Orbits are cool. Let’s see how, shall we?
When a satellite orbits the Earth in a perfect circle, it experiences a centripetal acceleration that depends on its orbital speed v (a constant) and its orbital radius r (also constant):
A centripetal acceleration changes only your direction of motion. A tangential acceleration changes your speed. In your car, your steering wheel produces a centripetal acceleration, and your gas and brake pedals produce a tangential acceleration.
The centripetal acceleration experienced by an orbiting satellite is due to the Earth’s gravitational field:
where G is the universal gravitational constant and M is the mass of the Earth. (If you’re going to build a universe, you must decide how strong the force of gravity will be. You do this by setting the value of G. In our universe, and perhaps others, G has the value 6.673 x 10-11 in SI units.)
Equating these expressions, we get
If we solve this expression for the orbital speed v, we get
This formula allows us to determine the orbital speed if we know the mass of the Earth and the orbital radius. It applies only to circular orbits. Since the orbital radius r appears in the denominator, we predict that orbits of larger radius yield slower orbital speeds. This leads to a somewhat paradoxical result in orbital mechanics. Suppose you have a spaceship orbiting the Earth in a circular path. The pilot fires the engines for a brief burn, accelerating the ship. The result? The ship slows down.
How does this happen? Generally, when a spaceship moving in a circular orbit changes speed, its orbit changes to an ellipse. In a circular orbit the speed is constant, but in an elliptical orbit an object’s speed constantly changes. The speed is greatest at pericenter, the point of closest approach to the parent body (the Earth in this case), and least at apocenter, the point of greatest distance from the parent body. When the pilot fires the engines and speeds up the ship, the ship’s current location becomes the pericenter of the new elliptical orbit. Very soon the engine burn is completed and the spaceship moves along its new orbit under the action of the Earth’s gravity. As it swings toward apocenter it slows down. So the result of the engine burn was to transfer the spaceship from a circular orbit to an elliptical orbit, and slow down its orbital speed. (After the spaceship passes through apocenter, it will speed up as it approaches pericenter.)
The opposite effect happens if the pilot hits the reverse thrusters and slows down the ship (briefly). The ship’s current location becomes the apocenter of the new elliptical orbit, and as it swings toward pericenter it speed increases.
Isn’t that the coolest thing ever? This is why physics is awesome.
Suppose you’re orbiting the Earth in a circular path, and your buddy’s spaceship is 5,000 km ahead of you on an identical orbit. You’ll never catch up, since each ship moves with the same speed. What should you do to catch up to your friends’ ship, fire the thrusters and speed up, or fire the reverse thrusters and slow down? (The answer is left to the reader as an exercise.)
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If we solve the orbital velocity expression for M, we get
This formula allows us to determine the mass of the Earth if we know the satellite’s orbital speed and radius. It also applies only to circular orbits.
Of course, the satellite’s orbital speed is also given by
where 2πr is the circumference of the orbit and T is the orbital period (the time needed to complete one orbit). Substituting into the mass expression, we get
We recognize this as Kepler’s third law, applied to circular orbits. Newton was able to derive a more general version of Kepler’s third law for elliptical orbits, a major triumph for his theory of gravity.
In the early 80′s, each time the Shuttle flew, I’d log on to a NASA bbs system and grab a set of orbital elements for it. I’d then program the relevant equations into my calculator and solve the orbit, plotting it with a grease pencil on an acetate sheet placed on top of a world map.
There was a certain thrill that went along with then tuning into NASA TV and seeing my work matching up with the plot as seen on the main screen in mission control.
Even better was when my plot of the shuttle’s orbit intersected my local horizon circle during either morning or evening twilight. Seeing that, of course, caused an immediate coordinate conversion to altitude and azimuth and a scheduled trip outside with my binoculars. There’s nothing like doing all those calculations, predicting its location at a given point in time, and proving correctness by eyeballing it directly.
“East takes you out, out takes you west, west takes you in, and in takes you east.” –Larry Niven, The Integral Trees. Here you are accelerating east (satellites normally go from west to east, seen from their own frame of reference, the same way the planet rotates) and moving out.