Higher-order symmetry energy and neutron star core-crust transition with Gogny forces

Background: An accurate determination of the core-crust transition is necessary in the modeling of neutron stars for astrophysical purposes. The transition is intimately related to the isospin dependence of the nuclear force at low baryon densities.

Purpose: To study the symmetry energy and the core-crust transition in neutron stars using the finite-range Gogny nuclear interaction and to examine the deduced crustal thickness and crustal moment of inertia.

Methods: The second-, fourth-, and sixth-order coefficients of the Taylor expansion of the energy per particle in powers of the isospin asymmetry are analyzed for Gogny forces. These coefficients provide information about the departure of the symmetry energy from the widely used parabolic law. The neutron star core-crust transition is evaluated by looking at the onset of thermodynamical instability of the liquid core. The calculation is performed with the exact Gogny equation of state (EoS) (i.e., the Gogny EoS with the full isospin dependence) for the $\beta$-equilibrated matter of the core, and also with the Taylor expansion of the Gogny EoS in order to assess the influence of isospin expansions on locating the inner edge of neutron star crusts.

Results: The properties of the core-crust transition derived from the exact EoS differ from the predictions of the Taylor expansion even when the expansion is carried through sixth order in the isospin asymmetry. Gogny forces, using the exact EoS, predict the ranges $0.094{\text{fm}}^{-3}\lesssim {\rho }_t\lesssim 0.118{\text{fm}}^{-3}$ for the transition density and $0.339\text{MeV}{\text{fm}}^{-3}\lesssim P_t\lesssim 0.665\text{MeV}{\text{fm}}^{-3}$ for the transition pressure. The transition densities show an anticorrelation with the slope parameter $L$ of the symmetry energy. The transition pressures are not found to correlate with $L$. Neutron stars obtained with Gogny forces have maximum masses below $1.74M_{\odot }$ and relatively small moments of inertia. The crustal mass and moment of inertia are evaluated and comparisons are made with the constraints from observed glitches in pulsars.

Conclusions: The finite-range exchange contribution of the nuclear force, and its associated nontrivial isospin dependence, is key in determining the core-crust transition properties. Finite-order isospin expansions do not reproduce the core-crust transition results of the exact EoS. The predictions of the Gogny D1M force for the stellar crust are overall in broad agreement with those obtained using the Skyrme-Lyon EoS.

Figure 1

Density dependence of the second-order symmetry energy coefficient $E_{\mathrm{sym},2}\left(\rho \right)$ for different Gogny interactions. Also represented are the symmetry energy constraints extracted from the analysis of data on isobaric analog states (IAS) and of IAS data combined with neutron skins (IAS+n.skin) [72], the constraints from the electric dipole polarizability in lead (${\alpha }_D$ in ${}^{208}\mathrm{Pb}$) [73], and from transport simulations of heavy-ion collisions of tin isotopes (HIC) [74].

Figure 2

Density dependence of the fourth-order symmetry energy coefficient $E_{\mathrm{sym},4}\left(\rho \right)$ for different Gogny interactions.

Figure 3

Density dependence of the sixth-order symmetry energy coefficient $E_{\mathrm{sym},6}\left(\rho \right)$ for different Gogny interactions.

Figure 4

Density dependence of the ratios $E_{\mathrm{sym},4}\left(\rho \right)/E_{\mathrm{sym},2}\left(\rho \right)$ (top panel) and $E_{\mathrm{sym},6}\left(\rho \right)/E_{\mathrm{sym},2}\left(\rho \right)$ (bottom panel) for different Gogny interactions.

Figure 5

Density dependence of the symmetry energy coefficient in the parabolic approximation [Eq. (19)] for different Gogny interactions.

Figure 6

Density dependence of the ratio $E_{\mathrm{sym}}^{\mathrm{PA}}\left(\rho \right)/E_{\mathrm{sym},2}\left(\rho \right)$ for different Gogny interactions.

Figure 7

Density dependence of the isospin asymmetry in $\beta$-stable matter calculated using the exact expression of the EoS or the expression in Eq. (2) up to second, fourth, and sixth order for the D1S and D280 interactions. The results of the parabolic approximation are also included.

Figure 8

Density dependence of the pressure in $\beta$-stable matter calculated using the exact expression of the EoS or the expression in Eq. (2) up to second, fourth, and sixth order for the D1S and D280 interactions. The results of the parabolic approximation are also included. The vertical axis is in logarithmic scale.

Figure 9

Density dependence of the thermodynamical potential $V_{\mathrm{ther}}\left(\rho \right)$ in $\beta$-stable matter calculated using the exact expression of the EoS (solid lines) or the expression in Eq. (2) up to second (dashed lines), fourth (dash-dotted lines), and sixth (dash-double-dotted lines) order for the D1S and D280 interactions. The results of the parabolic approximation are also included (double-dashed-dotted lines).

Figure 10

Core-crust transition asymmetry ${\delta }_t$ as a function of the slope parameter $L$ calculated using the exact expression of the EoS (crosses), and the approximations up to second (solid squares), fourth (solid diamonds), and sixth order (solid triangles). The parabolic approximation is also included (empty squares).

Figure 11

Core-crust transition density ${\rho }_t$ as a function of the slope parameter $L$. Symbols are defined as in Fig. 10.

Figure 12

Core-crust transition pressure, $P_t$, as a function of the slope parameter $L$. Symbols are defined as in Fig. 10.

Figure 13

Mass-radius relation for the neutron stars produced with the four stable Gogny functionals and with the unified SLy EoS [91]. We show physically excluded regions in the upper-left corner as well as the accurate $M\approx 2M_{\odot }$ mass measurement of Ref. [96].

Figure 14

(a) Neutron star moment of inertia as a function of mass for the four stable Gogny functionals and for the SLy EoS. (b) Dimensionless ratio $I/M{R}^2$ as a function of compactness for the EoSs. The shaded area enclosed by a solid line is the parametrization provided by Breu and Rezzolla [60]. The shaded area enclosed by a dashed line is that of Lattimer and Schutz [59].

Figure 15

(a) Crust thickness for the four stable Gogny functionals and the SLy EoS. (b) Mass enclosed by the crust. Short-dashed lines correspond to the approximation of Eq. (36). (c) Percentage fraction of the star's moment of inertia contained in the crust. Short-dashed lines correspond to the approximation of Eq. (37). Thick horizontal lines indicate the constraints of Refs. [37] (bottom line) and [9] (top line) to account for observed glitches in the Vela pulsar and other glitching sources.