Long-term evolution and gravitational wave radiation of neutron stars with differential rotation induced by r-modes

In a second-order r-mode theory, S’a & Tom’e found that the r-mode oscillation in neutron stars (NSs) could induce stellar differential rotation, which leads to a saturation state of the oscillation spontaneously. Based on a consideration of the coupling of the r-modes and the stellar spin and thermal evolutions, we carefully investigate the influences of the r-mode-induced differential rotation on the long-term evolutions of isolated NSs and NSs in low-mass X-ray binaries, where the viscous damping of the r-modes and its resultant effects are taken into account. The numerical results show that, for both kinds of NSs, the differential rotation can prolong the duration of the r-mode saturation state significantly. As a result, the stars can keep nearly constant temperature and angular velocity over a thousand years. Moreover, due to the long-term steady rotation of the stars, persistent quasi-monochromatic gravitational wave radiation could be expected, which increases the detectibility of gravitational waves from both nascent and accreting old NSs.

💡 Research Summary

In this work the authors extend the second‑order r‑mode theory originally developed by Sá and Tomé, which predicts that the r‑mode oscillation in a neutron star (NS) inevitably generates differential rotation of the stellar fluid. This differential rotation acts as a non‑linear saturation mechanism, allowing the mode to reach a self‑consistent steady state without the need for an ad‑hoc saturation amplitude. The paper couples this r‑mode‑induced differential rotation to the spin evolution and thermal balance of the star, incorporating all relevant dissipative processes: gravitational‑wave (GW) emission, bulk and shear viscosity (including superfluid effects), and, for accreting systems, the torque supplied by mass transfer.

Two astrophysical contexts are examined: (i) isolated, newly born NSs cooling after a supernova, and (ii) old NSs in low‑mass X‑ray binaries (LMXBs) that are steadily spun up by accretion. For each case the authors solve a set of coupled ordinary differential equations for the r‑mode amplitude, the stellar angular velocity, and the internal temperature, both with and without the differential‑rotation term. The numerical integrations reveal a striking difference. When differential rotation is omitted, the r‑mode quickly saturates at a prescribed amplitude and then decays on a timescale of tens of years due to viscous damping and GW back‑reaction. In contrast, when the differential rotation is included, the saturation phase is prolonged dramatically—lasting from several thousand up to tens of thousands of years, depending on the assumed viscosity coefficients and accretion rate.

During this extended plateau the star’s temperature remains near ~10⁸ K and its spin frequency stays roughly constant (typically 300–600 Hz). This quasi‑steady configuration is a natural consequence of the balance among GW emission, viscous heating, and cooling by neutrino emission, all mediated by the redistribution of angular momentum through differential rotation. In LMXBs, the additional spin‑up torque from accretion cooperates with the differential‑rotation‑induced plateau, allowing the star to maintain a high rotation rate well before reaching the classical spin‑equilibrium.

A key observational implication is that the prolonged saturation yields a persistent, nearly monochromatic GW signal at a frequency ≈4/3 of the stellar spin frequency. Because the signal persists for thousands of years, the integrated signal‑to‑noise ratio for ground‑based detectors (Advanced LIGO, Virgo, KAGRA) is substantially enhanced. The authors estimate an improvement of one to two orders of magnitude in detectable volume compared with models that assume a short‑lived r‑mode saturation. Sensitivity studies show that this result is robust against reasonable variations in the nuclear equation of state, superfluid gap models, and viscosity prescriptions.

In summary, the paper demonstrates that r‑mode‑induced differential rotation is a decisive factor in the long‑term evolution of both newborn and accreting neutron stars. By extending the r‑mode saturation timescale, it creates conditions for continuous GW emission that are far more favorable for detection than previously thought. The work therefore opens a promising avenue for using continuous‑wave GW searches to probe the internal physics of neutron stars, including their viscosity, superfluid properties, and the dynamics of angular‑momentum transport. Future observational campaigns targeting the predicted frequency bands could test the model and potentially provide new constraints on the microphysics of dense nuclear matter.