Kepler’s Laws Of Planetary Motion Part 3 – The Law of Periods

Reading Time – 8 Minutes, Difficulty Level – Challenging

The final post in a series of three covering Kepler’s Laws of Planetary Motion. Here we’ll look at law 3 or ‘The Law of Periods’ – The square of the period of any planet is proportional to the cube of the semimajor axis of its orbit.

Johannes Kepler’s Third Law of Planetary Motion, also known as the “harmonic law,” is a fundamental principle that relates the period of a planet’s orbit to its distance from the sun. This law helped to revolutionize our understanding of the solar system and has been a cornerstone of astronomy ever since.

In a nutshell

Kepler’s Third Law states that the square of the orbital period of a planet is proportional to the cube of its average distance from the sun. In other words, if you square the time it takes a planet to complete one orbit around the sun and divide it by the cube of its distance from the sun, you will always get the same value for all the planets in our solar system.

The detail

This law was a significant departure from the traditional belief at the time that planets moved in perfect circles around the sun at a constant speed. Kepler’s observations of the motion of planets had led him to conclude that their speed changed as they moved through their elliptical orbits, and their period depended on their distance from the sun. He then extended his analysis to other planets and found that their orbital periods followed this same relationship.

Kepler’s Third Law was crucial in determining the relative distances of the planets from the sun. By measuring the orbital periods of the planets and knowing their average distances from the sun, astronomers could calculate the distance of any planet from the sun. This allowed them to construct accurate models of the solar system and paved the way for further research into the nature of the universe.

One of the most significant implications of Kepler’s Third Law was that it helped establish a relationship between the size of a planet’s orbit and its distance from the sun. The farther a planet is from the sun, the longer it takes to complete one orbit. This relationship also helps explain why the outer planets in our solar system are much larger than the inner planets.

Neptune – The predicted planet

Kepler’s Third Law also played a critical role in the discovery of Neptune. In the mid-19th century, astronomers noticed that the orbit of Uranus did not follow the predicted path, suggesting that there was another planet beyond it. Using Kepler’s Third Law, astronomers were able to calculate the position of this hypothetical planet. In 1846, Neptune was discovered in the predicted location.

Today, Kepler’s Third Law remains a critical tool for astronomers in their study of the solar system and other celestial bodies. It allows us to calculate the distances of planets from their stars and has been used to discover hundreds of exoplanets in other solar systems.

Johannes Kepler’s Third Law of Planetary Motion, or the harmonic law, is a fundamental principle that relates the period of a planet’s orbit to its distance from the sun. This law helped to revolutionize our understanding of the solar system and has been a cornerstone of astronomy ever since. By establishing a relationship between the size of a planet’s orbit and its distance from the sun, Kepler’s Third Law has played a critical role in our understanding of the universe.

Kepler’s Third Law Mathematics

As with parts one and two of this series, it’s not vital to understand the mathematics of the third law to gain a high level understanding of the concept, but again I’ve included it here for you to read through if you would like to.

Kepler’s third law can be derived mathematically as follows:

Let T be the period of the planet’s orbit, which is the time it takes to complete one revolution around the sun. Let a be the semi-major axis of the planet’s elliptical orbit, which is the longest radius of the ellipse.

We can express Kepler’s third law as:

To derive this law, we start with Newton’s law of gravitation, which states that the force of attraction between two objects is proportional to the product of their masses and inversely proportional to the square of their distance. For the case of a planet orbiting the sun, we can write:

where G is the gravitational constant, M is the mass of the sun, m is the mass of the planet, and r is the distance between them.

Since the force of gravity is the centripetal force that keeps the planet in orbit, we can equate it to the centripetal force:

where v is the velocity of the planet.

Equating these two expressions for F, we get:

Simplifying, we get:

Now, we can express the semi-major axis a in terms of r as:

where e is the eccentricity of the elliptical orbit, which is a measure of how much it deviates from a circle.

Substituting this expression for r into the equation for v^2, we get:

We can rearrange this equation as:

This is Kepler’s third law in its final form, which states that the square of the orbital period of a planet is proportional to the cube of the semi-major axis of its elliptical orbit.

The constant of proportionality is 4π^2/GM, which depends only on the mass of the sun and the gravitational constant.

Further Reading

Kepler’s Laws of Planetary Motion Part 1 – The Law of Orbits
Kepler’s Laws of Planetary Motion Part 2 – The Law of Areas