Nonradial Oscillations of Neutron Stars with a Solid Crust -- Analysis in the Relativistic Cowling Approximation--

Nonradial oscillations of relativistic neutron stars with a solid crust are computed in the relativistic Cowling approximation, in which all metric perturbations are ignored. For the modal analysis, we employ three-component relativistic neutron star models with a solid crust, a fluid core, and a fluid ocean. As a measure for the relativistic effects on the oscillation modes, we calculate the relative frequency difference defined as $\delta\sigma/\sigma\equiv(\sigma_{GR}-\sigma_N)/\sigma_{GR}$, where $\sigma_{GR}$ and $\sigma_R$ are, respectively, the relativistic and the Newtonian oscillation frequencies. The relative difference $\delta\sigma/\sigma$ takes various values for different oscillation modes of the neutron star model, and the value of $\delta\sigma/\sigma$ for a given mode depends on the physical properties of the models. We find that $|\delta\sigma/\sigma|$ is less than $\sim0.1$ for most of the oscillation modes we calculate, although there are a few exceptions such as the fundamental (nodeless) toroidal torsional modes in the crust, the surface gravity modes confined in the surface ocean, and the core gravity modes trapped in the fluid core. We also find that the modal properties, represented by the eigenfunctions, are not strongly affected by introducing general relativity. It is however shown that the mode characters of the two interfacial modes, associated with the core/crust and crust/ocean interfaces, have been interchanged between the two through an avoided crossing when we move from Newtonian dynamics to general relativistic dynamics.