Results and Discussions

The differential equations are solved using the Runge-Kutta method, with integration from the shock surface to the white dwarf surface. The eigenvalues are obtained when the solution at the white dwarf surface matches the `stationary wall' boundary condition (i.e. ).

In Table 1, the real and imaginary parts of the eigenvalues ( and respectively) of the first eight harmonics for the cases , 1 and 10 are shown. For , only bremsstrahlung cooling is present; for , cyclotron cooling and bremsstrahlung cooling have the same efficiency at the shock surface; and for , cyclotron cooling is ten times more efficient than bremsstrahlung cooling at the shock surface, i.e. the accretion shock can be considered as dominated by cyclotron cooling.

 
Table 1: The eigenvalues for the cases 0, 1 and 10

 
Figure 1: The real part of the eigenvalue, as a function of the harmonic number n. The squares, diamonds and tiangles correspond to the cases 0, 1 and 10 respectively.

 
Figure 2: Same as Fig. 1 for .

In Figure 1, is plotted against n, the harmonic number. When bremsstrahlung is the only cooling process (), the first harmonic (n = 1) is linearly stable (), and the higher harmonics are unstable () (see also Chevalier & Imamura 1982).

For , the first harmonic is stable as for the case , but the second harmonic is also stable. The higher harmonics are, however, unstable, and the corresponding values for are smaller than those for the previous case. For a sufficiently high magnetic field, such as , the first eight harmonics we have obtained are stable. Thus, our perturbation analyses show the presence of cyclotron cooling tends to stabilise the shock oscillations, which is consistent with the results from the numerical studies of Chanmugam, Langer & Shaviv (1985).

In Figure 2, we plot against n. Clearly, there is a linear relationship that increases with n for all three cases, and is approximately given by

where 0.6087, 0.5666 and 0.4526 and -0.0153, 0.0077 and 0.0477 for 0, 1, and 10 respectively. This is similar to a pipe which is open at one end, for which the frequencies of its harmonics are given by , where and are constants.

It is worth noting that the smaller the value of , the steeper the slope of the vs n graph. If we define the cooling time as , then from the definition of we have (the imaginary part of ) proportional to , where . Thus, our calculations show that the imaginary part of the eigenvalue () is determined by both the cooling function and the boundary conditions. Although the real part () obviously depends on the details of the cooling function, whether the boundary condition is crucial still needs to be investigated.

Using the expression of given in Appendix A, one can estimate that for typical accretion parameters of AM Herculis systems, 50 MG when 10. The result that the accretion shock is stable for implies QPOs of Hz are suppressed when 50 MG. For AM Herculis systems with a weaker magnetic field ( MG), the higher oscillation modes are unstable, despite the lower oscillation modes being stable. However, the high oscillation modes have high frequencies and are difficult to excite. The null detection of 1 Hz QPOs in some AM Herculis systems may be due to insufficient instrumental sensitivity to detect the high frequency modes.

© Copyright Astronomical Society of Australia 1997