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Spin evolution of neutron stars in two modes: implication for millisecond pulsars

An understanding of spin frequency ($ν$) evolution of neutron stars in the low-mass X-ray binary (LMXB) phase is essential to explain the observed $ν$-distribution of millisecond pulsars (MSPs), and to probe the stellar and binary physics, including the possibility of continuous gravitational wave emission. Here, using numerical computations we conclude that $ν$ can evolve in two distinctly different modes, as $ν$ may approach a lower spin equilibrium value ($ν_{\rm eq,per}$) for persistent accretion for a long-term average accretion rate ($\dot{M}_{\rm av}$) greater than a critical limit ($\dot{M}_{\rm av,crit}$), and may approach a higher effective spin equilibrium value ($ν_{\rm eq,eff}$) for transient accretion for $\dot{M}_{\rm av} < \dot{M}_{\rm av,crit}$. For example, when $\dot{M}_{\rm av}$ falls below $\dot{M}_{\rm av,crit}$ for an initially persistent source, $ν$ increases considerably due to transient accretion, which is counterintuitive. We also find that, contrary to what was suggested, a fast or sudden decrease of $\dot{M}_{\rm av}$ to zero in the last part of the LMXB phase is not essential for the genesis of spin-powered MSPs, and neutron stars could spin up in this $\dot{M}_{\rm av}$-decreasing phase. Our findings imply that the traditional way of $ν$-evolution computation is inadequate in most cases, even for initially persistent sources, and may not even correctly estimate whether $ν$ increases or decreases.