Deformations of Accreting Neutron Star Crusts and Gravitational Wave Emission

Motivated by the remarkably narrow range of measured spin frequencies of ∼ 20 accreting (and weakly magnetic) neutron stars in the Galaxy, Bildsten (1998a) 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. Bildsten (1998a) 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, Bildsten (1998a) estimated that even e capture layers in the thin outer crust can develop the quadrupole necessary to balance accretion torque with gravitational waves, Q22 ∼ 10 37 − 10 g cm for accretion rates Ṁ ∼ 10 − 2× 10 M⊙ yr . 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 Q22 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 Q22 in the necessary range, as long as there are ∼ < 5% lateral temperature variations at densities in excess of 10 g cm, 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 Ṁ ∼ < 5 × 10 M⊙ yr −1 the induced quadrupole moments have approximately the same scaling, ∝ Ṁ, as that necessary to balance the accretion torque at the same spin frequency for all Ṁ . Temperature gradients in the deep crust lead to a modulation in the thermal emission from the surface of the star that is correlated with Q22. In addition, a ∼ 0.5% lateral variation in the nuclear charge-to-mass ratio in the crust will also result in a Q22 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 Q22 of the magnitude necessary to balance the accretion torque requires dimensionless strain σ ∼ 10 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 Q22 that the crust can sustain.