Lecture 5: Page 3

Page 3, History of Astronomy II [Previous Page] [Next Page] [Class Home Page]

Kepler's Three Laws of Planetary Motion (cont.)

Kepler's Second Law:

II. The line joining the planet to the Sun sweeps out equal areas in equal times as the planet travels around the ellipse.
So, the planet moves faster when nearer the Sun. A planet executes elliptical motion with constantly changing angular speed in its orbit. The point of nearest approach of the planet to the Sun is termed perihelion; the point of greatest separation is termed aphelion.

Kepler's Third Law:

III. The ratio of the squares of the revolutionary periods for two planets is equal to the ratio of the cubes of their semi-major axes:
In this equation P represents the period of revolution for a planet and R represents the length of its semi-major axis. The subscripts "1" and "2" distinguish quantities for planet 1 and 2 respectively. The period for a planet to orbit the Sun increases rapidly with the radius of its orbit. Thus, we find that Mercury, the innermost planet, takes only 88 days to orbit the Sun but the outermost planet (Pluto) requires 248 years to do the same.

Calculations using Kepler's Third Law
A convenient measurement for periods is Earth years. A convenient measurement for distances is the average separation of the Earth from the Sun, an astronomical unit (AU). If these units are used in Kepler's 3rd Law, the denominators in the preceding equation equal "1" (assume Planet "2" is Earth, with distance = 1 AU, period = 1 year) and it may be written in the simple form:

P(years)² = R(A.U.)³

With a bit of algebra, it can be expressed in years or distance:

P(years) = R(A.U.)^3/2 or R(A.U.) = P(years)^2/3

To calculate the "radius" of Mars orbit (semi-major axis) from the orbital period: Mars orbits the Sun in 1.88 Earth years:

R = P^2/3 = (1.88)^2/3 = 1.52 AU