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, $\sim 10^{16}-10^{17}g/s$. 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 $B_*\sim 5\times 10^{11}b_p^{-1/2}L_{x,36}^{1/2}P_{*,10}^{1/2}G$ where the magnetic pitch parameter $b_p\sim$ a few, $L_{x,36}$ is the X-ray luminosity in $10^{36} erg/s$, and $P_{*,10}$ is the pulsar spin period in 10s. Observed quasi-periodic oscillation (QPO) periods tightly constrain the model. For 4U 1626-67, ${\dot M}\approx 2.7\times 10^{16}g/s$ with $b_p^{1/2} B_*\approx 2\times 10^{12}G$. We estimate ${\dot M}\sim 6\times 10^{16} g/s$ and $b_p^{1/2}B_*\sim 5\times 10^{13}G$ for GX 1+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.