Finding Hidden Patterns in the Motions of Planet-Forming Disks

Measuring patterns rather than individual features

Protoplanetary disks are the birthplaces of planetary systems, but their gas motions can be difficult to decipher. After astronomers subtract a model of the disk’s background motion, the remaining velocity map may contain irregular patches rather than recognizable rings or spirals. Those residuals can mix physical structure, observational noise and imperfections in the background model.

In their preprint, Richard Teague and colleagues investigate the second-order structure function as a way to characterize these maps statistically. Their goal is to establish which properties can be recovered reliably, and where the method stops being informative.

The calculation averages squared velocity differences between pairs of locations at a chosen separation. If nearby locations have similar residual velocities, the result is small. As separation grows and the velocities become unrelated, it generally levels off. Comparing separations outward across the disk with separations around it reveals whether structures extend farther in one direction.

Separating signal from noise

The researchers first tested artificial random fields with known properties, including equal correlation lengths in both directions, lengths that grow with radius, and structures elongated around the disk. They measured amplitude, characteristic correlation lengths, their radial variation and directional elongation. They also fitted a Gaussian correlation model.

Crucially, the method removes an estimated noise contribution from the structure function, not from the original velocity map. The team generated ensembles of 1,000 noise realizations to construct that contribution and assess what noise alone could produce. Additional simulated signal-plus-noise fields estimated the scatter associated with sampling a finite number of structures.

Across the artificial noise conditions tested, excess velocity variance was detected in every case. Recovering its shape was harder. The clearest structural measurement was the correlation length around the disk in genuinely elongated fields. Radial changes in correlation length were not significantly distinguished from noise by the empirical measurements in those tests.

These results assume the noise model is correct. The artificial tests did not explore noise amplitudes greater than the intrinsic field amplitude, and they do not establish how reliably observers can infer the required noise properties from real data.

Three simulations, not three universal fingerprints

The authors then examined one simulation each of gravitational, magneto-rotational and vertical-shear instabilities. Their velocity fields contained, respectively, grand-design spirals, irregular spiral filaments and concentric ring-like features.

After adding noise and blurring the synthetic observations, the statistical shape measurements separated all three pairs. The reported separations were 8.4 standard deviations for gravitational versus magneto-rotational instability, 8.7 for magneto-rotational versus vertical-shear instability, and 4.1 for gravitational versus vertical-shear instability.

Those numbers describe these particular simulated fields under the adopted observing conditions. They are not evidence that the three mechanisms always have distinct statistical signatures. Only one snapshot per mechanism was used. Changing the noise model reduced the hardest pair’s separation from 4.1 to 2.8 standard deviations, illustrating the sensitivity of that comparison.

The Gaussian model also had important limitations. Coherent spirals and repeated rings can produce anticorrelations that its smoothly saturating structure function cannot reproduce. Allowing a spiral pitch angle did not solve that mismatch. For these simulations, the fitted shape parameters were not informative against the noise comparison; the simpler empirical measurements provided the useful distinctions.

What TW Hya reveals

For a real-data demonstration, the team analyzed carbon monoxide observations of the nearly face-on disk around TW Hya. They compared residual maps made with two background models: one retained a prominent large-scale feature, while the other removed the component described as m = 1.

Both maps showed significant correlated velocity structure. Across the disk radially, the characteristic width was about 95 milliarcseconds, corresponding to roughly six astronomical units at the adopted distance. Because that scale was close to the resolution limit, it provided an upper limit on the physical width rather than a well-resolved measurement.

Around the disk, the structure extended along resolved arcs. Removing the large-scale component shortened that coherence but did not eliminate the elongation. The removed component accounted for roughly half the variance above noise. The remaining pattern therefore was not simply the prominent feature visible by eye, but the analysis did not identify its physical cause.

The limits matter

Noise must undergo processing that reflects how the observations were made. In TW Hya, formal velocity-map uncertainties were not a reliable measure of pixel-to-pixel noise. More inclined disks introduce further complications from stretched resolution, uncertain geometry and projected motions.

Implemented in the eddy package, the framework offers a statistical description of disk velocity structure—not a unique diagnosis of turbulence or its driver.