Pasta phases in neutron stars under strong magnetic fields

In the present work, we consider nuclear matter in the innermost crust of neutron stars under the presence of a strong magnetic field within the framework of a relativistic mean-field description. Two models with a different slope of the symmetry energy are considered in order to discuss the density dependence of the equation of state on the crust structure. The nonhomogeneous matter in $\beta$ equilibrium is described within the coexisting phases method, and the effect of including the anomalous magnetic moment is discussed. Five different geometries for the pasta structures are considered. It is shown that strong magnetic fields cause an extension of the inner crust of the neutron stars, with the occurrence of a series of disconnected nonhomogeneous matter regions above the one existing for a null magnetic field. Moreover, we observed that in these disconnected regions, for some values of the magnetic field, all five different cluster geometrical shapes occur, and the gas density is close to the cluster density. Also, the pressure at the neutron star crust-core transition is much larger than the pressure obtained for a zero magnetic field. Another noticeable effect of the presence of strong magnetic fields is the increase of the proton fraction, favoring the appearance of protons in the gas background.

Figure 1

Radii of WS cell (red) and nucleus (green) for $\beta$-equilibrium matter using the NL3 parametrization without [first row, (a), (b) and (c)] and with [second row, (d), (e) and (f)] the inclusion of AMM for different magnetic field strengths $B^{*}=5\times {10}^3$ [left, (a), (d), (g), and (j)], ${10}^4$ [middle, (b), (e), (h), and (k)], $2\times {10}^4$ [right, (c), (f), (i), and (l)]. The no-field case is also shown with gray points as a reference. Growth rates obtained with a dynamical spinodal calculation in [48] are plotted with blue lines. The gas (blue), designated by L for low density, and the cluster (red), designated by H for high density, neutron [third row, (g), (h), (i)] and proton [fourth row, (j), (k), (l)] densities inside the WS cell are also plotted as a function of the density without (light colors) and with (dark colors) AMM.

Figure 2

The pasta phases at the largest-density disconnected region: radii of the WS cell (red) and cluster (green), proton density of high-density phase ${\rho }_p^H$ (blue) and low-density phase ${\rho }_p^L$ (light blue), neutron density of the high-density phase ${\rho }_n^H$ (purple), and low-density phase ${\rho }_n^L$ (orange) as a function of the density, for the NL3 parametrization, with (bottom) and without (top) AMM. The insets show that the low and high proton and neutron densities are similar but different. As before, H is for the cluster density and L is for the gas density.

Figure 3

The evolution of the pasta shapes as a function of the baryonic density for $\beta$-equilibrium matter and for different magnetic field strengths: $B^{*}=0$ (gray), $5\times {10}^3$ (red), ${10}^4$ (green), $2\times {10}^4$ (blue), for the NL3 model with (bottom) and without (top) AMM.

Figure 4

Radii of the WS cell (red) and nucleus (green) for $\beta$-equilibrium matter using the $\mathrm{NL}3\omega \rho$ parametrization without [first row, (a), (b) and (c)] and with [second row, (d), (e) and (f)] the inclusion of AMM for different magnetic field strengths $B^{*}=5\times {10}^3$ [left, (a), (d), (g) and (j)], ${10}^4$ [middle, (b), (e), (h) and (k)], $2\times {10}^4$ [right, (c), (f), (i) and (l)]. The no-field case is also shown with gray points as a reference. Growth rates obtained with a dynamical spinodal calculation in [48] are plotted with blue lines. The gas (L, blue) and the cluster (H, red) neutron [third row, (g), (h), (i)] and proton [fourth row, (j), (k), (l)] densities inside the WS cell are also plotted as a function of the density without (light colors) and with (dark colors) AMM.

Figure 5

The pasta phases at the largest-density disconnected region: radii of the WS cell (red) and cluster (green), proton density of high-density phase ${\rho }_p^H$ (blue) and low-density phase ${\rho }_p^L$ (light blue), neutron density of the high-density phase ${\rho }_n^H$ (purple), and low-density phase ${\rho }_n^L$ (orange) as a function of the density, for the $\mathrm{NL}3\omega \rho$ parametrization, with (bottom) and without (top) AMM. The insets show that the low and high proton and neutron densities are similar but different.

Figure 6

The evolution of the pasta shapes as a function of the baryonic density for $\beta$-equilibrium matter and for different magnetic field strengths: $B^{*}=0$ (gray), $5\times {10}^3$ (red), ${10}^4$ (green), $2\times {10}^4$ (blue), for the $\mathrm{NL}3\omega \rho$ model with (bottom) and without (top) AMM.

Figure 7

The difference between the finite-$B$ binding energy per nucleon of the pasta phases ($E_p$) and the zero-$B$ binding energy per nucleon of homogeneous matter ($E_h$), $\mathrm{\Delta }E=E_p(B)-E_h(B=0)$, as a function of the baryonic density for $\beta$-equilibrium matter using the NL3 (top) and $\mathrm{NL}3\omega \rho$ (bottom) parametrizations, considering different magnetic field strengths: $5\times {10}^3$ (red), ${10}^4$ (green), and $2\times {10}^4$ (blue). The results consider calculations with (dark colors) and without (light colors) AMM.

Figure 8

The proton fraction as a function of the baryonic density for $\beta$-equilibrium matter using the NL3 (top) and $\mathrm{NL}3\omega \rho$ (bottom) parametrizations, considering different magnetic field strengths: $B^{*}=0$ (gray), $5\times {10}^3$ (red), ${10}^4$ (green), and $2\times {10}^4$ (blue). The results consider calculations with (dark colors) and without (light colors) AMM.