Magnetic fields in late-stage proto-neutron stars

We explore the thermal and magnetic field structure of a late-stage proto-neutron star (proto-NS). We find the dominant contribution to the entropy in different regions of the star, from which we build a simplified equation of state (EOS) for the hot neutron star (NS). With this, we numerically solve the stellar equilibrium equations to find a range of models, including magnetic fields and rotation up to Keplerian velocity. We approximate the EOS as a barotrope, and discuss the validity of this assumption. For fixed magnetic field strength, the induced ellipticity increases with temperature; we give quantitative formulae for this. The Keplerian velocity is considerably lower for hotter stars, which may set a de facto maximum rotation rate for non-recycled NSs well below 1 kHz. Magnetic fields stronger than around 1014 G have qualitatively similar equilibrium states in both hot and cold NSs, with large-scale simple structure and the poloidal field component dominating over the toroidal one; we argue this result may be universal. We show that truncating magnetic field solutions at low multipoles leads to serious inaccuracies, especially for models with rapid rotation or a strong toroidal-field component.