Magnetized proto-neutron stars: structure and deformation

We investigate the evolution of magnetized protoneutron stars (PNSs) across 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 model strongly magnetized configurations ( $$B \sim 10^{17}\,\text {G}$$ B ∼ 10 17 G ) using axisymmetric XNS 4.0 with Equations of State (EoS) derived from relativistic mean-field theory calibrated by the DDME2 parameterization. We analyze the gravitational mass, equatorial radius, shape deformation, magnetic flux, and magnetic-to-binding energy ratio as functions of thermodynamic and compositional changes. Our results show that increasing entropy per baryon and decreasing lepton fraction raise core temperatures, which in turn 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, a process that ultimately determines the observable field strength of the mature NS. This work presents a systematic general-relativistic characterization of how thermal and compositional evolution reshapes magnetic-field structure, deformation, and magnetic energetics across multiple stages of PNS evolution.