# Kepler's Third Law Calculator

Created by Dominik Czernia, PhD candidate
Reviewed by Bogna Szyk and Jack Bowater
Last updated: Sep 24, 2022

This Kepler's third law calculator uses Kepler's third law equation to estimate the basic parameters of a planet's motion around the Sun, such as the orbital period and radius. It is based on the fact that the appropriate ratio of these parameters is constant for all planets in the same planetary system.

You can directly use our Kepler's third law calculator on the left-hand side or read on to find out what is Kepler's third law if you've just stumbled here. In the following article, you can learn about Kepler's third law equation, and we will present you with a Kepler's third law example involving all of the planets in our Solar system.

The orbital period calculator offers the estimation of the orbital period using two other equations. Be sure to check it!

## What is Kepler's third law?

The 17th century German astronomer, Johannes Kepler, made a number of astronomical observations. Upon the analysis of these observations, he found that the motion of every planet in the Solar system followed three rules. You can read more about them in our orbital velocity calculator. Here, we focus on the third one:

The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit.

And that's what Kepler's third law is. Simple, isn't it? This sentence reflects the relationship between the distance from the Sun of each planet in the Solar system and its corresponding orbital period (also known as the sidereal period that we described in the synodic period calculator). Note that, since the laws of physics are universal, the above statement should be valid for every planetary system!

## Kepler's third law equation

We can easily prove Kepler's third law of planetary motion using Newton's Law of gravitation. All we need to do is make two forces equal to each other: centripetal force and gravitational force (you can find more information about the latter in the gravitational force calculator). We obtain:

m * r * ω² = G * m * M / r²,

where,

• m - is the mass of the orbiting planet;
• r - is the orbital radius;
• ω - is the angular velocity, ω = v/r for circular motion (v - linear velocity);
• G - is the Gravitational constant, G = 6.67408 * 10⁻¹¹ m³ / (kg * s); and
• M - is the mass of the central star.

If we substitute ω with 2 * π / T (T - orbital period), and rearrange, we find that:

R³ / T² = 4 * π²/(G * M) = constant.

That's the basic Kepler's third law equation. There is also a more general derivation that includes the semi-major axis, a, instead of the orbital radius, or, in other words, it assumes that the orbit is elliptical. Since the derivation is more complicated, we will only show the final form of this generalized Kepler's third law equation here:

a³ / T² = 4 * π²/[G * (M + m)] = constant.

As you can see, the more accurate version of Kepler's third law of planetary motion also requires the mass, m, of the orbiting planet. To picture how small this correction is, compare, for example, the mass of the Sun M = 1.989 * 10³⁰ kg with the mass of the Earth m = 5.972 * 10²⁴ kg. That's a difference of six orders of magnitude!

Learn more about ellipses in the ellipse calculator that helps to analyze the properties of such mathematical figures.

## Kepler's third law calculator

The Kepler's third law calculator is straightforward to use, and it works in multiple directions. Just fill in two different fields, and we will calculate the third one automatically.

If you're interested in using the more exact form of Kepler's third law of planetary motion, then press the advanced mode button, and enter the planet's mass, m. Note, that the difference would be too tiny to notice, and you might need to change the units to a smaller measure (e.g., seconds, kilograms, or feet).

## Kepler's third law example

In the Kepler's third law calculator, we, by default, use astronomical units and Solar masses to express the distance and weight, respectively (you can always change it if you wish). It's very convenient since we can still operate with relatively low numbers. To test the calculator, try entering M = 1 Suns and T = 1 yrs, and check the resulting a. Is it another number one? It should be! That's proof that our calculator works correctly - this is the Earth's situation.

If you'd like to see some different Kepler's third law examples, take a look at the table below. Here, you can find all the planets that belong to our Solar system. Do they fulfill Kepler's third law equation?

planet

orbital period (yrs)

semi-major axis (au)

mass (Earth's masses)

Mercury

0.240

0.387

0.0553

Venus

0.615

0.723

0.815

Earth

1

1

1

Mars

1.881

1.524

0.107

Jupiter

11.862

5.203

317.83

Saturn

29.457

9.537

95.159

Uranus

84.021

19.191

14.536

Neptune

164.8

30.069

17.147

Pluto (dwarf planet)

247.94

39.482

0.0021

Dominik Czernia, PhD candidate
Star mass (M)
Suns
Planet period (T)
yrs
Semi-major axis (a)
au
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