Maximum elastic deformations of relativistic stars

We present a method for calculating the maximum elastic quadrupolar deformations of relativistic stars, generalizing the previous Newtonian Cowling approximation integral given by Ushomirsky et al. [Mon. Not. R. Astron. Soc. 319, 902 (2000)]. (We also present a method for Newtonian gravity with no-Cowling approximation.) We apply these methods to the $m=2$ quadrupoles most relevant for gravitational radiation in three cases: crustal deformations, deformations of crystalline cores of hadron-quark hybrid stars, and deformations of entirely crystalline color superconducting quark stars. In all cases, we find suppressions of the quadrupole due to relativity compared to the Newtonian Cowling approximation, particularly for compact stars. For the crust these suppressions are up to a factor of ~6, for hybrid stars they are up to ~4, and for solid quark stars they are at most ~2, with slight enhancements instead for low mass stars. We also explore ranges of masses and equations of state more than in previous work and find that for some parameters the maximum quadrupoles can still be very large. Even with the relativistic suppressions, we find that 1.4M_ stars can sustain crustal quadrupoles of a few×10³⁹ g cm² for the SLy equation of state or close to 10⁴⁰ g cm² for equations of state that produce less compact stars. Solid quark stars of 1.4M_ can sustain quadrupoles of around 10⁴⁴ g cm². Hybrid stars typically do not have solid cores at 1.4M_, but the most massive ones (~2M_) can sustain quadrupoles of a few×10⁴¹ g cm² for typical microphysical parameters and a few×10⁴² g cm² for extreme ones. All of these quadrupoles assume a breaking strain of 10^-1 and can be divided by 10⁴⁵ g cm² to yield the fiducial ``ellipticities'' quoted elsewhere.