Bondi–Hoyle–Lyttleton accretion

In a 1939 paper, For decades, it competed with many other theories to explain this phenomenon. In 2008, a review article found that there is no evidence that the motion of the Sun plays a significant role in climatic variations, ruling out the theory. Astrophysical applications Besides the application to the Earth's climate, the BHL model was also used in astrophysics, as in the paper by Bondi and Hoyle. Other applications of the model to astrophysics include the accretion of material by neutron stars and black holes, the accretion of material from the intergalactic medium, and the accretion of the stellar wind in binary star systems. Notably, the last application was used to show that barium stars can be created through this mechanism. ==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 (supersonic) velocity v with density \rho towards an accreting body with mass M. 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 v and the radial distance from the massive body R, 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 R. Thus, the condition for a particle to be accreted is :v This equation can also be written in terms of an accretion radius R_\text{acc}. Thus, all particles that pass through a circle of this radius R_\text{acc} around the massive body are accreted. This gives an accretion rate of :\dot M = \pi \rho v R_\text{acc}^2 = \frac{4 \pi (GM)^2 \rho}{v^3}. 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 (whose mass is M_\text{acc}) around the donor star (whose mass is M_\text{donor}), which is thus given by :v=\sqrt{v_w^2+v_\text{orb}^2}, where • v_w is the velocity of the stellar wind; • v_\text{orb}=\sqrt{G\frac{M_\text{donor}+M_\text{acc}}{a}} is the mean orbital velocity, where a is the semimajor axis of the orbit. Assuming that the stellar wind is emitted in a spherically symmetric way, it can be described by :\dot M_\text{donor}=4\pi v_w \rho(r)r^2, where • r is the distance from the donor star; • \rho(r) is the mass density of the stellar wind at distance r. Substituting these relations into the equation for the accretion rate, the accreted mass (per unit time) is given by :\dot M_\text{acc} = -\alpha\frac{1}{2a^2\sqrt{1-e^2}}\left[\frac{GM_\text{acc}}{v_w}\right]^2 \left[\frac{1}{1+(\frac{v_\text{orb}}{v_w})^2}\right]^{3/2}\dot M_\text{donor}, where e is the eccentricity of the orbit and where the r^2 was replaced by its average value a^2\sqrt{1-e^2} during the orbit. ==Accuracy==

Various hydrodynamical simulations of the process considered by BHL accretion have been conducted. These find that while the accretion line only exists temporarily, the values predicted by the BHL formalism agree fairly well with the numerical simulations; to within 10–20%. Binary star systems The BHL formalism is technically only applicable when the wind velocity is much larger than the orbital velocity. For the common scenario of systems containing a giant star, this is typically not the case. Instead, the wind velocity and the orbital velocity are often on the same order of magnitude. In particular, the BHL formalism tends to overpredict the efficiency of the mass transfer in cases where the wind velocity is lower than the orbital velocity. ==Notes==