The motion of satellites and planets

Why a satellite does not fall

A satellite in orbit is constantly in free fall towards the Earth: gravity pulls it towards the centre of the Earth. However, its horizontal speed is such that, whilst it is ‘falling’, the Earth’s surface curves away from it at the same speed. The result is a circular (or elliptical) trajectory rather than a vertical fall.

Speed of a satellite in circular orbit

For a satellite of mass m in a circular orbit of radius r around a celestial body of mass M, the force of gravity acts as the centripetal force. From this, we can derive the satellite’s speed: v = √(G * M / r)

This speed does not depend on the satellite’s mass m, but only on the mass M of the celestial body and the radius r of the orbit: the lower the orbit, the faster the satellite must travel to avoid falling.

The orbital period

The period T (the time taken for one complete orbit) is given by: T = 2 * π * r / v

Combining this with the expression for v, we arrive at Kepler’s third law: T² / r³ is a constant that is the same for all satellites of the same celestial body.

Example: the geostationary satellite

A geostationary satellite has a period T = 24 h (it always remains above the same point on the globe). This requires a precise orbital radius of approximately 42,200 km, which corresponds to an altitude of about 35,800 km above the ground.

Common pitfall

Do not confuse the orbital radius r (measured from the centre of the Earth) with the satellite’s altitude (measured from the surface). We always have r = R_Earth + altitude.