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.

Let a weak tangential deceleration $D>0$ act on a test body moving along a slowly evolving family of circular orbits. Write its circular speed as $V_c(r)>0$ and its specific angular momentum as $j=r{V}_c(r)$. The specific torque is $\dot{j}=-rD$. Defining $q=d\log V_c/d\log r$ givesThis derivation follows angular-momentum balance; simply setting ${\dot{V}}_c=-D$ would omit the angular momentum changed by radial migration.

An adiabatic inward drift along radially stable circular orbits requires $1+q>0$. Indeed, with $\Omega =V_c/r$, the radial epicyclic frequency satisfiesFor this stable branch, the positive local timescale defined as $r/(2|\dot{r}|)$ is $(1+q)V_c/(2D)$. It is $V_c/D$ in a uniform-mass density core with $q=1$, and $V_c/(2D)$ for a flat galaxy rotation curve with $q=0$.

At $q=-1$, the circular-orbit angular momentum is locally independent of radius, so division by $1+q$ fails. For $q<-1$, the algebraic drift formula points outward, but those circular orbits are radially unstable; it is not a stable slow-inspiral solution. The quasistatic approximation also requires the drift time to exceed an orbital time. Finally, this local timescale is not automatically the remaining time to the center, which requires integrating the radius-dependent drift.

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.