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 framehttp://www.livingreviews.org/lrr-2008-10 |
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