AKICIF: View of the Earth from 100,000 Miles

[jeriendhal]

Writing the next bit in Wake Up Call and I'm trying to visualize how much bigger the Earth would appear from the surface of the Ring compared to the Moon. Obviously bigger, probably terrifyingly huge given the shorter distance, but how much so? Just trying to find a good illustration without resorting to buying an astronomy program or downloading Orbiter.

Comments

[stoutfellow.livejournal.com]

The angular diameter of a circle of radius r from a point d units from its center is arctan(Sqrt(r^2/(d^2-r^2))). The Earth, as seen from the Ring, has angular diameter 2.4 times that of the Earth as seen from the Moon, so it would appear to be about 5.76 (=2.4^2) times as large. If you want the apparent size of the Moon from Earth vs the apparent size of the Earth from the Ring, the angular diameter of the Earth is about 8.85 times larger, and the ratio of the areas is about 78:1.

[stoutfellow.livejournal.com]

Gah. The correct figure for the angular diameter is twice what I gave; but since all of the diameters are off by the same factor, the ratios should be correct.

[stoutfellow.livejournal.com]

OK, one last try. The formula I gave you is correct, but unnecessarily complicated. The angular diameter is 2 arcsin(r/d), and if r is much smaller than d, arcsin(r/d) is very close to r/d, so you can use that as an approximation if you don't like arcsine.

[jeriendhal.livejournal.com]

I may have to settle for "Oh, dear god it's HUGE!" but thank you for figuring that out.

[allah-sulu.livejournal.com]

By my (admittedly rusty) math, using the Pythagorean Theorem and Law of Cosines, the Earth would appear to take up 4.6° of the “sky” as seen from the Ring. The moon, as seen from the Earth, takes up about 0.5° of the sky. This means that Earth-from-Ring would appear 9.2 times wider than Moon-from-Earth; or, in terms of area, Earth-from-Ring would appear about 84.64 times larger than Moon-from-Earth. My numbers may be different than stoutfellow's, but the final answer is still “Oh, dear god, it’s HUGE!”

[allah-sulu.livejournal.com]

“… 9.2 times wider …”

[jeriendhal.livejournal.com]

Thank you! May I post this image elsewhere as a useful illustration?

[allah-sulu.livejournal.com]

Go right ahead! And feel free to pass the blame on to me if anyone disproves my numbers.