Slowly Rotating Relativistic Stars: VII. Gravitational Radiation from the Quasi-Radial Modes

When a relativistic star rotates slowly and rigidly, centrifugal forces flatten it slightly, thereby catalyzing a small admixture of quadrupolar vibration into its radial modes and a damping of the resulting quasi-radial modes. The damping rate $1\mathrm{/}\tau$ of each quasi-radial mode divided by its frequency ${\sigma }^{(0)}$ is given by $(1\mathrm{/}\tau )\mathrm{/}{\sigma }^{(0)}=\beta ({\sigma }^{(0)}{)}^3{\mathrm{\Omega }}^4{R}^8\mathrm{/}M,$where $\mathrm{\Omega }$, $R$ and $M$ are the angular velocity, radius and mass of the star, and $\beta$ is a dimensionless number that depends on the mode, on relativistic corrections, and on the structure of the star, and is typically of order unity for the fundamental quasi-radial mode in the nonrelativistic limit. In this paper we develop equations and an algorithm for computing the emitted waves and the resulting damping factor $\beta$. The rotation is treated to second-order in the angular velocity and the pulsation amplitudes are assumed small and linearized, but no other approximations are made.