  We investigate the oscillations of slowly rotating superfluid stars, taking
into account the vortex mediated mutual friction force that is expected to be
the main damping mechanism in mature neutron star cores. Working to linear
order in the rotation of the star, we consider both the fundamental f-modes and
the inertial r-modes. In the case of the (polar) f-modes, we work out an
analytic approximation of the mode which allows us to write down a closed
expression for the mutual friction damping timescale. The analytic result is in
good agreement with previous numerical results obtained using an energy
integral argument. We extend previous work by considering the full range of
permissible values for the vortex drag, e.g. the friction between each
individual vortex and the electron fluid. This leads to the first ever results
for the f-mode in the strong drag regime. Our estimates provide useful insight
into the dependence on, and relevance of, various equation of state parameters.
In the case of the (axial) r-modes, we confirm the existence of two classes of
modes. However, we demonstrate that only one of these sets remains purely axial
in more realistic neutron star models. Our analysis lays the foundation for
companion studies of the mutual friction damping of the r-modes at second order
in the slow-rotation approximation, the first time evolutions for superfluid
neutron star perturbations and also the first detailed attempt at studying the
dynamics of superfluid neutron stars with both a relative rotation between the
components and mutual friction.
