APPENDIX A. The composite cooling function

We assume that the local total cooling function in the post-shock region is given by

where is bremsstrahlung cooling function, the effective cyclotron cooling function, the bremsstrahlung cooling timescale, the cyclotron cooling timescale, the subscript ``s'' denotes the values at the shock surface, and , and are constants to be determined. The bremsstrahlung cooling function is (eg. Rybicki & Lightman 1979), and the bremsstrahlung cooling timescale is

where and are the electron and proton number density respectively.

We assume that the optically thick cyclotron radiation has a Rayleigh-Jean spectrum up to a critical frequency , and the photons with frequencies beyond have insignificant contribution to the cooling process and can be neglected. The angle-averaged cyclotron luminosity is therefore

(Langer, Chanmugam & Shaviv 1982), where c is the speed of light, and A and x are the effective area and the thickness of the emission region respectively. For parameters appropriate for accretion shocks in magnetic cataclysmic variables

(Wada et al. 1980), where is the dimensionless plasma size parameter, the cyclotron frequency, the electron mass, and e the electron charge. The effective cyclotron cooling time scale is therefore

(see Langer, Chanmugam & Shaviv 1982). As ,

If we assume that , then and . It follows that

Since and , we have

Thus, , and , the ratio of the bremsstrahlung cooling timescale to the cyclotron cooling time scale at the shock surface. In terms of the parameters at the shock surface, the ratio is

(Wu, Chanmugam & Shaviv 1994), where B is the magnetic field, the shock temperature, the electron number density at the shock surface, and the shock height.

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