Core-crust transition in neutron stars with finite-range interactions: The dynamical method

The properties of the core-crust transition in neutron stars are investigated using effective nuclear forces of finite range. Special attention is paid to the so-called dynamical method for locating the transition point, which, apart from the stability of the uniform nuclear matter against clusterization, also considers contributions due to finite-size effects. In particular, contributions to the transition density and pressure from the direct and exchange energies are carefully analyzed. To this end, finite-range forces of Gogny, momentum-dependent interaction, and simple effective interaction types are used in the numerical applications. The results from the dynamical approach are compared with those from the popular thermodynamical method that neglects the surface and Coulomb effects in the stability condition. The dependence of the core-crust transition on the stiffness of the symmetry energy of the finite-range models is also addressed. Finally, we analyze the impact of the transition point on the mass, thickness, and fraction of the moment of inertia of the neutron star crust. Prominent differences in these crustal properties of the star are found between using the transition point obtained with the dynamical method or the thermodynamical method. It is concluded that the core-crust transition needs to be ascertained as precisely as possible in order to have realistic estimates of the observed phenomena where the crust plays a significant role.

Figure 1

Dynamical potential as a function of the density for the D1S, D1M, $\mathrm{D}1{\mathrm{M}}^{*}$, and D1N Gogny interactions.

Figure 2

Momentum dependence of the dynamical potential at two different densities, (a) $\rho =0.10{\mathrm{fm}}^{-3}$ and (b) $\rho =0.08{\mathrm{fm}}^{-3}$, for the D1S, D1M, and $\mathrm{D}1{\mathrm{M}}^{*}$ Gogny interactions. The results plotted with solid lines are obtained using the full expression of the dynamical potential, given in Eq. (26), while the dashed lines are the results of its $k^2$ approximation, given in Eq. (29).

Figure 3

Coefficient $\beta \left(\rho \right)$ of Eq. (29) as a function of the density for a set of Gogny interactions. The dashed line includes only contributions coming from the direct energy, the dash-dotted lines include contributions coming from the direct energy and from the from the gradient effects of the exchange energy. Finally, the solid lines include all contributions to $\beta \left(\rho \right)$.

Figure 4

(a) Transition density and (b) transition pressure as a function of the slope of the symmetry energy calculated using the thermodynamical and the dynamical methods for a set of different Gogny interactions.

Figure 5

Same as Fig. 4 but for a family of MDI interactions.

Figure 6

Same as Fig. 4 but for a family of SEI interactions of $\gamma =1/2$ ($K_0=237.5\mathrm{MeV}$).

Figure 7

Transition density as a function of the slope of the symmetry energy calculated using the dynamical method for a set of Skyrme, Gogny, MDI, and SEI interactions.

Figure 8

Transition density as a function of the slope of the symmetry energy calculated using the dynamical method for a set of Skyrme, Gogny, MDI, and SEI interactions.

Figure 9

(a) Transition density and (b) transition pressure as a function of the slope of the symmetry energy, calculated using the dynamical method for three SEI families with different nuclear matter incompressibilities.

Figure 10

Neutron star (a) crustal thickness, (b) crustal mass, and (c) crustal fraction of the moment of inertia against the total mass of the neutron star for some Gogny (left), MDI (center), and SEI (right) interactions. The core-crust transition has been determined using the thermodynamical potential (dashed lines) and the dynamical potential using all direct and ${\hslash }^2$ contributions (solid lines).