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

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2. Bulk viscosity in non-superfluid matter

2.1. Bulk viscosity in [FORMULA] matter

Consider the bulk viscosity produced by the direct Urca process (involving muons and electrons) in non-superfluid [FORMULA] matter.

Due to very frequent collisions between particles, dense stellar matter should very quickly (instantaneously on macroscopic time scales) achieve a quasi-equilibrium state with certain temperature T and chemical potentials [FORMULA] of different particle species [FORMULA]. Typically, all particle species are strongly degenerate. We assume that the matter is transparent for neutrinos, which therefore do not contribute to the thermodynamical quantities.

A quasi-equilibrium state described above does not mean full thermodynamic equilibrium. The latter assumes additionally the equilibrium with respect to the beta and muon decay and capture processes. We will call it the chemical equilibrium . Relaxation to the chemical equilibrium depends drastically on a given equation of state and local density of matter [FORMULA]. It is realized either through direct Urca or modified Urca processes (Sect. 1). Consideration of this subsection is valid for all Urca processes although the practical expressions (Sects. 2.3-2.4) will be obtained for the direct Urca processes.

The Urca processes of both types, direct and modified, are rather slow. Although the chemical relaxation rate depends strongly on temperature, in any case it takes much more time (from tens of seconds to much longer time intervals) than the rapid relaxation to a quasi-equilibrium state described above. Therefore, a neutron star can be in a quasi-equilibrium, but not in the chemical equilibrium, for a long time.

If the chemical equilibrium is achieved, then the chemical potentials satisfy the equalities [FORMULA] and [FORMULA], which imply [FORMULA]. Under these conditions, the rates of the direct and inverse reactions, [FORMULA] and [FORMULA] (l = e or µ), of any Urca process are equal.

Let us assume that the neutron star undergoes radial pulsations of frequency [FORMULA]. Associated temporal variations of the local baryon number density will be taken in the form [FORMULA], where [FORMULA] is the pulsation amplitude and [FORMULA] is the non-perturbed baryon number density ([FORMULA]). We assume further that [FORMULA] corresponds to the chemical equilibrium. This chemical equilibrium is violated slightly in pulsating matter. If the pulsation frequency [FORMULA] were much smaller than the chemical relaxation rates, the composition of matter would follow instantaneous values of [FORMULA], realizing the chemical equilibrium every moment of time.

In reality, the typical frequencies of the fundamental mode of the radial pulsations [FORMULA]-[FORMULA] s-1, are much higher than the chemical relaxation rates. As a result, the partial fractions [FORMULA] of all the constituents of dense matter are almost unaffected by pulsations (i.e., almost constant). Owing to the slowness of the Urca reactions, these fractions lag behind their instantaneous equilibrium values, producing non-zero differences of instantaneous [FORMULA]:

[EQUATION]

This causes an asymmetry of the direct and inverse direct Urca reactions, and, hence, slight deviations from the chemical equilibrium. The asymmetry, calculated in the linear approximation with respect to [FORMULA], is given by

[EQUATION]

where [FORMULA] are the coefficients specified in Sect. 2.4 for the direct Urca reactions. Microscopic calculation (Sect. 2.4) yields [FORMULA]. In this paper, we will restrict ourselves to the case [FORMULA] ([FORMULA]). Our definition of [FORMULA] is the same as was used by Sawyer (1989) for the case of npe matter. Notice that [FORMULA] defined in this way is negative.

The non-equilibrium Urca reactions provide the energy dissipation which causes damping of stellar pulsations. Accordingly, they contribute to the bulk viscosity of matter, [FORMULA]. Using the standard definition of the bulk viscosity, the energy dissipation rate per unit volume averaged over the pulsation period [FORMULA] can be written as

[EQUATION]

where [FORMULA] is the hydrodynamic velocity associated with the pulsations. The latter equality is obtained from continuity equation for baryons, [FORMULA] (pulsations do not change their total number), which yields [FORMULA].

The hydrodynamic matter flow implied by the stellar pulsations is accompanied by the time variations of the local pressure, [FORMULA]. The dissipation of the energy of the hydrodynamic motion is due to irreversibility of the periodic compression-decompression process. Averaged over the pulsation period, this dissipation rate in the unit volume is

[EQUATION]

For a strictly reversible process, [FORMULA]. However, in our case the quantities P and V follow variations of [FORMULA] in different ways. The specific volume [FORMULA] varies instantaneously as [FORMULA] varies, i.e., the oscillations of V and [FORMULA] are in phase but the pressure varies with certain phase shift. In [FORMULA] matter at quasi-equilibrium the pressure can be regarded as a function of four variables: [FORMULA], [FORMULA], [FORMULA], and T. Variations of T are insignificant, for our problem, and may be disregarded. Thus, it is sufficient to assume that [FORMULA]. Variations of the pressure contain the terms oscillating with shifted phases due to the lags of [FORMULA] and [FORMULA].

Let us evaluate the integral (6). We have [FORMULA]. Thus the only terms in P contributing into the energy dissipation are those which are proportional to [FORMULA]. At this stage it is convenient to use the formalism of complex variables and write [FORMULA], [FORMULA], where [FORMULA] and [FORMULA] are the equilibrium quantities while [FORMULA] and [FORMULA] are small complex amplitudes to be determined. We have

[EQUATION]

where all the derivatives are taken at equilibrium. The real part of P contains the terms with [FORMULA] provided the amplitudes [FORMULA] have imaginary part.

The change of the lepton fraction [FORMULA] is determined by the difference of the direct and inverse reaction rates given by Eq. (4). The quantity [FORMULA] in the latter equation varies near its equilibrium value [FORMULA] as [FORMULA], where

[EQUATION]

and all the derivatives are again taken at equilibrium. Combining the expression [FORMULA] with Eq. (4) and using the formalism of complex variables we obtain the two equations [FORMULA] (for [FORMULA] and µ). These two equations supplemented by Eq. (8) constitute a system of equations which solution is

[EQUATION]

where [FORMULA] and [FORMULA]. In analogy with Sawyer (1989) we have introduced the notations:

[EQUATION]

Note that all the derivatives are taken at equilibrium. In the absence of muons from Eq. (9) we have [FORMULA] and [FORMULA]. This is the well known limit considered by Sawyer (1989) and Haensel & Schaeffer (1992).

Generally, Eq. (9) is quite complicated. However, in practical applications stellar oscillations are always much more frequent than the beta and muon reaction rates ([FORMULA]) and it is sufficient to use the asymptotic form of the solution in the high-frequency limit . In this limit the imaginary part of [FORMULA] is related to the amplitude [FORMULA] as

[EQUATION]

Combining this equation with that for [FORMULA] (see above) and inserting into Eq. (6) we get the dissipation rate of mechanical energy

[EQUATION]

Finally, bearing in mind that [FORMULA], from Eqs. (5) and (12) we have the bulk viscosity

[EQUATION]

Here we have taken into account that [FORMULA] and [FORMULA] are negative and presented the viscosity in the form which clearly shows that [FORMULA] is positive. The expression for [FORMULA] was obtained by Sawyer (1989). Let us emphasize that the viscosity we deal with has meaning of a coefficient in the equation that determines the damping rate of stellar pulsations averaged over pulsation period (and it cannot generally be used in exact hydrodynamical equations of fluid motion).

Therefore, in the high frequency limit, which is the most important in practice, the bulk viscosity [FORMULA] is a sum of the partial viscosities [FORMULA] and [FORMULA] produced by the electron and muon Urca processes, respectively. This additivity rule greatly simplifies evaluation of [FORMULA]. The values of [FORMULA] and [FORMULA] are determined by an equation of state as described in Sects. 2.2 and 2.3. The factors [FORMULA] are studied in Sect. 2.4 for the direct Urca reactions. The results of analogous consideration for the modified Urca reactions will be published elsewhere.

2.2. Partial bulk viscosity

Let us discuss briefly how to calculate the partial bulk viscosity [FORMULA] of [FORMULA] matter for a given equation of state. All the quantities in this section and below are essentially (quasi)equilibrium values. Thus we will omit the index 0, for brevity.

Since the electrons and muons constitute almost ideal gases, the matter energy per baryon can be generally written as

[EQUATION]

where [FORMULA] is the nucleon energy per baryon, [FORMULA] is the proton fraction, and [FORMULA] is a lepton energy per one lepton (e or µ). The latter energy is determined by the lepton number density, [FORMULA]. Owing to charge neutrality, we have [FORMULA].

The neutron and proton chemical potentials are given by [FORMULA] and [FORMULA]. The derivatives should be taken using [FORMULA] and [FORMULA]. This gives [FORMULA]. The chemical potentials of electrons or muons are [FORMULA], where [FORMULA] is the Fermi momentum. Therefore, the difference of chemical potentials is

[EQUATION]

Now let us calculate [FORMULA] from Eq. (10). The derivative of [FORMULA] with respect to [FORMULA] is evaluated using [FORMULA]. The result is

[EQUATION]

Using Eq. (14) and the standard thermodynamic relations we obtain the pressure [FORMULA], where [FORMULA] is the nucleon pressure, while [FORMULA] and [FORMULA] are the well known partial pressures of free gases of e and µ, respectively. Direct calculations yields [FORMULA]. Inserting this derivative into Eq. (13) and using the definition of [FORMULA] we come to a very simple equation

[EQUATION]

Thus, a partial bulk viscosity [FORMULA] is expressed through the two factors, [FORMULA] and [FORMULA]. Calculation of [FORMULA] is discussed in Sect. 2.3, while [FORMULA] is analyzed in Sect. 2.4.

2.3. Illustrative model of [FORMULA] matter

For illustration, we use a phenomenological equation of state proposed by Prakash et al. (1988). According to these authors the nucleon energy is presented in the familiar form (neglecting small neutron-proton mass difference)

[EQUATION]

where [FORMULA] is the energy of the symmetric nuclear matter and [FORMULA] is the symmetry energy. From Eq. (15) at equilibrium ([FORMULA]) we immediately obtain [FORMULA], and from Eq. (16) we have

[EQUATION]

This is the practical expression for evaluating [FORMULA]. The factor [FORMULA] is not affected by a possible nucleon superfluidity which has a negligible effect on the equation of state. The relative effect of the superfluidity on the energy per nucleon is [FORMULA]-[FORMULA], where [FORMULA] is the superfluid energy gap, and [FORMULA] is the nucleon Fermi energy.

In accordance with Eqs. (19) and (17), the bulk viscosity is determined by the symmetry energy [FORMULA]. At the saturation density [FORMULA] fm-3 the symmetry energy [FORMULA] is measured rather reliably in laboratory (e.g., Moeller et al. 1988) but at higher [FORMULA] it is still unknown. Prakash et al. (1988) presented [FORMULA] in the form:

[EQUATION]

where [FORMULA], [FORMULA] MeV, and [FORMULA] satisfies the condition [FORMULA]. They proposed three theoretical models (I, II and III) for [FORMULA]:

[EQUATION]

and three models for [FORMULA] in Eq. (18). We do not discuss the latter models here because they are not required to calculate the particle fractions and the bulk viscosity as a function of [FORMULA]. Following Sawyer (1989) and Haensel & Schaeffer (1992) we will use models I and II, for illustration. Model I gives lower symmetry energy and accordingly lower excess of neutrons over protons. In contrast to the above authors we will allow for appearance of muons.

The three models for [FORMULA] correspond to three different values of the compression modulus of symmetric nuclear matter at saturation, [FORMULA]=120, 180 and 240 MeV. If, for instance, we take models I and II of [FORMULA] and the model of [FORMULA] with [FORMULA] MeV, we have two model equations of state of matter (models I and II) in the cores of neutron stars. The effective masses of nucleons, renormalized by the medium effects, will be set equal to 0.7 of their bare masses (the same values will be adopted in all numerical examples below). The equations of state I and II obtained in this way are moderately stiff. The maximum neutron star masses for models I and II are [FORMULA] and [FORMULA], respectively.

The equilibrium fractions of muons and electrons, [FORMULA] and [FORMULA], can be obtained as numerical solutions of the set of the chemical equilibrium equations at given [FORMULA]:

[EQUATION]

where [FORMULA]. If the muons are absent ([FORMULA]), the second equation should be disregarded, while the the first one determines the equilibrium composition of matter

[EQUATION]

where [FORMULA]. At given [FORMULA] the equilibrium fraction of electrons in [FORMULA]-matter is always smaller than it would be in npe matter, while the equilibrium fraction of protons is always higher. The threshold of muon appearance is determined by the condition [FORMULA]. For models I and II, the muons appear at the baryon number density 0.150 fm-3 and 0.152 fm-3, respectively.

In Fig. 1 we plot the factors [FORMULA] (solid lines) and [FORMULA] (dot-and-dash lines) which determine the bulk viscosity for models I and II. The dotted lines show [FORMULA] for the simplified models I and II in which appearance of muons is artificially forbidden. These results coincide with those obtained by Sawyer (1989) and Haensel & Schaeffer (1992). They coincide also with the solid lines at densities below the thresholds of muon appearance but go above the solid lines at higher densities (the presence of muons affects fractions of electrons and protons). As for the factor [FORMULA], it appears in a jump-like manner at the muon threshold, initially exceeds [FORMULA] and then tends to [FORMULA] with increasing density.

[FIGURE] Fig. 1. Linear response of chemical potential difference to baryon number density perturbation, [FORMULA], versus the baryon number density [FORMULA] for two model equations of state proposed by Prakash et al. (1988) and discussed in the text. Solid curves correspond to [FORMULA], while dot-and-dash curves to [FORMULA]. Dotted lines present [FORMULA] for the simplified models of matter without muons.

2.4. Practical expressions for the bulk viscosity of [FORMULA] matter

Now let us calculate the factor [FORMULA], which enters the bulk viscosity (17) and determines the asymmetry (4) of the rates of direct and inverse reactions of the direct Urca processes in [FORMULA] matter. In the absence of superfluidity, the rate of a direct reaction producing a lepton l is given by ([FORMULA])

[EQUATION]

where [FORMULA] is a nucleon momentum ([FORMULA] or 2), [FORMULA] and [FORMULA] are, respectively, the lepton momentum and energy, [FORMULA] and [FORMULA] are the neutrino momentum and energy, [FORMULA] and [FORMULA] are the delta functions, which conserve energy E and momentum [FORMULA] of the particles in initial (i) and final (f) states, [FORMULA] is the squared matrix element of the reaction, and [FORMULA] is an appropriate Fermi-Dirac function, [FORMULA]. Eq. (24) includes the instantaneous chemical potentials [FORMULA] ([FORMULA]) and does not generally require the chemical equilibrium.

For further analysis we introduce the dimensionless quantities:

[EQUATION]

where the chemical potential difference [FORMULA] is determined by Eq. (3). Thus, the delta function in Eq. (24) takes the form [FORMULA], where [FORMULA] for chemical equilibrium.

Multidimensional integrals in Eq. (24) are standard (see, e.g., Shapiro & Teukolsky 1983). Eq. (24) is simplified taking into account that nucleons and leptons l (e and µ) are strongly degenerate. The main contribution into the integral comes from the narrow vicinities of momentum space near the Fermi surfaces of these particles. The momenta of nucleons and leptons (e or µ) can be set equal to their Fermi momenta in all smooth functions. The squared matrix element summed over the spin states and averaged over orientations of the neutrino momenta is

[EQUATION]

Here, [FORMULA], [FORMULA] erg cm3 is the Fermi weak coupling constant, [FORMULA] is the vector normalization constant, [FORMULA] is the axial vector normalization constant, and [FORMULA] is the Cabibbo angle ([FORMULA]). Hence the squared matrix element is constant and can be taken out of the integral. Further procedure consists in the standard energy-momentum decomposition of the integration in Eq. (24). It yields:

[EQUATION]

We have transformed all the blocking factors [FORMULA] into the Fermi-Dirac functions [FORMULA] by replacing [FORMULA]. The prefactor [FORMULA] is given by (returning to the standard physical units)

[EQUATION]

Here, [FORMULA] is temperature in units of [FORMULA] K; [FORMULA] and [FORMULA] are, respectively, the effective masses of neutrons and protons in dense matter (which differ from the bare nucleon masses due to the in-medium effects). Moreover, we have defined [FORMULA] and [FORMULA], for leptons. The step function [FORMULA] equals 1 if the direct Urca process is switched on and equals 0 otherwise (Sect. 1). The direct Urca process of study is switched on if the Fermi momenta of the reacting particles satisfy the inequality [FORMULA] (Lattimer et al. 1991).

It easy to show that the rate [FORMULA] of the inverse reaction of the direct Urca process (lepton capture) differs from the rate of the direct reaction, given by Eq. (24), only by the argument of the delta function in the expression for I (one should replace [FORMULA] there). Therefore, the difference of the lepton production and capture rates (4) can be written as

[EQUATION]

where

[EQUATION]

In a non-superfluid matter, which we consider in this section, the function [FORMULA] is calculated analytically

[EQUATION]

We see that the difference (29) of the non-equilibrium rates of the direct Urca reactions in normal matter is determined solely by the parameter [FORMULA]. Moreover, the integral (30) is taken analytically for any [FORMULA]:

[EQUATION]

This relation, with account for Eqs. (4) and (30), gives the factor [FORMULA]:

[EQUATION]

In this paper, we do not consider large deviations from the chemical equilibrium. We restrict ourselves to the deviations [FORMULA] for which [FORMULA].

Finally, combining Eqs. (17) and (34), we obtain the partial bulk viscosity of [FORMULA] matter, [FORMULA] (subscript 0 refers to non-superfluid matter), induced by a non-equilibrium direct Urca process for [FORMULA]:

[EQUATION]

where [FORMULA]. Fig. 2 shows the total bulk viscosity [FORMULA] of non-superfluid matter versus nucleon density [FORMULA]. We have used models I and II of [FORMULA] matter described in Sect. 2.3. The dotted lines show the bulk viscosity for the simplified models in which the muons are absent (cf. with Fig. 1). The latter results coincide with those reported by Haensel & Schaeffer (1992).

[FIGURE] Fig. 2. The bulk viscosity [FORMULA] for models I and II of non-superfluid [FORMULA] matter (solid lines) induced by the non-equilibrium direct Urca processes involving electrons and muons versus the baryon number density [FORMULA], for [FORMULA] K and [FORMULA] s-1. Dotted lines show the same bulk viscosity but for the simplified models of matter without muons.

The results for models I and II are similar. The bulk viscosity due to the direct Urca processes is switched on in a jump-like manner at the threshold density at which the electron direct Urca process becomes operative (this happens at [FORMULA]= 0.414 fm-3 and 0.302 fm-3, respectively). The presence of muons lowers the threshold density (mainly due to increasing the number density and Fermi momenta of protons). On the other hand, the muons lower the bulk viscosity produced by the electron direct Urca process (by decreasing [FORMULA]). At larger [FORMULA], the total bulk viscosity suffers the second jump (at [FORMULA] 0.503 fm-3 and 0.358 fm-3 for models I and II, respectively). This time it is associated with switching on the muon direct Urca process, where muons participate by themselves. The contribution of the muon direct Urca into the bulk viscosity is even larger than the contribution of the electron direct Urca. The total bulk viscosity exceeds the bulk viscosity in npe matter. This is natural: the muons introduce additional non-equilibrium Urca process which makes stellar matter more viscous.

Finally, let us discuss practical calculation of the bulk viscosity. A user possesses usually number densities of different particles for any given equation of state of [FORMULA] matter. This information is sufficient to calculate factors [FORMULA] numerically from the second equality in Eq. (19). Other density dependent quantities which enter the expressions for the bulk viscosity are also expressed through the number densities. Thus, the evaluation of the bulk viscosity for any equation of state is not a problem.

For our illustrative equations of state the functions [FORMULA]=[FORMULA] [FORMULA], which enter Eq. (35), can be fitted by simple expressions

[EQUATION]

where [FORMULA], [FORMULA] fm-3, and the maximum error [FORMULA]%. The fit parameters are: [FORMULA], [FORMULA], [FORMULA], [FORMULA] for [FORMULA] and model I; [FORMULA], [FORMULA], [FORMULA], [FORMULA] for [FORMULA] and model I; [FORMULA], [FORMULA], [FORMULA], [FORMULA] for [FORMULA] and model II; [FORMULA], [FORMULA], [FORMULA], [FORMULA] for [FORMULA] and model II.

On the other hand, the baryon number number density as a function of mass density (for the model equations of state with the compression modulus [FORMULA] MeV) can be fitted as

[EQUATION]

where [FORMULA] and [FORMULA] for model I; [FORMULA] and [FORMULA] for model II. The maximum fit error is [FORMULA]%, and [FORMULA] g cm-3. These equations allow one to calculate the bulk viscosity as a function of mass density as required in practical applications.

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Online publication: June 5, 2000
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