Das ist die deutschsprachige Version. 
English versions will be available on en.yukterez.net and yukipedia. Last update: 1.7.2020
ungelinste (oben) und verzerrte Ansicht (unten) eines Kerr Newman SL mit Akkretionsscheibe (hier die geometrische Verzerrung ohne Rotverschiebung), FOV=90°×45°:
Codes (Syntax: Wolfram)
Aktuelle Version (all in one): Output der verzerrten Scheibengeometrie mit und ohne Kreisgeschwindigkeit derselben, Frequenzverschiebung, Hintergrund- und Horizontverzerrung. Als Layer für die umspannende Kugelschale werden Bilder im Plattkartenformat verwendet, andere Formate müssen zuerst in dieses Format transformiert werden. Für die Aberration kann die Lokalgeschwindigkeit des Beobachters eingegeben werden. Deren Betrag muss kleiner als 1 sein, und wird relativ zum ZAMO bemessen. Ein negatives vr steht für eine radiale Bewegung auf das schwarze Loch zu und ein positives von ihm weg, ein positives vφ für eine prograde und ein negatives für eine retrograde, und ein positives vθ im Bereich φ=-π/2..+π/2 für eine Bewegung in Richtung Südpol, und ein negatives in Richtung Nordpol (und umgekehrt in der gegenüberliegenden Kugelschalenhälfte).
Wenn der Beobachter nahe oder hinter dem Horizont ist empfehlen sich Raindrop Doran Koordinaten (für die Anwendung siehe hier):
Für den Umrechner der Lokalgeschwindigkeit vom einen System ins andere klick hier. Beispiel mit a=℧=0.3, θ=85°, r=50, ri=isco=4.826, ra=10:
Minkowski Raytracer:
Output (in diesem Beispiel mit unbewegter Scheibe und ruhendem Beobachter, r=100, θ=70°, rk=3, ri=3, ra=7):
Aberration mit vr=vθ=0, vφ=0.95 (ArcSin[vφ]=71.8°) auf θ=70°; rk=3, ri=4, ra=7; 1. Zeile: r=40, r=30, 2. Zeile: r=20, r=10:
Stereographische Projektion:
verwandte Themen:Kartographie, Kerr-Newman Metrik, Gravitationslinsen.
Code: Alles auswählen
(* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
(* > raytracing.yukterez.net | 07.04.2018 - 01.07.2020 | Version 18 | Simon Tyran, Vienna *)
(* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
Pause[1] (* Boyer Lindquist Koordinaten *)
wp = MachinePrecision;
mt0 = Automatic;
mt1 = {"StiffnessSwitching", Method-> {"ExplicitRungeKutta", Automatic}};
mta = mt0; (* mt0 for speed, mt1 for accuracy *)
kernels = 6; (* Parallelisierung *)
grain = 5; (* Subparallelisierung auf kernels*grain Streifen *)
rsp = "Nearest"; (* Resampling *)
breite = 60; (* Zielabmessungen in Pixeln *)
hoehe = 30; (* Höhe sollte ein ganzzahliges Vielfaches von kernels*grain sein *)
zoom = 1; (* doppelter Zoom ergibt halben Sichtwinkel *)
LaunchKernels[kernels]
wp = MachinePrecision; (* Genauigkeit *)
st = 0.1; (* Auflösung des Gradienten *)
pic1 = Import["http://www.yukterez.net/mw/2/flip70.png"]; (* Hintergrundpanorama laden *)
pic2 = Import["http://www.yukterez.net/mw/2/worldmap.png"]; (* Sphärenoberfläche laden *)
pic3 = Import["http://www.yukterez.net/mw/akkretionsscheibe.jpg"];(* Scheibentextur laden *)
pic4 = Import["http://www.yukterez.net/mw/disk.png"]; (* Scheibengeometrie laden *)
pic5 = Import["http://www.yukterez.net/mw/gradient1.png"]; (* Helligkeitsgradient laden *)
pic6 = Import["http://www.yukterez.net/mw/gradient2.png"]; (* Farbgradient laden *)
pic7 = Import["http://www.yukterez.net/mw/2/bw.png"]; (* Checkerboard laden *)
(* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
(* 1) Startbedingungen und Position des Beobachters ||||||||||||||||||||||||||||||||||||| *)
(* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
r0 = 10; (* Radialkoordinate des Beobachters *)
R0 = 500; (* Radius des umspannenden Kugelschalenpanoramas *)
R1 = Max[1, Re[1.0001 rA]]; (* Sphäre *)
rT = 10; (* Lichtlaufzeit Synchronisation *)
si = isco+0.1; (* Akkretionsscheibe Innenradius *)
sr = 7; (* Akkretionsscheibe Außenradius *)
θ0 = 70 π/180; (* Breitengrad *)
φ0 = 0; (* Längengrad *)
tmax = -100 R0; (* zeitlicher Integrationsbereich *)
a = 0.7; (* Spinparameter *)
℧ = 0.7; (* spezifische Ladung des schwarzen Lochs *)
v0 = 1; (* Geschwindigkeit der Photonen *)
q = 0; (* Ladung der Photonen *)
vr = -vя; (* Radiale Geschwindigkeit des Beobachters *)
vϑ = 0; (* Polare Geschwindigkeit des Beobachters *)
vφ = 0; (* Azimutale Geschwindigkeit des Beobachters: 0 für ZAMO, -й0 für stationär *)
vL = Sqrt[vr^2+vϑ^2+vφ^2];
hvs = 0; (* ArcSin[vφ/vL] *) (* horizontaler Versatz in Radianten *)
vvs = 0; (* ArcSin[vϑ/vL] *) (* vertikaler Versatz in Radianten *)
fmax = 2; fmin = 0; (* Doppler Frequenzbereich *)
(* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
(* 2) Bildreflektion |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
(* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
fpt[{x_, y_}] := {If[y<0, x+1, x], If[y<0, -y, y]}
pcr1 = ParallelTable[
ImageTransformation[pic1, fpt, DataRange->{{-1, 1}, {0, 1}},
PlotRange->{{-1, 1}, {-1+(x-1)/kernels, -1+x/kernels}}, Resampling->rsp, Padding->"Periodic"],
{x, 1, 2 kernels}];
pct1 = ImageAssemble[{Table[{pcr1[[x]]}, {x, 2 kernels, 1, -1}]}];
pcr2 = ParallelTable[
ImageTransformation[pic2, fpt, DataRange->{{-1, 1}, {0, 1}},
PlotRange->{{-1, 1}, {-1+(x-1)/kernels, -1+x/kernels}}, Resampling->rsp, Padding->"Periodic"],
{x, 1, 2 kernels}];
pct2 = ImageAssemble[{Table[{pcr2[[x]]}, {x, 2 kernels, 1, -1}]}];
pcr3 = ParallelTable[
ImageTransformation[pic7, fpt, DataRange->{{-1, 1}, {0, 1}},
PlotRange->{{-1, 1}, {-1+(x-1)/kernels, -1+x/kernels}}, Resampling->rsp, Padding->"Periodic"],
{x, 1, 2 kernels}];
pct3 = ImageAssemble[{Table[{pcr3[[x]]}, {x, 2 kernels, 1, -1}]}];
(* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
(* 3) Funktionen |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
(* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
gtt = (2r0-℧^2)/Σ-1;
grr = Σ/Δ;
gθθ = Σ;
gφφ = Χ/Σ Sin[θ0]^2;
gtφ =-a (2r0-℧^2) Sin[θ0]^2/Σ;
rA = 1+Sqrt[1-a^2-℧^2];
rE = 1+Sqrt[1-℧^2];
Σ = r0^2+a^2 Cos[θ0]^2;
Δ = r0^2-2 r0+a^2+℧^2;
Χ = (r0^2+a^2)^2-a^2 Sin[θ0]^2 Δ;
Σs[rs_] := rs^2;
Δs[rs_] := rs^2-2 rs+a^2+℧^2;
Χs[rs_] := (rs^2+a^2)^2-a^2 Δs[rs];
κs[rs_] := a;
Σj[rt_, θt_] := rt^2+a^2 Cos[θt]^2;
Δj[rt_, θt_] := rt^2-2 rt+a^2+℧^2;
Χj[rt_, θt_] := (rt^2+a^2)^2-a^2 Sin[θt]^2 Δj[rt, θt];
ωj[rt_, θt_] := (a(2 rt-℧^2))/Χj[rt, θt];
ς = [email protected][Χ/Δ/Σ]; ςr[rs_] := [email protected][Χs[rs]/Δs[rs]/Σs[rs]];
μ = If[Abs[[email protected]]==1.0, 0, -1];
j[v_] := Sqrt[1-v^2];
Ы[rs_] := Sqrt[Χs[rs]/Σs[rs]];
ωs[rs_] := (a (2 rs - ℧^2))/Χs[rs];
ε[rs_, vt] := Sqrt[Δs[rs] Σs[rs]/Χs[rs]]/j[vt]+Lz[rs, vt] ωs[rs];
Lz[rs_, vt] := vt Ы[rs]/j[vt];
nq[x_] := If[NumericQ[x], If[Element[x, Reals], x, 0], 0];
rl[x_] := If[Abs[Im[x]]>Abs[Re[x]], 1, x]
dΘF = Quiet[If[vL==0, 0, ArcCos[-vϑ/Sqrt[vr^2+vϑ^2+vφ^2]]]];
dθF = Quiet[If[vL==0, 0, If[vr>0, -dΘF, +dΘF]]];
dΦF = Quiet[If[vL==0, 0, ArcTan[Abs[vφ]/vr]]];
dφF = Quiet[If[vL==0, 0, If[vr!=0, -1, 1] If[vφ<0, +1, -1] If[NumericQ[dΦF], dΦF, π/2]]];
(* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
(* 4) Geschwindigkeitskomponenten ||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
(* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
εj[rt_, δt_, δr_, δθ_, δφ_] := δt (1-(2 rt-℧^2)/rt^2)+(a δφ (2 rt-℧^2))/rt^2;
vrj[rt_, δt_, δr_, δθ_, δφ_] := δr/Sqrt[Δj[rt, θt]] Sqrt[Σj[rt, π/2]];
vθj[rt_, δt_, δr_, δθ_, δφ_] := δθ Sqrt[Σj[rt, π/2]];
vφj[rt_, δt_, δr_, δθ_, δφ_] := (-(((a^2 Cos[(π/2)]^2+rt^2) (a^2+℧^2-2 rt+
rt^2) Sin[(π/2)] (-δφ-
(a q ℧ rt)/((a^2 Cos[(π/2)]^2+rt^2) (a^2+℧^2-2 rt+rt^2))+
(εj[rt, δt, δr, δθ, δφ] Csc[(π/2)]^2 (a (-a^2-℧^2+2 rt-rt^2) Sin[(π/2)]^2+a (a^2+
rt^2) Sin[(π/2)]^2))/((a^2 Cos[(π/2)]^2+rt^2) (a^2+℧^2-2 rt+rt^2))+(a q ℧ rt (a^2+
℧^2-2 rt+rt^2-a^2 Sin[(π/2)]^2))/((a^2 Cos[(π/2)]^2+rt^2)^2 (a^2+℧^2-2 rt+
rt^2))))/((a^2+℧^2-2 rt+rt^2-a^2 Sin[(π/2)]^2) Sqrt[((a^2+rt^2)^2-
a^2 (a^2+℧^2-2 rt+rt^2) Sin[(π/2)]^2)/(a^2 Cos[(π/2)]^2+rt^2)])));
vtj[rt_, δt_, δr_, δθ_, δφ_] := Sqrt[vrj[rt, δt, δr, δθ, δφ]^2+
vθj[rt, δt, δr, δθ, δφ]^2+vφj[rt, δt, δr, δθ, δφ]^2];
vφt[rt_, δt_, δr_, δθ_, δφ_] := vφj[rt, δt, δr, δθ, δφ]/vtj[rt, δt, δr, δθ, δφ];
shf[rt_, δt_, δr_, δθ_, δφ_] := Abs[ς/ςr[rt]/(1-vs[rt] vφt[rt, δt, δr, δθ, δφ])];
dφv[rt_, tt_] := (tt+rT) φs[rt]/ts[rt];
vθ =-vϑ;
θs = π/2;
θi =-θ0+π;
(* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
(* 5) Photonensphäre und ISCO ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
(* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
rp = ц/.Solve[4 a^2 (ц-℧^2)-(ц^2-3 ц+2 ℧^2)^2==0 && ц>=If[Element[rA, Reals], rA, 0], ц];
rP = 1.01 Min[rp]; Rp = 1.01 Max[rp];
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]]];
{"r horizon" -> [email protected], "r ergosphere" -> [email protected], "r isco" -> [email protected],
"r photon pro" -> [email protected][rp], "r photon ret" -> [email protected][rp], "r disk" -> [email protected],
"r observer" -> [email protected], "θ observer" -> [email protected]θ0 180/π}
(* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
(* 6) Geschwindigkeit und Zeitdilatation auf der Scheibe |||||||||||||||||||||||||||||||| *)
(* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
red[rs_] := Quiet[Reduce[
dt == (Lz[rs, vt] (-a (a^2+rs^2)+Δs[rs] κs[rs])+ε[rs, vt] ((a^2+rs^2)^2-
Δs[rs] κs[rs]^2))/(Δs[rs] Σs[rs])
&&
0 == ((a^2+(-2+rs) rs+℧^2) (16 a dt dΦ rs (rs-℧^2)+8 dt^2 rs (-rs+℧^2)+
dΦ^2 rs (8 rs (-a^2+rs^3)+a^2 (4 a^2+4 ℧^2-4 (a-℧) (a+℧)))))/(8 rs^6)
&&
dΦ == (Lz[rs, vt] (-a^2+Δs[rs])+ε[rs, vt] (a (a^2+rs^2)-Δs[rs] κs[rs]))/(Δs[rs] Σs[rs])
&&
vt > 0,
{vt, dΦ, dt},
Reals]];
vs = Interpolation[ParallelTable[{rr, If[[email protected][red[rr][[1, 2]]],
red[rr][[1, 2]], 0]}, {rr, 0, sr+st, st}]];
φs = Interpolation[ParallelTable[{rr, If[[email protected][red[rr][[2, 2]]],
red[rr][[2, 2]], 0]}, {rr, 0, sr+st, st}]];
ts = Interpolation[ParallelTable[{rr, If[[email protected][red[rr][[3, 2]]],
red[rr][[3, 2]], 0]}, {rr, 0, sr+st, st}]];
"Accretion disk:"
plot[func_, label_] := Plot[func, {rr, rP, sr},
GridLines -> {{nq[Min[rp]], nq[Max[rp]], nq[rA], nq[si], nq[isco], nq[rE], nq[sr]}, {}},
Frame -> True, ImagePadding -> {{40, 8}, {12, 8}}, ImageSize -> 340,
PlotLabel -> label, PlotRange->{{0, sr}, Automatic}]
Grid[{{
plot[Sqrt[Χs[rr]/Δs[rr]/Σs[rr]], "Gravitational time dilation: y=dt/dт, x=r"],
plot[φs[rr]/ts[rr], "Shapirodelayed angular velocity: y=dφ/dt, x=r"]},{
plot[ts[rr], "Total time dilation: y=dt/dτ, x=r"],
plot[φs[rr], "Coordinate speed: y=dφ/dτ, x=r"]}, {
plot[(a (2 rr-℧^2))/((a^2+rr^2)^2-a^2 (a^2-2 rr+rr^2+℧^2)), "Frame Dragging: y=dφ/dт, x=r"],
plot[vs[rr], "Local velocity: y=v=dl/dτ, x=r"]}}]
(* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
(* 7) Frame Dragging und Gammafaktor |||||||||||||||||||||||||||||||||||||||||||||||||||| *)
(* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
й0 = (a (2 r0-℧^2) Sin[θ0] Sqrt[((a^2+r0^2)^2-a^2 (a^2-2 r0+r0^2+℧^2) Sin[θ0]^2)/((a^2-
2 r0+r0^2+℧^2) (r0^2+a^2 Cos[θ0]^2))])/((r0^2+a^2 Cos[θ0]^2) Sqrt[((a^2+r0^2)^2-a^2 (a^2-
2 r0+r0^2+℧^2) Sin[θ0]^2)/(r0^2+a^2 Cos[θ0]^2)]);
vя = Sqrt[((a^2+r0^2)(℧^2-2r0))/(a^2 Sin[θ0]^2(a^2+(r0-2)r0+℧^2)-(a^2+r0^2)^2)];
vR = Sqrt[(2 r0-℧^2)/(a^2+r0^2)];
U = {+vr, +vθ, +vφ};
γ = 1/Sqrt[1-Norm[U]^2];
(* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
(* 8) Rotationsmatrix für die auf der Sichtebene eintreffenden Strahlen ||||||||||||||||| *)
(* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
Xyz[{x_, y_, z_}, α_] := {x Cos[α]-y Sin[α], x Sin[α]+y Cos[α], z};
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]};
(* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
(* 9) Raytracing Funktionscontainer ||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
(* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
raytracer[{Ф_, ϑ_}] :=
Quiet[Module[{DGL, sol, εj, pθi, pr0, Q, k, V, W, vw,
vr0i, vθ0i, vφ0i, vr0n, vθ0n, vφ0n, vr0a, vθia, vφ0a, vt0a,
t10, r10, Θ10, Φ10, t, r, θ, φ, τ,
т, т0, т1, т2, т3, т4, т5,
plunge, plunge0, plunge1, plunge2, plunge3,
plunge4, plunge5, plunge6, plunge7,
dθ0, dφ0, δφ0, δθ0, δr0, δt0, tt0, rt0, θt0, φt0,
dθ1, dφ1, δφ1, δθ1, δr1, δt1, tt1, rt1, θt1, φt1,
dθ2, dφ2, δφ2, δθ2, δr2, δt2, tt2, rt2, θt2, φt2,
dθ3, dφ3, δφ3, δθ3, δr3, δt3, tt3, rt3, θt3, φt3,
dθ4, dφ4, δφ4, δθ4, δr4, δt4, tt4, rt4, θt4, φt4,
dθ5, dφ5, δφ5, δθ5, δr5, δt5, tt5, rt5, θt5, φt5,
dθ6, dφ6, δφ6, δθ6, δr6, δt6, tt6, rt6, θt6, φt6,
X, Y, Z, ξ, stepsize, laststep, mtl, ft, fτ, varb,
ft0s, ft1s, ft2s, ft3s, ft4s,
ft0v, ft1v, ft2v, ft3v, ft4v,
ft0f, ft1f, ft2f, ft3f, ft4f,
ft5h, ft5b, ft5f, ft5x,
dt0y, dr0y, dθ0y, dφ0y,
initcon, shF},
vw=xyZ[Xyz[{0, 1, 0}, ϑ], Ф+π/2];
(* Übersetzung des Einfallswinkels in den lokalen Tetrad *)
vr0a = vw[[3]];
vφ0a = vw[[2]];
vθia = vw[[1]];
(* Betrag *)
vt0a = Sqrt[vr0a^2+vφ0a^2+vθia^2];
(* Normierung *)
vr0n = vr0a/vt0a;
vφ0n = vφ0a/vt0a;
vθ0n = vθia/vt0a;
(* Relativistische Geschwindigkeitsaddition *)
V={vr0n, vθ0n, vφ0n};
W=(U+V+γ/(1+γ)(U\[Cross](U\[Cross]V)))/(1+U.V);
(* Aberration *)
vr0i = W[[1]];
vθ0i = W[[2]];
vφ0i = W[[3]];
shF = Abs[If[vL==0, 1, Sqrt[1-vr^2-vϑ^2-vφ^2]/(1+vL (-Cos[If[vr>0, -dΘF, +dΘF]] Sin[ϑ]-
Sin[If[vr>0, -dΘF, +dΘF]] (Cos[-Ф-π/2+If[vr>0, -1, +1] ArcSin[vφ/vL]] Cos[ϑ] Cos[1/2 π]-
Cos[ϑ] Sin[-Ф-π/2+If[vr>0, -1, +1] ArcSin[vφ/vL]] Sin[1/2 π])))]];
initcon = TimeConstrained[NSolve[
dt0y == ς
&&
vr0i ==
Sqrt[(a^2 Cos[θi]^2+r0^2)/(a^2+℧^2-2 r0+r0^2)] Sqrt[1-μ^2 v0^2] dr0y
&&
vθ0i ==
Sqrt[a^2 Cos[θi]^2+r0^2] Sqrt[1-μ^2 v0^2] dθ0y
&&
vφ0i ==
(Sin[θi]^2 Sqrt[1-μ^2 v0^2] (a q μ^2 ℧ r0 v0^2+a (℧^2-
2 r0) dt0y+((a^2+r0^2)^2-a^2 (a^2+℧^2-
2 r0+r0^2) Sin[θi]^2) dφ0y))/((a^2 Cos[θi]^2+r0^2) Sqrt[(Sin[θi]^2 ((a^2+
r0^2)^2-a^2 (a^2+℧^2-2 r0+r0^2) Sin[θi]^2))/(a^2 Cos[θi]^2+r0^2)]),
{dt0y, dr0y, dθ0y, dφ0y}, Reals], 3];
DGL = If[NumericQ[dt0y /. initcon[[1]]] == False, {}, { (* Bewegungsgleichungen *)
t''[τ] == Re[-(1/((a^2 Cos[θ[τ]]^2+r[τ]^2)^2))((q ℧ (a^2 Cos[θ[τ]]^2-
r[τ]^2) (a^2+r[τ]^2) r'[τ])/(a^2+℧^2-2 r[τ]+r[τ]^2)-(2 (a^2 Cos[θ[τ]]^2+
℧^2 r[τ]-r[τ]^2) (a^2+r[τ]^2) r'[τ] t'[τ])/(a^2+℧^2-2 r[τ]+r[τ]^2)+
a^2 q ℧ r[τ] Sin[2 θ[τ]] θ'[τ]+a^2 (℧^2-2 r[τ]) Sin[2 θ[τ]] t'[τ] θ'[τ]+
(a (2 a^4 Cos[θ[τ]]^2+a^2 ℧^2 (3+Cos[2 θ[τ]]) r[τ]-a^2 (3+Cos[2 θ[τ]]) r[τ]^2+
4 ℧^2 r[τ]^3-6 r[τ]^4) Sin[θ[τ]]^2 r'[τ] φ'[τ])/(a^2+℧^2-2 r[τ]+r[τ]^2)-
2 a^3 Cos[θ[τ]] (℧^2-2 r[τ]) Sin[θ[τ]]^3 θ'[τ] φ'[τ])],
t'[0] == dt0y/.initcon[[1]],
t[0] == 0,
r''[τ] == Re[-(1/(2 (a^2 Cos[θ[τ]]^2+r[τ]^2)^4))((2 (-a^2 Cos[θ[τ]]^2 (-1+r[τ])+(a^2+
℧^2-r[τ]) r[τ]) (a^2 Cos[θ[τ]]^2+r[τ]^2)^3 r'[τ]^2)/(a^2+℧^2-2 r[τ]+r[τ]^2)+
q ℧ (a^2 Cos[θ[τ]]^2-r[τ]^2) (a^2+r[τ]^2) (a^2+2 ℧^2+a^2 Cos[2 θ[τ]]-4 r[τ]+
2 r[τ]^2) t'[τ]-2 a^2 q ℧ (℧^2-2 r[τ]) (a^2 Cos[θ[τ]]^2-r[τ]^2) Sin[θ[τ]]^2 t'[τ]-
2 (a^2 Cos[θ[τ]]^2+℧^2 r[τ]-r[τ]^2) (a^2 Cos[θ[τ]]^2+r[τ]^2) (a^2+℧^2-2 r[τ]+
r[τ]^2) t'[τ]^2-4 a^2 Cos[θ[τ]] (a^2 Cos[θ[τ]]^2+r[τ]^2)^3 Sin[θ[τ]] r'[τ] θ'[τ]-
2 r[τ] (a^2 Cos[θ[τ]]^2+r[τ]^2)^3 (a^2+℧^2-2 r[τ]+r[τ]^2) θ'[τ]^2-2 a q ℧ (℧^2-
2 r[τ]) (a^2 Cos[θ[τ]]^2-r[τ]^2) (a^2+r[τ]^2) Sin[θ[τ]]^2 φ'[τ]+
2 a q ℧ (a^2 Cos[θ[τ]]^2-r[τ]^2) (-(a^2+r[τ]^2)^2 Sin[θ[τ]]^2+a^2 (a^2+
℧^2-2 r[τ]+r[τ]^2) Sin[θ[τ]]^4) φ'[τ]+4 a (a^2 Cos[θ[τ]]^2+(℧^2-
r[τ]) r[τ]) (a^2 Cos[θ[τ]]^2+r[τ]^2) (a^2+℧^2-2 r[τ]+r[τ]^2) Sin[θ[τ]]^2 t'[τ] φ'[τ]+
2 (a^2 Cos[θ[τ]]^2+r[τ]^2) (a^2+℧^2-2 r[τ]+r[τ]^2) Sin[θ[τ]]^2 (-r[τ] (-a^4+r[τ]^4+
a^2 (a^2+℧^2-r[τ]) Sin[θ[τ]]^2)+Cos[θ[τ]]^2 (-2 a^2 r[τ] (a^2+r[τ]^2)+a^4 (-1+
r[τ]) Sin[θ[τ]]^2)) φ'[τ]^2)],
r'[0] == dr0y/.initcon[[1]],
r[0] == r0,
θ''[τ] == Re[-(1/(2 (a^2 Cos[θ[τ]]^2+r[τ]^2)^3))((a^2 (a^2 Cos[θ[τ]]^2+
r[τ]^2)^2 Sin[2 θ[τ]] r'[τ]^2)/(a^2+℧^2-2 r[τ]+r[τ]^2)+4 r[τ] (a^2 Cos[θ[τ]]^2+
r[τ]^2)^2 r'[τ] θ'[τ]-1/8 Sin[2 θ[τ]] (8 r[τ]^6 φ'[τ]^2+16 a r[τ]^3 φ'[τ] (q ℧-
2 t'[τ]+2 a Sin[θ[τ]]^2 φ'[τ])+8 a^2 r[τ]^4 (θ'[τ]^2+(2+Cos[2 θ[τ]]) φ'[τ]^2)+
a^2 (-8 ℧^2 t'[τ]^2+8 a^4 Cos[θ[τ]]^4 θ'[τ]^2+16 a ℧^2 t'[τ] φ'[τ]+a^2 (3 a^2-
5 ℧^2+4 (a^2+℧^2) Cos[2 θ[τ]]+(a^2+℧^2) Cos[4 θ[τ]]) φ'[τ]^2)+
r[τ]^2 (16 a^4 Cos[θ[τ]]^2 θ'[τ]^2+a φ'[τ] (16 ℧^2 t'[τ]+a (11 a^2-
8 ℧^2+4 (3 a^2+2 ℧^2) Cos[2 θ[τ]]+a^2 Cos[4 θ[τ]]) φ'[τ]))+
8 a^2 r[τ] (2 t'[τ]^2-2 t'[τ] (q ℧+2 a φ'[τ])+a φ'[τ] (2 q ℧+
a (3+Cos[2 θ[τ]]) Sin[θ[τ]]^2 φ'[τ]))))],
θ'[0] == dθ0y/.initcon[[1]],
θ[0] == θi,
φ''[τ] == Re[-(1/(4 (a^2 Cos[θ[τ]]^2+r[τ]^2)^2))((4 a q ℧ (a^2 Cos[θ[τ]]^2-
r[τ]^2) r'[τ])/(a^2+℧^2-2 r[τ]+r[τ]^2)-(8 a (a^2 Cos[θ[τ]]^2+℧^2 r[τ]-
r[τ]^2) r'[τ] t'[τ])/(a^2+℧^2-2 r[τ]+r[τ]^2)+8 a q ℧ Cot[θ[τ]] r[τ] θ'[τ]+
8 a Cot[θ[τ]] (℧^2-2 r[τ]) t'[τ] θ'[τ]+(1/(a^2+℧^2-2 r[τ]+r[τ]^2))(a^2 (3 a^2+
8 ℧^2+4 a^2 Cos[2 θ[τ]]+a^2 Cos[4 θ[τ]]) r[τ]-4 a^2 (3+Cos[2 θ[τ]]) r[τ]^2+
8 (a^2+℧^2+a^2 Cos[2 θ[τ]]) r[τ]^3-16 r[τ]^4+8 r[τ]^5+
2 a^4 Sin[2 θ[τ]]^2) r'[τ] φ'[τ]+Cot[θ[τ]] (a^2 (3 a^2-4 ℧^2+4 (a^2+
℧^2) Cos[2 θ[τ]]+a^2 Cos[4 θ[τ]])+16 a^2 Cos[θ[τ]]^2 r[τ]^2+8 r[τ]^4+
16 a^2 r[τ] Sin[θ[τ]]^2) θ'[τ] φ'[τ])],
φ'[0] == dφ0y/.initcon[[1]],
φ[0] == φ0,
WhenEvent[Abs[r[τ]] == N[R0]||Abs[r[τ]]>R0 &&
NumericQ[plunge6] == False,
(plunge6=τ) &&
(tt6=t[τ]) && (rt6=r[τ]) && (θt6=θ[τ]) && (φt6=φ[τ]);
"StopIntegration"],
WhenEvent[Abs[r[τ]] == N[R1] &&
NumericQ[plunge5] == False,
(plunge5=τ) && (tt5=t[τ]) && (rt5=r[τ]) && (θt5=θ[τ]) && (φt5=φ[τ]);
If[r0>R1, "StopIntegration"]],
WhenEvent[Mod[180/π Re[θ[τ]], 180] == 90.0 && r[τ]>N[si] &&
r[τ]<N[sr] && τ < -0.05 &&
NumericQ[plunge0] == False,
(plunge0=τ) &&
(tt0=t[τ]) && (rt0=r[τ]) && (θt0=θ[τ]) && (φt0=φ[τ]) &&
(δt0=t'[τ]) && (δr0=r'[τ]) && (δθ0=θ'[τ]) && (δφ0=φ'[τ])],
WhenEvent[Mod[180/π Re[θ[τ]], 180] == 90.0 && r[τ]>N[si] &&
r[τ]<N[sr] && Re[θ'[τ]]>0 && τ < -0.05 &&
NumericQ[plunge1] == False,
(plunge1=τ) &&
(tt1=t[τ]) && (rt1=r[τ]) && (θt1=θ[τ]) && (φt1=φ[τ]) &&
(δt1=t'[τ]) && (δr1=r'[τ]) && (δθ1=θ'[τ]) && (δφ1=φ'[τ])],
WhenEvent[Mod[180/π Re[θ[τ]], 180] == 90.0 &&
r[τ]>N[si] && r[τ]<N[sr] && Re[θ'[τ]]<0 && τ < -0.05 &&
NumericQ[plunge2] == False,
(plunge2=τ) &&
(tt2=t[τ]) && (rt2=r[τ]) && (θt2=θ[τ]) && (φt2=φ[τ]) &&
(δt2=t'[τ]) && (δr2=r'[τ]) && (δθ2=θ'[τ]) && (δφ2=φ'[τ])],
WhenEvent[Mod[180/π Re[θ[τ]], 180] == 90.0 && r[τ]>N[si] &&
r[τ]<N[sr] && Re[θ'[τ]]>0 &&
τ < plunge1-0.05 &&
NumericQ[plunge3] == False && NumericQ[plunge1] == True,
(plunge3=τ) &&
(tt3=t[τ]) && (rt3=r[τ]) && (θt3=θ[τ]) && (φt3=φ[τ]) &&
(δt3=t'[τ]) && (δr3=r'[τ]) && (δθ3=θ'[τ]) && (δφ3=φ'[τ])],
WhenEvent[Mod[180/π Re[θ[τ]], 180] == 90.0 && r[τ]>N[si] &&
r[τ]<N[sr] && Re[θ'[τ]]<0 &&
τ < plunge2-0.05 &&
NumericQ[plunge4] == False && NumericQ[plunge2] == True,
(plunge4=τ) &&
(tt4=t[τ]) && (rt4=r[τ]) && (θt4=θ[τ]) && (φt4=φ[τ]) &&
(δt4=t'[τ]) && (δr4=r'[τ]) && (δθ4=θ'[τ]) && (δφ4=φ'[τ])]
}];
(* Integrator *)
sol = If[NumericQ[dt0y /. initcon[[1]]] == False, {},
NDSolve[DGL, {t, r, θ, φ}, {τ, 0, tmax},
WorkingPrecision-> wp,
Method-> mta,
MaxSteps-> Infinity,
StepMonitor :> (laststep=plunge7; plunge7=τ;
stepsize=plunge7-laststep;), Method->{"EventLocator",
"Event" :> (If[stepsize<1*^-4, 0, 1])}
]];
ft5h=If[NumericQ[plunge5], {φt5-(tt5+r0) ωs[R1], θt5+π/2}, {0, -π/2}];
ft5b=If[NumericQ[plunge6], If[rt6<If[Element[rA, Reals], rA, 0], {0, -π/2},
If[rt6>4 R0, {0, -π/2}, {φt6-π, θt6+π/2}]], {0, -π/2}];
ft5f={0, Min[fmax, ς shF]};
ft5x=If[NumericQ[plunge6], {0, 1}, {0, 0}];
ft0s=If[NumericQ[plunge0], {φt0, rt0}, {π, sr}];
ft1s=If[NumericQ[plunge1], {φt1, rt1}, {π, sr}];
ft2s=If[NumericQ[plunge2], {φt2, rt2}, {π, sr}];
ft3s=If[NumericQ[plunge3], {φt3, rt3}, {π, sr}];
ft4s=If[NumericQ[plunge4], {φt4, rt4}, {π, sr}];
ft0v=If[NumericQ[plunge0], {-dφv[rt0, tt0], 0}, {0, 0}];
ft1v=If[NumericQ[plunge1], {-dφv[rt1, tt1], 0}, {0, 0}];
ft2v=If[NumericQ[plunge2], {-dφv[rt2, tt2], 0}, {0, 0}];
ft3v=If[NumericQ[plunge3], {-dφv[rt3, tt3], 0}, {0, 0}];
ft4v=If[NumericQ[plunge4], {-dφv[rt4, tt4], 0}, {0, 0}];
ft0f=If[NumericQ[plunge0], {0, Min[fmax, shF shf[rt0, δt0, δr0, δθ0, δφ0]]}, {0, 0}];
ft1f=If[NumericQ[plunge1], {0, Min[fmax, shF shf[rt1, δt1, δr1, δθ1, δφ1]]}, {0, 0}];
ft2f=If[NumericQ[plunge2], {0, Min[fmax, shF shf[rt2, δt2, δr2, δθ2, δφ2]]}, {0, 0}];
ft3f=If[NumericQ[plunge3], {0, Min[fmax, shF shf[rt3, δt3, δr3, δθ3, δφ3]]}, {0, 0}];
ft4f=If[NumericQ[plunge4], {0, Min[fmax, shF shf[rt4, δt4, δr4, δθ4, δφ4]]}, {0, 0}];
{
ft0s, ft1s, ft2s, ft3s, ft4s,
ft0s+ft0v, ft1s+ft1v, ft2s+ft2v, ft3s+ft3v, ft4s+ft4v,
ft0f, ft1f, ft2f, ft3f, ft4f,
ft5h, ft5b, ft5f, ft5x
}]];
(* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
(* 10) Memoryfunktion ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
(* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
mem : raytrace[{Ф_, ϑ_}] := mem = Quiet[Re[raytracer[{Ф, ϑ}]]];
ray01[{Ф_, ϑ_}] := raytrace[{Ф, ϑ}][[01]];
ray02[{Ф_, ϑ_}] := raytrace[{Ф, ϑ}][[02]];
ray03[{Ф_, ϑ_}] := raytrace[{Ф, ϑ}][[03]];
ray04[{Ф_, ϑ_}] := raytrace[{Ф, ϑ}][[04]];
ray05[{Ф_, ϑ_}] := raytrace[{Ф, ϑ}][[05]];
ray06[{Ф_, ϑ_}] := raytrace[{Ф, ϑ}][[06]];
ray07[{Ф_, ϑ_}] := raytrace[{Ф, ϑ}][[07]];
ray08[{Ф_, ϑ_}] := raytrace[{Ф, ϑ}][[08]];
ray09[{Ф_, ϑ_}] := raytrace[{Ф, ϑ}][[09]];
ray10[{Ф_, ϑ_}] := raytrace[{Ф, ϑ}][[10]];
ray11[{Ф_, ϑ_}] := raytrace[{Ф, ϑ}][[11]];
ray12[{Ф_, ϑ_}] := raytrace[{Ф, ϑ}][[12]];
ray13[{Ф_, ϑ_}] := raytrace[{Ф, ϑ}][[13]];
ray14[{Ф_, ϑ_}] := raytrace[{Ф, ϑ}][[14]];
ray15[{Ф_, ϑ_}] := raytrace[{Ф, ϑ}][[15]];
ray16[{Ф_, ϑ_}] := raytrace[{Ф, ϑ}][[16]];
ray17[{Ф_, ϑ_}] := raytrace[{Ф, ϑ}][[17]];
ray18[{Ф_, ϑ_}] := raytrace[{Ф, ϑ}][[18]];
ray19[{Ф_, ϑ_}] := raytrace[{Ф, ϑ}][[19]];
(* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
(* 11) Proportionen ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
(* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
width1 = ImageDimensions[pic1][[1]]; height1 = ImageDimensions[pic1][[2]];
width2 = ImageDimensions[pic2][[1]]; height2 = ImageDimensions[pic2][[2]];
width3 = ImageDimensions[pic3][[1]]; height3 = ImageDimensions[pic3][[2]];
width4 = ImageDimensions[pic4][[1]]; height4 = ImageDimensions[pic4][[2]];
width5 = ImageDimensions[pic5][[1]]; height5 = ImageDimensions[pic5][[2]];
width6 = ImageDimensions[pic7][[1]]; height6 = ImageDimensions[pic7][[2]];
hzoom = If[breite>2 hoehe, 1/zoom, 1/zoom/2/hoehe*breite];
vzoom = If[breite>2 hoehe, 1/zoom*2 hoehe/breite, 1/zoom];
"approximate time remaining" -> 1.2 (AbsoluteTiming[Do[raytracer[{
RandomReal[{-π, π}]/zoom, RandomReal[{-π/2, π/2}]/zoom
}], {ü, 1, 50}]][[1]])/50*hoehe*breite/kernels/60 "minutes"
FOV->{360.0 hzoom "degree", 180.0 vzoom "degree"}
(* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
(* 12) Output ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
(* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
img = ParallelTable[{
(* 1 Hintergrundpanorama *)
[email protected][pct1, ray17, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{-π, π-2π/width1},
{-π/2, 3π/2}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+x+vvs/vzoom, -π/2+x+vvs/vzoom+π/kernels/grain} vzoom
},
Resampling->rsp, Padding->"Periodic"],
(* 2 Sphäre *)
[email protected][pct2, ray16, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{-π, π-2π/width2},
{-π/2, 3π/2}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+x+vvs/vzoom, -π/2+x+π/kernels/grain+vvs/vzoom} vzoom
},
Resampling->rsp, Padding->"Periodic"],
(* 3 Scheibe Textur komplett *)
[email protected][pic3, ray01, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{0, 2π-2π/width3},
{si, sr+(sr-si)/height3}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+vvs/vzoom+x, -π/2+vvs/vzoom+x+π/kernels/grain} vzoom
},
Resampling->rsp, Padding->"Periodic"],
(* 4 Scheibe Textur Front *)
[email protected][pic3, ray02, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{0, 2π-2π/width3},
{si, sr+(sr-si)/height3}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+vvs/vzoom+x, -π/2+vvs/vzoom+x+π/kernels/grain} vzoom
},
Resampling->rsp, Padding->"Periodic"],
(* 5 Scheibe Textur Echo 1 *)
[email protected][pic3, ray03, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{0, 2π-2π/width3},
{si, sr+(sr-si)/height3}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+vvs/vzoom+x, -π/2+vvs/vzoom+x+π/kernels/grain} vzoom
},
Resampling->rsp, Padding->"Periodic"],
(* 6 Scheibe Textur Echo 2 *)
[email protected][pic3, ray04, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{0, 2π-2π/width3},
{si, sr+(sr-si)/height3}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+vvs/vzoom+x, -π/2+vvs/vzoom+x+π/kernels/grain} vzoom
},
Resampling->rsp, Padding->"Periodic"],
(* 7 Scheibe Textur Echo 3 *)
[email protected][pic3, ray05, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{0, 2π-2π/width3},
{si, sr+(sr-si)/height3}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+vvs/vzoom+x, -π/2+vvs/vzoom+x+π/kernels/grain} vzoom
},
Resampling->rsp, Padding->"Periodic"],
(* 8 Scheibe Geometrie still komplett *)
[email protected][pic4, ray01, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{0, 2π-2π/width4},
{si, sr}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+vvs/vzoom+x, -π/2+vvs/vzoom+x+π/kernels/grain} vzoom
},
Resampling->rsp, Padding->"Periodic"],
(* 9 Scheibe Geometrie still Front *)
[email protected][pic4, ray02, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{0, 2π-2π/width4},
{si, sr}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+vvs/vzoom+x, -π/2+vvs/vzoom+x+π/kernels/grain} vzoom
},
Resampling->rsp, Padding->"Periodic"],
(* 10 Scheibe Geometrie still Echo 1 *)
[email protected][pic4, ray03, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{0, 2π-2π/width4},
{si, sr}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+vvs/vzoom+x, -π/2+vvs/vzoom+x+π/kernels/grain} vzoom
},
Resampling->rsp, Padding->"Periodic"],
(* 11 Scheibe Geometrie still Echo 2 *)
[email protected][pic4, ray04, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{0, 2π-2π/width4},
{si, sr}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+vvs/vzoom+x, -π/2+vvs/vzoom+x+π/kernels/grain} vzoom
},
Resampling->rsp, Padding->"Periodic"],
(* 12 Scheibe Geometrie still Echo 3 *)
[email protected][pic4, ray05, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{0, 2π-2π/width4},
{si, sr}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+vvs/vzoom+x, -π/2+vvs/vzoom+x+π/kernels/grain} vzoom
},
Resampling->rsp, Padding->"Periodic"],
(* 13 Scheibe Geometrie rotierend komplett *)
[email protected][pic4, ray06, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{0, 2π-2π/width4},
{si, sr}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+vvs/vzoom+x, -π/2+vvs/vzoom+x+π/kernels/grain} vzoom
},
Resampling->rsp, Padding->"Periodic"],
(* 14 Scheibe Geometrie rotierend Front *)
[email protected][pic4, ray07, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{0, 2π-2π/width4},
{si, sr}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+vvs/vzoom+x, -π/2+vvs/vzoom+x+π/kernels/grain} vzoom
},
Resampling->rsp, Padding->"Periodic"],
(* 15 Scheibe Geometrie rotierend Echo 1 *)
[email protected][pic4, ray08, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{0, 2π-2π/width4},
{si, sr}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+vvs/vzoom+x, -π/2+vvs/vzoom+x+π/kernels/grain} vzoom
},
Resampling->rsp, Padding->"Periodic"],
(* 16 Scheibe Geometrie rotierend Echo 2 *)
[email protected][pic4, ray09, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{0, 2π-2π/width4},
{si, sr}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+vvs/vzoom+x, -π/2+vvs/vzoom+x+π/kernels/grain} vzoom
},
Resampling->rsp, Padding->"Periodic"],
(* 17 Scheibe Geometrie rotierend Echo 3 *)
[email protected][pic4, ray10, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{0, 2π-2π/width4},
{si, sr}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+vvs/vzoom+x, -π/2+vvs/vzoom+x+π/kernels/grain} vzoom
},
Resampling->rsp, Padding->"Periodic"],
(* 18 Frequenzverschiebung komplett SW *)
[email protected][pic5, ray11, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{-1, 1},
{fmin, fmax}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+vvs/vzoom+x, -π/2+vvs/vzoom+x+π/kernels/grain} vzoom
},
Resampling->rsp, Padding->"Fixed"],
(* 19 Frequenzverschiebung Front SW *)
[email protected][pic5, ray12, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{0, 2π},
{fmin, fmax}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+vvs/vzoom+x, -π/2+vvs/vzoom+x+π/kernels/grain} vzoom
},
Resampling->rsp, Padding->"Fixed"],
(* 20 Frequenzverschiebung Echo 1 SW *)
[email protected][pic5, ray13, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{0, 2π},
{fmin, fmax}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+vvs/vzoom+x, -π/2+vvs/vzoom+x+π/kernels/grain} vzoom
},
Resampling->rsp, Padding->"Fixed"],
(* 21 Frequenzverschiebung Echo 2 SW *)
[email protected][pic5, ray14, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{0, 2π},
{fmin, fmax}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+vvs/vzoom+x, -π/2+vvs/vzoom+x+π/kernels/grain} vzoom
},
Resampling->rsp, Padding->"Fixed"],
(* 22 Frequenzverschiebung Echo 3 SW *)
[email protected][pic5, ray15, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{0, 2π},
{fmin, fmax}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+vvs/vzoom+x, -π/2+vvs/vzoom+x+π/kernels/grain} vzoom
},
Resampling->rsp, Padding->"Fixed"],
(* 23 Frequenzverschiebung komplett Farbe *)
[email protected][pic6, ray11, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{0, 2π},
{fmin, fmax}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+vvs/vzoom+x, -π/2+vvs/vzoom+x+π/kernels/grain} vzoom
},
Resampling->rsp, Padding->"Fixed"],
(* 24 Frequenzverschiebung Front Farbe *)
[email protected][pic6, ray12, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{0, 2π},
{fmin, fmax}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+vvs/vzoom+x, -π/2+vvs/vzoom+x+π/kernels/grain} vzoom
},
Resampling->rsp, Padding->"Fixed"],
(* 25 Frequenzverschiebung Echo 1 Farbe *)
[email protected][pic6, ray13, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{0, 2π},
{fmin, fmax}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+vvs/vzoom+x, -π/2+vvs/vzoom+x+π/kernels/grain} vzoom
},
Resampling->rsp, Padding->"Fixed"],
(* 26 Frequenzverschiebung Echo 2 Farbe *)
[email protected][pic6, ray14, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{0, 2π},
{fmin, fmax}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+vvs/vzoom+x, -π/2+vvs/vzoom+x+π/kernels/grain} vzoom
},
Resampling->rsp, Padding->"Fixed"],
(* 27 Frequenzverschiebung Echo 3 Farbe *)
[email protected][pic6, ray15, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{0, 2π},
{fmin, fmax}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+vvs/vzoom+x, -π/2+vvs/vzoom+x+π/kernels/grain} vzoom
},
Resampling->rsp, Padding->"Fixed"],
(* 28 Hintergrundpanorama *)
ImageTransformation[pct3, ray17, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{-π, π-2π/width6},
{-π/2, 3π/2}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+x+vvs/vzoom, -π/2+x+vvs/vzoom+π/kernels/grain} vzoom
},
Resampling->rsp, Padding->"Periodic"],
(* 29 Sphäre *)
ImageTransformation[pct3, ray16, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{-π, π-2π/width6},
{-π/2, 3π/2}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+x+vvs/vzoom, -π/2+x+π/kernels/grain+vvs/vzoom} vzoom
},
Resampling->rsp, Padding->"Periodic"],
(* 30 Scheibe Textur komplett *)
ImageTransformation[pic7, ray01, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{0, 2π-2π/width6},
{si, sr+(sr-si)/height6}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+vvs/vzoom+x, -π/2+vvs/vzoom+x+π/kernels/grain} vzoom
},
Resampling->rsp, Padding->"Periodic"],
(* 31 Frequenzverschiebung Hintergrund Farbe *)
[email protected][pic6, ray18, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{0, 2π},
{fmin, fmax}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+vvs/vzoom+x, -π/2+vvs/vzoom+x+π/kernels/grain} vzoom
},
Resampling->rsp, Padding->"Fixed"],
(* 32 Frequenzverschiebung Hintergrund SW *)
[email protected][pic5, ray18, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{0, 2π},
{fmin, fmax}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+vvs/vzoom+x, -π/2+vvs/vzoom+x+π/kernels/grain} vzoom
},
Resampling->rsp, Padding->"Fixed"],
(* 33 Kontur BH *)
[email protected][pic5, ray19, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{0, 2π},
{0, 1}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+vvs/vzoom+x, -π/2+vvs/vzoom+x+π/kernels/grain} vzoom
},
Resampling->rsp, Padding->"Fixed"]
}, {x, 0, π-π/kernels/grain, π/kernels/grain}];
image[num_] := ImageAssemble[{Table[{img[[x, num]]}, {x, kernels grain, 1, -1}]}];
fi[x_] := ColorNegate[x];
(* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
(* 13) Composite |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
(* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
ParallelTable[image[n], {n, 1, 33, 1}]
ds1 = image[8]
ds2 = ImageMultiply[[email protected][09], [email protected][10]];
ds3 = ImageMultiply[[email protected][11], [email protected][12]];
ds4 = [email protected][ds2, ds3]
ds5 = image[13]
ds6 = ImageMultiply[[email protected][14], [email protected][15]];
ds7 = ImageMultiply[[email protected][16], [email protected][17]];
ds8 = [email protected][ds6, ds7]
bf1 = image[31]
bf2 = image[32]
bf3 = image[33]
bl1 = image[23]
bl2 = ImageMultiply[[email protected][24], [email protected][25]];
bl3 = ImageMultiply[[email protected][26], [email protected][27]];
bl4 = [email protected][bl2, bl3]
gr1 = image[18]
gr2 = ImageMultiply[[email protected][19], [email protected][20]];
gr3 = ImageMultiply[[email protected][21], [email protected][22]];
gr4 = [email protected][gr2, gr3]
sea = ImageMultiply[image[04], image[19]];
seb = ImageMultiply[image[05], image[20]];
sec = ImageMultiply[image[06], image[21]];
sed = ImageMultiply[image[07], image[22]];
se1 = image[03]
se2 = ImageMultiply[[email protected][04], [email protected][05]];
se3 = ImageMultiply[[email protected][06], [email protected][07]];
se4 = [email protected][se2, se3]
se5 = ImageMultiply[[email protected], [email protected]];
se6 = ImageMultiply[[email protected], [email protected]];
se7 = [email protected][se5, se6]
se8 = [email protected][[email protected], [email protected]]
se9 = [email protected][[email protected], [email protected]]
bkg = image[1]
hrz = image[2]
cb1 = image[28]
cb2 = image[29]
cb3 = image[30]
Quit[]
(* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
(* 14) Alternative initcon dt/dτ falls Partikel statt Photonen verwendet werden ||||||||| *)
(* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
(* initcon = NSolve[
-μ == -(((r0^2+a^2 Cos[θ0]^2) dr0y^2)/(a^2-2 r0+r0^2+℧^2))+((a^2-2 r0+
r0^2+℧^2-a^2 Sin[θ0]^2) (dt0y)^2)/(r0^2+a^2 Cos[θ0]^2)+(-r0^2-
a^2 Cos[θ0]^2) dθ0y^2+(2 a (2 r0-℧^2) Sin[θ0]^2 dt0y dφ0y)/(r0^2+
a^2 Cos[θ0]^2)+((-(a^2+r0^2)^2 Sin[θ0]^2+a^2 (a^2-2 r0+r0^2+
℧^2) Sin[θ0]^4) dφ0y^2)/(r0^2+a^2 Cos[θ0]^2)
&&
dt0y > 0
&&
vr0i ==
Sqrt[(a^2 Cos[θi]^2+r0^2)/(a^2+℧^2-2 r0+r0^2)] Sqrt[1-μ^2 v0^2] dr0y
&&
vθ0i ==
Sqrt[a^2 Cos[θi]^2+r0^2] Sqrt[1-μ^2 v0^2] dθ0y
&&
vφ0i ==
(Sin[θi]^2 Sqrt[1-μ^2 v0^2] (a q μ^2 ℧ r0 v0^2+a (℧^2-
2 r0) dt0y+((a^2+r0^2)^2-a^2 (a^2+℧^2-
2 r0+r0^2) Sin[θi]^2) dφ0y))/((a^2 Cos[θi]^2+r0^2) Sqrt[(Sin[θi]^2 ((a^2+
r0^2)^2-a^2 (a^2+℧^2-2 r0+r0^2) Sin[θi]^2))/(a^2 Cos[θi]^2+r0^2)]),
{dt0y, dr0y, dθ0y, dφ0y}, Reals]; *)
Wenn der Beobachter nahe oder hinter dem Horizont ist empfehlen sich Raindrop Doran Koordinaten (für die Anwendung siehe hier):
Code: Alles auswählen
(* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
(* > raytracing.yukterez.net | 07.04.2018 - 01.07.2020 | Version 18 | Simon Tyran, Vienna *)
(* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
Pause[1] (* Raindrop Doran Koordinaten *)
wp = MachinePrecision;
mt0 = Automatic;
mt1 = {"StiffnessSwitching", Method-> {"ExplicitRungeKutta", Automatic}};
mta = mt0; (* mt0 for speed, mt1 for accuracy *)
kernels = 6; (* Parallelisierung *)
grain = 5; (* Subparallelisierung auf kernels*grain Streifen *)
rsp = "Nearest"; (* Resampling *)
breite = 60; (* Zielabmessungen in Pixeln *)
hoehe = 30; (* Höhe sollte ein ganzzahliges Vielfaches von kernels*grain sein *)
zoom = 1; (* doppelter Zoom ergibt halben Sichtwinkel *)
LaunchKernels[kernels]
wp = MachinePrecision; (* Genauigkeit *)
st = 0.1; (* Auflösung des Gradienten *)
pic1 = Import["http://www.yukterez.net/mw/2/flip70.png"]; (* Hintergrundpanorama laden *)
pic2 = Import["http://www.yukterez.net/mw/2/worldmap.png"]; (* Sphärenoberfläche laden *)
pic3 = Import["http://www.yukterez.net/mw/akkretionsscheibe.jpg"];(* Scheibentextur laden *)
pic4 = Import["http://www.yukterez.net/mw/disk.png"]; (* Scheibengeometrie laden *)
pic5 = Import["http://www.yukterez.net/mw/gradient1.png"]; (* Helligkeitsgradient laden *)
pic6 = Import["http://www.yukterez.net/mw/gradient2.png"]; (* Farbgradient laden *)
pic7 = Import["http://www.yukterez.net/mw/2/bw.png"]; (* Checkerboard laden *)
(* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
(* 1) Startbedingungen und Position des Beobachters ||||||||||||||||||||||||||||||||||||| *)
(* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
r0 = 10; (* Radialkoordinate des Beobachters *)
R0 = 500; (* Radius des umspannenden Kugelschalenpanoramas *)
R1 = Max[1, Re[1.0001 rA]]; (* Sphäre *)
rT = 10; (* Lichtlaufzeit Synchronisation *)
si = isco+0.1; (* Akkretionsscheibe Innenradius *)
sr = 7; (* Akkretionsscheibe Außenradius *)
θ0 = 70 π/180; (* Breitengrad *)
φ0 = 0; (* Längengrad *)
tmax = -100 R0; (* zeitlicher Integrationsbereich *)
a = 0.7; (* Spinparameter *)
℧ = 0.7; (* spezifische Ladung des schwarzen Lochs *)
v0 = 1; (* Geschwindigkeit der Photonen *)
vr = 0; (* Radiale Geschwindigkeit relativ zum Raindrop *)
vϑ = 0; (* Polare Geschwindigkeit des Beobachters *)
vφ = 0; (* Azimutale Geschwindigkeit des Beobachters: 0 für ZAMO, -й0 für stationär *)
vL = Sqrt[vr^2+vϑ^2+vφ^2];
hvs = 0; (* ArcSin[vφ/vL] *) (* horizontaler Versatz in Radianten *)
vvs = 0; (* ArcSin[vϑ/vL] *) (* vertikaler Versatz in Radianten *)
fmax = 2; fmin = 0; (* Doppler Frequenzbereich *)
(* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
(* 2) Bildreflektion |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
(* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
fpt[{x_, y_}] := {If[y<0, x+1, x], If[y<0, -y, y]}
pcr1 = ParallelTable[
[email protected][pic1, fpt, DataRange->{{-1, 1}, {0, 1}},
PlotRange->{{-1, 1}, {-1+(x-1)/kernels, -1+x/kernels}},
Resampling->rsp, Padding->"Periodic"],
{x, 1, 2 kernels}];
pct1 = ImageAssemble[{Table[{pcr1[[x]]}, {x, 2 kernels, 1, -1}]}];
pcr2 = ParallelTable[
[email protected][pic2, fpt, DataRange->{{-1, 1}, {0, 1}},
PlotRange->{{-1, 1}, {-1+(x-1)/kernels, -1+x/kernels}},
Resampling->rsp, Padding->"Periodic"],
{x, 1, 2 kernels}];
pct2 = ImageAssemble[{Table[{pcr2[[x]]}, {x, 2 kernels, 1, -1}]}];
pcr3 = ParallelTable[
ImageTransformation[pic7, fpt, DataRange->{{-1, 1}, {0, 1}},
PlotRange->{{-1, 1}, {-1+(x-1)/kernels, -1+x/kernels}}, Resampling->rsp, Padding->"Periodic"],
{x, 1, 2 kernels}];
pct3 = ImageAssemble[{Table[{pcr3[[x]]}, {x, 2 kernels, 1, -1}]}];
(* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
(* 3) Funktionen |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
(* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
g11 = gtt = 1+(-4 r0+2 ℧^2)/(a^2+2 r0^2+a^2 Cos[2 θ0]);
g22 = grr = -((a^2+2 r0^2+a^2 Cos[2 θ0])/(2 (a^2+r0^2)));
g33 = gθθ = -r0^2-a^2 Cos[θ0]^2;
g44 = gφφ = (-(a^2+r0^2)^2 Sin[θ0]^2+a^2 (a^2+(-2+r0) r0+℧^2) Sin[θ0]^4)/(r0^2+a^2 Cos[θ0]^2);
g12 = gtr = -(Sqrt[2 r0-℧^2]/Sqrt[a^2+r0^2]);
g14 = gtφ = (a (2 r0-℧^2) Sin[θ0]^2)/(r0^2+a^2 Cos[θ0]^2);
g24 = grφ = (a Sqrt[2 r0-℧^2] Sin[θ0]^2)/Sqrt[a^2+r0^2];
g13 = g23 = g34 = 0;
rA = 1+Sqrt[1-a^2-℧^2];
rE = 1+Sqrt[1-℧^2];
q = 0;
Σ = r0^2+a^2 Cos[θ0]^2;
Δ = r0^2-2 r0+a^2+℧^2;
Χ = (r0^2+a^2)^2-a^2 Sin[θ0]^2 Δ;
Σs[rs_] := rs^2;
Δs[rs_] := rs^2-2 rs+a^2+℧^2;
Χs[rs_] := (rs^2+a^2)^2-a^2 Δs[rs];
κs[rs_] := a;
Σj[rt_, θt_] := rt^2+a^2 Cos[θt]^2;
Δj[rt_, θt_] := rt^2-2 rt+a^2+℧^2;
Χj[rt_, θt_] := (rt^2+a^2)^2-a^2 Sin[θt]^2 Δj[rt, θt];
ωj[rt_, θt_] := (a(2 rt-℧^2))/Χj[rt, θt];
ς = [email protected][Χ/Δ/Σ]; ςr[rs_] := [email protected][Χs[rs]/Δs[rs]/Σs[rs]];
μ = If[Abs[[email protected]]==1.0, 0, -1];
j[v_] := Sqrt[1-v^2];
Ы[rs_] := Sqrt[Χs[rs]/Σs[rs]];
ωs[rs_] := (a (2 rs - ℧^2))/Χs[rs];
ε[rs_, vt_] := Sqrt[Δs[rs] Σs[rs]/Χs[rs]]/j[vt]+Lz[rs, vt] ωs[rs];
Lz[rs_, vt_] := vt Ы[rs]/j[vt];
nq[x_] := If[NumericQ[x], If[Element[x, Reals], x, 0], 0];
dΘF = Quiet[If[vLF==0, 0, ArcCos[-vθF/Sqrt[vrF^2+vθF^2+vφF^2]]]];
dθF = Quiet[If[vLF==0, 0, If[vrF>0, -dΘF, +dΘF]]];
dΦF = Quiet[If[vLF==0, 0, ArcTan[Abs[vφF]/vrF]]];
dφF = Quiet[If[vLF==0, 0, If[vrF!=0, -1, 1] If[vφF<0, +1, -1] If[NumericQ[dΦF], dΦF, π/2]]];
(* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
(* 4) Geschwindigkeitskomponenten ||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
(* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
εj[rt_, δt_, δr_, δθ_, δφ_] := (δt-δr Sqrt[(2rt-℧^2)/(a^2+rt^2)]/(1-(2rt-℧^2)/(a^2+
rt^2))) (1-(2 rt-℧^2)/rt^2)+(a (δφ-δr a Sqrt[(2rt-℧^2)/(a^2+rt^2)]/(1-(2rt-℧^2)/(a^2+
rt^2))/(a^2+rt^2)) (2 rt-℧^2))/rt^2;
vrj[rt_, δt_, δr_, δθ_, δφ_] := δr/Sqrt[Δj[rt, θt]] Sqrt[Σj[rt, π/2]];
vθj[rt_, δt_, δr_, δθ_, δφ_] := δθ Sqrt[Σj[rt, π/2]];
vφj[rt_, δt_, δr_, δθ_, δφ_] := (-(((a^2 Cos[(π/2)]^2+rt^2) (a^2+℧^2-2 rt+
rt^2) Sin[(π/2)] (-(δφ-δr a Sqrt[(2rt-℧^2)/(a^2+rt^2)]/(1-(2rt-℧^2)/(a^2+rt^2))/(a^2+rt^2))-
(a q ℧ rt)/((a^2 Cos[(π/2)]^2+rt^2) (a^2+℧^2-2 rt+rt^2))+
(εj[rt, δt, δr, δθ, δφ] Csc[(π/2)]^2 (a (-a^2-℧^2+2 rt-rt^2) Sin[(π/2)]^2+a (a^2+
rt^2) Sin[(π/2)]^2))/((a^2 Cos[(π/2)]^2+rt^2) (a^2+℧^2-2 rt+rt^2))+(a q ℧ rt (a^2+
℧^2-2 rt+rt^2-a^2 Sin[(π/2)]^2))/((a^2 Cos[(π/2)]^2+rt^2)^2 (a^2+℧^2-2 rt+
rt^2))))/((a^2+℧^2-2 rt+rt^2-a^2 Sin[(π/2)]^2) Sqrt[((a^2+rt^2)^2-
a^2 (a^2+℧^2-2 rt+rt^2) Sin[(π/2)]^2)/(a^2 Cos[(π/2)]^2+rt^2)])));
vtj[rt_, δt_, δr_, δθ_, δφ_] := Sqrt[vrj[rt, δt, δr, δθ, δφ]^2+
vθj[rt, δt, δr, δθ, δφ]^2+vφj[rt, δt, δr, δθ, δφ]^2];
vφt[rt_, δt_, δr_, δθ_, δφ_] := vφj[rt, δt, δr, δθ, δφ]/vtj[rt, δt, δr, δθ, δφ];
shf[rt_, δt_, δr_, δθ_, δφ_] := Abs[ς/ςr[rt]/(1-vs[rt] vφt[rt, δt, δr, δθ, δφ])];
dφv[rt_, tt_] := (tt+r0) φs[rt]/ts[rt];
vsolve = Block[{r=r0, θ=θ0, μ=-1, q=0},
ini = NSolve[
vr ==
(Sqrt[2] Sqrt[1-μ^2 vL^2] ((q μ^2 ℧ r Sqrt[-℧^2+2 r] vL^2)/((a^2+℧^2+
(-2+r) r) Sqrt[a^2+r^2])+((a^2+a^2 Cos[2 θ]+2 r^2) dR)/(2 (a^2+r^2))+(Sqrt[-℧^2+2 r] (dT-
a Sin[θ]^2 dΦ))/Sqrt[a^2+r^2]))/Sqrt[(a^2+a^2 Cos[2 θ]+2 r^2)/(a^2+r^2)]
&&
vϑ ==
Sqrt[a^2 Cos[θ]^2+r^2] Sqrt[1-μ^2 vL^2] dΘ
&&
vφ ==
(Sin[θ]^2 Sqrt[1-μ^2 vL^2] (a q μ^2 ℧ r Sqrt[a^2+r^2] vL^2-1/2 a Sqrt[-℧^2+2 r] (a^2+
a^2 Cos[2 θ]+2 r^2) dR+Sqrt[a^2+r^2] (a (℧^2-2 r) dT+((a^2+r^2)^2-a^2 (a^2+℧^2-2 r+
r^2) Sin[θ]^2) dΦ)))/(Sqrt[a^2+r^2] (a^2 Cos[θ]^2+r^2) Sqrt[(Sin[θ]^2 ((a^2+r^2)^2-
a^2 (a^2+℧^2-2 r+r^2) Sin[θ]^2))/(a^2 Cos[θ]^2+r^2)])
&&
-μ ==
-(((a^2+2 r^2+
a^2 Cos[2 θ]) (dR)^2)/(2 (a^2+r^2)))-(2 Sqrt[2 r-℧^2] dR dT)/Sqrt[a^2+r^2]+(1+(-4 r+
2 ℧^2)/(a^2+2 r^2+a^2 Cos[2 θ])) (dT)^2+(-r^2-a^2 Cos[θ]^2) dΘ^2+(2 a Sqrt[2 r-
℧^2] Sin[θ]^2 dR dΦ)/Sqrt[a^2+r^2]+(2 a (2 r-℧^2) Sin[θ]^2 dT dΦ)/(r^2+a^2 Cos[θ]^2)+
((-(a^2+r^2)^2 Sin[θ]^2+a^2 (a^2+(-2+r) r+℧^2) Sin[θ]^4) (dΦ)^2)/(r^2+a^2 Cos[θ]^2)
&&
dT > 0,
{dT, dR, dΘ, dΦ},
Reals];
β = -Sqrt[(2r-℧^2)/(a^2+r^2)];
drf = (dR/.ini[[1]]);
dθf = (dΘ/.ini[[1]]);
dφf = (dΦ/.ini[[1]])+drf a β/(1-β^2)/(a^2+r^2);
NSolve[
vrf ==
Sqrt[(a^2 Cos[θ]^2+r^2)/(a^2+℧^2-2 r+r^2)] Sqrt[1-μ^2 v0f^2] drf
&&
vϑf == Sqrt[a^2 Cos[θ]^2+r^2] Sqrt[1-μ^2 v0f^2] dθf
&&
vφf == (Sin[θ]^2 Sqrt[1-μ^2 v0f^2] (a q μ^2 ℧ r v0f^2+a (℧^2-2 r) dtf+((a^2+r^2)^2-
a^2 (a^2+℧^2-2 r+r^2) Sin[θ]^2) dφf))/((a^2 Cos[θ]^2+r^2) Sqrt[(Sin[θ]^2 ((a^2+
r^2)^2-a^2 (a^2+℧^2-2 r+r^2) Sin[θ]^2))/(a^2 Cos[θ]^2+r^2)])
&&
v0f == Sqrt[vrf^2+vϑf^2+vφf^2]
&&
-μ == -(((r^2+a^2 Cos[θ]^2) drf^2)/(a^2-2 r+r^2+℧^2))+((a^2-2 r+r^2+℧^2-
a^2 Sin[θ]^2) (dtf)^2)/(r^2+a^2 Cos[θ]^2)+(-r^2-a^2 Cos[θ]^2) dθf^2+(2 a (2 r-
℧^2) Sin[θ]^2 dtf dφf)/(r^2+a^2 Cos[θ]^2)+((-(a^2+r^2)^2 Sin[θ]^2+a^2 (a^2-2 r+
r^2+℧^2) Sin[θ]^4) dφf^2)/(r^2+a^2 Cos[θ]^2)
&&
dtf>0,
{dtf, vrf, vϑf, vφf, v0f},
Reals]];
vrF = vrf/.vsolve[[1]]; (* Geschwindigkeit relativ zu den Fixsternen *)
vθF = vϑf/.vsolve[[1]];
vφF = vφf/.vsolve[[1]];
vLF = Sqrt[vrF^2+vθF^2+vφF^2];
vθ =-vϑ;
θs = π/2;
θi =-θ0+π;
(* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
(* 5) Photonensphäre und ISCO ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
(* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
rp = ц/.Solve[4 a^2 (ц-℧^2)-(ц^2-3 ц+2 ℧^2)^2==0 && ц>=If[Element[rA, Reals], rA, 0], ц];
rP = 1.01 Min[rp]; Rp = 1.01 Max[rp];
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]]];
{"r horizon" -> [email protected], "r ergosphere" -> [email protected], "r isco" -> [email protected],
"r photon pro" -> [email protected][rp], "r photon ret" -> [email protected][rp], "r disk" -> [email protected],
"r observer" -> [email protected], "θ observer" -> [email protected]θ0 180/π}
(* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
(* 6) Geschwindigkeit und Zeitdilatation auf der Scheibe |||||||||||||||||||||||||||||||| *)
(* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
red[rs_] := Quiet[Reduce[
dt == (Lz[rs, vt] (-a (a^2+rs^2)+Δs[rs] κs[rs])+ε[rs, vt] ((a^2+rs^2)^2-
Δs[rs] κs[rs]^2))/(Δs[rs] Σs[rs])
&&
0 == ((a^2+(-2+rs) rs+℧^2) (16 a dt dΦ rs (rs-℧^2)+8 dt^2 rs (-rs+℧^2)+
dΦ^2 rs (8 rs (-a^2+rs^3)+a^2 (4 a^2+4 ℧^2-4 (a-℧) (a+℧)))))/(8 rs^6)
&&
dΦ == (Lz[rs, vt] (-a^2+Δs[rs])+ε[rs, vt] (a (a^2+rs^2)-Δs[rs] κs[rs]))/(Δs[rs] Σs[rs])
&&
vt > 0,
{vt, dΦ, dt},
Reals]];
vs = Interpolation[ParallelTable[{rr, If[[email protected][red[rr][[1, 2]]],
red[rr][[1, 2]], 0]}, {rr, 0, sr+st, st}]];
φs = Interpolation[ParallelTable[{rr, If[[email protected][red[rr][[2, 2]]],
red[rr][[2, 2]], 0]}, {rr, 0, sr+st, st}]];
ts = Interpolation[ParallelTable[{rr, If[[email protected][red[rr][[3, 2]]],
red[rr][[3, 2]], 0]}, {rr, 0, sr+st, st}]];
"Accretion disk:"
plot[func_, label_] := Plot[func, {rr, rP, sr},
GridLines -> {{nq[Min[rp]], nq[Max[rp]], nq[rA], nq[si], nq[isco], nq[rE], nq[sr]}, {}},
Frame -> True, ImagePadding -> {{40, 8}, {12, 8}}, ImageSize -> 340,
PlotLabel -> label, PlotRange->{{0, sr}, Automatic}]
Grid[{{
plot[Sqrt[Χs[rr]/Δs[rr]/Σs[rr]], "Gravitational time dilation: y=dt/dт, x=r"],
plot[φs[rr]/ts[rr], "Shapirodelayed angular velocity: y=dφ/dt, x=r"]},{
plot[ts[rr], "Total time dilation: y=dt/dτ, x=r"],
plot[φs[rr], "Coordinate speed: y=dφ/dτ, x=r"]}, {
plot[(a (2 rr-℧^2))/((a^2+rr^2)^2-a^2 (a^2-2 rr+rr^2+℧^2)), "Frame Dragging: y=dφ/dт, x=r"],
plot[vs[rr], "Local velocity: y=v=dl/dτ, x=r"]}}]
(* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
(* 7) Frame Dragging und Gammafaktor |||||||||||||||||||||||||||||||||||||||||||||||||||| *)
(* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
й0 = (a (2 r0-℧^2) Sin[θ0] Sqrt[((a^2+r0^2)^2-a^2 (a^2-2 r0+r0^2+℧^2) Sin[θ0]^2)/((a^2-
2 r0+r0^2+℧^2) (r0^2+a^2 Cos[θ0]^2))])/((r0^2+a^2 Cos[θ0]^2) Sqrt[((a^2+r0^2)^2-a^2 (a^2-
2 r0+r0^2+℧^2) Sin[θ0]^2)/(r0^2+a^2 Cos[θ0]^2)]);
vя = Sqrt[((a^2+r0^2)(℧^2-2r0))/(a^2 Sin[θ0]^2(a^2+(r0-2)r0+℧^2)-(a^2+r0^2)^2)];
vR = Sqrt[(2 r0-℧^2)/(a^2+r0^2)];
U = {+vr, +vθ, +vφ};
γ = 1/Sqrt[1-Norm[U]^2];
(* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
(* 8) Rotationsmatrix für die auf der Sichtebene eintreffenden Strahlen ||||||||||||||||| *)
(* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
Xyz[{x_, y_, z_}, α_] := {x Cos[α]-y Sin[α], x Sin[α]+y Cos[α], z};
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]};
(* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
(* 9) Raytracing Funktionscontainer ||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
(* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
raytracer[{Ф_, ϑ_}] :=
Quiet[Module[{DGL, sol, εj, pθi, pr0, Q, k, V, W, vw,
vr0i, vθ0i, vφ0i, vr0n, vθ0n, vφ0n, vr0a, vθia, vφ0a, vt0a,
t10, r10, Θ10, Φ10, t, r, θ, φ, τ,
т, т0, т1, т2, т3, т4, т5,
plunge, plunge0, plunge1, plunge2, plunge3,
plunge4, plunge5, plunge6, plunge7,
dθ0, dφ0, δφ0, δθ0, δr0, δt0, tt0, rt0, θt0, φt0,
dθ1, dφ1, δφ1, δθ1, δr1, δt1, tt1, rt1, θt1, φt1,
dθ2, dφ2, δφ2, δθ2, δr2, δt2, tt2, rt2, θt2, φt2,
dθ3, dφ3, δφ3, δθ3, δr3, δt3, tt3, rt3, θt3, φt3,
dθ4, dφ4, δφ4, δθ4, δr4, δt4, tt4, rt4, θt4, φt4,
dθ5, dφ5, δφ5, δθ5, δr5, δt5, tt5, rt5, θt5, φt5,
dθ6, dφ6, δφ6, δθ6, δr6, δt6, tt6, rt6, θt6, φt6,
X, Y, Z, ξ, stepsize, laststep, mtl, ft, fτ, varb,
ft0s, ft1s, ft2s, ft3s, ft4s,
ft0v, ft1v, ft2v, ft3v, ft4v,
ft0f, ft1f, ft2f, ft3f, ft4f,
ft5h, ft5b, ft5f, ft5x,
dt0y, dr0y, dθ0y, dφ0y,
initcon, shF},
vw=xyZ[Xyz[{0, 1, 0}, ϑ], Ф+π/2];
(* Übersetzung des Einfallswinkels in den lokalen Tetrad *)
vr0a = vw[[3]];
vφ0a = vw[[2]];
vθia = vw[[1]];
(* Betrag *)
vt0a = Sqrt[vr0a^2+vφ0a^2+vθia^2];
(* Normierung *)
vr0n = vr0a/vt0a;
vφ0n = vφ0a/vt0a;
vθ0n = vθia/vt0a;
(* Relativistische Geschwindigkeitsaddition *)
V={vr0n, vθ0n, vφ0n};
W=(U+V+γ/(1+γ)(U\[Cross](U\[Cross]V)))/(1+U.V);
(* Aberration *)
vr0i = W[[1]];
vθ0i = W[[2]];
vφ0i = W[[3]];
shF = Abs[If[vLF==0, 1, Sqrt[1-vLF^2]/(1+vLF (-Cos[If[vrF>0, -dΘF, +dΘF]] Sin[ϑ]-
Sin[If[vrF>0, -dΘF, +dΘF]] (Cos[-Ф-π/2+If[vrF>0, -1,
1] ArcSin[vφF/vLF]] Cos[ϑ] Cos[1/2 π If[vφF<0,
1, -1] If[vrF!=0,-1,1]]-Cos[ϑ] Sin[-Ф-π/2+
If[vrF>0, -1, +1]ArcSin[vφF/vLF]] Sin[1/2 π])))]];
initcon = TimeConstrained[NSolve[
vr0i ==
N[(Sqrt[2] Sqrt[1-μ^2 v0^2] ((q μ^2 ℧ r0 Sqrt[-℧^2+2 r0] v0^2)/((a^2+℧^2+(-2+
r0) r0) Sqrt[a^2+r0^2])+((a^2+a^2 Cos[2 θi]+2 r0^2) dr0y)/(2 (a^2+r0^2))+(Sqrt[-℧^2+
2 r0] (dt0y-a Sin[θi]^2 dφ0y))/Sqrt[a^2+r0^2]))/Sqrt[(a^2+a^2 Cos[2 θi]+2 r0^2)/(a^2+r0^2)]]
&&
vθ0i ==
N[Sqrt[a^2 Cos[θi]^2+r0^2] Sqrt[1-μ^2 v0^2] dθ0y]
&&
vφ0i ==
N[(Sin[θi]^2 Sqrt[1-μ^2 v0^2] (a q μ^2 ℧ r0 Sqrt[a^2+r0^2] v0^2-1/2 a Sqrt[-℧^2+2 r0] (a^2+
a^2 Cos[2 θi]+2 r0^2) dr0y+Sqrt[a^2+r0^2] (a (℧^2-2 r0) dt0y+((a^2+r0^2)^2-a^2 (a^2+℧^2-
2 r0+r0^2) Sin[θi]^2) dφ0y)))/(Sqrt[a^2+r0^2] (a^2 Cos[θi]^2+r0^2) Sqrt[(Sin[θi]^2 ((a^2+
r0^2)^2-a^2 (a^2+℧^2-2 r0+r0^2) Sin[θi]^2))/(a^2 Cos[θi]^2+r0^2)])]
&&
-μ ==
N[-(((a^2+2 r0^2+a^2 Cos[2 θi]) (dr0y)^2)/(2 (a^2+r0^2)))-(2 Sqrt[2 r0-
℧^2] dr0y dt0y)/Sqrt[a^2+r0^2]+(1+(-4 r0+2 ℧^2)/(a^2+2 r0^2+a^2 Cos[2 θi])) (dt0y)^2+
(-r0^2-a^2 Cos[θi]^2) dθ0y^2+(2 a Sqrt[2 r0-℧^2] Sin[θi]^2 dr0y dφ0y)/Sqrt[a^2+
r0^2]+(2 a (2 r0-℧^2) Sin[θi]^2 dt0y dφ0y)/(r0^2+a^2 Cos[θi]^2)+((-(a^2+r0^2)^2 Sin[θi]^2+
a^2 (a^2+(-2+r0) r0+℧^2) Sin[θi]^4) (dφ0y)^2)/(r0^2+a^2 Cos[θi]^2)]
&&
dt0y > 0,
{dθ0y, dr0y, dt0y, dφ0y}, Reals], 3];
DGL = If[NumericQ[dt0y /. initcon[[1]]] == False, {}, { (* Bewegungsgleichungen *)
t''[τ] == Re[1/(8 (a^2 Cos[θ[τ]]^2+r[τ]^2)^3) (8 q ℧ (-a^4 Cos[θ[τ]]^4+r[τ]^4) r'[τ]+
(8 (a^2 Cos[θ[τ]]^2+℧^2 r[τ]-r[τ]^2) (a^2 Cos[θ[τ]]^2+r[τ]^2)^2 r'[τ]^2)/(Sqrt[-℧^2+
2 r[τ]] Sqrt[a^2+r[τ]^2])+8 q ℧ Sqrt[-℧^2+2 r[τ]] Sqrt[a^2+r[τ]^2] (-a^2 Cos[θ[τ]]^2+
r[τ]^2) t'[τ]+16 (a^2 Cos[θ[τ]]^2+℧^2 r[τ]-r[τ]^2) (a^2 Cos[θ[τ]]^2+r[τ]^2) r'[τ] t'[τ]+
8 Sqrt[-℧^2+2 r[τ]] (a^2 Cos[θ[τ]]^2+℧^2 r[τ]-r[τ]^2) Sqrt[a^2+r[τ]^2] t'[τ]^2-
8 a^2 q ℧ r[τ] (a^2 Cos[θ[τ]]^2+r[τ]^2) Sin[2 θ[τ]] θ'[τ]+(8 a^2 Sqrt[-℧^2+
2 r[τ]] (a^2 Cos[θ[τ]]^2+r[τ]^2)^2 Sin[2 θ[τ]] r'[τ] θ'[τ])/Sqrt[a^2+r[τ]^2]-8 a^2 (℧^2-
2 r[τ]) (a^2 Cos[θ[τ]]^2+r[τ]^2) Sin[2 θ[τ]] t'[τ] θ'[τ]+8 r[τ] Sqrt[-℧^2+
2 r[τ]] Sqrt[a^2+r[τ]^2] (a^2 Cos[θ[τ]]^2+r[τ]^2)^2 θ'[τ]^2-8 a q ℧ Sqrt[-℧^2+
2 r[τ]] Sqrt[a^2+r[τ]^2] (-a^2 Cos[θ[τ]]^2+r[τ]^2) Sin[θ[τ]]^2 φ'[τ]-
16 a (a^2 Cos[θ[τ]]^2+(℧^2-r[τ]) r[τ]) (a^2 Cos[θ[τ]]^2+r[τ]^2) Sin[θ[τ]]^2 r'[τ] φ'[τ]-
16 a Sqrt[-℧^2+2 r[τ]] (a^2 Cos[θ[τ]]^2+℧^2 r[τ]-r[τ]^2) Sqrt[a^2+
r[τ]^2] Sin[θ[τ]]^2 t'[τ] φ'[τ]+16 a^3 Cos[θ[τ]] (℧^2-2 r[τ]) (a^2 Cos[θ[τ]]^2+
r[τ]^2) Sin[θ[τ]]^3 θ'[τ] φ'[τ]+Sqrt[-℧^2+2 r[τ]] Sqrt[a^2+r[τ]^2] (a^4+
a^4 Cos[4 θ[τ]] (-1+r[τ])+3 a^4 r[τ]+4 a^2 ℧^2 r[τ]-4 a^2 r[τ]^2+8 a^2 r[τ]^3+
8 r[τ]^5+4 a^2 Cos[2 θ[τ]] r[τ] (a^2-℧^2+r[τ]+2 r[τ]^2)) Sin[θ[τ]]^2 φ'[τ]^2)],
t'[0] == dt0y/.initcon[[1]],
t[0] == 0,
r''[τ] == Re[1/(8 (a^2 Cos[θ[τ]]^2+r[τ]^2)^3) (-8 q ℧ Sqrt[-℧^2+2 r[τ]] Sqrt[a^2+
r[τ]^2] (-a^2 Cos[θ[τ]]^2+r[τ]^2) r'[τ]+(8 a^2 q ℧ Sqrt[-℧^2+2 r[τ]] (-a^2 Cos[θ[τ]]^2+
r[τ]^2) Sin[θ[τ]]^2 r'[τ])/Sqrt[a^2+r[τ]^2]+(4 (a^2 Cos[θ[τ]]^2+
r[τ]^2)^2 (a^2 Cos[2 θ[τ]] (-1+r[τ])-a^2 (1+r[τ])+2 r[τ] (-℧^2+r[τ])) r'[τ]^2)/(a^2+
r[τ]^2)-4 q ℧ (2 a^2 Cos[θ[τ]]^2-2 r[τ]^2) (a^2+r[τ]^2) (1+(℧^2-
2 r[τ])/(a^2 Cos[θ[τ]]^2+r[τ]^2)) t'[τ]+(8 a^2 q ℧ (℧^2-2 r[τ]) (a^2 Cos[θ[τ]]^2-
r[τ]^2) Sin[θ[τ]]^2 t'[τ])/(a^2 Cos[θ[τ]]^2+r[τ]^2)-(16 Sqrt[-℧^2+
2 r[τ]] (a^2 Cos[θ[τ]]^2+℧^2 r[τ]-r[τ]^2) (a^2 Cos[θ[τ]]^2+
r[τ]^2) r'[τ] t'[τ])/Sqrt[a^2+r[τ]^2]+8 (a^2 Cos[θ[τ]]^2+℧^2 r[τ]-r[τ]^2) (a^2+
℧^2-2 r[τ]+r[τ]^2) t'[τ]^2+8 a^2 (a^2 Cos[θ[τ]]^2+r[τ]^2)^2 Sin[2 θ[τ]] r'[τ] θ'[τ]+
8 r[τ] (a^2 Cos[θ[τ]]^2+r[τ]^2)^2 (a^2+℧^2-2 r[τ]+r[τ]^2) θ'[τ]^2+(8 a q ℧ (℧^2-
2 r[τ]) (a^2 Cos[θ[τ]]^2-r[τ]^2) (a^2+r[τ]^2) Sin[θ[τ]]^2 φ'[τ])/(a^2 Cos[θ[τ]]^2+
r[τ]^2)-(4 a q ℧ (2 a^2 Cos[θ[τ]]^2-2 r[τ]^2) (-(a^2+r[τ]^2)^2 Sin[θ[τ]]^2+a^2 (a^2+
℧^2+(-2+r[τ]) r[τ]) Sin[θ[τ]]^4) φ'[τ])/(a^2 Cos[θ[τ]]^2+r[τ]^2)-(8 a Sqrt[-℧^2+
2 r[τ]] (a^2 Cos[θ[τ]]^2+r[τ]^2) (a^2 (-1+r[τ])+a^2 Cos[2 θ[τ]] (-1+r[τ])+
2 r[τ] (-℧^2+r[τ]+r[τ]^2)) Sin[θ[τ]]^2 r'[τ] φ'[τ])/Sqrt[a^2+r[τ]^2]-
16 a (a^2 Cos[θ[τ]]^2+(℧^2-r[τ]) r[τ]) (a^2+℧^2-2 r[τ]+
r[τ]^2) Sin[θ[τ]]^2 t'[τ] φ'[τ]+(a^2+℧^2-2 r[τ]+r[τ]^2) (a^4+a^4 Cos[4 θ[τ]] (-1+
r[τ])+3 a^4 r[τ]+4 a^2 ℧^2 r[τ]-4 a^2 r[τ]^2+8 a^2 r[τ]^3+8 r[τ]^5+
4 a^2 Cos[2 θ[τ]] r[τ] (a^2-℧^2+r[τ]+2 r[τ]^2)) Sin[θ[τ]]^2 φ'[τ]^2)],
r'[0] == dr0y/.initcon[[1]],
r[0] == r0,
θ''[τ] == Re[-1/(2 (a^2 Cos[θ[τ]]^2+r[τ]^2)^4) ((2 a^2 Cos[θ[τ]] (a^2 Cos[θ[τ]]^2+
r[τ]^2)^3 Sin[θ[τ]] r'[τ]^2)/(a^2+r[τ]^2)-2 a^2 q ℧ (℧^2-2 r[τ]) r[τ] Sin[2 θ[τ]] t'[τ]+
a^2 q ℧ r[τ] (a^2+2 ℧^2+a^2 Cos[2 θ[τ]]-4 r[τ]+2 r[τ]^2) Sin[2 θ[τ]] t'[τ]+
2 a^2 Cos[θ[τ]] (℧^2-2 r[τ]) (a^2 Cos[θ[τ]]^2+r[τ]^2) Sin[θ[τ]] t'[τ]^2+
4 r[τ] (a^2 Cos[θ[τ]]^2+r[τ]^2)^3 r'[τ] θ'[τ]-2 a^2 Cos[θ[τ]] (a^2 Cos[θ[τ]]^2+
r[τ]^2)^3 Sin[θ[τ]] θ'[τ]^2-4 a^3 q ℧ Cos[θ[τ]] (℧^2-2 r[τ]) r[τ] Sin[θ[τ]]^3 φ'[τ]+
4 a q ℧ Cot[θ[τ]] r[τ] (-(a^2+r[τ]^2)^2 Sin[θ[τ]]^2+a^2 (a^2+℧^2+(-2+
r[τ]) r[τ]) Sin[θ[τ]]^4) φ'[τ]+(2 a Sqrt[-℧^2+2 r[τ]] (a^2 Cos[θ[τ]]^2+
r[τ]^2)^3 Sin[2 θ[τ]] r'[τ] φ'[τ])/Sqrt[a^2+r[τ]^2]-2 a (℧^2-2 r[τ]) (a^2+
r[τ]^2) (a^2 Cos[θ[τ]]^2+r[τ]^2) Sin[2 θ[τ]] t'[τ] φ'[τ]+(a^2 Cos[θ[τ]]^2+
r[τ]^2) (2 a^2 Cos[θ[τ]] Sin[θ[τ]]^3 (-(a^2+r[τ]^2)^2+a^2 (a^2+℧^2+(-2+
r[τ]) r[τ]) Sin[θ[τ]]^2)+(a^2 Cos[θ[τ]]^2+r[τ]^2) (4 a^2 Cos[θ[τ]] (a^2+℧^2+
(-2+r[τ]) r[τ]) Sin[θ[τ]]^3-(a^2+r[τ]^2)^2 Sin[2 θ[τ]])) φ'[τ]^2)],
θ'[0] == dθ0y/.initcon[[1]],
θ[0] == θi,
φ''[τ] == Re[1/(4 (a^2 Cos[θ[τ]]^2+r[τ]^2)^3) ((4 a q ℧ (-a^4 Cos[θ[τ]]^4+r[τ]^4) r'[τ])/(a^2+
r[τ]^2)+(4 a (a^2 Cos[θ[τ]]^2+℧^2 r[τ]-r[τ]^2) (a^2 Cos[θ[τ]]^2+
r[τ]^2)^2 r'[τ]^2)/(Sqrt[-℧^2+2 r[τ]] (a^2+r[τ]^2)^(3/2))+(4 a q ℧ Sqrt[-℧^2+
2 r[τ]] (-a^2 Cos[θ[τ]]^2+r[τ]^2) t'[τ])/Sqrt[a^2+r[τ]^2]+(8 a (a^2 Cos[θ[τ]]^2+(℧^2-
r[τ]) r[τ]) (a^2 Cos[θ[τ]]^2+r[τ]^2) r'[τ] t'[τ])/(a^2+r[τ]^2)+(4 a Sqrt[-℧^2+
2 r[τ]] (a^2 Cos[θ[τ]]^2+℧^2 r[τ]-r[τ]^2) t'[τ]^2)/Sqrt[a^2+r[τ]^2]-
8 a q ℧ Cot[θ[τ]] r[τ] (a^2 Cos[θ[τ]]^2+r[τ]^2) θ'[τ]+(8 a Cot[θ[τ]] Sqrt[-℧^2+
2 r[τ]] (a^2 Cos[θ[τ]]^2+r[τ]^2)^2 r'[τ] θ'[τ])/Sqrt[a^2+r[τ]^2]-8 a Cot[θ[τ]] (℧^2-
2 r[τ]) (a^2 Cos[θ[τ]]^2+r[τ]^2) t'[τ] θ'[τ]+(4 a r[τ] Sqrt[-℧^2+2 r[τ]] (a^2 Cos[θ[τ]]^2+
r[τ]^2)^2 θ'[τ]^2)/Sqrt[a^2+r[τ]^2]-(4 a^2 q ℧ Sqrt[-℧^2+2 r[τ]] (-a^2 Cos[θ[τ]]^2+
r[τ]^2) Sin[θ[τ]]^2 φ'[τ])/Sqrt[a^2+r[τ]^2]+(8 (a^2 Cos[θ[τ]]^2+
r[τ]^2) (a^4 Cos[θ[τ]]^2 (-1+r[τ])-r[τ] (a^2 (a^2+℧^2-r[τ])+2 a^2 Cot[θ[τ]]^2 (a^2+
r[τ]^2)+Csc[θ[τ]]^2 (-a^4+r[τ]^4))) Sin[θ[τ]]^2 r'[τ] φ'[τ])/(a^2+r[τ]^2)-
(8 a^2 Sqrt[-℧^2+2 r[τ]] (a^2 Cos[θ[τ]]^2+℧^2 r[τ]-
r[τ]^2) Sin[θ[τ]]^2 t'[τ] φ'[τ])/Sqrt[a^2+r[τ]^2]-Cot[θ[τ]] (a^2 Cos[θ[τ]]^2+
r[τ]^2) (a^2 (3 a^2-4 ℧^2+4 (a^2+℧^2) Cos[2 θ[τ]]+a^2 Cos[4 θ[τ]])+
16 a^2 Cos[θ[τ]]^2 r[τ]^2+8 r[τ]^4+16 a^2 r[τ] Sin[θ[τ]]^2) θ'[τ] φ'[τ]+
(4 a Sqrt[-℧^2+2 r[τ]] Sin[θ[τ]]^2 (r[τ] (-a^4+r[τ]^4+a^2 (a^2+℧^2-r[τ]) Sin[θ[τ]]^2)+
Cos[θ[τ]]^2 (2 a^2 r[τ] (a^2+r[τ]^2)-a^4 (-1+r[τ]) Sin[θ[τ]]^2)) φ'[τ]^2)/Sqrt[a^2+r[τ]^2])],
φ'[0] == dφ0y/.initcon[[1]],
φ[0] == φ0,
WhenEvent[Abs[r[τ]] == N[R0]||Abs[r[τ]] > R0 &&
NumericQ[plunge6] == False,
(plunge6=τ) &&
(tt6=t[τ]) && (rt6=r[τ]) && (θt6=θ[τ]) && (φt6=φ[τ]);
"StopIntegration"],
WhenEvent[Abs[r[τ]] == N[R1] &&
NumericQ[plunge5] == False,
(plunge5=τ) && (tt5=t[τ]) && (rt5=r[τ]) && (θt5=θ[τ]) && (φt5=φ[τ]);
If[r0>R1, "StopIntegration"]],
WhenEvent[Mod[180/π Re[θ[τ]], 180] == 90.0 && r[τ]>N[si] &&
r[τ]<N[sr] && τ < -0.05 &&
NumericQ[plunge0] == False,
(plunge0=τ) &&
(tt0=t[τ]) && (rt0=r[τ]) && (θt0=θ[τ]) && (φt0=φ[τ]) &&
(δt0=t'[τ]) && (δr0=r'[τ]) && (δθ0=θ'[τ]) && (δφ0=φ'[τ])],
WhenEvent[Mod[180/π Re[θ[τ]], 180] == 90.0 && r[τ]>N[si] &&
r[τ]<N[sr] && Re[θ'[τ]]>0 && τ < -0.05 &&
NumericQ[plunge1] == False,
(plunge1=τ) &&
(tt1=t[τ]) && (rt1=r[τ]) && (θt1=θ[τ]) && (φt1=φ[τ]) &&
(δt1=t'[τ]) && (δr1=r'[τ]) && (δθ1=θ'[τ]) && (δφ1=φ'[τ])],
WhenEvent[Mod[180/π Re[θ[τ]], 180] == 90.0 && r[τ]>N[si] &&
r[τ]<N[sr] && Re[θ'[τ]]<0 && τ < -0.05 &&
NumericQ[plunge2] == False,
(plunge2=τ) &&
(tt2=t[τ]) && (rt2=r[τ]) && (θt2=θ[τ]) && (φt2=φ[τ]) &&
(δt2=t'[τ]) && (δr2=r'[τ]) && (δθ2=θ'[τ]) && (δφ2=φ'[τ])],
WhenEvent[Mod[180/π Re[θ[τ]], 180] == 90.0 && r[τ]>N[si] &&
r[τ]<N[sr] && Re[θ'[τ]]>0 &&
τ < plunge1-0.05 &&
NumericQ[plunge3]==False && NumericQ[plunge1] == True,
(plunge3=τ) &&
(tt3=t[τ]) && (rt3=r[τ]) && (θt3=θ[τ]) && (φt3=φ[τ]) &&
(δt3=t'[τ]) && (δr3=r'[τ]) && (δθ3=θ'[τ]) && (δφ3=φ'[τ])],
WhenEvent[Mod[180/π Re[θ[τ]], 180] == 90.0 && r[τ]>N[si] &&
r[τ]<N[sr] && Re[θ'[τ]]<0 &&
τ < plunge2-0.05 &&
NumericQ[plunge4] == False && NumericQ[plunge2] == True,
(plunge4=τ) &&
(tt4=t[τ]) && (rt4=r[τ]) && (θt4=θ[τ]) && (φt4=φ[τ]) &&
(δt4=t'[τ]) && (δr4=r'[τ]) && (δθ4=θ'[τ]) && (δφ4=φ'[τ])]
}];
(* Integrator *)
sol = If[NumericQ[dt0y /. initcon[[1]]] == False, {},
NDSolve[DGL, {t, r, θ, φ}, {τ, 0, tmax},
WorkingPrecision-> wp,
Method-> mta,
MaxSteps-> Infinity,
InterpolationOrder-> All,
StepMonitor :> (laststep=plunge7; plunge7=τ;
stepsize=plunge7-laststep;), Method->{"EventLocator",
"Event" :> (If[stepsize<1*^-4, 0, 1])}
]];
ft5h=If[NumericQ[plunge5], {φt5-π, θt5+π/2}, {0, -π/2}];
ft5b=If[NumericQ[plunge6], {φt6-π, θt6+π/2}, {0, -π/2}];
ft5f={0, Min[fmax, ς shF]};
ft5x=If[NumericQ[plunge6], {0, 1}, {0, 0}];
ft0s=If[NumericQ[plunge0], {φt0, rt0}, {π, sr}];
ft1s=If[NumericQ[plunge1], {φt1, rt1}, {π, sr}];
ft2s=If[NumericQ[plunge2], {φt2, rt2}, {π, sr}];
ft3s=If[NumericQ[plunge3], {φt3, rt3}, {π, sr}];
ft4s=If[NumericQ[plunge4], {φt4, rt4}, {π, sr}];
ft0v=If[NumericQ[plunge0], {-dφv[rt0, tt0], 0}, {0, 0}];
ft1v=If[NumericQ[plunge1], {-dφv[rt1, tt1], 0}, {0, 0}];
ft2v=If[NumericQ[plunge2], {-dφv[rt2, tt2], 0}, {0, 0}];
ft3v=If[NumericQ[plunge3], {-dφv[rt3, tt3], 0}, {0, 0}];
ft4v=If[NumericQ[plunge4], {-dφv[rt4, tt4], 0}, {0, 0}];
ft0f=If[NumericQ[plunge0], {0, Min[fmax, shF shf[rt0, δt0, δr0, δθ0, δφ0]]}, {0, 0}];
ft1f=If[NumericQ[plunge1], {0, Min[fmax, shF shf[rt1, δt1, δr1, δθ1, δφ1]]}, {0, 0}];
ft2f=If[NumericQ[plunge2], {0, Min[fmax, shF shf[rt2, δt2, δr2, δθ2, δφ2]]}, {0, 0}];
ft3f=If[NumericQ[plunge3], {0, Min[fmax, shF shf[rt3, δt3, δr3, δθ3, δφ3]]}, {0, 0}];
ft4f=If[NumericQ[plunge4], {0, Min[fmax, shF shf[rt4, δt4, δr4, δθ4, δφ4]]}, {0, 0}];
{
ft0s, ft1s, ft2s, ft3s, ft4s,
ft0s+ft0v, ft1s+ft1v, ft2s+ft2v, ft3s+ft3v, ft4s+ft4v,
ft0f, ft1f, ft2f, ft3f, ft4f,
ft5h, ft5b, ft5f, ft5x
}]];
(* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
(* 10) Memoryfunktion ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
(* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
mem : raytrace[{Ф_, ϑ_}] := mem = Quiet[Re[raytracer[{Ф, ϑ}]]];
ray01[{Ф_, ϑ_}] := raytrace[{Ф, ϑ}][[01]];
ray02[{Ф_, ϑ_}] := raytrace[{Ф, ϑ}][[02]];
ray03[{Ф_, ϑ_}] := raytrace[{Ф, ϑ}][[03]];
ray04[{Ф_, ϑ_}] := raytrace[{Ф, ϑ}][[04]];
ray05[{Ф_, ϑ_}] := raytrace[{Ф, ϑ}][[05]];
ray06[{Ф_, ϑ_}] := raytrace[{Ф, ϑ}][[06]];
ray07[{Ф_, ϑ_}] := raytrace[{Ф, ϑ}][[07]];
ray08[{Ф_, ϑ_}] := raytrace[{Ф, ϑ}][[08]];
ray09[{Ф_, ϑ_}] := raytrace[{Ф, ϑ}][[09]];
ray10[{Ф_, ϑ_}] := raytrace[{Ф, ϑ}][[10]];
ray11[{Ф_, ϑ_}] := raytrace[{Ф, ϑ}][[11]];
ray12[{Ф_, ϑ_}] := raytrace[{Ф, ϑ}][[12]];
ray13[{Ф_, ϑ_}] := raytrace[{Ф, ϑ}][[13]];
ray14[{Ф_, ϑ_}] := raytrace[{Ф, ϑ}][[14]];
ray15[{Ф_, ϑ_}] := raytrace[{Ф, ϑ}][[15]];
ray16[{Ф_, ϑ_}] := raytrace[{Ф, ϑ}][[16]];
ray17[{Ф_, ϑ_}] := raytrace[{Ф, ϑ}][[17]];
ray18[{Ф_, ϑ_}] := raytrace[{Ф, ϑ}][[18]];
ray19[{Ф_, ϑ_}] := raytrace[{Ф, ϑ}][[19]];
(* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
(* 11) Proportionen ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
(* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
width1 = ImageDimensions[pic1][[1]]; height1 = ImageDimensions[pic1][[2]];
width2 = ImageDimensions[pic2][[1]]; height2 = ImageDimensions[pic2][[2]];
width3 = ImageDimensions[pic3][[1]]; height3 = ImageDimensions[pic3][[2]];
width4 = ImageDimensions[pic4][[1]]; height4 = ImageDimensions[pic4][[2]];
width5 = ImageDimensions[pic5][[1]]; height5 = ImageDimensions[pic5][[2]];
width6 = ImageDimensions[pic7][[1]]; height6 = ImageDimensions[pic7][[2]];
hzoom = If[breite>2 hoehe, 1/zoom, 1/zoom/2/hoehe*breite];
vzoom = If[breite>2 hoehe, 1/zoom*2 hoehe/breite, 1/zoom];
"approximate time remaining" -> 1.2 (AbsoluteTiming[Do[raytracer[{
RandomReal[{-π, π}]/zoom, RandomReal[{-π/2, π/2}]/zoom
}], {ü, 1, 50}]][[1]])/50*hoehe*breite/kernels/60 "minutes"
FOV->{360.0 hzoom "degree", 180.0 vzoom "degree"}
(* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
(* 12) Output ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
(* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
img = ParallelTable[{
(* 1 Hintergrundpanorama *)
[email protected][pct1, ray17, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{-π, π-2π/width1},
{-π/2, 3π/2}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+x+vvs/vzoom, -π/2+x+vvs/vzoom+π/kernels/grain} vzoom
},
Resampling->rsp, Padding->"Periodic"],
(* 2 Sphäre *)
[email protected][pct2, ray16, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{0, 2π-2π/width2},
{-π/2, 3π/2}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+x+vvs/vzoom, -π/2+x+π/kernels/grain+vvs/vzoom} vzoom
},
Resampling->rsp, Padding->"Periodic"],
(* 3 Scheibe Textur komplett *)
[email protected][pic3, ray01, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{0, 2π-2π/width3},
{si, sr+(sr-si)/height3}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+vvs/vzoom+x, -π/2+vvs/vzoom+x+π/kernels/grain} vzoom
},
Resampling->rsp, Padding->"Periodic"],
(* 4 Scheibe Textur Front *)
[email protected][pic3, ray02, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{0, 2π-2π/width3},
{si, sr+(sr-si)/height3}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+vvs/vzoom+x, -π/2+vvs/vzoom+x+π/kernels/grain} vzoom
},
Resampling->rsp, Padding->"Periodic"],
(* 5 Scheibe Textur Echo 1 *)
[email protected][pic3, ray03, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{0, 2π-2π/width3},
{si, sr+(sr-si)/height3}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+vvs/vzoom+x, -π/2+vvs/vzoom+x+π/kernels/grain} vzoom
},
Resampling->rsp, Padding->"Periodic"],
(* 6 Scheibe Textur Echo 2 *)
[email protected][pic3, ray04, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{0, 2π-2π/width3},
{si, sr+(sr-si)/height3}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+vvs/vzoom+x, -π/2+vvs/vzoom+x+π/kernels/grain} vzoom
},
Resampling->rsp, Padding->"Periodic"],
(* 7 Scheibe Textur Echo 3 *)
[email protected][pic3, ray05, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{0, 2π-2π/width3},
{si, sr+(sr-si)/height3}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+vvs/vzoom+x, -π/2+vvs/vzoom+x+π/kernels/grain} vzoom
},
Resampling->rsp, Padding->"Periodic"],
(* 8 Scheibe Geometrie still komplett *)
[email protected][pic4, ray01, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{0, 2π-2π/width4},
{si, sr}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+vvs/vzoom+x, -π/2+vvs/vzoom+x+π/kernels/grain} vzoom
},
Resampling->rsp, Padding->"Periodic"],
(* 9 Scheibe Geometrie still Front *)
[email protected][pic4, ray02, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{0, 2π-2π/width4},
{si, sr}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+vvs/vzoom+x, -π/2+vvs/vzoom+x+π/kernels/grain} vzoom
},
Resampling->rsp, Padding->"Periodic"],
(* 10 Scheibe Geometrie still Echo 1 *)
[email protected][pic4, ray03, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{0, 2π-2π/width4},
{si, sr}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+vvs/vzoom+x, -π/2+vvs/vzoom+x+π/kernels/grain} vzoom
},
Resampling->rsp, Padding->"Periodic"],
(* 11 Scheibe Geometrie still Echo 2 *)
[email protected][pic4, ray04, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{0, 2π-2π/width4},
{si, sr}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+vvs/vzoom+x, -π/2+vvs/vzoom+x+π/kernels/grain} vzoom
},
Resampling->rsp, Padding->"Periodic"],
(* 12 Scheibe Geometrie still Echo 3 *)
[email protected][pic4, ray05, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{0, 2π-2π/width4},
{si, sr}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+vvs/vzoom+x, -π/2+vvs/vzoom+x+π/kernels/grain} vzoom
},
Resampling->rsp, Padding->"Periodic"],
(* 13 Scheibe Geometrie rotierend komplett *)
[email protected][pic4, ray06, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{0, 2π-2π/width4},
{si, sr}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+vvs/vzoom+x, -π/2+vvs/vzoom+x+π/kernels/grain} vzoom
},
Resampling->rsp, Padding->"Periodic"],
(* 14 Scheibe Geometrie rotierend Front *)
[email protected][pic4, ray07, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{0, 2π-2π/width4},
{si, sr}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+vvs/vzoom+x, -π/2+vvs/vzoom+x+π/kernels/grain} vzoom
},
Resampling->rsp, Padding->"Periodic"],
(* 15 Scheibe Geometrie rotierend Echo 1 *)
[email protected][pic4, ray08, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{0, 2π-2π/width4},
{si, sr}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+vvs/vzoom+x, -π/2+vvs/vzoom+x+π/kernels/grain} vzoom
},
Resampling->rsp, Padding->"Periodic"],
(* 16 Scheibe Geometrie rotierend Echo 2 *)
[email protected][pic4, ray09, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{0, 2π-2π/width4},
{si, sr}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+vvs/vzoom+x, -π/2+vvs/vzoom+x+π/kernels/grain} vzoom
},
Resampling->rsp, Padding->"Periodic"],
(* 17 Scheibe Geometrie rotierend Echo 3 *)
[email protected][pic4, ray10, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{0, 2π-2π/width4},
{si, sr}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+vvs/vzoom+x, -π/2+vvs/vzoom+x+π/kernels/grain} vzoom
},
Resampling->rsp, Padding->"Periodic"],
(* 18 Frequenzverschiebung komplett SW *)
[email protected][pic5, ray11, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{-1, 1},
{fmin, fmax}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+vvs/vzoom+x, -π/2+vvs/vzoom+x+π/kernels/grain} vzoom
},
Resampling->rsp, Padding->"Fixed"],
(* 19 Frequenzverschiebung Front SW *)
[email protected][pic5, ray12, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{0, 2π},
{fmin, fmax}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+vvs/vzoom+x, -π/2+vvs/vzoom+x+π/kernels/grain} vzoom
},
Resampling->rsp, Padding->"Fixed"],
(* 20 Frequenzverschiebung Echo 1 SW *)
[email protected][pic5, ray13, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{0, 2π},
{fmin, fmax}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+vvs/vzoom+x, -π/2+vvs/vzoom+x+π/kernels/grain} vzoom
},
Resampling->rsp, Padding->"Fixed"],
(* 21 Frequenzverschiebung Echo 2 SW *)
[email protected][pic5, ray14, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{0, 2π},
{fmin, fmax}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+vvs/vzoom+x, -π/2+vvs/vzoom+x+π/kernels/grain} vzoom
},
Resampling->rsp, Padding->"Fixed"],
(* 22 Frequenzverschiebung Echo 3 SW *)
[email protected][pic5, ray15, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{0, 2π},
{fmin, fmax}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+vvs/vzoom+x, -π/2+vvs/vzoom+x+π/kernels/grain} vzoom
},
Resampling->rsp, Padding->"Fixed"],
(* 23 Frequenzverschiebung komplett Farbe *)
[email protected][pic6, ray11, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{0, 2π},
{fmin, fmax}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+vvs/vzoom+x, -π/2+vvs/vzoom+x+π/kernels/grain} vzoom
},
Resampling->rsp, Padding->"Fixed"],
(* 24 Frequenzverschiebung Front Farbe *)
[email protected][pic6, ray12, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{0, 2π},
{fmin, fmax}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+vvs/vzoom+x, -π/2+vvs/vzoom+x+π/kernels/grain} vzoom
},
Resampling->rsp, Padding->"Fixed"],
(* 25 Frequenzverschiebung Echo 1 Farbe *)
[email protected][pic6, ray13, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{0, 2π},
{fmin, fmax}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+vvs/vzoom+x, -π/2+vvs/vzoom+x+π/kernels/grain} vzoom
},
Resampling->rsp, Padding->"Fixed"],
(* 26 Frequenzverschiebung Echo 2 Farbe *)
[email protected][pic6, ray14, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{0, 2π},
{fmin, fmax}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+vvs/vzoom+x, -π/2+vvs/vzoom+x+π/kernels/grain} vzoom
},
Resampling->rsp, Padding->"Fixed"],
(* 27 Frequenzverschiebung Echo 3 Farbe *)
[email protected][pic6, ray15, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{0, 2π},
{fmin, fmax}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+vvs/vzoom+x, -π/2+vvs/vzoom+x+π/kernels/grain} vzoom
},
Resampling->rsp, Padding->"Fixed"],
(* 28 Hintergrundpanorama *)
ImageTransformation[pct3, ray17, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{-π, π-2π/width6},
{-π/2, 3π/2}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+x+vvs/vzoom, -π/2+x+vvs/vzoom+π/kernels/grain} vzoom
},
Resampling->rsp, Padding->"Periodic"],
(* 29 Sphäre *)
ImageTransformation[pct3, ray16, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{-π, π-2π/width6},
{-π/2, 3π/2}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+x+vvs/vzoom, -π/2+x+π/kernels/grain+vvs/vzoom} vzoom
},
Resampling->rsp, Padding->"Periodic"],
(* 30 Scheibe Textur komplett *)
ImageTransformation[pic7, ray01, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{0, 2π-2π/width6},
{si, sr+(sr-si)/height6}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+vvs/vzoom+x, -π/2+vvs/vzoom+x+π/kernels/grain} vzoom
},
Resampling->rsp, Padding->"Periodic"],
(* 31 Frequenzverschiebung Hintergrund Farbe *)
[email protected][pic6, ray18, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{0, 2π},
{fmin, fmax}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+vvs/vzoom+x, -π/2+vvs/vzoom+x+π/kernels/grain} vzoom
},
Resampling->rsp, Padding->"Fixed"],
(* 32 Frequenzverschiebung Hintergrund SW *)
[email protected][pic5, ray18, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{0, 2π},
{fmin, fmax}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+vvs/vzoom+x, -π/2+vvs/vzoom+x+π/kernels/grain} vzoom
},
Resampling->rsp, Padding->"Fixed"],
(* 33 Kontur BH *)
[email protected][pic5, ray19, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{0, 2π},
{0, 1}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+vvs/vzoom+x, -π/2+vvs/vzoom+x+π/kernels/grain} vzoom
},
Resampling->rsp, Padding->"Fixed"]
}, {x, 0, π-π/kernels/grain, π/kernels/grain}];
image[num_] := ImageAssemble[{Table[{img[[x, num]]}, {x, kernels grain, 1, -1}]}];
fi[x_] := ColorNegate[x];
(* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
(* 13) Composite |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
(* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
ParallelTable[image[n], {n, 1, 33, 1}]
ds1 = image[8]
ds2 = ImageMultiply[[email protected][09], [email protected][10]];
ds3 = ImageMultiply[[email protected][11], [email protected][12]];
ds4 = [email protected][ds2, ds3]
ds5 = image[13]
ds6 = ImageMultiply[[email protected][14], [email protected][15]];
ds7 = ImageMultiply[[email protected][16], [email protected][17]];
ds8 = [email protected][ds6, ds7]
bf1 = image[31]
bf2 = image[32]
bf3 = image[33]
bl1 = image[23]
bl2 = ImageMultiply[[email protected][24], [email protected][25]];
bl3 = ImageMultiply[[email protected][26], [email protected][27]];
bl4 = [email protected][bl2, bl3]
gr1 = image[18]
gr2 = ImageMultiply[[email protected][19], [email protected][20]];
gr3 = ImageMultiply[[email protected][21], [email protected][22]];
gr4 = [email protected][gr2, gr3]
sea = ImageMultiply[image[04], image[19]];
seb = ImageMultiply[image[05], image[20]];
sec = ImageMultiply[image[06], image[21]];
sed = ImageMultiply[image[07], image[22]];
se1 = image[03]
se2 = ImageMultiply[[email protected][04], [email protected][05]];
se3 = ImageMultiply[[email protected][06], [email protected][07]];
se4 = [email protected][se2, se3]
se5 = ImageMultiply[[email protected], [email protected]];
se6 = ImageMultiply[[email protected], [email protected]];
se7 = [email protected][se5, se6]
se8 = [email protected][[email protected], [email protected]]
se9 = [email protected][[email protected], [email protected]]
bkg = image[1]
hrz = image[2]
cb1 = image[28]
cb2 = image[29]
cb3 = image[30]
Quit[]
Für den Umrechner der Lokalgeschwindigkeit vom einen System ins andere klick hier. Beispiel mit a=℧=0.3, θ=85°, r=50, ri=isco=4.826, ra=10:
Minkowski Raytracer:
Code: Alles auswählen
(* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
(* > raytracing.yukterez.net | 07.04.2018 - 01.07.2020 | Version 18 | Simon Tyran, Vienna *)
(* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
Pause[1] (* Minkowski Koordinaten *)
wp = MachinePrecision;
mt0 = Automatic;
mt1 = {"StiffnessSwitching", Method-> {"ExplicitRungeKutta", Automatic}};
mta = mt0; (* mt0 for speed, mt1 for accuracy *)
kernels = 6; (* Parallelisierung *)
grain = 5; (* Subparallelisierung auf kernels*grain Streifen *)
rsp = "Nearest"; (* Resampling *)
breite = 60; (* Zielabmessungen in Pixeln *)
hoehe = 30; (* Höhe sollte ein ganzzahliges Vielfaches von kernels*grain sein *)
zoom = 1; (* doppelter Zoom ergibt halben Sichtwinkel *)
LaunchKernels[kernels]
wp = MachinePrecision; (* Genauigkeit *)
pic1 = Import["http://www.yukterez.net/mw/2/flip70.png"]; (* Hintergrundpanorama laden *)
pic2 = Import["http://www.yukterez.net/mw/2/worldmap.png"]; (* Sphärenoberfläche laden *)
pic3 = Import["http://www.yukterez.net/mw/akkretionsscheibe.jpg"];(* Scheibentextur laden *)
pic4 = Import["http://www.yukterez.net/mw/disk.png"]; (* Scheibengeometrie laden *)
pic7 = Import["http://www.yukterez.net/mw/2/bw.png"]; (* Checkerboard laden *)
(* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
(* 1) Startbedingungen und Position des Beobachters ||||||||||||||||||||||||||||||||||||| *)
(* |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| *)
r0 = 10; (* Radialkoordinate des Beobachters *)
R0 = 500; (* Radius des umspannenden Kugelschalenpanoramas *)
R1 = 1+1/(5 Sqrt[2]); (* Sphäre *)
si = 1.6367824; (* Akkretionsscheibe Innenradius *)
sr = 7; (* Akkretionsscheibe Außenradius *)
θ0 = 70 π/180; (* Breitengrad *)
φ0 = 0; (* Längengrad *)
tmax =-3 R0; (* zeitlicher Integrationsbereich *)
vr = 0; (* Radiale Geschwindigkeit des Beobachters *)
vϑ = 0; (* Polare Geschwindigkeit des Beobachters *)
vφ = 0; (* Azimutale Geschwindigkeit des Beobachters *)
vL = Sqrt[vr^2+vϑ^2+vφ^2];
hvs = 0; (* ArcSin[vφ/vL] *) (* horizontaler Versatz in Radianten *)
vvs = 0; (* ArcSin[vϑ/vL] *) (* vertikaler Versatz in Radianten *)
fpt[{x_, y_}] := {If[y<0, x+1, x], If[y<0, -y, y]}
pcr1 = ParallelTable[
ImageTransformation[pic1, fpt, DataRange->{{-1, 1}, {0, 1}},
PlotRange->{{-1, 1}, {-1+(x-1)/kernels, -1+x/kernels}}, Padding->"Periodic"],
{x, 1, 2 kernels}];
pct1 = ImageAssemble[{Table[{pcr1[[x]]}, {x, 2 kernels, 1, -1}]}];
pcr2 = ParallelTable[
ImageTransformation[pic2, fpt, DataRange->{{-1, 1}, {0, 1}},
PlotRange->{{-1, 1}, {-1+(x-1)/kernels, -1+x/kernels}}, Padding->"Periodic"],
{x, 1, 2 kernels}];
pct2 = ImageAssemble[{Table[{pcr2[[x]]}, {x, 2 kernels, 1, -1}]}];
pcr3 = ParallelTable[
ImageTransformation[pic7, fpt, DataRange->{{-1, 1}, {0, 1}},
PlotRange->{{-1, 1}, {-1+(x-1)/kernels, -1+x/kernels}}, Resampling->rsp, Padding->"Periodic"],
{x, 1, 2 kernels}];
pct3 = ImageAssemble[{Table[{pcr3[[x]]}, {x, 2 kernels, 1, -1}]}];
a = 0; ℧ = 0; v0 = 1; q = 0; st = 0.1; vϑ = 0;
vθ =-vϑ;
θs = π/2;
θi =-θ0+π;
θ1 = θi;
dφv[rt_, tt_] := 0;
shf[rt_, δt_, δr_, δθ_, δφ_] := 1;
ωφ = -vφ Csc[θ1]/r0;
ωθ = -vϑ/r0;
gtt = -1;
grr = +1;
gθθ = +r0^2;
gφφ = +r0^2 Sin[θ1]^2;
gtφ = +0;
ς = 1;
j[v_] := Sqrt[1-v^2];
nq[x_] := If[NumericQ[x], If[Element[x, Reals], x, 0], 0];
U = {+vr, +vθ, +vφ};
γ = 1/Sqrt[1-Norm[U]^2];
Xyz[{x_, y_, z_}, α_] := {x Cos[α]-y Sin[α], x Sin[α]+y Cos[α], z};
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]};
raytracer[{Ф_, ϑ_}] :=
Quiet[Module[{DGL, sol, εj, pθi, pr0, Q, k, V, W, vw,
vr0i, vθ0i, vφ0i, vr0n, vθ0n, vφ0n, vr0a, vθia, vφ0a, vt0a,
t10, r10, Θ10, Φ10, t, r, θ, φ, τ,
т, т0, т1, т2, т3, т4, т5,
plunge, plunge0, plunge1, plunge2, plunge3,
plunge4, plunge5, plunge6, plunge7,
dθ0, dφ0, δφ0, δθ0, δr0, δt0, tt0, rt0, θt0, φt0,
dθ1, dφ1, δφ1, δθ1, δr1, δt1, tt1, rt1, θt1, φt1,
dθ2, dφ2, δφ2, δθ2, δr2, δt2, tt2, rt2, θt2, φt2,
dθ3, dφ3, δφ3, δθ3, δr3, δt3, tt3, rt3, θt3, φt3,
dθ4, dφ4, δφ4, δθ4, δr4, δt4, tt4, rt4, θt4, φt4,
dθ5, dφ5, δφ5, δθ5, δr5, δt5, tt5, rt5, θt5, φt5,
dθ6, dφ6, δφ6, δθ6, δr6, δt6, tt6, rt6, θt6, φt6,
X, Y, Z, ξ, stepsize, laststep, mtl, ft, fτ, varb,
ft0s, ft1s, ft2s, ft3s, ft4s,
ft0v, ft1v, ft2v, ft3v, ft4v,
ft0f, ft1f, ft2f, ft3f, ft4f,
ft5h, ft5b, ft5f},
vw=xyZ[Xyz[{0, 1, 0}, ϑ], Ф+π/2];
(* Übersetzung des Einfallswinkels in den lokalen Tetrad *)
vr0a = vw[[3]];
vφ0a = vw[[2]];
vθia = vw[[1]];
(* Betrag *)
vt0a = Sqrt[vr0a^2+vφ0a^2+vθia^2];
(* Normierung *)
vr0n = vr0a/vt0a;
vφ0n = vφ0a/vt0a;
vθ0n = vθia/vt0a;
(* Relativistische Geschwindigkeitsaddition *)
V={vr0n, vθ0n, vφ0n};
W=(U+V+γ/(1+γ)(U\[Cross](U\[Cross]V)))/(1+U.V);
(* Aberration *)
vr0i = W[[1]];
vθ0i = W[[2]];
vφ0i = W[[3]];
DGL = { (* Kerr Newman Bewegungsgleichungen *)
t''[τ] == 0,
t'[0] == 1,
t[0] == 0,
r''[τ] == r[τ](θ'[τ]^2+Sin[θ[τ]]^2 φ'[τ]^2),
r'[0] == vr0i,
r[0] == r0,
θ''[τ] == Sin[θ[τ]] Cos[θ[τ]] φ'[τ]^2-2 θ'[τ] r'[τ]/r[τ],
θ'[0] == vθ0i/r0,
θ[0] == θi,
φ''[τ] == -2 φ'[τ] (r'[τ]+r[τ] θ'[τ] Cot[θ[τ]])/r[τ],
φ'[0] == vφ0i Csc[θ1]/r0,
φ[0] == φ0,
WhenEvent[Mod[θ[τ], π] == π/2.0 && r[τ]>si && r[τ]<sr &&
NumericQ[plunge0] == False,
(plunge0=τ) &&
(tt0=t[τ]) && (rt0=r[τ]) && (θt0=θ[τ]) && (φt0=φ[τ]) &&
(δt0=t'[τ]) && (δr0=r'[τ]) && (δθ0=θ'[τ]) && (δφ0=φ'[τ])],
WhenEvent[Mod[θ[τ], π] == π/2.0 && r[τ]>si && r[τ]<sr && Re[θ'[τ]]>0 &&
NumericQ[plunge1] == False,
(plunge1=τ) &&
(tt1=t[τ]) && (rt1=r[τ]) && (θt1=θ[τ]) && (φt1=φ[τ]) &&
(δt1=t'[τ]) && (δr1=r'[τ]) && (δθ1=θ'[τ]) && (δφ1=φ'[τ])],
WhenEvent[Mod[θ[τ], π] == π/2.0 && r[τ]>si && r[τ]<sr && Re[θ'[τ]]<0 &&
NumericQ[plunge2] == False,
(plunge2=τ) &&
(tt2=t[τ]) && (rt2=r[τ]) && (θt2=θ[τ]) && (φt2=φ[τ]) &&
(δt2=t'[τ]) && (δr2=r'[τ]) && (δθ2=θ'[τ]) && (δφ2=φ'[τ])],
WhenEvent[Mod[θ[τ], π] == π/2.0 && r[τ]>si && r[τ]<sr && Re[θ'[τ]]>0 &&
τ < plunge1-0.05 &&
NumericQ[plunge3]==False && NumericQ[plunge1] == True,
(plunge3=τ) &&
(tt3=t[τ]) && (rt3=r[τ]) && (θt3=θ[τ]) && (φt3=φ[τ]) &&
(δt3=t'[τ]) && (δr3=r'[τ]) && (δθ3=θ'[τ]) && (δφ3=φ'[τ])],
WhenEvent[Mod[θ[τ], π] == π/2.0 && r[τ]>si && r[τ]<sr && Re[θ'[τ]]<0 &&
τ < plunge2-0.05 &&
NumericQ[plunge4] == False && NumericQ[plunge2] == True,
(plunge4=τ) &&
(tt4=t[τ]) && (rt4=r[τ]) && (θt4=θ[τ]) && (φt4=φ[τ]) &&
(δt4=t'[τ]) && (δr4=r'[τ]) && (δθ4=θ'[τ]) && (δφ4=φ'[τ])],
WhenEvent[Abs[r[τ]] == [email protected] &&
NumericQ[plunge5] == False,
(plunge5=τ) && (tt5=t[τ]) && (rt5=r[τ]) && (θt5=θ[τ]) && (φt5=φ[τ]);
If[r0>R1, "StopIntegration"]],
WhenEvent[Abs[r[τ]] == R0||r[τ]>R0 &&
NumericQ[plunge6] == False,
(plunge6=τ) &&
(tt6=t[τ]) && (rt6=r[τ]) && (θt6=θ[τ]) && (φt6=φ[τ]);
"StopIntegration"]
};
(* Integrator *)
sol = NDSolve[DGL, {t, r, θ, φ}, {τ, 0, tmax},
WorkingPrecision-> wp,
Method-> mta,
MaxSteps-> Infinity,
StepMonitor :> (laststep=plunge7; plunge7=τ;
stepsize=plunge7-laststep;), Method->{"EventLocator",
"Event" :> (If[stepsize<1*^-4, 0, 1])}
];
ft0s=If[NumericQ[plunge0], If[rt0>sr, {π, sr},
If[rt0<si, {π, sr}, {φt0+(tt0+r0) ωφ, rt0}]], {π, sr}];
ft1s=If[NumericQ[plunge1], If[rt1>sr, {π, sr},
If[rt1<si, {π, sr}, {φt1+(tt1+r0) ωφ, rt1}]], {π, sr}];
ft2s=If[NumericQ[plunge2], If[rt2>sr, {π, sr},
If[rt2<si, {π, sr}, {φt2+(tt2+r0) ωφ, rt2}]], {π, sr}];
ft3s=If[NumericQ[plunge3], If[rt3>sr, {π, sr},
If[rt3<si, {π, sr}, {φt3+(tt3+r0) ωφ, rt3}]], {π, sr}];
ft4s=If[NumericQ[plunge4], If[rt4>sr, {π, sr},
If[rt4<si, {π, sr}, {φt4+(tt4+r0) ωφ, rt4}]], {π, sr}];
ft0v=If[NumericQ[plunge0], If[rt0>sr, {0, 0},
If[rt0<si, {0, 0}, {-dφv[rt0, tt0], 0}]], {0, 0}];
ft1v=If[NumericQ[plunge1], If[rt1>sr, {0, 0},
If[rt1<si, {0, 0}, {-dφv[rt1, tt1], 0}]], {0, 0}];
ft2v=If[NumericQ[plunge2], If[rt2>sr, {0, 0},
If[rt2<si, {0, 0}, {-dφv[rt2, tt2], 0}]], {0, 0}];
ft3v=If[NumericQ[plunge3], If[rt3>sr, {0, 0},
If[rt3<si, {0, 0}, {-dφv[rt3, tt3], 0}]], {0, 0}];
ft4v=If[NumericQ[plunge4], If[rt4>sr, {0, 0},
If[rt4<si, {0, 0}, {-dφv[rt4, tt4], 0}]], {0, 0}];
ft0f=If[NumericQ[plunge0], If[rt0>sr, {0, 0},
If[rt0<si, {0, 0}, {0, Min[2, shf[rt0, δt0, δr0, δθ0, δφ0]]}]], {0, 0}];
ft1f=If[NumericQ[plunge1], If[rt1>sr, {0, 0},
If[rt1<si, {0, 0}, {0, Min[2, shf[rt1, δt1, δr1, δθ1, δφ1]]}]], {0, 0}];
ft2f=If[NumericQ[plunge2], If[rt2>sr, {0, 0},
If[rt2<si, {0, 0}, {0, Min[2, shf[rt2, δt2, δr2, δθ2, δφ2]]}]], {0, 0}];
ft3f=If[NumericQ[plunge3], If[rt3>sr, {0, 0},
If[rt3<si, {0, 0}, {0, Min[2, shf[rt3, δt3, δr3, δθ3, δφ3]]}]], {0, 0}];
ft4f=If[NumericQ[plunge4], If[rt4>sr, {0, 0},
If[rt4<si, {0, 0}, {0, Min[2, shf[rt4, δt4, δr4, δθ4, δφ4]]}]], {0, 0}];
ft5h=If[NumericQ[plunge5], {φt5+(tt5+r0) ωφ, θt5+π/2+(tt5+r0) ωθ}, {0, -π/2}];
ft5b=If[NumericQ[plunge6], If[rt6<If[Element[R1, Reals], R1, 0], {0, -π/2},
If[rt6>4 R0, {0, -π/2}, {φt6-π, θt6+π/2}]], {0, -π/2}];
{
ft0s, ft1s, ft2s, ft3s, ft4s,
ft0s+ft0v, ft1s+ft1v, ft2s+ft2v, ft3s+ft3v, ft4s+ft4v,
ft0f, ft1f, ft2f, ft3f, ft4f,
ft5h, ft5b
}]];
mem : raytrace[{Ф_, ϑ_}] := mem = Quiet[Re[raytracer[{Ф, ϑ}]]];
ray01[{Ф_, ϑ_}] := raytrace[{Ф, ϑ}][[01]];
ray02[{Ф_, ϑ_}] := raytrace[{Ф, ϑ}][[02]];
ray03[{Ф_, ϑ_}] := raytrace[{Ф, ϑ}][[03]];
ray04[{Ф_, ϑ_}] := raytrace[{Ф, ϑ}][[04]];
ray05[{Ф_, ϑ_}] := raytrace[{Ф, ϑ}][[05]];
ray06[{Ф_, ϑ_}] := raytrace[{Ф, ϑ}][[06]];
ray07[{Ф_, ϑ_}] := raytrace[{Ф, ϑ}][[07]];
ray08[{Ф_, ϑ_}] := raytrace[{Ф, ϑ}][[08]];
ray09[{Ф_, ϑ_}] := raytrace[{Ф, ϑ}][[09]];
ray10[{Ф_, ϑ_}] := raytrace[{Ф, ϑ}][[10]];
ray11[{Ф_, ϑ_}] := raytrace[{Ф, ϑ}][[11]];
ray12[{Ф_, ϑ_}] := raytrace[{Ф, ϑ}][[12]];
ray13[{Ф_, ϑ_}] := raytrace[{Ф, ϑ}][[13]];
ray14[{Ф_, ϑ_}] := raytrace[{Ф, ϑ}][[14]];
ray15[{Ф_, ϑ_}] := raytrace[{Ф, ϑ}][[15]];
ray16[{Ф_, ϑ_}] := raytrace[{Ф, ϑ}][[16]];
ray17[{Ф_, ϑ_}] := raytrace[{Ф, ϑ}][[17]];
width1 = ImageDimensions[pic1][[1]]; height1 = ImageDimensions[pic1][[2]];
width2 = ImageDimensions[pic2][[1]]; height2 = ImageDimensions[pic2][[2]];
width3 = ImageDimensions[pic3][[1]]; height3 = ImageDimensions[pic3][[2]];
width4 = ImageDimensions[pic4][[1]]; height4 = ImageDimensions[pic4][[2]];
width5 = ImageDimensions[pic7][[1]]; height5 = ImageDimensions[pic7][[2]];
hzoom = If[breite>2 hoehe, 1/zoom, 1/zoom/2/hoehe*breite];
vzoom = If[breite>2 hoehe, 1/zoom*2 hoehe/breite, 1/zoom];
"approximate time remaining" -> 1.2 (AbsoluteTiming[Do[raytracer[{
RandomReal[{-π, π}]/zoom, RandomReal[{-π/2, π/2}]/zoom
}], {ü, 1, 50}]][[1]])/50*hoehe*breite/kernels/60 "minutes"
FOV->{360.0 hzoom "degree", 180.0 vzoom "degree"}
img = ParallelTable[{
(* 1 Hintergrundpanorama *)
ImageTransformation[pct1, ray17, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{-π, π-2π/width1},
{-π/2, 3π/2}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+x+vvs/vzoom, -π/2+x+vvs/vzoom+π/kernels/grain} vzoom
},
Resampling->rsp, Padding->"Periodic"],
(* 2 Sphäre *)
ImageTransformation[pct2, ray16, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{-π, π-2π/width2},
{-π/2, 3π/2}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+x+vvs/vzoom, -π/2+x+π/kernels/grain+vvs/vzoom} vzoom
},
Resampling->rsp, Padding->"Periodic"],
(* 3 Scheibe Textur *)
ImageTransformation[pic3, ray01, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{0, 2π-2π/width3},
{si, sr+(sr-si)/height3}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+vvs/vzoom+x, -π/2+vvs/vzoom+x+π/kernels/grain} vzoom
},
Resampling->rsp, Padding->"Periodic"],
(* 4 Scheibe Geometrie *)
ImageTransformation[pic4, ray01, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{0, 2π-2π/width4},
{si, sr}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+vvs/vzoom+x, -π/2+vvs/vzoom+x+π/kernels/grain} vzoom
},
Resampling->rsp, Padding->"Periodic"],
(* 5 Hintergrundpanorama *)
ImageTransformation[pct3, ray17, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{-π, π-2π/width5},
{-π/2, 3π/2}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+x+vvs/vzoom, -π/2+x+vvs/vzoom+π/kernels/grain} vzoom
},
Resampling->rsp, Padding->"Periodic"],
(* 6 Sphäre *)
ImageTransformation[pct3, ray16, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{-π, π-2π/width5},
{-π/2, 3π/2}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+x+vvs/vzoom, -π/2+x+π/kernels/grain+vvs/vzoom} vzoom
},
Resampling->rsp, Padding->"Periodic"],
(* 7 Scheibe Textur komplett *)
ImageTransformation[pic7, ray01, {breite, Ceiling[hoehe/kernels/grain]},
DataRange->{
{0, 2π-2π/width5},
{si, sr+(sr-si)/height5}
},
PlotRange->{
{-π+hvs/hzoom, π+hvs/hzoom} hzoom,
{-π/2+vvs/vzoom+x, -π/2+vvs/vzoom+x+π/kernels/grain} vzoom
},
Resampling->rsp, Padding->"Periodic"]
}, {x, 0, π-π/kernels/grain, π/kernels/grain}];
image[num_] := ImageAssemble[{Table[{img[[x, num]]}, {x, kernels grain, 1, -1}]}];
fi[x_] := ColorNegate[x];
Grid[{Table[image[n], {n, 1, 7, 1}]}]
Quit[]
Output (in diesem Beispiel mit unbewegter Scheibe und ruhendem Beobachter, r=100, θ=70°, rk=3, ri=3, ra=7):
Aberration mit vr=vθ=0, vφ=0.95 (ArcSin[vφ]=71.8°) auf θ=70°; rk=3, ri=4, ra=7; 1. Zeile: r=40, r=30, 2. Zeile: r=20, r=10:
Stereographische Projektion:
Code: Alles auswählen
Eq2St[{x_,y_}]:={0+ArcTan[-y,x],-2ArcTan[2,Sqrt[x^2+y^2]]+π/2};
St=ImageTransformation[pic,Eq2St,DataRange->{{-π,π},{-π/2,π/2}},PlotRange->{{-2π,2π},{-2π,2π}}]verwandte Themen:
Kartographie, Kerr-Newman Metrik, Gravitationslinsen.
