Astronomy and astrophysics Stellar dynamics Proposition

Dynamical friction

Builds on Newton's law of universal gravitation Maxwell–Boltzmann distribution Error function Stellar dynamics

Statement

Let a body of mass $M$ move with velocity $v_M$ through an infinite, homogeneous background of stars of mass $m \ll M$ . The bodies interact by Newtonian gravity with gravitational constant $G$ . Let the background have an isotropic velocity distribution $f(v)$ , normalized so that $4\pi \int_0^\infty f(v) v^2 \, dv = n$ is the number density, and let $\rho = n m$ be its mass density. Treat the encounters as independent two-body encounters with impact parameters up to a maximum $b_{\max}$ , and let $\ln \Lambda$ be the Coulomb logarithm, with $\Lambda \approx b_{\max} v_M^2 / (G M)$ . Then, up to terms smaller by a factor $1/\ln \Lambda$ , the body decelerates according to Chandrasekhar's dynamical friction formula $\frac{d v_M}{dt} = -16 \pi^2 G^2 M m \ln \Lambda \left( \int_0^{\abs{v_M}} f(v) v^2 \, dv \right) \frac{v_M}{\abs{v_M}^3}.$ For a Maxwellian background with one-dimensional dispersion $\sigma$ , and $X = \abs{v_M}/(\sqrt{2} \sigma)$ , this becomes $\frac{d v_M}{dt} = -\frac{4 \pi G^2 M \rho \ln \Lambda}{\abs{v_M}^3} \left( \operatorname{erf}(X) - \frac{2X}{\sqrt{\pi}} e^{-X^2} \right) v_M,$ where $\operatorname{erf}$ is the error function.

Remarks

Each encounter deflects a background star and slightly changes the velocity of $M$ ; averaging over impact parameters and over the background velocities gives a mean change opposite to $v_M$ . Only stars slower than $M$ contribute to the leading term, as for the gravitational field of a uniform spherical shell, which pulls only on points outside it. In another picture, $M$ focuses the background stars into a denser wake behind it, and the wake pulls $M$ back.

The drag is proportional to $M$ , not to $m$ , so heavy bodies sink fastest. For a slow body, $X \ll 1$ , the drag is linear in $v_M$ ; for a fast body, $X \gg 1$ , it falls like $\abs{v_M}^{-2}$ . This sets the time in which globular clusters and satellite galaxies spiral into the centre of their host galaxy. It also sets the time in which massive black holes sink to the centre of a merged galaxy.

Going deeper

For a student

The formula comes from the theory of two-body encounters that also gives the relaxation time in stellar dynamics. It assumes an infinite uniform medium. In a real galaxy $b_{\max}$ is of the order of the size of the orbit, and the result is accurate only up to the uncertainty in $\ln \Lambda$ . Apply it to a body on a circular orbit in a singular isothermal sphere. It gives an inspiral time proportional to the square of the initial radius and inversely proportional to $M$ .

Reading path

To understand this page from the ground up, read these 47 pages in order. The foundations come first.

Derivative concept
Differential equation concept
Error function concept
Euclidean space object
Euclidean vector concept
Function concept
Interval concept
Maxwell–Boltzmann distribution concept
Ordinary differential equation concept
Stellar dynamics concept
Velocity concept
Fréchet derivative definition
Ordinary differential equation definition
Cross product definition
Function of several variables concept
Vector-valued function concept
Lipschitz condition definition
Classical mechanics concept
Limit of a function of several variables concept
Momentum concept
Partial derivative concept
Solution of an ordinary differential equation definition
Inertial frame of reference concept
Continuity in R^n definition
Differentiable function of several variables definition
Partial derivative definition
Initial value problem definition
Newton's laws of motion statement
Differentiable on an open set definition
Gradient definition
Higher partial derivatives definition
Picard–Lindelöf hypotheses definition
Iterated derivative definition
Picard operator definition
C^k function characterised by continuous derivatives statement
Integral equation form of an initial value problem statement
An iterate of the Picard operator is a contraction statement
Vector field definition
Picard–Lindelöf theorem statement
Conservative vector field definition
Classical mechanics definition
Momentum definition
Potential energy definition
Angular momentum definition
Central force definition
Newton's law of universal gravitation statement
Dynamical friction statement

Tag	`MGMHMC7E5J`
URI	`m2://MGMHMC7E5J`
Kind	statement, proposition
Writing	W0 stub
MSC	85A05, 70F10
Status	This copy of the site has no build data. Run `m2 status MGMHMC7E5J`.
About	Stellar dynamics
Uses	Newton's law of universal gravitation, Maxwell–Boltzmann distribution, Error function, Stellar dynamics
Changed	3 November 2026