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Astron. Astrophys. 343, 650-660 (1999)

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3. Efficiency of Cooper-pair neutrinos

Eqs. (22), (29) and (34) enable one to calculate the neutrino emissivity Q due to Cooper pairing of nucleons or hyperons. For illustration, we use a moderately stiff equation of state in a neutron star core proposed by Prakash et al. (1988) (the version with the compression modulus [FORMULA] MeV, and with the same simplified symmetry factor [FORMULA] that has been used by Page & Applegate, 1992). According to this equation of state, dense matter consists of neutrons with admixture of protons and electrons (no hyperons). We set the effective nucleon masses [FORMULA], for simplicity. Since the critical temperatures [FORMULA] and [FORMULA] of the neutron and proton superfluids are model dependent (e.g., Tamagaki 1970, Amundsen & Ostgaard 1985, Baldo et al. 1992, Takatsuka & Tamagaki 1993, 1970), we do not use any specific microscopic superfluid model but treat [FORMULA] and [FORMULA] as free parameters.

Fig. 1 shows temperature dependence of the emissivity Q produced by Cooper pairing of neutrons in the neutron star core at [FORMULA] g cm-3. The critical temperature is assumed to be [FORMULA] K. Given [FORMULA] is typical for the transition from the singlet-state to the triplet-state pairing (Sect. 1). Thus various superfluid types are possible according to different microscopic theories. We present the curves for all three superfluid types (A), (B) and (C) discussed in Sect. 2.2. A growth of the emissivity with decreasing T below [FORMULA] is very steep. The main neutrino emission occurs in the temperature range [FORMULA]. In this range, the emissivity depends weakly on the superfluid type and is rather high, of the order of or even higher than in the modified Urca reaction (1) in non-superfluid matter. This indicates that Cooper-pair neutrinos are important for neutron star cooling (Sect. 4).

[FIGURE] Fig. 1. Temperature dependence of the neutrino emissivity due to Cooper pairing of neutrons at [FORMULA] g cm-3 and [FORMULA] K for superfluidity (A) (solid line), (B) (dashed line) and (C) (dots).

Fig. 2 shows temperature dependence of various neutrino energy generation rates in a neutron star core at [FORMULA]. The neutron critical temperature is assumed to be [FORMULA] K, while the proton critical temperature is [FORMULA] K. We show the contributions from the modified Urca reaction (1) (sum of the proton and neutron branches), nucleon-nucleon bremsstrahlung (2) (sum of nn, np and pp contributions), Cooper pairing of nucleons (sum of n and p reactions), and also the total neutrino emissivity (solid line). The direct Urca process (3) is forbidden at given [FORMULA]. Here and in what follows the rates of the reactions (1)-(3) are taken as described in Levenfish & Yakovlev (1996), with proper account of suppression of the reactions by neutron and/or proton superfluidity (Levenfish & Yakovlev 1994a, Yakovlev & Levenfish 1995). The neutron superfluid is assumed to be of type (B) and the proton superfluid of type (A). A large bump of the total emissivity at [FORMULA] K is produced by the Cooper pairing of neutrons. If neutrino emission due to this pairing were absent the total neutrino emissivity at [FORMULA] would be 2-4 orders of magnitude smaller owing to the strong suppression by the nucleon superfluidity. The Cooper pairing can easily turn suppression into enhancement.

[FIGURE] Fig. 2. Temperature dependence of the neutrino emissivities in different reactions at [FORMULA] for neutron superfluid (B) with [FORMULA] K, and proton superfluid (A) with [FORMULA] K

Fig. 3 shows a joint effect of the neutron and proton superfluids onto the total neutrino emissivity Q at [FORMULA] and [FORMULA] K. Presentation of Q as a function of [FORMULA] and [FORMULA] allows us to display all the effects of superfluidity onto the neutrino emission. If [FORMULA] and [FORMULA] we have Q in nonsuperfluid matter (a plateau at small [FORMULA] and [FORMULA]). For other T, the neutron and/or proton superfluidity affects the neutrino emission. It is seen that the neutron pairing at [FORMULA] may enhance the neutrino energy losses. A similar effect of the proton pairing at [FORMULA] is much weaker as explained above. In a strongly superfluid matter (highest [FORMULA] and [FORMULA]) the neutrino emission is drastically suppressed by the superfluidity.

[FIGURE] Fig. 3. Total neutrino emissivity versus [FORMULA] and [FORMULA] in a neutron star core at [FORMULA] and [FORMULA] K.

Fig. 4 shows temperature dependence of some partial and total neutrino emissivities for the same [FORMULA] and [FORMULA] as in Fig. 2 but for higher density [FORMULA]. The equation of state we adopt opens the powerful direct Urca process at [FORMULA] g cm[FORMULA]. We take into account all major neutrino generation reactions (1)-(4). For simplicity, we do not show all partial emissivities since the total emissivity is mainly determined by the interplay between the direct Urca and Cooper pairing reactions.

[FIGURE] Fig. 4. Temperature dependence of the neutrino emissivities in main reactions at [FORMULA] for neutron superfluid (B) with [FORMULA] K, and proton superfluid (A) with [FORMULA] K

The effect of Cooper-pair neutrinos is seen to be less important than for the standard neutrino reactions in Fig. 2 but, nevertheless quite noticeable. It is especially pronounced if [FORMULA]. In this case, the strong proton superfluid suppresses greatly the direct Urca process, and the emission due to Cooper pairing of neutrons can be significant.

Finally notice that while calculating neutron star cooling one often assumes the existence of one dominant neutrino generation mechanism, for instance, the direct Urca process for the enhanced cooling or the modified Urca process for the standard cooling. This is certainly true for non-superfluid neutron-star cores, but wrong in superfluid matter. In the latter case, different mechanisms can dominate (Fig. 5) at various cooling stages depending on temperature, [FORMULA], [FORMULA], and density.

[FIGURE] Fig. 5a and b. Domains of [FORMULA] and [FORMULA], where different neutrino emission mechanisms dominate (different shaded regions). In addition to the mechanisms (1)-(4) we include also the neutrino emission due to ee bremsstrahlung (Kaminker et al. 1997). a  corresponds to the standard neutrino emission at [FORMULA] and [FORMULA] K, while b refers to the emission enhanced by the direct Urca process at [FORMULA] and [FORMULA] K. Dashed lines correspond to [FORMULA] or [FORMULA].

In particular, the Cooper-pair neutrinos dominate the standard neutrino energy losses at [FORMULA] K for a moderate neutron superfluidity ([FORMULA]). This parameter range is important for cooling theories. At the early cooling stages, when [FORMULA] K, the Cooper-pair neutrinos can also be important but in a narrower range around [FORMULA] especially in the presence of the proton superfluid. As mentioned above, Cooper-pair neutrinos can dominate also in the regime of rapid neutrino emission if the nucleons of one species are strongly superfluid but the other nucleons are moderately or weakly superfluid (see Fig. 5b).

Figs. 5a and b display the domains of [FORMULA] and [FORMULA], where different neutrino mechanisms are dominant. Fig. 5a refers to the standard cooling at [FORMULA] and [FORMULA] K. Fig. 5b corresponds to the rapid cooling at [FORMULA] and [FORMULA] K. Dashes show the lines of [FORMULA] and [FORMULA], which separate [FORMULA]-[FORMULA] planes into four regions. In the left down corners enclosed by these lines, matter is nonsuperfluid. The right down corners correspond to superfluidity of n alone; the left upper corners to superfluidity of p alone, and the right upper corners to joint n and p superfluidity. One can observe a variety of dominant cooling mechanisms regulated by the superfluidity. If both n and p superfluids are very strong, they switch off all the neutrino emission mechanisms involving nucleons (and discussed here). In this regime, a slow neutrino emission due to ee-bremsstrahlung (Kaminker et al. 1997) survives and dominates. This mechanism is rather insensitive to the superfluid state of dense matter.

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© European Southern Observatory (ESO) 1999

Online publication: March 1, 1999
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