On the Interpretation of Velocity Residuals in Protoplanetary Disks

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 $δh(r)$, inclination $δi(r)$, and position angle $δ\mathrm{PA}(r)$. 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 ($\sin3φ$ and $\cos3φ$). Moreover, we show that the radial profiles of $δh(r)$, $δi(r)$, and $δ\mathrm{PA}(r)$ 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.