Accretion and propeller torque in the spin-down phase of neutron stars: The case of transitional millisecond pulsar PSR J1023+0038

The spin-down rate of PSR J1023+0038, one of the three confirmed transitional millisecond pulsars, was measured in both radio pulsar (radio millisecond pulsar, RMSP) and X-ray pulsar (low-mass X-ray binary, LMXB) states. The spin-down rate in the LMXB state is only about 27 per cent greater than the spin-down rate in the RMSP state. The inner disc radius, r(in), obtained recently by Ertan for the propeller phase, which is close to the co-rotation radius, r(co), and insensitive to the mass-flow rate, can explain the observed torques together with the X-ray luminosities, L-x. The X-ray pulsar and radio pulsar states correspond to accretion with spin-down (weak-propeller) and strong-propeller situations, respectively. A several times increase in the disc mass-flow rate takes the source from the strong propeller with a low L-x to the weak propeller with a higher L-x powered by accretion on to the star. The resultant decrease in r(in) increases the magnetic torque slightly, explaining the observed small increase in the spin-down rate. We have found that the spin-up torque exerted by accreting material is much smaller than the magnetic spin-down torque exerted by the disc in the LMXB state.