Physics · Ch 6 — Gravitation
Kepler's Laws of Planetary Motion
Kepler's Laws of Planetary Motion
Kepler's three laws, extracted purely from Tycho Brahe's observational data (with no underlying force law behind them yet), completely describe planetary orbits.
1. Law of orbits. Every planet moves around the Sun in an elliptical orbit with the Sun at one of the two foci of the ellipse, not at the centre. The point of closest approach to the Sun is the perihelion; the farthest point is the aphelion. If is the semi-major axis and the semi-minor axis, both Copernicus and Ptolemy had assumed circular orbits -- Kepler's real discovery was that the true orbits are elliptical, with the Sun off-centre at a focus.
2. Law of areas. The radial vector joining the Sun to a planet sweeps out equal areas in equal intervals of time. Because the Sun is off-centre, this forces the planet to move faster when it is close to the Sun (near perihelion) and slower when it is far away (near aphelion), so that the swept-out area per unit time stays the same throughout the orbit. This law is really a statement of conservation of angular momentum about the Sun.
3. Law of periods. The square of a planet's orbital period is directly proportional to the cube of its semi-major axis:
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What this figure shows. A planet traces an ellipse with the Sun sitting at one of the two foci, not at the centre. The point on the orbit closest to the Sun is labelled perihelion (P) and the farthest point is aphelion (A); the long axis through both is the major axis, and half of it is the semi-major axis 'a'. The short axis is the minor axis, half of which is the semi-minor axis 'b'. Because the Sun sits off-centre at one focus, the planet's distance from the Sun keeps changing as it goes around, which is exac …
| Planet | a (10^10 m) | T (years) | T^2/a^3 |
|---|---|---|---|
| Mercury | 5.79 | 0.24 | 2.95 |
| Venus | 10.8 | 0.615 | 3.00 |
| Earth | 15.0 | 1 | 2.96 |
| Mars | 22.8 | 1.88 | 2.98 |
| Jupiter | 77.8 | 11.9 | 3.01 |
| Saturn | 143 | 29.5 | 2.98 |
What this figure shows. The radial line (called the radial vector) joining the Sun to a planet is shown sweeping out several shaded regions as the planet moves along its elliptical path over equal one-month intervals. Every shaded region has exactly the same area, even though the arcs of the orbit they correspond to are clearly different lengths -- the arc is longer (and the planet moves faster) near the Sun, and shorter (planet moves slower) far from the Sun, s …