Orbital speed and period of a satellite

Falling around the Earth

Newton imagined a cannon on a mountain top. Fire the ball faster and faster, and it lands farther away; at one speed, the ground curves away beneath it as fast as it falls, and it never lands. It is in orbit: always falling, never arriving.

Orbital speed

In a circular orbit of radius $r$, gravity provides the centripetal force:

The satellite's mass cancels. Near the surface, $r \approx R$ and $v = \sqrt{gR} \approx 7.9\ \text{km/s}$.

Period

One orbit is $2\pi r$ long, so

which is Kepler's third law, $T^2 \propto r^3$, for satellites. Near the surface it is about $85$ minutes.

You should find that a higher orbit is slower, but its period grows even faster, reaching $24$ hours at about $36\,000\ \text{km}$.

Watch out. Always use $r = R + h$, the distance from the Earth's centre, not the height.

A worked example

The International Space Station orbits about $420\ \text{km}$ up. ($GM = gR^2 = 4.01 \times 10^{14}\ \text{m}^3/\text{s}^2$.)

1. Radius. $r = 6400\ \text{km} + 420\ \text{km} = 6820\ \text{km}$.
2. Speed. $v = \sqrt{\dfrac{4.01 \times 10^{14}\ \text{m}^3/\text{s}^2}{6.82 \times 10^{6}\ \text{m}}} = 7.67\ \text{km/s}$.
3. Period. $T = \dfrac{2\pi \times 6.82 \times 10^{6}\ \text{m}}{7670\ \text{m/s}} = 5590\ \text{s}$, about $93$ minutes: some $15$ orbits a day.

Think first. $v$ and $T$ depend only on $r$ and the planet's mass, never on the satellite.

Practise with new numbers

Try it in Python

The program finds the speed and period of four orbits.

Remember these

• $v = \sqrt{\dfrac{GM}{r}}$, $r = R + h$; independent of the satellite's mass.
• $T = 2\pi\sqrt{\dfrac{r^3}{GM}}$; $T^2 \propto r^3$.
• Close to the Earth: $7.9\ \text{km/s}$, $85$ minutes.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A satellite moves to an orbit of four times the radius. How do its speed and period change?

2. Two satellites, one heavy and one light, share the same orbit. Do they keep pace?

3. Why do satellites launched eastward need less fuel?

4. A moon orbits a planet of mass $M$ at radius $r$. What would its speed be around a planet of mass $4M$ at the same radius?