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

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1. Introduction

The dissipative processes in neutron stars play an important role for some dynamical properties of these unique objects. Shear viscosity damps differential rotation of neutron stars, leading to their uniform rigid-body rotation. Quite generally, the viscosity of neutron star matter implies damping of pulsations of neutron stars. Such pulsations could be excited during the process of neutron star formation. They could also be triggered by instabilities appearing during neutron star evolution, or by some external perturbations. The corresponding damping timescales involve the shear and bulk viscosities of neutron star interior. Both viscosities depend on density, temperature and composition of dense matter. Calculations of damping timescales of pulsations for various models of neutron star interiors have been done by Cutler et al. (1990). Viscous damping of pulsations of newly born hot neutron stars turns out to be due to the bulk viscosity.

Another role of the viscosity of neutron star matter is that it can damp gravitational radiation driven instabilities in rotating neutron stars and, therefore, could be important for determination of the maximum rotation frequency of neutron stars. In the absence of viscosity all rotating neutron stars would be driven unstable by the emission of gravitational waves. Viscous damping timescales enter explicitly the criteria for the appearance of these instabilities. Similarly, as in pulsating non-rotating neutron stars, viscous damping of gravitational radiation driven instabilities in rapidly rotating newly born neutron stars is dominated by bulk viscosity of neutron star interiors (e.g., Lindblom 1995, Zdunik 1996, Lindblom et al. 1998).

In this paper, we focus on the viscosity of matter in the neutron star cores (which extend from the layers of density [FORMULA] g cm-3 to the stellar centers). It is well known that the cores consist of baryons (neutrons n, protons p and possibly hyperons) and leptons (electrons e and muons µ). All constituents of matter are strongly degenerate fermions. The electrons and muons form almost ideal Fermi gases. The electrons are ultrarelativistic while the muons may be non-relativistic or relativistic depending on density. The nucleons are, to a good approximation, non-relativistic and constitute strongly interacting Fermi liquid. At the densities close to the normal nuclear density (baryon number density [FORMULA] fm-3 which corresponds to the mass density [FORMULA] g cm-3), neutron star matter is composed of n, p, e, and µ. At higher densities [[FORMULA]-[FORMULA]] some models of dense matter predict appearance of hyperons. At still higher densities, some calculations indicate possible presence of exotic phases (pion condensate, kaon condensate, deconfined quark matter). We will not consider the hyperonic or exotic phases but restrict ourselves to the study of the [FORMULA] matter.

Our analysis is additionally complicated by possible superfluidity of nucleons in the neutron star cores. The superfluidity is thought to be produced by Cooper pairing of nucleons due to attractive parts of nucleon-nucleon interaction. The superfluidity of nucleons in the neutron star cores has been the subject of numerous papers (as reviewed, for instance, by Yakovlev et al. 1999). Various microscopic theories predict very different superfluid gaps (critical temperatures [FORMULA] and [FORMULA]) of neutrons and protons depending on specific model of strong interaction employed and specific many-body theory used to account for medium effects. However, all these results have important common features. In particular, the proton pairing occurs mainly in the 1S0-state since the pp interaction is attractive in this state everywhere in the neutron star core due to not too high number density of protons. As for the neutrons, their interaction in the 1S0 state turns from attraction to repulsion at densities [FORMULA] but the interaction in the 3P2 state may be attractive and may lead to Cooper pairing. The critical temperatures [FORMULA] and [FORMULA] in the neutron star cores predicted by different microscopic theories depend on density and scatter in a wide range from about [FORMULA] to [FORMULA] K. Under these conditions we will not rely on any specific microscopic theory of nucleon superfluidity, but will treat [FORMULA] and [FORMULA] as free parameters.

The viscosity, we are interested in, is well known to consist of the shear viscosity and bulk viscosity. The standard source of the shear viscosity of the neutron star matter is scattering between its constituents. Classical calculations of shear viscosity for the npe model of non-superfluid matter were done by Flowers & Itoh (1979). Their results were used in the studies of damping of neutron star pulsations by Cutler et al. (1990). In the superfluid core of a rotating neutron star, there is an additional viscous mechanism, called mutual friction, resulting from the scattering of electrons off the magnetic field trapped in the cores of superfluid neutron vortices (Lindblom & Mendell 1995).

The bulk viscosity may partly be determined by particle scattering. However, this component of bulk viscosity is usually much smaller than the shear viscosity (e.g., Baym & Pethick 1991). The main contribution into the bulk viscosity of sufficiently hot [FORMULA] matter comes from the neutrino processes of Urca type associated with electron and muon emission and capture by nucleons. We will focus on such bulk viscosity. Generally, the neutrino processes in question are divided into the direct Urca and the modified Urca processes. A direct Urca process is a sequence of two reactions (direct and inverse one) and can be written as


where l is either electron or muon, and [FORMULA] is an associated neutrino. The most important is the process (Lattimer et al. 1991) involving electrons ([FORMULA]); it consist of the beta decay of neutron and subsequent beta capture. It should be emphasized that the both direct Urca processes are forbidden by momentum conservation of reacting particles for the simplest model of dense matter as a gas of noninteracting Fermi particles (e.g., Shapiro & Teukolsky 1983) at any density [FORMULA] in the neutron star cores. Nevertheless they are allowed (Lattimer et al. 1991) for some realistic equations of state at densities higher than some threshold densities (of several [FORMULA]). Thus, the direct Urca processes may be open in the inner cores of rather massive neutron stars. The threshold density for the muon process is always higher than for the electron one.

If allowed, the direct Urca processes produce the most powerful neutrino emission from the neutron star cores (Lattimer et al. 1991). Corresponding neutrino emissivities were calculated by Lattimer et al. (1991) and used in numerous simulations of the neutron star cooling as reviewed, for instance, by Yakovlev et al. (1999). In the absence of nucleon superfluidity, the direct Urca processes lead to the fast cooling of neutron stars. If allowed, the direct Urca processes produce the main contribution into the bulk viscosity we are interested in.

However, for many equations of state the direct Urca processes are forbidden by momentum conservation at any density in the neutron star cores. Moreover, they are prohibited at [FORMULA] for the majority of equations of state. In such cases, they do not operate in the low and medium-mass neutron stars and in the outer cores of all neutron star models constructed using these equations of state. If so, the bulk viscosity is determined by the the reactions of the modified Urca processes


where N is an additional nucleon required to conserve momentum of the reacting particles. For instance, in npe matter one has two modified Urca processes corresponding to [FORMULA] and [FORMULA], respectively, which can be referred to as the neutron and proton branches of the modified Urca process (e.g., Friman & Maxwell 1979, Yakovlev & Levenfish 1995). The rates of the modified Urca processes are typically 3-5 orders of magnitude lower than the rates of the direct Urca processes. The modified Urca processes either have no density threshold (as the neutron branch in npe matter) or have much lower density thresholds than the direct Urca processes. Thus they operate in the entire neutron star core. If the direct Urca processes are forbidden and matter is non-superfluid, the modified Urca processes produce the main neutrino emission from the neutron star cores leading to slow (standard ) cooling of neutron stars. Their role in the neutron star cooling theory has been studied in many papers (see, e.g., Yakovlev et al. 1999, for review).

Thus, the problem of calculating the bulk viscosity due to neutrino processes is quite complicated: there are several neutrino processes involved influenced by possible nucleon superfluidity. So far the bulk viscosity has been studied only for non-superfluid npe matter. The viscosity due to the neutron branch of the modified Urca process was analyzed by Sawyer (1989) while the viscosity produced by the nucleon direct Urca process was considered by Haensel & Schaeffer (1992). The effects of superfluidity have not been analyzed for the problem of bulk viscosity but studied thoroughly for the neutrino emissivity produced in different reactions (e.g., Yakovlev et al. 1999, and references therein).

The relative importance of the bulk viscosity produced by neutrino reactions with respect to the shear viscosity produced by collisions can be estimated by comparing the results by Sawyer (1989) and Haensel & Schaeffer (1992) with the values of the shear viscosity calculated by Flowers & Itoh (1979). The comparison shows that the neutrino bulk viscosity dominates in the neutron star cores for temperatures [FORMULA] K if the direct Urca processes are switched on and for [FORMULA] K if the direct Urca processes are forbidden. In superfluid matter, the bulk viscosity can be even more important.

In this paper, we consider the bulk viscosity produced by the direct Urca processes in [FORMULA] matter of the neutron star cores. In analogy with the effects of superfluidity on the neutrino emissivity, we will analyze the effects of superfluidity of nucleons on the bulk viscosity.

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Online publication: June 5, 2000