A&A 381, 1080-1089 (2002)
DOI: 10.1051/0004-6361:20011532
P. Haensel^{1} - K. P. Levenfish^{2} - D. G. Yakovlev^{2}
1 - N. Copernicus Astronomical Center,
Bartycka 18, 00-716 Warszawa, Poland
2 -
Ioffe Physical Technical Institute, Politekhnicheskaya 26,
194021 St.-Petersburg, Russia
Received 16 October 2001 / Accepted 29 October 2001
Abstract
The bulk viscosity of neutron star cores containing hyperons
is studied taking into account
non-equilibrium weak process
.
The rapid growth of the bulk viscosity
within the neutron star core associated with
switching on new reactions (modified Urca process, direct
Urca process, hyperon reactions) is analyzed.
The suppression of the bulk viscosity by superfluidity of
baryons is considered and found out to be very
important.
Key words: stars: neutron - dense matter
The bulk viscosity of matter in the cores of neutron stars has recently attracted great attention in connection with damping of neutron star pulsations and gravitational radiation driven instabilities, particularly - in damping of r-modes (e.g., Andersson & Kokkotas 2001). It is well known that the bulk viscosity is caused by energy dissipation associated with weak-interaction non-equilibrium reactions in a pulsating dense matter. The reactions and the bulk viscosity itself depend sensitively on the composition of matter.
In the outermost part of the outer neutron star core
composed mainly of neutrons n with admixture
of protons p, electrons e, and possibly muons bulk viscosity is mainly determined by the reactions
of non-equilibrium modified Urca process,
Deeper in the core, at densities
of a few (
g cm^{-3} is the
saturated nuclear matter density), direct Urca process
may be open (Lattimer et al. 1991)
At about the same densities, hyperons may appear in
the neutron star cores (first of all,
and
hyperons, and then ,
,
).
To be specific, we will mainly consider
and hyperons. Once appeared,
the hyperons may also initiate their own
direct Urca processes (Prakash et al. 1992)
giving additional contribution to
the bulk viscosity, nearly as high as
that due to nucleon direct Urca process (2).
However, direct non-leptonic hyperon collisions
which go via weak interaction
(with strangeness non-conservation) such as
Calculation of the bulk viscosity limited by non-leptonic processes in hyperon matter is a complicated problem. There are a number of processes of comparable efficiency. The matrix elements can easily be calculated in the approximation of bare particles and exact SU(3) symmetry, and appear to be nonzero for some processes, e.g., (3), but are zero for the others, e.g., (4). However, experimental data on the lifetime of in massive hypernuclei indicate (e.g., Jones 2001b and references therein) that process (4) (with N = n) is nearly as efficient as "bare-particle'' process (3). Calculation of the matrix elements for "dressed'' particles is complicated and model dependent; additional complications arise - even in the in-vacuum case - due to the SU(3) symmetry breaking (Savage & Walden 1997).
Another complication is introduced by superfluidity of neutron-star matter. It is well known that neutrons, protons and other baryons may be superfluid due to attractive part of strong baryon-baryon interaction. Superfluidity of neutrons and protons has been studied in numerous papers (as reviewed, for instance, by Yakovlev et al. 1999; Lombardo & Schulze 2001). Hyperons can also be in superfluid state as discussed, e.g., by Balberg & Barnea (1998). Critical temperatures of baryon superfluidities are very sensitive to the model of strong interaction and to many-body theory employed in microscopic calculations. Their typical values range from 10^{8} to 10^{10} K. They are density dependent, and they mainly decrease with at densities higher than several .
The effects of superfluidity of nucleons on the bulk viscosity associated with direct and modified Urca processes in matter were considered in Papers I and II. It was shown that the superfluidity may drastically reduce the bulk viscosity and, hence, the damping of neutron star pulsations.
In this paper we propose a simple solvable model of the bulk viscosity in hyperonic matter (Sect. 2) due to process (3) and study (Sect. 3) the effects of possible superfluidity of n, p, and on this bulk viscosity. In Sect. 4 we discuss density and temperature dependence of the bulk viscosity in non-superfluid and superfluid neutron star cores.
Consider non-superfluid hyperonic stellar matter in the core of a neutron star pulsating with a typical frequency s^{-1}. In the presence of hyperons the contribution of direct Urca and modified Urca processes, (2) and (1), into the bulk viscosity may be neglected. It is sufficient to include the non-leptonic weak-interaction processes (3) and (4). For the sake of simplicity, let us take into account process (3) alone although we assume that matter may contain not only but other hyperons. The advantage of this model is that it can be solved analytically. We will compare it with other models in Sect. 2.4.
Let us start with the matrix element M in the
"bare-particle'' approximation. The process
is described by two diagrams with the states of two neutrons
interchanged. Accordingly,
,
and (
)
Using the standard technique in the limit of
non-relativistic baryons we sum |M|^{2} over particle
spin states and obtain
Due to very frequent interparticle collisions, dense stellar matter almost instantaneously (on microscopic time scales) achieves a quasiequilibrium state with certain temperature T and chemical potentials of various particle species i. Relaxation to the full thermodynamic ("chemical'') equilibrium lasts much longer since it realizes through much slower weak interaction processes.
In the case of process (3) the chemical equilibrium implies
.
In the chemical equilibrium
the rates
s^{-1}] of the direct and
inverse reactions of the process are balanced,
.
In a pulsating star, the chemical equilibrium is
violated (
)
which can be described by the lag
of instantaneous chemical potentials,
Let us calculate the rate
of the direct reaction,
,
of the process.
In the non-relativistic approximation
we have (
):
Evaluation of
is standard (e.g. Shapiro & Teukolsky 1983)
and takes advantage of strong degeneracy of reacting particles
in neutron star matter.
The multidimensional integral
is decomposed into the energy and
angular integrals. All momenta
are placed on the appropriate Fermi spheres
wherever possible.
Introducing the dimensionless quantities
The integral I, Eq. (12), is:
The bulk viscosity
due to
the hyperon process (3) is calculated in
analogy with that due to the modified
or direct Urca process (Sawyer 1989a;
Haensel & Schaeffer 1992). The result is
The bulk viscosity depends on the frequency
of neutron star pulsations.
Using the results of Sect. 2.3 the dynamical parameter a can be
written as
In the absence of hyperons the bulk viscosity is determined by direct or modified Urca processes (Sect. 1). These processes are much slower than hyperonic ones. They can certainly be described in the high-frequency approximation in which partial bulk viscosities due to various processes are summed together into the total bulk viscosity (e.g., Papers I and II). Thus we will add contributions from direct and modified Urca processes whenever necessary in our numerical examples in Sect. 4.
Note that all the studies of bulk viscosity of hyperonic matter performed so far are approximate. The subject was introduced by Langer & Cameron (1969) who estimated dumping of neutron star vibrations but did not calculate the bulk viscosity itself. Jones (1971,2001a) calculated effective hyperon relaxation times and estimated the bulk viscosity but did not evaluate it exactly for any selected model of dense matter. Recently Jones (2001b) analyzed the bulk viscosity of hyperonic matter taking into account a number of hyperonic processes but also restricted himself to the order-of-magnitude estimates.
Our approach is also simplified since we take into account the only one hyperonic process (3) and neglect the others. Even in this case we are forced to introduce the phenomenological parameter (Sect. 2.2) to describe the reaction rate. The advantage of our model is that, once this parameter is specified, we can easily calculate the bulk viscosity (as illustrated in Sect. 4) and introduce the effects of superfluidity (Sects. 3 and 4). Technically, it would be easy to incorporate the contribution of process (4) as well as of other hyperonic processes (Sect. 1). However, for any new process we need its own phenomenological parameter (similar to ) which is currently unknown. Generally, in the presence of several hyperonic processes, the bulk viscosity cannot be described by a simple analytical expression analogous to Eq. (17). Nevertheless, in the high-frequency limit the contributions from different processes are additive and it will be sufficient to add new contributions to that given by Eq. (20). Thus we prefer to use our simplified model rather than extend it introducing large uncertainties.
Now consider the effects of baryon superfluidity on the bulk viscosity associated with process (3). According to microscopic theories (reviewed, e.g., by Yakovlev et al. 1999 and Lombardo & Schulze 2001) at supranuclear densities (at which hyperons appear in dense matter) neutrons may undergo triplet-state (^{3}P_{2}) Cooper pairing while protons may undergo singlet-state (^{1}S_{0}) pairing. As discussed in Sect. 1 microscopic calculations of the nucleon gaps (critical temperatures) are very model dependent. Current knowledge of hyperon interaction in dense matter is poor and therefore microscopic theory of hyperon pairing is even much more uncertain. Since the number density of hyperons is typically not too large it is possible to expect that such a pairing, if available, is produced by singlet-state hyperon interaction. Some authors (e.g., Balberg & Barnea 1998) calculated singlet-state gaps for hyperons. We assume also singlet-state pairing of hyperons and consider the bulk viscosity of matter in which n, p and may form three superfluids. Since the critical temperatures , and are uncertain we will treat these temperatures as arbitrary parameters.
Microscopically, superfluidity introduces a gap
into momentum dependence of the baryon energy,
.
Near the Fermi surface
(
)
we have
It is convenient to introduce the dimensionless quantities
We consider the effects of superfluidity on the bulk viscosity
in the same manner as in Papers I and II and omit technical
details described in these papers.
Following Papers I and II we assume that all constituents of matter
participate in stellar pulsations with the same macroscopic
velocity (as in the first-sound waves). Then
the damping of pulsations is described by one coefficient of bulk viscosity .
The effects of superfluidity are included
by introducing superfluid gaps into the reaction rates,
and
,
Eq. (10),
through the dispersion relations, Eq. (21).
These effects influence mainly the only parameter in Eq. (17). Quite generally, we can write
Using Eqs. (17) and (25)
we can write the hyperon bulk viscosity in
superfluid matter in the form
In the high-frequency limit (Sect. 2.4),
which is often realized in neutron star matter,
we have
,
i.e.,
Under our assumptions superfluidity modifies only the integral in the factor
given by Eq. (16).
To generalize
to the superfluid case it is sufficient to
replace
in the all functions
under the integral in Eq. (12).
Then R can be written as
We have composed a code which calculates R numerically in the presence of all three superfluids. The results will be presented in Sect. 4. Here we mention some limiting cases in which evaluation of R is simplified.
The cases in which either protons or
hyperons are superfluid
are similar. Let, for example, neutrons and be normal while protons
undergo ^{1}S_{0} Cooper pairing.
Accordingly,
depends on the only parameter
.
For a strong superfluidity (
,
)
the asymptote is
If neutrons are normal but protons and
hyperons are
superfluid
depends on
and
.
We have determined
the asymptote of
at large
and .
Let Y be the larger gap,
,
and
.
At
the asymptote reads
Equation (31) becomes invalid at . In this case , where is described below.
Now let neutrons be superfluid while protons and
hyperons
not. For a strong superfluidity (
,
)
we get
Figure 1: Density dependence of partial bulk viscosities associated with various processes (indicated near the curves) at T=10^{9} K and s^{-1}in non-superfluid matter. Dotted and dashed lines refer to Urca processes involving electrons and muons, respectively; dot-and-dashed line refers to hyperon process (3). Thick solid line is the total bulk viscosity. | |
Open with DEXTER |
Figure 1 shows the partial bulk viscosities and the total bulk viscosity versus at T=10^{9} K for stellar vibration frequency s^{-1}. One can see three density intervals where the bulk viscosity is drastically different.
At low densities,
fm^{-3}, the bulk viscosity
is determined by modified Urca processes (Paper II). For
fm^{-3} it is produced by neutron and proton
branches of Urca process involving electrons
(processes (1) with
or p and with ).
At higher
muons are created and muonic modified Urca processes
(1) (again with
or p but now with )
introduce comparable contribution. Note that Eq. (29)
of Paper II for the angular integral
of the
proton branch of modified Urca process is actually valid at not
too high densities, as long as
.
For higher
densities, it is replaced with
At intermediate densities (0.227 fm fm^{-3}) the main contribution into the bulk viscosity comes from direct Urca processes (Paper I). As long as fm^{-3} the only one direct Urca process (2) operates with while at higher the other one with is switched on; it makes comparable contribution. We see that direct Urca processes at intermediate densities amplify the bulk viscosity by more than five orders of magnitude as compared to the low-density case.
Finally, at high densities ( fm^{-3}), according to the results of Sect. 2, the bulk viscosity increases further by about four orders of magnitude under the action of non-leptonic process (3) involving hyperons. These values of the bulk viscosity are in qualitative agreement with those reported by Jones (2001b). If our model of bulk viscosity were more developed and incorporated the contributions of process (4) involving hyperons then the high-density regime would start to operate at somewhat earlier density, at the hyperon threshold, fm^{-3}. The associated bulk viscosity is expected to be of nearly the same order of magnitude as produced by hyperons (Jones 2001b). Actually, in the presence of hyperons, some contribution into the bulk viscosity comes from modified and direct Urca processes involving hyperons (e.g., Prakash et al. 1992; Yakovlev et al. 2001). This contribution is not shown in Fig. 1. It is expected to be smaller than the contributions from nucleon modified and direct Urca processes (1) and (2) displayed in the figure.
Figure 1 refers to one value of
temperature, T=10^{9} K, and one value of the
vibration frequency,
s^{-1}.
Nevertheless one can easily rescale
to other T and
in non-superfluid matter
in the high-frequency regime.
Forthe modified
Urca processes (M), direct Urca processes (D),
and hyperonic process ()
we obtain
the estimates:
Figure 2: Schematic representation of temperature dependence of viscous relaxation time scale in non-superfluid neutron star cores with different compositions of matter at stellar vibration frequency s^{-1}. Three dashed lines show the relaxation due to high-frequency bulk viscosity associated either with modified Urca processes, or with direct Urca processes, or with hyperonic processes. The dotted line presents the relaxation due to shear viscosity. Solid lines refer to the total (bulk+shear) viscous relaxation for the three regimes. | |
Open with DEXTER |
Now we can
estimate viscous dissipation time scales
of
neutron star vibrations.
A standard estimate based on hydrodynamic momentum-diffusion
equation yields
,
where
km is a radius of the neutron
star core, and
is a typical density.
For the leading processes of three types in
the high-frequency regime we have
Great difference of possible bulk-viscosity scales is in striking contrast with the shear viscosity limited by interparticle collisions. The shear viscosity should be rather insensitive to composition of matter being of the same order of magnitude as in npe matter (Flowers & Itoh 1979), i.e., g cm^{-1} s^{-1}. It is independent of the pulsation frequency . The damping time of stellar pulsations via shear viscosity in a non-superfluid stellar core is yrs. It is shown in Fig. 2 by the dotted line. This damping dominates at low Twhile the damping by bulk viscosity dominates at higher T. The total viscous damping time ( ) is displayed in Fig. 2 by the solid lines (for the three high-frequency bulk-viscosity damping regimes). One can easily show that damping by bulk viscosity associated with modified Urca processes dominates at K. For direct Urca processes it dominates at K, and for hyperonic processes at K.
Finally, let us mention the validity of high-frequency bulk viscosity regime. As follows from Eq. (17) it is valid as long as , i.e., , where the threshold frequency . From Eq. (19) for the hyperon bulk viscosity we have s^{-1}. Using the results of Papers I and II we obtain s^{-1} and s^{-1} for modified and direct Urca processes. Therefore, we always have the high-frequency regime for modified and direct Urca processes at typical temperatures K and pulsation frequencies s^{-1}. The same is true for hyperon bulk viscosity excluding possibly the case of very hot plasma, K. Notice that in the low-frequency (static) limit and the temperature dependence of the bulk viscosity is inverted with respect to the high-frequency case.
As discussed in detail in Papers I and II superfluidity
of nucleons can strongly suppress the bulk viscosity
produced by direct and modified Urca processes.
Now let us use the results of Sect. 3 and illustrate
superfluid suppression of hyperon bulk viscosity.
Figure 3: Temperature dependence of bulk viscosity at fm^{-3}and s^{-1} in the presence of proton superfluidity with and neutron superfluidity with . Dot-and-dashed lines (from up to down): partial bulk viscosities due to hyperonic, direct Urca and modified Urca processes, respectively, in non-superfluid matter. Associated solid and dashed lines: the same bulk viscosities in superfluid matter. Vertical dotted lines show and . | |
Open with DEXTER |
Figure 3 shows this suppression
at
fm^{-3} and
s^{-1}. We present partial
bulk viscosities produced by hyperonic processes, as well as by
direct and modified Urca processes.
The straight dot-and-dashed lines are the partial bulk
viscosities in non-superfluid matter.
The striking difference of these
bulk viscosities is discussed in Sect. 4.1.
Solid and dashed lines show partial bulk viscosities
in matter with superfluid protons (
)
and neutrons (
). At
K
the high-frequency approximation for the
hyperon bulk viscosity is violated. One can see the tendency
of inversion of the temperature dependence
of
at
K
associated with the transition to
the low-frequency regime (Sect. 4.1).
At
superfluidity
reduces all partial bulk viscosities.
In the temperature range
,
where protons are superfluid alone,
all the three partial bulk viscosities are suppressed
in about the same manner. This is natural
(e.g., Yakovlev et al. 1999) since
the reactions responsible for the partial bulk viscosities
contain the same number of superfluid particles
(one proton). Indeed, there is one proton in hyperonic
reaction (3) and direct Urca reaction (2), as well as in the neutron
branch
of
modified Urca reactions (1).
At lower temperatures,
,
where neutrons
become superfluid in addition to protons, the suppression
is naturally stronger and becomes qualitatively different
for different partial bulk viscosities since the leading
reactions involve
different numbers of neutrons. Evidently, the suppression
is stronger for larger number of superfluid particles
(as well as
for higher critical temperatures ).
Figure 4: Same as in Fig. 2 but in the presence of proton, neutron and superfluids with , and . | |
Open with DEXTER |
Figure 4 exhibits the same temperature dependence of the partial bulk viscosities, as Fig. 3, but in the presence of superfluidity of n, p, and ( , , ). One can see that superfluidity of hyperons strongly reduces the partial bulk viscosity associated with hyperonic process. As a result, at K the total bulk viscosity is determined by direct Urca processes. In this regime one should generally take into account the contribution from direct Urca processes with hyperons (Sects. 1, 4.1). However, under the conditions displayed in Fig. 4 this contribution can be neglected.
Therefore, sufficiently strong superfluidity of baryons may reduce the high-frequency bulk viscosity by many orders of magnitude. This reduction will suppress very efficient viscous damping of neutron star pulsations in the presence of hyperons (Sect. 4.1). Accordingly, tuning critical temperatures of different baryon species one can obtain drastically different viscous relaxation times.
Note that relaxation in superfluid neutron star cores may also be produced by a specific mechanism of mutual friction (e.g., Alpar et al. 1984; Lindblom & Mendell 2000, and references therein). If the neutron star core is composed of n, p, e (and possibly ), this mechanism requires superfluidity of neutrons and protons, as well as rapid stellar rotation. The fact that the (conserved) particle currents are, in the case of a mixture of superfluids, not simply proportional to the superfluid velocities, implies non-dissipative drag (called also entrainment) of protons by neutrons. Dissipation (mutual friction) is caused by the scattering of electrons (and muons) off the magnetic field induced by proton drag within the neutron vortices. The relaxation (damping) time associated with mutual friction, , depends on the type of stellar pulsations and the physical conditions within the superfluid neutron star core, in particular - on the poorly known superfluid drag coefficient. Its typical value varies from s to s. One can expect that similar mechanisms may operate in the superfluid hyperon core of a rapidly rotating neutron star. If so these mechanisms will produce efficient damping of stellar pulsations. Note, however, that theoretical description of mutual friction is complicated and contains many uncertainties.
We have proposed a simple solvable model (Sect. 2) of the bulk viscosity of hyperonic matter in the neutron star cores as produced by process (3) involving hyperons. We have analyzed (Sect. 3) the hyperonic bulk viscosity in the presence of superfluids of neutrons, protons, and hyperons. We have presented illustrative examples (Sect. 4) of the bulk viscosity in non-superfluid and superfluid neutron star cores using the equation of state of matter proposed by Glendenning (1985). In particular, we emphasized the existence of three distinct layers of the core (outer, intermediate and inner ones), where the bulk viscosity in non-superfluid matter is very different (in agreement with the earlier results of Jones 1971,2001a,2001b). This leads to very different viscous damping times of neutron star vibrations for different neutron star models (the presence or absence of hyperons; the presence or absence of direct Urca process). If we used another equation of state of hyperonic matter the threshold densities of the appearance of muons and hyperons, and the fractions of various particles would be different but the principal conclusions would remain the same. As seen from the results of this paper and Papers I and II, the high-frequency bulk viscosities in all three layers may be strongly reduced by superfluidity of baryons. A strong superfluidity may smear out large difference of the bulk viscosities in different layers. In addition, it relaxes the conditions of the high-frequency regime.
Our consideration of the bulk viscosity in hyperonic matter is approximate since we include only one hyperonic process (3) (Sects. 1, 2.1, 2.4) characterized by one phenomenological constant . It would be interesting to undertake microscopic calculations of . Analogous problem of quenching the axial-vector constant of weak interaction in dense matter has been considered recently by Carter & Prakash (2001). It would also be important to determine analogous constants for other hyperonic reactions (e.g., for (4)) in the dressed-particle approximation. This would allow one to perform accurate microscopic calculations of the bulk viscosity of hyperonic matter.
In Sect. 4.1 we have presented simple estimates
of typical bulk viscosities and associated damping time scales
of neutron star vibrations in different non-superfluid
neutron star models. Let us stress that the
actual decrements
or increments of neutron star pulsations
have to be determined numerically by solving an appropriate
eigenvalue problem taking into account various
dissipation and amplification mechanisms
(e.g., bulk and shear viscosities;
mutual friction; gravitational radiation)
in all neutron star layers, proper boundary conditions,
etc. (e.g., Andersson & Kokkotas 2001).
In principle, the vibrational motion of various superfluids
may be partially decoupled. If so our
analysis of superfluid suppression of the bulk viscosity
must be modified (Sect. 3.2).
Nevertheless, the presented estimates
and the theory of superfluid suppression show
that one can reach drastically different conclusions
on the dynamical evolution of neutron star vibrations
by adopting different equations of state in the neutron
star cores (with hyperons or without), different
superfluid models and neutron stars models
(different central densities, allowing or forbidding
the appearance of hyperons and/or operation of direct
Urca processes). We expect that the results of this paper
combined with the results of Papers I and II will be useful
one to analyze this wealth of theoretical scenarios.
Note added at the final submission. After submitting this paper to publication we became aware of the paper by Lindblom & Owen (2001) devoted to the effects of hyperon bulk viscosity on neutron-star r-modes. The authors analyzed the bulk viscosity taking into account two hyperonic reactions, Eqs. (3) and (4), and superfluidity of and hyperons (but considering non-superfluid nucleons). Their treatment of the bulk viscosity in non-superfluid matter is more general than in the present paper since they include the reaction (4). They treat the superfluid effects using simplified reduction factors which is less accurate (a comparison of analogous exact and simplified reduction factors is discussed, for instance, by Yakovlev et al. 1999). The principal conclusions on the main properties of the bulk viscosity in hyperonic non-superfluid and superfluid matter are the same.
Acknowledgements
Two of the authors (KPL and DGY) acknowledge hospitality of N. Copernicus Astronomical Center in Warsaw. The authors are grateful to M. Gusakov who noticed a new form of the angular integral, Eq. (34). This work was supported in part by the RBRF (grant No. 99-02-18099), and the KBN (grant No. 5 P03D 020 20).