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

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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 [FORMULA] of nn-pair moment onto a quantization axis z (see, e.g., Amundsen and Ostgaard 1985): [FORMULA]. 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 [FORMULA]. We will consider the 3P2-state neutron superfluidity either with [FORMULA] or with [FORMULA]. In these two cases the effect of superfluidity on the bulk viscosity is qualitatively different. Consideration of the superfluidity based on mixed [FORMULA] states is much more complicated and goes beyond the scope of the present paper.

Thus we will study three different superfluidity types: 1S0, 3P2 ([FORMULA]) and 3P2 ([FORMULA]) 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

Microscopically, superfluidity introduces an energy gap [FORMULA] in momentum dependence of the nucleon energy, [FORMULA]. Near the Fermi level ([FORMULA]), this dependence can be written as


where [FORMULA] and [FORMULA] 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 [FORMULA], where [FORMULA] is the amplitude which describes temperature dependence of the gap; [FORMULA] specifies dependence of the gap on the angle [FORMULA] between the particle momentum and the z axis (Table 1). In case A the gap is isotropic, and [FORMULA]. In cases B and C the gap depends on [FORMULA]. Note that in case C the gap vanishes at the poles of the Fermi sphere at any temperature: [FORMULA].

The gap amplitude [FORMULA] is derived from the standard equation of the BCS theory (see, e.g., Yakovlev et al. 1999). The value of [FORMULA] determines the critical temperature [FORMULA]. The values of [FORMULA] 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 y can be presented in the form:


The dimensionless gap amplitude v depends only on [FORMULA]. 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 [FORMULA]) on the nucleon Fermi surface. The dependence of v on [FORMULA] can be fitted as (Levenfish & Yakovlev 1994):


The mean errors of these fits are [FORMULA] for all [FORMULA].

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 ([FORMULA] 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 [FORMULA] is a partial bulk viscosity of non-superfluid matter, Eq. (35), and [FORMULA] 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 [FORMULA] and reproduce the results of Sect. 2.

Thus the problem consists in calculating the reduction factors [FORMULA]. Each factor depends generally on two parameters, [FORMULA] and [FORMULA], 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 [FORMULA] in the expression for the bulk viscosity, Eq. (35). As seen from Eq. (34), the main factor affected by the superfluidity in [FORMULA] is the integral [FORMULA], 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 [FORMULA] the integrand of this equation is [FORMULA], where [FORMULA] 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 [FORMULA].

Generalization of [FORMULA] 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 y is given by Eq. (39). In the absence of superfluidity, we have [FORMULA] and [FORMULA].

Let the index [FORMULA] correspond to a nucleon which can suffer superfluidity of type A while [FORMULA] 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 [FORMULA] for [FORMULA] and 2 [in [FORMULA] and in the delta function] and introduce averaging over orientations of [FORMULA] (analogous procedure is considered in detail by Levenfish & Yakovlev 1994 for the problem of superfluid reduction of the neutrino emissivity). Then the factor [FORMULA] can be written as


where [FORMULA] refers to the non-superfluid case, [FORMULA] is the reduction factor in question, and


with [FORMULA]. Here, [FORMULA] is the solid angle element in the direction of [FORMULA].

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 [FORMULA], and we can set [FORMULA] in Eqs. (45) and (46). Integration over [FORMULA] and [FORMULA] in Eq. (46) reduces to well-known integrals of the theory of Fermi liquids and yields:


where [FORMULA]. For [FORMULA], one has [FORMULA]. If superfluidity is strong ([FORMULA]), 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 [FORMULA] can be obtained from Eq. (47):


Note that the factors [FORMULA] and [FORMULA] are suppressed exponentially with decreasing temperature, whereas [FORMULA] varies as [FORMULA]. 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 [FORMULA]) and reproduce the asymptotes (48)-(50):


Here, [FORMULA], [FORMULA] and [FORMULA] in the factors [FORMULA], [FORMULA] and [FORMULA], respectively. Using Eqs. (41) and (51)-(53), one can easily calculate the reduction factors R for any [FORMULA]. These factors are shown in Fig. 3 versus [FORMULA]. 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 [FORMULA] we obtain [FORMULA], [FORMULA] and [FORMULA].

[FIGURE] Fig. 3. Reduction factors R of the bulk viscosity versus [FORMULA] for three superfluidity types A, B and C (Table 1).

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 [FORMULA] and obtain


Notice that [FORMULA] as [FORMULA], and [FORMULA] as [FORMULA].

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 [FORMULA] and [FORMULA]. It is evident that [FORMULA]. We have also derived the asymptote of [FORMULA] in the limit of strong superfluidity. Furthermore, we have calculated the factor [FORMULA] 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 [FORMULA] const as a function of [FORMULA] and [FORMULA]. This visualizes the reduction the bulk viscosity for any T, [FORMULA] and [FORMULA].

[FIGURE] Fig. 4. Isolevels of the reduction factor R of the bulk viscosity by nucleon superfluidity of type AA versus [FORMULA] and [FORMULA]. In the domain [FORMULA], [FORMULA] nucleons 1 and 2 are normal and [FORMULA]. In the domain [FORMULA], [FORMULA] nucleons 1 are superfluid and nucleons 2 normal, while in the domain [FORMULA], [FORMULA] nucleons 1 are normal and 2 superfluid; in these domains R depends on one parameter ([FORMULA] or [FORMULA]). In the domain [FORMULA], [FORMULA] both nucleons 1 and 2 are superfluid at once.

One can observe (Fig. 4) one important property of the reduction factor R. If both superfluidities are strong, [FORMULA], the factor R is mainly determined by the larger of the two gaps (by the strongest superfluidity):


Here, [FORMULA] and [FORMULA] 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 [FORMULA] 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 [FORMULA] and [FORMULA] in the limit of strong superfluidity of both nucleon species ([FORMULA], [FORMULA]).

The factors [FORMULA] and [FORMULA] can be calculated easily in a wide range of [FORMULA] and [FORMULA]. The calculation reduces to the one-dimensional integration in Eq. (45) of the function [FORMULA] fitted by Eq. (A3). The results are exhibited in Figs. 5 and 6. One can see that the dependence of the factors [FORMULA] and [FORMULA] on [FORMULA] and [FORMULA] has much in common with the dependence of [FORMULA] but [FORMULA] and [FORMULA]. 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 [FORMULA] and [FORMULA] at [FORMULA]. Accordingly, for [FORMULA], the reduction factor [FORMULA] exceeds greatly [FORMULA] and [FORMULA].

[FIGURE] Fig. 5. Same as in Fig. 4 but for the case in which the superfluidity of nucleons 1 is of type A while the superfluidity of nucleons 2 is of type B.

[FIGURE] Fig. 6. Same as in Fig. 5 but for the case in which superfluidity of nucleons 2 is of type C.

For practical calculations of the bulk viscosity in superfluid matter, one needs to know how to evaluate [FORMULA], [FORMULA] and [FORMULA]. 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 [FORMULA] can be determined easily from the fit Eq. (A3). As for the reduction factors [FORMULA] and [FORMULA], 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 [FORMULA] matter with decreasing temperature by superfluidity of neutrons of type B or protons of type A for [FORMULA] and [FORMULA] 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 [FORMULA] indicated near the curves. The dot-and-dashed line shows the effect of neutron superfluidity ([FORMULA] K) for normal protons. We see that the superfluid reduction of the bulk viscosity depends drastically on temperature, superfluidity type, and critical temperatures [FORMULA] and [FORMULA]. One can hardly expect [FORMULA] and [FORMULA] higher than [FORMULA] K for [FORMULA] as large as [FORMULA] (e.g., Yakovlev et al. 1999). If so, the superfluid reduction cannot be very large, say, for [FORMULA] 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 [FORMULA] K at [FORMULA] K. It grows exponentially with further decrease of T.

[FIGURE] Fig. 7. Bulk viscosity [FORMULA] of superfluid [FORMULA] matter (model II) produced by the electron and muon direct Urca processes at the baryon number density [FORMULA] and [FORMULA] s-1 as a function of temperature for non-superfluid matter (thick solid line), for matter with superfluid protons (solid curves, [FORMULA], [FORMULA], [FORMULA] and [FORMULA]) and normal neutrons, and for matter with superfluid neutrons (dash-and-dotted curve, [FORMULA] K) and normal protons.

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Online publication: June 5, 2000