On the Interpretation of Velocity Residuals in Protoplanetary Disks

TL;DR

This work develops a first-order analytical framework to interpret line-of-sight velocity residuals in flared, nearly axisymmetric protoplanetary disks by incorporating a rigorous projection-deprojection mapping and perturbations to disk height, inclination, and position angle. The residual field is shown to decompose into azimuthal harmonics up to the third order, enabling unique recovery of the radial profiles of $\delta h(r)$, $\delta i(r)$, and $\delta\mathrm{PA}(r)$ through a linear inverse problem. The authors validate the model against numerical calculations and demonstrate improvements over previous approaches that neglected projection effects and height perturbations. They also outline a Bayesian framework for joint inference across annuli, with closed-form posteriors and analytic hyperparameter marginalization, and discuss extensions to other observables and more complex physics. The methodology provides a robust pathway to diagnose warps and other geometric perturbations in disks, with potential to refine interpretations of ALMA and exoALMA velocity data.

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 $δh(r)$, 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.

Figures (4)

• Figure 1: Illustration of the coordinate system used in our model, as described in Section \ref{['sec:coordinate']}. The upper panels show face-on and observer's views of the disk surfaces, along with both the disk-plane coordinates $(l,m,n)$ and the sky-plane coordinates $(x,y,z)$. The lower panels present velocity fields for counterclockwise ($\epsilon=0$) and clockwise ($\epsilon=1$) rotations. We adopt $\mathrm{PA}_{\rm geo} = 7 \pi /4$.
• Figure 2: Sky-plane line-of-sight velocity (left) and residuals (right) from numerical calculations for a height perturbation of $\delta h(r)=0.1\,h(r)$.
• Figure 3: Comparison of line-of-sight velocity residuals from numerical calculation (left column) and analytical solution (right column) for three perturbations: $\delta h(r)=0.1\,h(r)$ (top row), $\delta i=1^\circ$ (middle row), and $\delta\mathrm{PA}=1^\circ$ (bottom row).
• Figure 4: (a-c) Radial profiles of the key Fourier coefficients in Equation (\ref{['eq:delta_final_clean']}) for perturbations in (a) disk height $\delta h(r)$, (b) inclination $\delta i$, and (c) position angle $\delta\mathrm{PA}$. (d) Fiducial disk surface height profile $h_0(r)$ (solid) and its radial derivative $h_0'(r)$ (dashed).