In this work, we interpreted the high braking index of PSR J1640−4631 with a combination of the magnetodipole radiation and dipole magnetic field decay models. By introducing a mean rotation energy conversion coefficient z , 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 17 44 0~( -) ms, corresponding to the moment of inertia I 0.8 2.1 10 45( -) g cm 2 .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 D t , and the initial surface dipole magnetic field at the pole B 0 p ( ) 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 dt 1.66 3.85 10 p 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.

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By introducing Dirac's δ-function in superhigh magnetic fields, we deduce a general formula for the pressure of degenerate and relativistic electrons, P e , which is suitable for superhigh magnetic fields, discuss the quantization of Landau levels of electrons, and consider the quantum electrodynamic(QED) effects on the equations of states (EOSs) for different matter systems. The main conclusions are as follows: the stronger the magnetic field strength, the higher the electron pressure becomes; compared with a common radio pulsar, a magnetar could be a more compact oblate spheroid-like deformed neutron star due to the anisotropic total pressure; and an increase in the maximum mass of a magnetar is expected because of the positive contribution of the magnetic field energy to the EoS of the star. Since this is an original work in which some uncertainties could exist, modifications and improvements of our theory should be considered in our future studies.

Due to the lack of long term pulsed emission in quiescence and the strong timing noise, it is impossible to directly measure the braking index n of a magnetar. Based on the estimated ages of their potentially associated supernova remnants (SNRs), we estimate the values of the mean braking indices of eight magnetars with SNRs, and find that they cluster in a range of 1 ∼42. Five magnetars have smaller mean braking indices of 1 < n < 3, and we interpret them within a combination of magneto-dipole radiation and wind aided braking, while the larger mean braking indices of n > 3 for other three magnetars are attributed to the decay of external braking torque, which might be caused by magnetic field decay. We estimate the possible wind luminosities for the magnetars with 1 < n < 3, and the dipolar magnetic field decay rates for the magnetars with n > 3 within the updated magneto-thermal evolution models. Although the constrained range of the magnetars' braking indices is tentative, due to the uncertainties in the SNR ages, which come from distance uncertainties and the unknown conditions of the expanding shells, our method provides an effective way to constrain the magnetars' braking indices if the measurements of the SNRs' ages are reliable, which can be improved by future observations.

Magnetic braking (MB) likely plays a vital role in the evolution of low-mass X-ray binaries (LMXBs). However, there is still uncertainty around the physics of MB, and there are various proposed scenarios for MB in the literature. To examine and discriminate the efficiency of MB, we investigate the LMXB evolution with five proposed MB laws. Combining detailed binary evolution calculation with binary population synthesis, we obtain the expected properties of LMXBs and their descendant binary millisecond pulsars. We then discuss the strength and weakness of each MB law by comparing the calculated results with observations. We conclude that the τ-boosted MB law seems to best match the observational characteristics.

In this paper, we investigate the electron Landau level stability and its influence on the electron Fermi energy, [Formula: see text], in the circumstance of magnetars, which are powered by magnetic field energy. In a magnetar, the Landau levels of degenerate and relativistic electrons are strongly quantized. A new quantity [Formula: see text], the electron Landau level stability coefficient is introduced. According to the requirement that [Formula: see text] decreases with increasing the magnetic field intensity [Formula: see text], the magnetic field index [Formula: see text] in the expression of [Formula: see text] must be positive. By introducing the Dirac-[Formula: see text] function, we deduce a general formulae for the Fermi energy of degenerate and relativistic electrons, and obtain a particular solution to [Formula: see text] in a superhigh magnetic field (SMF). This solution has a low magnetic field index of [Formula: see text], compared with the previous one, and works when [Formula: see text] and [Formula: see text] Gauss. By modifying the phase space of relativistic electrons, a SMF can enhance the electron number density [Formula: see text], and decrease the maximum of electron Landau level number, which results in a redistribution of electrons. According to Pauli exclusion principle, the degenerate electrons will fill quantum states from the lowest Landau level to the highest Landau level. As [Formula: see text] increases, more and more electrons will occupy higher Landau levels, though [Formula: see text] decreases with the Landau level number [Formula: see text]. The enhanced [Formula: see text] in a SMF means an increase in the electron Fermi energy and an increase in the electron degeneracy pressure. The results are expected to facilitate the study of the weak-interaction processes inside neutron stars and the magnetic-thermal evolution mechanism for magnetars.

PSR J1640+2224 is a binary millisecond pulsar (BMSP) with a white dwarf (WD) companion. Recent observations indicate that the WD is very likely to be a ∼ 0.7 M ⊙ CO WD. Thus the BMSP should have evolved from an intermediatemass X-ray binary (IMXB). However, previous investigations on IMXB evolution predict that the orbital periods of the resultant BMSPs are generally < 40 days, in contrast with the 175 day orbital period of PSR J1640+2224. In this paper, we explore the influence of the mass of the neutron star (NS) and the chemical compositions of the companion star on the formation of BMSPs. Our results show that, the final orbital period becomes longer with increasing NS mass, and the WD mass becomes larger with decreasing metallicity. In particular, to reproduce the properties of PSR J1640+2224, the NS was likely born massive (> 2.0 M ⊙ ).Here the total rate of change in the orbital angular momentum is determined bẏThe three terms on the right-hand-side of Eq. (6) represent angular momentum losses caused by GR, MB, and mass loss, respectively. The GR-induced rateJ gr is calculated with the standard formula (Landau & Lifshitz 1959;Faulkner 1971) J gr J orb = − 32G 3 5c 5

Young pulsars are thought to be highly magnetized neutron stars (NSs). The crustal magnetic field of a NS usually decays at different timescales in the forms of Hall drift and Ohmic dissipation. The magnetization parameter ω B τ is defined as the ratio of the Ohmic timescale τ O h m to the Hall drift timescale τ H a l l . During the first several million years, the inner temperature of the newly born neutron star cools from T = 10 9 K to T = 1.0 × 10 8 K, and the crustal conductivity increases by three orders of magnitude. In this work, we adopt a unified equations of state for cold non-accreting neutron stars with the Hartree–Fock–Bogoliubov method, developed by Pearson et al. (2018), and choose two fiducial dipole magnetic fields of B = 1.0 × 10 13 G and B = 1.0 × 10 14 G, four different temperatures, T, and two different impurity concentration parameters, Q, and then calculate the conductivity of the inner crust of NSs and give a general expression of magnetization parameter for young pulsars: ω B τ ≃ ( 1 − 50 ) B 0 / ( 10 13 G) by using numerical simulations. It was found when B ≤ 10 15 G, due to the quantum effects, the conductivity increases slightly with the increase in the magnetic field, the enhanced magnetic field has a small effect on the matter in the low-density regions of the crust, and almost has no influence the matter in the high-density regions. Then, we apply the general expression of the magnetization parameter to the high braking-index pulsar PSR J1640-4631. By combining the observed arrival time parameters of PSR J1640-4631 with the magnetic induction equation, we estimated the initial rotation period P 0 , the initial dipole magnetic field B 0 , the Ohm dissipation timescale τ O h m and Hall drift timescale τ H a l l . We model the magnetic field evolution and the braking-index evolution of the pulsar and compare the results with its observations. It is expected that the results of this paper can be applied to more young pulsars.