Some basic principles
Without going into the mathematics [go here for that] the velocity of a spacecraft in a circular orbit around the Earth is given by
where G is the Universal Gravitational Constant, ME is the mass of the Earth and R is the distance between the centre of the Earth and the centre of the spacecraft. We show this formula to draw some important conclusions about orbital motion:
- The velocity decreases with increasing altitude above the Earth. Hence higher satellites move slower than satellites in lower orbits.
- Velocity is independent of the mass of the spacecraft. This is a very important conclusion because it means that e.g. at the altitude of the ISS any object will move with exactly the same orbital velocity no matter how heavy or light. So the ISS itself (500 tons) will move exactly the same as any of the astronauts inside (or outside) or as any other object, say a paperclip inside the spacecraft.
- We can calculate that at low orbit (marginally above the Earth’s atmosphere) the velocity is about 7.9 km/s.
- With a little more calculation (knowing that the circumference of a circle is 2πR, we can calculate the altitude of a geostationary satellite (42,240 km) that has an orbital period of 24 hours, i.e. it orbits at the same angular velocity as the Earth itself. These satellites are extremely important in particular for telecommunication as they are always in the same position in the sky as seen from the surface of the Earth.
Johannes Kepler (1571 – 1630) was the first to realise that planetary orbits in general are ellipses. The circle we suggested above is just a special case of an ellipse with zero eccentricity. Kepler formulated his famous three laws (in about 1618) that describe orbital motion.
It was one of Newton’s great accomplishments that he could mathematically prove Kepler’s laws, what Kepler had never been able to do, because Kepler did not have access to the required mathematics, in particular Calculus.
When an object is moving in an elliptical orbit it has a velocity that is always directed along the tangent to the orbit at every point. It also has an acceleration in the direction of the focal point where the central mass is. This acceleration is due to the gravitational attraction between the object and the central mass.
Now get a feeling for the elliptic orbit and Kepler’s Laws by spending some time with this great animation.
(Choose “Newtonian features” to switch on the v and a vectors). Show ellipse features, e.g. centre, focal points, semi-major and –minor axis, eccentricity, etc. Also note that the ellipse becomes a circle at zero eccentricity.