Bondi–Hoyle–Lyttleton accretion


Bondi–Hoyle–Lyttleton 'accretion' is a mathematical model for the accretion of a uniform gas by a massive body. It is a general model of accretion with multiple applications, such as accretion of the interstellar medium by neutron stars and black holes, or wind mass transfer in binary star systems.
It is named after Hermann Bondi, Fred Hoyle, and Raymond Lyttleton. An initial estimate of the accretion rate for a supersonic wind was derived by Hoyle and Lyttleton in 1939 to explain variations in the climate of the earth. They proposed that these variations could be explained by the luminosity of the sun changing due to the accretion of interstellar material. Bondi and Hoyle showed in 1944 that, when taking some of the effects of the pressure of the gas into account, the previously derived accretion rate was only a maximum. Bondi later studied the complementary case of spherically symmetric accretion of a stationary gas. For the intermediate case, he proposed an interpolation formula that is now the generally accepted form of BHL accretion.
The accretion rate of the BHL formalism is given by
where

Derivation of Hoyle–Lyttleton Accretion

Hoyle–Lyttleton accretion is a simplified version of BHL accretion which treats the gas as being supersonic.
Hoyle–Lyttleton accretion assumes a homogeneous flow of incoming particles traveling with a velocity with density towards an accreting body with mass. The particles flow around the massive body, by which they are deflected towards the accretion line that lies behind it. At the accretion line, the particles collide, which cancels their momenta in the radial direction.
Depending on the initial velocity and the radial distance from the massive body, a particle may either be gravitationally bound to the body or not. A bound particle will then be accreted, while an unbound particle will escape. The initial velocity needed to escape from the massive body is given by its escape velocity at the distance. Thus, the condition for a particle to be accreted is
This equation can also be written in terms of an accretion radius. Thus, all particles that pass through a circle of this radius around the massive body are accreted. This gives an accretion rate of
When taking into account some limited pressure effects and combining the resulting formula with Bondi accretion through an interpolation formula, the canonical formula for the BHL accretion rate can be found.

Application to Binary Star Systems

BHL accretion is used to model mass transfer in binary star systems, such as barium stars. For this, the velocity of the incoming flow is set to the relative velocity between the stellar wind and the accreting star around the donating star, which is thus given by
where
Assuming that the stellar wind is emitted in a spherically symmetric way, it can be described by
where is the distance from the donor star.
Substituting these relations into the equation for the accretion rate, the accreted mass is given by
where is the eccentricity of the orbit and where the was replaced by its average value during the orbit.