The Kepler’s Third Law Calculator helps determine the relationship between the orbital period and the semi-major axis of an orbit. This law is crucial in astronomy for understanding the motion of planets and satellites.

## Inputs

**Orbital Period (T)**: The time taken for one complete orbit, measured in years or seconds.**Semi-Major Axis (a)**: The average distance from the orbiting body to the center of mass of the system, measured in astronomical units (AU) or meters.

## Formulas and Calculations

Kepler’s Third Law states that the square of the orbital period (T) is directly proportional to the cube of the semi-major axis (a) of the orbit:

[ T^2 \propto a^3 ]

For calculations involving the orbits around the Sun (in our Solar System), Kepler’s Third Law can be simplified to:

[ T^2 = a^3 ]

where:

- ( T ) is the orbital period in years.
- ( a ) is the semi-major axis in astronomical units (AU).

## 1. Calculate Semi-Major Axis

If the orbital period (T) is known, the semi-major axis (a) can be calculated as:

[ a = \sqrt[3]{T^2} ]

## 2. Calculate Orbital Period

If the semi-major axis (a) is known, the orbital period (T) can be calculated as:

[ T = \sqrt{a^3} ]

## Example Calculation

**Inputs:**

- Orbital Period (T): 8 years
- Semi-Major Axis (a): 4 AU

**Calculate Semi-Major Axis:**

If the orbital period is known:

[ a = \sqrt[3]{T^2} ]

[ a = \sqrt[3]{8^2} = \sqrt[3]{64} \approx 4 \text{ AU} ]

**Calculate Orbital Period:**

If the semi-major axis is known:

[ T = \sqrt{a^3} ]

[ T = \sqrt{4^3} = \sqrt{64} = 8 \text{ years} ]

## Summary

Kepler’s Third Law provides a simple relationship between the orbital period and the semi-major axis of an orbit. For an object with an orbital period of 8 years, the semi-major axis of its orbit is approximately 4 AU. Conversely, for an object with a semi-major axis of 4 AU, the orbital period is 8 years.

## Table of Example Calculations

Orbital Period (T) | Semi-Major Axis (a) |
---|---|

8 years | 4 AU |

1 year | 1 AU |

27 years | 9 AU |

64 years | 16 AU |

## Conclusion

Understanding the relationship between the orbital period and the semi-major axis of an orbit is fundamental in the field of astronomy. The Kepler’s Third Law Calculator provides a straightforward way to perform these calculations, making it an essential tool for anyone involved in studying celestial mechanic.