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Astron. Astrophys. 357, 1157-1169 (2000)

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Appendix

Consider the case in which both nucleon species are superfluid at once. The asymptotic behaviour of the reduction factor R given by Eq. (54) in the limit of strong superfluidity depends on the gap amplitudes [FORMULA] and [FORMULA] and on the type of superfluidities.

Case AA

Let both nucleon superfluidities be of type A. If the superfluidities are strong ([FORMULA]) and [FORMULA] we obtain:

[EQUATION]

where [FORMULA]. In the limiting case [FORMULA] Eq. (A2) yields [FORMULA] and reproduces the asymptote given by Eq. (48). In the opposite limit [FORMULA] one has [FORMULA]. In the intermediate region [FORMULA] the asymptote (A2) becomes invalid. One can show that [FORMULA] for [FORMULA].

We have calculated [FORMULA] in a wide range of arguments [FORMULA] and [FORMULA] and proposed the fit expression valid for [FORMULA]:

[EQUATION]

Here,

[EQUATION]

with [FORMULA] and [FORMULA].

Case AB

Let nucleons of species 1 suffer pairing of type A and nucleons of species 2 suffer pairing of type B. In this case [FORMULA], see Eq. (40). There are three domains in the [FORMULA]-plane, in which the asymptotes of the reduction factor [FORMULA] in the limit of strong superfluidity are different.

The first domain corresponds to [FORMULA], i.e., to [FORMULA] for all [FORMULA]. If the both superfluidities are strong ([FORMULA] and [FORMULA]) and [FORMULA], the asymptote of [FORMULA] can be obtained by averaging Eq. (A1) over [FORMULA] after making a formal replacement [FORMULA] and [FORMULA]. Since [FORMULA] the main contribution into the integral comes from the region in which [FORMULA]. This allows us to put [FORMULA] in all smooth functions under the integral. In this way we obtain

[EQUATION]

The second domain corresponds to [FORMULA]. We have [FORMULA] for all [FORMULA] in this domain. Then the asymptote of [FORMULA] is derived by direct averaging over [FORMULA] of the asymptote [FORMULA] given by Eq. (A1), after replacing formally [FORMULA]. We have:

[EQUATION]

where [FORMULA].

The third domain corresponds to [FORMULA]. While averaging over [FORMULA] we can split the integral into two terms. The first term represents integration over the region [FORMULA], in which [FORMULA], while the second term contains integration over the region [FORMULA], in which [FORMULA], with [FORMULA]. The latter term can be taken as an asymptote of the factor [FORMULA], since the main contribution into the integral comes, again, from [FORMULA]. We have

[EQUATION]

Note that the asymptotes given by Eqs. (A5) and (A7) are invalid at [FORMULA]. If [FORMULA], Eq. (A5) reproduces Eq. (48) for [FORMULA]. For [FORMULA], Eq. (A5) transforms into Eq. (A6). Eqs. (A6) and (A7) coincide at [FORMULA]. If [FORMULA], the reduction factor can be estimated as an [FORMULA].

Case AC

Let superfluidity of nucleons 1 be of type A, as before, while superfluidity of nucleons 2 be of type C. In this case [FORMULA]. For deriving [FORMULA] in the limit of strong superfluidity we will use the asymptote of the function H defined by Eq. (55). If [FORMULA], the main contribution into the integral (54) comes from the region, in which [FORMULA]. In this case the asymptote is

[EQUATION]

If [FORMULA], this asymptote reproduces the asymptote (48) for [FORMULA]. For [FORMULA], we have [FORMULA]. Finally, at [FORMULA] the asymptote of [FORMULA] is

[EQUATION]

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© European Southern Observatory (ESO) 2000

Online publication: June 5, 2000
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