Let us remark that the definition of neutrons that have to be counted as “free” is not unique and there is some arbitrariness in the above model. Nonetheless, it can be shown that the 4-momentum co-vector of the neutron superfluid is invariant under such “chemical” readjustments and the above superfluidity conditions are well defined . Note also that these conditions are valid for both relativistic and nonrelativistic superfluids.
As discussed by Chamel & Carter , there are two cases, which are consistent with the nondissipative models considered here. The first possibility is that the neutron vortices are free and are co-moving with the superfluid, i.e. . On a sufficiently short time scale, it may be further assumed that the free neutron current is conserved, which implies from Equations (221) and (253) that the force vanishes. Since no external force is supposed to be exerted on the system, the force acting on the confined nucleons also vanishes. However, on longer time scales, as discussed in Section 10.2, it would be more appropriate to replace Equation (240) by the equilibrium condition Equation (241). In this case, there will still be a force acting on the superfluid (hence, also a force, acting on the confined nucleons) owing to the conversion of free neutrons into confined protons and vice versa. The other possibility is that the vortices are pinned to the crust, so that . As shown by Chamel & Carter , the pinning condition is equivalent to imposing that the individual vortices be subject to the corresponding Magnus force.
The dynamics of the neutron superfluid vortices in the crust play a major role in the understanding of pulsar glitches and post-glitch relaxation. Due to entrainment effects, the distribution of vortices is not simply given by Equation (163), but also depends on the angular velocity of the crust according to the following formula derived by Chamel & Carter :decrease the surface density of neutron vortices in the liquid core).
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