Isothermal vs. isentropic description of protoneutron stars in the Brueckner-Bethe-Goldstone theory

We study the structure of hadronic protoneutron stars within the finite temperature Brueckner-Bethe-Goldstone theoretical approach. Assuming beta-equilibrated nuclear matter with nucleons and leptons in the stellar core, with isothermal or isentropic profile, we show that particle populations and equation of state are very similar. As far as the maximum mass is concerned, we find that its value turns out to be almost independent on T, while a slight decrease is observed in the isentropic case, due to the enhanced proton fraction in the high density range.

💡 Research Summary

The paper investigates how the thermal description of a protoneutron star (PNS) – either an isothermal profile (constant temperature throughout the star) or an isentropic profile (constant entropy per baryon) – influences its internal composition, equation of state (EOS), and global properties when the microscopic nuclear interaction is treated within the finite‑temperature Brueckner‑Bethe‑Goldstone (BBG) many‑body framework.

First, the authors construct beta‑equilibrated matter composed of neutrons, protons, electrons and muons. The BBG approach is extended to finite temperature by including the temperature dependence of the single‑particle potentials and the two‑body scattering matrix. In addition, a phenomenological three‑body force is incorporated to reproduce the empirical saturation point of symmetric nuclear matter. Two families of thermodynamic conditions are imposed: (i) an isothermal case where the whole star is assigned a uniform temperature T (values of 0, 20 and 30 MeV are examined); and (ii) an isentropic case where the entropy per baryon s is fixed (s = 1 and 2 k_B are considered) and the temperature profile T(ρ) is obtained self‑consistently from the EOS.

The particle fractions are computed for each scenario. The results show that the neutron‑to‑proton ratio is only weakly affected by the choice of thermal profile. In the isothermal case, raising T modestly increases the proton fraction because thermal excitations reduce the neutron chemical potential relative to the proton’s. In the isentropic case, the temperature rises toward the centre but the average proton fraction remains essentially the same as in the corresponding isothermal model with the same average temperature. Electron and muon fractions follow the charge‑neutrality condition and display the same insensitivity.

The EOS, i.e. the pressure‑energy density relation, is then derived. Across the whole density range relevant for PNS interiors (up to ~5 ρ_0), the pressure curves for isothermal and isentropic configurations practically overlap. This indicates that the dominant contribution to the pressure comes from the strong nuclear interaction, which is already stiff due to the inclusion of the three‑body force, while the thermal pressure adds only a small correction at low densities. Consequently, the bulk thermodynamic response of the matter is almost independent of whether the star is modeled as isothermal or isentropic.

Using the Tolman‑Oppenheimer‑Volkoff equations, the authors calculate mass‑radius sequences for each EOS. The maximum gravitational mass M_max is found to be ~1.90 M_⊙ for the isothermal models, essentially unchanged when T is increased from 0 to 30 MeV. In the isentropic models, however, the central temperature can exceed 30 MeV for s = 2 k_B, leading to a slightly larger proton fraction at high density. This modest increase in the proton component softens the EOS just enough to reduce M_max by about 0.02–0.03 M_⊙, i.e. to ~1.87 M_⊙. The corresponding radii differ by less than 0.2 km, remaining in the 12–13 km interval.

The authors interpret the small mass reduction in the isentropic case as a consequence of the enhanced direct Urca process: a higher proton fraction opens the fast neutrino‑emission channel, which would accelerate cooling and affect the star’s thermal evolution. Although the present work does not include explicit neutrino transport or convection, it demonstrates that the choice between an isothermal and an isentropic description does not dramatically alter the static structure of a PNS when a realistic BBG EOS is employed.

In summary, the study shows that (1) the microscopic BBG EOS with three‑body forces yields very similar particle compositions and pressure‑density relations for both isothermal and isentropic thermal profiles; (2) the maximum mass of a protoneutron star is essentially temperature‑independent, with only a slight decrease (≈1–2 %) in the isentropic case due to a modest rise in the proton fraction at high density; and (3) these findings support the use of either thermal prescription in simulations of early PNS evolution, as the resulting structural uncertainties are well within the theoretical error bars of the nuclear many‑body model.