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Orbits under gravity: spoiler

Q: In Kepler's Laws "radius" means the angular-averaged radius, but what's the time-averaged radius?

A: (1 + 0.5 e2) a where e = eccentricity, a = semi-major axis.

Q: What are the speed and period of a satellite orbiting at the earth's surface?

A: Assume the earth has mass 6e24 kg and radius 6.4e6 m. Then the speed of a satellite orbiting the earth at its surface is
V = 7940 m/s = 17800 mph
and its orbital period is
P = 5070 s = 84.4 min.
Escape velocity at the earth's surface is
E = √2 V = 11200 m/s = 25200 mph.

Some more facts about orbits:

If you dig a straight tunnel between any two points A and B on the earth's surface and drop a vehicle down it, it will oscillate between the ends of the tunnel with period P. If the tunnel passes through the earth's centre O then its velocity there will be V. If the acceleration of the vehicle is magically reversed between O and B then it will have speed E at B.

The time taken to fall through a tunnel in a planet of given density is independent of the radius, by dimensional analysis alone M. Pilloff, rec.puzzles.

P is also the period of a Shuler pendulum, ie a pendulum of length the radius of the earth whose bob is at the earth's surface. If the bob is released from the same height as the pivot it will pass through its lowest point with speed V.

Q: If a (light) satellite in a circular orbit is suddenly stopped, how long does it take to hit the (point) mass it's orbiting?

A: 1 / (4 √2) = 17.7% of its orbital period, so for example 4h 15m for a geostationary satellite, 4d 19h for the moon. The time doesn't depend on anything else.


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