ]+t'[τ])^2-8 r[τ] (a^2 Cos[θ[τ]]^2+r[τ]^2)^2 r'[τ] θ'[τ]+2 a^2 (a^2 Cos[θ[τ]]^2+r[τ]^2)^2 Sin[2 θ[τ]] θ'[τ]^2-8 a ((Cos[θ[τ]] (a^2 Cos[θ[τ]]^2+r[τ]^2)^2 Sin[θ[τ]]+r[τ] (a^2+r[τ]^2) Sin[2 θ[τ]]) r'[τ]+r[τ] (a^2+r[τ]^2) Sin[2 θ[τ]] t'[τ]) φ'[τ]+(2 a^6 Cos[θ[τ]]^4+r[τ] (a^4 Cos[θ[τ]]^2 (5+Cos[2 θ[τ]]) r[τ]+2 a^2 (2+Cos[2 θ[τ]]) r[τ]^3+2 r[τ]^5+2 a^2 (a^2 (3+Cos[2 θ[τ]])+4 r[τ]^2) Sin[θ[τ]]^2)) Sin[2 θ[τ]] φ'[τ]^2)/(4 (a^2 Cos[θ[τ]]^2+r[τ]^2)^3), θ'[0] == dθ, θ[0] == θ0, φ''[τ] == If[θ[τ]==0, 0, (4 (a^3 Cos[θ[τ]]^2-a r[τ]^2) r'[τ]^2+4 (a^3 Cos[θ[τ]]^2-a r[τ]^2) t'[τ]^2+4 a r[τ] (a^2 Cos[θ[τ]]^2+r[τ]^2)^2 θ'[τ]^2-8 (a^2 Cos[θ[τ]]^2+r[τ]^2) (Cot[θ[τ]] (a^2 Cos[θ[τ]]^2+r[τ]^2)^2+a^2 r[τ] Sin[2 θ[τ]]) θ'[τ] φ'[τ]+a Sin[θ[τ]]^2 (4 r[τ] ((a^2 Cos[θ[τ]]^2+r[τ]^2)^2-a^2 r[τ] Sin[θ[τ]]^2)+a^4 Sin[2 θ[τ]]^2) φ'[τ]^2+8 a t'[τ] (2 Cot[θ[τ]] r[τ] (a^2 Cos[θ[τ]]^2+r[τ]^2) θ'[τ]+a (-a^2 Cos[θ[τ]]^2+r[τ]^2) Sin[θ[τ]]^2 φ'[τ])+8 r'[τ] ((a^3 Cos[θ[τ]]^2-a r[τ]^2) t'[τ]+a Cot[θ[τ]] (a^2 Cos[θ[τ]]^2+r[τ]^2) (a^2 Cos[θ[τ]]^2+r[τ] (2+r[τ])) θ'[τ]-(r[τ] (a^2 Cos[θ[τ]]^2+r[τ]^2)^2+a^2 (a^2 Cos[θ[τ]]^2-r[τ]^2) Sin[θ[τ]]^2) φ'[τ]))/(4 (a^2 Cos[θ[τ]]^2+r[τ]^2)^3)], φ'[0] == dΦ, φ[0] == φk, str'[τ]== vd[τ]/Max[1*^-16, Abs[Sqrt[1-vd[τ]^2]]], str[0] == 0, vlt'[τ]== vd[τ], vlt[0] == 0 }; DGL = If[crd==1, DG1, DG2]; (* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *) (* |||||||| 8) INTEGRATION |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *) (* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *) sol= NDSolve[DGL, {t, r, θ, φ, str, vlt, pr, pθ}, {τ, 0, tmax}, WorkingPrecision-> wp, MaxSteps-> Infinity, Method-> mta, InterpolationOrder-> All, StepMonitor :> (laststep=plunge; plunge=τ; stepsize=plunge-laststep;), Method->{"EventLocator", "Event" :> (If[stepsize<1*^-4, 0, 1])}]; (* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *) (* |||||||| 9) KOORDINATEN |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *) (* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *) XBL[τ_] := Evaluate[Sqrt[r[τ]^2+a^2] Sin[θ[τ]] Cos[φ[τ]]/.sol][[1]]; YBL[τ_] := Evaluate[Sqrt[r[τ]^2+a^2] Sin[θ[τ]] Sin[φ[τ]]/.sol][[1]]; Z[τ_] := Evaluate[r[τ] Cos[θ[τ]]/.sol][[1]]; XKS[τ_] := Evaluate[((r[τ] Cos[φ[τ]]+a Sin[φ[τ]]) Sin[θ[τ]])/.sol][[1]]; YKS[τ_] := Evaluate[((r[τ] Sin[φ[τ]]-a Cos[φ[τ]]) Sin[θ[τ]])/.sol][[1]]; X[τ_] := If[crd==1, XBL[τ], XKS[τ]]; Y[τ_] := If[crd==1, YBL[τ], YKS[τ]]; xBL[τ_] := Evaluate[Sqrt[r[τ]^2+A^2] Sin[θ[τ]] Cos[φ[τ]]/.sol][[1]]; yBL[τ_] := Evaluate[Sqrt[r[τ]^2+A^2] Sin[θ[τ]] Sin[φ[τ]]/.sol][[1]]; z[τ_] := Z[τ]; xKS[τ_] := Evaluate[((r[τ] Cos[φ[τ]]+A Sin[φ[τ]]) Sin[θ[τ]])/.sol][[1]]; yKS[τ_] := Evaluate[((r[τ] Sin[φ[τ]]-A Cos[φ[τ]]) Sin[θ[τ]])/.sol][[1]]; x[τ_] := If[crd==1, xBL[τ], xKS[τ]]; y[τ_] := If[crd==1, yBL[τ], yKS[τ]]; XYZ[τ_] := Sqrt[X[τ]^2+Y[τ]^2+Z[τ]^2]; XY[τ_] := Sqrt[X[τ]^2+Y[τ]^2]; (* kartesisches R *) Xyz[{x_, y_, z_}, α_] := {x Cos[α]-y Sin[α], x Sin[α]+y Cos[α], z}; (* Rotationsmatrix *) xYz[{x_, y_, z_}, β_] := {x Cos[β]+z Sin[β], y, z Cos[β]-x Sin[β]}; xyZ[{x_, y_, z_}, ψ_] := {x, y Cos[ψ]-z Sin[ψ], y Sin[ψ]+z Cos[ψ]}; (* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *) (* |||||||| 10) PLOT EINSTELLUNGEN |||||||||||||||||||||||||||||||||||||||||||||||||||||| *) (* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *) PR = 1.2 r1; (* Plot Range *) d1 = 2; (* Schweiflänge *) plp = 50; (* Flächenplot Details *) Plp = Automatic; (* Kurven Details *) Mrec = 100; (* Parametric Plot Subdivisionen *) mrec = 10; imgsize = 380; (* Bildgröße *) w1l = 0; (* Startperspektiven, Winkel *) w2l = 0; w1r = 0; w2r = 0; s[text_]:=Style[text, FontFamily->"Consolas", FontSize->11]; (* Anzeigestil *) (* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *) (* |||||||| 11) PLOT NACH EIGENZEIT ||||||||||||||||||||||||||||||||||||||||||||||||||||| *) (* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *) display[T_] := Grid[{ {If[μ==0, s[" affineP"], s[" τ propr"]], " = ", s[n0[tp]], s["GM/c³"], s[dp]}, {s[" t coord"], " = ", s[n0[т[tp]]], s["GM/c³"], s[dp]}, (* {s[" т coord"], " = ", s[n0[т[tp]+Sign[1.5-dsp] 2 NIntegrate[R[Ti]/(R[Ti]^2-2 R[Ti]+a^2),{Ti,0,T}]]], s["GM/c³"], s[dp]}, *) {s[" ṫ total"], " = ", s[n0[If[dsp==1, ю[tp]-2 If[crd==1, 0, R'[tp] R[tp]/(R[tp]^2-2 R[tp]+a^2)], ю[T]]]], s[If[dsp==1, "dt/dτ", "dт/dτ"]], s[dp]}, {s[" ς gravt"], " = ", s[n0[ς[tp]]], s["dt/dτ"], s[dp]}, {s[" γ kinet"], " = ", If[μ==0, s[n0[ς[tp]]], s[n0[1/Sqrt[1-v[tp]^2]]]], s["dt/dτ"], s[dp]}, {s[" R carts"], " = ", s[n0[XYZ[tp]]], s["GM/c²"], s[dp]}, {s[" x carts"], " = ", s[n0[X[tp]]], s["GM/c²"], s[dp]}, {s[" y carts"], " = ", s[n0[Y[tp]]], s["GM/c²"], s[dp]}, {s[" z carts"], " = ", s[n0[Z[tp]]], s["GM/c²"], s[dp]}, {s[" s dstnc"], " = ", s[n0[dst[tp]]], s["GM/c²"], s[dp]}, {s[" r coord"], " = ", s[n0[R[tp]]], s["GM/c²"], s[dp]}, {s[" φ longd"], " = ", s[n0[Φ[tp] 180/π]], s["deg"], s[dp]}, {s[" θ lattd"], " = ", s[n0[Θ[tp] 180/π]], s["deg"], s[dp]}, {s[" d¹r/dτ¹"], " = ", s[n0[R'[tp]]], s["c"], s[dp]}, {s[" d¹φ/dτ¹"], " = ", s[n0[Φ'[tp]]], s["c\.b3/G/M"], s[dp]}, {s[" d¹θ/dτ¹"], " = ", s[n0[Θ'[tp]]], s["c\.b3/G/M"], s[dp]}, {s[" d\.b2r/dτ\.b2"], " = ", s[n0[R''[tp]]], s["c⁴/G/M"], s[dp]}, {s[" d\.b2φ/dτ\.b2"], " = ", s[n0[Φ''[tp]]], s["c⁶/G\.b2/M\.b2"], s[dp]}, {s[" d\.b2θ/dτ\.b2"], " = ", s[n0[Θ''[tp]]], s["c⁶/G\.b2/M\.b2"], s[dp]}, {s[" a SpinP"], " = ", s[n0[a]], s["GM²/c"], s[dp]}, {s[" Ř crc.φ"], " = ", s[n0[яi[tp]]], s["GM/c²"], s[dp]}, {s[" Σ crc.θ"], " = ", s[n0[Sqrt[Σi[tp]]]], s["GM/c²"], s[dp]}, {s[" M irred"], " = ", s[N[mirr]], s["M"], s[dp]}, {s[" E kinet"], " = ", s[n0[ekin[tp]]], s["mc²"], s[dp]}, {s[" E poten"], " = ", s[n0[epot[tp]]], s["mc²"], s[dp]}, {s[" E total"], " = ", s[n0[ε]], s["mc²"], s[dp]}, {s[" CarterQ"], " = ", s[n0[Q]], s["GMm/c"], s[dp]}, {s[" L axial"], " = ", s[n0[Lz]], s["GMm/c"], s[dp]}, {s[" L polar"], " = ", s[n0[If[crd==1, pΘ[T], pΘks[T]]]], s["GMm/c"], s[dp]}, {s[" p r.mom"], " = ", s[n0[If[crd==1, pR[tp], pRks[tp]]]], s["mc"], s[dp]}, {s[" ω fdrag"], " = ", s[n0[ω[tp]]], s["c³/G/M"], s[dp]}, {s[" v fdrag"], " = ", s[n0[й[tp]]], s["c"], s[dp]}, {s[" Ω fdrag"], " = ", s[n0[Ω[tp]]], s["c"], s[dp]}, {s[" v propr"], " = ", s[n0[v[tp]/Sqrt[1-v[tp]^2]]], s["c"], s[dp]}, {s[" v escpe"], " = ", s[n0[ж[tp]]], s["c"], s[dp]}, {s[" v obsvd"], " = ", s[n0[ß[tp]]], s["c"], s[dp]}, {s[" v r,loc"], " = ", s[n0[vr[tp]]], s["c"], s[dp]}, {s[" v θ,loc"], " = ", s[n0[vθ[tp]]], s["c"], s[dp]}, {s[" v φ,loc"], " = ", s[n0[vφ[tp]]], s["c"], s[dp]}, {s[" v local"], " = ", s[n0[v[tp]]], s["c"], s[dp]}, {s[" "], s[" "], s[" "], s[" "]}}, Alignment-> Left, Spacings-> {0, 0}]; plot1b[{xx_, yy_, zz_, tk_, w1_, w2_}] := (* Animation *) Show[ Graphics3D[{ {PointSize[0.011], Red, Point[ Xyz[xyZ[{x[tp], y[tp], z[tp]}, w1], w2]]}}, ImageSize-> imgsize, PlotRange-> PR, SphericalRegion->False, ImagePadding-> 1], horizons[A, None, w1, w2], If[A==0, {}, If[a==0, {}, ParametricPlot3D[ Xyz[xyZ[{ Sin[prm] a, Cos[prm] a, 0}, w1], w2], {prm, 0, 2π}, PlotStyle -> {Thickness[0.005], Orange}]]], If[crd==1, If[a==0, {}, Graphics3D[{{PointSize[0.009], Purple, Point[ Xyz[xyZ[{ Sin[-φ0-ω0 т[tp]+π/2] Sqrt[x0[A]^2+y0[A]^2], Cos[-φ0-ω0 т[tp]+π/2] Sqrt[x0[A]^2+y0[A]^2], z0[A]}, w1], w2]]}}]], If[a==0, {}, Graphics3D[{{PointSize[0.009], Purple, Point[ Xyz[xyZ[{ Sin[-φ0-ц0 a Ξ/χ т[tp]+π/2] Sqrt[x0[A]^2+y0[A]^2], Cos[-φ0-ц0 a Ξ/χ т[tp]+π/2] Sqrt[x0[A]^2+y0[A]^2], z0[A]}, w1], w2]]}}]]], If[crd==1, If[tk==0, {}, If[a==0, {}, ParametricPlot3D[ Xyz[xyZ[{ Sin[-φ0-ω0 т[tt]+π/2] Sqrt[x0[A]^2+y0[A]^2], Cos[-φ0-ω0 т[tt]+π/2] Sqrt[x0[A]^2+y0[A]^2], z0[A]}, w1], w2], {tt, Max[0, д[т[tp]-1/2 π/ω0]], tp}, PlotStyle -> {Thickness[0.001], Dashed, Purple}, PlotPoints-> Automatic, MaxRecursion-> 12]]], If[tk==0, {}, If[a==0, {}, ParametricPlot3D[ Xyz[xyZ[{ Sin[-φ0-ц0 a Ξ/χ т[tt]+π/2] Sqrt[x0[A]^2+y0[A]^2], Cos[-φ0-ц0 a Ξ/χ т[tt]+π/2] Sqrt[x0[A]^2+y0[A]^2], z0[A]}, w1], w2], {tt, Max[0, д[т[tp]-1/2 π/ω0]], tp}, PlotStyle -> {Thickness[0.001], Dashed, Purple}, PlotPoints-> Automatic, MaxRecursion-> 12]]]], If[tk==0, {}, Block[{$RecursionLimit = Mrec}, ParametricPlot3D[ Xyz[xyZ[{x[tt], y[tt], z[tt]}, w1], w2], {tt, If[tp<0, Min[0, tp+d1], Max[0, tp-d1]], tp}, PlotStyle-> {Thickness[0.004]}, ColorFunction-> Function[{x, y, z, t}, Hue[0, 1, 0.5, If[tp<0, Max[Min[(+tp+(-t+d1))/d1, 1], 0], Max[Min[(-tp+(t+d1))/d1, 1], 0]]]], ColorFunctionScaling-> False, PlotPoints-> Automatic, MaxRecursion-> mrec]]], If[tk==0, {}, Block[{$RecursionLimit = Mrec}, ParametricPlot3D[ Xyz[xyZ[{x[tt], y[tt], z[tt]}, w1], w2], {tt, 0, If[tp<0, Min[-1*^-16, tp+d1/3], Max[1*^-16, tp-d1/3]]}, PlotStyle-> {Thickness[0.004], Opacity[0.6], Darker@Gray}, PlotPoints-> Plp, MaxRecursion-> mrec]]], ViewPoint-> {xx, yy, zz}]; Quiet[Do[ Print[Rasterize[Grid[{{ plot1b[{0, -Infinity, 0, tp, w1l, w2l}], plot1b[{0, 0, +Infinity, tp, w1r, w2r}], display[tp] }, {" ", " ", " "} }, Alignment->Left]]], {tp, 0, tMax, tMax/1}]] (* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *) (* |||||||| 12) PLOT NACH KOORDINATENZEIT ||||||||||||||||||||||||||||||||||||||||||||||| *) (* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *) display[T_] := Grid[{ {s[" t coord"], " = ", s[n0[tk]], s["GM/c³"], s[dp]}, (* {s[" т coord"], " = ", s[n0[tk+Sign[1.5-dsp] 2 NIntegrate[R[Ti]/(R[Ti]^2-2 R[Ti]+a^2),{Ti,0,T}]]], s["GM/c³"], s[dp]}, *) {If[μ==0, s[" affineP"], s[" τ propr"]], " = ", s[n0[T]], s["GM/c³"], s[dp]}, {s[" ṫ total"], " = ", s[n0[If[dsp==1, ю[T]-2 If[crd==1, 0, R'[T] R[T]/(R[T]^2-2 R[T]+a^2)], ю[T]]]], s[If[dsp==1, "dt/dτ", "dт/dτ"]], s[dp]}, {s[" ς gravt"], " = ", s[n0[ς[T]]], s["dt/dτ"], s[dp]}, {s[" γ kinet"], " = ", If[μ==0, s[n0[ς[T]]], s[n0[1/Sqrt[1-v[T]^2]]]], s["dt/dτ"], s[dp]}, {s[" R carts"], " = ", s[n0[XYZ[T]]], s["GM/c²"], s[dp]}, {s[" x carts"], " = ", s[n0[X[T]]], s["GM/c²"], s[dp]}, {s[" y carts"], " = ", s[n0[Y[T]]], s["GM/c²"], s[dp]}, {s[" z carts"], " = ", s[n0[Z[T]]], s["GM/c²"], s[dp]}, {s[" s dstnc"], " = ", s[n0[dst[T]]], s["GM/c²"], s[dp]}, {s[" r coord"], " = ", s[n0[R[T]]], s["GM/c²"], s[dp]}, {s[" φ longd"], " = ", s[n0[Φ[T] 180/π]], s["deg"], s[dp]}, {s[" θ lattd"], " = ", s[n0[Θ[T] 180/π]], s["deg"], s[dp]}, {s[" d¹r/dτ¹"], " = ", s[n0[R'[T]]], s["c"], s[dp]}, {s[" d¹φ/dτ¹"], " = ", s[n0[Φ'[T]]], s["c\.b3/G/M"], s[dp]}, {s[" d¹θ/dτ¹"], " = ", s[n0[Θ'[T]]], s["c\.b3/G/M"], s[dp]}, {s[" d\.b2r/dτ\.b2"], " = ", s[n0[R''[T]]], s["c⁴/G/M"], s[dp]}, {s[" d\.b2φ/dτ\.b2"], " = ", s[n0[Φ''[T]]], s["c⁶/G\.b2/M\.b2"], s[dp]}, {s[" d\.b2θ/dτ\.b2"], " = ", s[n0[Θ''[T]]], s["c⁶/G\.b2/M\.b2"], s[dp]}, {s[" a SpinP"], " = ", s[n0[a]], s["GM²/c"], s[dp]}, {s[" Ř crc.φ"], " = ", s[n0[яi[T]]], s["GM/c²"], s[dp]}, {s[" Σ crc.θ"], " = ", s[n0[Sqrt[Σi[T]]]], s["GM/c²"], s[dp]}, {s[" M irred"], " = ", s[N[mirr]], s["M"], s[dp]}, {s[" E kinet"], " = ", s[n0[ekin[T]]], s["mc²"], s[dp]}, {s[" E poten"], " = ", s[n0[epot[T]]], s["mc²"], s[dp]}, {s[" E total"], " = ", s[n0[ε]], s["mc²"], s[dp]}, {s[" CarterQ"], " = ", s[n0[Q]], s["GMm/c"], s[dp]}, {s[" L axial"], " = ", s[n0[Lz]], s["GMm/c"], s[dp]}, {s[" L polar"], " = ", s[n0[If[crd==1, pΘ[T], pΘks[T]]]], s["GMm/c"], s[dp]}, {s[" p r.mom"], " = ", s[n0[If[crd==1, pR[T], pRks[T]]]], s["mc"], s[dp]}, {s[" ω fdrag"], " = ", s[n0[Abs[ω[T]]]], s["c³/G/M"], s[dp]}, {s[" v fdrag"], " = ", s[n0[Abs[й[T]]]], s["c"], s[dp]}, {s[" Ω fdrag"], " = ", s[n0[Abs[Ω[T]]]], s["c"], s[dp]}, {s[" v propr"], " = ", s[n0[v[T]/Sqrt[1-v[T]^2]]], s["c"], s[dp]}, {s[" v escpe"], " = ", s[n0[ж[T]]], s["c"], s[dp]}, {s[" v obsvd"], " = ", s[n0[ß[T]]], s["c"], s[dp]}, {s[" v r,loc"], " = ", s[n0[vr[T]]], s["c"], s[dp]}, {s[" v θ,loc"], " = ", s[n0[vθ[T]]], s["c"], s[dp]}, {s[" v φ,loc"], " = ", s[n0[vφ[T]]], s["c"], s[dp]}, {s[" v local"], " = ", s[n0[v[T]]], s["c"], s[dp]}, {s[" "], s[" "], s[" "], s[" "]}}, Alignment-> Left, Spacings-> {0, 0}]; plot1a[{xx_, yy_, zz_, tk_, w1_, w2_}]:= (* Animation *) Show[ Graphics3D[{ {PointSize[0.011], Red, Point[ Xyz[xyZ[{x[T], y[T], z[T]}, w1], w2]]}}, ImageSize-> imgsize, PlotRange-> PR, SphericalRegion->False, ImagePadding-> 1], horizons[A, None, w1, w2], If[A==0, {}, If[a==0, {}, ParametricPlot3D[ Xyz[xyZ[{ Sin[prm] a, Cos[prm] a, 0}, w1], w2], {prm, 0, 2π}, PlotStyle -> {Thickness[0.005], Orange}]]], If[crd==1, If[a==0, {}, Graphics3D[{{PointSize[0.009], Purple, Point[ Xyz[xyZ[{ Sin[-φ0-ω0 tk+π/2] Sqrt[x0[A]^2+y0[A]^2], Cos[-φ0-ω0 tk+π/2] Sqrt[x0[A]^2+y0[A]^2], z0[A]}, w1], w2]]}}]], If[a==0, {}, Graphics3D[{{PointSize[0.009], Purple, Point[ Xyz[xyZ[{ Sin[-φ0-ц0 a Ξ/χ tk+π/2] Sqrt[x0[A]^2+y0[A]^2], Cos[-φ0-ц0 a Ξ/χ tk+π/2] Sqrt[x0[A]^2+y0[A]^2], z0[A]}, w1], w2]]}}]]], If[crd==1, If[tk==0, {}, If[a==0, {}, ParametricPlot3D[ Xyz[xyZ[{ Sin[-φ0-ω0 tt+π/2] Sqrt[x0[A]^2+y0[A]^2], Cos[-φ0-ω0 tt+π/2] Sqrt[x0[A]^2+y0[A]^2], z0[A]}, w1], w2], {tt, Max[0, tk-1/2 π/ω0], tk}, PlotStyle -> {Thickness[0.001], Dashed, Purple}, PlotPoints-> Automatic, MaxRecursion-> mrec]]], If[tk==0, {}, If[a==0, {}, ParametricPlot3D[ Xyz[xyZ[{ Sin[-φ0-ц0 a Ξ/χ tt+π/2] Sqrt[x0[A]^2+y0[A]^2], Cos[-φ0-ц0 a Ξ/χ tt+π/2] Sqrt[x0[A]^2+y0[A]^2], z0[A]}, w1], w2], {tt, Max[0, tk-1/2 π/ω0], tk}, PlotStyle -> {Thickness[0.001], Dashed, Purple}, PlotPoints-> Automatic, MaxRecursion-> mrec]]]], Block[{$RecursionLimit = Mrec}, If[tk==0, {}, ParametricPlot3D[ Xyz[xyZ[{x[tt], y[tt], z[tt]}, w1], w2], {tt, If[TMax<0, Min[0, T+d1], Max[0, T-d1]], T}, PlotStyle-> {Thickness[0.004]}, ColorFunction-> Function[{x, y, z, t}, Hue[0, 1, 0.5, If[TMax<0, Max[Min[(+T+(-t+d1))/d1, 1], 0], Max[Min[(-T+(t+d1))/d1, 1], 0]]]], ColorFunctionScaling-> False, PlotPoints-> Automatic, MaxRecursion-> mrec]]], If[tk==0, {}, Block[{$RecursionLimit = Mrec}, ParametricPlot3D[ Xyz[xyZ[{x[tt], y[tt], z[tt]}, w1], w2], {tt, 0, If[Tmax<0, Min[-1*^-16, T+d1/3], Max[1*^-16, T-d1/3]]}, PlotStyle-> {Thickness[0.004], Opacity[0.6], Darker@Gray}, PlotPoints-> Plp, MaxRecursion-> mrec]]], ViewPoint-> {xx, yy, zz}]; Quiet[Do[ Print[Rasterize[Grid[{{ plot1a[{0, -Infinity, 0, tk, w1l, w2l}], plot1a[{0, 0, Infinity, tk, w1r, w2r}], display[Quiet[д[tk]]] }, {" ", " ", " "} }, Alignment->Left]]], {tk, 0, TMax, TMax/1}]] (* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *) (* |||||||| 13) EXPORTOPTIONEN |||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *) (* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *) (* Export als HTML Dokument *) (* Export["Y:\\export\\dateiname.html", EvaluationNotebook[], "GraphicsOutput" -> "PNG"] *) (* Export direkt als Bildsequenz *) (* Do[Export["Y:\\export\\dateiname" <> ToString[tk] <> ".png", Rasterize[...] *) (* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *) (* |||||||||||||||| http://kerr.yukerez.net ||||| Simon Tyran, Vienna ||||||||||||||||||| *) (* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)