## 3. Bulk viscosity of superfluid matter## 3.1. Superfluid gapsSuperfluidity 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 Thus we will study three different superfluidity types:
Microscopically, superfluidity introduces an energy gap in momentum dependence of the nucleon energy, . Near the Fermi level (), this dependence can be written as where and
are the Fermi momentum and Fermi
velocity of the nucleon, respectively, and 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. For further analysis it is convenient to introduce the dimensionless quantities: The dimensionless gap The dimensionless gap amplitude The mean errors of these fits are for all . ## 3.2. Superfluid reduction factorsNow 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 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 . Generalization of to the superfluid case can be achieved by introducing the neutron and proton energy gaps into Eq. (43). For convenience, let us define the dimensionless quantities where 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 where refers to the non-superfluid case, is the reduction factor in question, and 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 ## 3.3. Superfluidity of neutrons or protonsConsider the superfluidity of nucleon of one species, for instance,
of species 2. In this case 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 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 Here, ,
and
in the factors
,
and , respectively. Using Eqs. (41)
and (51)-(53), one can easily calculate the reduction factors
## 3.4. Superfluidity of neutrons and protonsLet both nucleons, 1 and 2, be superfluid at once, and let the
superfluidity of nucleon 1 be of type A. In this case Notice that as , and as . First consider the case in which the superfluidities of nucleons 1 and 2 are of type A. Using Eqs. (40) and (45) we get 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
One can observe (Fig. 4) one important property of the reduction
factor 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 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
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
© European Southern Observatory (ESO) 2000 Online publication: June 5, 2000 |