    NEWTON'S VERSION OF KEPLER'S THIRD LAW
Johannes Kepler discovered a simple relationship between the average distance of a planet from the Sun (called its semi-major axis, A, measured in Astronomical Units) and the amount of time it takes a planet to orbit the Sun once (called its orbital period, P, measured in years). For objects orbiting the Sun, the semi-major axis to the third power equals the period squared:

A3 = P2

There were two problems with this relation. First, Kepler did not know how it worked, he just knew it did. Second, the relation does not work for objects which are not orbiting the Sun, for example, the Moon orbiting the Earth. Isaac Newton solved both these problems with his Theory of Gravity, and discovered that the masses of the orbiting bodies also play a part. Newton developed a more general form of what was called Kepler's Third Law that could apply to any two objects orbiting a common center of mass. This is called Newton's Version of Kepler's Third Law:

M1 + M2 = A3 / P2

Special units must be used to make this equation work. If the data are not given in the proper units, they must be converted.

The masses must be measured in solar masses, where one solar mass is 1.99 X 1033 grams, or 1.99 X 1030 kilograms.

The semi-major axis must be measured in Astronomical Units, where 1 AU is 149,600,000 kilometers, or 93,000,000 miles.

The orbital period must be measured in years, where 1 year is 365.25 days.

This relation has many uses: determining the mass of a planet by looking at its moon(s), studying binary star systems, even determining the mass of the Galaxy!

There is a problem, however, with the way the equation is written above. Often, we are not able to determine to a high degree of accuracy the average distance between, say, two binary stars. We must use a modified version of NVK3L for very distant objects.

To achieve this modification, we must first introduce an equation for velocity, how fast an object is traveling. Everybody who has driven a car has encountered the formula for velocity. The speedometer on a car measures velocity in miles per hour, or kilometers per hour. Now miles or kilometers are ways of measuring distance, hours are what we use to measure time, and "per" is a word signaling division. Therefore, the formula for velocity is

Velocity = Distance traveled / Time to travel

How does this relate to NVK3L? Remember that our real problem is often that we do not know the average distance between the two objects that are orbiting each other. Many times, we can only clearly see one of the objects that is orbiting!  But velocity is something we can measure, as long as we can see one of the partners, using the Doppler Effect.

Technically what we are measuring is the orbital velocity of the visible partner, which can be related to the distance traveled by the visible partner in its orbit and the time it takes the visible partner to orbit once.  That time is simply  the orbital period P, which is generally easy to observe.  What we usually don't know is the distance traveled around the orbit by the visible partner, called the circumference of the orbit.  This circumference is related to the average distance, A, by the formula

Circumference = C = 2 (pi) A

So the velocity equation becomes

Velocity = V = C / P = 2 (pi) A / P

Remember that we can compute velocity using  the Doppler Effect.  We can observe the orbital period easily.  It is the value of A that is typically very hard to find.  So we turn the equation above around, and solve for A:

A = V P / 2 (pi)

We can now take this value of A and plug it in to Newton's Version of Kepler's Third Law to get an equation involving knowable things, like V and P:

M1 + M2 = V3P3 / 23(pi)3P2

M1 + M2 = V3P / 8(pi)3

What this equation is basically telling us is, the more mass there is in a system, the faster the components of that system are moving as they orbit each other.  We shall not use this more complicated version of NVK3L for homework calculations, but we will use the concept in our discussion of black holes.

## Sample Calculations

To see some sample calculations with Newton's Version of Kepler's Third Law, click on the examples below Updated 8/27/99
By James E. Heath

Copyright Ó 1999 Austin Community College