Role of nuclear correlations and kinematic effects on neutrino emission from the modified Urca processes

The neutrino emissivity from the proton branch of the modified Urca process is calculated. Angular integrations in momentum space are performed numerically without recourse to widely used, but unjustified in this specific case, approximations. To deal with nuclear correlation effects, the independent-pair approximation for nucleon quasiparticles is assumed. The realistic nuclear correlation effects are extracted from a modified extended version of the lowest-order constrained variational method in asymmetric nuclear matter in $\beta$ equilibrium with electrons and muons that allows for three-body nucleon interactions and fits all semiempirical saturation parameters of nuclear matter quite well. Two-body nucleon interaction is modeled by a realistic Argonne AV18 potential and the three-body potential used is a phenomenological Urbana UIX, generalizing parametrization of its component strengths in a manner suitable for application in asymmetric nuclear matter. Limiting to the two-body forces only, we find that, at fixed temperature, neutrino emissivity is a (weakly) decreasing function of density. This is also the case for the neutron branch of the modified Urca process analyzed by us in a recent paper. Inclusion of the modified three-body force effects changes this behavior. Quantitatively the value of the emissivity at each density increases and qualitatively the overall effect of the modified three-body force addition is such that dependence on density becomes weak in a density range from $n_0$ to $3.5n_0$. An updated version of neutron branch results obtained using our modified three-body force model is also presented. This is done using exact treatment of the angular integrations of nuclear transition rate in momentum space.

Figure 1

Proton fractions predicted by the LOCV method using 2BF potential only (solid line), 2BF+$3{\mathrm{BF}}^{\left(\mathrm{M}\right)}$ (dotted line), and 2BF+$3{\mathrm{BF}}^{\left(\mathrm{I}\right)}$ (dashed line) results obtained in paper I.

Figure 2

Different components of $n$-branch Murca process ${\mathcal{R}}_n$ factor calculated with LOCV density-dependent correlation functions in $\beta$-stable matter with two-body interaction considered only, obtained using the triangle approximation procedure and the exact numerical one. Dashed lines present the central component and the solid lines are the central plus tensor part.

Figure 3

Different components of the $n$-branch Murca process ${\mathcal{R}}_n$ factor calculated with LOCV density-dependent correlation functions in $\beta$-stable matter with and without three-body forces. Solid lines are the results of the central plus tensor part. Dashed lines represent the central component.

Figure 4

The same as Fig. 3 but for the Murca-$p$ process ${\mathcal{R}}_p$ factor. The cut of the 2BF curve at 0.1 ${\mathrm{fm}}^{-3}$ is due to Murca-$p$ being forbidden at lower densities.

Figure 5

Electron neutrino emissivity of $\beta$-stable matter at $T=3\times {10}^8\mathrm{K}$, obtained using 2BF only (dashed lines) and two-body plus $3{\mathrm{BF}}^{\left(\mathrm{M}\right)}$ (solid lines) for the angle-dependence considered $n$- and $p$-branch Murca processes. For the explanation of the low-density behavior of the $p$-branch emissivity with 2BF interaction only, see the caption to Fig. 4.

Figure 6

Density dependence of the electron plus muon neutrino emissivity from the Murca-$n$ and Murca-$p$ branches (LOCV, 2BF+$3{\mathrm{BF}}^{\left(\mathrm{M}\right)}$) compared with results obtained using the OPE approximation in Ref. [2] (label Yak2001). The range of density is the same as in Fig. 12 of Ref. [2].

Figure 7

Angles $\theta$ in Eq. (A3).