Constraining the shear modulus of a polycrystalline neutron star crust: Hashin-Shtrikman variational approach

The elastic properties of the neutron star crust are thought to play a crucial role in various phenomena of neutron stars (glitches, oscillations, gravitational wave emission) and should be described quantitatively to model these phenomena. The fundamental problem of this description is associated with the polycrystalline nature of the crust: similar to terrestrial materials, the elastic moduli, strictly speaking, depend on the shape and orientation of crystallites, but for the crust, they are unknown. As a result, some assumptions are generally required to predict the elastic properties or constrain their possible range. In this paper, we follow the commonly believed assumption that the crust is (locally) isotropic, which allows us to describe elastic properties by two (effective) parameters: bulk and shear moduli. The bulk modulus is well determined by the Voigt-Reuss bounds, and we constrain the shear modulus by applying, for the first time in astrophysics of compact stars, the variational Hashin-Shtrikman approach, based on the additional assumption that there are no correlations in the orientation of crystallites. We analyse the Hashin-Shtrikman bounds for the one-component crust taking into account the electron screening and the motion of the nuclei, and for two-component static crystals. In particular, we demonstrate that within applied assumptions the effective shear modulus should be lower than the Voigt estimate, typically applied in the astrophysical literature.