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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

[EQUATION]

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

[EQUATION]

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]

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
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