A primer to numerical simulations: the perihelion motion of Mercury

Abstract

Numerical simulations are playing an increasingly important role in modern science. In this work it is suggested to use a numerical study of the famous perihelion motion of the planet Mercury (one of the prime observables supporting Einsteins General Relativity) as a test case to teach numerical simulations to high school students. The paper includes details about the development of the code as well as a discussion of the visualization of the results. In addition a method is discussed that allows one to estimate the size of the effect as well as the uncertainty of the approach a priori. At the same time this enables the students to double check the results found numerically. The course is structured into a basic block and two further refinements which aim at more advanced students.

An elementary approach to simulating the perihelion of Mercury

“…Although the work of Koerber et al is ambitious, we do not think that it achieves its goal of being accessible to high school students (or even most undergraduates) and that the treatment in terms of the coding is unnecessarily complicated. Obviously, there is the important exception of many European countries where the use of Python and writing of computer programs is an obligatory part of the physics curriculum [10]. In their work, it was stated that they wished to communicate the beauty of theoretical physics to beginner students with an important historical example, but the amount of detail provided was perhaps overwhelming for students approaching this topic for the first time.…”

“…In common with [10], we have shown that Maple can be used to visualise the precession of the orbit of a light body due to relativistic corrections. However, we can now go further and make a quantitative connection with the precession of Mercury (in [10] this part of the analysis was considered to be for advanced students only). To begin, we need to find the angular positions f 1 , f 2 , f 3 and f 4 of the first four perihelia in figure 1 (right).…”

“…In this article, we discuss the pedagogical utility of the perihelion of Mercury as an introduction to general relativity and computer solutions of differential equations. This has been discussed in the literature before by Koerber et al, who carried out a very thorough study of the perihelion [10]. However, although the work of Koerber et al claims to be accessible to advanced high school students, this seems questionable considering the amount of mathematics that they use, as well as a large amount of sophisticated Python code which might be confusing even to undergraduates.…”

“…However, although the work of Koerber et al claims to be accessible to advanced high school students, this seems questionable considering the amount of mathematics that they use, as well as a large amount of sophisticated Python code which might be confusing even to undergraduates. We should emphasise that students in certain European countries are required to learn steps for programs to be implemented in Python, in which case [10] would fit well into the curriculum. For this reason, we view our approach as complementary to that of Koerber et al, with a set of aims which are different and simpler.…”

“…In section 4, we show that our code can reproduce the value for the precession of the perihelion of Mercury when suitable input parameters are introduced. An advantage of our approach is that this does not involve any additional complexity, whereas extracting quantitative information from the code of Koerber et al was considered to be for more advanced students [10]. We should note that the simulation we discuss in this article is most suitable for general undergraduate students, whereas the topic of general relativity is often designated for advanced undergraduates or even beginner graduate students.…”