Planet-driven spirals in protoplanetary discs: Limitations of the semi-analytical theory for observations

¹ Univ. Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France
² Université Côte d’Azur, Observatoire de la Côte d’Azur, CNRS, Laboratoire Lagrange, France
e-mail: daniele.fasano@oca.eu
³ Dipartimento di Fisica, Università degli Studi di Milano, Via Celoria 16, Milano, Italy
⁴ School of Physics and Astronomy, Monash University, Clayton, Vic 3800, Australia
⁵ European Southern Observatory (ESO), Karl-Schwarzschild-Strasse 2, 85748 Garching bei Munchen, Germany
⁶ INAF, Osservatorio Astrofisico di Arcetri, 50125 Firenze, Italy
⁷ Leiden Observatory, Leiden University, PO Box 9513, 2300 RA Leiden, The Netherlands
⁸ Department of Astronomy, University of Florida, Gainesville, FL 32611, USA

Received: 22 March 2024
Accepted: 27 May 2024

Abstract

Context. Detecting protoplanets during their formation stage is an important but elusive goal of modern astronomy. Kinematic detections via the spiral wakes in the gaseous disc are a promising avenue to achieve this goal.

Aims. We aim to test the applicability of a commonly used semi-analytical model for planet-induced spiral waves to observations in the low and intermediate planet mass regimes. In contrast to previous works that proposed using the semi-analytical model to interpret observations, in this study we analyse for the first time both the structure of the velocity and density perturbations. Methods. We ran a set of FARGO3D hydrodynamic simulations and compared them with the output of the semi-analytic model in the code WAKEFLOW. We divided the disc into two regions. We used the density and velocity fields from the simulation in the linear region, where density waves are excited. In the non-linear region, where density waves propagate through the disc, we then solved Burgers’ equation to obtain the density field, from which we computed the velocity field.

Results. We find that the velocity field derived from the analytic theory is discontinuous at the interface between the linear and nonlinear regions. After ~0.2 r_p from the planet, the behaviour of the velocity field closely follows that of the density perturbations. In the low mass limit, the analytical model is in qualitative agreement with the simulations, although it underestimates the azimuthal width and the amplitude of the perturbations, predicting a stronger decay but a slower azimuthal advance of the shock fronts. In the intermediate regime, the discrepancy increases, resulting in a different pitch angle between the spirals of the simulations and the analytic model.

Conclusions. The implementation of a fitting procedure based on the minimisation of intensity residuals is bound to fail due to the deviation in pitch angle between the analytic model and the simulations. In order to apply this model to observations, it needs to be revisited so that it can also account for higher planet masses.