Magnetized Proto-Neutron Stars: Structure and Stability

TL;DR

This work investigates how thermal and compositional evolution reshapes the structure and energetics of strongly magnetized proto-neutron stars across four evolutionary stages, using a quasi-static, general-relativistic framework. By employing the XNS 4.0 code with DDME2-based equations of state and axisymmetric magnetic topologies (poloidal, toroidal, and twisted-torus) at a fixed baryonic mass, the authors quantify how entropy and lepton content influence radius, mass, deformation, magnetic flux, and the magnetic-to-binding energy ratio. They find that hotter, lepton-rich stages enhance deformation and flux confinement, while the cold, catalyzed NS is more compact and magnetically rigid, with decay timescales strongly sensitive to core temperature and magnetic geometry. The results highlight the coupled roles of thermodynamics, magnetic topology, and stellar structure in shaping the early magnetic evolution of neutron stars, and they establish a baseline for future work incorporating magnetic flux conservation and flux amplification during cooling.

Abstract

We investigate the evolution of magnetized protoneutron stars (PNSs) through four schematic stages: neutrino trapped, deleptonization, neutrino transparent, and the final cold, catalyzed neutron star (NS). Using a quasi static approximation on the Kelvin Helmholtz timescale, we construct strongly magnetized configurations (magnetic field strengths up to 1e17 G) with the axisymmetric XNS 4.0 code, employing equations of state derived from relativistic mean field theory calibrated with the DDME2 parameter set. We analyze the evolution of the gravitational mass, equatorial radius, stellar deformation, magnetic flux, and the ratio of magnetic to gravitational binding energy as functions of thermodynamic and compositional changes. We find that increasing entropy per baryon and decreasing lepton fraction lead to higher core temperatures, which enhance magnetic deformation, flux confinement, and the magnetic to binding energy ratio. Magnetic field dissipation is most efficient during the deleptonization and neutrino transparent stages, and this process largely determines the observable magnetic field strength of the mature neutron star. This work provides the first general relativistic characterization of how the thermal and compositional evolution of protoneutron stars reshapes magnetic field deformation and energetics across poloidal, toroidal, and mixed field configurations at fixed baryonic mass.

Figures (5)

• Figure 1: The plot shows the relationship between the radius and the gravitational mass, $M_{\odot}$ at different stages of PNS evolution, up to the final stage when the star becomes cold and catalyzed at $T=0$, calculated for unmagnetized case with TOV. The steel blue area indicates the constraints obtained from the binary components of GW170817, with their respective 90% and 50% credible intervals. Additionally, the plot includes the 1 $\sigma$ (68%) CI for the 2D mass-radius posterior distributions of the millisecond pulsars PSR J0030 + 0451 (in cyan and yellow color) riley2019Miller:2019cac and PSR J0740 + 6620 (in orange and peru color)riley2021Miller:2021qha, based on NICER X-ray observations. Furthermore, we display the latest NICER measurements for the mass and radius of PSR J0437-4715 Choudhury:2024xbk (lilac color). The supernova remnant HESS J1731$-$347 2022NatAs...6.1444D is shown in red, with the outer contour representing the 90% CL and the inner contour representing the 50% CL.
• Figure 2: Meridional cross-sections of PNSs with a purely toroidal magnetic field at four evolutionary stages: (a) neutrino-trapped ($s_B=1$, $Y_l=0.4$), (b) deleptonizing ($s_B=2$, $Y_l=0.2$), (c) neutrino-transparent ($s_B=2$, $Y_{\nu_e}=0$), and (d) cold and catalyzed ($T=0$). The color map shows the isocontours of magnetic field strength ($|B|$) in Gauss, while the solid cyan and dashed white lines indicate the surfaces of the magnetized and unmagnetized reference stars, respectively. All configurations share the same baryonic mass ($1.92\,M_{\odot}$) and maximum magnetic field ($B_{\text{max}} = 5.67 \times 10^{17}$ G).
• Figure 3: Meridional cross-sections of PNSs with a purely poloidal magnetic field at four evolutionary stages: (a) neutrino-trapped ($s_B=1$, $Y_l=0.4$), (b) deleptonizing ($s_B=2$, $Y_l=0.2$), (c) neutrino-transparent ($s_B=2$, $Y_{\nu_e}=0$), and (d) cold and catalyzed ($T=0$). The color map shows the isocontours of magnetic field strength ($|B|$) in Gauss, while the solid cyan and dashed white lines indicate the surfaces of the magnetized and unmagnetized reference stars, respectively. All configurations share the same baryonic mass ($1.92\,M_{\odot}$) and maximum magnetic field ($B_{\text{max}} = 5.67 \times 10^{17}$ G).
• Figure 4: Meridional cross-sections of PNSs with a mixed magnetic field at four evolutionary stages: (a) neutrino-trapped ($s_B=1$, $Y_l=0.4$), (b) deleptonizing ($s_B=2$, $Y_l=0.2$), (c) neutrino-transparent ($s_B=2$, $Y_{\nu_e}=0$), and (d) cold and catalyzed ($T=0$). The color map shows the total magnetic field strength ($|B|$) in Gauss and the white lines trace the poloidal magnetic field lines, while the solid cyan and dashed white lines indicate the surfaces of the magnetized and unmagnetized reference stars, respectively. All configurations share the same baryonic mass ($1.92\,M_{\odot}$) and were computed with magnetic parameters $a=0.5$ and $k_{\text{pol}}=0.2$.
• Figure 5: Time evolution of the maximum magnetic field strength for different magnetic field configurations in PNSs. The upper left and upper right panels show the decay of the poloidal and toroidal magnetic fields, respectively, while the lower panels depict the evolution of mixed field configurations: the lower left for $a = 1$ and the lower right for $k_{pol} = 0.04$. Each curve represents a distinct PNS model: $s_B=1$, $Y_l=0.4$ (blue), $s_B=2$, $Y_l=0.2$ (green), $s_B=2$, $Y_{\nu_e}=0$ (orange), and a cold deleptonized $T=0$ model (red).