Technically Speaking: Recovering Kepler's Equation

Notice what happens in Newton's new equation if one of the masses is very large compared with the other. In particular, suppose the Sun is labeled as mass 1, and its mass is much larger than the mass for any of the planets. Then the sum of the two masses is always approximately equal to the mass of the Sun, and if we take ratios of Kepler's third law for two different planets the masses cancel from the ratio and we are left with the original form of Kepler's third law:

P12 / P22 = R13 / R23

where P and R refer to the period and semimajor axes.

Newton's Modification of Kepler's Third Law

Because for every action there is an equal and opposite reaction, Newton realized that in the planet-Sun system the planet does not orbit around a stationary Sun. Instead, Newton proposed that both the planet and the Sun orbited around the common center of mass for the planet-Sun system.

He then modified Kepler's third law to read:

(m1 + m2)P2 = (d1 + d2)3 = R3

Effect of the Center of Mass

Here is a Java applet that implements Newton's modified form of Kepler's third law for two objects (planets or stars) revolving around their common center of mass. By making one mass much larger than the other in this interactive animation, you can recover Kepler's original form of his third law where a less massive object appears to revolve around a massive object fixed at one focus of an ellipse.

where P is the planetary orbital period and the other quantities have the meanings described above, with the Sun as one mass and the planet as the other mass.

Validity of Kepler's Original Law
Kepler's third law in Kepler's original form is approximately valid for the Solar System because the Sun is much more massive than any of the planets and therefore Newton's correction is small. The data Kepler had access to were not good enough to show this small effect. However, detailed observations made after Kepler show that Newton's modified form of Kepler's third law is in better accord with the data than Kepler's original form. Here is a Kepler's laws calculator that allows you to make simple calculations for periods, separations, and masses for Kepler's laws as modified by Newton to include the effect of the center of mass.
Limiting Case 1: One Mass Very Large Relative to Other
We can gain further insight by considering the position of the center of mass in two limits. First consider the example addressed in the top right box, where one mass is much larger than the other. Then, we see that the center of mass for the system essentially coincides with the center of the massive object:

m2 / m1 ~ 0 means that d1 = (m2 / m1)d2 ~ 0

This is the situation in the Solar System: the Sun is so massive compared with any of the planets that the center of mass for a Sun-planet pair is always very near the center of the Sun. In this case, for all practical purposes the Sun is almost motionless at the center of the system, as Kepler originally thought.

More Playground Physics

These limiting cases for the location of the center of mass are perhaps familiar from our aforementioned playground experience. If persons of equal weight are on a see-saw, the fulcrum must be placed in the middle to balance, but if one person weighs much more than the other person, the fulcrum must be placed close to the heavier person to achieve balance. These are just the limiting cases 2 and 1, respectively, discussed in the main text.

Limiting Case 2: Equal Masses
However, now consider the other limiting case where the two masses are equal to each other. Then it is easy to see that the center of mass lies equidistant from the two masses. In this case, if they are gravitationally bound each mass orbits the common center of mass for the system that lies midway between them. If m1 = m2, then

d1 = (m2 / m1)d2 = d2

This situation occurs commonly with binary stars (two stars bound gravitationally to each other so that they revolve around their common center of mass). In many binary star systems the masses of the two stars are similar and Newton's correction to Kepler's third law is very large. The Java applet in the above left box illustrates.