Introduction

Theoretical studies (e.g. Langer, Chanmugam & Shaviv 1981, Chevalier & Imamura 1982, Imamura 1985) have shown that accretion onto white dwarfs is thermally unstable if bremsstrahlung radiation is the cooling process. The accretion shock, which is formed near the white dwarf surface when supersonic accretion matter becomes subsonic, is thereby driven to oscillate, with a typical oscillation timescale similar to the cooling timescale of the shock-heated matter. These studies stimulated searches for fast photometric variabilities in accreting white dwarfs and led to the subsequent discovery of optical quasi-periodic-oscillations (QPOs) in the AM Herculis systems V834 Cen, AN UMa, EF Eri and VV Pup (Middleditch 1982, Mason et al. 1983, Larsson 1985, 1987). (AM Herculis systems are binaries consisting of a Roche-lobe-filling red dwarf and a magnetic white dwarf both in synchronous rotation with the orbit [See review by Cropper 1990].) While these four systems were found to show QPOs, null detections were reported for other AM Herculis systems.

The typical shock temperature of accreting white dwarfs is  K, and the white dwarf magnetic fields in AM Herculis systems . Cyclotron cooling is therefore also an important process in the post-shock emission regions. For sufficiently low accretion rates, cyclotron cooling may dominate bremsstrahlung cooling (Lamb & Masters 1979, King & Lasota 1979) and hence alter the stability properties of the accretion shock. The effects of cyclotron cooling were investigated by Langer, Chanmugam & Shaviv (1982) and Chanmugam, Langer & Shaviv (1985). Their numerical simulations show damping of the shock oscillations when cyclotron cooling is present. Moreover, the oscillation frequency increases with the magnetic field strength (Wu, Chanmugam & Shaviv 1992).

Although the numerical studies have been successful in describing the instabilities in the accretion flow, they fail to explain why the oscillations are excited and why certain modes can persist while others cannot. There are also discrepancies in the results obtained by calculations using different numerical formulations (cf. Langer, Chanmugam & Shaviv 1982, Imamura 1985).

Here we investigate the stability properties of accreting shocks with cyclotron cooling by considering an analytic approach. Perturbation analyses of accretion shocks have been carried by many authors (e.g. Chevalier & Imamura 1982, Bertschinger 1986, Houck & Chevalier 1992, Toth & Draine 1993, Dgani & Soker 1994). In these studies, the cooling function is generally taken to be a single power-law form. Although it is adequate for the situation where bremsstrahlung cooling is the only process, it is insufficient for the study of the competing effects of two cooling processes with different stability properties. Here, we consider a formulation generalised from that given in Chevalier & Imamura (1982), such that the total cooling is represented by a composite function consisting of two power-law terms. The first term is the bremsstrahlung cooling function, which has the form , where and T are density and temperature respectively. The second term is the effective cyclotron cooling function, mimicked by a function of the form , where a and b are power-law indices to be determined (Appendix A, see also Wu, Chanmugam & Shaviv 1994). With this composite function, we obtain the hydrodynamic equations and carry out a perturbation analysis to determine the oscillation modes and their stability properties.

© Copyright Astronomical Society of Australia 1997