Leptodermic corrections to the TOV equations and nuclear astrophysics within the effective surface approximation

TL;DR

The paper introduces an effective-surface, leptodermic expansion to extend the Tolman–Oppenheimer–Volkoff framework by incorporating gradient (surface) terms in the energy density for neutron stars. It derives a modified TOV (MTOV) with first-order surface corrections in the small parameter $a/R$, using a general energy-density functional $\mathcal{E}(\rho)=\mathcal{A}(\rho)+\mathcal{C}(\nabla\rho)^2$ and a density profile $\rho(r)=\bar{\rho} y((r-R)/a)$. The leading zero-order solution recovers the standard TOV results, while the first-order corrections, quantified by $\mathcal{D}_1$ and scaled by the incompressibility $K_G$ and gravity, modify the pressure mainly near the NS surface and grow with NS mass. This approach provides analytic insights into how crustal surface effects influence the NS equation of state and global structure, with potential extensions to multi-component and rotating stars. The findings offer a principled framework to incorporate surface physics into NS modeling, complementing traditional bulk EoS approaches.

Abstract

The macroscopic model for a neutron star (NS) as a liquid drop at the equilibrium is used to extend the Tolman-Oppenheimer-Volkoff (TOV) equations taking into account the gradient terms responsible for the system surface. The parameters of the Schwarzschild metric in the spherical case are found with these surface corrections to the known leading (zero) order of the leptodermic approximation $a/R<<1$, where $a$ is the NS effective-surface (ES) thickness, and $R$ is the effective NS radius. The energy density $\mathcal{E}$ is considered in a general form including the functions of the particle number density and of its gradient terms. The macroscopic gravitational component $Φ(ρ)$ of the energy density is taken into account in the simplest form as expansion in powers of $ρ-\overlineρ $, where $\overlineρ$ is the saturation density, up to second order, in terms of its contributions to the separation particle energy and incompressibility. Density distributions $ρ$ across the NS ES in the normal direction to the ES, which are derived in the simple analytical form at the same leading approximation, was used for the derivation of the modified TOV (MTOV) equations by accounting for their NS surface corrections. The MTOV equations are analytically solved at first order and the results are compared with the standard TOV approach of the zero order.

Figures (9)

• Figure 1: Particle number density, $\rho$, in units of the saturation value, $\overline{\rho}$, as function of the radial variable $r$ (in km) for a neutron star in a simple compressed liquid-drop model at the stable equilibrium. Solid line is related to the asymmetric solution, Eq. (32)$^\ast$ [the asterisk means the equation taken from Table \ref{['table-1']} of Ref. MM24, line "1"], which is the leading approximation over a small leptodermic parameter, $a/R\ll 1$. Dashed line presents the same but for the Wilets symmetric solution wiletsMM24 (line "2"). Parameters, for example: the effective radius $R=10$ km, diffuseness of the NS crust $a=1$ km, and temperature $T=0$; see, e.g., Refs. ST04HPY07MM24. The dots ES1 and ES2 show the ES at the effective radius $R$ for the solid and dashed lines, respectively (from Ref. MM24).
• Figure 2: Contour plots for the pressure $P(r)$, Eq. (\ref{['solTOVplus']}), in units of the central value, $P(r=0)$ as function of the radial coordinate $r$ and the dimensionless parameter $A_{\rm S}$ of the Schwarzschild metric, Eq. (\ref{['Schwarz']}). The numbers in squares show the values of this pressure. White color presents regions where we have in-determination, infinity by infinity, with the finite limit 1 at $r \rightarrow 0$. Red color shows negative values of the ratio $P(r)/P(r=0)$ [a positive pressure $P(r)$]. The value "0.00" displays the zero value in the horizontal and vertical coordinate lines on right of plots. The effective NS radius $R=10$ km is the same as in Fig. \ref{['fig1']} (from Ref. MM24).
• Figure 3: The pressure $P(r)$, Eq. (\ref{['solTOVplus']}), in units of the central value, $P(r=0)$, as function of the radial coordinate $r$ for the Schwarzschild metric, Eq. (\ref{['Schwarz']}), at two values of the parameter $A_{\rm S}$, $A_{\rm S}=1.15$ (solid line) and $A_{\rm S} \approx 0.96$ (dashed line). The latter corresponds to the two values for the NS masses, $M=1.4 M_\odot$ and $M=2.0 M_\odot$ ($R_{\rm S} \approx 13.0$ km and $15.6$ km, respectively; see Refs. GR21CC20TR21OL20 and text). The effective NS radius $R=10$ km is the same as in Figs. \ref{['fig1']} and \ref{['fig2']}.
• Figure 4: Pressure $P$, Eq. (60)$^\ast$, in units of $\overline{\rho} K_G/9$ through the NS diffuse surface as function of the density variable $y$, (a), for the universal Equation of State, and the radial coordinate $r$, (b), through the particle number density, $\rho=\overline{\rho}y((r-R)/a)$. The density $y(x)$ is given, e.g., by Eq. (32)$^\ast$ (solid lines). Dashed and dotted lines show the surface (S) and volume (V) components, $P_{\rm S}$ [Eq. (62)$^\ast$] and $P_{\rm V}$ [Eq. (61)$^\ast$], respectively. The crust thickness $a=1.0$ km and the effective radius parameter $R=10$ km are the same as in Figs. \ref{['fig1']} and \ref{['fig2']}. The full dots and arrows present the ES for $y=y_0$, (a), and $r=R$, (b).
• Figure 5: Contour plots for the dimensionless pressure $p^{}_0(r,A_{\rm S})$, Eq. (\ref{['p0cal']}), as function of the radial coordinate $r$ and the dimensionless parameter $A_{\rm S}$ of the Schwarzschild metric, Eq. (\ref{['Schwarz']}). Other notations and parameters are the same as in Fig. \ref{['fig2']}.