Simulation of the orbit and spin period evolution of the double pulsars PSR J0737-3039 from their birth to coalescence induced by the gravitational wave radiation

The complete orbital and sin period evolutions of the double neutron star (NS) system PSR J0737-3039 are simulated from the birth to coalescence, which include the two observed radio pulsars classified as primary NS PSR J0737-3039A and companion NS PSR J0737-3039B. By employing the characteristic age of PSR J0737-3039B to constrain the true age of the double pulsar system, the initial orbital period and initial binary separation are obtained as 2.89 hrs and $1.44 \times 10^{6}$ km, respectively, and the coalescence age or the lifetime from the birth to merger of PSR J0737-3039 is obtained to be $1.38 \times 10^{8}$ yr. At the last minute of coalescence, corresponding to that the gravitational wave frequency changes from 20 Hz to 1180 Hz, we present the binary separation of PSR J0737-3039 to be from 442 km to 30 km, while the spin periods of PSR J0737-3039A and PSR J0737-3039B are 27.10 ms and 4.63 s, respectively. From the standard radio pulsar emission model, before the system merged, the primary NS could still be observed by the radio telescope, while the companion NS has crossed the death line in the pulsar magnetic-field versus period ($B-P$) diagram which is usually considered to cease the life as a pulsar. It is for the first time that the whole life evolutionary simulation of the orbit and spin periods for double NS system is presented, which provides the useful information for observing the primary NS at the coalescence stage.