7.2.4

# Orbits of Planets & Satellites

Test yourself

When an object orbits a more massive body it has a set period and radius which depend on each other.

### Orbital period

• The time taken for an object to do one full orbit is called the period.
• Even if the orbit is elliptical the period will remain constant.

• The orbital radius is the average distance between the centre of the body and the centre of the object.
• For the circle, the radius is always the same.
• For an elliptical orbit, the radius changes.

### Relationship

• Orbital period and radius have the following relationship:
• The period squared is proportional to the radius cubed.
• $T^2\; \alpha \; r^3$
• The constant of proportionality can be found by finding the gradient of a graph of period squared against radius cubed.
• This is Kepler's third law.

## Energies of Orbiting Objects

Sometimes, considering the total energy of a system, such as a satellite orbiting a planet, can be much easier than thinking about the resultant force and acceleration of an object.

### Circular orbits

• In a circular orbit around a planet, the satellite is always on the same equipotential and so the total energy of an orbiting satellite is constant.
• The planet does no work on the satellites, so there is no loss in potential and no loss in gravitational potential energy (GPE).
• The radius of the orbit does not change.
• The satellite does not change kinetic energy (KE) and so has a constant speed.

### Non-circular orbits

• This approach also works for non-circular orbits such as ellipses and parabolas.
• This is because we can show that the total energy of an orbiting satellite is always equal to half of the gravitational potential energy of the satellite.
• This is because gravitational field strength follows an inverse-square law.