Contributions to from various neutrino emission mechanisms (except for the Cooper-pair mechanism, which will be considered later in this section) versus are plotted in Figures 61, 62 and 63.
We start with the crust of a very young neutron star, with a temperature (age 1 year), Figure 61. For density , the contribution is dominant. However, with increasing density, electrons become degenerate, and positrons disappear in the matter, so that is strongly suppressed at . We also notice that is never important in the inner crust, because of the strong electron degeneracy. from the plasmon decay gives the dominant contribution to from down to the bottom of the crust. We notice also that behaves differently than the other contributions. Namely, at its density dependence is very weak, and scales approximately with the magnetic field as . Finally, one notices jumps of and , which result from jumps in and in the ground-state matter. As we will see, this feature is even more pronounced at lower temperatures .
Let us now consider the case of a colder crust at , Figure 62. Except for , which is just scaled down due to the decrease of temperature, there is a dramatic change in the overall landscape. For a magnetic field , dominates in the lowest-density region. On the contrary, is of marginal importance, and is influenced by (increases with ). Moreover, contribution of is negligible. Neglecting the effect of magnetic fields, one concludes that dominates in the outer crust, while dominates in the inner crust. Let us notice that reaches its maximum near and then decreases by four orders of magnitude when the density falls below ; this characteristic behavior is due to the factor, Equation (265). On the contrary, rises steadily with increasing density.
Finally, in Figure 63 we consider an even colder crust at . Pair and photoneutrino constributions have disappeared completely, while dominates for , whereas at higher densities, is the main source of neutrino emission. At the magnetic field , characteristic of magnetars, synchrotron radiation dominates in the density interval near , but then at , becomes the strongest neutrino radiation mechanism.
Two general remarks are in order. First, as we have already mentioned, jumps in and are due to specific factors involving and and reflect the jumps in and in the layered crust. For the other mechanisms, the electron chemical potential with its smooth dependence on plays the role of the crucial plasma parameter, and therefore no jumps are seen. Secondly, were the magnetic field , would be overall dominant for and .
The Cooper-pair mechanism of neutrino radiation differs fundamentally from the other mechanisms of neutrino cooling, discussed above, and therefore we consider it separately. depends sensitively on the interplay between temperature and the pairing gap of the dripped neutrons. The gap itself depends on , rising from zero at to the asymptotic value for (see Section 8.2.2). As we already discussed in Section 8.2.1, the dependence of on the free neutron density, , is very poorly understood, and this introduces a large uncertainty in the calculated values of . Notice that the relation , needed to get , depends on the model of the inner crust.
Figure 64 refers to , a selected model of neutron superfluidity, and a selected model of the inner crust. In the BCS theory (Section 8.2.1), the maximum of , denoted by , corresponds to the maximum of , given by (Section 8.2.2). In the case presented in Figure 64, is significantly larger than . The Cooper-pair mechanism is efficient only within a narrow range of temperature below , namely for , and is strongly damped outside this region. In view of this, usually has two maxima, around and , which are the two solutions of (remembering the bell shape of the pairing gap as a function of density). Only the lower-density maximum can be seen in Figure 64. Because , the localization of the peaks (at 1012 g cm–3 and at 1014 g cm–3) does not change much with decreasing temperature. However, the heights of the peaks decrease very fast. For the selected superfluidity model, and at , in the peak region dominates over all other neutrino emission mechanisms. Let us notice that a proper inclusion of the in-medium modification of the weak interactions could significantly decrease the maximum value of by about two orders of magnitude [368, 245].
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