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Astron. Astrophys. 321, 1024-1026 (1997)

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6. The overlap integrals

Using the [FORMULA] and [FORMULA] decomposition of Eq. (8) for [FORMULA] gives


By Eq. (12), V is a spherical harmonic of order 2. Therefore, only the normal modes belonging to [FORMULA] will contribute to the overlap integral, that is, [FORMULA] Substituting for [FORMULA] and V and carrying out integrations over [FORMULA], [FORMULA] gives


For numerical calculations the following steps were taken.

1) A Rayleigh-Ritz variational method was employed to obtain the eigenfrequencies and eigenfunctions for various [FORMULA] and [FORMULA] modes(Sobouti & Silverman 1978). The method consisted of expanding the [FORMULA] and [FORMULA] potentials of Eqs. (8) in power series of r, substituting the resulting [FORMULA] 's in Eq. (6) and finding the expansion coefficients by variational calculations.

2) The information thus obtained was used to extract the [FORMULA] potential for each of the [FORMULA] and [FORMULA] modes and to calculate the overlap integrals of Eq. (19a), and eventually the cross sections and the energy absorption rates. Numerical values for polytropic structures are summarized in Table 1.


Table 1. Cross sections for different modes of polytropic indeces 1.5, 2, 2.5

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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998