Establish whether resonant cavity modes exist and are unstable

Establish, using an analysis appropriate for algebraically growing responses such as Laplace transforms, whether the resonant cavity modes of reflexively forced accretion discs exist and whether they are unstable.

Background

The analytical treatment identifies frequencies at which the boundary conditions cannot be satisfied by ordinary Fourier solutions. The authors interpret these frequencies as possible resonant cavity modes whose disc response grows algebraically in time because the forcing repeatedly injects energy into waves of the same phase.

The feedback of the disc response on the central object's motion could potentially convert algebraic growth into exponential instability. However, the paper does not analyze these modes conclusively because the relevant problem requires Laplace transforms rather than the Fourier-mode framework used for most of the study.

References

Unfortunately, we cannot prove this convincingly. Indeed, to investigate algebraic instabilities, one needs to use Laplace transforms instead of Fourier transforms. This is why we cannot conclusively affirm that resonant modes exist, nor that they are unstable. We can only highlight their potential.

— Reflex instabilities II: The reflex motion of the central object drives at least seven instabilities in accretion discs  (2610.08472 - Magnan et al., 6 Oct 2026) in Section 4.1, “Resonant cavity modes”

The ${n = 2}$ mode, however, is always unstable with a growth rate that scales with $h{2}$. We pushed our exploration all the way down to \smash{${ h = 10{-6} }$} (where LAADS hits its machine precision limit) but could not find any hint of a collision in this mode's low-temperature history. We therefore conjecture that it is not a collisional mode.

— Reflex instabilities II: The reflex motion of the central object drives at least seven instabilities in accretion discs  (2610.08472 - Magnan et al., 6 Oct 2026) in Section 6.3, “Standard cavity modes”

Indeed, we did not explain the physical mechanism of some instabilities, we do not know how they respond to additional physics like magnetohydrodynamics or thermodynamics, we need to study whether and how they generalise to 3D, and we are yet to study their non-linear outcome.

— Reflex instabilities II: The reflex motion of the central object drives at least seven instabilities in accretion discs  (2610.08472 - Magnan et al., 6 Oct 2026) in Section 11, Discussion and Conclusion