Motivated by the remarkably narrow range of measured spin frequencies of \sim 20 accreting ( and weakly magnetic ) neutron stars in the Galaxy , \citet Bildsten98 : GWs conjectured that their spin-up had been halted by the emission of gravitational waves .
If so , then the brightest persistent X-ray source on the sky , Scorpius X-1 , should be detected by gravitational wave interferometers within ten years . \citet Bildsten98 : GWs pointed out that small nonaxisymmetric temperature variations in the accreted crust will lead to “ wavy ” electron capture layers , and the resulting horizontal density variations near e ^ { - } capture layers create a mass quadrupole moment .
Neglecting the elastic response of the crust , \citet Bildsten98 : GWs estimated that even e ^ { - } capture layers in the thin outer crust can develop the quadrupole necessary to balance accretion torque with gravitational waves , Q _ { 22 } \sim 10 ^ { 37 } -10 ^ { 38 } g cm ^ { -2 } for accretion rates \dot { M } \sim 10 ^ { -10 } -2 \times 10 ^ { -8 } M _ { \odot } yr ^ { -1 } .

We present a full calculation of the crust ’ s elastic adjustment to the density perturbations induced by the temperature-sensitive e ^ { - } capture reactions .
We find that , due to the tendency of the denser material to sink rather than spread sideways , neglecting the elastic response of the crust overestimates , by a factor of 20 - 50 , the Q _ { 22 } that results from a wavy capture layer in the thin outer crust .
However , we find that this basic picture , when applied to capture layers in the deep inner crust , can still generate Q _ { 22 } in the necessary range , as long as there are \lesssim 5 \% lateral temperature variations at densities in excess of 10 ^ { 12 } { g cm ^ { -3 } } , and as long as the crustal breaking strain is high enough .
By calculating the thermal flow throughout the core and the crust , we find that temperature gradients this large are easily maintained by asymmetric heat sources or lateral composition gradients in the crust .
If the composition or heating asymmetries are independent of the accretion rate , then for \dot { M } \lesssim 5 \times 10 ^ { -9 } M _ { \odot } yr ^ { -1 } the induced quadrupole moments have approximately the same scaling , \propto \dot { M } ^ { 1 / 2 } , as that necessary to balance the accretion torque at the same spin frequency for all \dot { M } .
Temperature gradients in the deep crust lead to a modulation in the thermal emission from the surface of the star that is correlated with Q _ { 22 } .
In addition , a \sim 0.5 \% lateral variation in the nuclear charge-to-mass ratio in the crust will also result in a Q _ { 22 } sufficient to halt spin-up from accretion even in the absence of a lateral temperature gradient .

We also derive a general relation between the stresses and strains in the crust and the maximum quadrupole moment they can generate .
We show under quite general conditions that maintaining a Q _ { 22 } of the magnitude necessary to balance the accretion torque requires dimensionless strain \sigma \sim 10 ^ { -2 } at near-Eddington accretion rates , of order the breaking strain of conventional materials .
This leads us to speculate that accreting neutron stars reach the same equilibrium spin because they all are driven to the maximum Q _ { 22 } that the crust can sustain .