Càlcul de trajectòries interplanetàries de satèl·lits artificials

Abstract

Since ancient times humans have tried to comprehend what was happening in the sky. They created many different myths trying to explain what held world up, or what caused the sun to rise. They realised that they could navigate using the sun and the stars so tried to calculate their movements. However, they found it extremely difficult to describe them as they incorrectly believed the Earth as the centre of the universe which made the movement of every planet very complicated. Although some philosophers had already conjectured that the sun was actually the centre of the solar system, since they didn’t posses the necessary scientific methods, they couldn’t prove or refute a theory in the way we can now. This changed with Galileo, the father of the scientific method, it is due to him that we are now able to conclusively prove or disprove hypotheses. Based on observation of the moons of Jupiter, he claimed that the sun was the centre of the universe. At the same time, Johannes Kepler formulated his three laws of the movement of the planets around the Sun. It was then Newton who was first to produce a concrete scientific theory that would explain the movement of the planets. After he published his famous Principia most of the great mathematicians of history devoted part of their time trying to describe the movements of celestial objects using the Newtonian theory. Despite all of the theory developed, it wasn’t until the beginning of the twentieth that it occurred to the scientific community that a man-made object could be sent to the space and kept there. However, it was not until the end of the Second World War, that astrodynamics truly began to develop. Before then, scientists were unable to develop a vehicle capable of producing the great amount of velocity needed to escape the gravitational effects of the Earth. It was after the Second World War when the USSR and the USA started the space race that large amounts of money were invested with the goal of sending probes into space. Since this moment, satellites have gained more and more importance in our lives. As every satellite needs somebody able to design and control its trajectory, astrodynamics has become incredibly important. To exactly predict the trajectory of a satellite we need to take into account many factors, such as the gravitational pull of the other planets in the solar system or the effects of radiation pressure. However, the resulting equations do not have an analytic solution, and can only be solved by integrating them numerically. Finding a trajectory numerically requires a lot of trial and error, and without a good first approximation the time needed to find it would be prohibitive. Thus we need to find ways to approximate trajectories; this essay is devoted to the study of two of these approximations. The first approximation we are going to deal with is the two body problem. This is the simplest approximation and is rather easy to study. The equations for the two body problem can be integrated, but the solution is given by a transcendent equation which cannot be solved using algebraic methods. For this reason we use numerical integration to find a solution. We can make a complete qualitative study that will allow us to analytically find the trajectory of the satellite for most possible initial conditions. Hence, not only will we try to explain the principles of this method, but we will also apply it to compute transfers from the earth to the moon. Moreover, once we have a method to systematically compute these trajectories, we will try to find the most energy efficient way to do such transfers. Once we have computed the minimum energy needed to perform a transfer, we will introduce an alternative and more complicated model to find the approximated trajectory of the satellite. This is the restricted circular and planar three body problem (RCPTBP). We will try to describe how the use of this allows us to approach a lunar transfer using less energy than the minimum found using the method derived from studying the two body problem. Due to the complexity of this model, we will focus only on the study of its equation of motion and the study of the basic dynamics introduced by them. Whilst completing this project the main challenges I have had to face, apart from the obvious mathematical ones, have been the physical concepts such as dealing with units, or changing coordinates frames as well as learning how to use Latex and writing in a correct and understandable way.