Tycho Brahe (1546-1601) was a Danish astronomer who spent most of his life studying the planets and the stars. Because of the accuracy of his unaided-eye observations (a telescope had not yet been invented), Brahe is considered to have been one of the greatest practical astronomers of modern times. Johannes Kepler (1571-1630) was a German mathematician and astronomer who used Brahe’s data on the positions of the planets and stars to propose the three laws that have made him famous. These laws describe planetary motion within a heliocentric (Sun-centered) framework. In effect, he dethroned the Earth as the center of the universe and the solar system. Here is the modern version of his laws:

1. Law of ellipses -- All planets move in elliptical orbits around the Sun, with the sun at one focal point of the ellipse.

2. Law of areas -- The line from the Sun to any planet sweeps out equal areas of space in equal times.

3. Law of Harmony -- The period T and the average distance R of a planet from the Sun are related by T²/R³=constant, the same for all planets.

A convenient unit of measurement for periods is in Earth years, and a convenient unit of measurement for distances is the average separation of the Earth from the Sun, which is termed an astronomical unit, abbreviated as AU. If these units are used in Kepler's 3rd Law, the denominators in the preceding equation are numerically equal to unity and it may be written in the simple form

P² =R³  where P is given in years and R is expressed in AU .

Although Kepler’s formulation was mostly empirical (i.e., having no theoretical foundation), today his laws can be quite simply derived using Newton’s law of gravitation plus a bit of math.

The orbits of the planets are primarily determined by the gravitational interaction between the Sun and the planet (two-body approximation).  Recall from Newton's gravitational law that the force of interaction between any two massive objects is only proportional to product of their mass and inversely proportional to square of distance between them.  Sun is so massive compared with every other object in the Solar System (Sun is about 300,000 times more massive than the Earth, and about 100 times more massive than the largest planet), and objects outside the Solar System are so distant that the their gravitational interactions with the planets are negligible. Thus, the product of the mass of a planet and the mass of the Sun is always much larger than the product of the masses of any two planets, and it is a good initial approximation to neglect all interactions except that of the planet and the Sun.

Example: Mars orbits the Sun at an average distance of 2.3x10¹¹ m, greater than that of the Earth by a factor of about 1.53. Use Kepler’s third law to calculate the period of the Martian orbit (i.e., the length of the Martian year).

Solution: According to Kepler’s third law, the quantity T₂/R₃ should be the same for any two planets in the solar system, and, in particular, for Earth and Mars. Thus, since T_M2/R_M3=T_E2/R_E3 or T_M2=T_E2· (R_M3/R_E3)=1·(1.53)³ or T_M=1.89 years. (The actual number is 1.88 years or 687 days).

Example: Kepler’s third law is obeyed by all planets, not just by the six on which he based his conclusions. Since Kepler’s time, three new planets have been discovered with the aid of a telescope. All are beyond Saturn and are not bright enough to be seen with the naked eye. Their periods can be determined by simple observation and are given in the table below. Verify that the average distance from the Sun for each planet is given correctly in the table.

Planet	Average Distance from the Sun (AU)	Period (years)
Uranus	19.18	84.0
Neptune	30.06	164.8
Pluto	39.52	248.4

Solution: The simplest approach is to use astronomical units (AU) and Kepler’s third law, in the form T=R^3/2. For example, (19.18)^3/2=84 years.

Further Reading:

Mark Twain's Comet 

Playing Balls

For an account of Kepler's life and his discoveries read "The Sleepwalker", by A. Koestler, Hutchinson & Co., London, 1959.

  See an excellent animation of Kepler's laws of motion by NASA.