Following the optical imaging of exoplanet candidate Fomalhaut b ( Fom b ) , we present a numerical model of how Fomalhaut ’ s debris disk is gravitationally shaped by a single interior planet .
The model is simple , adaptable to other debris disks , and can be extended to accommodate multiple planets .
If Fom b is the dominant perturber of the belt , then to produce the observed disk morphology it must have a mass M _ { pl } < 3 M _ { J } , an orbital semimajor axis a _ { pl } > 101.5 { AU } , and an orbital eccentricity e _ { pl } = 0.11 –0.13 .
If the planet ’ s orbit is apsidally aligned with the belt ’ s , our model predicts M _ { pl } = 0.5 M _ { J } , a _ { pl } = 115 { AU } , and e _ { pl } = 0.12 .
These conclusions are independent of Fom b ’ s photometry .
To not disrupt the disk , a greater mass for Fom b demands a smaller orbit farther removed from the disk ; thus , future astrometric measurement of Fom b ’ s orbit , combined with our model of planet-disk interaction , can be used to determine the mass more precisely .
The inner edge of the debris disk at a \approx 133 { AU } lies at the periphery of Fom b ’ s chaotic zone , and the mean disk eccentricity of e \approx 0.11 is secularly forced by the planet , supporting predictions made prior to the discovery of Fom b .
However , previous mass constraints based on disk morphology rely on several oversimplifications .
We explain why our constraint is more reliable .
It is based on a global model of the disk that is not restricted to the planet ’ s chaotic zone boundary .
Moreover , we screen disk parent bodies for dynamical stability over the system age of \sim 100 Myr , and model them separately from their dust grain progeny ; the latter ’ s orbits are strongly affected by radiation pressure and their lifetimes are limited to \sim 0.1 Myr by destructive grain-grain collisions .
The single planet model predicts that planet and disk orbits be apsidally aligned .
Fomalhaut b ’ s nominal space velocity does not bear this out , but the astrometric uncertainties may be large .
If the apsidal misalignment proves real , our calculated upper mass limit of 3 M _ { J } still holds .
Parent bodies are evacuated from mean-motion resonances with Fom b ; these empty resonances are akin to the Kirkwood gaps opened by Jupiter .
The belt contains at least 3 M _ { \oplus } of solids that are grinding down to dust , their velocity dispersions stirred so strongly by Fom b that collisions are destructive .
Such a large mass in solids is consistent with Fom b having formed in situ .