Kepler's Third Law Calculator

What is Kepler's Third Law?

Kepler's Third Law, also known as the Law of Periods, is a fundamental principle in astronomy that describes the relationship between the orbital period of a planet and its average distance from the sun. This law applies to any system where one body orbits another under the influence of gravity.

Formula

Kepler's Third Law is expressed mathematically as:

$$\frac{T^2}{a^3}=\frac{4{\pi }^2}{GM}$$

Where:

• $T$ is the orbital period of the planet (in seconds)
• $a$ is the semi-major axis of the orbit (in meters)
• $G$ is the gravitational constant ($6.67430\times {10}^{-11}{\text{ m}}^3{\text{ kg}}^{-1}{\text{ s}}^{-2}$)
• $M$ is the mass of the central body (in kilograms)

Calculation Steps

Let's calculate the orbital period of Earth around the Sun:

1. Given:
   • Semi-major axis of Earth's orbit ($a$) = $1.496\times {10}^{11}\text{ m}$
   • Mass of the Sun ($M$) = $1.989\times {10}^{30}\text{ kg}$
2. Rearrange Kepler's Third Law to solve for $T$: $$T=\sqrt{\frac{4{\pi }^2{a}^3}{GM}}$$
3. Substitute the known values: $$T=\sqrt{\frac{4{\pi }^2(1.496\times {10}^{11}\text{ m}{)}^3}{(6.67430\times {10}^{-11}{\text{ m}}^3{\text{ kg}}^{-1}{\text{ s}}^{-2})(1.989\times {10}^{30}\text{ kg})}}$$
4. Perform the calculation: $$T\approx 3.156\times {10}^7\text{ s}$$
5. Convert to years: $$T\approx 1.000\text{ year}$$

Example and Visual Representation

Let's visualize Kepler's Third Law for the inner planets of our solar system:

This diagram illustrates:

• The Sun at the center (yellow)
• The orbits of Mercury (red), Venus (green), Earth (blue), and Mars (orange)
• As the orbital distance increases, the orbital period increases more rapidly (proportional to a³/²)