## 2. Bulk viscosity in non-superfluid matter## 2.1. Bulk viscosity in matterConsider the bulk viscosity produced by the direct Urca process (involving muons and electrons) in non-superfluid 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
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 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
and , which imply
. Under these conditions, the rates
of the direct and inverse reactions,
and ( Let us assume that the neutron star undergoes radial pulsations of frequency . Associated temporal variations of the local baryon number density will be taken in the form , where is the pulsation amplitude and is the non-perturbed baryon number density (). We assume further that corresponds to the chemical equilibrium. This chemical equilibrium is violated slightly in pulsating matter. If the pulsation frequency were much smaller than the chemical relaxation rates, the composition of matter would follow instantaneous values of , realizing the chemical equilibrium every moment of time. In reality, the typical frequencies of the fundamental mode of the
radial pulsations
- s 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 where are the coefficients
specified in Sect. 2.4 for the direct Urca reactions. Microscopic
calculation (Sect. 2.4) yields . In
this paper, we will restrict ourselves to the case
().
Our definition of is the same as was
used by Sawyer (1989) for the case of The non-equilibrium Urca reactions provide the energy dissipation which causes damping of stellar pulsations. Accordingly, they contribute to the bulk viscosity of matter, . Using the standard definition of the bulk viscosity, the energy dissipation rate per unit volume averaged over the pulsation period can be written as where is the hydrodynamic velocity associated with the pulsations. The latter equality is obtained from continuity equation for baryons, (pulsations do not change their total number), which yields . The hydrodynamic matter flow implied by the stellar pulsations is accompanied by the time variations of the local pressure, . 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 For a strictly reversible process,
. However, in our case the quantities
Let us evaluate the integral (6). We have
. Thus the only terms in where all the derivatives are taken at equilibrium. The real part
of The change of the lepton fraction is determined by the difference of the direct and inverse reaction rates given by Eq. (4). The quantity in the latter equation varies near its equilibrium value as , where and all the derivatives are again taken at equilibrium. Combining
the expression with Eq. (4) and
using the formalism of complex variables we obtain the two equations
(for
and where and . In analogy with Sawyer (1989) we have introduced the notations: Note that all the derivatives are taken at equilibrium. In the absence of muons from Eq. (9) we have and . 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 ()
and it is sufficient to use the asymptotic form of the solution in the
Combining this equation with that for (see above) and inserting into Eq. (6) we get the dissipation rate of mechanical energy Finally, bearing in mind that , from Eqs. (5) and (12) we have the bulk viscosity Here we have taken into account that and are negative and presented the viscosity in the form which clearly shows that is positive. The expression for 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 is a sum of the partial viscosities and produced by the electron and muon Urca processes, respectively. This additivity rule greatly simplifies evaluation of . The values of and are determined by an equation of state as described in Sects. 2.2 and 2.3. The factors 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 viscosityLet us discuss briefly how to calculate the partial bulk viscosity of 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 where is the nucleon energy per
baryon, is the proton fraction, and
is a lepton energy per one lepton
( The neutron and proton chemical potentials are given by and . The derivatives should be taken using and . This gives . The chemical potentials of electrons or muons are , where is the Fermi momentum. Therefore, the difference of chemical potentials is Now let us calculate from Eq. (10). The derivative of with respect to is evaluated using . The result is Using Eq. (14) and the standard thermodynamic relations we obtain
the pressure , where
is the nucleon pressure, while
and
are the well known partial
pressures of free gases of Thus, a partial bulk viscosity is expressed through the two factors, and . Calculation of is discussed in Sect. 2.3, while is analyzed in Sect. 2.4. ## 2.3. Illustrative model of matterFor 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) where is the energy of the symmetric nuclear matter and is the symmetry energy. From Eq. (15) at equilibrium () we immediately obtain , and from Eq. (16) we have This is the practical expression for evaluating . The factor 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 -, where is the superfluid energy gap, and is the nucleon Fermi energy. In accordance with Eqs. (19) and (17), the bulk viscosity is
determined by the symmetry energy .
At the saturation density
fm where , MeV, and satisfies the condition . They proposed three theoretical models (I, II and III) for : and three models for 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 . 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 correspond to three different values of the compression modulus of symmetric nuclear matter at saturation, =120, 180 and 240 MeV. If, for instance, we take models I and II of and the model of with 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 and , respectively. The equilibrium fractions of muons and electrons, and , can be obtained as numerical solutions of the set of the chemical equilibrium equations at given : where . If the muons are absent (), the second equation should be disregarded, while the the first one determines the equilibrium composition of matter where . At given
the equilibrium fraction of
electrons in -matter is always smaller
than it would be in In Fig. 1 we plot the factors (solid lines) and (dot-and-dash lines) which determine the bulk viscosity for models I and II. The dotted lines show 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 , it appears in a jump-like manner at the muon threshold, initially exceeds and then tends to with increasing density.
## 2.4. Practical expressions for the bulk viscosity of matterNow let us calculate the factor ,
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 matter. In the absence
of superfluidity, the rate of a direct reaction producing a lepton
where is a nucleon momentum
( or 2),
and
are, respectively, the lepton
momentum and energy, and
are the neutrino momentum and
energy, and
are the delta functions, which
conserve energy For further analysis we introduce the dimensionless quantities: where the chemical potential difference is determined by Eq. (3). Thus, the delta function in Eq. (24) takes the form , where 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 Here, ,
erg cm We have transformed all the blocking factors into the Fermi-Dirac functions by replacing . The prefactor is given by (returning to the standard physical units) Here, is temperature in units of K; and 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 and , for leptons. The step function 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 (Lattimer et al. 1991). It easy to show that the rate 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 In a non-superfluid matter, which we consider in this section, the function is calculated analytically 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 . Moreover, the integral (30) is taken analytically for any : This relation, with account for Eqs. (4) and (30), gives the factor : In this paper, we do not consider large deviations from the chemical equilibrium. We restrict ourselves to the deviations for which . Finally, combining Eqs. (17) and (34), we obtain the partial bulk viscosity of matter, (subscript 0 refers to non-superfluid matter), induced by a non-equilibrium direct Urca process for : where . Fig. 2 shows the total bulk viscosity of non-superfluid matter versus nucleon density . We have used models I and II of 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).
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 =
0.414 fm 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 matter. This information is sufficient to calculate factors 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 = , which enter Eq. (35), can be fitted by simple expressions where ,
fm 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 MeV) can be fitted as where and
for model I;
and
for model II. The maximum fit error
is %, and
g cm © European Southern Observatory (ESO) 2000 Online publication: June 5, 2000 |