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11.9 Neutrinos from the crust – summary in the T − ρ plane

In this section we will summarize results for neutrino emission from a neutron star crust. We will limit ourselves to densities ρ ≳ 107 g cm −3, so that electrons will always be ultra-relativistic (see Section 3).

Contributions to Q ν from various neutrino emission mechanisms (except for the Cooper-pair mechanism, which will be considered later in this section) versus ρ are plotted in Figures 61View Image, 62View Image and 63View Image.

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Figure 61: Neutrino emissivities associated with different mechanisms of neutrino emission acting in a neutron star crust, versus ρ, at temperature T = 3 × 109 K. Numbers in parentheses indicate log10B. Effect of B = 1014 G (and a fortiori – effect of a lower B) on Qpaνir is insignificant. Qplνas, QBνrem, and Qphνot were calculated at B = 0. Label “syn (14)” – synchrotron radiation by electrons in constant magnetic field B = 1014 G, etc. Ground-state composition of the crust is assumed: Haensel & Pichon [183] model for the outer crust, and Negele & Vautherin [303] model for the inner crust. For further explanations see the text. From [428Jump To The Next Citation Point].

We start with the crust of a very young neutron star, with a temperature 9 T = 3 × 10 K (age ≲ 1 year), Figure 61View Image. For density ρ ≲ 108 g cm −3, the contribution Qpaνir is dominant. However, with increasing density, electrons become degenerate, and positrons disappear in the matter, so that Qpaνir is strongly suppressed at ρ > 109 g cm −3. We also notice that Qphot ν is never important in the inner crust, because of the strong electron degeneracy. plas Q ν from the plasmon decay gives the dominant contribution to Q ν from 9 −3 ρ ≈ 10 g cm down to the bottom of the crust. We notice also that syn Qν behaves differently than the other contributions. Namely, at ρ ≳ 109 g cm− 3 its density dependence is very weak, and Qsyνn scales approximately with the magnetic field B as ∝ B2. Finally, one notices jumps of QBrem ν and Qplas ν, which result from jumps in Z and A in the ground-state matter. As we will see, this feature is even more pronounced at lower temperatures T.

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Figure 62: Same as for Figure 61View Image, but at T = 109 K. pair Q ν is increased by B, as shown by the labels (14), (13), (12). Notice, that pair Q ν at B = 0 is too low to be seen. From [428Jump To The Next Citation Point].

Let us now consider the case of a colder crust at 9 T = 10 K, Figure 62View Image. Except for syn Q ν, which is just scaled down due to the decrease of temperature, there is a dramatic change in the overall landscape. For a magnetic field B = 1014 G, Qsyνn dominates in the lowest-density region. On the contrary, Qpair ν is of marginal importance, and is influenced by BBB (increases with B). Moreover, contribution of phot Qν is negligible. Neglecting the effect of magnetic fields, one concludes that plas Q ν dominates in the outer crust, while Brem Q ν dominates in the inner crust. Let us notice that Qplνas reaches its maximum near 1010.5 g cm −3 and then decreases by four orders of magnitude when the density falls below ρ ∼ 1013 g cm− 3; this characteristic behavior is due to the exp (− T ∕T ) pe factor, Equation (265View Equation). On the contrary, QBrem ν rises steadily with increasing density.

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Figure 63: Same as in Figure 61View Image, but calculated at T = 3 × 108 K. pair Q ν and phot Qν are too small to be seen. From [428Jump To The Next Citation Point].
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Figure 64: Neutrino emissivity from the Cooper pair formation mechanism, calculated for strong (uniform) neutron superfluidity with the maximum critical temperature max 10 Tcn = 1.6 × 10 K (a model from [231Jump To The Next Citation Point]). For comparison, Qν from two most efficient other mechanisms, plasmon decay and electron Bremsstrahlung, are also plotted. The smooth composition ground-state crust model of Kaminker et al. [231] is used; it predicts a specific ρn (ρ) in the inner crust. This crust model is described in detail in Appendix B.2 of Haensel et al. [184Jump To The Next Citation Point]). From [428Jump To The Next Citation Point].

Finally, in Figure 63View Image we consider an even colder crust at 8 T = 3 × 10 K. Pair and photoneutrino constributions have disappeared completely, while Qplνas dominates for ρ < 109 g cm− 3, whereas at higher densities, QBrνem is the main source of neutrino emission. At the magnetic field B = 1014 G, characteristic of magnetars, synchrotron radiation dominates in the density interval near 10 −3 ∼ 10 g cm, but then at 11 − 3 ρ ≳ 10 g cm, Brem Q ν becomes the strongest neutrino radiation mechanism.

Two general remarks are in order. First, as we have already mentioned, jumps in QBrνem and Qplνas are due to specific factors involving Z2 and A and reflect the jumps in Z and A in the layered crust. For the other mechanisms, the electron chemical potential μ e with its smooth dependence on ρ plays the role of the crucial plasma parameter, and therefore no jumps are seen. Secondly, were the magnetic field 15 B ≥ 10 G, syn Q ν would be overall dominant for 9 T < 10 K and ρ > 109 g cm −3.

The Cooper-pair mechanism of neutrino radiation differs fundamentally from the other mechanisms of neutrino cooling, discussed above, and therefore we consider it separately. QCP ν depends sensitively on the interplay between temperature T and the 1 S0 pairing gap ΔF of the dripped neutrons. The gap itself depends on T, rising from zero at T = Tcn to the asymptotic value Δ0 ≡ ΔF (T = 0) for T ≪ Tcn (see Section 8.2.2). As we already discussed in Section 8.2.1, the dependence of Δ0 on the free neutron density, ρn, is very poorly understood, and this introduces a large uncertainty in the calculated values of QCP ν. Notice that the relation ρ (ρ) n, needed to get QCP (ρ) ν, depends on the model of the inner crust.

Figure 64View Image refers to 9 T = 10 K, 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 Δ0(ρ), denoted by Δma0x, corresponds to the maximum of Tcn(ρ), given by T max = 0.5669Δmax ∕kB cn 0 (Section 8.2.2). In the case presented in Figure 64View Image, T max = 1.6 × 1010 K cn is significantly larger than T. The Cooper-pair mechanism is efficient only within a narrow range of temperature below Tcn, namely for 0.7 ≲ T∕Tcn < 1, and is strongly damped outside this region. In view of this, QCPν usually has two maxima, around ρ1 and ρ2, which are the two solutions of Tcn(ρ) = T (remembering the bell shape of the pairing gap as a function of density). Only the lower-density maximum can be seen in Figure 64View Image. Because Tmax ≫ T cn, 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 T = 109 K, QCνP 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 QCP ν by about two orders of magnitude [368245].


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