Inertial modes in slowly rotating stars: An evolutionary description

Abstract

We present a new hydro code based on spectral methods using spherical coordinates. The first version of this code aims at studying time evolution of inertial modes in slowly rotating neutron stars. In this article, we introduce the anelastic approximation, developed in atmospheric physics, using the mass conservation equation to discard acoustic waves. We describe our algorithms and some tests of the linear version of the code, and also some preliminary linear results. We show, in the Newtonian framework with differentially rotating background, as in the relativistic case with the strong Cowling approximation, that the main part of the velocity quickly concentrates near the equator of the star. Thus, our time evolution approach gives results analogous to those obtained by Karino et al. [1] within a calculation of eigenvectors. Furthermore, in agreement with the work of Lockitch et al.[2], we found that the velocity seems to always get a non-vanishing polar part.

Constrained scheme for the Einstein equations based on the Dirac gauge and spherical coordinates

“…We are also made confident by existing constrained schemes for vector equations which have proved to be successful: the divergence-free hydro scheme of Ref. [56] (the constraint being that the velocity field is divergence-free) and some MHD schemes in cylindrical coordinates [64] (the constraint being that the magnetic field is divergence-free).…”

Rotating Stars in Relativity

“…The 3D simulations need to achieve significantly higher resolutions before definite conclusions can be reached, while the Arras et al work could be extended to rapidly rotating relativistic stars (in which case the mode frequencies and eigenfunctions could change significantly, compared to the slowly rotating Newtonian case, which could affect the nonlinear coupling coefficients). Spectral methods can be used for achieving high accuracy in mode calculations; first results have been obtained by Villain and Bonazzolla [316] for inertial modes of slowly rotating stars in the relativistic Cowling approximation.…”

Continuum and Discrete Initial-Boundary Value Problems and Einstein’s Field Equations

“…The use of spectral methods in relativistic evolutions can be traced back to pioneering work in the mid-1980s [66] (see also [67, 68, 213]). Over the last decade they have gained popularity, with applications in scenarios as diverse as relativistic hydrodynamics [313, 427, 428], characteristic evolutions [43], absorbing and/or constraint-preserving boundary conditions [314, 369, 365, 363], constraint projection [244], late time “tail” behavior of black-hole perturbations [382, 420], cosmological studies [19, 49, 50], extreme-mass-ratio inspirals within perturbation theory and self-forces [112, 162, 111, 425, 114, 113, 123] and, prominently, binary black-hole simulations (see, for example, [384, 329, 71, 381, 132, 288, 402, 131, 90, 289]) and black-hole-neutron-star ones [150, 168]. The method of lines (Section 7.3) is typically used with a small enough timestep so that the time integration error is smaller than the one due to the spatial approximation and spectral convergence is observed.…”