8.3 Orbits and satellites

Orbits

The Moon goes round the Earth, the Earth goes round the Sun, and hundreds of artificial satellites go round the Earth. All of these are orbits, and they all happen for the same reason: gravity.

Gravity provides the centripetal force

In an earlier lesson, you saw that an object moving in a circle needs a resultant force towards the centre, called the centripetal force. For a planet or a satellite, this force is provided by gravity.

• The gravitational force always points towards the centre of the object being orbited.
• The velocity of the satellite is along the tangent, at right angles to the force.
• Because the force is at right angles to the motion, it changes the direction of the satellite, but not its speed.

So in a circular orbit, the speed is constant, but the velocity is changing, because the direction is always changing. The satellite is accelerating, towards the centre, even though it never gets any closer. An orbiting satellite is actually falling continually towards the Earth, but its sideways velocity means it keeps missing it.

Speed and orbital radius

For a stable circular orbit, the speed and the radius are linked:

• closer to the Earth (a smaller orbit), the gravitational force is stronger, so the satellite must move faster to stay in orbit;
• further from the Earth (a larger orbit), gravity is weaker, so the satellite moves more slowly, and takes longer to go round.

If a satellite's speed changes, its orbital radius must change. If the speed is just right, the orbit is circular. The same thing happens with the planets: Mercury moves much faster than Neptune.

Calculating the speed. The speed is the distance divided by the time. For one orbit, the distance is the circumference 2 \pi r and the time is the orbital period T:

v = \frac{2 \pi r}{T}

Worked example 1: the International Space Station. The ISS orbits at r = 6.8 \times 10^6 \text{ m} from the centre of the Earth and takes 93 minutes (5580 s) per orbit.

v = \frac{2 \pi \times 6.8 \times 10^6}{5580} = 7700 \text{ m/s} (about 7.7 km/s, or 28 000 km/h)

Worked example 2: a geostationary satellite. The orbital radius is 4.2 \times 10^7 \text{ m} and the period is 24 hours (86 400 s).

v = \frac{2 \pi \times 4.2 \times 10^7}{86\,400} = 3100 \text{ m/s}

This is much slower than the ISS, as the orbit is much bigger.

Natural and artificial satellites

• Natural satellites: the Moon orbits the Earth, and the planets orbit the Sun (the Sun is orbited by planets, so they are also satellites in this sense).
• Artificial satellites are put into orbit by rockets. The type of orbit depends on the job:

Weight and weightlessness

The astronauts on the ISS appear to float, but gravity has not gone: at the height of the ISS, the Earth's gravitational field strength is still about 90% of its value on the ground. The astronauts feel "weightless" because they are in free fall together with the station: both are falling at the same rate around the Earth, so there is no contact force between them.

(Edexcel) The gravitational field strength g is different on different bodies: about 9.8 N/kg on the Earth, 1.6 N/kg on the Moon and 3.7 N/kg on Mars. The weight of an object is W = m g, so the same mass weighs less on the Moon.