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Millisecond pulsars (MSPs) are a kind of radio pulsars with short spin periods, playing a key role in many aspects of stellar astrophysics. In recent years, some more MSPs with wide orbits ($>30\,\rm d$) have been discovered, but their origin is still highly unclear. In the present work, according to an adiabatic power-law assumption for the mass-transfer process, we carried out a large number of complete binary evolution computations for the formation of MSPs with wide orbits through the iron core-collapse supernova (CCSN) channel, in which a neutron star (NS) originating from a CCSN accretes matter from a red-giant (RG) star and spun up to millisecond periods. We found that this channel can form the observed MSPs with wide orbits in the range of $30-1200\,{\rm d}$, in which the WD companions have masses in the range of $0.28-0.55\,\rm M_{\odot}$. We also found that almost all the observed MSPs can be reproduced by this channel in the WD companion mass versus orbital period diagram. We estimate that the Galactic numbers of the resulting MSPs from the CCSN channel are in the range of $\sim 4.8-8.5\times10^{5}$. Compared with the accretion-induced collapse channel, the CCSN channel provides a main way to produce MSPs with wide orbits., Comment: 19 pages, 8 figures, 3 tables. Accepted for publication in MNRAS

Evolution of a large part of low-mass X-ray binaries (LMXBs) leads to the formation of rapidly rotating pulsars with a helium white dwarf (He WD) companion. Observations indicate that some He WDs in binary pulsar systems are ultracool (with the effective temperatures $T_{\rm eff}\lesssim$ 4000\, K). It is hard to cool down a He WD to such low temperatures within the Hubble time, because a thick hydrogen envelope was left behind around the He core after the mass transfer process. A possible mechanism that can accelerate the WD cooling is the evaporative wind mass loss from the He WD driven by the high-energy radiation from the recycled pulsar. In this paper, we evolve a large number of LMXBs and investigate the influence of the pulsar's high-energy radiation on the WD cooling with different input parameters, including the neutron star's spin-down luminosity, the evaporation efficiency and the metallicity of the companion star. By comparing our results with observations we note that, for relatively hot He WDs (with $T_{\rm eff}> 7000$ K), standard WD cooling without evaporation considered is able to reproduce their temperatures, while evaporation is probably required for the He WDs with relatively low temperatures ($T_{\rm eff}$ <5000 K)., Comment: 10 pages, 4 figures, accepted for publication in MNRAS

Ultra-compact X-ray binaries (UCXBs) are low-mass X-ray binaries with hydrogen-deficient mass-donors and ultra-short orbital periods. They have been suggested to be the potential Laser Interferometer Space Antenna (LISA) sources in the low-frequency region. Several channels for the formation of UCXBs have been proposed so far. In this article, we carried out a systematic study on the He star donor channel, in which a neutron star (NS) accretes matter from a He main-sequence star through Roche-lobe overflow, where the mass-transfer is driven by gravitational wave radiation. Firstly, we followed the long-term evolution of the NS+He main-sequence star binaries by employing the stellar evolution code Modules for Experiments in Stellar Astrophysics, and thereby obtained the initial parameter spaces for the production of UCXBs. We then used these results to perform a detailed binary population synthesis approach to obtain the Galactic rates of UCXBs through this channel. We estimate the Galactic rates of UCXBs appearing as LISA sources to be $\sim3.1-11.9\, \rm Myr^{-1}$ through this channel, and the number of such UCXB-LISA sources in the Galaxy can reach about $1-26$ calibrated by observations. The present work indicates that the He star donor channel may contribute significantly to the Galactic UCXB formation rate. We found that the evolutionary tracks of UCXBs through this channel can account for the location of the five transient sources with relatively long orbital periods quite well. We also found that such UCXBs can be identified by their locations in the mass-transfer rate versus the orbital period diagram., Comment: 21 pages, 9 figures, 1 table, accepted for publication in MNRAS