  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.
