KEPLER'S 3rd LAW

The ratio of the squares of the revolutionary periods for two planets is equal to the ratio of the cubes of their semimajor axes

In this equation "P" represents the period of the orbit for a planet and "a" represents the length of its semimajor axis. The subscripts "1" and "2" distinguish quantities for planet 1 and 2 respectively. The period is how much time it takes to do one orbit.  Note that both periods must be in the same time units and both semimajor axis lengths must be in the same distance units.

Kepler's Third Law implies that 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 distant dwarf planet Pluto requires 248 years to do the same.

This graphic illustrates the third law as it applies to the Solar System. Watch how quickly the inner planets move with respect to the outer planets.

This applet illustrates Kepler's Third Law and allows you to visualize orbits in the Solar System. Hit the Fast Forward button to start the objects moving. You can view the orbits from any angle using the slide bars at the bottom and right of the animation, and the zoom slide bar allows you to focus on the inner planets, or zoom out to the orbit of the dwarf planet Eris. You can also change the speed of the animation by changing the time increment to 1 month, for example. And you can run it backwards if you want.

Sir Isaac Newton developed an understanding of forces in general and showed that Kepler's Laws can be generalized to apply to all objects moving under the influence of gravity including those outside our own solar system. The ratio of a3 to p2 for objects orbiting our Sun is equal to 1, when a is expressed in AU and p in years. The equation below represents this relationship but says M = a3 / p2 where M is equal to 1 solar mass or the mass of our Sun. We can find the mass of other stars in units of the Sun's mass using this equation if we know the semi-major axis and periods of the planets that orbit them. M is the total mass of both bodies, star and planet, but the star is usually almost all of the mass.


 

For Example: Let's look at the extrasolar data sheet from the link below and find a planet orbiting the star 51 Peg. Open the link and select "Candidate planets around main sequence stars" and scroll down to 51 Peg. The one planet listed has a semi-major axis of 0.052 AU and a period of 4.23 days.

To calculate for M you first need to find a3. The cube of 0.052 is 1.4e-4. Then we need p2 . But first we must convert p to years. 4.23 days is equal to 0.0116 years ( 4.23 / 365 ). Now 0.0116 squared is equal to 1.34e-4. So the value for M is 1.4e-4 / 1.34e-4 = 1.04. This tells us that 51 Peg is 1.04 solar masses or almost the same size as our Sun.
How about the star Tau Boo? What is its mass in solar masses?

Kepler's Laws apply to all objects moving under the influence of gravity. Click Here to get data on extrasolar planets (you will need this link when filling in the answer page).