  We study the time-independent scattering of a planar gravitational wave
propagating in the curved spacetime of a compact body with a polytropic
equation of state. We begin by considering the geometric-optics limit, in which
the gravitational wave propagates along null geodesics of the spacetime; we
show that a wavefront passing through a neutron star of tenuity $R/M = 6$ will
be focussed at a cusp caustic near the star's surface. Next, using the
linearized Einstein Field Equations on a spherically-symmetric spacetime, we
construct the metric perturbations in the odd and even parity sectors; and,
with partial-wave methods, we numerically compute the gravitational scattering
cross section from helicity-conserving and helicity-reversing amplitudes. At
long wavelengths, the cross section is insensitive to stellar structure and, in
the limit $M \omega \rightarrow 0$, it reduces to the known low-frequency
approximation of the black hole case. At higher frequencies $M \omega \gtrsim
1$, the gravitational wave probes the internal structure of the body. In
essence, we find that the gravitational wave cross section is similar to that
for a massless scalar field, although with subtle effects arising from the
non-zero helicity-reversing amplitude, and the coupling in the even-parity
sector between the gravitational wave and the fluid of the body. The cross
section exhibits \emph{rainbow scattering} with an Airy-type oscillation
superposed on a Rutherford cross section. We show that the rainbow angle, which
arises from a stationary point in the geodesic deflection function, depends on
the polytropic index. In principle, rainbow scattering provides a diagnostic of
the equation of state of the compact body; but, in practice, this requires a
high-frequency astrophysical source of gravitational waves.

General Relativity and Quantum Cosmology