Rotational Brownian motion (astronomy)
Encyclopedia
In astronomy, rotational Brownian motion refers to the random walk
in orientation of a binary star
's orbital plane, induced by gravitational perturbations from passing stars.
and 
be the masses of the two components of the binary whose total mass is 
. A field star that approaches the binary with impact parameter
and velocity 
passes a distance 
from the binary, where
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
, is approximately 
where 
and 
are the number density and velocity dispersion of the field stars and 
is the semi-major axis
of the binary.
As it passes near the binary, the field star experiences a change in velocity of order
,
where
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,
, is then Δl ≈ a Vbin. Conservation of angular momentum implies that the binary's angular momentum changes by Δlbin ≈ -(m/μ12)Δl where m is the mass of a field star and μ12 is the binary reduced mass
. Changes in the magnitude of lbin correspond to changes in the binary's orbital eccentricity via the relation e = 1 - lb2/GM12μ12a. Changes in the direction of lbin correspond to changes in the orientation of the binary, leading to rotational diffusion. The rotational diffusion coefficient is
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
If <Δξ2>, a, ρ and σ are constant in time, this becomes
where μ = cos θ and τ is the time in units of the relaxation time trel, where
The solution to this equation states that the expectation value of μ decays with time as
Hence, trel is the time constant for the binary's orientation to be randomized by torques from field stars.
s at the centers of galaxies. 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 in N-body simulation
s of galaxies containing binary black holes. The massive binary sinks to the center of the galaxy via dynamical 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 the gravitational slingshot
. It can be shown that the rms change in the binary's orientation, from the time the binary forms until the two black holes collide, is roughly
In a real galaxy, the two black holes would eventually coalesce due to emission of gravitational wave
s. The spin axis of the coalesced hole will be aligned with the orbital angular momentum 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.
Random walk
A random walk, sometimes denoted RW, is a mathematical formalisation of a trajectory that consists of taking successive random steps. For example, the path traced by a molecule as it travels in a liquid or a gas, the search path of a foraging animal, the price of a fluctuating stock and the...
in orientation of a binary star
Binary star
A binary star is a star system consisting of two stars orbiting around their common center of mass. The brighter star is called the primary and the other is its companion star, comes, or secondary...
's orbital plane, induced by gravitational perturbations from passing stars.
Theory
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 and be the masses of the two components of the binary whose total mass is . A field star that approaches the binary with impact parameterImpact parameter
The impact parameter b is defined as the perpendicular distance between the path of a projectile and the center of the field U created by an object that the projectile is approaching...
and velocity passes a distance from the binary, wherethe 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
, is approximately where and are the number density and velocity dispersion of the field stars and is the semi-major axisSemi-major axis
The major axis of an ellipse is its longest diameter, a line that runs through the centre and both foci, its ends being at the widest points of the shape...
of the binary.
As it passes near the binary, the field star experiences a change in velocity of order
,where
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,
, is then Δl ≈ a Vbin. Conservation of angular momentum implies that the binary's angular momentum changes by Δlbin ≈ -(m/μ12)Δl where m is the mass of a field star and μ12 is the binary reduced massReduced mass
Reduced mass is the "effective" inertial mass appearing in the two-body problem of Newtonian mechanics. This is a quantity with the unit of mass, which allows the two-body problem to be solved as if it were a one-body problem. Note however that the mass determining the gravitational force is not...
. Changes in the magnitude of lbin correspond to changes in the binary's orbital eccentricity via the relation e = 1 - lb2/GM12μ12a. Changes in the direction of lbin correspond to changes in the orientation of the binary, leading to rotational diffusion. The rotational diffusion coefficient is
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
If <Δξ2>, a, ρ and σ are constant in time, this becomes
where μ = cos θ and τ is the time in units of the relaxation time trel, where
The solution to this equation states that the expectation value of μ decays with time as
Hence, trel is the time constant for the binary's orientation to be randomized by torques from field stars.
Applications
Rotational Brownian motion was first discussed in the context of binary supermassive black holeSupermassive black hole
A supermassive black hole is the largest type of black hole in a galaxy, in the order of hundreds of thousands to billions of solar masses. Most, and possibly all galaxies, including the Milky Way, are believed to contain supermassive black holes at their centers.Supermassive black holes have...
s at the centers of galaxies. 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 in N-body simulation
N-body simulation
An N-body simulation is a simulation of a dynamical system of particles, usually under the influence of physical forces, such as gravity . In cosmology, they are used to study processes of non-linear structure formation such as the process of forming galaxy filaments and galaxy halos from dark...
s of galaxies containing binary black holes. The massive binary sinks to the center of the galaxy via dynamical friction
Dynamical friction
Dynamical friction is a term in astrophysics related to loss of momentum and kinetic energy of moving bodies through a gravitational interaction with surrounding matter in space...
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 the gravitational slingshot
Gravitational slingshot
In orbital mechanics and aerospace engineering, a gravitational slingshot, gravity assist maneuver, or swing-by is the use of the relative movement and gravity of a planet or other celestial body to alter the path and speed of a spacecraft, typically in order to save propellant, time, and expense...
. It can be shown that the rms change in the binary's orientation, from the time the binary forms until the two black holes collide, is roughly
In a real galaxy, the two black holes would eventually coalesce due to emission of gravitational wave
Gravitational wave
In physics, gravitational waves are theoretical ripples in the curvature of spacetime which propagates as a wave, traveling outward from the source. Predicted to exist by Albert Einstein in 1916 on the basis of his theory of general relativity, gravitational waves theoretically transport energy as...
s. The spin axis of the coalesced hole will be aligned with the orbital angular momentum 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.
External links
- Gravitational Scattering Review article on the dynamics of encounters between binaries and single stars.