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 Incoming links: Circular orbit

Chandrasekhar dynamical friction is the drag exerted by the overdense gravitational wake that a massive body creates in a background of lighter particles. It transfers orbital energy and angular momentum to the host, causing satellites, star clusters, and massive black holes to spiral inward and promoting galaxy mergers. In a homogeneous isotropic Maxwellian background its force is

F_{df} = - 4 π G^{2} M^{2} ρ lo g Λ [erf (X) - \frac{2 X}{π} e^{- X^{2}}] \frac{v}{v ^{3}}, X = \frac{v}{2 σ},

where

lo g Λ

is the Coulomb logarithm in stellar dynamics. Its scaling can be reconstructed from the strong-deflection impact parameter

b_{cr} \sim GM / v^{2}

: the encountered mass rate is

\sim ρ v π b_{cr}^{2}

, and multiplying by momentum change

\sim v

gives

F_{df} \sim G^{2} M^{2} ρ / v^{2}

M_{h} (< r) = \frac{v _{c}^{2} r}{G}, ρ_{h} (r) = \frac{v _{c}^{2}}{4 π G r ^{2}} .

This is a singular isothermal sphere with one-dimensional dispersion

σ = v_{c} / 2

, so

X = 1

C = erf (1) - \frac{2 e ^{- 1}}{π} ≃ 0.428.

For a constant-mass satellite on a circular orbit, the drag magnitude and its torque are

F_{df} = \frac{CG M _{s}^{2} l o g Λ}{r ^{2}}, \frac{d}{d t} (M_{s} v_{c} r) = - r F_{df} .

\overset{r}{˙} = - \frac{CG M _{s} l o g Λ}{v _{c} r}, t_{df, 0} = \frac{v _{c} r _{0}^{2}}{2 CG M _{s} lo g Λ} ≃ \frac{1.17 v _{c} r _{0}^{2}}{G M _{s} lo g Λ} .

The quadratic radius dependence and inverse mass dependence explain why massive nearby satellites merge much faster than light or distant ones.

Let the satellite also have a flat internal rotation curve of speed

v_{s}

. Equating its edge density to the host density gives the tidal radius

\frac{v _{s}^{2}}{4 π G R _{t}^{2}} = \frac{v _{c}^{2}}{4 π G r ^{2}}, R_{t} (r) = \frac{v _{s}}{v _{c}} r .

Because

M_{s} (< R_{t}) = v_{s}^{2} R_{t} / G

, the bound mass decreases linearly:

M_{s} (r) = M_{s} (r_{0}) \frac{r}{r _{0}} .

Under the question's literal closure that the derivative of the remaining satellite's total orbital angular momentum equals the frictional torque,

\frac{d}{d t} [M_{s} (r) v_{c} r] = - \frac{CG M _{s} ( r ) ^{2} l o g Λ}{r},

and hence

\overset{r}{˙} = - \frac{CG M _{s} ( r _{0} ) l o g Λ}{2 v _{c} r _{0}}, t_{df, strip} = \frac{2 v _{c} r _{0}^{2}}{CG M _{s} ( r _{0} ) lo g Λ} = 4 t_{df, 0} .

If stripped material is explicitly assigned the satellite's instantaneous specific orbital angular momentum, the balance for the bound remnant is instead

M_{s} d (v_{c} r) / d t = - r F_{df}

; that convention gives

t_{df, strip} = 2 t_{df, 0}

. Both treatments show the robust point: tidal stripping weakens the drag as the orbit shrinks and substantially delays coalescence. In less idealized profiles the mass can fall faster than linearly, producing dynamical-friction stalling by tidal stripping.