Kepler's laws
Planets move in ellipses with the Sun at one focus, the line between planet and Sun sweeps out equal areas in equal times with , and the period of the orbit is proportional to .
- K1
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Planets move in ellipses with the Sun at one focus.
- K2
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The line between the planet and the Sun sweeps out equal areas in equal times, and
- K3
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The period of of the orbit is proportional to .
Proof
For K2, a small sweep of the Sun–planet line satisfies , so
which is constant by the conservation of angular momentum; this holds for any central force.
For K3, dimensional analysis applies because the only parameter in Newton’s equation is : matching gives and , so . There is no unique radius associated to an ellipse, but taking any will do.
More precisely, integrating over the full period,
using , where . This is for the average radius of the ellipse.
Related
Stated in
- Proposition 3.2 (Kepler's Laws)§3.5 Kepler’s Laws