On the Interpretation of Velocity Residuals in Protoplanetary Disks

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 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 ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>sin</mml:mi> <mml:mn>3</mml:mn> <mml:mi>ϕ</mml:mi> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>cos</mml:mi> <mml:mn>3</mml:mn> <mml:mi>ϕ</mml:mi> </mml:math> ). Moreover, we show that the radial profiles of δh ( r ), δi ( r ), and δ 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.