Unified equations of state for cold nonaccreting neutron stars with Brussels-Montreal functionals. IV. Role of the symmetry energy in pasta phases

Our previous investigation of neutron-star crusts, based on the functional BSk24, led to a substantial reduction of the pasta mantle when Strutinsky integral and pairing corrections were added on top of the fourth-order extended Thomas-Fermi (ETF) method. Here, our earlier calculations are widened to a larger set of functionals within the same family and we find that the microscopic corrections weaken significantly the influence of the symmetry energy. In particular, the correlation observed at the pure ETF level between the density for the onset of pasta formation and the symmetry energy vanishes, not only for the $L$ coefficient but also for the symmetry-energy values at the relevant densities. Moreover, the inclusion of microscopic corrections results in a much lower abundance of pasta for all functionals.

Figure 1

Symmetry energy (3) (in MeV) as a function of the baryon density $n$ (in ${\mathrm{fm}}^{-3}$) for the following Brussels-Montreal nuclear energy-density functionals: BSk22 (green dash-dotted), BSk24 (blue solid), BSk25 (red dashed), and BSk26 (purple dotted).

Figure 2

ETF energy per nucleon $e$ (in keV) for different pasta shapes with the energy per nucleon of spheres subtracted versus average baryon number density $\overline{n}$ (in ${\mathrm{fm}}^{-3}$). Each of the four panels shows results for a given BSk functional belonging to the same family but differing in the prediction for the symmetry energy defined by Eq. (4). The energies of spaghetti, lasagna, bucatini, Swiss cheese and homogeneous matter are represented by red dashed lines, yellow solid lines, purple dash-dotted lines, green dotted lines and pink solid lines, respectively. The horizontal histograms illustrate the sequence of pasta phases with the same color coding, adding a blue area for spheres.

Figure 3

ETFSI and ETF energy per nucleon (in keV) for different pasta shapes with ETF energy of spheres subtracted versus average baryon number density $\overline{n}$ (in ${\mathrm{fm}}^{-3}$). ETF results are plotted by curves with the same legend as in Fig. 2. ETFSI results are displayed by symbols. Solid blue circles and yellow squares correspond to solutions with bound protons for spheres and for lasagna, respectively, while empty red crosses to solutions for spaghetti with dripped protons (see text for details). The horizontal histograms show the sequence of pasta phases with microscopic corrections.

Figure 4

Relative abundances of pasta phases in the crust of neutron stars for four nuclear functionals within the ETF and ETFSI approaches. Blue regions represent spheres, red correspond to spaghetti, yellow are for lasagna, purple indicate bucatini, and finally green are for Swiss cheese.