Gravitational wave asteroseismology of accreting neutron stars in a steady state

An accreting neutron star is potentially the gravitational wave source. In this study, we examine the gravitational wave frequencies from such an object in the steady state, adopting the Cowling approximation. We can derive the empirical relations independently of the mass accretion rate for the frequencies of the fundamental and 1st pressure modes multiplied by the stellar mass as a function of the stellar compactness, together with those for the 1st and 2nd gravity mode frequencies. So, once one simultaneously observes the fundamental (or 1st pressure) and gravity mode frequencies, one could constrain the neutron star mass and radius. In addition, we find that the luminosity can be well characterized by the mass accretion rate independently of the stellar mass and equation of state, if the direct Urca does not work inside the star. Since the luminosity from the neutron star with the direct Urca can deviate from this characterization, one could identify whether the direct Urca process works or not inside the star by observing the luminosity. Both information obtained from the gravitational waves and luminosity help us to understand the equation of state for neutron star matter.

💡 Research Summary

In this paper the authors investigate the gravitational‑wave (GW) asteroseismology of accreting neutron stars (NSs) that have reached a thermal steady state under continuous mass accretion. Using a one‑dimensional general‑relativistic stellar evolution code, they construct a series of NS models with two representative equations of state (EOS): the relatively stiff TM1e EOS and the softer Togashi EOS. For each EOS they consider three stellar masses (1.4, 1.8 and 2.1 M⊙) and a wide range of mass‑accretion rates (ṁ = 10⁻¹¹–10⁻⁷ M⊙ yr⁻¹). The thermal structure is obtained by solving the Tolman‑Oppenheimer‑Volkoff equations together with energy‑transport equations that include heating from accretion‑induced nuclear reactions, compressional heating, and several neutrino‑cooling processes (modified Urca, bremsstrahlung, pair creation, etc.). In addition, the direct Urca (DU) process is switched on for models whose central density exceeds the EOS‑dependent threshold (e.g., the 2.1 M⊙ TM1e model).

The authors first examine the surface luminosity L∞ as a function of ṁ. When DU is absent, L∞ is essentially independent of the stellar mass and EOS; it follows a simple quadratic fit in log ṁ:
log₁₀ L∞ = 49.4518 + 2.6307 log₁₀ ṁ + 0.097019 (log₁₀ ṁ)² (L∞ in erg s⁻¹, ṁ in M⊙ yr⁻¹).
If DU operates, the core cools efficiently, the temperature gradient in the crust changes, and L∞ deviates markedly from this relation. Hence, measuring L∞ together with an estimate of ṁ can reveal whether DU is active, providing indirect information about the EOS‑dependent critical mass for DU.

Next, the paper turns to oscillation modes. The Cowling approximation (fixed background metric) is employed to compute fluid eigenfrequencies for the fundamental (f) mode, the first pressure (p₁) mode, and the first two gravity (g₁, g₂) modes. The key result is that the products M f and M p₁ depend only on the compactness C = M/R, not on ṁ or the detailed thermal profile. Similarly, the g‑mode frequencies can be expressed as simple functions of C. This universality arises because the steady‑state temperature profile, while altered by ṁ, does not significantly affect the restoring forces governing f and p modes, and the buoyancy that sets g‑mode frequencies is primarily set by the composition gradient, which is fixed in the models.

Because the f (or p₁) frequency and a g‑mode frequency can be measured simultaneously in a GW signal, one can invert the empirical relations to obtain both M and R (or equivalently C and M). The authors demonstrate that the presence of DU does not appreciably shift the f or p₁ frequencies, but it does lower the g‑mode frequencies slightly due to the cooler core.

The paper therefore proposes a two‑pronged observational strategy: (i) detect GW signals from an accreting NS and identify at least one fluid mode of the f/p family together with a g‑mode; use the derived empirical relations to infer the star’s mass and radius, thereby constraining the EOS. (ii) Measure the steady‑state X‑ray/UV luminosity L∞ and the accretion rate (e.g., from the observed X‑ray flux and binary parameters). Compare L∞ with the universal ṁ–L∞ relation to test for the operation of the direct Urca process. If DU is active, the inferred critical mass provides an additional EOS constraint.

In summary, the study shows that accreting neutron stars in a thermal steady state constitute a promising continuous GW source whose oscillation spectrum, combined with electromagnetic luminosity measurements, can yield tight, complementary constraints on the dense‑matter EOS and on the presence of fast neutrino cooling (direct Urca). The work paves the way for future multi‑messenger observations and for more sophisticated calculations that relax the Cowling approximation and include superfluid effects.