  We present an analytic description of the $r$-mode instability in newly-born
neutron stars, using the approximation of uniform density. Our computation is
consistently accurate to second order in the angular velocity of the star. We
obtain formulae for the growth-time of the instability due to
gravitational-wave emission, for both current and mass multipole radiation and
for the damping timescale, due to viscosity. The $l=m=2$ current-multipole
radiation dominates the timescale of the instability. We estimate the deviation
of the second order accurate results from the lowest order approximation and
show that the uncertainty in the equation of state has only a small effect on
the onset of the $r$-mode instability. The viscosity coefficients and the
cooling process in newly-born neutron stars are, at present, uncertain and our
analytic formaulae enables a quick check of such effects on the development of
the instability.

Astrophysics;General Relativity and Quantum Cosmology