  We establish spectral rigidity for spherically symmetric manifolds with
boundary and interior interfaces determined by discontinuities in the metric
under certain conditions. Rather than a single metric, we allow two distinct
metrics in between the interfaces enabling the consideration of two wave types,
like P- and S-polarized waves in isotropic elastic solids. Terrestrial planets
in our solar system are approximately spherically symmetric and support
toroidal and spheroidal modes. Discontinuities typically correspond with phase
transitions in their interiors. Our rigidity result applies to such planets as
we ensure that our conditions are satisfied in generally accepted models in the
presence of a fluid outer core. The proof is based on a novel trace formula. We
also prove that the length spectrum of the Euclidean ball is simple.
