Constraints on Torque-Reversing Accretion-Powered X-ray Pulsars

The observed abrupt torque reversals in X-ray pulsars, 4U 1626-67, GX 1+4, and OAO 1657-415, can be explained by transition in accretion flow rotation from Keplerian to sub-Keplerian, which takes place at a critical accretion rate, $\sim {10}^{16}-{10}^{17}g\mathrm{/}s$. When a pulsar system spins up near equilibrium spin before the transition, the system goes into spin-down after transition to sub-Keplerian. If a system is well into the spin-up regime, the transition can cause a sharp decrease in spin-up rate but not a sudden spin-down. These observable types of abrupt torque change are distinguished from the smooth torque variation caused by change of $\dot{M}$ in the Keplerian flow. The observed abrupt torque reversal is expected when the pulsar magnetic field $B_{\ast }\sim 5\times {10}^{11}{b}_p^{-1\mathrm{/}2}{L}_{x,36}^{1\mathrm{/}2}{P}_{\ast ,10}^{1\mathrm{/}2}G$ where the magnetic pitch parameter $b_p\sim$ a few, $L_{x,36}$ is the X-ray luminosity in ${10}^{36}erg\mathrm{/}s$, and $P_{\ast ,10}$ is the pulsar spin period in 10s. Observed quasi-periodic oscillation (QPO) periods tightly constrain the model. For 4U 1626-67, $\dot{M}\approx 2.7\times {10}^{16}g\mathrm{/}s$ with $b_p^{1/2} B_\approx 2\times 10^{12}G$\mathrm{.}Weestimate${\dot M}\sim 6\times 10^{16} g/s$and$b_p^{1/2}B_\sim 5\times 10^{13}G$forGX1+4,and${\dot M} \sim 1\times 10^{17} g/s$and$b_p^{1/2}B_*\sim 2\times 10^{13}G$ for OAO 1657-415. Reliable detection of QPOs before and after torque reversal could directly test the model.