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 $\ensuremath{\sim}6$, for hybrid stars they are up to $\ensuremath{\sim}4$, and for solid quark stars they are at most $\ensuremath{\sim}2$, 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 $1.4{M}_{\ensuremath{\bigodot}}$ stars can sustain crustal quadrupoles of a $\mathrm{\text{few}}\ifmmode\times\else\texttimes\fi{}{10}^{39}\text{ }\text{ }\mathrm{g}\text{ }{\mathrm{cm}}^{2}$ for the SLy equation of state or close to ${10}^{40}\text{ }\text{ }\mathrm{g}\text{ }{\mathrm{cm}}^{2}$ for equations of state that produce less compact stars. Solid quark stars of $1.4{M}_{\ensuremath{\bigodot}}$ can sustain quadrupoles of around ${10}^{44}\text{ }\text{ }\mathrm{g}\text{ }{\mathrm{cm}}^{2}$. Hybrid stars typically do not have solid cores at $1.4{M}_{\ensuremath{\bigodot}}$, but the most massive ones ($\ensuremath{\sim}2{M}_{\ensuremath{\bigodot}}$) can sustain quadrupoles of a $\mathrm{\text{few}}\ifmmode\times\else\texttimes\fi{}{10}^{41}\text{ }\text{ }\mathrm{g}\text{ }{\mathrm{cm}}^{2}$ for typical microphysical parameters and a $\mathrm{\text{few}}\ifmmode\times\else\texttimes\fi{}{10}^{42}\text{ }\text{ }\mathrm{g}\text{ }{\mathrm{cm}}^{2}$ for extreme ones. All of these quadrupoles assume a breaking strain of ${10}^{\ensuremath{-}1}$ and can be divided by ${10}^{45}\text{ }\text{ }\mathrm{g}\text{ }{\mathrm{cm}}^{2}$ to yield the fiducial ``ellipticities'' quoted elsewhere.