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

Technical Summary: Evolution of Proto-Neutron Stars to Pulsars, Magnetars, and Central Compact Objects

Problem Statement
The origin of the ultra-strong magnetic fields observed in magnetars ($B \sim 10^{14}–10^{15}$ G) remains a subject of debate. While the "fossil field" hypothesis (flux conservation from progenitor stars) can explain the fields of conventional neutron stars and Central Compact Objects (CCOs), it struggles to account for magnetar fields without invoking rare, extreme progenitor conditions. Conversely, the dynamo hypothesis, requiring rapid rotation and vigorous convection during the proto-neutron star (PNS) phase, is widely accepted for magnetars but faces a challenge: the inferred birth rate of magnetars is comparable to the core-collapse supernova rate, suggesting magnetar formation is not as rare as extreme parameter requirements imply. Furthermore, standard flux conservation calculations suggest that even with a collapsing iron core, the resulting fields ($B \sim 10^{10}$ G) are insufficient to explain typical pulsar fields ($B \sim 10^{12}$ G), implying a need for a dynamo mechanism even for ordinary pulsars. The central problem is to reconcile the high birth rates of various neutron star types with their diverse magnetic field strengths using a unified evolutionary framework.

Methodology
The authors employ a reduced, one-dimensional shear-driven $\alpha-\Omega$ dynamo model, adapted from Wickramasinghe et al. (2014) who originally applied it to white dwarf mergers. The model tracks the evolution of poloidal ($B_p$) and toroidal ($B_\phi$) magnetic fields during the PNS phase. Key features of the methodology include:

• Governing Equations: The model solves coupled differential equations for $B_p$ and $B_\phi$. The toroidal field is generated via the $\Omega$-effect (differential rotation acting on the poloidal field), while the poloidal field is regenerated via the $\alpha$-effect (turbulent convection acting on the toroidal field).
• PNS Evolution: Unlike static models, this framework incorporates the dynamic contraction of the PNS from an initial radius $R_0 \sim 40$ km to a final radius $R_\infty \approx 12$ km. The radius evolution is parameterized based on Camelio et al. (2017) for various masses and equations of state (EoS).
• Mass and Angular Momentum: The model accounts for the decrease in gravitational mass due to neutrino cooling and the conservation of angular momentum, which drives the spin-up of the PNS.
• Viscosity and Shear Damping: The damping of the differential rotation (shear rate $q$) is modeled via a combination of magnetorotational instability (MRI) driven viscosity and convective viscosity. The shear rate decays over time, eventually terminating the dynamo.
• Flux Loss and Stability: The model includes a reconnection term to account for magnetic flux loss (set initially at 20%) and incorporates stability criteria (Braithwaite 2009) where field instabilities determine the decay timescales of $B_p$ and $B_\phi$.
• Parameter Space Scan: The authors numerically integrate the equations using the Runge-Kutta method, systematically varying initial parameters including the shear rate ($q_0$), initial mass ($M_0$), initial radius ($R_0$), initial seed fields, and the final rotational period ($P_\infty$).

Key Results
The simulations reveal that the final magnetic field configuration is highly sensitive to the initial rotation rate and shear, allowing a single dynamo mechanism to produce the full spectrum of observed neutron star types:

1. Shear Rate Threshold: The saturation value of the toroidal field ($B_{\phi, \infty}$) depends strongly on the initial shear rate $q_0$. A critical threshold exists at $q_0 \simeq 0.23$.

   • Below Threshold ($q_0 < 0.23$): The $\alpha$-effect is not sufficiently triggered. The poloidal field remains near the value dictated by flux conservation ($B_p \approx 2.5 \times 10^{10}$ G), and the toroidal field does not grow significantly. This regime corresponds to Central Compact Objects (CCOs).
   • Above Threshold: The dynamo becomes active. Higher shear rates lead to stronger toroidal fields.

2. Formation of Different Populations:

   • Magnetars: For the shortest initial periods (rapid rotation, e.g., $P_\infty \sim 2.5$ ms), the model produces magnetar-like strengths with $B_p \simeq 10^{15}$ G and $B_\phi \simeq 10^{16}$ G.
   • Ordinary Pulsars: For slower initial rotators, the model yields typical pulsar fields ($B_p \simeq 10^{12}$ G, $B_\phi \simeq 10^{14}$ G).
   • Low-B Magnetars/High-B Pulsars: Intermediate rotation rates produce fields in the range of low-B magnetars or high-field pulsars.

3. Role of Flux Conservation: The initial poloidal field strength is largely determined by flux conservation during the collapse. However, the dynamo process (specifically the $\alpha$-effect) is required to amplify these fields to observed pulsar and magnetar levels. The toroidal field is generically dominated by the $\Omega$-effect.

4. Central Compact Objects (CCOs): The model suggests CCOs do not undergo a vigorous dynamo process due to their slow initial rotation ($P_0 \sim 100$ ms). Their poloidal fields are set by flux conservation, but they may still acquire toroidal fields amplified solely by the $\Omega$-effect, explaining their small dipole fields but potentially significant internal toroidal components.

Significance and Claims
The paper claims to reconcile the apparent discrepancy between the high birth rates of magnetars and the extreme conditions often required by formation scenarios. By demonstrating that a dynamo process operating during the PNS phase can naturally produce a continuum of magnetic field strengths depending on the initial rotation rate, the authors propose that:

• Both ordinary pulsars and magnetars likely pass through a dynamo phase; the difference lies in the efficiency of this process driven by initial rotation.
• The "fossil field" hypothesis is insufficient to explain typical pulsar fields, necessitating a dynamo even for non-magnetar neutron stars.
• CCOs represent the low-rotation end of this spectrum where the dynamo is quenched, leaving fields dominated by flux conservation.

The authors emphasize that their one-dimensional model, while simplified, captures the essential dependencies on rotation, shear, and buoyancy, and qualitatively aligns with more complex three-dimensional simulations (e.g., Raynaud et al. 2020). They conclude that the $\alpha-\Omega$ dynamo provides a unified framework for understanding the diversity of young neutron star magnetic fields without requiring rare, extreme progenitor conditions for magnetars.