Statistical analysis of the effect of the symmetry energy on the crust-core transition density and pressure in neutron stars

There are generally two factors on which the form of the symmetry energy depends. The first one is the model involved in determining the nuclear matter equation of state, and the second the accuracy of the approximation of the symmetry energy function given by a Taylor series. Including the fourth-order term in the Taylor series accounts for a more reasonable representation of the symmetry energy. This paper focuses on understanding the symmetry energy influence on the neutron star crust-core phase boundary characteristics in terms of the above factors. All calculations were based on selected models of the relativistic mean field theory. The analysis begins with determining the analytical form of the fourth-order symmetry energy. The applied method allows for deriving the potential part of the fourth-order symmetry energy when the model contains various nonlinear couplings between mesons. The presence of the fourth-order term in the symmetry energy description affects the neutron star crust-core phase boundary characteristics by changing the transition density $n_t$, the corresponding pressure $P_t$, and the equilibrium proton fraction $Y_p^{\mathrm{eq}}\left(n_t\right)$ value. The performed statistical analysis clarifies the role of the second- and fourth-order symmetry energy in determining the relationships between the transition density $n_t$ and the leading coefficients characterizing the density dependence of the symmetry energy $[E_{\mathrm{sym}}\left(n_0\right),$ $L_{\mathrm{sym}},$ $K_{\mathrm{sym}}]$, where $L_{\mathrm{sym}}$ is the slope and $K_{\mathrm{sym}}$ is the curvature. It is shown that the regression analysis makes it possible to identify $(P_t,L_{\mathrm{sym}},K_{\mathrm{sym}})$ as the group of factors that significantly influence the variability of the transition density $n_t$. The results also allow for estimating the effect of the fourth-order symmetry energy term inclusion. Additionally, the correlation analysis points to the individual role of $L_{\mathrm{sym}}$ and $K_{\mathrm{sym}}$. Only when the fourth-order term is included are both variables $K_{\mathrm{sym}}$ and $L_{\mathrm{sym}}$ equally anticorrelated with $n_t$, leading to the increasing role of $L_{\mathrm{sym}}$ in analyzing the variability of $n_t$.

Figure 1

This figure illustrates, for the chosen density, the influence of the fourth-order symmetry energy term on the solution of the equation defining the equilibrium concentration of protons.

Figure 2

The dependence on the density of the symmetry energy $E_{\mathrm{sym},2}$, obtained in the case of the parabolic approximation for three distinguished groups of models.

Figure 3

Density dependence of the fourth-order symmetry energy potential part. The results obtained for the three distinguished groups of models point to the third group as the one having the most significant contribution of the potential part.

Figure 4

The factor $x$ as a function of density for group I (left panel) and II (right panel); it vanishes for group III.

Figure 5

The dependence on the density of the symmetry energy $E_{\mathrm{sym},4}$ for three distinguished groups of models.

Figure 6

The equilibrium proton fraction $Y_p^{\mathrm{eq}}$ dependence on the density for three distinguished groups of models. The lowest equilibrium concentration of protons characterizes the third group, with the highest value of fourth-order symmetry energy potential contribution.

Figure 7

The dependence $L_{\mathrm{sym}}$ on $E_{\mathrm{sym}}\left(n_0\right)$. In the left panel, dashed lines represent the regression lines obtained for individual groups, whereas the black straight line corresponds to the regression analysis results performed for the entire sample of models. The right panel shows analogous results after considering fourth-order terms in the Taylor expansion.

Figure 8

The dependence $K_{\mathrm{sym}}$ on $L_{\mathrm{sym}}$. In the left panel, dashed lines represent the regression lines obtained for individual groups. The right panel shows analogous results after considering fourth-order terms in the Taylor expansion. Due to the lack of correlation when the whole sample of models is considered, the regression line, in this case, is not marked.

Figure 9

The dependence of the transition density $n_t$ on the corresponding transition pressure $P_t$. In the left panel, dashed lines represent the regression lines obtained for individual groups. The right panel shows analogous results with fourth-order terms in the Taylor expansion.