We present a method for calculating the maximum elastic quadrupolar deformations of relativistic stars , generalizing the previous Newtonian , Cowling approximation integral given by [ G. 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 \sim 6 , for hybrid stars they are up to \sim 4 , and for solid quark stars they are at most \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 _ { \odot } stars can sustain crustal quadrupoles of \text { a few } \times 10 ^ { 39 } \text { g cm } ^ { 2 } for the SLy equation of state , or close to 10 ^ { 40 } \text { g cm } ^ { 2 } for equations of state that produce less compact stars .
Solid quark stars of 1.4 M _ { \odot } can sustain quadrupoles of around 10 ^ { 44 } \text { g cm } ^ { 2 } .
Hybrid stars typically do not have solid cores at 1.4 M _ { \odot } , but the most massive ones ( \sim 2 M _ { \odot } ) can sustain quadrupoles of \text { a few } \times 10 ^ { 41 } \text { g cm } ^ { 2 } for typical microphysical parameters and \text { a few } \times 10 ^ { 42 } \text { g cm } ^ { 2 } for extreme ones .
All of these quadrupoles assume a breaking strain of 10 ^ { -1 } and can be divided by 10 ^ { 45 } \text { g cm } ^ { 2 } to yield the fiducial “ ellipticities ” quoted elsewhere .