Accretion onto Magnetic White Dwarfs

The height of accretion shocks (i.e. the thickness of post-shock regions), , of magnetic white dwarfs are generally , where is the white dwarf radius. For typical accretion rates (), the ram pressure of the accretion matter near the white dwarf surface is less than the magnetic stress. As the accretion flow is channelled by the field lines, it can be approximated by a one-dimensional planar model. The time-dependent continuity and momentum equations are therefore

and

where P is the pressure and v is the velocity. If we assume that the temperature of the accretion matter is proportional to and the matter has an adiabatic index of 5/3, the energy equation is then



(cf. Wu 1994), where is the effective ratio of the bremsstrahlung cooling timescale to the cyclotron cooling timescale at the shock surface. In the above equation and hereafter, the subscript ``s'' denotes parameters and variables at the shock surface. The coefficient corresponding to bremsstrahlung cooling, where is the Boltzmann constant, h the Planck constant, c the velocity of light, e the electron charge, the electron mass, the proton mass, the mean molecular weight of the gas and the Gaunt factor (see Rybicki & Lightman 1979). The numerical value of is in c.g.s. units, assuming that and . For typical parameters of AM Herculis systems, and (Appendix A).

We assume a `stationary wall' boundary condition at the white dwarf surface, so that the lower boundary condition is v = 0. At the shock surface, we assume the strong shock condition, so that , and , where is the free-fall velocity at the white dwarf surface and is density above the shock. With the boundary conditions specified, a closed-form steady state solution to equations (1), (2) and (3) and a steady state shock height can be obtained. (Wu, Chanmugam & Shaviv 1994, see also Aizu 1973 and Chevalier & Imamura 1982 for the cases of single power-law form cooling functions).

© Copyright Astronomical Society of Australia 1997