## 6. The overlap integralsUsing the and decomposition of Eq. (8) for gives By Eq. (12), V is a spherical harmonic of order 2. Therefore, only the normal modes belonging to will contribute to the overlap integral, that is, Substituting for and V and carrying out integrations over , 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
and modes(Sobouti & Silverman 1978). The
method consisted of expanding the and
potentials of Eqs. (8) in power series of
2) The information thus obtained was used to extract the potential for each of the and 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.
© European Southern Observatory (ESO) 1997 Online publication: June 30, 1998 |