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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 [79Jump To The Next Citation Point] (see also Chamel & Carter [94Jump To The Next Citation Point]). 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 [90Jump To The Next Citation Point]. 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 [76Jump To The Next Citation Point], 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 [93Jump To The Next Citation Point]. In Newtonian theory, the 4-velocities are defined by

μ dx μ uX = ----, (226 ) dt
t being the “universal” time. The components of the 4-velocity vectors have the form u0X = 1, u iX = viX 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 n0 = n X X, while the space components are those of the current 3-vector i i nX = nXvX (using Latin letters i,j,... for the space coordinate indices).

The basic variables of the two-fluid model considered here are the particle 4-current vectors nμc, nμf and the number density nN of nuclear clusters, which accounts for stratification effects. For clusters with mass number A, we have the relation nc = AnN. In the following we will neglect the small neutron-proton mass difference and we will write simply m for the nucleon mass (which can be taken as the atomic mass unit, for example). The total mass density is thus given by ρ = m (n + n ) c f. The Lagrangian density Λ, which contains the microphysics of the system, has been derived by Carter, Chamel & Haensel [79Jump To The Next Citation Point78].

The general dynamic equations (221View Equation) are given, in this case, by

nνfϖfνμ + πfμ∇ νnνf = ffμ, (227 )
ν c c ν c ncϖ νμ + π μ∇ νnc = fμ . (228 )
The time components of the 4-momentum co-vectors πX μ 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 X pi, defined by
X -∂Λ- pi = ∂ni . (229 ) X
In general, as a result of entrainment effects [20], the momentum co-vector pXi can be decomposed into a purely kinetic part and a chemical part,
pXi = mXvXi + χXi . (230 )
The chemical momentum X χi arises from interactions between the particles constituting the fluids, and is defined by
X ∂Λint χi = ---i-. (231 ) ∂n X
In this case, Λint is the internal contribution to the Lagrangian density defined by
Λ = Λ − Λ , (232 ) int kin
where
1 ∑ 2 Λkin = -- mXnXv X . (233 ) 2 X
According to the Galilean invariance, Λint can only depend on the relative velocities between the fluids, which implies the following Noether identity
∑ n χX γij = 0 , (234 ) X j X
where γij is the Euclidean space metric. Consequently, unlike the individual momenta (230View Equation), the total momentum density is simply given by the sum of the kinetic momenta
∑ n pXγij = ∑ n m vi . (235 ) X j X X X X X
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 nN. It can thus be written as N π μ = − μNtμ, where tμ = ∇ μt is the gradient of the universal time t and μN is a cluster potential, whose gradient leads to stratification effects. The dynamic equation of the nuclear clusters therefore reduces to

n ∇ μ − t ∇ (nν μ ) = fN , (236 ) N μ N μ ν N N μ
where nμN = nNu μc. The space components of the 4-force density co-vectors fμf, fcμ and f N μ 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 [76Jump To The Next Citation Point]. In the nondissipative model considered here, the total external force density co-vector vanishes:
f f+ fc + fN = 0. (237 ) ν ν ν
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 [94Jump To The Next Citation Point]. 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:

∇ μn μN = 0 , (238 )
∇ nμ + ∇ nμ = 0 . (239 ) μ f μ c
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
∇ μnμf = 0 . (240 )
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 [428Jump To The Next Citation Point] 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
𝒜 = 0 , (241 )
where the chemical affinity 𝒜 [76] is defined by the chemical potential difference in the crust rest frame
𝒜 = u ν(πf − πc ). (242 ) c ν ν

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