Nonradial superfluid modes in oscillating neutron stars

For the first time nonradial oscillations of superfluid nonrotating stars are self-consistently studied at finite stellar temperatures. We apply a realistic equation of state and realistic density dependent model of critical temperature of neutron and proton superfluidity. In particular, we discuss three-layer configurations of a star with no neutron superfluidity at the centre and in the outer region of the core but with superfluid intermediate region. We show, that oscillation spectra contain a set of modes whose frequencies can be very sensitive to temperature variations. Fast temporal evolution of the pulsation spectrum in the course of neutron star cooling is also analysed.

💡 Research Summary

This paper presents the first self‑consistent study of non‑radial oscillations of superfluid (SFL) non‑rotating neutron stars (NSs) at finite stellar temperatures. Previous works on SFL NS oscillations have largely assumed zero temperature, employed simplified equations of state (EOS), and used toy models for nucleon superfluidity. In contrast, the authors adopt a realistic EOS (the APR model) and density‑dependent critical temperature profiles for neutrons (T_cⁿ) and protons (T_cᵖ) that are compatible with microscopic calculations and with observations of the cooling neutron star in the Cassiopeia A supernova remnant.

The theoretical framework builds on the relativistic two‑fluid formalism. The baryon current is split into normal and superfluid components, with four‑vectors w^μ(k) (k = n, p) describing the motion of the superfluid neutrons and protons relative to the normal fluid. The entrainment matrix Y_ik, which depends on temperature, couples the two superfluid species. By imposing electric charge neutrality and vanishing electric current, the proton superfluid velocity is expressed through the neutron one (Eq. 2). The coupling parameter s, which measures the interaction between normal and superfluid modes, is found to be small (|s| ≈ 0.02) for the APR EOS, allowing the authors to set s = 0 and treat the two families of modes as completely decoupled.

In the s = 0 limit the metric and pressure remain unperturbed for SFL modes, and the dynamics reduces to a single second‑order differential equation (Eq. 5) for the perturbed chemical‑potential imbalance δμ_l(r). This equation contains the temperature‑dependent parameter y(T/T_cⁿ, T/T_cᵖ) and the effective superfluid sound speed v_sf, both of which are functions of the red‑shifted temperature T_∞ (assumed constant throughout the core because of high thermal conductivity). Boundary conditions enforce regularity at the centre, vanishing superfluid current at the crust–core interface, and appropriate matching at the moving boundaries of the superfluid region when the temperature is such that only part of the core is superfluid.

The authors construct a 1.4 M_⊙, R = 12.2 km neutron‑star model using the APR EOS. The neutron critical temperature T_cⁿ(r) peaks at ≈5.1 × 10⁸ K around 0.75 R and drops toward the centre and crust, while the proton critical temperature T_cᵖ(r) is high (≈2 × 10⁹ K) and essentially makes protons superfluid throughout the core shortly after birth. As the star cools, the superfluid region evolves from a thin spherical shell (when T_∞ is just below the maximum of T_cⁿ) to a thick layer that eventually engulfs the centre when T_∞ falls below T_cⁿ(0) ≈ 2 × 10⁸ K.

Numerical solutions of Eq. 5 are obtained for multipolarities l = 0–3 and for radial node numbers n = 0–4. The eigenfrequencies ω_ln are presented in units of \tilde{ω}=c/R≈2.5 × 10⁴ s⁻¹. Several key trends emerge:

1. Temperature Sensitivity: For modes with n ≥ 1 the frequencies are remarkably insensitive to T_∞ over a broad range (10⁸–3 × 10⁸ K). This is because two opposing effects—expansion of the superfluid region (which tends to lower ω) and increase of v_sf as temperature drops (which raises ω)—almost cancel. In contrast, the fundamental n = 0 modes show a strong linear decrease of ω as T_∞ approaches the local neutron critical temperature; ω_l0 vanishes exactly at T_∞ = T_cⁿ(r) where the superfluid disappears.

2. Analytic Limits: At high temperatures (T_∞ ≈ 4 × 10⁸ K) the superfluid region is a thin shell, and the term h in Eq. 5 is dominated by its inverse. The eigenfunctions become proportional to Legendre polynomials P_n, yielding ω_ln ∝ n(n+1) and an almost l‑independent spectrum. At lower temperatures (T_∞ ≲ 3 × 10⁸ K) the eigenfunctions become nearly constant across the superfluid region, and the fundamental frequencies scale as ω_l0 ∝ l(l+1) v_sf/r_maxⁿ · (1 − T_∞/T_cⁿ), vanishing when T_∞ reaches the neutron critical temperature.

3. Temporal Evolution: By coupling the cooling history of a neutron star (using standard cooling curves) to the temperature‑dependent eigenfrequencies, the authors demonstrate that as the star ages from ≈10³ yr to ≈10⁴ yr the quadrupole (l = 2) superfluid mode frequencies shift from a few hundred Hz up to several kHz. The most dramatic changes occur when the superfluid region first reaches the centre (at age t₁), after which higher‑order modes (n > 1) experience a modest decline due to the vanishing of v_sf near the centre.

The paper emphasizes that the qualitative behavior of the spectrum—particularly the existence of temperature‑sensitive low‑order modes and temperature‑insensitive higher‑order modes—is robust against variations in the microphysical input, provided the maximum of T_cⁿ lies between the centre and the crust‑core boundary. This makes the results broadly applicable to realistic neutron‑star models.

In summary, the work delivers a comprehensive, relativistic treatment of superfluid non‑radial oscillations at finite temperature, incorporating realistic nuclear physics and thermal structure. It predicts a distinct set of superfluid modes whose frequencies can serve as probes of the internal temperature and superfluid gap profile of neutron stars. The strong temperature dependence of the fundamental modes suggests that future observations of quasi‑periodic oscillations, gravitational‑wave emission, or timing irregularities could be used to infer the onset and evolution of neutron superfluidity in cooling neutron stars.