 Spherical ray invariant

For a light ray in a spherically stratified refractive index $n(r)$, $p=n(r)r\sin \psi$ is constant, where $\psi$ is the angle to the radius. Approximate the medium by concentric homogeneous shells. Straight propagation within each shell conserves $r\sin \psi$, the line's distance from the centre; Snell's law at each shell boundary conserves $nr\sin \psi$. Passing to the smooth limit proves the result. In an exterior region with $n=1$, $p$ is the distance of the asymptotic straight light ray from the centre. Consequently a surface-tangent light ray at radius $R$ and refractive index $n_s$ has vacuum asymptote at altitude $(n_s-1)R$, provided the light ray reaches the exterior without trapping.