Kepler's Laws

Kepler's Laws

Johannes Kepler, working with data painstakingly collected by Tycho Brahe without the aid of a telescope, developed three laws which described the motion of the planets across the sky.

1. The Law of Orbits: All planets move in elliptical orbits, with the sun at one focus.

2. The Law of Areas: A line that connects a planet to the sun sweeps out equal areas in equal times.

3. The Law of Periods: The square of the period of any planet is proportional to the cube of the semimajor axis of its orbit.

Kepler's laws were derived for orbits around the sun, but they apply to satellite orbits as well.

The Law of Orbits

All planets move in elliptical orbits, with the sun at one focus.

This is one of Kepler's laws. The elliptical shape of the orbit is a result of the inverse square force of gravity. The eccentricity of the ellipse is greatly exaggerated here

Orbit Eccentricity The eccentricity of an ellipse can be defined as the ratio of the distance between the foci to the major axis of the ellipse. The eccentricity is zero for a circle. Of the planetary orbits, only Pluto has a large eccentricity. Examples of Ellipse Eccentricity Planetary orbit eccentricities Mercury .206 Venus .0068 Earth .0167 Mars .0934 Jupiter .0485 Saturn .0556 Uranus .0472 Neptune .0086 Pluto .25 The Law of Areas A line that connects a planet to the sun sweeps out equal areas in equal times. This is one of Kepler's laws.This empirical law discovered by Kepler arises fromconservation of angular momentum. When the planet is closer to the sun, it moves faster, sweeping through a longer path in a given time.

The Law of Periods

The square of the period of any planet is proportional to the cube of the semimajor axis of its orbit.

This is one of Kepler's laws.This law arises from the law of gravitation. Newton first formulated the law of gravitation from Kepler's 3rd law.

Kepler's Law of Periods in the above form is an approximation that serves well for the orbits of the planets because the Sun's mass is so dominant. But more precisely the law should be written

In this more rigorous form it is useful for calculation of the orbital period of moons or other binary orbits like those of binary stars.

Data: Law of Periods

Data confirming Kepler's Law of Periods comes from measurements of the motion of the planets.

Planet	Semimajor axis (1010m)	Period T (y)	T2/a3 (10-34y2/m3)
Mercury	5.79	0.241	2.99
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
Uranus	287	84	2.98
Neptune	450	165	2.99
Pluto	590	248	2.99

The quantity T²/a³ depends upon the sum of the masses of the Sun and the planet, but since the mass of the Sun is so great, adding the mass of the planet makes very little difference.

Data from Halliday, Resnick, Walker, Fundamentals of Physics 4th Ed Extended.