# Kepler's Laws

## Brahe and Kepler - A feud worthy of Shakespeare...

### In Science, students at times can feel that scientists lead dull, isolated lives with very little real excitement. The reality is, scientists lives at times can be in fact full of sworn enemies, royal intrigue, running swordfights, disfiguring injuries and even bizarre death scenes more worthy of fiction... ## Kepler's Laws

### The diagram below shows a mathematical representation of an Ellipse :- ## Newton's Law of Universal Gravitation

### As has been shown previously, the Gravitational Field Strength varies greatly between locations. In 1687, Isaac Newton published 'The Principia', containing his work on Gravity. Within this publication, Newton had devised a general rule which would allow the calculation of the Force of attraction due to Gravity between any two objects at any separation. This formula is called Newton's Law of Universal Gravitation :- ## Example 1 - Gravitational Attraction to the Earth

### A pupil of mass 50 kg is sitting on the Earth's surface and is attracted to the Earth which has a mass of 5.97x1024 kg. What is the Gravitational Force that the pupil experiences if the radius of the Earth is 6.38x106 m ? ## Example 2 - Gravitational Attraction between pupils

### What is the Gravitational Force between two pupils sitting 2 m apart if both pupils have a mass of 50 kg ? ## Gravitational Field Diagrams

### The Gravitational Field around an isolated Mass can be modelled as shown below :- ### However, when taken to a very small section of the field, the field lines are approximately parallel and as such, on a small scale the Field Strength can be approximated as a constant. This allows the use of formulae such as Ep = mgh for small height changes, without requiring a changing value for "g" :- ### If the Mass in question is not an isolated Mass, but is interacting with another Gravitational Field, the field lines distort as shown in the diagram below :- ### The location of this zero point between two masses depends upon the relative field strengths of the masses involved. For example, the diagram below shows the Gravitational Field pattern around the Earth - Moon System :- ## Gravitational Field Strength and Altitude

### As stated earlier, as the distance from an object increases, the Gravitational Field Strength gets smaller. Due to this, the value of 'g' is reduced at higher altitude. The graph below shows how the value of 'g' varies in relation to height :- ### The value of 'g' below the surface reduces due to the fact that the deeper into a planet the object is, the less mass below to contribute to 'g'. This gives a value of 'g' of zero at the planet's core. The diagram below shows how the value of "g" varies throughout the inside of the Earth :- ## Orbital Period of a Satellite

### Gravitational Force :- ### Centripetal Force :- ### By combining the above formulae together and by then substituting Velocity, the Period can be found as follows :- ### Note - The above formula is a variation on Kepler's third law. It is in fact a reversal of this derivation that Newton used to calculate G in the first place, using Kepler's data for the planets. Kepler's Third Law is generally written in the following form, showing the relationship between T2 and r3 to be a constant :- ## Example 3 -

### If the orbital Period of a Geostationary satellite is equal to 1 day ( 23.93 hours ), what is the radius of its orbit, if the mass of the Earth is equal to 5.97x1024 kg? 