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

Spherically symmetric terrestrial planets with discontinuities are spectrally rigid