Chandrasekhar dynamical friction

Chandrasekhar dynamical friction is the drag on a massive body produced by its gravitational wake in a background of lighter particles. Its magnitude is proportional to the body's mass and the background density.

For a slowly evolving circular orbit with tangential deceleration $D>0$, the specific angular momentum is $j=r{V}_c(r)$ and its torque is $\dot{j}=-rD$. Differentiating $j$ gives the displayed drift when $1+q\ne 0$. The positive local decay time $r/(2|\dot{r}|)$ is $(1+q)V_c/(2D)$. In a uniform-density core $q=1$, so it is $V_c/D$; for a flat galaxy rotation curve $q=0$, it is $V_c/(2D)$. Simply setting ${\dot{V}}_c=-D$ neglects the radial contribution to angular momentum change.

For an isotropic, integrable background phase-space distribution function, rotational symmetry makes the integral radial in velocity space. Its divergence with respect to $\mathbf{V}$ is $4\pi f(V)$ by the Dirac delta identity for the divergence of the inverse-square vector field. Integrating over a velocity-space sphere and using the divergence theorem proves the displayed identity. Exterior velocity shells have zero contribution, just as in the spherical shell theorem. This explains why the leading-log Chandrasekhar dynamical friction integral contains only particles slower than the test body.

For a test mass $M$ and isotropic background particles of mass $m$, the hyperbolic two-body problem uses the relative deflection scale $b_{90}=G(M+m)/w^2$. The test body's longitudinal velocity change in one encounter is $-2mw/[(M+m)(1+b^2/b_{90}^2)]$. Integrating the encounter rate over impact parameter gives a logarithm; replacing its velocity-dependent value by a common Coulomb logarithm in stellar dynamics gives the displayed systematic drag. The approximation $M+m\simeq M$ requires $M\gg m$, not merely $M>m$. Velocity diffusion and exact cutoff dependence are distinct effects omitted by this leading-log drag model.