On the Interpretation of Velocity Residuals in Protoplanetary Disks

TL;DR

This paper develops an analytical model to interpret velocity residuals in protoplanetary disks, enabling the extraction of disk surface and orientation perturbations from observational data.

Contribution

It introduces a first-order analytical framework that relates velocity residuals to disk surface height, inclination, and position angle perturbations, allowing their unique recovery from observations.

Findings

Velocity residuals exhibit Fourier components up to third harmonic.

Radial profiles of perturbations can be uniquely recovered.

Framework can be extended to other observables.

Abstract

We present a first-order analytical model for line-of-sight velocity residuals, defined as the difference between observed velocities and those predicted by a fiducial model, assuming a flared, nearly axisymmetric disk with the perturbations in disk surface height $\delta h(r)$, inclination $\delta i(r)$, and position angle $\delta\mathrm{PA}(r)$. Introducing projection-deprojection mapping between sky-plane and disk-frame coordinates, we demonstrate that the normalized velocity residuals exhibit Fourier components up to the third harmonic ($\sin3\phi$ and $\cos3\phi$). Moreover, we show that the radial profiles of $\delta h(r)$, $\delta i(r)$, and $\delta\mathrm{PA}(r)$ can be uniquely recovered from the data by solving a linear inverse problem. For comparison, we highlight factors that are not considered in previous models. We also outline how our framework can be extended beyond the first-order residuals and applied to additional observables, such as line intensities and widths.