Modeling isolated magnetar spin-down evolution and implications for long-period radio transients

Long-period radio transients (LPTs) are a new class of radio sources characterized by long spin periods ($P_{\text{spin}}>10^3$ s) and highly variable radio emission. While known magnetars are relatively young ($\tau<10^5$ yrs) with spin periods clustered between $1-10$ sec, it has been proposed that LPTs may be linked to a missing population of older magnetars. In this paper, we present an extensive parametric analysis of isolated magnetar spin evolution using various propeller spin-down models. In general, at higher initial magnetar B-fields ($B_0>\sim10^{15}$ G) and larger ambient densities ($n_0>\sim10^2$ cm$^{-3}$), magnetars will transition to the propeller phase earlier, and they start accreting gas from the ISM or molecular clouds after $\tau\sim10^8$ yrs. We found that a transition from the pulsar to the propeller phase is required to reach the observed LPT period range of $P>10^3$ s. More specifically, our population synthesis study based on Monte-Carlo simulations shows that two propeller models can account for most of the observed LPT periods ($P\sim1-400$ [min]) and their period derivative constraints ($\dot{P}<10^{-9}$ s s$^{-1}$). Our spin-down models predict that (1) nearby radio-quiet neutron stars with the estimated dipole $B$-field range of $B\sim(1-5)\times10^{13}$ G will transition to the propeller phase eventually after $\tau>\sim10^7$ yrs; (2) thermal X-ray emission from accretion-phase magnetars becomes too faint for detection after traveling ($d>\sim10$ kpc) from their birth places; (3) sporadic radio outbursts observed from LPTs may not be explained by regular radio pulsar and magnetar emission mechanisms that operate during the propeller phase.