MULTIPOLE MOMENTS OF STELLAR OSCILLATION MODES

The oscillating mass 2l-pole moment, M(nl), of a star in a given (norm alized) oscillation mode determines the energy that can be absorbed by the mode in a tidal interaction and the power radiated by the mode in gravitational waves, both of which are proportional to \M(nl)\2. The coefficients in the expansion of the vector fields del[r(l)Y(lm)(theta , phi)] in terms of the displacement fields of modes of given l and m are proportional to M(nl). This expansion leads to a sum rule SIGMA(n) \M(nl)\2 = constant. For stars of weak to moderate central condensatio n (such as neutron stars), the f-mode is well approximated by the vect or field being expanded, and therefore it takes the lion's share of th e sum. Thus the multipole moments of all other modes must be small. In their numerical evaluation, it is necessary to know the shape of the eigenfunctions quite precisely, since a small f-mode contamination can significantly increase the obtained values. This contamination occurs in some ''hybrid'' numerical computations of neutron star oscillation s with relativistic equilibrium stars and Newtonian dynamics (e.g., Mc Dermott et al. 1988). In this case, it is due to a slight inconsistenc y in the models and leads to a large overestimate of the power radiate d in gravitational waves by modes other than the f-mode, although thei r oscillation periods are nearly unaffected.