In the Mood for Love: A Mode-sum approach to Tidal deformation of slowly Rotating Stars

Abstract

In this 5th year MS thesis, we analyze the dynamical tidal deformations of a slowly rotating star using a mode-sum approach. We analyze this problem in the Newtonian, the post-Newtonian and the Relativistic frameworks. First, we build up Newtonian and postNewtonian Lagrangian perturbation formalisms for a slowly rotating Newtonian star and estimate the energy associated with such perturbations. We then devise new normalization conditions and mode decomposition conventions to simplify our equations. We then describe our Newtonian and post-Newtonian tidal environments, commenting on their parity sectors. Next, we calculate the dynamical mode amplitudes and the Overlap integrals associated with each tidal driving term present externally. We also calculate the boundary conditions associated with each of the potentials and parameters present inside the star, using multipole moments, virial theorems and spherical harmonics decomposition to simplify our problem. We estimate the tidal Love numbers for these stars, both in the Fourier domain and in the Time domain, and end up deriving new structures like the tidal Love tensors and the tidal Love matrices. We consider the cases of rotating and non-rotating stars separately, while also analyzing the limiting cases of our results. We study a similar calculation for a compact star being driven by relativistic tidal fields, although by using a different approach to solve the problem. We then estimate the tidal phasing of the gravitational waves emitted by a binary inspiral having a tidally deformed, slowly rotating Newtonian star. We also calculate the post-Newtonian corrections to this tidal phasing embedded in the gravitational waves. Our end-goal is to incorporate the Equation of State of a star in this tidal phasing through the use of tidal Love numbers. A few future directions, either analytical or computational, to be taken after this thesis are also discussed.