Properties of the neutron star crust: Quantifying and correlating uncertainties with improved nuclear physics

A compressible liquid-drop model (CLDM) is used to correlate uncertainties associated with the properties of the neutron star (NS) crust with theoretical estimates of the uncertainties associated with the equation of state (EOS) of homogeneous neutron and nuclear matter. For the latter, we employ recent calculations based on Hamiltonians constructed using chiral effective-field theory ($\chi \mathrm{EFT}$). Fits to experimental nuclear masses are employed to constrain the CLDM further, and we find that they disfavor some of the $\chi \mathrm{EFT}$ Hamiltonians. The CLDM allows us to study the complex interplay between bulk, surface, curvature, and Coulomb contributions, and their impact on the NS crust. It also reveals how the curvature energy alters the correlation between the surface energy and the bulk symmetry energy. Our analysis quantifies how the uncertainties associated with the EOS of homogeneous matter implies significant uncertainties for the composition of the crust, its proton fraction, and the volume fraction occupied by nuclei. We find that the finite-size effects impact the crust composition but have a negligible effect on the net isospin asymmetry of matter. The isospin asymmetry is largely determined by the bulk properties and the isospin dependence of the surface energy. The most significant uncertainties associated with matter properties in the densest regions of the crust, the precise location of the crust-core transition, are found to be strongly correlated with uncertainties associated with the Hamiltonians. By adopting a unified model to describe the crust and the core of NSs, we tighten the correlation between their global properties such as their mass-radius relationship, moment of inertia, crust thickness, and tidal deformability with uncertainties associated with the nuclear Hamiltonians.

Figure 1

[(a) and (c)] Energy per particle as a function of the baryon density, in NM and SM. [(b) and (d)] Energy per particle normalized to the free Fermi gas energy [$E_{\mathrm{FFG}}=(3/5)E_F$ with $E_F={\hslash }^2{k}_F^2/\left(2m\right)$ and $k_F$ the Fermi momentum] in SM and NM function of the neutron Fermi momentum $k_{F_n}$ for the six Hamiltonians H1-H7 (except H6). Panels (a) and (b) include DHS Hamiltonians for comparison. Data from the original model are plotted in dots (squares) for H1–H5 and H7 (${\mathrm{DHS}}_{L59}$ and ${\mathrm{DHS}}_{L69}$), with the same color as the MM version. Bottom panels include the nucleon effective mass.

Figure 2

(a) Symmetry energy with respect to the baryon density for H1–H5 and H7, ${\mathrm{DHS}}_{L59}$, ${\mathrm{DHS}}_{L69}$, and SLy4. Yellow band shows constraint from neutron skin experiments by PREX-II. Blue bands show constrains from isobaric-analog state IAS (IAS $+$ neutron skin, $\mathrm{\Delta }{r}_{np}$). (b) Symmetry energy normalized to the free Fermi gas symmetry energy $[E_{\mathrm{sym},\mathrm{FFG}}=(3/5)(E_{F,\mathrm{NM}}-E_{F,\mathrm{SM}})]$ with respect to the Fermi momentum $k_F$. Continuous (dashed) lines consider bare (effective) nucleon mass.

Figure 3

Pressure as function of the baryon density in $\beta$-equilibrium for the six Hamiltonians, ${\mathrm{DHS}}_{L59}$, ${\mathrm{DHS}}_{L69}$, and SLy4. Continuous (dashed) lines consider bare (effective) nucleon mass. Error bar shows the constraint for the pressure at $2n_{\mathrm{sat}}$ inferred by the LIGO-Virgo Collaboration with the GW170817 observation at $90\%$ credible level.

Figure 4

Sum of surface and curvature energies normalized by $4\pi r_{0,\mathrm{sat}}^2{A}_{\mathrm{cl}}^{2/3}$ function of the cluster asymmetry $I_{\mathrm{cl}}$ in isolated nuclei. (a) Shows a comparison between the four finite-size models in the case in which they are fixed by the standard values given in Table 4 (solid lines) and in the case in which they are fit to the nuclear masses (dotted lines). (b) Shows the influence of the surface parameter $p_{\mathrm{surf}}$ at large $I_{\mathrm{cl}}$, after the fit to the nuclear masses.

Figure 5

Loss function ${\mathrm{\Delta }}_E$ fit to the total energy (a) and ${\mathrm{\Delta }}_{E/A}$ fit to the energy per particle (b) with respect to ${\epsilon }_{\mathrm{sat}}$ for FS1–FS4 for H1–H5 and H7 (${\mathrm{DHS}}_{L59}$ and ${\mathrm{DHS}}_{L69}$) in lines (symbols). Solid lines stand for the bare mass while dashed lines stand for the effective mass.

Figure 6

(a) The isoscalar surface tension parameter ${\sigma }_{\mathrm{surf},\mathrm{sat}}$ is represented against $E_{\mathrm{sym}}$ for the six Hamiltonians H1–H5 and H7, and for the FS models FS1 to FS4. The two Hamiltonians DHS are included with dots (squares) for the minimization to the total energy (energy per particle). (b) Same for ${\sigma }_{\mathrm{sym}}$. Silver band shows the values for the two Hamiltonians which best reproduce nuclear masses, H2 and H3.

Figure 7

Neutron gas and electron contribution to the pressure on the NS crust. (a) Shows the difference originated by the four finite-size models (with standard parameters, see Table 4). (b) Shows the influence of the different optimization procedures (see Sec. 3c).

Figure 8

Crust composition (a)–(d), asymmetry (e) and (f), and volume fraction occupied by the cluster (g) and (h), for the different ingredients in the CLDM within the $\mathrm{SLy}{4}_{\mathrm{MM}}$ interaction. FS1 (red), FS2 (magenta), FS3 (green), and FS4 (black) are as explained in Table 5. Dotted lines in panel (c) represent the magic numbers $Z_{\mathrm{cl}}=28$ and $Z_{\mathrm{cl}}=40$. Continuous lines represent the parameters fit to the total energy, while dashed lines represent the fitting to the energy per particle. Left panels show outer and inner crust. Right panels show zoom at crust-core transition.

Figure 9

Crust composition (a)–(d), asymmetry (e) and (f), and volume fraction occupied by the cluster (g) and (h), for the different ingredients in the CLDM. In the left panels we show the 6 H from outer to inner crust, on the right we show the 4 H allowed by the finite nuclei analysis with a zoom close the crust-core transition. Dotted lines in panel (c) represent the magic numbers $Z_{\mathrm{cl}}=28$, $Z_{\mathrm{cl}}=40$, and $Z=50$. Dashed lines include the nucleon effective mass. All figures use $\mathrm{FS}{4}^{\mathrm{\Delta }E}$ model.

Figure 10

Same as Fig. 9. Comparison of three different values of the surface parameters $p_{\mathrm{surf}}$, within the $\mathrm{H}{2}_{{{\mathrm{MM}}_{{\mathrm{m}}^{*}}}^{\mathrm{FS}4,{\mathrm{\Delta }}_E}}$ and $\mathrm{H}{3}_{{{\mathrm{MM}}_{{\mathrm{m}}^{*}}}^{\mathrm{FS}4,{\mathrm{\Delta }}_E}}$ models.

Figure 11

Mass and radius relation. Observational constrains from NICER to the pulsar J0030 $+$ 0451 and J0740 $+$ 6620, and the constraint from Capano et al. [66] combining multimessenger signals with nuclear physics are shown in error bars. Wheat-colored band shows the lowest mass NS observed with a mass of $1.17M_{\odot }$. Marks on curves represent the density where causality is broken for a given EOS. Top panels show the results for H1–H4 and (a) the influence of the optimization procedure and (b) the effect of the inclusion of the nucleon effective mass. Bottom panels show the results for the two Hamiltonians that better reproduce nuclear masses, H2 and H3, (c) the impact of finite-size terms, and (d) the impact of the surface parameter $p_{\mathrm{surf}}$. See text for more details.

Figure 12

Correlation of crust thickness for a $1.0M_{\odot }$, $1.2M_{\odot }$, and $1.4M_{\odot }$ NS and the symmetry energy evaluated at $n_{\mathrm{sat}}/2$. Dashed lines include the nucleon effective mass.