Abstract
We present a geometrical derivation of the relativistic dynamics of the superfluid inner crust of a neutron star. The resulting model is analogous to the Hall-Vinen-Bekarevich-Khalatnikov hydrodynamics for a single-component superfluid at finite temperature, but particular attention should be paid to the fact that some fraction of the neutrons is locked to the motion of the protons in nuclei. This gives rise to an ambiguity in the definition of the two currents (the normal and the superfluid one) on which the model is built, a problem that manifests itself as a chemical gauge freedom of the theory. To ensure chemical gauge covariance of the hydrodynamic model, the phenomenological equation of motion for a quantized vortex should contain an extra transverse force, that is the relativistic version of the Iordanskii force discussed in the context of superfluid Helium. Hence, we extend the mutual friction model of Langlois et al. (1998) to account for the possible presence of this Iordanskii-like force. Furthermore, we propose that a better understanding of the (still not completely settled) controversy around the presence of the Iordanskii force in superfluid Helium, as well as in neutron stars, may be achieved by considering that the different incompatible results present in the literature pertain to two, opposite, dynamical regimes of the fluid system.
Highlights
The presence of superfluidity has a significant impact on the behaviour and evolution of neutron stars, both on short timescales and on secular timescales
We have analyzed some formal aspects of the effective two-fluid hydrodynamic description of the inner crust of a neutron star initiated by Langlois et al [8], which is based on the simplifying assumptions of absence of elastic stresses, viscosity and heat conduction
As a consequence of these assumptions, it is easy to establish a connection between this relativistic two-fluid effective description and the relativistic HVBK hydrodynamics for a single superfluid at finite temperature of Gusakov [10]
Summary
The presence of superfluidity has a significant impact on the behaviour and evolution of neutron stars, both on short timescales (where superfluidity leads to additional modes of oscillation, e.g., [1,2,3]) and on secular timescales (where superfluidity affects the nuclear reactions responsible for neutrino cooling [4]). We allow for a special feature of the inner crust layers, namely the clustering of some nucleons into ions, which gives rise to a fundamental ambiguity on the number of free neutrons, as well as on the number of confined nucleons that behave as an effectively normal fluid in the inner crust This ambiguity was discussed in depth by Carter et al [18], and takes the form of a “chemical gauge freedom” of the macroscopic hydrodynamic model, see [19]. Throughout the paper we adopt the spacetime signature (−, +, +, +), choose units with the speed of light c = 1 and Newton’s constant G = 1, use greek letters ν, ρ, σ.
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