5.4.3 Planetary motion

Kepler had three different laws to help us with planetary motion:

• K1 - the orbit of a planet is an ellipse with the Sun at one of two foci
• In most cases orbits have low eccentricity so their orbits are modelled as circles. E.g the difference is distance from aphelion (furthest point from the Sun) and perihelion (closest point from the sun) is just 3%
• K2 - A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time
• As plants move on their elliptical orbit around the Sun their speed is not constant. When a planet is closer to the Sun it moves faster. This explains why we don't see great comets - their orbits are highly elliptical and when they get close to the sun they move fast and spend much less time on this part of their orbit. They also spend most of their time too far from the Sun to be visible.
• K3 - The square of the orbital period T of a planet is directly proportional to the cube of its average distance r from the sun.
• T² ∝ r³ ∴ T²/r³ = K

Most planets in the Solar System have almost circular orbits so we can use circular motion mathematics along with Newton's law of gravitation to relate the orbital period 'T' of a planet to it's distance 'r' from the Sun, justifying K3...

If we think about it, a planet of mass m orbiting the Sun at a distance r will have a speed of v and an orbital period of T. The mass of the sun is M (M_☉) and the centripetal force on the planet is provided by the gravitational force between it and the Sun. Therefore the gravitational force 'F' on the planet is equal to the centripetal force:

F = mv²/r = GMm/r²

v² = GM/r

The planet is moving in a circle. This means the speed of the planet can be determined by dividing the circumference of its orbit by its orbital period (v = 2πr/T). We can substitute this into the above equation:

4π²r²/T² = GM/r

T² = (4π²/GM)r³

This equation can be used to determine the mass of an object, the orbital period of a planet, or its distance from the Sun. From this we can make the following statements:

• The ratio T²/r³ is equal to 4π²/GM
• The gradient of the graph of T² against r³ is equal to 4π²/GM

It is important to note that Kepler's laws also apply to any smaller objects in orbit around a larger one (e.g satellites and moons in orbits around planets) and not just planets in our Solar System.

Geostationary satellites

Satellites orbiting the Earth obey Kepler's laws of planetary motion. Satellites always fall to Earth due to the fact that the only force acting on them is the gravitational attraction between it and the Earth. However it is travelling at such a great distance that as it falls the Earth curves away beneath it so it stays at the same height above the surface. All satellites must be at a given height and speed to keep a stable orbit. We can rearrange the above equation to form an equation for the speed 'v' of a satellite in stable orbit at a distance 'r' from the centre of the Earth:

v = √(GM/r)

NOTE: the mass 'm' of the satellite is not present meaning all satellites placed in a given orbit at a given height will travel at the same speed.Once launched they are usually above the atmosphere so they experience no air resistance to slow them down so their speed remains constant.

There are a few uses of satellites which are pretty useful (duh):

• communications: satellite phones (NOT mobile phones!!), TV, some types of satellite radio
• military uses: reconnaissance
• scientific research: looking down on Earth to monitor crops/pollution/vegetation etc, looking outwards to study the Universe
• weather and climate: predicting/monitoring the weather across the globe. Also monitoring long-term changes in climate
• global positioning

Depending on the use, different satellites have different orbits. Satellites placed in the equator can be geostationary satellites. As the height of these satellites increase, so does their period. A geostationary satellite is placed in a geostationary orbit so it remains above the same point of the Earth whilst the Earth rotates. To be a geostationary satellite the satellite must:

• be in an orbit above the Earth's equator
• rotate in the same direction as the Earth's rotation
• have an orbital period of 24 hours