Importance of Perturbed Gravitational Potentials in Differentially Rotating Newtonian Stars
It is usually believed that the Cowling approximation can give satisfactory solutions if the stars %are highly compressible have soft equations of state and/or if the strongly general relativistic stars are treated in the case of rigid rotation. Since, however, there have been no systematic studies about the accuracy of the Cowling approximation for differentially rotating compressible stars, we investigate eigenfrequencies and eigenfunctions of the oscillation modes in {\it rapidly} and {\it differentially} rotating compressible stars by employing the exact method including full-perturbations and the Cowling approximation. We have found that the Cowling approximation for f-mode oscillations is not a good approximation in rapidly and differentially rotating stars, although rapid rotation makes this approximation better for rigidly rotating stars. This result suggests that we must be careful when we apply the Cowling approximation to differentially rotating stars even in the framework of general relativity. On the other hand, the approximation will work well for r-modes even if the star is rotating differentially. Therefore, the Cowling approximation can be used as a strong tool for the investigation of r-mode oscillations in the general relativistic framework that it is difficult to compute including the perturbations of gravity.