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 [79]. Note also that these conditions are valid for both relativistic and
nonrelativistic superfluids.
As discussed by Chamel & Carter [94], 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 [94
], 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 [94
]:
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