Gap Clearing by Planets in a Collisional Debris Disk

We apply our 3D debris disk model, SMACK, to simulate a planet on a circular orbit near a ring of planetesimals that are experiencing destructive collisions. Previous simulations of a planet opening a gap in a collisionless debris disk have found that the width of the gap scales as the planet mass to the 2/7th power ($α$=2/7). We find that the gap sizes in a collisional disk still obey a power law scaling with planet mass, but that the index alpha of the power law depends on the age of the system t relative to the collisional timescale $t_{coll}$ of the disk by $α= 0.32(t/t_{coll})^{-0.04}$, with inferred planet masses up to five times smaller than those predicted by the classical gap law. The increased gap sizes likely stem from the interaction between the collisions and the mean motion resonances near the chaotic zone. We investigate the effects of the initial eccentricity distribution of the disk particles and find a negligible effect on the gap size at Jovian planet masses, since collisions tend to erase memory of the initial particle eccentricity distributions. Finally, we find that the presence of Trojan analogs is a potentially powerful diagnostic of planets in the mass range ~1-10 $M_{jup}$. We apply our model to place new upper limits on planets around Fomalhaut, HR 4796 A, HD 202628, HD 181327, and $β$ Pictoris.

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