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 similar to 6, for hybrid stars they are up to similar to 4, and for solid quark stars they are at most similar to 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(circle dot) stars can sustain crustal quadrupoles of a few x 10(39) g cm(2) for the SLy equation of state or close to 10(40) g cm(2) for equations of state that produce less compact stars. Solid quark stars of 1.4M(circle dot) can sustain quadrupoles of around 10(44) g cm(2). Hybrid stars typically do not have solid cores at 1.4M(circle dot), but the most massive ones (similar to 2M(circle dot)) can sustain quadrupoles of a few x 10(41) g cm(2) for typical microphysical parameters and a few x 10(42) g cm(2) for extreme ones. All of these quadrupoles assume a breaking strain of 10(-1) and can be divided by 10(45) g cm(2) to yield the fiducial ellipticities quoted elsewhere.