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

TL;DR

This work demonstrates that a shear-driven $α$–$Ω$ dynamo operating during the proto-neutron star phase, including contraction and flux-loss processes, can account for the emergence of magnetar-like fields as well as ordinary pulsar fields from a common physical mechanism. By solving reduced dynamical equations for the poloidal and toroidal fields, along with angular-momentum transport and evolving PNS structure, the authors show a threshold in the initial shear, $q_0 \approx 0.23$, that separates poloidal-dominated from toroidal-dominated outcomes, and they find magnetar strengths ($B_p \sim 10^{15}$ G, $B_\phi \sim 10^{16}$ G) for rapid rotation and ordinary pulsar fields ($B_p \sim 10^{12}$ G, $B_\phi \sim 10^{14}$ G) for slower rotators. Central compact objects are predicted to acquire toroidal fields primarily via the $Ω$-effect with poloidal fields set by flux conservation, aligning with their observed properties. The results qualitatively agree with three-dimensional simulations and offer a unified scenario for the diverse young neutron star populations, while highlighting the need for improved convection transport prescriptions in more detailed models.

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 $α$--$Ω$ dynamo model that includes PNS contraction. The dynamo generically produces toroidal-dominated fields set mainly by the $Ω$-effect. The evolution of the poloidal field is first dominated by flux conservation during collapse and then by the $α$-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 ($B_{\rm p} \simeq 10^{15}\,{\rm G}$, $B_φ\simeq 10^{16}\,{\rm G}$), while for the slower rotators it yields ordinary pulsar fields ($B_{\rm p} \simeq 10^{12}\,{\rm G}$, $B_φ\simeq 10^{14}\,{\rm G}$). We also argue that the central compact objects can acquire toroidal fields amplified solely by the $Ω$-effect; lacking the $α$-effect, their poloidal fields are not shaped by the dynamo effect.

Figures (7)

• Figure 1: Dependence of the exponential decay time-scale of the radius of the PNS on mass and equation of state. We inferred this result from the data tables given in cam+17. Different curves correspond to different EoS (see text).
• Figure 2: Evolution of the magnetic fields, $B_{\rm p}$ (red line), $B_{\phi}$ (blue line) and differential rotation, $q$ (green line). Here the initial conditions are $B_{\phi}(t=0) = B_{\rm p}(t=0) = 3\times 10^{9}\,{\rm G}$. $q_0 =0.4$ and $P_0 = 29.5\,{\rm ms}$ ($P_{\infty} = 2.5\,{\rm ms}$). We assumed $a=10^{3}$.
• Figure 3: Effect of the initial shear rate, $q_{0}$, on the saturation value of magnetic fields, $B_{\rm p, \infty}$ (red line) and $B_{\phi, \infty}$ (blue line). Here the initial fields are $B_{\phi}(t=0) = 3\times 10^{9}\,{\rm G}$ and $B_{\rm p}(t=0) = 3\times 10^{9}\,{\rm G}$. We assumed $a=10^{3}$.
• Figure 4: Here $B_{\phi}(t=0) = 3\times 10^{9}\,{\rm G}$, $B_{\rm p}(t=0) = 3\times 10^{9}\,{\rm G}$ and $q_0 = 0.4$. (Left panel) Effect of the flux loss rate on the saturated value of the poloidal field, $B_{\rm p, \infty}$ (red line) and the toroidal field, $B_{\phi, \infty}$ (blue line) with $a=10^3$. (Right panel) Effect of buoyancy factor, $a$, on the saturated value of the poloidal field, $B_{\rm p, \infty}$ (red line) and the toroidal field, $B_{\phi, \infty}$ (blue line).
• Figure 5: Here $B_{\phi}(t=0) = 3\times 10^{9}\,{\rm G}$, $B_{\rm p}(t=0) = 3\times 10^{9}\,{\rm G}$, $q_0 = 0.4$, $a = 10^{3}$. (Left panel) Effect of the initial mass of the PNS, $M_{0}$, on the saturated value of the poloidal field, $B_{\rm p, \infty}$ (red line) and the toroidal field, $B_{\phi, \infty}$ (blue line) with the initial radius of $R_{0} = 40\,{\rm km}$. (Right panel) Effect of the initial radius of the PNS, $R_{0}$, on the saturated value of the poloidal field, $B_{\rm p}$ (red line) and the toroidal field, $B_{\phi}$ (blue line) with the initial mass of $M_{0} = 1.55 M_{\sun}$.