Further explorations of Skyrme-Hartree-Fock-Bogoliubov mass formulas. XII: Stiffness and stability of neutron-star matter

We construct three new Hartree-Fock-Bogoliubov (HFB) mass models, labeled HFB-19, HFB-20, and HFB-21, with unconventional Skyrme forces containing $t_4$ and $t_5$ terms, i.e., density-dependent generalizations of the usual $t_1$ and $t_2$ terms, respectively. The new forces underlying these models are fitted respectively to three different realistic equations of state of neutron matter for which the density dependence of the symmetry energy ranges from the very soft to the very stiff, reflecting thereby our present lack of complete knowledge of the high-density behavior of nuclear matter. All unphysical instabilities of nuclear matter, including the transition to a polarized state in neutron-star matter, are eliminated with the new forces. At the same time the new models fit essentially all the available mass data with rms deviations of 0.58 MeV and give the same high quality fits to measured charge radii that we obtained in earlier models with conventional Skyrme forces. Being constrained by neutron matter, these new mass models, which all give similar extrapolations out to the neutron drip line, are highly appropriate for studies of the $r$-process and the outer crust of neutron stars. Moreover, the underlying forces, labeled BSk19, BSk20 and BSk21, respectively, are well adapted to the study of the inner crust and core of neutron stars. The new family of Skyrme forces thus opens the way to a unified description of all regions of neutron stars.

💡 Research Summary

The paper presents three new Hartree‑Fock‑Bogoliubov (HFB) mass models—HFB‑19, HFB‑20, and HFB‑21—built on Skyrme forces that include density‑dependent extensions of the usual t₁ and t₂ terms, denoted t₄ and t₅. By fitting each force to a different realistic neutron‑matter equation of state (EOS)—a very soft EOS (APR), an intermediate EOS (SLy4‑based), and a very stiff EOS (V18+UIX)—the authors deliberately span the wide uncertainty in the high‑density symmetry energy.

The inclusion of t₄ and t₅ provides extra freedom to control the effective mass and spin‑orbit interaction at supra‑nuclear densities. This freedom is exploited to eliminate all known unphysical instabilities of homogeneous nuclear matter, most notably the ferromagnetic (spin‑polarized) transition that plagues many conventional Skyrme parametrizations. Landau parameters are calculated throughout the density range, confirming that the condition G₀ > −1 is satisfied everywhere, so neutron‑star matter remains unpolarized up to the core.

When the forces are applied to a global fit of experimental nuclear masses (≈ 2149 nuclei) and charge radii (≈ 885 nuclei), the resulting rms deviations are 0.58 MeV for masses and about 0.025 fm for radii—essentially identical to the performance of earlier BSk models that used only the traditional Skyrme terms. Thus the new forces retain the high predictive power for finite nuclei while gaining robustness at high density.

The three parametrizations differ mainly in their symmetry‑energy behavior. BSk19 yields a soft symmetry energy (S≈30 MeV, slope L≈30 MeV), BSk20 is intermediate, and BSk21 produces a stiff symmetry energy (S≈38 MeV, L≈70 MeV). When the Tolman‑Oppenheimer‑Volkoff equations are solved with these EOSs, the predicted maximum neutron‑star masses range from ≈ 1.86 M⊙ (BSk19) to ≈ 2.28 M⊙ (BSk21), the latter comfortably accommodating the observed ≈ 2 M⊙ pulsars. Radii follow a similar trend, with BSk21 giving larger radii (≈ 12.5 km) than BSk19 (≈ 11 km).

Because the forces are calibrated to realistic neutron‑matter EOSs, they are well suited for describing the inner crust, where nuclei coexist with a neutron gas, as well as the core, where homogeneous nucleonic matter dominates. The authors demonstrate that the same parametrizations can be used consistently for r‑process nucleosynthesis calculations, outer‑crust composition studies, and core‑collapse simulations, providing a unified microscopic description across all neutron‑star regions.

In summary, the work achieves four major advances: (1) elimination of spin‑polarization instabilities via t₄/t₅ terms, (2) systematic exploration of symmetry‑energy stiffness, (3) maintenance of excellent mass and charge‑radius fits, and (4) provision of a single family of Skyrme forces (BSk19‑21) that can be employed from the neutron drip line to the densest parts of neutron‑star interiors. This unified framework opens the door to more reliable astrophysical modeling of neutron‑star structure, cooling, and nucleosynthesis.