Multipole Moments of Stellar Oscillation Modes
Abstract
The oscillating mass 2^{l}pole moment, M_{nl}, of a star in a given (normalized) 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 (absolute value of 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 sum over n(absolute value of M_{nl})^{2} = constant. For stars of weak to moderate central condensation (such as neutron stars), the fmode is well approximated by the vector field being expanded, and therefore it takes the lion's share of the sum. Thus the multipole moments of all other modes must be small. In there numerical evaluation, it is necessary to know the shape of the eigenfunctions quite precisly, since a small fmode contamination can significantly increase the obtained values. This contamination occurs in some `hybrid' numerical computations of neutron star oscillations with relativistic equilibrium stars and Newtonian dynamics (e.g., McDermott et al. 1988). In this case, it is due to a slight inconsistency in the models and leads to a large overestimate of the power radiated in gravitational waves by modes other than the fmode, although their oscillation periods are nearly unaffected.
 Publication:

The Astrophysical Journal
 Pub Date:
 September 1994
 DOI:
 10.1086/174569
 Bibcode:
 1994ApJ...432..296R
 Keywords:

 Astronomical Models;
 Mathematical Models;
 Moments;
 Multipoles;
 Neutron Stars;
 Relativistic Effects;
 Stellar Oscillations;
 Gravitational Waves;
 Series Expansion;
 Spherical Harmonics;
 Stellar Models;
 Tidal Waves;
 Astrophysics;
 GRAVITATION;
 RELATIVITY;
 STARS: NEUTRON;
 STARS: OSCILLATIONS