Kepler Third Law Semi-Major Axis

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

Variables

asemi-major axis of the orbit (m)

Ggravitational constant (m³/kg·s²)

Mmass of the central body (kg)

mmass of the orbiting body (kg)

Torbital period (s)

Description

What is this formula?

This formula calculates the semi-major axis of an orbit using Kepler's third law in Newtonian form. It relates the size of an orbit to the orbital period and the total mass of the two-body system.

When to use it

Use this formula when the orbital period and masses of the two bodies are known and the orbital size needs to be determined.

Example

A satellite orbits Earth once every 90 minutes.

Given:

T=5400 s

G=6.67430×10^-11 m³/kg·s²

M=5.972×10^24 kg

m=1000 kg

Formula:

a=((G*(M+m)*T²)/(4*pi²))^(1/3)

Substitution:

a=((6.67430×10^-11*(5.972×10^24+1000)*(5400)²)/(4*pi²))^(1/3)

Result:

a≈6.65×10^6 m

The satellite's semi-major axis is approximately 6650 km from Earth's center.

Applications

- Satellite orbit design

- Planetary orbit calculation

- Space mission planning

- Binary star analysis

- Celestial mechanics