The Crusts of Neutron Stars Revisited: Approximations within a Polytropic Equation of State Approach

TL;DR

This work assesses whether detailed crust modeling is necessary for neutron star radii by revisiting thin-crust approximations against exact $TOV$ solutions across multiple equations of state, including SINPA with pasta. It develops a four-tier crust-approximation ladder (Simplest, Newtonian thin crust, Relativistic thin crust, and GR-corrected relativistic thin crust) and demonstrates that, for many cases, the crust can be treated with simple semi-analytic methods provided the core solution and the crust–core interface density $\epsilon_{cc}$ are known. Across EoSs MPA1, AP4, MS1, and SINPA, the relativistic approximations reproduce the exact $M$–$R$ relations within tens to hundreds of meters, while some EoSs yield larger discrepancies, underscoring persistent degeneracies with gravity theory and additional physics. The study emphasizes that, unless radii are measured with $\sim 100$ m precision, disentangling subnuclear EoS details from crustal and gravitational modeling remains challenging, and it advocates incorporating glitches and alternative gravity/dark matter effects to break degeneracies.

Abstract

In this work, we revisit several thin-crust approximations presented in the literature and compare them with the exact solutions of the Tolman--Oppenheimer--Volkoff (TOV) equations. In addition, we employ three different equations of state (EoSs), including one with a pasta phase, each based on a distinct theoretical framework: the variational method, relativistic Brueckner--Hartree--Fock theory, and relativistic mean-field theory. We emphasize that these approximations require only the TOV solutions for the core and the EoS properties at the core--crust interface; in our approach, only the energy density is needed. Finally, the relativistic approximation, as well as the Newtonian approximation with corrections, shows good agreement with the exact solutions. This indicates that a simple treatment of the crust is sufficient for structural purposes, independently of the uncertainties in the sub-nuclear equation of state, which are not very large. The unified EoS SINPA (relativistic mean-field theory), including the pasta phase, was used to study the thin-crust approximation, while degeneracy in the $M$--$R$ relation is demonstrated through: (i) anisotropic pressure in the modified TOV equations, (ii) the $f(R, L_m, T)$ gravity model, and (iii) dark matter admixture. As demonstrated, modifications to the description of gravitation introduce degeneracies in the mass--radius relation that are challenging to disentangle or quantify precisely.

Figures (11)

• Figure 1: Results for the EoS MPA1. Upper, density profiles for the $M=1.4 ~M_\odot$ and $M=2.08~M_\odot$ up to the surface. The dashed horizontal line marks the core-crust transition density. Middle, the simple approximation and the relativistic approximation is the most suitable for this EoS. As it stands, the simple approximation for $M=2.08~M_\odot$ has more deviation from the exact result than the one for $M=1.4 ~M_\odot$. Bottom, the blue solid line represents the exact TOV's solutions for the MPA1 EoS. The dashed green line, represents the thin crust approximation (relativistic). Even for an inner crust fit with just a polytropic EoS, the result is excellent. The red dashed line is the thin crust approximation for the Newtonian case with corrections. It is possible to see that even with correction, the results become accurate above $\sim \, M=1.4 ~M_\odot$.
• Figure 2: Results for the EoS AP4. Upper, the same as in Fig. \ref{['fig:mpa1_all']}. Note that the values for $\epsilon(r=0)$ for each NS mass are higher. Middle, the same as in Fig. \ref{['fig:mpa1_all']}, but now the simple approximation for $M=1.4 ~M_\odot$ produces more difference in the radius than the same approximation for the $M=2.08 ~M_\odot$ model. Note that the radius of the cores are the same for this choice of the supranuclear equation of state. Bottom, the same as in Fig. \ref{['fig:mpa1_all']}.
• Figure 3: Comparison of results for the EoS MS1. Upper, the same as in Fig. \ref{['fig:mpa1_all']}. Middle, the same as in Fig. \ref{['fig:mpa1_all']}. Bottom, the same as in Fig. \ref{['fig:mpa1_all']}. The calculated radii are too large to provide a fair representation of the stars measured with the NICER data.
• Figure 4: Results obtained with the SINPA EoS. Upper, same as Fig. \ref{['fig:mpa1_all']}, but this EoS does not allow neutron stars with $M > 2.0\,M_{\odot}$ and the inner crust treated without a pasta phase (BBP fit). Middle, the same as Fig. \ref{['fig:mpa1_all']}. However, the radius difference between exact TOV - with pasta - solutions and relativistic thin-crust approximations (BBP fit for inner crust with no pasta phase) remains below 400 m. Bottom, the same as Fig. \ref{['fig:mpa1_all']}. Mass-radius relation: exact TOV solution (solid blue), TOV without pasta (BPS+BBP+SINPA core; solid magenta), relativistic thin-crust approximation without pasta (BBP fit; green dashed), Newtonian thin-crust approximation without pasta (BBP fit; red dashed), and relativistic thin-crust approximation using $\Theta$ value from Equation \ref{['rc']} (yellow dashed). Since $\Theta$ is fixed from the SINPA EoS for both core and full model, the approximation closely matches the exact TOV result (solid blue line).
• Figure 5: The mass of the crust as a function of neutron star mass for the four equations of state, calculated using Eq. \ref{['mcrust']}: MPA1 (blue), MS1 (orange), AP4 (green), and unified SINPA (red). Our results agree with those shown in Fig. 1(a) in the Ref. Dutra2021 at comparable transition pressures $p_t$. For the unified SINPA equation of state, the present approximation reproduces the crustal mass at $2\,M_{\odot}$ reported in Table III in Ref. PhysRevD.106.023031.