Planetary motion: (c) the equation T² = (4π²/GM)r³

The idea

This equation links a satellite's orbital period T to its orbital radius r around a central mass M. It comes from equating gravitational force GMm/r² to centripetal force m(4π²/T²)r. The mass m cancels, giving T² = (4π²/GM)r³. The equation shows T² is directly proportional to r³ for a given central body. For example, a satellite orbiting Earth at radius r has M equal to Earth's mass, not the satellite's mass. Rearranging gives M = 4π²r³/(GT²), which allows astronomers to find the mass of a planet or star from the orbit of a moon or satellite. The equation assumes a circular orbit and that the central mass is much larger than the orbiting mass.

What earns the marks

• Equate gravitational force GMm/r² to centripetal force m(4π²/T²)r.
• Cancel the orbiting mass m to obtain T² = (4π²/GM)r³.
• Identify M as the mass of the central body, not the orbiting body.
• Use M = 4π²r³/(GT²) to determine the central mass from orbital data.
• Recognise that T² ∝ r³ for a fixed central mass.

Common mistakes and how to put them right

• Using the satellite's mass for M instead of the central body's mass.

  Put it right by using the mass of the body being orbited.

• Forgetting to square the period when substituting into the equation.

  Put it right by ensuring T is in seconds and squared.

• Mixing units, such as using km for r and hours for T.

  Put it right by converting r to metres and T to seconds before calculation.

Exam tips

• Write down the equation exactly as given before substituting numbers.
• Check that r is the orbital radius from the centre of the central body, not the altitude above its surface.
• Use standard SI units: metres for r, seconds for T, kilograms for M.