## 2018 IPFW Student Research and Creative Endeavor Symposium

Jeff Anderson

Mathematics

Engineering

#### Abstract

Once, humans thought that the moon’s movement was simple and elegant –a perfect circle. Then during the 17th century, Johannes Kepler published his stunning observation that the motion of the moon, as well as the planets, were in fact best described by ellipses – still elegant, but a tad more complicated to analyze. Isaac Newton later published his theory of gravity that strongly supported this observation, as Newton himself demonstrated that if the motion of the Moon were indeed governed by his theory, then it would follow the path of an ellipse. However, the mystery surrounding the motion of the Moon does not stop there. It turns out that the ellipse the moon follows is not stationary, but is observed to rotate slowly, but noticeably over time. This observation perplexed Newton greatly as he tried to reconcile his theory with reality. All was not lost, however, because in his derivation of an elliptical orbit, Newton only accounted for the attraction of the main body(the Earth) on the Moon. He tried to account for the attraction of other entities, especially the Sun, to see whether these other forces could “perturb” the Moon’s ellipse over time and cause it to gradually rotate. Newton failed, however, because what he was attempting is what mathematicians now call a three-body problem, which does not have a closed-formed, exact solution. Newton attempted some approximations, and his result fell short. Now, with the advent of modern computers, this task becomes significantly easier. In this research, we introduce an approximation method using Taylor series expansion to solve the differential equations posed in an n-body problem and utilize the program Matlab to carry out the numerical approximation. We will use this procedure to find solutions to certain configurations of the three-body problem, most notably the Earth -- Moon – Sun system, in an attempt to explain the observed motion of the Moon. As we shall see, the Earth – Moon – Sun system actually has a much simpler and more elegant solution than we would otherwise expect a general three-body problem to have. We also extend the scope of our study to other planets. Specifically, we carried out a simulation on the entire solar system and were able to accurately model another phenomenon that also

#### Disciplines

Engineering | Mathematics

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