Gravitational radiation limit on the spin of young neutron stars

A newly discovered instability in rotating neutron stars, driven by gravitational radiation reaction acting on the stars' $r$-modes, is shown here to set an upper limit on the spin rate of young neutron stars. We calculate the timescales for growth of linear perturbations due to gravitational radiation reaction, and for dissipation by shear and bulk viscosity, working to second order in a slow-rotation expansion within a Newtonian polytropic stellar model. The results are very temperature-sensitive: in hot neutron stars ($T>10^9$ K), the lowest-order $r$-modes are unstable, while in colder stars they are damped by viscosity. These calculations have a number of interesting astrophysical implications. First, the $r$-mode instability will spin down a newly born neutron star to a period close to the initial period inferred for the Crab pulsar, probably between 10 and 20 ms. Second, as an initially rapidly rotating star star spins down, an energy equivalent to roughly 1% of a solar mass is radiated as gravitational waves, which makes the process an interesting source for detectable gravitational waves. Third, the $r$-mode instability rules out the scenario whereby millisecond pulsars are formed by accretion-induced collapse of a white dwarf: the new star would be hot enough to spin down to much slower rates.