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9.4 Elliptical Orbits; Kepler’s Laws 285

Kepler’s Third Law relates the period of the orbit to the size of the orbit:

The square of the period is proportional to the cube of the semimajor axis of the
Kepler’s Third Law
planetary orbit.

This Third Law, or law of periods, is nothing but a generalization of Eq. (9.13) to
elliptical orbits.
Table 9.1 lists the orbital data for the planets of the Solar System. The mean
distance listed in this table is defined as the average of the perihelion and aphelion
distances; that is, it is the semimajor axis of the ellipse. The difference between the
perihelion and aphelion distances gives an indication of the elongation of the ellipse.

TABLE 9.1 THE PLANETS

MEAN DISTANCE
FROM SUN PERIHELION APHELION
PLANET (a) MASS (SEMIMAJOR AXIS) DISTANCE DISTANCE PERIOD
6
23
6
6
Mercury 3.30 10 kg 57.9 10 km 45.9 10 km 69.8 10 km 0.241 yr
Venus 4.87 10 24 108 107 109 0.615
Earth 5.98 10 24 150 147 152 1.00
Mars 6.42 10 23 228 207 249 1.88
Jupiter 1.90 10 27 778 740 816 11.9
Saturn 5.67 10 26 1430 1350 1510 29.5
Uranus 8.70 10 25 2870 2730 3010 84.0
Neptune 1.03 10 26 4500 4460 4540 165
Pluto 1.50 10 22 5890 4410 7360 248

JOHANNES KEPLER (1571–1630) German astronomer
and mathematician. Kepler relied on the theoretical framework
of the Copernican system, and he extracted his three laws by a
(a) A photomontage of the planets in sequence from meticulous analysis of the observational data on planetary
Mercury (top left, partly hidden) to Pluto (bottom left). motions collected by the great Danish astronomer Tycho Brahe.