Neutron star radii and crusts: Uncertainties and unified equations of state

The uncertainties in neutron star radii and crust properties due to our limited knowledge of the equation of state are quantitatively analyzed. We first demonstrate the importance of a unified microscopic description for the different baryonic densities of the star. If the pressure functional is obtained matching a crust and a core equation of state based on models with different properties at nuclear matter saturation, the uncertainties can be as large as $\sim 30$ % for the crust thickness and 4% for the radius. Necessary conditions for causal and thermodynamically consistent matchings between the core and the crust are formulated and their consequences examined. A large set of unified equations of state for purely nucleonic matter is obtained based on twenty-four Skyrme interactions and nine relativistic mean-field nuclear parametrizations. In addition, for relativistic models fifteen equations of state including a transition to hyperonic matter at high density are presented. All these equations of state have in common the property of describing a $2M_{\odot }$ star and of being causal within stable neutron stars. Spans of $\sim 3$ and $\sim 4$ km are obtained for the radius of, respectively, $1.0M_{\odot }$ and $2.0M_{\odot }$ stars. Applying a set of nine further constraints from experiment and ab initio calculations the uncertainty is reduced to $\sim 1$ and 2 km, respectively. These residual uncertainties reflect lack of constraints at large densities and insufficient information on the density dependence of the equation of state near the nuclear matter saturation point. The most important parameter to be constrained is shown to be the symmetry energy slope $L$. Indeed, this parameter exhibits a linear correlation with the stellar radius, which is particularly clear for small mass stars around $1.0M_{\odot }$. The other equation-of-state parameters do not show clear correlations with the radius, within the present uncertainties. Potential constraints on $L$, the neutron star radius, and the equation of state from observations of thermal states of neutron stars are also discussed. The unified equations of state are made available in the Supplemental Materials and via the CompOSE database.

Figure 1

Mass versus radius (left) and crust thickness $l^{\mathrm{cr}}$ versus mass (right) for the relativistic mean field model GM1, using different matching procedures (see text).

Figure 2

Pressure $P$ versus chemical potential $\mu$, for the NL3 EOS for the core (black solid line) and DH for the crust EOS (red solid line). The presented situation corresponds to a matching between $n_1=0.09{\mathrm{fm}}^{-3}$ and $n_2=n_0=0.16{\mathrm{fm}}^{-3}$. The dashed lines correspond to the condition of thermodynamical consistency and are given by Eqs. (14) and (15). The dotted lines are given by the causality limit, Eqs. (16) and (17). The area defined by these lines corresponds to the shaded region and is a bit smaller than the region allowed for a thermodynamically consistent matching between points 1 and 2 (see insets).

Figure 3

Pressure $P$ versus chemical potential $\mu$, for the NL3 EOS for the core (black solid line) and DH for the crust EOS (red solid line). The points A2 and B2 correspond to two different values of $n_2$: $n_{\mathrm{t}}$, the core-crust transition density, and $n_0/2$, respectively. The dashed lines indicate the condition of thermodynamical consistency and the dotted lines mark the causality limit. They are almost indistinguishable. The points A1 and B1 correspond to the higher limits on $\mu$ or equivalently $n$, such that a thermodynamically consistent gluing with the core at the points A2 and B2 exists.

Figure 4

Linear matching between core and crust. Upper panel, solid line: direct connection between $({\rho }_1,P_1)$ and $({\rho }_2,P_2)$ which is thermodynamically inconsistent $\left(a=0.088\right)$; dotted and dashed lines: thermodynamically consistent gluing for $a=0.33$ (dotted) and $a=a_{\min }=0.0913$ (dashed) accompanied by a density jump (jumps) defined by ${\rho }_1^{\prime },{\rho }_2^{\prime }$ (dotted line). Bottom panel: $M\left(R\right)$ for these matchings; thermodynamically consistent solutions give very similar radii for $M>1.0M_{\odot }$. Stars with central pressure equal to $P_2$ have masses $\sim 0.6M_{\odot }$.

Figure 5

Mass-radius relation for the various RMF models: noY, Y, and Yss. The horizontal lines indicate the constraints set by the pulsars PSR J0348+0432 and PSR J1614− 2230.

Figure 6

$(n,p,e)$ matter in $\beta$ equilibrium. Upper panel: total pressure versus total energy density for the unified EOS (solid lines) and using experimental masses when available (dashed lines). Lower panel: relative deviation of the NS radius as a function of mass with the two prescriptions shown on the top part. The SkI4 functional is used.

Figure 7

Mass-radius relations for Skyrme models. As in Fig. 5, the horizontal lines indicate the constraints set by the pulsars PSR J0348+0432 and PSR J1614−2230.

Figure 8

Mass as a function of the radius for the BSk20 and BSk21 functionals. Full lines: full microscopic HFB calculation from [84]. Dashed lines: our model for the unified EOS. As in Fig. 5, the horizontal lines indicate the constraints set by the pulsars PSR J0348+0432 and PSR J1614−2230.

Figure 9

Radii of purely nucleonic NS as a function of the symmetry energy slope $L$ (upper plots) and the incompressibility of symmetric matter $K$ (lower plots) for different masses ($1.0M_{\odot },1.4M_{\odot }$, and $1.8M_{\odot }$ from left to right). The red dots indicate Skyrme models, the black ones RMF models. The blue symbols (stars for RMF and a pentagon for SLy9) correspond to models which are at the intersection of all nuclear constraints in the $L\text{−}J$ plane; see Fig. 13. The shaded areas and the solid violet line indicate the result of a linear regression with and without, respectively, taking into account the error bars in the fitted parameters. The correlation coefficient $r$ is indicated in each plot.

Figure 10

Mass vs crust thickness relation for the various RMF models: noY, Y, and Yss. Line styles correspond to the ones used in Fig. 5.

Figure 11

Energy per particle of neutron matter as a function of density $n$ for Skyrme models and constraints by [93] and [94].

Figure 12

Energy per particle of neutron matter as a function of density $n$ for RMF models and constraints by [93] and [94].

Figure 13

$L$ and $J$ parameters of all our EOS compared to various nuclear constraints (see text for details). The white crossed region corresponds to the intersection of all constraints. EOS fulfilling all constraints are indicated by blue symbols: a pentagon for the unique Skyrme model and a star for the three RMF ones.

Figure 14

Mass-radius relations for the models fulfilling all nuclear constraints. As in Fig. 5, the horizontal lines indicate the constraints set by the pulsars PSR J0348+0432 and PSR J1614−2230.

Figure 15

Density threshold $n_{\mathrm{DU}}$ (upper plot) and mass threshold $M_{\mathrm{DU}}$ (lower plot) for the nucleonic DUrca process to operate in purely nucleonic NS versus the slope of the symmetry energy $L$. The convention for colors and symbols is the same as in Fig. 9. Empty symbols next to the upper $x$ axis indicate EOS for which $n_{\mathrm{DU}}$ is larger than the central density of the most-massive NS. In the lower plot, the error bars indicate the mass range over which the DUrca process operates.