Rotational Brownian motion (astronomy)

Consider a binary that consists of two massive objects (stars, black holes etc.) and that is embedded in a stellar system containing a large number of stars. Let ${\displaystyle M_{1}}$ and ${\displaystyle M_{2}}$ be the masses of the two components of the binary whose total mass is ${\displaystyle M_{12}=M_{1}+M_{2}}$. A field star that approaches the binary with impact parameter ${\displaystyle p}$ and velocity ${\displaystyle V}$ passes a distance ${\displaystyle r_{p}}$ from the binary, where

${\displaystyle p^{2}=r_{p}^{2}\left(1+2GM_{12}/V^{2}r_{p}\right)\approx 2GM_{12}r_{p}/V^{2};}$

the latter expression is valid in the limit that gravitational focusing dominates the encounter rate. The rate of encounters with stars that interact strongly with the binary, i.e. that satisfy ${\displaystyle r_{p}<a}$, is approximately ${\displaystyle n\pi p^{2}\sigma =2\pi GM_{12}na/\sigma }$ where ${\displaystyle n}$ and ${\displaystyle \sigma }$ are the number density and velocity dispersion of the field stars and ${\displaystyle a}$ is the semi-major axis of the binary.

As it passes near the binary, the field star experiences a change in velocity of order

${\displaystyle \Delta V\approx V_{\rm {bin}}={\sqrt {GM_{12}/a}}}$,

where ${\displaystyle V_{\rm {bin}}}$ is the relative velocity of the two stars in the binary. The change in the field star's specific angular momentum with respect to the binary, ${\displaystyle l}$, is then Δl ≈ a V_bin. Conservation of angular momentum implies that the binary's angular momentum changes by Δl_bin ≈ -(m/μ₁₂)Δl where m is the mass of a field star and μ₁₂ is the binary reduced mass. Changes in the magnitude of l_bin correspond to changes in the binary's orbital eccentricity via the relation e = 1 - l_b²/GM₁₂μ₁₂a. Changes in the direction of l_bin correspond to changes in the orientation of the binary, leading to rotational diffusion. The rotational diffusion coefficient is

${\displaystyle \langle \Delta \xi ^{2}\rangle =\langle \Delta l_{\rm {bin}}^{2}\rangle /l_{\rm {bin}}^{2}\approx \left({m \over M_{12}}\right)^{2}\langle \Delta l^{2}\rangle /GM_{12}a\approx {m \over M_{12}}{G\rho a \over \sigma }}$

where ρ = mn is the mass density of field stars.

Let F(θ,t) be the probability that the rotation axis of the binary is oriented at angle θ at time t. The evolution equation for F is ^[1]

${\displaystyle {\partial F \over \partial t}={1 \over \sin \theta }{\partial \over \partial \theta }\left(\sin \theta {\langle \Delta \xi ^{2}\rangle \over 4}{\partial F \over \partial \theta }\right).}$

If <Δξ²>, a, ρ and σ are constant in time, this becomes

${\displaystyle {\partial F \over \partial \tau }={1 \over 2}{\partial \over \partial \mu }\left[(1-\mu ^{2}){\partial F \over \partial \mu }\right]}$

where μ = cos θ and τ is the time in units of the relaxation time t_rel, where

${\displaystyle t_{\rm {rel}}\approx {M_{12} \over m}{\sigma \over G\rho a}.}$

The solution to this equation states that the expectation value of μ decays with time as

${\displaystyle {\overline {\mu }}={\overline {\mu }}_{0}e^{-\tau }.}$

Hence, t_rel is the time constant for the binary's orientation to be randomized by torques from field stars.