  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 corotation torques are, which impacts how
planets can survive fast orbital migration. We determine numerically how the
average surface density inside the gap, sigma_gap, depends on planet-to-star
mass ratio q, Shakura-Sunyaev viscosity parameter alpha, and disk
height-to-radius aspect ratio h/r. Our results are derived from our new
GPU-accelerated Lagrangian hydrodynamical code PEnGUIn, and are verified by
independent simulations with ZEUS90. For Jupiter-like planets, we find
sigma_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 inside
and outside gaps can be as large as 10^4, even when the planet does not
accrete. 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 low
Neptune-like masses, and use discrepancies to highlight areas for progress.
