2025ApJ...980..232T

PSR J2021+4026 is a gamma-ray pulsar having variations in its spin-down rate and gamma-ray flux. Its variations in timing and emission are correlated, e.g., a larger spin-down rate for a low gamma-ray flux. We show that the mode change in PSR J2021+4026 can be understood in the precession scenario. In the precession model, the inclination angle is modulated due to precession. At the same time, the wobble angle may decay with time. This results in damping of the precession. Combined with the magnetospheric torque model and the outer gap model, the damped precession can explain that (1) when the inclination angle is larger, the spin-down rate will be larger, accompanied by a lower gamma-ray flux. (2) The variation in amplitude of the gamma-ray flux and spin-down rate is smaller than previous results due to the damping of the precession. The modulation period is becoming shorter due to a smaller wobble angle. In the end, we propose that there are two kinds of modulations in pulsars. Long-term modulations in pulsars may be due to precession. Short-term modulations may be of magnetospheric origin.