Relativistic dynamics of superfluid-superconducting mixtures in the presence of topological defects and an electromagnetic field with application to neutron stars

Abstract

The relativistic dynamic equations are derived for a superfluid-superconducting mixture coupled to the electromagnetic field. For definiteness, and bearing in mind possible applications of our results to neutron stars, it is assumed that the mixture is composed of superfluid neutrons, superconducting protons, and normal electrons. Proton superconductivity of both I and II types is analysed, and possible presence of neutron and proton vortices (or magnetic domains in the case of type-I proton superconductivity) is allowed for. The derived equations neglect all dissipative effects except for the mutual friction dissipation and are valid for arbitrary temperatures (i.e. they do not imply that all nucleons are paired), which is especially important for magnetar conditions. It is demonstrated that these general equations can be substantially simplified for typical neutron stars, for which a kind of magnetohydrodynamic approximation is justified. Our results are compared to the nonrelativistic formulations existing in the literature and a number of discrepancies are found. In particular, it is shown that, generally, the electric displacement D D D does not coincide with the electric field E E E, contrary to what is stated in the previous works. The relativistic framework developed here is easily extendable to account for more sophisticated microphysics models and it provides the necessary basis for realistic modelling of neutron stars.

On the magnetic field evolution time-scale in superconducting neutron star cores

“…Surprisingly, although the momentum equations given by Glampedakis et al (2011) and Gusakov & Dommes (2016) are equivalent, Dommes & Gusakov (2017) find that their evolution timescales differ by several orders of magnitude from the results of Glampedakis et al (2011). We now discuss the cause of this discrepancy.…”

Evolution of the magnetic field in neutron stars

“…where n n,th ≡ n n −µ n Y nn is the number density of (normal) neutron thermal excitations and Y nn is the nn component of the relativistic entrainment matrix [42][43][44][45][46] (all other components of this matrix vanish when protons are normal).…”

“…(55) describes the motion of the superfluid neutron component (see, e.g., Refs. [42,45,46]). As in Sec.…”

“…where H H H c1 is the vector directed along B B B, whose absolute value equals the lower critical magnetic field for a simplified model of non-interacting proton vortices [46,47]. 8 This equation can be easily solved [40] for ∆µ e and δµ ∞ n , similar to how it is done in Sec.…”

“…The complex dynamic equations describing these effects have been (partly) formulated in Refs. [46,47]; full account is given in Ref. [48].…”