# Kepler's Laws of Orbital Motion

by Ron Kurtus (updated 18 January 2022)

The first of Kepler's Laws of Orbital Motion states that the planets travel in ellipses around the Sun, which is situated at one of the focal points. The second law states that the planet or orbiting satellite speeds up when it is closer to the main focal point of the ellipse. The third law gives the relationship between the period of motion and the distance to the Sun.

The background to these laws starts in the 16th century, when Polish astronomer Nicolaus Copernicus determined that the Earth and the planets orbit around the Sun. Previously, scientists thought everything revolved around the Earth. Copernicus thought the orbits were circles.

Then about 75 years later, German mathematician Johannes Kepler found that the orbits were not circles, but ellipses. He formulated three laws as to how planets and other space objects travel, when in an orbit. These became known as Kepler's Laws.

Questions you may have include:

• What is Kepler's first law?
• What is Kepler's second law?
• What is Kepler's third law?

This lesson will answer those questions.

## Kepler's first law

The first law is that the orbit of an object moving around another in space is elliptical with the stationary object located at one of the focal points of the ellipse. In other words, the Earth travels around the Sun in an ellipse, and the Sun is at a focal point of that ellipse. The same is true for a space satellite traveling around the Earth. It is possible for a satellite to travel in a circular orbit, but that is a special case.

An ellipse is a geometric shape similar to an oval. It has two focus points, as seen in the picture below. Typical ellipse with two focal points

A circle is a special case of an ellipse where both focal points are at the same point—the center of the circle.

## Kepler's second law

Kepler's second law states that the orbiting satellite will speed up when it gets closer to the object at the focus. This is caused by the increased effect of gravity on the orbiting object as it gets closer to what it is orbiting around.

The mathematical statement of the law is that the area swept by the planet or orbiting object in a giving time is the same, independent of the distance to the object at the focus. Areas swept in a given time are equal

Since the areas are equal, the arc that is further away is shorter, meaning that the speed will be slower. This is true for most objects in space.

## Kepler's third law

This law shows the relationship for the time required for a planet to move around the Sun and the average distance from the Sun. The relationship is that the time squared is proportional to the distance cubed. Thus, if you knew the time it took to go around the Sun and the distance for one planet, you could find values for another.

Consider comparing the times and distances of two planets. The equation is written as:

t²/T² = d³/D³

where

• t is the time is takes the first planet to go around the Sun
• T is the time is takes the second planet to go around the Sun
• d is the average distance of the first planet from the Sun
• D is the average distance of the second planet from the Sun
• is t*t and is d*d*d
• is T*T and is D*D*D

### Example

Let's see if we can use this equation to verify how long it takes the planet Jupiter to go around the Sun.

Suppose we say that T is the time it takes the Earth to go around the Sun one time. Thus T = 1 year. The distance the Earth is from the Sun is about 92 million miles. Astronomers defined that distance as 1 astronomical unit or 1 AU. Thus D = 1 AU. Since T = 1, then T² = 1. Also since D = 1, thus D³ = 1.

Now Jupiter's distance from the Sun is about 5 times as far as the distance of the Earth to the Sun. Thus, d = 5 AU. That means that d³ = 5³ = 125. Let's put the values into the equation t²/T² = d³/D³ to find what t equals:

t²/1 = 125/1

t² = 125

Take the square root of both sides of the equation, and find that t = 11.2 Earth years, which is close to the actual length of a Jupiter year according to astronomical measurements

## Summary

Kepler's three laws explain orbital motion. The laws are: (1) Orbits are elliptical in shape, (2) the area swept in a given time is constant for a given ellipse, and (3) the relationship for the time required for a planet to move around the Sun and the average distance from the Sun is the time squared is proportional to the distance cubed.

Solve problems by breaking them down into smaller pieces

## Resources and references

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### Websites

Kepler's Laws - HyperPhysics

Kepler's laws of planetary motion - Wikipedia

Kepler's Three Laws - The Physics Classroom

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