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

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 and and on the type of superfluidities.

Case AA

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

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

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

Here,

with and .

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 , see Eq. (40). There are three domains in the -plane, in which the asymptotes of the reduction factor in the limit of strong superfluidity are different.

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

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

where .

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

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

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 . For deriving in the limit of strong superfluidity we will use the asymptote of the function H defined by Eq. (55). If , the main contribution into the integral (54) comes from the region, in which . In this case the asymptote is

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

© European Southern Observatory (ESO) 2000

Online publication: June 5, 2000
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