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.

We analyse the oscillations of general relativistic superfluid hyperon stars, following the approach suggested by Gusakov & Kantor and Gusakov et al. and generalizing it to the nucleon-hyperon matter. We show that the equations governing the oscillations can be split into two weakly coupled systems with the coupling parameters s e , s µ , and s str . The approximation s e = s µ = s str = 0 (decoupling approximation) allows one to drastically simplify the calculations of stellar oscillation spectra. An efficiency of the presented scheme is illustrated by the calculation of sound speeds in the nucleonhyperon matter composed of neutrons (n), protons (p), electrons (e), muons (µ), as well as Λ, Ξ − , and Ξ 0 -hyperons. However, the gravity oscillation modes (g-modes) cannot be treated within this approach, and we discuss them separately. For the first time we study the composition g-modes in superfluid hyperon stars with the npeµΛ core and show that there are two types of g-modes ('muonic' and 'Λ-hyperonic') in such stars. We also calculate the g-mode spectrum and find out that the eigenfrequencies ν of the superfluid g-modes can be exceptionally large (up to ν ≈ 742 Hz for a considered stellar model).

Buoyancy of proton vortices is considered as one of the important mechanisms of magnetic field expulsion from the superconducting interiors of neutron stars. Here we show that the generally accepted expression for the buoyancy force is not correct and should be modified. The correct expression is derived for both neutron and proton vortices. It is argued that this force is already contained in the coarse-grained hydrodynamics of Bekarevich & Khalatnikov and its various multifluid extensions, but is absent in the hydrodynamics of Hall. Some potentially interesting buoyancy-related effects are briefly discussed.

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We calculate the spectrum of inertial oscillation modes in a slowly rotating superfluid neutron star, including, for the first time, both the effects of finite temperatures and entrainment between superfluid neutrons and protons. We work in the Newtonian limit and assume minimal core composition (neutrons, protons and electrons). We also developed an approximate method that allows one to calculate the superfluid r-mode analytically. Finally, we derive and analyze dispersion relations for inertial modes in the superfluid NS matter in the short wavelength limit.