Stability of Accretion Shocks with a Composite Cooling Function

Curtis J. Saxton , Kinwah Wu , Helen Pongracic, PASA, 14 (2), in press.

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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 tex2html_wrap_inline643 are obtained when the solution at the white dwarf surface matches the `stationary wall' boundary condition (i.e. tex2html_wrap_inline635).

In Table 1, the real and imaginary parts of the eigenvalues (tex2html_wrap_inline651 and tex2html_wrap_inline653 respectively) of the first eight harmonics for the cases tex2html_wrap_inline655, 1 and 10 are shown. For tex2html_wrap_inline655, only bremsstrahlung cooling is present; for tex2html_wrap_inline659, cyclotron cooling and bremsstrahlung cooling have the same efficiency at the shock surface; and for tex2html_wrap_inline661, 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 tex2html_wrap_inline663 0, 1 and 10

Figure 1: The real part of the eigenvalue, tex2html_wrap_inline651 as a function of the harmonic number n. The squares, diamonds and tiangles correspond to the cases tex2html_wrap_inline663 0, 1 and 10 respectively.

Figure 2: Same as Fig. 1 for tex2html_wrap_inline653.

In Figure 1, tex2html_wrap_inline651 is plotted against n, the harmonic number. When bremsstrahlung is the only cooling process (tex2html_wrap_inline655), the first harmonic (n = 1) is linearly stable (tex2html_wrap_inline701), and the higher harmonics are unstable (tex2html_wrap_inline703) (see also Chevalier & Imamura 1982).

For tex2html_wrap_inline659, the first harmonic is stable as for the case tex2html_wrap_inline655, but the second harmonic is also stable. The higher harmonics are, however, unstable, and the corresponding values for tex2html_wrap_inline651 are smaller than those for the previous case. For a sufficiently high magnetic field, such as tex2html_wrap_inline661, 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 tex2html_wrap_inline653 against n. Clearly, there is a linear relationship that tex2html_wrap_inline653 increases with n for all three cases, and tex2html_wrap_inline653 is approximately given by
where tex2html_wrap_inline723 0.6087, 0.5666 and 0.4526 and tex2html_wrap_inline725 -0.0153, 0.0077 and 0.0477 for tex2html_wrap_inline663 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 tex2html_wrap_inline729, where tex2html_wrap_inline731 and tex2html_wrap_inline733 are constants.

It is worth noting that the smaller the value of tex2html_wrap_inline563, the steeper the slope of the tex2html_wrap_inline653 vs n graph. If we define the cooling time as tex2html_wrap_inline741, then from the definition of tex2html_wrap_inline643 we have tex2html_wrap_inline745 (the imaginary part of tex2html_wrap_inline619) proportional to tex2html_wrap_inline749, where tex2html_wrap_inline751. Thus, our calculations show that the imaginary part of the eigenvalue (tex2html_wrap_inline653) is determined by both the cooling function and the boundary conditions. Although the real part (tex2html_wrap_inline651) obviously depends on the details of the cooling function, whether the boundary condition is crucial still needs to be investigated.

Using the expression of tex2html_wrap_inline563 given in Appendix A, one can estimate that for typical accretion parameters of AM Herculis systems, tex2html_wrap_inline759 50 MG when tex2html_wrap_inline663 10. The result that the accretion shock is stable for tex2html_wrap_inline763 implies QPOs of tex2html_wrap_inline765 Hz are suppressed when tex2html_wrap_inline767 50 MG. For AM Herculis systems with a weaker magnetic field (tex2html_wrap_inline769 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.

Next Section: Conclusions
Title/Abstract Page: Stability of Accretion Shocks
Previous Section: Perturbation Analysis
Contents Page: Volume 14, Number 2

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