r-mode instability: Analytical solution with gravitational radiation reaction

Analytical r-mode solutions are investigated within the linearized theory in the case of a slowly rotating, Newtonian, barotropic, nonmagnetized, perfect-fluid star in which the gravitational radiation (GR) reaction force is present. For the GR reaction term we use the 3.5 post-Newtonian order expansion of the GR force, in order to include the contribution of the current quadrupole moment. We find the explicit expression for the r-mode velocity perturbations and we conclude that they are sinusoidal with the same frequency as the well-known GR force-free linear r-mode solution, and that the GR force drives the r-modes unstable with a growth time scale that agrees with the expression first found by Lindblom, Owen, and Morsink. We also show that the amplitude of these velocity perturbations is corrected, relatively to the GR force-free case, by a term of order Ω⁶, where Ω is the angular velocity of the star.