In astronomy,rotational Brownian motion is therandom walk in orientation of abinary star's orbital plane, induced by gravitational perturbations from passing stars.

Consider a binary that consists of two massive objects (stars, black holes etc.) and that is embedded in astellar 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 withimpact parameter${\displaystyle p}$ and velocity${\displaystyle V}$ passes a distance${\displaystyle r_p}$ from the binary, where

${\displaystyle p^2=r_p^2\left(1+2G{M}_{12}/V^2{r}_p\right)\approx 2G{M}_{12}{r}_p/V^2;}$

the latter expression is valid in the limit thatgravitational focusing dominates the encounter rate. The rate of encounters with stars that interact strongly with the binary, i.e. that satisfy, is approximately${\displaystyle n\pi p^2\sigma =2\pi G{M}_{12}na/\sigma }$ where${\displaystyle n}$ and${\displaystyle \sigma }$ are thenumber density andvelocity dispersion of the field stars and${\displaystyle a}$ is thesemi-major axis of the binary.

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

${\displaystyle \mathrm{\Delta }V\approx V_{\mathrm{b}\mathrm{i}\mathrm{n}}=\sqrt{G{M}_{12}/a}}$,

where${\displaystyle V_{\mathrm{b}\mathrm{i}\mathrm{n}}}$ is the relative velocity of the two stars in the binary.The change in the field star'sspecific angular momentum with respect to the binary,${\displaystyle l}$, is then Δl ≈aV_bin. Conservation of angular momentum implies that the binary's angular momentum changes by Δl_bin ≈ -(m/μ₁₂)Δl wherem is the mass of a field star and μ₁₂ is the binaryreduced mass. Changes in the magnitude ofl_bin correspond to changes in the binary'sorbital eccentricity via the relatione = 1 -l_b²/GM₁₂μ₁₂a. Changes in the direction ofl_bin correspond to changes in the orientation of the binary, leading to rotational diffusion. The rotational diffusion coefficient is

${\displaystyle ⟨\mathrm{\Delta }{\xi }^2⟩=⟨\mathrm{\Delta }{l}_{\mathrm{b}\mathrm{i}\mathrm{n}}^2⟩/l_{\mathrm{b}\mathrm{i}\mathrm{n}}^2\approx {\left(\frac{m}{M_{12}}\right)}^2⟨\mathrm{\Delta }{l}^2⟩/G{M}_{12}a\approx \frac{m}{M_{12}}\frac{G\rho a}{\sigma }}$

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

LetF(θ,t) be the probability that the rotation axis of the binary is oriented at angle θ at timet. The evolution equation forF is^[1]

${\displaystyle \frac{\mathrm{\partial }F}{\mathrm{\partial }t}=\frac{1}{\sin \theta }\frac{\mathrm{\partial }}{\mathrm{\partial }\theta }\left(\sin \theta \frac{⟨\mathrm{\Delta }{\xi }^2⟩}{4}\frac{\mathrm{\partial }F}{\mathrm{\partial }\theta }\right).}$

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

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

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

${\displaystyle t_{\mathrm{r}\mathrm{e}\mathrm{l}}\approx \frac{M_{12}}{m}\frac{\sigma }{G\rho a}.}$

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.

Rotational Brownian motion was first discussed in the context of binarysupermassive black holes at the centers of galaxies.^[2] Perturbations from passing stars can alter the orbital plane of such a binary, which in turn alters the direction of the spin axis of the single black hole that forms when the two coalesce.

Rotational Brownian motion is often observed inN-body simulations ofgalaxies containing binary black holes.^[3][4] The massive binary sinks to the center of the galaxy viadynamical friction where it interacts with passing stars. The same gravitational perturbations that induce a random walk in the orientation of the binary, also cause the binary to shrink, via thegravitational slingshot. It can be shown^[2] that the rms change in the binary's orientation, from the time the binary forms until the two black holes collide, is roughly

${\displaystyle \delta \theta \approx \sqrt{20m/M_{12}}.}$

In a real galaxy, the two black holes would eventually coalesce due to emission ofgravitational waves. The spin axis of the coalesced hole will be aligned with the angular momentum axis of the orbit of the pre-existing binary. Hence, a mechanism like rotational Brownian motion that affects the orbits of binary black holes can also affect the distribution of black hole spins. This may explain in part why the spin axes of supermassive black holes appear to be randomly aligned with respect to their host galaxies.^[5]

• Gravitational Scattering Review article on the dynamics of encounters between binaries and single stars.

• Astrophysics
• Celestial mechanics
• Supermassive black holes

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