PICUP Exercise Sets: 3 body problem - Code for visualization of 2D Earth-Moon-3rd body trajectories

Exercise Sets » 3 body problem - Code for visualization of 2D Earth-Moon-3rd body trajectories

3 body problem - Code for visualization of 2D Earth-Moon-3rd body trajectories

Developed by Fabio Daniel Saccone - Published September 5, 2024

DOI: 10.1119/PICUP.Exercise.3BodyProblem

The problem of Three Bodies interacting by gravitational forces among them does not have an exact solution, as it can be found in a Two-Body problem. For example, when we analyze the trajectory of two bodies with very different masses interacting by gravitational forces, we find as a result that the lighter one describes an orbit that corresponds to an elliptic trajectory, while the more massive body is located on the focus. From the center-of-mass reference system, both bodies move on an elliptic orbit. This activity is a simulation exploration of a three-body problem of the Earth, the Moon, and an unknown third body. The trajectories observed are strongly dependent of the initial conditions and mass of the 3rd body.

Subject Area	Mechanics
Level	First Year
Available Implementation	IPython/Jupyter Notebook
Learning Objectives	The learning objectives of this exercise are: Apply Gravitational law to solve a hypothetical 3-body problem with a mass entering to the Earth and Moon orbit (Exercises 1-4) Because of no exact solution is available for this problem, a visualization code of trajectories and their dependence with initial conditions is implemented (Exercises 1-4) Also helpful, it is to visualize the bodies orbits from the center of mass reference frame. (Exercise 5)
Time to Complete	30 min

These exercises are not tied to a specific programming language. Example implementations are provided under the Code tab, but the Exercises can be implemented in whatever platform you wish to use (e.g., Excel, Python, MATLAB, etc.).

It is requested a code to visualize the trajectories of Earth. Moon and the 3rd body, employing gravitational law frame and the Euler Algorithm to calculate it.

The following selections, for the 3-body problem, are suggested:

Exercise 1)

A Moon’s trapped orbit (i. e. 95% of Moon to Earth distance for the 3rd body)

Exercise 2)

An Earth’s trapped orbit (i. e. A half a Moon mass and a half distance to Earth for the 3rd body)

Exercise 3)

A very small 3rd body near to Moon

Exercise 4)

A chaotic Moon trajectory (hint: 90 % of both, $m_{Moon}$ and $r_{Moon-Earth}$ for the 3rd body

Exercise 5):

Solve all previous cases from a center of mass reference frame. (See Theory section for definition of $\vec{r_{MC}}$ and $\vec{v_{MC}}$ , using them to recalculate positions and velocities of each body)

Free exercise

Solve for arbitrary selection of $x$ and $f$ parameters, seen their trajectories and also the orbits from MC.

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Credits and Licensing

Fabio Daniel Saccone, "3 body problem - Code for visualization of 2D Earth-Moon-3rd body trajectories," Published in the PICUP Collection, September 2024, https://doi.org/10.1119/PICUP.Exercise.3BodyProblem.

DOI: 10.1119/PICUP.Exercise.3BodyProblem

The exercises are released under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 license

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