Restricted three-body problem (Toomre & Toomre 1972)
Each galaxy is a massive extended body (a Plummer sphere carrying the dark halo). Thousands of massless test stars orbit inside each disc and feel the gravity of both bodies. The stars do not pull on each other or on the cores, so this is the classic “Galactic Bridges and Tails” model, not a full N-body code.
a(x) = −Σj G mj (x−Xj) / (|x−Xj|² + aj²)3/2
Discs start on circular orbits, vc² = G m r²/(r²+a²)3/2, plus 3.5% random motion. Integration is a drift-kick-drift leapfrog (symplectic, second order) at 1 Myr steps.
The orbit
r = p/(1+e cos f), p = rp(1+e), v² = GM(2/r − (1−e)/rp)
e < 1 is bound, e = 1 parabolic (typical of a first encounter), e > 1 a hyperbolic fly-by. The dashed curves are this analytic orbit; the solid tracks are what the integrator did. The actual pericentre differs a little because the bodies are softened and lose energy to dynamical friction.
Why prograde makes tails
A star is torn away efficiently when the perturber sweeps past it at about the star’s own orbital angular speed. For a prograde disc the encounter lasts longer at resonance, so the tidal force acts coherently and flings the outer disc into a long tail (near side) and a bridge (far side). A retrograde disc sees a fast, jerky passage: the perturbation averages out and the disc is only slightly distorted. The tidal (Hill) radius rt = rp (m/3M)1/3 says which stars can be stripped; the spin direction decides whether they actually are.
Mass ratio
In a 1:1 merger both discs are shredded and the remnant is a rounder, elliptical-like spheroid. In a 1:10 minor merger the big disc survives and the small galaxy is stretched into a stream; on a nearly radial orbit its stars pile up at the turning points as shells.
Dynamical friction
dv/dt = −4π G² m ρ lnΛ χ(X) / v²
Chandrasekhar’s drag (with χ(X)=erf X − 2X e−X²/√π) is applied between the two bodies using the Plummer density of each, because the point-like cores here would otherwise never feel each other’s halo. The slider scales it. This is the main simplification that makes the cores merge.
Star-formation proxy
Gas is assumed to follow the stars. A 3-D grid counts particles per cell. The rate is Σcells n1.4 (Kennicutt–Schmidt law) plus a cloud-collision term 4 Σ (nA nB)0.7 where gas of the two galaxies overlaps and shocks. It is shown relative to the isolated discs; pink particles sit in over-dense or overlapping cells. It is a proxy only: there is no gas physics.
Black-hole pair (toy)
Each core hosts a black hole of 0.05% of its mass. Before the cores merge the BH separation is the core separation. After that a sub-grid model takes over: dynamical friction down to the hard-binary radius ah = Gμ/4σ², stellar hardening da/dt = −H Gρ a²/σ, then Peters (1964) gravitational-wave decay da/dt = −(64/5) G³ m1 m2 (m1+m2) / (c&sup5; a³) on a circular orbit. The nuclear density and efficiencies are assumed, so the merger delay is illustrative.
What is simplified
- Test particles: no self-gravity, so no bars or spiral waves, and no gas.
- Rigid Plummer potentials: halos do not deform or get stripped.
- Dynamical friction is an analytic drag, not an emergent effect.
- Masses and sizes are in kpc, km/s and 1010 M☉ (galaxy A = 1011 M☉).