Inferring Planet Mass from Spiral Structures in Protoplanetary Disks

Recent observations of protoplanetary disk have reported spiral structures that are potential signatures of embedded planets, and modeling efforts have shown that a single planet can excite multiple spiral arms, in contrast to conventional disk-planet interaction theory. Using two and three-dimensional hydrodynamics simulations to perform a systematic parameter survey, we confirm the existence of multiple spiral arms in disks with a single planet, and discover a scaling relation between the azimuthal separation of the primary and secondary arm, $\phi_{\rm sep}$, and the planet-to-star mass ratio $q$: $\phi_{\rm sep} = 102^{\circ} (q/0.001)^{0.2}$ for companions between Neptune mass and 16 Jupiter masses around a 1 solar mass star, and $\phi_{\rm sep} = 180^{\circ}$ for brown dwarf mass companions. This relation is independent of the disk's temperature, and can be used to infer a planet's mass to within an accuracy of about 30% given only the morphology of a face-on disk. Combining hydrodynamics and Monte-Carlo radiative transfer calculations, we verify that our numerical measurements of $\phi_{\rm sep}$ are accurate representations of what would be measured in near-infrared scattered light images, such as those expected to be taken by Gemini/GPI, VLT/SPHERE, or Subaru/SCExAO in the future. Finally, we are able to infer, using our scaling relation, that the planet responsible for the spiral structure in SAO 206462 has a mass of about 6 Jupiter masses.