tokens: rate of change in the orbital separation:Here the total rate of change in the orbital angular momentum is determined byThe three terms on the right side of Equation ( 6 ) represent angular momentum losses caused by GR, MB, and mass loss, respectively. The GR-induced rate is calculated with the standard formula (Landau Lifshitz 1959 ; Faulkner 1971 )where G and c are the gravitational constant and the speed of light, respectively. The prescription of Verbuant Zwaan ( 1981 ) is adopted to calculate the angular momentum loss due to MB,For IMXBs and wide LMXBs, the mass-transfer rate may be higher than the Eddington-limit accretion rate of the NS. Thus, the accretion rate of the NS is evaluated usingand the mass loss rate from the binary system isIn the case of super-Eddington accretion, we adopt the isotropic reemission model, assuming that the extra material leaves the binary in the form of isotropic wind from the NS. Therefore, the angular momentum loss rate due to mass loss can be derived to bewhere a 1 is the distance between the NS and the center of mass of the binary. 3. Results of Evolution Calculations We have performed calculations of the binary evolution for thousands of I/LMXB systems. We choose the initial NS mass and the initial donor mass , and set the initial orbital period 1.0 day days. We use exponential diffusive overshooting for the donor stars. Note that there exists a bifurcation period for the initial orbital period day (Pylyser Savonije 1988 , 1989 ). If the initial orbital period is below P bf , the binary systems will evolve with shrinking orbits, possibly forming ultra-compact binaries. This kind of evolution cannot reproduce PSR J1640+2224, and will not be considered here. In addition, we exclude the evolutions with the final orbital periods exceeding 500 days. 3.1. A Typical Example for the IMXB Evolution We present the evolutionary sequences for a typical IMXB system that consists of a 1.4 M ⊙ NS and a 3.5 M ⊙ donor star with an initial orbital period P orb = 6 days in Figures 1 and 2 . The donor star starts to fill its RL and commence mass transfer at the age of 240.6 Myr. The mass-transfer process lasts about 1.9 Myr, at a rate higher than the Eddington accretion rate (the maximum mass-transfer rate >10 −5 M ⊙ yr −1 ). In this case, most of the transferred material from the donor is lost from the binary system. Thus, after the mass transfer, the donor evolves into a CO WD of mass 0.543 M ⊙ , while the NS has accreted only 0.034 M ⊙ mass. The orbital period becomes smaller in the former stage of the mass-transfer process due to the relatively high mass ratio (>1); when the mass ratio is reversed, the orbit begins to expand. The final orbital period is P orb = 17.8 days. Figure 1. Example evolution of a typical IMXB consisting of a NS and a donor star with initial masses M 1 = 1.4 M ⊙ and M 2 = 3.5 M ⊙ respectively, and orbital period P orb = 6 days. The blue and red solid curves denote the evolutionary tracks for
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