## 1.2 Kepler’s Laws of Planetary Motion

**Question:** What are the three laws of planetary orbits stated by Kepler? How did he derive them?

- Law of ellipses: The orbit of each planet is an ellipse, with the Sun located at one of its foci.
- Law of equal areas: A line drawn between the Sun and the planet sweeps out equal areas in equal times as the planet orbits the Sun.
- Harmonic Law: The square of the sidereal period of a planet (the time taken for a planet to complete one revolution about the Sun relative to the stars) is directly proportional to the cube of the semimajor axis of the planet’s orbit.

**Question:** What is Newton’s contribution to the study of planetary orbit?

Kepler derived the above laws empirically, i.e. using historical data. Newton was able to derive Kepler’s laws using a theoretical framework, specifically the law of gravitation and laws of mechanics. Newton was able to show that the physics governing the falling of an apple is the same as that for planetary motion.

Solving Newton’s equations for a two-body system can be done analytically (see lecture notes). It is however quite difficult and daunting for some. Fortunately there is another way to solve the differential equations and at the same time visualise the solution with an animated output. This is done using a computer programme, which is the ultimate objective of this worksheet.

But before that, we need to build up a bit of knowledge in science (and math). So…