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Gap clearing by giant planets has been proposed to explain the optically thincavities observed in many protoplanetary disks. How much material remains inthe gap determines not only how detectable young planets are in their birthenvironments, but also how strong corotation torques are, which impacts howplanets can survive fast orbital migration. We determine numerically how theaverage surface density inside the gap, sigma_gap, depends on planet-to-starmass ratio q, Shakura-Sunyaev viscosity parameter alpha, and diskheight-to-radius aspect ratio h/r. Our results are derived from our newGPU-accelerated Lagrangian hydrodynamical code PEnGUIn, and are verified byindependent simulations with ZEUS90. For Jupiter-like planets, we findsigma_gap \propto q^-2.2 alpha^1.4 (h/r)^6.6, and for near brown dwarf masses,sigma_gap \propto q^-1 alpha^1.3 (h/r)^6.1. Surface density contrasts insideand outside gaps can be as large as 10^4, even when the planet does notaccrete. We derive a simple analytic scaling, sigma_gap \propto q^-2 alpha^1(h/r)^5, that compares reasonably well to empirical results, especially at lowNeptune-like masses, and use discrepancies to highlight areas for progress.

How Empty are Disk Gaps Opened by Giant Planets?