          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 