Neutrino mean free path for proto–neutron star matter within the Skyrme model

The neutrino mean free path is evaluated in hot proto–neutron star matter under a strong magnetic field. We consider densities in the range $0.04\le \rho \le 0.4{\mathrm{fm}}^{-3}$, several proton fractions from symmetric matter up to pure neutron matter, temperatures up to 30 MeV, and two magnetic field strengths $B={10}^{17}$ and ${10}^{18}$ G. Polarized proto–neutron star matter is described within the nonrelativistic Hartree-Fock model using the LNS Skyrme interaction, where the proper treatment for instabilities for low densities and temperatures is implemented. Under the same conditions, the degree of polarization of protons is stronger than the one for neutrons. For the neutrino mean free path we consider three reactions: the neutrino-neutron and neutrino-proton scattering, $\nu +n\to {\nu }^{\prime }+n^{\prime }$ and $\nu +p\to {\nu }^{\prime }+p^{\prime }$, respectively, and the neutrino absorption reaction $\nu +n\to e^{-}+p$. The magnetic field induces an asymmetry in the mean free path which favors the flux of neutrinos parallel to the magnetic field in the case of neutron scattering and the absorption reaction, whereas it is antiparallel in the case of proton scattering. For most of the conditions the absorption reaction is the dominant one. The dependence of the neutrino mean free path on the magnetic field, the temperature, and the proton fraction is different for each reaction. As a representative case of our results, the asymmetry in the mean free path is $\approx 21\%$ at saturation density for $B={10}^{18}$ G, $T=15$ MeV, and symmetric matter, while we have $\approx -1\%$ for $B={10}^{17}$ G and the same values of all the other conditions.

Figure 1

The spin asymmetry with and without the instability construction for both protons and neutrons. The magnetic field intensity is $B={10}^{18}$ G, $T=5$ MeV, and we consider two values of the isospin asymmetry $\omega$. [(a), (b)] The parameter $\alpha$ [see Eq. (21)]. [(c), (d)] The continuous line is the final result for $W_n$ and $W_p$, while the dash-dotted line is the result without the instability construction. In these same panels we have plotted $W_i^a$ and $W_i^b$ by dashed lines. The final value for $W_i$ is obtained with $\alpha$ and these quantities using Eq. (27). Finally, in (e) and (f), we have $A_n$ and $A_p$, where the continuous (dash-dotted) lines include (do not include) the instability construction. In all panels a vertical dotted line indicates the upper limit of the instability. The lower limit is very close to the origin and it is not shown. Units for $W_i$ are ${\mathrm{fm}}^{-3}$.

Figure 2

Density dependence of the proton and neutron spin asymmetries for $T=5$ MeV, as defined in Eq. (15). The magnetic field is (a) $B={10}^{17}$ and (b) $B={10}^{18}$. Both panels give results for different values of the isospin asymmetry $\omega$. In this figure the Gibbs or Maxwell construction has been already implemented.

Figure 3

Proton and neutron spin asymmetries for different values of the isospin asymmetry.

Figure 4

Density dependence of the proton and neutron spin asymmetries for different temperatures and for two isospin asymmetries. Only for $T=5$ MeV the instability construction shows a sizable effect.

Figure 5

The proton scattering NMFP as a function of the density and for three different values for the neutrino incoming angle, ${\theta }_{\nu }$. Results for a magnetic field intensity (a) $B={10}^{17}$ G and (b) $B={10}^{18}$ G. For both panels, the momentum of the incoming neutrino is $|{\vec{p}}_{\nu }|=3T, T=15$ MeV, and $\omega =0.5$.

Figure 6

The proton scattering NMFP as a function of the density and for three different values for the temperature for ${\theta }_{\nu }=\pi /2$ and $\omega =0.5$. As in Fig. 3, results for a magnetic field intensity (a) $B={10}^{17}$ and (b) $B={10}^{18}$ G, respectively, using the same approximation for the momentum of the incoming neutrino.

Figure 7

Effect of the instability over the NMFP. In all curves the dotted lines represent the result without the instability construction. We have employed $B={10}^{18}$ G, $T=5$ MeV, $\omega =0$, and three values for the incoming neutrino angle ${\theta }_{\nu }$. The momentum model for the incoming neutrino is the same as in Fig. 5.

Figure 8

The NMFP for different values of the isospin asymmetry $\omega$. For $\omega =0$ the curve is divided into a solid line and a dotted line, which represent the result with and without the instability construct, respectively. In this figure we have used $B={10}^{18}$ G and $T=15$ MeV. For the incoming neutrino, we have employed ${\theta }_{\nu }=\pi /2$ and the same model for the momentum as in Fig. 5.

Figure 9

The total NMFP for different values of the isospin asymmetry $\omega$ and for three different angles of the incident neutrino. In this figure we have used $B={10}^{17}$ G and $T=15$ MeV. We have employed the same model for the momentum as in Fig. 5.

Figure 10

The total NMFP for different values of the isospin asymmetry $\omega$ and for three different angles of the incident neutrino. In this figure we have used $B={10}^{18}$ G and $T=15$ MeV. We have employed the same model for the momentum as in Fig. 5.