tilt i: 0° = prograde (spin along the orbit), 90° = polar, 180° = retrograde. ω turns the disc about the orbit normal.
The science
Restricted three-body method. Toomre & Toomre (1972, ApJ 178, 623) treated each galaxy as a single massive core and each disc star as a massless test particle that feels both cores but pulls on nothing. The cores follow a two-body orbit; the stars simply trace the resulting time-dependent field. That is why tens of thousands of stars are cheap here, and why the model shows the geometry of tails and bridges very well but not self-gravity, bars, gas or star formation.
Potentials. Each galaxy has two Plummer (1911) components, a compact baryonic core and an extended halo:
Φ(r) = −G M / √(r² + a²) acore=3 kpc, ahalo=20 kpc
The circular speed follows from the enclosed mass, vc² = G M r² / (r²+a²)3/2 summed over components; stars start on those circular orbits (plus 2 % random velocity). The mutual potential energy of two Plummer spheres is exactly a Plummer form with a² = a1² + a2², so the two cores interact through four such softened terms. G = 4.30091×10−6 kpc (km/s)² / M☉; 1 kpc/(km/s) = 977.8 Myr. Masses are in 108 M☉.
Orbit set-up. From the pericentre rp and eccentricity e the conic p = rp(1+e), h = √(μp), μ = G(M1+M2) fixes the starting position and velocity (e = 1 parabolic, < 1 bound, > 1 hyperbolic). Because the haloes overlap the real pericentre differs from the point-mass value; the measured one is shown in the HUD.
Integrator. Kick-drift-kick leapfrog (symplectic, second order) with a fixed 0.5 Myr step. The checks below measure its energy and angular-momentum conservation.
Dynamical friction. A Chandrasekhar (1943) drag slows the secondary inside the primary’s halo, with equal and opposite force on the primary so momentum is conserved:
a = −4π G² Ms ρ(r) lnΛ [erf X − (2X/√π) e−X²] v / v³, X = v/(√2 σ)
with σ = vc/√2, lnΛ = 2.5 × (strength slider). This is a crude single-parameter treatment (no mass loss, no tidal truncation, primary halo only); it is what lets orbits decay and merge. It dissipates orbital energy by design, so the ΔE readout is only a conservation test with friction switched off.
What to look for. A disc rotating in the same sense as the orbit (prograde) is resonantly kicked by the tide and throws out long thin tails plus a bridge toward the companion; a retrograde disc does not resonate and only wobbles (Toomre & Toomre 1972; Barnes & Hernquist 1992). The experiment button measures this ratio in the page. The Antennae (NGC 4038/9), the Mice (NGC 4676) and M51 are the classic real-sky analogues; the presets are illustrative, not fits to those systems.
Honest limits. No disc or halo self-gravity, rigid non-responsive haloes (so no true merger remnant relaxation or shells), no gas, stars or star formation, tails are test particles only. Colours are artistic: warm = primary’s stars, blue = secondary’s; “density” mode colours by local 3D star count in a grid.
Context. The Milky Way and Andromeda approach at about 110 km/s; earlier work (van der Marel et al. 2012) put a merger about 4–5 Gyr away, and a 2025 re-analysis (Sawala et al.) gives roughly a coin-flip chance within 10 Gyr. Disk art and sky were rendered in Blender 5.2 (Cycles); the dynamics are computed live in JavaScript.
Drag to orbit, wheel / pinch to zoom, double-tap to re-frame. URL options: ?q=low, ?embed=true.