3. Bulk viscosity of superfluid matter
3.1. Superfluid gaps
Superfluidity of nucleons in a neutron star core may strongly affect the bulk viscosity. Neutrons are believed to form Cooper pairs due to their interaction in the triplet state, while protons suffer singlet-state pairing (Sect. 1). While studying the triplet-state neutron pairing one should distinguish the cases of different projections of nn-pair moment onto a quantization axis z (see, e.g., Amundsen and Ostgaard 1985): . The actual (energetically most favorable) state of nn-pairs is not known being extremely sensitive to the (still unknown) details of nn interaction. One cannot exclude that this state varies with density and is a superposition of states with different . We will consider the 3P2-state neutron superfluidity either with or with . In these two cases the effect of superfluidity on the bulk viscosity is qualitatively different. Consideration of the superfluidity based on mixed states is much more complicated and goes beyond the scope of the present paper.
Thus we will study three different superfluidity types: 1S0, 3P2 () and 3P2 () denoted as A, B and C, respectively (Table 1). The superfluidity of type A may be attributed to any protons, while superfluidity of types B and C may be attributed to neutrons.
Table 1. Studied type of superfluidity
where and are the Fermi momentum and Fermi velocity of the nucleon, respectively, and µ is the nucleon chemical potential. In the cases of study one has , where is the amplitude which describes temperature dependence of the gap; specifies dependence of the gap on the angle between the particle momentum and the z axis (Table 1). In case A the gap is isotropic, and . In cases B and C the gap depends on . Note that in case C the gap vanishes at the poles of the Fermi sphere at any temperature: .
The gap amplitude is derived from the standard equation of the BCS theory (see, e.g., Yakovlev et al. 1999). The value of determines the critical temperature . The values of for cases A, B and C are given in Table 1.
The dimensionless gap amplitude v depends only on . In case A the quantity v coincides with the isotropic dimensionless gap, while in cases B and C it represents, respectively, the minimum and maximum gap (as a function of ) on the nucleon Fermi surface. The dependence of v on can be fitted as (Levenfish & Yakovlev 1994):
The mean errors of these fits are for all .
3.2. Superfluid reduction factors
Now let us consider the effects of nucleon superfluidity on the bulk viscosity. The dynamics of superfluid is generally much more complicated than the dynamics of ordinary fluids. Even the motion of matter which consists of particles of one species is described by the equations of two-fluid hydrodynamics (normal and superfluid components), and viscous dissipation of the normal component is determined by three coefficients of the second (bulk) viscosity (Landau & Lifshitz 1987). Our main assumption is that stellar pulsations represent fluid motion of the first-sound type (particularly, temperature variations are neglected) in which all constituents of matter move with the same hydrodynamical velocity. In this case the hydrodynamical equations reduce to the equation of one-fluid hydrodynamics with one coefficient of the bulk viscosity ( in the notation of Landau & Lifshitz, 1987).
We will see that superfluidity reduces the bulk viscosity due to the appearance of energy gaps in the nucleon dispersion relation, Eq. (38). Quite generally, the bulk viscosity can be presented in the form
where is a partial bulk viscosity of non-superfluid matter, Eq. (35), and is a factor which describes reduction of the partial bulk viscosity by superfluidity of nucleons 1 and 2 involving into a corresponding direct Urca process. If both nucleons, 1 and 2, belong to non-superfluid component of matter, we have and reproduce the results of Sect. 2.
Thus the problem consists in calculating the reduction factors . Each factor depends generally on two parameters, and , which are dimensionless gap amplitudes of nucleons 1 and 2 (and on the type of superfluidity of these nucleons). Let us study the effect of superfluidity on the partial bulk viscosity. For this purpose let us reconsider derivation of the bulk viscosity (Sect. 2.1). If all constituents of matter have the same macroscopic velocity, the superfluidity affects noticeably only the factor in the expression for the bulk viscosity, Eq. (35). As seen from Eq. (34), the main factor affected by the superfluidity in is the integral , Eq. (30), which describes the asymmetry of the lepton production and capture rates in the direct and inverse reactions of the direct Urca process. At the integrand of this equation is , where is given by Eq. (31). Thus, at small deviations from the equilibrium one can transform Eq. (30) to:
Here, the index "0" refers to the non-superfluid case, in which we have obtained .
where y is given by Eq. (39). In the absence of superfluidity, we have and .
Let the index correspond to a nucleon which can suffer superfluidity of type A while correspond to a nucleon which can suffer any superfluidity, A, B or C. In order to account for superfluidity in Eq. (43) it is sufficient to replace for and 2 [in and in the delta function] and introduce averaging over orientations of (analogous procedure is considered in detail by Levenfish & Yakovlev 1994 for the problem of superfluid reduction of the neutrino emissivity). Then the factor can be written as
with . Here, is the solid angle element in the direction of .
Thus, we have derived explicit Eqs. (45) and (46) for calculating the reduction factor R. Calculation is quite similar (and in fact, simpler) to that done for the factor which describes superfluid reduction of the neutrino emissivity in the direct Urca process (Levenfish & Yakovlev 1994, Yakovlev et al. 1999). The effect of superfluidity on the bulk viscosity has also much in common with the effect on the emissivity. Thus we omit technical details and present only the results and their brief discussion.
3.3. Superfluidity of neutrons or protons
Consider the superfluidity of nucleon of one species, for instance, of species 2. In this case R depends on the only parameter , and we can set in Eqs. (45) and (46). Integration over and in Eq. (46) reduces to well-known integrals of the theory of Fermi liquids and yields:
where . For , one has . If superfluidity is strong (), the direct Urca process is drastically suppressed by large superfluid gap in the nucleon spectrum and reduces the bulk viscosity. The asymptotic expressions of R for can be obtained from Eq. (47):
Note that the factors and are suppressed exponentially with decreasing temperature, whereas varies as . The latter fact is associated with the presence of gap nodes at the Fermi surface (Levenfish & Yakovlev 1994).
In addition, we have calculated the reduction factors R numerically in a wide range of v and propose the expressions which fit the numerical results (with a mean error of ) and reproduce the asymptotes (48)-(50):
Here, , and in the factors , and , respectively. Using Eqs. (41) and (51)-(53), one can easily calculate the reduction factors R for any . These factors are shown in Fig. 3 versus . We see that the reduction can be quite substantial. The strongest reduction is provided by superfluidity A and the weakest by superfluidity C. For instance, at we obtain , and .
3.4. Superfluidity of neutrons and protons
Let both nucleons, 1 and 2, be superfluid at once, and let the superfluidity of nucleon 1 be of type A. In this case R can be calculated from Eqs. (45) and (46). Using the delta function, we remove the integration over and obtain
Notice that as , and as .
where and . It is evident that . We have also derived the asymptote of in the limit of strong superfluidity. Furthermore, we have calculated the factor and derived the fit expression which reproduces the numerical results and the asymptotes. Both, the asymptotes and fits, are given by the complicated expressions presented in the Appendix. In Fig. 4 we show the curves const as a function of and . This visualizes the reduction the bulk viscosity for any T, and .
One can observe (Fig. 4) one important property of the reduction factor R. If both superfluidities are strong, , the factor R is mainly determined by the larger of the two gaps (by the strongest superfluidity):
Here, and are the reduction factors for the superfluidity of nucleons of one species. The weaker superfluidity (with smaller energy gap) produces some additional reduction of the viscosity which is relatively small; this is confirmed by the asymptote given in the Appendix. The same effect takes place for the reduction of the neutrino emissivity (e.g., Yakovlev et al. 1999).
Eq. (45) can be used also to evaluate R for the case in which the nucleons of species 1 suffer pairing of type A, while the nucleons of species 2 suffer pairing of types B or C. In the Appendix we present the asymptotes of the factors and in the limit of strong superfluidity of both nucleon species (, ).
The factors and can be calculated easily in a wide range of and . The calculation reduces to the one-dimensional integration in Eq. (45) of the function fitted by Eq. (A3). The results are exhibited in Figs. 5 and 6. One can see that the dependence of the factors and on and has much in common with the dependence of but and . The simple estimate (57) turns out to be valid in cases AB and AC as well. However since the superfluidity of type C reduces the factor R in a much weaker way than the superfluidities of types A or B, the transition from one dominating superfluidity to the other takes place in a rather wide region of and at . Accordingly, for , the reduction factor exceeds greatly and .
For practical calculations of the bulk viscosity in superfluid matter, one needs to know how to evaluate , and . Corresponding expressions for superfluidity of one nucleon species are given in Sect. 3.3. If nucleons 1 and 2 are superfluid at once, the reduction factor can be determined easily from the fit Eq. (A3). As for the reduction factors and , we have generated their extensive tables. These tables and numerical code which generates them are freely distributed.
Finally, Fig. 7 illustrates reduction of the bulk viscosity of matter with decreasing temperature by superfluidity of neutrons of type B or protons of type A for and s-1. Thick solid line shows the viscosity of non-superfluid matter (cf. with Fig. 2). Thin solid lines exhibit the bulk viscosity suppressed by the proton superfluidity at several selected critical temperatures indicated near the curves. The dot-and-dashed line shows the effect of neutron superfluidity ( K) for normal protons. We see that the superfluid reduction of the bulk viscosity depends drastically on temperature, superfluidity type, and critical temperatures and . One can hardly expect and higher than K for as large as (e.g., Yakovlev et al. 1999). If so, the superfluid reduction cannot be very large, say, for K, but it can reach five orders of magnitude in the case of superfluid protons (or six orders of magnitude if n and p are superfluid at once, see Fig. 5) for K at K. It grows exponentially with further decrease of T.
© European Southern Observatory (ESO) 2000
Online publication: June 5, 2000