The braking index of PSR B0540-69 and the associated pulsar wind nebula emission after spin-down rate transition

In 2011 December, PSR B054-69 experienced a spin-down rate transition (SRT), after which the spin-down power of the pulsar increased by $\sim 36{{\ \rm per\ cent}}$. About 1000 d after the SRT, the X-ray luminosity of the associated pulsar wind nebula (PWN) was found to brighten by $32\pm 8{{\ \rm per\ cent}}$. After the SRT, the braking index n of PSR B0540-69 changes from n = 2.12 to 0.03 and then keeps this value for about five years before rising to n = 0.9 in the following years. We find that most of the current models have difficulties in explaining the measured braking index. One exceptive model of the braking index evolution is the increasing dipole magnetic field of PSR B0540-69. We suggest that the field increase may result from some instabilities within the pulsar core that enhance the poloidal component at the price of toroidal component of the magnetic field. The increasing dipole magnetic field will result in the X-ray brightening of the PWN. We fit the PWN X-ray light curve by two models: one assumes a constant magnetic field within the PWN during the brightening and the other assumes an enhanced magnetic field proportional to the energy density of the PWN. It appears that the two models fit the data well, though the later model seems to fit the data a bit better. This provides marginal observational evidence that magnetic field in the PWN is generated by the termination shock. Future high-quality and high-cadence data are required to draw a solid conclusion.