Equation of State of Highly Asymmetric Neutron-Star Matter from Liquid Drop Model and Meson Polytropes

We present a unified description of dense matter and neutron-star structure based on simple but physically motivated models. Starting from the thermodynamics of degenerate Fermi gases, we construct an equation of state for cold, catalyzed matter by combining relativistic fermion statistics with the liquid drop model of nuclear binding. The internal stratification of matter in the outer crust is described by $β$-equilibrium, neutron drip and a gradual transition to supranuclear matter. Short-range repulsive interactions inspired by Quantum Hadrodynamics are incorporated at high densities in order to ensure stability and causality. The resulting equation of state is used as input to the Tolman–Oppenheimer–Volkoff equations, yielding self-consistent neutron-star models. We compute macroscopic stellar properties including the mass-radius relation, compactness and surface redshift that can be compared with recent observational data. Despite the simplicity of the underlying microphysics, the model produces neutron-star masses and radii compatible with current observational constraints from X-ray timing and gravitational-wave measurements. This work demonstrates that physically transparent models can already capture the essential features of neutron-star structure and provide valuable insight into the connection between dense-matter physics and astrophysical observables while they can also be used as easy to handle models to test the impact of more complicated phenomena and variations in neutron stars.

💡 Research Summary

The paper presents a unified, physically motivated framework for constructing the equation of state (EOS) of highly asymmetric neutron‑star matter and for calculating neutron‑star structure. Starting from the thermodynamics of degenerate fermions, the authors derive the general expressions for the energy density and pressure of a relativistic Fermi gas, introducing the dimensionless Fermi momentum x = p_F/(mc) to write compact formulas that smoothly interpolate between the non‑relativistic and ultra‑relativistic limits.

The low‑density regime (ρ ≲ 10⁸ g cm⁻³) is described by a fully ionized electron gas in β‑equilibrium with protons and neutrons. The liquid‑drop model (LDM) is employed to capture the bulk, surface, Coulomb and symmetry contributions to nuclear binding. Using empirical coefficients (E_u = 15.6 MeV, E_σ = 17.1 MeV, E_c = 0.71 MeV, E_s = 23.4 MeV), the total energy per baryon is written as a function of the mass number A and proton fraction Y_p. Minimization of the energy yields coupled equations for A and Y_p, which are solved numerically as density increases, showing the growth of larger, more neutron‑rich nuclei.

At the neutron‑drip point (ρ ≈ 4 × 10¹¹ g cm⁻³) free neutrons appear. The authors introduce a free‑neutron fraction Y_n and write the total energy density as a sum of nuclear, electron, and neutron contributions. Equilibrium conditions are obtained by setting the partial derivatives of the total energy with respect to A, Y_p and Y_n to zero, leading to a set of algebraic relations that determine the composition throughout the outer crust.

When the density exceeds roughly 5 × 10¹³ g cm⁻³, nuclei begin to overlap, forming “nuclear pasta” where surface and Coulomb terms become negligible. The composition is then governed by β‑equilibrium and the condition that the neutron chemical potential equals the electron Fermi energy. An effective interpolation Y_n = 1 − ρ₀/ρ is adopted near nuclear saturation (ρ₀ ≈ 2.7 × 10¹⁴ g cm⁻³) to account for the fraction of matter that does not contribute to pressure support.

Because the pressure from pure fermionic degeneracy is insufficient to stabilize massive neutron stars, a short‑range repulsive component is added. The authors adopt a Quantum Hadrodynamics (QHD) mean‑field model in which nucleons interact via σ, ω and ρ meson exchange. The interaction Lagrangian is given, and in uniform matter the meson fields are replaced by their mean values. The resulting mesonic pressure has a simple quadratic (polytropic) form:

 P_meson = −½ m_σ² σ² + ½ m_ω² ω² + ½ m_ρ² ρ².

The scalar σ field reduces the effective neutron mass (m*_n = m_n − g_σ σ), while the vector fields provide a repulsive contribution. Non‑linear σ self‑interactions are omitted; instead, a density‑dependent effective mass is used to reproduce saturation properties while keeping the model transparent. The mesonic pressure is expressed as a polytrope P = K (Y_n ρ/ρ₀)^γ, where K and the adiabatic index γ are calibrated to satisfy causality (sound speed < c) and to match observational constraints.

The complete EOS, built from the low‑density LDM‑based crust, the neutron‑drip interpolation, and the high‑density meson polytrope, is continuous and thermodynamically consistent across all density regimes. It is then fed into the Tolman‑Oppenheimer‑Volkoff (TOV) equations to compute stellar models. The resulting mass‑radius curve reproduces the observed two‑solar‑mass pulsars and the radius estimates (≈10–14 km) from NICER and gravitational‑wave events. The authors also calculate the compactness C = GM/Rc² and the surface redshift z = (1 − 2GM/Rc²)⁻¹⁄² − 1, finding agreement with recent X‑ray timing and GW170817 analyses.

A sensitivity study shows that modest variations in the polytropic index γ or the constant K change the maximum mass by less than 5 %, demonstrating the robustness of the approach. The simplicity of the model makes it an ideal baseline for exploring more sophisticated physics, such as hyperons, deconfined quarks, or strong magnetic fields, by simply adjusting the mesonic sector or adding extra terms.

In conclusion, the paper demonstrates that a transparent, phenomenological EOS built from degenerate fermion thermodynamics, the liquid‑drop model, and a QHD‑inspired meson polytrope can capture the essential features of neutron‑star structure while remaining computationally inexpensive. This framework provides a useful test‑bed for assessing how macroscopic observables depend on the stiffness of the EOS and for benchmarking more elaborate microscopic calculations.