Polar Ice Cap

Question: How much of the Earth's sunlight hits land under the icecaps?

First approximation: Circular Antarctica, no axial tilt

Antarctica is 14.6E6 km2, or 0.0286 of the 5.1E8 km2 of the earth. We can approximate this as a circular cap with the northern edge at latitude \phi = 1.331 radians = 76.26°S . The area is 2\pi R^2(1-sin(\phi)) . The portion of the sun-facing disk occupied by this cap is 0.5 - ( \phi + 0.5 \sin{ 2\phi } / \pi