How empty are disk gaps opened by giant planets?

Gap clearing by giant planets has been proposed to explain the optically thin cavities observed in many protoplanetary disks. How much material remains in the gap determines not only how detectable young planets are in their birth environments, but also how strong co-rotation torques are, which impacts how planets can survive fast orbital migration. We determine numerically how the average surface density inside the gap, Σ_gap, depends on planet-to-star mass ratio q, Shakura-Sunyaev viscosity parameter α, and disk height-to-radius aspect ratio h/r. Our results are derived from our new graphics processing unit accelerated Lagrangian hydrodynamical code PEnGUIn and are verified by independent simulations with ZEUS90. For Jupiter-like planets, we find Σ_gap∝q ^–2.2α^1.4(h/r)^6.6, and for near brown dwarf masses, Σ_gap∝q ^–1α^1.3(h/r)^6.1. Surface density contrasts inside and outside gaps can be as large as 10⁴, even when the planet does not accrete. We derive a simple analytic scaling, Σ_gap∝q ^–2α¹(h/r)⁵, that compares reasonably well to empirical results, especially at low Neptune-like masses, and use discrepancies to highlight areas for progress.