Bulk viscosity of superfluid neutron stars
Bulk viscosity of superfluid neutron stars
Mikhail E. Gusakov
Ioffe Physical Technical Institute, Politekhnicheskaya 26, 194021 Saint-Petersburg, Russia
(Dated:)
The hydrodynamics, describing dynamical effects in superfluid neutron stars, essentially differs
from the standard one-fluid hydrodynamics. In particular, we have four bulk viscosity coefficients in
the theory instead of one. In this paper we calculate these coefficients, for the first time, assuming
they are due to non-equilibrium beta-processes (such as modified or direct Urca process). The results
of our analysis are used to estimate characteristic damping times of sound waves in superfluid neutron
stars. It is demonstrated that all four bulk viscosity coefficients lead to comparable dissipation of
sound waves and should be considered on the same footing.
PACS numbers: 97.60.Jd, 26.60.+c, 47.37.+q, 47.75.+f
I. INTRODUCTION
The matter in pulsating neutron stars is not (even lo-
cally) in chemical equilibrium. Particles of different kinds
turn into one another so that the system evolves to equi-
librium. If a deviation from the equilibrium is small then
the processes of mutual transformations of particles can
be described in terms of an effective bulk viscosity (see,
e.g., Ref. [1]). This viscosity influences the ‘instability
windows’, that are the regions of physical parameters
(e.g., the rotation period and temperature of a star) at
which the neutron star becomes unstable against the ra-
diation of gravitational waves [2, 3, 4, 5]. The bulk vis-
cosity, generated by non-equilibrium processes of particle
transformations, was calculated in a series of papers for
neutron-star matter of various composition (for example,
for matter composed of neutrons, protons, and electrons
with an admixture of muons; for hyperon or quark mat-
ter). A short review and references to these papers can
be found in Ref. [6].
It is generally agreed that the stellar matter becomes
superfluid at a certain stage of neutron star thermal evo-
lution [7, 8, 9]. A lot of attention has been paid to the
question how superfluidity affects the bulk viscosity (see,
e.g., Refs. [10, 11, 12, 13, 14, 15]). In analogy with the or-
dinary hydrodynamics of non-superfluid liquid, the only
one ‘standard’ bulk viscosity coefficient has been calcu-
lated and analyzed in all these papers. Meanwhile, it is
well known [1, 16, 17] that a superfluid liquid, composed
of identical particles, is generally described by the four
bulk viscosity coefficients. So, what can be expected from
neutron stars, which contain a mixture of many super-
fluid species?
In this paper we show that non-equilibrium processes
of particle transformations lead to the appearance of at
least four bulk viscosity coefficients. Each of them is im-
portant for analyzing dissipative processes in superfluid
neutron stars. To be specific, we consider the simplest
model of stellar matter composed of neutrons, protons,
and electrons (npe-matter). In this case the bulk vis-
cosity is associated with the non-equilibrium direct or
modified Urca process.
The paper is organized as follows. In Sec. II we phe-
nomenologically obtain the general form of the dissipa-
tive corrections to the equations of relativistic hydrody-
namics [18, 19], describing a superfluid liquid composed
of identical particles. In Sec. III this dissipative hydro-
dynamics is generalized to describe superfluid mixtures
and applied to npe-matter. In Sec. IV we calculate and
analyse all four bulk viscosity coefficients provided by
non-equilibrium beta-processes. For illustration of these
results, in Sec. V we calculate the characteristic damping
times of sound waves in superfluid npe-matter. Sec. VI
presents summary.
II. THE DISSIPATIVE RELATIVISTIC
HYDRODYNAMICS OF ONE-COMPONENT
SUPERFLUID LIQUID
In this section we obtain the general form of dissipa-
tive terms entering the equations of relativistic superfluid
hydrodynamics of electrically neutral liquid composed of
identical particles. For that purpose, we need to choose
a version of non-dissipative hydrodynamics. There is a
number of equivalent formulations of non-dissipative rel-
ativistic superfluid hydrodynamics [18, 20, 21, 22, 23, 24].
The most elegant and general amongst them seems to be
the formulation of Carter [20, 21, 22] in which the hydro-
dynamic equations follow from a convective variational
principle. Most of the relativistic calculations (see Refs.
[25, 26, 27]) modelling pulsations of superfluid neutron
stars have been made within this approach (see, however,
[19]).
In this paper, we will not use the Carter’s hydrody-
namics because it is an essentially phenomenological the-
ory and it does not allow easy interpretation in terms of
quantities calculated from microscopic theory. Since our
main goal is the calculation of bulk viscosity coefficients,
we will employ the hydrodynamics of Son [18]. It was
initially proposed in the context of heavy-ion collisions
and is derived directly from microscopic theory. There-
fore, it is straightforward to relate various parameters of
this hydrodynamics to microphysics. Using the notations
of Ref. [19] it can be rewritten in a particularly simple
form, which is a natural relativistic generalization of the
http://arxiv.org/abs/0704.1071v2
standard non-relativistic superfluid hydrodynamics pio-
neered by Tisza [28], Landau [29, 30], and Khalatnikov
[31].
Although the Son’s description is ideal for comparing
with microphysics, it has some serious disadvantages. In
contrast to the Carter’s hydrodynamics, in which the ba-
sic fluid variables are the particle number density current
and the entropy density current, the Son’s hydrodynam-
ics is a hybrid in a sense that its fluid variables are the
rescaled entropy density current and the rescaled momen-
tum of a particle (or a Cooper pair) from the condensate
(in the literature it is traditionally and somewhat confus-
edly referred to as ‘the superfluid velocity’). As a con-
sequence, the Landau-type hydrodynamics of Son has a
lower symmetry than that of Carter. However, it can be
simply translated into the hydrodynamics of Carter as
was demonstrated, for example, in the review paper by
Andersson and Comer [32] (see their section 16.2).
Below we will use the hydrodynamics of Son [18], em-
ploying the notations of Ref. [19], convenient for our
problem.
Unless otherwise stated, the speed of light is set equal
to c = 1.
The hydrodynamic equations take the standard form
µ = 0, (1)
µν = 0, (2)
where ∂µ ≡ ∂/∂xµ (space-time indices are denoted by
Greek letters). Neglecting dissipation, the particle cur-
rent density jµ and the energy-momentum tensor T µν
can be presented in the form [19]
jµ = nuµ + Y wµ, (3)
T µν = (P + ε)uµuν + Pηµν
+Y (wµwν + µwµuν + µwνuµ) . (4)
Here n, P , ε, and µ (not to be confused with the space-
time index µ!) are the number density, pressure, energy
density, and relativistic chemical potential of particles,
respectively; ηµν = diag(−1,+1,+1,+1) is the special-
relativistic metric; Y is the relativistic analogue of su-
perfluid density ρs. In the non-relativistic limit, we have
Y = ρs/m
2, where m is the mass of a free particle. Fur-
ther, uµ is the four-velocity of normal (non-superfluid)
liquid component, normalized so that
µ = −1; (5)
wµ is the four-velocity, which describes the motion of
superfluid liquid component. It can be expressed through
some scalar function φ,
wµ = ∂µφ− µuµ. (6)
It is easy to verify that φ is related to the wave function
phase Φ of the Cooper-pair condensate by the equality
▽▽▽φ = (~/2) ▽▽▽Φ (see Ref. [19]). In the non-relativistic
limit spatial components of the four-vectors uµ and wµ
are equal to
uuu = VVV q, www = m(VVV s −VVV q), (7)
where VVV s = ▽▽▽φ/m and VVV q are, respectively, the su-
perfluid and normal velocities of the well known non-
relativistic theory of superfluid liquids (see, e.g., Ref.
[17]).
The four-velocity wµ must satisfy one additional con-
straint. This constraint fixes a comoving frame, that is
the frame where we measure (and define) all the thermo-
dynamic quantities. Choosing the constraint in the form
µ = 0, (8)
from Eqs. (3) and (4) one obtains that in this particu-
lar case comoving is the frame where four-velocity equals
uµ = (1, 0, 0, 0). It is straightforward to show that in this
frame the basic thermodynamic quantities n, ▽▽▽φ, and ε
are defined by
j0 = n, (9)
jjj = Y ▽▽▽φ, (10)
T 00 = ε. (11)
Using some equation of state and taking into account the
second law of thermodynamics (T is the temperature, S
is the entropy density)
dε = T dS + µ dn+
d (wµwµ) , (12)
as well as the definition of the pressure
P ≡ −ε+ µn+ TS, (13)
we can express all other thermodynamic quantities as
functions of ε, n, and▽▽▽φ. Eqs. (1)–(13) fully describe the
non-dissipative relativistic hydrodynamics of uncharged
superfluid liquid. As a consequence of these formulae, one
can easily derive the continuity equation for the entropy,
∂µ(Su
µ) = 0. (14)
Now let us include dissipation in the hydrodynam-
ics described above. As in the non-dissipative case,
we assume that in the comoving frame [in which
uµ = (1, 0, 0, 0)], the basic thermodynamic quantities
ε, n, and ▽▽▽φ are still defined by Eqs. (9)–(11). Being
written in a relativistically invariant form, the conditions
(9) and (10) imply that the particle current density jµ is
still given by Eq. (3), where some four-velocity wµ sat-
isfies the constraint (8) [as in the non-dissipative case].
In view of Eqs. (9) and (10), the most general expression
for the four-velocity wµ is
wµ ≡ ∂µφ− (µ+ κ)uµ. (15)
Here a scalar κ is a small dissipative correction to be
determined below. If we neglect dissipation, then κ = 0
and Eq. (15) coincides naturally with (6).
Without any loss of generality, the expression for the
energy-momentum tensor T µν can be presented in the
form:
T µν = (P + ε)uµuν + Pηµν
+Y (wµwν + µwµuν + µwνuµ) + τµν . (16)
Here τµν is an unknown dissipative tensor. In view of
Eqs. (8) and (11), τµν satisfies the constraint
uµuντ
µν = 0. (17)
Let us determine the dissipative corrections τµν and κ
assuming they are linear in small gradients of hydrody-
namic variables. For this aim we need to derive an en-
tropy generation equation. It can be easily obtained from
Eqs. (1)–(2) if we make use of Eqs. (3), (8), (12), (13),
(15), and (16),
µ = −
∂µ (Y w
µ)− τµν ∂µ
+Y wµ
∂µT + u
ν Y wµ
∂νuµ. (18)
Here the entropy current density Sµ is given by
Sµ = Suµ −
τµν −
Y wµ, (19)
and satisfies the natural constraint uµS
µ = −S.
Let us analyze the last two terms in Eq. (18). In ad-
dition to a quadratic dependence on small gradients of
hydrodynamic variables, they also depend on the four-
velocity wµ. As follows from Eq. (7), the spatial part of
the four-vector wµ is proportional to the difference be-
tween the superfluid and normal velocities. In a large
variety of problems this difference is small (as a conse-
quence, the time component w0 is also small because of
the constraint 8). In particular, it cannot exceed some
(not very large) critical value ∆VVV cr at which superfluidity
breaks down (see Sec. IV). For instance, if we study small
perturbations of matter which is initially at rest (in ther-
modynamic equilibrium with uµ = 0 and wµ = 0), then
the last two terms in Eq. (18) are much smaller than the
first two terms (a typical example of such situation is pro-
vided by sound waves in superfluid npe-matter, see Sec.
V). Below, we will neglect the last two terms in Eq. (18)
when obtaining the dissipative corrections τµν and κ.
Moreover, in the dissipative corrections we will also
neglect small dissipative terms, explicitly depending on
wµ. For example, we will neglect the terms of the form
wµ∂µT and u
µwν∂γu
γ in the expressions for κ and τµν ,
respectively. An inclusion of these small terms would
result in 13 kinetic coefficients describing dissipation in
superfluid liquid. In the non-relativistic case the same
approximation is used, for instance, in the textbook by
Landau and Lifshitz [1] (see their §140) and in the mono-
graph by Khalatnikov [17].
Since the entropy does not decrease, the right-hand
side of Eq. (18) must be positive. This requirement puts
certain restrictions on a general form of τµν and κ (linear
in gradients). The standard consideration (see, e.g., Ref.
[33]) shows that
τµν = −κ (Hµγ uν +Hνγ uµ)
∂γT + Tu
δ ∂δuγ
− η Hµγ Hνδ
∂δuγ + ∂γuδ −
ηγδ ∂εu
− ξ1 H
µν ∂γ (Y w
γ)− ξ2 H
µν ∂γu
γ , (20)
κ = −ξ3 ∂µ (Y w
µ)− ξ4 ∂µu
µ. (21)
In these equations Hµν ≡ ηµν + uµuν ; κ and η are, re-
spectively, the thermal conductivity and shear viscosity
coefficients; ξ1,. . .,ξ4 are the bulk viscosity coefficients.
From the Onsager symmetry principle it follows that
ξ1 = ξ4. (22)
For the positive definiteness of the quadratic form in the
right-hand side of Eq. (18) it is necessary to have the
kinetic coefficients κ, η, ξ2, and ξ3 positive and the coef-
ficient ξ1 satisfying the inequality
ξ21 ≤ ξ2ξ3. (23)
In the non-relativistic limit the dissipative hydrody-
namics proposed here coincides with the well known the-
ory of Khalatnikov (see, e.g., Refs. [1, 17, 34]). For illus-
tration, let us indicate how the bulk viscosity coefficients
ξKh1,. . .,ξKh4 of Khalatnikov are related to those intro-
duced in this paper. It is easy to demonstrate that
ξKh1 =
, ξKh2 = ξ2, (24)
ξKh3 =
, ξKh4 =
. (25)
Thus, in this section we have constructed the relativistic
dissipative hydrodynamics of a superfluid liquid, com-
posed of identical particles. It should be noted, that the
dissipation was first included into the hydrodynamics [18]
by Pujol and Davesne [35]. However, it is difficult to
use their dissipative hydrodynamics in applications. The
point is that the authors do not specify the comoving
frame, where they define thermodynamic quantities. It
is easy to verify that the frame which is defined as co-
moving in our paper, cannot serve as comoving in Ref.
[35].
III. VISCOSITY IN SUPERFLUID MIXTURES
Let us apply the general formulae obtained in Sec. II
to neutron star matter. As already mentioned in Sec.
I, we consider the simplest model of neutron star cores
composed of neutrons (n), protons (p), and electrons (e).
All results of this and following sections can be easily
generalized to the case of matter with more complicated
composition (e.g., npe-matter with admixture of muons
or hyperon matter).
It is generally agreed that as a neutron star cools down
the neutrons and protons become superfluid in its core.
In such a system we have three velocity fields (instead
of two, as in the previous section). They are superfluid
velocity of neutrons, superfluid velocity of protons, and
normal velocity uµ of neutron and proton Bogoliubov
quasiparticles and electrons. In this section we do not
consider the dissipative effects related to the diffusion of
particles. Neglecting the diffusion, nucleon Bogoliubov
excitations and electrons move with the same velocity uµ.
In analogy with Sec. II, it is convenient to introduce four-
velocities w
and w
instead of superfluid velocities of
neutrons and protons, respectively. The Son’s version of
non-dissipative hydrodynamics was extended to the case
of npe-mixture in Ref. [19] (for earlier formulations see,
e.g., Refs. [25, 36, 37, 38, 39, 40]). The main goal of this
section is to include the viscous dissipative terms into
the hydrodynamics [19]. Below the subscripts i and k
refer to nucleons, i, k = n, p. Unless otherwise stated,
the summation is assumed over repeated nucleon indices
i and k.
The full set of hydrodynamic equations describing su-
perfluid mixtures consists of i) energy-momentum con-
servation law (2) with the energy-momentum tensor T µν
given by
T µν = (P + ε)uµuν + Pηµν
wν(k) + µiw
uν + µk w
+ τµν ; (26)
ii) particle conservation laws written for neutrons, pro-
tons, and electrons (l = n, p, e),
= 0, j
= niu
µ + Yikw
= neu
µ; (27)
iii) constraints on the four-velocities w
= 0, (28)
and iv) the second law of thermodynamics,
dε = T dS + µi dni + µe dne +
w(k)µ
. (29)
To take into account potentiality of superfluid motion,
four-velocities w
should be expressed through some
scalar functions φi and presented in the form
= ∂µφi − qiA
µ − (µi + κi)u
µ. (30)
Note that one can avoid introduction of these new func-
tions φi in the hydrodynamics of superfluid mixtures
if one formulates the potentiality condition (30) in the
equivalent way
+ qiA
µ + (µi + κi)u
wν(i) + qiA
ν + (µi + κi)u
. (31)
Below we will use the latter formulation because it is
more suitable for our purpose. In this approach four-
velocities w
should be treated as independent hydro-
dynamic variables.
In Eqs. (26)–(31) µl and nl are, respectively, the rel-
ativistic chemical potential and the number density of
particle species l = n, p, e; Aµ is the four-potential of
the electromagnetic field; qi is the electric charge of nu-
cleon species i. Furthermore, Yik = Yki is a 2 × 2 sym-
metric matrix which naturally appears in the theory as
a generalization of the superfluid density to the case of
superfluid mixtures. In the non-relativistic limit this ma-
trix is related to the entrainment matrix ρik (see Refs.
[19, 41, 42, 43]) by the equality Yik = ρik/(mimk), where
mi is the mass of nucleon species i. The pressure P is
defined in the same way as for non-superfluid npe-matter
(compare with Eq. 13),
P ≡ −ε+ µini + µene + TS. (32)
The dissipative hydrodynamics formulated above differs
from the hydrodynamics [19], describing superfluid mix-
tures, only by the dissipative terms τµν , κn, and κp.
The general form of these terms can be found from the
entropy generation equation which is analogous to Eq.
(18) [see Sec. II],
µ = −
− τµν ∂µ
+Yikw
∂µT + u
ν Yikw
∂νuµ, (33)
where the entropy density current Sµ is
Sµ = Suµ −
τµν −
. (34)
Using the requirement that the entropy does not de-
crease, one can easily obtain the dissipative terms τµν ,
κn, and κp from Eq. (33),
τµν = −κ (Hµγ uν +Hνγ uµ)
∂γT + Tu
δ ∂δuγ
− η Hµγ Hνδ
∂δuγ + ∂γuδ −
ηγδ ∂εu
− ξ1i H
µν ∂γ
− ξ2 H
µν ∂γu
γ , (35)
κn = −ξ3i ∂µ
− ξ4n ∂µu
µ, (36)
κp = −ξ5i ∂µ
− ξ4p ∂µu
µ. (37)
Here, as in Sec. II, we omit small dissipative terms, ex-
plicitly depending on w
and, in addition, we neglect
particle diffusion. An inclusion of these terms would
result in 19 kinetic coefficients as it has been recently
demonstrated by Andersson and Comer [40] for the case
of npe-matter (they obtained their result using a non-
relativistic version of the Carter’s hydrodynamics).
In Eqs. (35)–(37) ξ1i, ξ2, ξ3i, ξ4i, and ξ5i are the bulk
viscosity coefficients (i = n, p). Some of them are related
by the Onsager symmetry principle,
ξ1i = ξ4i, ξ3p = ξ5n. (38)
In addition, for positive definiteness of the quadratic form
in the right-hand side of Eq. (33) one needs the following
inequalities
κ ≥ 0, η ≥ 0, ξ3n ≥ 0, ξ5p ≥ 0, ξ2 ≥ 0,
ξ5pξ2 ≥ ξ
1p, ξ3nξ2 ≥ ξ
1n, ξ3nξ5p ≥ ξ
2ξ1nξ1pξ3p + ξ2ξ3nξ5p − ξ
1pξ3n − ξ
3pξ2 − ξ
1nξ5p ≥ 0. (39)
Equations (33)–(39) are derived under the assump-
tion that electrons and protons can move independently.
However, this is not the case since they are charged. Any
macroscopic motion of electrons is accompanied by that
of protons to ensure quasineutrality condition (see, e.g.,
[19]),
ne = np. (40)
One can obtain then from the continuity equations (27)
for protons and electrons (neglecting small ‘diffusive’
terms),
= 0. (41)
In principle (if we are not interested in the distribution
of the electromagnetic field, which couples together elec-
trons and protons), we can use this equation instead of
the constraints (28) and (31) for protons.
Equations (33)–(39) should be modified to take into
account the conditions (40) and (41). As follows from Eq.
(41), the only bulk viscosity coefficients which contribute
to τµν and κn (see Eqs. 35 and 36) are ξ1n, ξ2, ξ3n, and
ξ4n. Since we neglect the last two terms in the right-
hand side of the entropy generation equation (33), only
these four coefficients are responsible for dissipation of
mechanical energy of macroscopic motion in superfluid
npe-matter.
IV. NON-EQUILIBRIUM BETA-PROCESSES
AND THE CALCULATION OF BULK
VISCOSITY
In Sec. III we phenomenologically considered npe-
matter and found the viscous terms in the relativistic
hydrodynamic equations for superfluid mixtures. When
doing this, we have ignored the fact that because of non-
equilibrium beta-processes, the number of particles in the
system is not conserved. In this section we demonstrate
that the effect of non-equilibrium beta-processes is equiv-
alent to the appearance of four effective bulk viscosity
coefficients ξ1n, ξ2, ξ3n, and ξ4n in the hydrodynamic
equations.
In the npe-matter we have two types of beta-processes
responsible for beta-equilibration. They are the direct
Urca process and the modified Urca process. The power-
ful direct Urca process is open only for some equations of
state with large symmetry energy (and at large enough
densities, when pFn ≤ pFp + pFe, pFn, pFp, and pFe being
the Fermi momenta of neutrons, protons, and electrons,
respectively). The direct and inverse reactions of this
process have the form (see, e.g., Refs. [7, 9, 44])
n → p + e + ν̄, p + e → n + ν. (42)
Here ν and ν̄ stand for neutrino and antineutrino, re-
spectively. If the direct Urca process is forbidden by
momentum conservation, then the main mechanism of
beta-equilibration is the modified Urca process [7, 9, 44]
N +n → N +p + e+ ν̄, N +p+ e → N +n+ ν. (43)
An additional nucleon N = n, p here is needed to take an
excess of momentum away and open the process.
The full thermodynamic equilibrium includes beta-
equilibrium and we have [44]
δµ ≡ µn − µp − µe = 0. (44)
In this case the number of direct reactions in a matter ele-
ment per unit time is equal to the number of inverse reac-
tions. Thus, the total number of particles of any species
remains constant. In particular, the electron generation
rate ∆Γ (that is the net number of electrons generated
in beta-reactions in a unit volume per unit time) is zero,
∆Γ = 0.
If we perturb the system, the condition (44) will not
necessarily hold (δµ 6= 0) and the direct and inverse reac-
tions will not precisely compensate each other (∆Γ 6= 0).
In this paper we assume that the deviation from the equi-
librium is small, δµ ≪ kBT . Then ∆Γ can be presented
in the form [10, 11, 12, 45, 46]
∆Γ = λ δµ. (45)
Here λ is a function of various thermodynamic quanti-
ties defined in the equilibrium state (e.g., of the temper-
ature and the particle number densities). For superfluid
npe-matter this function was calculated by Haensel, Lev-
enfish, and Yakovlev [10, 11]. Note, that these authors
neglected the dependence of ∆Γ on the scalars wα
w(k)α
though (in principle) they can be non-zero in thermody-
namic equilibrium. However, it seems that the results of
Refs. [10, 11] are accurate as long as (as an example, we
take the case of i = k = n)
wµ(n)w
∼ mn (VVV sn −VVV q)
≪ kBT, (46)
where VVV sn is the superfluid velocity of neutrons. A nu-
merical estimate of Eq. (46) gives
|∆VVV n| ≡ |VVV sn −VVV q| ≪ 0.01 c
. (47)
For clarity, we introduce the velocity of light c in this
condition. On the other hand, as follows from the Landau
criterion (see, e.g., Ref. [47]), superfluidity of neutrons
breaks down if |∆VVV n| > |∆VVV cr|, where
|∆VVV cr| ∼
kBTcn
≈ 0.0003 c
Here ∆n is the energy gap in the neutron dispersion re-
lation; Tcn is the critical temperature of neutron super-
fluidity onset; n0 = 0.16 fm
−3 is the number density of
nucleons in saturated nuclear matter. It is worth not-
ing that the criterion (48) is actually an upper limit on
|∆VVV cr|. In reality, a superfluid state can be destroyed
at much lower |∆VVV cr| due to the formation of vortices
in superfluid matter [16, 48]. Comparing Eqs. (47) and
(48) one can see that if the neutrons are superfluid then
at not very low temperatures the condition (47) is always
justified.
To calculate the bulk viscosity coefficients we will use
the non-dissipative hydrodynamics of superfluid mixtures
[19] (see also Sec. III). The energy-momentum tensor
T µν for such a hydrodynamics is given by Eq. (26), while
the four-velocity w
satisfies the conditions (28) and
(31). Notice, that the dissipative components τµν and
κi in Eqs. (26) and (31) should be taken zero, τ
µν = 0
and κi = 0. Further, we assume that the quasineutrality
condition (40) is fulfilled in both equilibrated and non-
equilibrated matter. Using Eq. (40), the second law of
thermodynamics for mixtures (29) can be rewritten in
the form:
dε = T dS + µndnb − δµ dne +
w(k)µ
. (49)
Finally, let us assume that the four-velocities w
= 0 in
equilibrium.
To take the non-equilibrium beta-processes into con-
sideration it is necessary to add corresponding sources
in the right-hand sides of the continuity equations for
electrons, protons, and neutrons,
∂µ (neu
µ) = ∆Γ, (50)
µ + Ypkw
= ∆Γ, (51)
µ + Ynkw
= −∆Γ. (52)
When writing Eqs. (50)–(52) we bear in mind that every
neutron decay is accompanied by the appearance of an
electron and a proton (see the reactions 42 and 43).
Taking into account the quasineutrality condition (40),
one gets from Eqs. (50) and (51) the equality (41). Thus,
Eq. (41) remains the same as in the absence of beta-
processes. It is more convenient to use the continuity
equation for baryons instead of Eqs. (51) and (52), be-
cause non-equilibrium beta-processes do not influence the
total number of baryons per unit volume, nb = nn + np.
Summing together Eqs. (51) and (52) and using Eq. (41),
one obtains
µ + Ynkw
= 0. (53)
Let us assume that npe-matter is slightly perturbed
out of thermodynamic equilibrium so that deviations
from the equilibrium are small and one can linearize
the hydrodynamic equations. Below we will work in
the comoving (at one particular moment) frame asso-
ciated with some element of npe-matter. In such a
frame uµ = (1, 0, 0, 0). We can assume further that
perturbations in the comoving frame depend on time t
as exp(iωct), where ωc is the frequency of perturbation
(measured in this frame).
From the normalization condition (5) and Eq. (28) it
follows that
w0(i) = 0, (54)
0 = 0, ∂tw
(i) = 0. (55)
Using these equalities, one gets from Eqs. (50) and (53)
∂tne + div (neuuu) = ∆Γ, (56)
∂tnb + div
nbuuu+ Ynkwww(k)
= 0. (57)
Here uuu and www(k) are the spatial components of four-
vectors uµ and w
, respectively. The number densi-
ties of electrons ne and baryons nb can be presented as
ne = ne0 + δne, nb = nb0 + δnb, where ne0 and nb0
are the equilibrium number densities, while δne and δnb
are small non-equilibrium terms depending on time as
exp(iωct). Here and hereafter the thermodynamic quan-
tities related to the equilibrium state will be denoted by
the subscript ‘0’. Using these notations as well as formula
(45) and linearizing Eqs. (56)–(57), we get
δne =
[λ δµ− ne0 div(uuu)] , (58)
δnb = −
nb0 div(uuu) + div
Ynkwww(k)
. (59)
Notice, that the chemical potential disbalance δµ in Eq.
(58) depends on δne and δnb. Actually, δµ can gener-
ally be presented as a function of nb, ne, T , and the
scalars wµ(i)w
(the proton number density is equal to
the electron one, see the quasineutrality condition 40).
One can neglect the temperature dependence of δµ in a
strongly degenerate npe-matter (see, e.g., Refs. [49, 50]).
Moreover, since the scalars wµ(i)w
are of the second
order smallness, their contribution to δµ is also negligi-
ble (we recall that w
= 0 in equilibrium). Expanding
δµ(nb, ne) in Taylor series in the vicinity of its equilib-
rium value (which is zero, δµ(nb0, ne0) = 0, see Eq. 44),
one obtains in the first approximation
δµ(nb, ne) =
∂δµ(nb0, ne0)
δnb +
∂δµ(nb0, ne0)
Analogous formulae can be written for perturbations of
pressure δP ≡ P (nb, ne) − P0, neutron chemical poten-
tial δµn ≡ µn(nb, ne) − µn0, and energy density δε ≡
ε(nb, ne)− ε0,
∂P (nb0, ne0)
δnb +
∂P (nb0, ne0)
δne, (61)
δµn =
∂µn(nb0, ne0)
δnb +
∂µn(nb0, ne0)
δne, (62)
∂ε(nb0, ne0)
δnb. (63)
In the last equation we have neglected the term of the
form [∂ε(nb0, ne0)/∂ne0] δne. From the second law of
thermodynamics (49) we have ∂ε(nb0, ne0)/∂ne0 = −δµ.
Therefore, this term is quadratically small and can be
omitted.
Using Eqs. (58)–(60) one finds
δne =
i ne0 ωc div(uuu) +
∂δµ(nb0, ne0)
nb0 λ div(uuu)
∂δµ(nb0, ne0)
λ div
Ynkwww(k)
, (64)
where F ≡ ω2c + i ωc λ∂δµ(nb0, ne0)/∂ne0. For non-
equilibrium Urca-processes and typical (for neutron
stars) pulsation frequencies ωc ∼ 10
3 − 104 s−1, we have
(see, e.g., Refs. [10, 11]) ωc ≫ λ |∂δµ(nb0, ne0)/∂ne0|. In
this case Eq. (64) can be simplified by keeping only terms
linear in λ. The result can be written as
δne = δne1 + δne2, (65)
where the first term equals
δne1 =
div(uuu) (66)
and describes compression and decompression of the pul-
sating matter. This term remains the same even in the
absence of non-equilibrium beta-processes. The second
term δne2 is due to non-equilibrium beta-processes
δne2 =
∂δµ(nb0, xe0)
div(uuu)
∂δµ(nb0, ne0)
Ynkwww(k)
. (67)
Notice, that in this formula the partial derivative
∂δµ(nb0, xe0)/∂nb0 is taken at constant value of xe0 ≡
ne0/nb0. When obtaining Eq. (67) we used the identity
∂Ψ(nb0, xe0)
= nb0
∂Ψ(nb0, ne0)
∂Ψ(nb0, ne0)
where Ψ is an arbitrary function of nb0 and ne0.
Our further strategy is as follows. We take the energy-
momentum tensor T µν for superfluid mixtures (with
τµν=0) from Eq. (26) and expand all the thermodynamic
quantities (e.g., the pressure P and the energy density
ε), which determine this tensor, around their equilib-
rium values. Restricting ourselves to linear perturbation
terms, we obtain for the tensor T µν (in the comoving
frame),
T 00 = ε0 + δε,
T 0j = T j0 = µi0Yik w
T jm = (P0 + δP ) δjm. (69)
Here, the spatial indices j and m are equal to 1, 2, 3; the
relativistic entrainment matrix Yik is taken in equilib-
rium; δP and δε are given by Eqs. (61) and (63), respec-
tively.
Let us assume for a while that there are no non-
equilibrium beta-processes in the matter. In this case
δne2 = 0 (see Eq. 67) and the matter is reversibly pulsat-
ing around the equilibrium. Then the mechanical energy
is not dissipating, and the entropy is conserved. Thus
it is obvious that dissipative are only those terms in the
tensor T µν which are directly related to δne2. Writing
out these terms in the form of a separate tensor τ
bulk, we
τ00bulk = 0,
bulk = τ
bulk = 0,
bulk =
∂P (nb0, ne0)
δne2 δjm. (70)
This tensor can be easily rewritten in an arbitrary frame
if we take into account Eq. (67),
bulk =
∂P (nb0, ne0)
∂δµ(nb0, ne0)
∂δµ(nb0, xe0)
. (71)
Comparing the tensor τ
bulk with the phenomenological
dissipative tensor τµν (see Eq. 35), we find the expres-
sions for the effective bulk viscosity coefficients ξ1n and
ξ2, generated by non-equilibrium beta-processes
ξ1n = −
∂P (nb0, ne0)
∂δµ(nb0, ne0)
, (72)
ξ2 = −
∂P (nb0, ne0)
∂δµ(nb0, xe0)
. (73)
Let us do the same with the potentiality condition (31)
on the four-velocity of neutrons w
. As a result, we
obtain in the comoving frame the dissipative component
κn, appearing because of non-equilibrium beta-processes,
∂µn(nb0, ne0)
δne2. (74)
In a fully covariant form, this component is given by Eq.
(36) where the effective bulk viscosity coefficients ξ3n and
ξ4n are
ξ3n = −
∂µn(nb0, ne0)
∂δµ(nb0, ne0)
, (75)
ξ4n = −
∂µn(nb0, ne0)
∂δµ(nb0, xe0)
, (76)
and, in addition, the condition (41) is taken into account.
Thus, we have calculated the four effective bulk viscos-
ity coefficients ξ1n, ξ2, ξ3n, and ξ4n. As will be shown in
Sec. V, each of them makes a comparable contribution to
characteristic damping times of mechanical energy. No-
tice, that only the coefficient ξ2 is usually analyzed in
the literature devoted to non-equilibrium beta-processes
in superfluid matter. The expression (73) for ξ2 coincides
with earlier results (see, e.g., Refs. [10, 11]).
Not all of the coefficients (72)–(73) and (75)–(76) are
independent. The coefficients ξ1n and ξ4n are equal be-
cause of the Onsager principle (38). This can be shown
if one applies the following relation for npe-matter (see,
e.g., Ref. [10]),
∂P (nb0, xe0)
= −n2b0
∂δµ(nb0, xe0)
. (77)
Furthermore, it is easy to verify, that instead of one of
the inequalities (39) relating the coefficients ξ1n, ξ2, and
ξ3n, we have the strict equality
ξ21n = ξ2ξ3n. (78)
It is fulfilled only for those non-equilibrium processes,
for which the expansion (65) is valid. Therefore, we have
only two independent bulk viscosity coefficients.
To prove Eq. (78) it is instructive to consider the en-
tropy generation equation. Neglecting all the dissipative
processes except for the non-equilibrium beta-processes
(e.g., neglecting thermal conductivity, diffusion, shear
viscosity), one can obtain from the hydrodynamics dis-
cussed in this section
T ∂µS
µ = δµ∆Γ = λ δµ2. (79)
Here we made use of Eq. (45). Since we are interested
only in terms linear in λ, we can substitute δne1 for δne
into Eq. (60) which determines δµ.
On the other hand, the entropy generation equation in
terms of the effective bulk viscosities takes the form (see
Eqs. 33, (35)–(37), and 41),
T ∂µS
µ = ξ3n
+(ξ1n + ξ4n) ∂µ
µ + ξ2 (∂µu
. (80)
Let us compare the right-hand sides of Eqs. (79) and
(80). It follows from Eqs. (60) and (79) that for any given
uµ it is always possible to choose four-velocities w
such a way, that δµ = 0 and the entropy generation rate
vanishes (at some point and at some particular moment).
In terms of the bulk viscosity formalism this means that
one can vanish the quadratic form in the right-hand side
of Eq. (80) by an appropriate choice of these velocities.
This is possible only if the equality (78) is satisfied.
V. DAMPING OF SOUND WAVES IN
SUPERFLUID NPE-MATTER
Let us illustrate the results of previous sections by
calculating characteristic damping times of sound waves
propagating in a homogeneous superfluid npe-matter.
For simplicity, we consider only the damping due to the
effective bulk viscosity. Neglecting dissipation, the sound
modes of superfluid npe-matter have been thoroughly in-
vestigated starting from the pioneering paper by Epstein
[51] in which he argued that there would be two types of
sound modes in neutron stars (see, e.g., Refs. [19, 38, 39]).
In particular, Gusakov and Andersson [19] were the first
who considered in full relativity sound modes in npe-
matter at finite temperatures. Here we closely follow
their analysis. The pulsation equations (81) and (82)
of Ref. [19] can be used to describe sound waves tak-
ing into account dissipation. Thus, there is no need to
derive these equations from the hydrodynamics of super-
fluid mixtures (Secs. III and IV) once again. Instead,
we will rewrite them using the notations adopted in our
paper. The result is
(P0 + ε0)uuu+ µn0Ynkwww(k)
= −▽▽▽δP, (81)
µn0uuu+www(n)
= −▽▽▽δµn. (82)
The first equation is a consequence of the relativistic Eu-
ler equation, which can be derived from Eq. (2) in a stan-
dard way (see, e.g., Ref. [1]). The second equation follows
from the condition (31) written for neutrons. To fully
define the system, Eqs. (81) and (82) should be supple-
mented by the condition (41). Using Eqs. (54) and (55),
this condition can be presented in the form
Ypkwww(k)
= 0. (83)
Now assuming that all the perturbations are plane waves
proportional to exp(iωt− ikkkrrr), one obtains the following
compatibility condition for Eqs. (81)–(83) [s = ω/k is the
velocity of sound in units of c]
y s4 + C1s
2 + C2 + δA = 0, (84)
where
Ypp nb0
µn0 (YnnYpp − YnpYpn)
− 1, (85)
µn0nb0
(β1 − γ1 − γ1y) + γ2 − β2
, (86)
µn0nb0
(β2γ1 − β1γ2) , (87)
∂P (nb0, xe0)
, γ2 =
∂µn(nb0, xe0)
, (88)
∂P (nb0, ne0)
, β2 =
∂µn(nb0, ne0)
. (89)
A small complex term δA appears in the compatibility
condition (84) because of the bulk viscosity. It is given
δA = −
µ2n0 nb0
A1 + s
, (90)
A1 = µn0nb0 γ2 ξ1n − µn0 β2 ξ2
−P0nb0 γ1 ξ3n + P0 β1 ξ4n, (91)
A2 = µn0
y ξ2 + ξ2 + n
b0 ξ3n − nb0 ξ1n − nb0 ξ4n
.(92)
We remind that the bulk viscosity coefficients (and the
quantities A1 and A2) depend on the frequency ω, A1,2 ∼
ω−2. The biquadratic equation (84) has two non-trivial
solutions for two possible sound velocities. Neglecting
dissipation, these modes have been analyzed in details in
Ref. [19]. In particular, the sound velocities s
1 and s
have been calculated there for the first and second modes.
The dissipation leads to the appearance of small complex
corrections δs1,2 to the velocities s
1,2 and consequently
to decrements of sound waves. Since δA is small in com-
parison with other terms in Eq. (84), one can use the
perturbation theory in deriving the characteristic damp-
ing times τ1,2. The parameters τ1 and τ2 are e-folding
times of the pulsation amplitude for the first and second
sound modes, respectively,
τ1,2 ≈
k δs1,2
2 i s
2 y s
1,2 + C1
. (93)
As follows from Eqs. (90)–(92), they are independent of ω.
At T → Tcn we have Ynn, Ynp, Ypn → 0 and y ≈
nb0/(µn0Ynn) → ∞ (see Ref. [19] for a more detailed dis-
cussion). In this limit the characteristic damping times
2P0 γ1
, (94)
τ2 ≈ −
2µn0P0 γ1 D
ω2 (γ1 A1 + µn0 D ξ2)
. (95)
Here we introduce the parameter D ≡ β2γ1 − β1γ2. At
T > Tcn the neutrons are non-superfluid. In this case the
second mode does not exist [formally, s
2 = 0], while the
first mode is the usual sound wave. The characteristic
FIG. 1: Characteristic damping times τ1,2 of sound waves
versus temperature T for the first mode (two upper curves)
and for the second mode (two lower curves). The solid curves
demonstrate the damping of sound taking into account all
four bulk viscosity coefficients. The dashed curves are calcu-
lated assuming that only ξ2 is non-zero. The neutron critical
temperature is indicated by the vertical dot-dashed line. The
baryon number density is nb = 3n0 = 0.48 fm
damping time for an ordinary sound wave is given by
Eq. (94). As expected, its damping is governed by the
only one bulk viscosity coefficient ξ2.
For illustration, in Fig. 1 we present the character-
istic damping times τ1,2 of sound waves (in years) as a
function of temperature T for two sound modes. The
figure is plotted for npe-matter with the baryon number
density nb0 = 3n0. The critical temperature of neutrons
is taken to be Tcn = 10
9 K. The protons are assumed
to be non-superfluid. When calculating thermodynamic
quantities and their derivatives we employed the equa-
tion of state from Ref. [52]. It opens the direct Urca
process at baryon number density of 5.84n0. Therefore,
the process is forbidden for nb0 = 3n0. In this case,
the main mechanism of energy dissipation is the non-
equilibrium modified Urca process. To calculate the func-
tion λ, which enters Eqs. (72)–(73) and (75)–(76) for the
bulk viscosity coefficients, we have used the results of Ref.
[11]. For calculating the relativistic entrainment matrix
Yik we have employed the BJ v6 nucleon-nucleon poten-
tial [19, 43, 53]. Actually, the microphysics input we use
here to plot the figure is taken from Ref. [19] (see this
reference for more details).
The two upper curves correspond to the first sound
mode, while the two lower curves – to the second mode.
The solid curves are plotted taking into account all four
bulk viscosity coefficients. The dashed curves are ob-
tained under the assumption that all coefficients but ξ2
are equal to zero, ξ1n = ξ3n = ξ4n ≡ 0.
As seen from the figure, the characteristic damping
times of sound waves increase as the temperature de-
creases. This is natural, because when the neutrons are
superfluid, the Urca processes (and hence the bulk vis-
cosity) are exponentially suppressed at T ≪ Tcn. Let
us emphasize that at temperatures T <∼ 5 × 10
8 K, the
shear viscosity of electrons can exceed the bulk viscosity
generated by the non-equilibrium modified Urca process.
As a result, the damping of sound waves will be mainly
due to the shear viscosity.
The first mode turns into the ordinary sound at T >
Tcn. As follows from the figure, in the vicinity of neutron
critical temperature the dissipation is primarily deter-
mined by the bulk viscosity coefficient ξ2 (in accordance
with Eq. 94). Consequently, near the transition point the
solid and the dashed curves for the first mode coincide.
On the contrary, the difference between the solid and the
dashed curves for the second mode remains significant at
any T < Tcn. The characteristic damping times for these
two curves differ approximately by a factor of 3.
It is worth noting that we would come to the similar
conclusions if we considered sound waves in denser mat-
ter, where the direct Urca process is open. In that case
the characteristic damping times would be 6–7 orders of
magnitude smaller, but the relative difference between
the solid and the dashed curves will be approximately
the same.
Therefore, the main result of the present section is
that the bulk viscosity coefficients ξ1n, ξ3n, and ξ4n es-
sentially influence the dissipative properties of superfluid
npe-matter and cannot be ignored. All four bulk viscos-
ity coefficients should be considered on the same footing.
VI. SUMMARY
We performed a self-consistent analysis of the influ-
ence of non-equilibrium beta-processes on dissipation of
mechanical energy in superfluid matter of neutron stars.
We start with the Son’s version of non-dissipative one-
fluid relativistic hydrodynamics to describe superfluid
mixtures (see Refs. [18, 19]). We determined the gen-
eral form of dissipative terms entering the equations of
this hydrodynamics. For simplicity, the effects of parti-
cle diffusion were ignored. The equations of dissipative
hydrodynamics were applied to the matter composed of
neutrons, protons, and electrons (npe-matter). In this
case the hydrodynamic equations contain four bulk vis-
cosity coefficients rather than one, as in non-superfluid
matter.
It was demonstrated, that non-equilibrium beta-
processes generate all four bulk viscosity coefficients, and
only two of them are independent. The other two coef-
ficients can be expressed through the first two by Eqs.
(38) and (78). It is worth to emphasize that only the
bulk viscosity coefficient ξ2 has been considered in the
astrophysical literature so far. The expression (73) for ξ2
coincides with similar expressions of previous works (see,
e.g., Refs. [10, 11, 45, 46]).
To illustrate the results obtained in the present paper
we considered a problem of damping of sound waves via
the bulk viscosity due to non-equilibrium beta-processes
in superfluid homogeneous npe-matter. It was shown
that all four bulk viscosity coefficients make compara-
ble contributions to the characteristic damping times of
sound waves.
Our results can be important for the analysis of vari-
ous gravitational-driven instabilities in neutron stars, in
particular, the r-mode instability (see, e.g., Ref. [4]). The
additional bulk viscosity coefficients lead to a more effec-
tive damping of these instabilities. Moreover, the results
can be applied to the problems of rotochemical and gravi-
tochemical heating of millisecond pulsars with superfluid
cores. In the absence of superfluidity these problems were
carefully analyzed in Refs. [49, 54, 55, 56]. The first at-
tempt to discuss qualitatively the effects of superfluidity
has been made in Ref. [57].
In conclusion let us note that the method of the bulk
viscosity calculation, used here in the simple case of npe-
matter, can be extended to matter with more compli-
cated composition (npe-matter with admixture of muons,
hyperon or quark matter). Such a generalization is be-
yond the scope of the present study and will be consid-
ered elsewhere.
Acknowledgments
The author is grateful to D.P. Barsukov and E.M. Kan-
tor for discussions, to A.I. Chugunov for technical assis-
tance, to D.G. Yakovlev for reading the manuscript and
critical comments, and to A. Reisenegger and anonymous
referee for very useful remarks. This research was sup-
ported by RFBR (grants 05-02-16245 and 05-02-22003)
and by the Federal Agency for Science and Innovations
(grant NSh 9879.2006.2).
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http://arxiv.org/abs/astro-ph/0012209
http://arxiv.org/abs/astro-ph/0607643
