A small satellite galaxy orbiting inside a larger host's dark-matter halo drags a gravitational "wake" of background particles behind it. That wake pulls back on the satellite — a braking force called dynamical friction, first derived by Chandrasekhar (1943):
F_df = -4πG² M_sat² ρ(r) lnΛ / v³ · v_vec
Here ρ(r) is the local background density, v is the satellite's orbital speed, and lnΛ (the Coulomb logarithm) accounts for the range of gravitational encounters. For a singular isothermal sphere halo with flat rotation speed vc, the density is ρ(r) = vc² / (4πG r²), so the drag acceleration on the satellite simplifies to:
a_df = -(G M_sat v_c² lnΛ) / (r² v³) · v_vec
Combined with the host's own (inward, centripetal) gravity, this drag steadily removes orbital angular momentum, so the satellite spirals inward instead of orbiting forever. The standard order-of-magnitude sinking timescale (Binney & Tremaine) is:
t_df ≈ (1.17 / lnΛ) · (r_i² v_c) / (G M_sat)
- Satellite mass ratio — heavier satellites raise a bigger wake and sink faster (t_df ∝ 1/M_sat).
- Coulomb logarithm — larger lnΛ (bigger host, smaller satellite) also speeds up the sink.
- Initial radius — sinking time grows as r_i², so satellites starting further out take much longer to merge.
- This is exactly how the Milky Way is currently swallowing the Sagittarius and Large Magellanic Cloud dwarf galaxies, and how galaxy clusters build up their central giant ellipticals over cosmic time — distinct from a fast, head-on tidal-tail collision between two comparable-mass galaxies.
- The 1.17 prefactor is Binney & Tremaine's own order-of-magnitude estimate — a direct numerical integration of the drag law above lands within roughly a factor of two of it, which this simulator's live t vs. predicted tdf readout lets you check yourself.