  We present an inversion method based on Bayesian analysis to constrain the
interior structure of terrestrial exoplanets, in the form of chemical
composition of the mantle and core size. Specifically, we identify what parts
of the interior structure of terrestrial exoplanets can be determined from
observations of mass, radius, and stellar elemental abundances. We perform a
full probabilistic inverse analysis to formally account for observational and
model uncertainties and obtain confidence regions of interior structure models.
This enables us to characterize how model variability depends on data and
associated uncertainties. We test our method on terrestrial solar system
planets and find that our model predictions are consistent with independent
estimates. Furthermore, we apply our method to synthetic exoplanets up to 10
Earth masses and up to 1.7 Earth radii as well as to exoplanet Kepler-36b.
Importantly, the inversion strategy proposed here provides a framework for
understanding the level of precision required to characterize the interior of
exoplanets. Our main conclusions are: (1) observations of mass and radius are
sufficient to constrain core size; (2) stellar elemental abundances (Fe, Si,
Mg) are key constraints to reduce degeneracy in interior structure models and
to constrain mantle composition; (3) the inherent degeneracy in determining
interior structure from mass and radius observations does not only depend on
measurement accuracies but also on the actual size and density of the
exoplanet. We argue that precise observations of stellar elemental abundances
are central in order to place constraints on planetary bulk composition and to
reduce model degeneracy. [...]
