Tkachenko modes in rotating neutron stars: The effect of compressibility and implications for pulsar timing noise

Long wavelength oscillations (Tkachenko waves) of the triangular lattice of quantized vortices in superfluid neutron stars have been suggested as one of the possible explanations for the timing noise observed in many radio pulsars, in particular, for the 100-1000 day variations in the spin of PSR 1828-11. Most studies to date have, however, been based on the hydrodynamics developed for superfluid Helium. In this paper we extend the formulation to a two-fluid neutron and proton system, relevant for neutron star interiors and include the effect of chemical coupling, compressibility and mutual friction between the components. In particular we find that chemical coupling and compressibility can have a drastic effect on the mode structure. However, for the slower pulsars rotating at 1-10 Hz (such as PSR B1828-11), most choices of parameters in the equation of state lead to Tkachenko oscillations with frequencies in the correct range to explain the timing noise. We also investigate the case of more rapidly rotating pulsars (above 100 Hz) for which we find that there is a vast portion of parameter space in which there are no Tkachenko modes, but only modified sound waves at much higher frequencies.