Kepler's Third Law

The square of the period of time it takes a planet to complete an orbit of the Sun is proportional to the cube of its mean distance from the Sun.

Kepler's Third Law, also known as the Harmonic Law, is one of the most powerful statements of physical law that we find in astronomy. This law can be generalized to describe the motions of bodies other than the planets about the Sun, of course, and if we do a bit of mathematical gymnastics, it can be used to divulge some of the more interesting secrets of the Universe.

When Kepler first published his third law in 1619, the mathematical statement of it looked like this:

          (1)

P represents the period of the orbit in seconds and a is the semi-major axis of the orbit in meters; k is a constant which is unique for every body under consideration. For example, the value of k for the Sun--using Earth as the orbiting body--is 2.95 x 10^-19. Using Jupiter as the orbiting body gives a k of 2.97 x 10^-19, very close to the other value. Kepler knew that k was unique for each body, but he didnÍt know what k represented. It would take a man of Newton's genius to figure out that k was a description of gravitational force.

Newton reformulated the Harmonic Law, and changed equation 1 into this:

          (2)

G represents the universal gravitational constant (G=6.67206 x 10^-11 m³/kg s²), M₁ is the mass of the primary body, and M₂ is the mass of the body orbiting the primary. If we can estimate the masses of a star and a planet in orbit around it, and we know roughly what the semi-major axis of the orbit is, we can use equation 2 to predict the period of the orbit. We can also manipulate the equation and turn it into this:

          (3)

If we know the semi-major axis of the orbit and the orbital period, we can use equation 3 to estimate the total masses of the primary and the body in orbit around it. This is where the real power of this law resides--we can estimate mass based on two easily measurable orbital properties.

In the case of our solar system, M₁ obviously represents the Sun and M₂ represents the mass of a planet. Since the Sun is so tremendously large in comparison to the planets, we can ignore M₂ in the equation without much loss of accuracy.

The semi-major axis of Earth's orbit is a = 1.5 x 10¹¹ m, and the period of the orbit is P = 3.16 x 10⁷ s. Plugging these values into equation 3 and ignoring the mass of the Earth gives M₂ = 1.99 x 10³⁰ kg for the mass of the Sun. (As with all of my articles, please confirm the calculations yourself).

To determine the mass of the Earth, we will use orbital data for two different objects to show that the resulting masses are about the same. Space Shuttle orbiters have orbital properties of P = 5400 s and a=6,678,000. (We have to measure the distance from the center of the Earth, not the surface.) For the Moon, P = 2360592 s and a = 384,400,000 m. Calculate these yourself; both answers should be in close agreement. Try the following to determine the mass of Jupiter:

Remember to convert to meters and seconds before doing the calculations. The mass for Jupiter should be about 1.90 x 10²⁷kg.

View this MPEG movie (176K) to see a planet moving around the Sun in an elliptical orbit. The movie was created at the University of Oregon.

Astronomers can use Kepler's Third Law to determine the masses of stars in binary systems. We will still use equation 3, but we will not be able to ignore the mass of the secondary since it is often the case that the masses are similar. If we know where the center of mass of the system is located, however, we can determine the masses of the individual stars.

Alpha Centauri is the closest star system to our own. It is really a multiple star system, with two stars pretty close together and a third star (Proxima Centauri) at a tremendous distance. The two main stars, a Centauri A and B, are about 23 AU apart and orbit each other with a period of 80 years.

The total mass of these two stars given by equation 3 is 3.817 x 10³⁰ kg, or almost 2 solar masses. But how do we determine the mass of each star?

In a binary star system, the components do not really orbit each other; they orbit the center of mass of the system. The more massive the star, the closer it orbits to the center of mass. The ratio of the distances from the stars to the center of mass is the same ratio as the masses of the stars. The exact location of the center of mass can be determined observationally if the star shows motion across our line of sight (proper motion). As the two stars orbit the center of mass, they will seem to corkscrew around one another. They do not move across our line of sight in a straight line, but the center of mass does.

If we monitor the stars through several orbits and track their proper motion, we can determine the position of the center of mass. Once we know where the center of mass is, we can determine the relative distances for each star and solve for the mass of each. Note that this only works if we know how far away the stars are because we need to convert the angular distance between them into the actual separation in meters.

For Alpha Centauri, the A component is 1.3 times closer to the center of mass than the B component. Thus, A is 1.3 times more massive than B. Since the total mass of both stars is 3.817 x 10³⁰ kg, this represents the mass of B plus 1.3 times the mass of B. So, we have

and

So A is about 1.1 solar masses, and B is about 0.83 solar masses. These values are pretty close to the generally accepted values.

Kepler's Third Law is a powerful tool for exploring the Universe because it gives us a way to measure the absolute masses of stars and planets. Once we determine the mass of a star, we can make estimates for intrinsic luminosity, density, size, etc. of stars. Thus Kepler's Third Law can help to determine information that is vital to our understanding of stars. Double star observations are extremely important. (If you are interested in double star observing, I suggest that you subscribe to the journal Double Star Observer. It is published quarterly by Ronald C. Tanguey, 306 Reynolds Drive, Saugus, MA, 01906-1533, or call (781) 231-1158. Subscriptions in the US cost $6 per year.)