Satellites and their energy

A satellite is a body that keeps going round a planet: the moon is the earth's natural satellite, and the earth is itself a satellite of the Sun. The first artificial satellite, Sputnik 1, was launched in 1957; India's first, Aryabhata, in 1975.

For a circular orbit of radius r = R_E + h, gravity provides the centripetal force: GMm/r² = mv²/r. So the orbital speed is v = √(GM/r). It does not depend on the satellite's mass, and it falls as the orbit gets higher.

Close to the surface, v ≈ √(gR_E) ≈ 7.9 km/s. Compare the escape speed: v_e = √2 × v, about 11.2 km/s.

The period is T = 2πr/v, which gives T² = (4π²/GM)r³: Kepler's third law again, now for satellites. A satellite just above the surface goes round in about 85 minutes. At 1000 km up (r = 7.38 × 10⁶ m), v ≈ 7.4 km/s and T ≈ 105 minutes.

Turned round, this weighs the central body: M = 4π²r³/(GT²). Time any moon or planet, measure its orbit, and you know the mass of what it goes round: the Sun, the earth, Mars or Jupiter.

A satellite at about 36 000 km takes exactly 24 hours, the same time the earth takes to turn, so it hangs over one spot. These geostationary satellites carry television and weather pictures.

The energies in a circular orbit:

• Kinetic: K = ½mv² = GMm/2r.
• Potential: V = –GMm/r.
• Total: E = K + V = –GMm/2r. It is negative (the satellite is bound to the planet), and E = –K = V/2.

So a higher orbit has less kinetic energy but more total energy: you must supply energy to raise a satellite, and it ends up moving more slowly.

Astronauts float in a space station not because gravity has vanished (it is about 90% of its surface value there), but because they and their ship are falling freely together. Nothing pushes up on them, so they feel weightless.