-Here is a new going further sequence. Once again, I will display the same shortcomings by going back in time and looking at history. This time we will talk about an amazing character, Johannes Kepler. Johannes Kepler lived between the 16th and the 17th century. He was a mathematician, an astronomer and an astrologer. But he is probably best known for his talents as a mathematician and astronomer. Indeed, the natal charts he may have done for Emperor Rudolf II never reached our era. Johannes Kepler is indeed known for the three laws that explain planet movements around the Sun. But we can also use them to explain satellite movements around the Earth. We will see this in the next illustrations. Furthermore, he is really important for the space field. The proof is that one of the ATV vessels, these automatic transfer vehicles for the International Space Station, was named after him. Let us have a look at the three laws. They were written between 1602 and 1618. I will give a verbatim quote of each law on the illustrations and then rephrase them to apply to a satellite revolving around the Earth. The first law says that the orbit of a planet is an ellipse with the Sun at one of the two focal points. The first law indeed specifies that the orbit of a satellite is elliptic and that the Earth is at one on the focal points of the ellipse. This allowed us at the beginning of the sequence to represent the orbit in two dimensions. The second law specifies that a line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time. This means that during an equal interval of time, the segment linking the satellite to the Earth's center will sweep an equal surface. We only dealt with circular orbits and saw that this speed was constant. So the swept surface is indeed constant. What about elliptic orbits? This is where it gets interesting. Here is the pattern of an elliptic orbit. Let us have a look at the perigee, that is to say when the satellite is the closest to the Earth. We see that with a constant interval of time, the two S surfaces must be equal. But the distance that has to be covered at the perigee is much longer than at the apogee. So the satellite's speed will have to increase at that moment. Do not get me wrong. When I say that the satellite's speed has to increase, it simply means that the satellite will automatically revolve faster. This was the second law. Now, the third law. The third law says that the square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit. Until now, we have not seen a relation between the orbital period and the altitude of the satellite because we established a relation between the satellite's speed and its altitude. But we can very easily link orbital period and speed as long as, using the speed and the perimeter of the orbit's journey, we are able to calculate the orbital period. I will let you do this calculation at home. It will allow you to check that the third law is absolutely consistent with what we told you.