Energy of an orbiting satellite

The energy account of an orbit

From $\dfrac{mv^2}{r} = \dfrac{GMm}{r^2}$, the kinetic energy is

The potential energy is $U = -\dfrac{GMm}{r} = -2K$. So the total energy is

The total is negative: the satellite is bound to the Earth. To escape, it would need $E \ge 0$.

You should find all three energies shrinking toward zero far out, with $U$ always twice $K$ in size.

Moving to a higher orbit

To climb from $r_1$ to $r_2$, the satellite needs

It gains potential energy, but loses kinetic energy (higher orbits are slower), so the cost is only half the change in $U$.

Watch out. A satellite slowed by thin air speeds up: losing total energy drops it into a lower orbit, where $K = -E$ is larger. Drag makes satellites go faster as they sink.

A worked example

A $1200\ \text{kg}$ satellite orbits at $3R$ from the Earth's centre. ($gR = 6.27 \times 10^{7}\ \text{J/kg}$.)

1. Kinetic. $K = \dfrac{mgR}{6} = \dfrac{1200\ \text{kg} \times 6.27 \times 10^{7}\ \text{J/kg}}{6} = 1.25 \times 10^{10}\ \text{J}$.
2. Potential. $U = -2K = -2.51 \times 10^{10}\ \text{J}$.
3. Total. $E = -K = -1.25 \times 10^{10}\ \text{J}$; that much more would let it escape.

Think first. Find $K$ first; then $U = -2K$ and $E = -K$ follow at once.

Practise with new numbers

Try it in Python

The program lists the energies of a $1000\ \text{kg}$ satellite in four orbits.

Remember these

• $K = \dfrac{GMm}{2r}$, $U = -2K$, $E = -K$.
• Negative total energy means a bound orbit.
• Raising an orbit costs $\Delta E$, half of $\Delta U$.

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's total energy is $-3.0 \times 10^{9}\ \text{J}$. What are its kinetic and potential energies?

2. How much more energy does a satellite in orbit need to escape?

3. Why does a satellite in a lower orbit move faster?

4. What happens to a satellite's speed as air drag slowly lowers its orbit?