Study of the High Braking Index of Pulsar PSR J1640–4631 by the Combination of Magnetic Dipole and Gravitational Wave Radiation

The braking index of a pulsar is a key parameter for understanding its radiation characteristics and kinetic energy loss mechanisms. The magnetic dipole radiation (MDR) model predicts a constant value for the braking index \(n = 3\), as described by a power-law form of the stellar spin-down between the spin angular frequency (\(\Omega \)) and its derivative as \( - \dot {\Omega } \propto {{\Omega }^{{n = 3}}}\). However, the timing observations indicate that the pulsar PSR J1640–4631 has an unusually high braking index of \(n = 3.15 \pm 0.03\), which is unlike the other pulsars with precisely measured index in between 1 and 3. Therefore, the spin-down of this pulsar should not be controlled by the standard MDR model itself, thus we consider the gravitational wave radiation (GWR) induced by the deformed neutron star to have a contribution, however, which predicts the braking index \(n = 5\). Thus, we applied the combination of MDR and GWR to explain the higher braking index than 3, and then found that the \(n\) value is not a constant, but evolves from 5 to 3 with time. We also derived the evolution formula of the braking index and spin period (\(P = 2\pi {\text{/}}\Omega \)), and their evolution simulations are also presented by assuming the initial spin period of this pulsar to be 1, 10 and 20 ms, respectively. Furthermore, the particular properties of the pulsar PSR J1640–4631 are discussed, and as a comparison, the stellar spin evolution with the constant spin-down power-law index \(n = 3.15\) is also thoroughly investigated.