Here's how it is expressed:
T² = (4π²/GM) * r³
Where:
* T is the orbital period (time to complete one orbit)
* G is the gravitational constant (approximately 6.674 x 10⁻¹¹ m³ kg⁻¹ s⁻²)
* M is the mass of the central object (e.g., the Sun, Earth)
* r is the average orbital radius (the semi-major axis of the elliptical orbit)
Key differences from Kepler's Third Law:
* Kepler's Third Law only applies to planets orbiting the Sun. Newton's version applies to any two objects orbiting each other, including planets around stars, moons around planets, or even stars in binary systems.
* Kepler's Third Law states that the square of the orbital period is proportional to the cube of the orbital radius. Newton's version adds the proportionality constant (4π²/GM), which is a more precise relationship.
* Newton's version accounts for the mass of both objects. Kepler's Third Law assumes the mass of the planet is negligible compared to the Sun.
In essence, Newton's version of Kepler's Third Law demonstrates the fundamental relationship between gravity, mass, and orbital motion. This law has become a cornerstone of celestial mechanics and has been used to calculate everything from the mass of planets to the distance to distant galaxies.
