# Kepler's second law

How fast does the planet move in its orbit?

As the planet moves in its orbit, a line from the sun to the planet sweeps out equal areas in equal times.

This means that the planet moves faster when it is near the sun, slower when it is far away.

The eccentricity of the earth's orbit is 0.017. This means that the distance from the earth to the sun when we are nearest to the sun is ( 1.000 - 0.017) AU = 0.983 AU. Similarly, the distance from the earth to the sun when we are farthest from the sun is ( 1.000 + 0.017) AU = 1.017 AU.

This is not much of a difference, but using Kepler's second law, we can deduce something. Call the time of day when the sun crosses the great circle in the sky that contains the north celestial pole and the zenith ``local noon.'' When the earth is nearest the sun, is the time from one ``local noon'' and the next ``local noon'' bigger or less than 24 hours?

A comment:
It's no wonder that Ptolemy had trouble.

• Suppose planet orbits were exactly circles centered on the sun.
• Then according to Kepler, they move around their orbits at a constant speed.
• Now just shift your view put the earth at the center.
• The planets then move on little circles whose centers move on big circles.
• The motion around both circles is at a constant speed.
• You could always expand the size of one of the big circles and its little circle by, say, 70%, without affecting the prediction for what you see in the sky.
• This is exactly the Ptolemaic system.
• But the orbits aren't exactly circles.
• Also, according to Kepler, the speeds aren't exactly constant.
• Now the motion from the point of view of the earth is very complicated.
• You can almost get it right with complications like the ``equant,'' but not exactly.

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Davison E. Soper, Institute of Theoretical Science, University of Oregon, Eugene OR 97403 USA soper@bovine.uoregon.edu