Dynamical Stability of Imaged Planetary Systems in Formation: Application to HL Tau
Tamayo et al
We present a general and simple framework for understanding the dynamical stability of planets embedded in a protoplanetary nebula over typical disk lifetimes, and provide estimates for the maximum allowable planetary masses. We collect these easily evaluated dynamical constraints into a workflow that can help guide the design and interpretation of new observational campaigns and numerical simulations of gap opening in such systems. We argue that the locations of resonances should be significantly shifted from integer period ratios in massive disks like HL Tau, and that theoretical uncertainties in the exact shift, together with observational errors, imply a large uncertainty in the dynamical state and stability in such disks. This renders our results largely insensitive to an improved determination of the gaps' orbital radii, and presents an important barrier to using systems like HL Tau as a proxy for the initial conditions following planet formation. An important observational avenue to breaking this degeneracy is to search for eccentric gaps, which could implicate resonantly interacting planets. Unfortunately, massive disks like HL Tau should induce swift pericenter precession that would smear out any such eccentric features of planetary origin. This strongly motivates pushing toward more typical, less massive disks. For a nominal non-resonant model of the HL Tau system with five planets, we find a maximum mass for the outer three bodies of approximately 2 Neptune masses. In a resonant configuration, these planets can reach at least the mass of Saturn. The inner two planets' masses are unconstrained by dynamical stability arguments.