Evolution of proto-neutron stars to pulsars, magnetars and central compact objects

Abstract

Some young neutron stars, the magnetars, have ultra-strong magnetic fields, yet their inferred birth rate is comparable to the core-collapse supernova rate, challenging scenarios that require rare, extreme conditions. We propose that this discrepancy can be reconciled if both pulsars and magnetars pass through a dynamo process during the proto-neutron star (PNS) phase. We employ a shear-driven $\alpha$--$\Omega$ dynamo model that includes PNS contraction. The dynamo generically produces toroidal-dominated fields set mainly by the $\Omega$-effect. The evolution of the poloidal field is first dominated by flux conservation during collapse and then by the $\alpha$-effect. The saturated toroidal field depends strongly on the initial value of the shear, with a threshold at $q_0 \simeq 0.23$; below this, the poloidal field remains near the value obtained by the flux-conservation ($\approx 2.5\times10^{10}\,{\rm G}$). For the shortest initial periods, the model leads to magnetar-like strengths ($B_{\rm p} \simeq 10^{15}\,{\rm G}$, $B_\phi \simeq 10^{16}\,{\rm G}$), while for the slower rotators it yields ordinary pulsar fields ($B_{\rm p} \simeq 10^{12}\,{\rm G}$, $B_\phi \simeq 10^{14}\,{\rm G}$). We also argue that the central compact objects can acquire toroidal fields amplified solely by the $\Omega$-effect; lacking the $\alpha$-effect, their poloidal fields are not shaped by the dynamo effect.