 Now, Kepler's first law states that a planet orbits the sun in an ellipse with the sun at one focus of the ellipse. Let's take a quick look at ellipses to start with. So we know that a circle has a centre and a constant radius. Those are basically the two things that uniquely define a circle. If we consider an ellipse on the other hand, an ellipse has two focus points, which have similar mathematical characteristics to a circle's centre and are placed symmetrically along the long axis. Ellipses have both a long radius and a short radius, called the semi-major axis and the semi-mine axis respectively. So ellipses are basically elongated circles. You can quantify how elongated an ellipse is with a number known as the eccentricity, which is between zero and one. A higher eccentricity means the ellipse is more elongated. And it turns out circles are actually a special case of ellipses and have an eccentricity of zero. In circles, the semi-major axis is equal to the semi-mine axis, which gives us a constant radius. Now let's look at Kepler's first law, which tells us that planets have elliptical orbits. We've already modelled circular orbits in our uniform circular motion, and this immediately agrees with Kepler's first law. In this case, the orbit is a circle, and the sun being at one focus of the ellipse is just the sun being at the centre of the circle of orbit. The most surprising result of Kepler's first law is that for non-circular elliptical orbits, the sun is at the focus rather than the centre of the orbit. This idea was rejected and ignored by many astronomers at the time because they believed that the sun should naturally be in the centre of any orbit. This incorrect belief was easier to hold because most planet orbits in our solar system are only slightly elliptical, and therefore modelling these as circular orbits gives a fairly accurate representation of their behaviour. Kepler's second law tells us the line segment joining a planets and the sun sweeps out equal areas during equal intervals of time. So let's draw a diagram of a circular path. For a circular path, we immediately know from our understanding of uniform circular motion that the planet will travel at a constant speed, and therefore for the same interval of time a planet will sweep out equal areas. However, for an elliptical path, the planets will not be travelling at a constant speed, so it's not obvious that equal areas would be swept out, but this is exactly what Kepler's second law tells us. A planet must also travel further along the path when it is closer to the sun to sweep out an equal area. Therefore, Kepler's second law also tells us that a planet moves faster when it is closer to the sun. We already know this is true from conservation of energy. As the planet moves further away from the sun, the kinetic energy decreases as the potential energy increases. Kepler's third law tells us that the square of a planet's period is proportional to the cube of the semi-major axis of that planet's orbit. And because the mass of the sun is so much larger than the mass of the planets, then for all planets in our solar system, the proportional factor is approximately the same. So let's consider a circular orbit and see if Kepler's third law holds. Kepler's third law states that the square of the period of the orbit will be proportional to the cube of the radius of the orbit. Let's consider our satellite equation. This tells us that t squared on r cubed is equal to 4 pi squared divided by big g times m. Now we know that the right-hand side, 4 pi squared on big g m, is a constant. We'll call this k, and therefore the square of the period is proportional to the cube of the radius, and we've proved Kepler's third law for circular orbits. More generally, if we consider different orbits, we can find a relationship between the distance of a planet from the sun and the period of that planet's orbit. So, for example, if we consider a planet with a semi-major axis that is half that of another planet, then we can calculate the period of the closer planet. We'll denote the semi-major axis of the planets as r. Using Kepler's third law, we can calculate the period of the closer planet with half the radius. We'll have a period that is 35.4% of the further planet's orbit. And from Kepler's third law, we know that this result is independent of the eccentricity of the ellipse. And therefore, even if we were to consider a circular orbit that had a radius equal to the semi-major axis of the previous planet, then this orbit, too, would have the same period. In fact, regardless of the eccentricity of planets' orbits, the period will be approximately the same as long as the semi-major axis is the same. Thank you.