Exact solution of equations for proton localization in neutron star matter

The rigorous treatment of proton localization phenomenon in asymmetric nuclear matter is presented. The solution of proton wave function and neutron background distribution is found by the use of the extended Thomas-Fermi approach. The minimum of energy is obtained in the Wigner-Seitz approximation of a spherically symmetric cell. The analysis of four different nuclear models suggests that the proton localization is likely to take place in the interior of a neutron star.

Figure 1

The proton fraction for different models used in the calculation: APR, SLy4, AV14+UVII and UV14+TNI. The squares indicate the proton localization threshold and the full dots indicate the central density of a star with maximum mass.

Figure 2

The subsequent steps in the relaxation method for proton wave function $\psi \left(r\right)$ (upper panel) and neutron density $n_n\left(r\right)$ (lower panel) in the nuclear model AV14+UVII. The solid curve represents the final results after 9 steps when the accuracy equal to ${10}^{-4}$ was achieved for ground energy $E_p$.

Figure 3

The evolution of proton wave function (solid) and neutron background (dashed) with increasing mean baryon density $n$ for the two nuclear models. Vertical lines indicate the position of $R_{WS}$. In the case of UV14+TNI, the distribution is truncated because for densities above $0.7{\mathrm{fm}}^{-3}$ the $R_{WS}$ was greater than $3{\mathrm{fm}}^{-3}$.