Roton instabilities in the superfluid outer core of neutron stars

We study the superfluid dynamics of the outer core of neutron stars by means of a generalized hydrodynamic model made of a neutronic superfluid and a protonic superconductor, coupled by both the dynamic entrainment and the Skyrme SLy4 nucleon-nucleon interactions. The resulting nonlinear equations of motion are probed in the search for dynamical instabilities triggered by the relative motion of the superfluids that could be of potential relevance to observational features of neutron stars. Through linear analysis, the origin and expected growth of the instabilities is explored for varying nuclear-matter density. Differently from previous findings, the dispersion of linear excitations in our model shows rotonic structures below the pair-breaking energy threshold, which lies at the origin of the dynamical instabilities and could eventually lead to emergent vorticity along with modulations of the superfluid density.

Figure 1

(a) Asymmetric density, (b) bulk chemical potential, and (c) particle current density as a function of the total density of nuclear matter in $\beta$ and electric-charge equilibrium, as given by the effective nuclear interaction SLy4. The chemical potentials are represented for static and dynamical equilibrium with relative velocity $2\left|\mathbf{V}\right|=0.2c$ between proton and neutron superfluids.

Figure 2

Linear excitation energies in a uniform state with total density $\rho =0.16{\mathrm{fm}}^{-3}$ and several relative superfluid velocities ${\mathbf{v}}_{pn}=2\left|\mathbf{V}\right|$. The dispersion contains only stable, real frequencies for $\left|\mathbf{V}\right|=\mathit{0}$ and $0.075c$ (top panel), whereas it includes complex frequencies (unstable excitations) for $\left|\mathbf{V}\right|=0.125c$ (middle and bottom panels). The dashed lines in the top panel trace the speed of sound of superfluid neutrons (higher slope) and superfluid protons (lower slope) when considered alone (see text). The insets show enlargements of the rotonic regions, below (top-panel inset) and beyond (middle-panel inset) the relative velocity threshold for the onset of instabilities.

Figure 3

(a) Rotonic structure in the dispersion of linear excitations of a uniform state with relative superfluid velocity ${\mathbf{v}}_{pn}=2\left|\mathbf{V}\right|=0.15c$. The roton minimum belongs to the lowest positive-energy branch and takes place at $k_r=0.31{\mathrm{fm}}^{-1}$. Simultaneously there exists a rotonic structure in the highest negative-energy branch that presents a maximum. (b) Range of unstable uniform states (yellow) in the presence of relative superfluid velocity ${\mathbf{v}}_{pn}=2\left|\mathbf{V}\right|$. The region encircled by the continuous line indicates the presence of unstable modes with wave number $k=0$. The staircase boundary lines are due to numerical resolution.

Figure 4

Excitation modes of constant density states with nuclear density $\rho =0.16{\mathrm{fm}}^{-3}$, and relative velocity (a) $\left|\mathbf{V}\right|=\mathit{0}$, top panels, and (b) $\left|\mathbf{V}\right|=0.075c$, bottom panels, corresponding to the dispersion relations depicted in the top panel of Fig. 2. The label ${\omega }_l$ (${\omega }_h$) indicates the correspondence of the mode branch with the lowest-energy (highest-energy) branches of the dispersion; analogously, the label ${\omega }_r$ refers to the rotonic branch of the dispersion. The continuation of this latter branch in the velocity component panels at high $k>1.2$, at high positive values $\delta {\mathbf{v}}_{\mathbf{p}}/\delta {\mathbf{v}}_{\mathbf{n}}>0$, is not captured by the graph scale. Since the systems are dynamically stable, all modes are chosen to have only real components. The insets zoom in to the low wave number range in order to see the in- and out-of-phase character between neutron and proton modes.

Figure 5

Same as Fig. 4 for an unstable stationary state at total density $\rho =0.16{\mathrm{fm}}^{-3}$ and relative superfluid motion with $\left|\mathbf{V}\right|=0.125c$. The represented modes correspond to the dispersion relations shown in the middle and bottom panels of Fig. 2. Inside the region of unstable wave numbers $k\in [0.16,0.49]{\mathrm{fm}}^{-1}$, the modes are described by complex components. The main panels depict the real part of the mode components versus wave number, whereas the two insets within each main panel represent their real (top inset) and imaginary (bottom inset) parts.