Kepler laws explained with worked examples

Kepler laws explained gives you a practical way to read orbital motion. Johannes Kepler published his first two laws in 1609 and his third in 1619, and Isaac Newton later showed that they follow from universal gravitation when one body is much more massive than the other.

The three laws in plain language

Kepler’s first law says that planets move in ellipses, with the Sun at one focus. In a perfect circle, every point is the same distance from the centre. In an ellipse, the distance changes, so a planet has a closest point, called perihelion, and a farthest point, called aphelion.

Kepler’s second law says that a line from the Sun to the planet sweeps out equal areas in equal times. A planet moves faster near perihelion and slower near aphelion. This is why Earth is slightly faster around 3 January, when it is near perihelion, than around 4 July, when it is near aphelion.

Kepler’s third law links orbital size and orbital period. For bodies orbiting the same central mass, the square of the period is proportional to the cube of the semi-major axis:

\[P^2 \propto a^3\]

For the Solar System, if you measure \(P\) in years and \(a\) in astronomical units, the relation becomes \(P^2 = a^3\) for a small object orbiting the Sun.

Where Kepler laws explained comes from in Newtonian gravity

Newton’s law of gravitation says that two masses attract each other with force

\[F = \frac{G M m}{r^2}\]

where \(G\) is the gravitational constant, \(M\) is the central mass, \(m\) is the orbiting mass, and \(r\) is their separation. This inverse-square force points toward the central body, so the motion is a central-force problem.

Two key results follow. First, angular momentum is conserved because the force points along the radius. That leads directly to Kepler’s second law: equal areas in equal times. Second, solving Newton’s equations for an inverse-square force gives conic sections. For bound orbits, the path is an ellipse, so Kepler’s first law appears as a special case of Newtonian gravity.

Newton’s version of Kepler’s third law is

\[P^2 = \frac{4\pi^2}{G(M+m)}a^3\]

When the orbiting body is much lighter than the central one, \(m\) can be ignored. For planets around the Sun, this is usually an excellent approximation. For binary stars, both masses matter.

A worked example with Earth and Mars

Take Earth, with \(a = 1.000\,\text{AU}\), and Mars, with \(a = 1.524\,\text{AU}\). Using \(P^2 = a^3\) for Solar System orbits around the Sun, Mars should have

\[P = \sqrt{1.524^3} \approx \sqrt{3.54} \approx 1.88\,\text{years}\]

The measured orbital period of Mars is about 1.88 years, or 687 Earth days. That agreement is one reason Kepler’s third law was so powerful before Newton explained the physics behind it.

You can also check the second law qualitatively. Mars moves faster near perihelion and slower near aphelion, so its angular speed changes even though the areal speed stays constant.

A moon and an exoplanet example

The same equations work for moons. Europa orbits Jupiter at a mean distance of about \(671{,}100\,\text{km}\) and takes about 3.55 days to complete one orbit. Because Jupiter is so massive, the Moon-Jupiter system follows the same Newtonian form, with Jupiter’s mass replacing the Sun’s mass.

For exoplanets, the mass of the star matters. Suppose a planet orbits a Sun-like star at \(0.05\,\text{AU}\). Then

\[P = \sqrt{0.05^3}\,\text{years} = \sqrt{1.25\times10^{-4}}\,\text{years} \approx 0.0112\,\text{years}\]

That is about 4.1 days. This is the sort of short-period orbit found among hot Jupiters, which are easy to detect because they complete many orbits in a short time. For a star of different mass, use the full Newtonian form with \(M\) inside the equation.

What the laws do and do not say

Kepler’s laws describe ideal two-body motion. Real orbits are affected by other planets, the shape of the central body, tides, and relativistic effects. Mercury, for example, has a small perihelion advance that Newtonian gravity alone does not fully explain. Even so, Kepler’s laws are still the starting point for many orbital calculations.

They are also a practical tool. Astronomers use them to estimate masses, compare orbital scales, and plan observations. If you know a period and a semi-major axis, you can estimate the mass of the system. If you know the mass and one orbital size, you can predict the period.

For more orbital practice, see astronomy olympiad practice problems with worked solutions, how astronomers find planets around other stars, and tides, the Moon and the slow lengthening of the day. The best match in the masterclass is The Sky and Celestial Mechanics, and the formula sheet at Important equations is useful for the algebra.

Reliable background sources include NASA’s overview of Kepler’s laws, NASA’s basics of space flight on orbit mechanics, ESA’s material on the Solar System, and a classic review of celestial mechanics.

Frequently asked questions

Why is an orbit an ellipse and not a circle?

In Newtonian gravity, a bound orbit under an inverse-square force is generally an ellipse. A circle is just the special case with zero eccentricity.

Why does a planet move faster near the Sun?

As the planet falls inward, gravitational potential energy changes into kinetic energy. Conservation of angular momentum also means the planet must move faster when it is closer in.

Do Kepler’s laws work for exoplanets?

Yes. The same physics applies to any star-planet pair, with the star’s mass setting the period for a given orbit size.