Long term evolution of CFS-unstable neutron stars and role of differential rotation on short time-scales

Abstract

I consider differential rotation, associated with radiation-driven Chandrasekhar-Friedman-Schutz (CFS) instability and its possible observational evidences. I focus on the evolution of the apparent spin frequency, which is typically associated with the motion of a specific point on the stellar surface (e.g., polar cap). I start from long-term evolution (on the timescale when instability significantly changes the spin frequency). For this case, I reduce evolution equations to one differential equation and demonstrate that it can be directly derived from energy conservation law. This equation governs evolution rate through sequence of thermally equilibrium states and provides linear coupling for the cooling power and rotation energy losses via gravitational wave emission. In particular, it shows that differential rotation do not affect long-term spin-down. On the contrary, on the short timescales, differential rotation can significantly modify the apparent spin-down, if one examines a strongly unstable star with very small initial amplitude of unstable mode. This statement is confirmed by consideration of Newtonian non-magnetized perfect fluid and dissipative stellar models as well as magnetized stellar model. For example, despite that widely applied evolution equations predict effective spin to be constant in absence of dissipation, the CFS-unstable star should be observed as spinning-down. However, effects of differential rotation on apparent spin-down are negligible for realistic models of neutron star recycling, unless: (1) the neutron star is not magnetized, and (2) r-mode amplitude is modulated faster than the shear viscosity dissipation timescale, being large enough that spin-down can be measured on modulation timescale.