10.2 Two-fluid model of neutron star crust
In this section, we will review the simple model for neutron star crust developed by Carter, Chamel &
Haensel [79
] (see also Chamel & Carter [94
]). The crust is treated as a two-fluid mixture containing a
superfluid of free neutrons (index f) and a fluid of nucleons confined inside nuclear clusters
(index c), in a uniform background of degenerate relativistic electrons. This model includes
the effects of stratification (variation of the crust structure and composition with depth; see
Section 3) and allows for entrainment effects (Section 8.3.7) that have been shown to be very
strong [90
]. However, this model does not take into account either the elasticity of the crust
or magnetic fields. For simplicity, we will restrict ourselves to the case of zero temperature
and we will not consider dissipative processes (see, for instance, Carter & Chamel [76
], who
have discussed this issue in detail and have proposed a three-fluid model for hot neutron star
crust).
This model has been developed in the Newtonian framework, since relativistic effects are expected to be
small in crust layers, but using a 4D fully-covariant formulation in order to facilitate the link with
relativistic models of neutron star cores [93
]. In Newtonian theory, the 4-velocities are defined by
being the “universal” time. The components of the 4-velocity vectors have the form
,
in “Aristotelian” coordinates (representing the usual kind of 3+1 spacetime decomposition with
respect to the rest frame of the star). This means that the time components of the 4-currents are simply
equal to the corresponding particle number densities
, while the space components are
those of the current 3-vector
(using Latin letters
for the space coordinate
indices).
The basic variables of the two-fluid model considered here are the particle 4-current vectors
,
and the number density
of nuclear clusters, which accounts for stratification effects. For clusters with
mass number
, we have the relation
. In the following we will neglect the small
neutron-proton mass difference and we will write simply
for the nucleon mass (which can be taken as
the atomic mass unit, for example). The total mass density is thus given by
. The
Lagrangian density
, which contains the microphysics of the system, has been derived by Carter,
Chamel & Haensel [79
, 78].
The general dynamic equations (221) are given, in this case, by
The time components of the 4-momentum co-vectors
are interpretable as the opposite
of the chemical potentials of the corresponding species in the Aristotelian frame, while the
space components coincide with those of the usual 3-momentum co-vectors
, defined by
In general, as a result of entrainment effects [20], the momentum co-vector
can be decomposed into a
purely kinetic part and a chemical part,
The chemical momentum
arises from interactions between the particles constituting the fluids, and is
defined by
In this case,
is the internal contribution to the Lagrangian density defined by
where
According to the Galilean invariance,
can only depend on the relative velocities between the fluids,
which implies the following Noether identity
where
is the Euclidean space metric. Consequently, unlike the individual momenta (230), the total
momentum density is simply given by the sum of the kinetic momenta
Let us stress that entrainment is a nondissipative effect and is different from drag.
The cluster 4-momentum co-vector is purely timelike since the Lagrangian density depends only on
. It can thus be written as
, where
is the gradient of the universal time
and
is a cluster potential, whose gradient leads to stratification effects. The dynamic equation of the
nuclear clusters therefore reduces to
where
. The space components of the 4-force density co-vectors
,
and
coincide with those of the usual 3-force density co-vectors while the time components
are related to the rate of energy loss as discussed in more detail by Carter & Chamel [76
]. In
the nondissipative model considered here, the total external force density co-vector vanishes:
At this point, let us remark that, in general, the total force density co-vector may not vanish due to elastic
crustal stresses, as shown by Chamel & Carter [94
]. Moreover for a secular evolution of pulsars, it would
also be necessary to account for the external electromagnetic torque.
Both the cluster number and baryon number have to be conserved:
On a short time scale, relevant for pulsar glitches or high frequency oscillations, it can be assumed that the
composition of the crust remains frozen, i.e., the constituents are separately conserved so that we have the
additional condition
However, on a longer time scale, the free and confined nucleon currents may not be separately conserved
owing to electroweak interaction processes, which transform neutrons into protons and vice versa. The
relaxation times are strongly dependent on temperature [428
] and on superfluidity [414]. A more realistic
assumption in such cases is therefore to suppose that the system is in equilibrium, which can be expressed
by
where the chemical affinity
[76] is defined by the chemical potential difference in the crust rest frame