Braking index of the frequently glitching PSR J0537$-$6910
The pulsar J0537$-$6910 undergoes spin-up glitches more frequently than any other known pulsar, at a rate of roughly thrice per year. Its glitches are typically large and accompanied by spin-down rate changes $\Delta\dot\nu$ that partially recover with a nearly constant positive frequency second derivative $\ddot\nu$ for the post-glitch intervals. The long-term value of $\ddot\nu$, however, is negative because $\dot\nu$ has decreased over the years of observations. We wish to determine if permanent shifts (non-relaxing parts of the glitch change $\Delta\dot\nu$ in the spin-down rate, like those observed in the Crab pulsar) can explain the long-term enhancement of the spin-down rate which results in an effective negative braking index. We demonstrate, as a proof of concept, that the actual braking index associated with the pulsar's braking torque can be n~3 if the internal superfluid torque and permanent shifts are considered. We use published RXTE and NICER data to calculate the average permanent shift per glitch needed to bring an underlying braking index $n$ to the effective long-term value n' =-1.2 inferred from the data. We use this average value as the actual permanent shift in each glitch and extract the contributions of the internal and external torques to $\ddot\nu$, under the assumption that the next glitch occurs when all glitch-induced offsets to internal torques are fully restored. We find that if the braking index of the magnetospheric torque is close to n~3, moderate permanent changes of the spin-down rate are required, similar to those inferred for the Crab pulsar. The natural mechanism to produce such permanent changes is crustquakes. Crustal failure associated with PSR J0537$-$6910 glitches can have interesting and potentially observable consequences, such as transient changes of the X-ray emission, activation of radio emission, or emission of gravitational waves.