Unified treatment of subsaturation stellar matter at zero and finite temperature

The standard variational derivation of stellar-matter structure in the Wigner-Seitz approximation is generalized to the finite-temperature situation where a wide distribution of different nuclear species can coexist in the same density and proton fraction condition, possibly out of $\beta$ equilibrium. The same theoretical formalism is shown to describe on one side the single-nucleus approximation (SNA), currently used in most core-collapse supernova simulations and on the other side the nuclear statistical equilibrium (NSE) approach, routinely employed in $r$- and $p$-process explosive nucleosynthesis problems. In particular, we show that in-medium effects have to be accounted for in NSE to have a theoretical consistency between the zero-temperature and the finite-temperature modeling. The bulk part of these in-medium effects is analytically calculated in the local density approximation and shown to be different from a Van der Waals excluded-volume term. This unified formalism allows controlling quantitatively the deviations from the SNA in the different thermodynamic conditions, as well as having a NSE model which is reliable at any arbitrarily low value of the temperature, with potential applications for neutron-star cooling and accretion problems. We present different illustrative results with several mass models and effective interactions, showing the importance of accounting for the nuclear species distribution even at temperatures lower than 1 MeV.

Figure 1

Surface of the energy per baryon $E_{\mathrm{WS}}/A_{\mathrm{WS}}$ at ${\rho }_B={10}^{-7} \text{fm}{}^{-3}$ and different values of proton fractions. The cluster energy is calculated according to FRDM [74] (a) and using the experimental database [67] (b).

Figure 2

Outer crust composition at $T=0$: Baryonic (a) and atomic (b) numbers of the $\beta$-equilibrium nucleus as a function of the baryonic density. BPS corresponds to predictions by BPS [1]; FRDM and $\exp +\mathrm{SLY}4$ (SKMs) stand for present model predictions when nuclear masses are calculated according to finite-range droplet model of Ref. [74] and, respectively, atomic mass data of Ref. [67] +LDM-SLY4 (SKMs) model of Ref. [68]. The vertical lines mark the drip line in the stellar medium.

Figure 3

Crust composition at $T=0$: Baryonic (a) and atomic (b) numbers of the equilibrium nucleus as a function of the baryonic density. NV stands for predictions by Negele and Vautherin [8]; $\mathrm{BPS}+\mathrm{BBP}$ corresponds to predictions by BPS [1] and BBP [2]; $\exp +\mathrm{SLY}4$ (SKMs) stand for present model predictions when nuclear masses are calculated according to atomic mass data of Ref. [67] + LDM-SLY4 (SKMs) model of Ref. [68]. The inset in (b) depicts the evolution with baryonic density of the total proton fraction.

Figure 4

Behavior of the cluster mass number as a function of the baryonic density in the inner crust using two different Skyrme functionals (SKMs [79] and SLY4 [9]) for both the free neutron energy density and the nuclear masses according to the LDM-Skyrme model of Ref. [68]. The total mass of the cluster is compared to the bound part of the cluster, obtained by simply subtracting the number of free neutrons according to Eq. (3).

Figure 5

Total $\left(\mathrm{baryonic}+\mathrm{leptonic}\right)$ energy density and pressure as a function of baryonic density corresponding to the neutron-star crust ($T=0,\beta$ equilibrium). Experimental [67] and LDM binding energies [68] have been used in the outer and, respectively, inner crust. The employed nuclear effective interaction are SLY4 [9] and SKMs [79]. Present results are confronted with those of NV [8] and BBP [2].

Figure 6

Constrained energy densities as a function of density. (a) Outer crust as obtained using FRDM. The numbers accompanying the curves are in MeV and stand for ${\lambda }_B$. (b) Crust (thick line) and $\beta$-equilibrated homogeneous matter (thin line) at $T=0$, corresponding to SLY4 and ${\lambda }_B=11.25$ MeV.

Figure 7

Constrained energy density [Eq. (44)] in the thermodynamic condition corresponding to the liquid-gas phase transition of symmetric matter given by ${\lambda }_B=-15.97$ MeV and ${\lambda }_3=0$. The different curves represent different values of the total proton density ${\rho }_p$. Solid (dashed) curves correspond to constrained baryonic $\left(\mathrm{baryonic}+\mathrm{leptonic}\right)$ energy density. The considered effective interaction is SLY4.

Figure 8

Mass (a) and atomic (b) numbers of the unique nucleus of the WS cell as a function of baryonic density for $Y_p=0.2$ and different temperatures $T=0,1,2.5,5$ MeV. Predictions of present SNA model (solid lines) are compared with the LS results [39] corresponding to LS220 as calculated in Ref. [90] (dotted lines) as well as with the NSE prediction for the most probable cluster (dashed lines). The LDM-SKMs model of Ref. [68] is used for the cluster energy functional.

Figure 9

The same as in Fig. 8 for $\beta$ equilibrium and different temperatures $T=0,0.5,2.5$ MeV. Predictions of the present SNA model (solid lines) are compared with the LS results [39] corresponding to LS220 as calculated in Ref. [90] (dotted lines), as well as with the NSE predictions for the most probable cluster (dashed lines). The LDM-SKMs model of Ref. [68] is used for the cluster energy functional.

Figure 10

Contours of cluster constrained free energy $F_{\beta }^e-{\lambda }_{B}A-{\lambda }_3(A-2Z)$ (in MeV). The considered thermodynamical conditions are ${\rho }_B={10}^{-3} {\text{fm}}^{-3},Y_p=0.39$, and $T=0.5$ MeV (a) and, respectively, 2.4 MeV (b). The values of the external isovector and isoscalar chemical potentials ($\lambda ,{\lambda }_3$) are (−10.06 MeV, 3.98 MeV) and, respectively, (−11.03 MeV, 5.86 MeV). Experimental values [67] and predictions of the ten-parameter mass model of Duflo-Zuker [91] have been used for the binding energies.

Figure 11

Constrained cluster free energy for ${\rho }_B={10}^{-3} {\text{fm}}^{-3},T=1.5$ MeV in $\beta$ equilibrium with chemical potentials corresponding to the NSE (solid line) and SNA (dashed line) models for a fixed cluster proton fraction corresponding to the minimum of the constrained free energy. The arrow gives the SNA solution. The dotted line gives the NSE multiplicity distribution in arbitrary units.

Figure 12

Structure of the (P)NS crust at $\beta$ equilibrium as a function of temperature for different values of the baryonic density ${\rho }_B={10}^{-6},{10}^{-4},{10}^{-3},{10}^{-2} {\text{fm}}^{-3}$. Solid and dashed lines correspond to predictions of SNA and, respectively, the most probable (mp) cluster in NSE. Dotted lines correspond to the average (av.) heavy $\left(A\ge 20\right)$ cluster atomic mass in NSE. Standard NSE predictions are plotted against predictions of a modified NSE (NSEm), where no cluster lighter than $A=20$ is allowed to exist. In SNA Skyrme-LDM [68] binding energies have been used. In NSE, experimental data [67] have been used for the binding energies of nuclides with $A<16$ and Skyrme-LDM [68] predictions otherwise. The considered effective interaction is SLY4.

Figure 13

The same as in Fig. 12 for the unbound nucleons mass fraction.

Figure 14

NSE results at $\beta$ equilibrium for different densities and temperatures (expressed in MeV and listed in the key legend). (a), (b) Average mass and atomic numbers of clusters heavier than $A=20$. (c) Corresponding mass fraction. Experimental and DZ10 [91] nuclear masses have been considered for clusters. The SLY4 effective interaction was used for the unbound nucleon component.

Figure 15

NSE mass fractions of unbound nucleons and ${}^2\mathrm{H},{}^3\mathrm{H},{}^3\mathrm{He},{}^4\mathrm{He},{}^{A\ge 4}\text{H}$, and ${}^{A\ge 5}\text{He}$ at $\beta$ equilibrium for different densities and temperatures expressed in MeV and listed in the key legend. Experimental [67] and DZ10 [91] data have been used for the binding energies. The SLY4 effective interaction was used for the unbound nucleon component. Note that the $x$-axis range is not the same in all panels.

Figure 16

Behavior of the constrained free-energy density in $\beta$ equilibrium $\left({\mu }_L=0\right)$ for the clusterized phase (solid lines) and the homogeneous phase (dashed lines) at different values of the baryonic chemical potential ${\lambda }_B$ mentioned in the key legend (in MeV). Experimental binding energies and predictions of the ten-parameter mass model of Duflo-Zuker are used for the nuclear clusters. For the unbound nucleon gas the SLY4 effective interaction has been used.

Figure 17

Fragment mass distributions corresponding to $\beta$ equilibrium, $T=2$ MeV, and different baryonic densities (expressed in ${\text{fm}}^{-3}$) as listed in the key legend. The numbers next to the peaks specify the charge number of the most abundant nucleus.