The Dipole Magnetic Field and Spin-down Evolutions of the High Braking Index Pulsar PSR J1640-4631

In this work, we interpreted the high braking index of PSR J1640-4631 with a combination of the magneto-dipole radiation and dipole magnetic field decay models. By introducing a mean rotation energy conversion coefficient \overline{\zeta }, the ratio of the total high-energy photon energy to the total rotation energy loss in the whole life of the pulsar, and combining the pulsar’s high-energy and timing observations with a reliable nuclear equation of state, we estimate the pulsar’s initial spin period, {P}₀∼ (17{--}44) ms, corresponding to the moment of inertia I∼ (0.8{--}2.1)× {10}⁴⁵ g cm². Assuming that PSR J1640-4631 has experienced a long-term exponential decay of the dipole magnetic field, we calculate the true age {t}_{age}, the effective magnetic field decay timescale {τ }_{{D}}, and the initial surface dipole magnetic field at the pole {B}_p(0) of the pulsar to be 2900-3100 yr, 1.07(2)× {10}⁵ yr, and (1.84{--}4.20)× {10}¹³ G, respectively. The measured braking index of n=3.15(3) for PSR J1640-4631 is attributed to its long-term dipole magnetic field decay and a low magnetic field decay rate, {{dB}}_{{p}}/{dt}∼ -(1.66{--}3.85)× {10}⁸ G yr^-1. Our model can be applied to both the high braking index (n> 3) and low braking index (n< 3) pulsars, tested by the future polarization, timing, and high-energy observations of PSR J1640-4631.