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Astron. Astrophys. 331, 485-492 (1998)

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Appendix

In the special case of a weak and purely radial perturbation, [FORMULA], it is easy to show the dependence of the local oscillation frequency on the position of the particle within the density wave. While this is quite different from the potential arising from a shearing density wave, one can readily verify by numerical integration of the orbits that the oscillation frequency of mass points in a potential of the type of Eq. (2) roughly follows the behaviour predicted here with minor deviations resulting from the time-dependence of the wavelength of the perturbation (Fig. 4).

The assumption of a purely radial perturbation allows a closed integration of Eq. (1b). Inserting the result into Eq. (1a) yields

[EQUATION]

where [FORMULA] is an integration constant. To the author's knowledge, a closed solution of this differential equation does not exist. We therefore chose to investigate its solutions using the method of harmonic balance (e.g., Bogoliubov & Mitropolski 1961). We assume a quasi-harmonic solution of the form

[EQUATION]

where [FORMULA]. Inserting this into Eq. (A1) and Fourier analyzing the nonlinear term leads to

[EQUATION]

where [FORMULA] and [FORMULA] denote the Bessel functions of order 0 and 1, respectively. By comparing the 'leading' terms proportional to [FORMULA] one obtains

[EQUATION]

A straightforward application of (A3) to the perturbation used in our simulations - justified a posteriori by Fig. 4 - can be done by identifying [FORMULA] with the distance of the guiding center

of a particle from the wave crest. Since the epicycle sizes are small compared to the wavelegth [FORMULA] of the perturbation, one may furthermore approximate [FORMULA]. This leads to Eq. (3). These approximations obviously break down for epicycle sizes of the order of a typical stellar epicycle size where the amplitude of the oscillation in [FORMULA] has to be zero.

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© European Southern Observatory (ESO) 1998

Online publication: February 16, 1998
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