Galactic Gravity Simulations

Here is a simulation of a galaxy using Newton’s law of universal gravitation. Click here to see a 3D rotatable model of the Milky Way Galaxy.

(* runtime: 8 seconds *)
n = 275; G = 1; SeedRandom[0]; image = Table[0, {n}, {n}]; nstar = 500;dt = 0.001; theta = Pi/4; Trot = {{1, 0, 0}, {0, Cos[theta], Sin[theta]}, {0, -Sin[theta], 1}};
stars = Table[r = 0.5 Random[]; theta = 2Pi Random[]; {Trot.{r Cos[theta], r Sin[theta],0.1(2Random[] - 1)Exp[-4r]} + {0, 0, 1}, Sqrt[G/r]{-Sin[theta], Cos[theta], 0}}, {nstar}];
Do[image *= 0.9; stars = Map[({r, v} = #; v -= G r dt/(r.r)^1.5; r += v dt; {x, y, z} = r; {j, i} = Round[n({x, y}/z + 0.5)]; If[0 < i <= n && 0 < j <= n, image[[i, j]] += 0.5/z]; {r, v}) &, stars]; ListDensityPlot[image, Mesh -> False, Frame -> False,PlotRange -> {0, 1}], {50}];

Links

• Milky Way and Andromeda Galaxy Collision – amazing simulations by John Dubinski
• Millennium Simulation – interesting video, see also Dark Matter Simulation
• Type Ia Supernova – simulations
• Simulation of Newtonian Gravity – Java applet by Mathew Tizard
• Hyperion Spiral Galaxy Demo – cool-looking but I don’t know how accurate this is
• Colliding Galaxies – fun Java applet, originally by Uli Siegmund
• galaxy – nice-looking Java applet
• Universe Holocube – 100 Megaparsec scale laser-etched crystal cube
• Milky Way Galaxy Holocube – scale laser-etched crystal cube
• Astronomy Picture of the Day – amazing astronomy pictures, some of my favorites are Crab Pulsar, Cat’s Eye Nebula, Eta Carinae, Variable star V838 Monocerotis, and Hourglass Nebula
• Cosmographica – impressive astronomical art gallery by Don Dixon
• Breakthrough Propulsion Physics (BPP) – is warp drive possible?

Halo Orbit

In 1772, Joseph Lagrange showed that a small satelite moving in the Earth-Moon system can have 5 balance points, called Lagrange points. Here is a small halo orbit around Lagrange point L4. This is interesting because L4 is off to the side, so it looks like the satelite is orbiting nothing! The velocity components have been represented by the third axis and color.
(* runtime = 6 seconds *)
Clear[x, y, t, p, v]; mu = 0.0123001; x1 = -mu; x2 = 1 - mu; r1 = Sqrt[(x - x1)^2 + y^2]; r2 = Sqrt[(x - x2)^2 + y^2];
{x0, y0} = {x, y} /. Last[NSolve[{-x == -x2(x - x1)/r1^3 + x1(x - x2)/r2^3, -y == -(x2/r1^3 - x1/r2^3) y}, {x, y}]];
JacobianMatrix[p_, q_] := Outer[D, p, q];
vlist = Eigenvectors[JacobianMatrix[{xdot, -x2(x - x1)/r1^3 + x1(x - x2)/r2^3 + 2ydot + x, ydot, -(x2/r1^3 - x1/r2^3)y - 2xdot + y}, {x, xdot, y, ydot}] /. {x -> x0, xdot -> 0, y -> y0, ydot -> 0}];
r1 = Sqrt[(x[t] - x1)^2 + y[t]^2]; r2 = Sqrt[(x[t] - x2)^2 + y[t]^2];
tmax = 355; p0 = {x0, y0, 0, 0} + Re[0.5 vlist[[1]] + vlist[[3]]]/3.84400*^5;
p[t_] = {x[t], y[t]} /. NDSolve[{x''[t] - 2y'[t] - x[t] == -x2(x[t] - x1)/r1^3 + x1(x[t] - x2)/r2^3, y''[t] + 2x'[t] - y[t] == -(x2/r1^3 - x1/r2^3)y[t], x[0] == p0[[1]], y[0] == p0[[2]], x'[0] == p0[[3]], y'[0] == p0[[4]]}, {x[t], y[t]}, {t, 0, tmax}, MaxSteps -> 100000, AccuracyGoal -> 10][[1]];
v[t_] = D[p[t], t];
ParametricPlot3D[Append[p[t], v[t][[1]]], {t, 0, tmax}, Compiled -> False, PlotPoints -> 1000, BoxRatios -> {1, 1, 1}, ViewPoint -> {0, 10, 0}]

Links

• Interplanetary Superhighway – by Shane Ross
• Stable 3-Body Orbits – Java applet by Robert Vanderbei, be sure to see the Ducati 3-Body Orbit

Gravitational Lensing of a Black Hole

Gravitational Lensing of a Black Hole : According to Einstein’s Theory of Relativity, the intense gravity of a black hole can bend light into a circle called an Einstein Ring. See also Event horizon, Schwarzschild radius, photon sphere. The following code is only an approximation:
(* runtime: 30 seconds *)
Clear[stars]; SeedRandom[535]; stars[{x_, y_}] = Sum[Exp[-500((x - Random[])^2 + (y - Random[])^2)/Random[]^2], {10}];
DensityPlot[Module[{r = x^2 + y^2}, If[r 275, PlotRange -> {0, 1}, Mesh -> False, Frame -> False]

Links

• “Monster of the Milky Way” – Nova video about black holes
• Black Holes – BBC Video with Sam Neill
• ART project – relativity simulations, see the movies “Tackling the Riddles of Gravity” and “Numerical Relativity”
• Werner Benger – Black Hole Raytracing, Einstein Ring Animation, and simulation of Earth as a black hole
• Virtual Trips to Black Holes – simulations by Robert Nemiroff
• Black Hole Simulation – by Phil Armitage
• Black Hole Flight Simulator – screenshots and Java applet, by Andrew Hamilton