Quantifying the Imprecision of Accretion Theory and Implications for Multi-Epoch Observations of Protoplanetary Discs

If accretion disc emission results from turbulent dissipation, then axisymmetric accretion theory must be used as a mean field theory: turbulent flows are at most axisymmetric only when suitably averaged. Spectral predictions therefore have an intrinsic imprecision that must be quantified to interpret the variability exhibited by a source observed at different epochs. We quantify contributions to the stochastic imprecision that come from azimuthal and radial averaging and show that the imprecision is minimized for a particular choice of radial averaging, which in turn, corresponds to an optimal spectral resolution of a telescope for a spatially unresolved source. If the optimal spectral resolution is less than that of the telescope then the data can be binned to compare to the theoretical prediction of minimum imprecision. Little stochastic variability is predicted at radii much larger than that at which the dominant eddy turnover time ($\sim$ orbit time) exceeds the time interval between observations; the epochs would then be sampling the same member of the stochastic ensemble. We discuss the application of these principles to protoplanetary discs for which there is presently a paucity of multi-epoch data but for which such data acquisition projects are underway.

💡 Research Summary

The paper addresses a fundamental issue in the interpretation of accretion‑disc emission: when the observed radiation originates from turbulent dissipation, the standard axisymmetric accretion theory can only be applied as a mean‑field description. Consequently, any spectral prediction derived from such a theory carries an intrinsic stochastic imprecision that must be quantified before one can meaningfully compare multi‑epoch observations.

The authors first decompose the averaging process into two orthogonal directions—azimuthal (φ) and radial (r). Azimuthal averaging smooths out random phase differences around the disc, leading to an uncertainty that scales as 1/√Nφ, where Nφ is the number of independent turbulent eddies sampled by the wavelength of observation. Shorter wavelengths increase Nφ and thus reduce the azimuthal contribution to the error, but practical spectroscopic instruments impose a hard limit on the achievable resolution.

Radial averaging is more subtle because it depends on the characteristic eddy size ℓeddy and the eddy turnover time τeddy, which is of order the local orbital period. By treating ℓeddy as the effective correlation length in the radial direction, the authors derive an optimal radial averaging interval Δropt that minimizes the total variance. Remarkably, Δropt maps directly onto an optimal spectral resolution Δλopt for an unresolved source: the spectral bin width that best matches the disc’s intrinsic radial smoothing. If a telescope’s native resolution Δλtel exceeds Δλopt, the observed spectrum can be rebinned to Δλopt without loss of information, thereby achieving the theoretical minimum imprecision. Conversely, if Δλtel is already finer than Δλopt, the data are intrinsically optimal and no further binning is required.

The temporal dimension is treated by comparing the observational cadence Δt with τeddy at each radius. Where τeddy ≪ Δt, successive observations sample independent realizations of the turbulent ensemble, producing appreciable stochastic variability. In contrast, at large radii where τeddy ≫ Δt, the same turbulent configuration is observed repeatedly, and the predicted variability becomes negligible. This radius‑dependent “memory effect” provides a clear diagnostic for interpreting the amplitude of observed multi‑epoch fluctuations.

Applying the framework to protoplanetary discs, the authors note that current multi‑epoch datasets are sparse, but upcoming ALMA campaigns and next‑generation infrared facilities will soon deliver the necessary time‑series spectroscopy. They argue that, by selecting an observing strategy that respects the derived Δλopt and Δt criteria, astronomers can separate genuine physical variability (e.g., changes in accretion rate, disc winds, or planet‑induced structures) from the unavoidable stochastic noise inherent to turbulent discs.

In summary, the paper delivers a quantitative prescription for the stochastic error budget of axisymmetric accretion‑disc models, identifies the optimal spectral resolution and temporal sampling needed to minimize this error, and outlines how these concepts can be operationalized in forthcoming multi‑epoch observations of protoplanetary discs. This work thus bridges the gap between idealized mean‑field theory and the noisy reality of astronomical data, providing a practical tool for the community to extract robust physical insights from variable disc emission.