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We study the time-independent scattering of a planar gravitational wavepropagating in the curved spacetime of a compact body with a polytropicequation of state. We begin by considering the geometric-optics limit, in whichthe gravitational wave propagates along null geodesics of the spacetime; weshow that a wavefront passing through a neutron star of tenuity $R/M = 6$ willbe focussed at a cusp caustic near the star's surface. Next, using thelinearized Einstein Field Equations on a spherically-symmetric spacetime, weconstruct the metric perturbations in the odd and even parity sectors; and,with partial-wave methods, we numerically compute the gravitational scatteringcross section from helicity-conserving and helicity-reversing amplitudes. Atlong wavelengths, the cross section is insensitive to stellar structure and, inthe limit $M \omega \rightarrow 0$, it reduces to the known low-frequencyapproximation of the black hole case. At higher frequencies $M \omega \gtrsim1$, the gravitational wave probes the internal structure of the body. Inessence, we find that the gravitational wave cross section is similar to thatfor a massless scalar field, although with subtle effects arising from thenon-zero helicity-reversing amplitude, and the coupling in the even-paritysector between the gravitational wave and the fluid of the body. The crosssection exhibits \emph{rainbow scattering} with an Airy-type oscillationsuperposed on a Rutherford cross section. We show that the rainbow angle, whicharises from a stationary point in the geodesic deflection function, depends onthe polytropic index. In principle, rainbow scattering provides a diagnostic ofthe equation of state of the compact body; but, in practice, this requires ahigh-frequency astrophysical source of gravitational waves.

Rainbow scattering of gravitational plane waves by a compact body