A phenomenological model of the magnetic field re-emergence in magnetars and discrepancy between the kinematic and characteristic ages

Abstract

Robust age measurements for isolated neutron stars (NSs) are not easily available. That is why, often the characteristic age $\tau_\mathrm{ch}=P/2\dot P$ is used as a proxy. Here $P$ is the spin period of the NS and $\dot P$ is the time derivative of $P$. Additional assumptions related to the initial properties and spin-down evolution are made to derive $\tau_\mathrm{ch}$. As a result, it is expected that $\tau_\mathrm{ch}$ is an upper limit for the real age $\tau_\mathrm{real}$. Recently, Chrimes et al. presented measurements of kinematic ages $\tau_\mathrm{kin}$ for several magnetars. Surprisingly, for the majority of these sources $\tau_\mathrm{kin}>\tau_\mathrm{ch}$. We present a simple model including a realistic approximation for the magnetic field decay in magnetars and a simple phenomenological description of the field re-emergence after an episode of fallback after the birth of a NS. We demonstrate that this simple model can explain the observed relation $\tau_\mathrm{kin}>\tau_\mathrm{ch}$ for realistic sets of parameters.