3-Body Problem Simulation

A Python implementation of the 3-Body Problem, a fundamental problem in astrodynamics that describes the motion of three celestial bodies interacting with each other.

Table of Contents

1. Introduction
2. Dependencies
3. Variables
4. Functions
5. Code Structure
6. Example Use Cases
7. Notes

Introduction

The 3-Body Problem is a classic problem in astrodynamics that describes the motion of three celestial bodies interacting with each other. This code simulates the 3-Body Problem using numerical integration and visualizes the trajectories of the three planets.

Dependencies

• NumPy (numpy)
• Matplotlib (matplotlib.pyplot)
• SciPy (scipy.integrate.solve_ivp)

Variables

Variable	Description	Units
m1, m2, m3	Masses of the three planets	kg
initial_position_1, initial_position_2, initial_position_3	Initial positions of the three planets	m
initial_velocity_1, initial_velocity_2, initial_velocity_3	Initial velocities of the three planets	m/s
t_s, t_e	Time span for the simulation	s
t_points	Time points for the numerical integration	s
solution	Solution of the 3-Body Problem	-

Functions

system_odes

• Description: Defines the system of ordinary differential equations (ODEs) describing the motion of the three planets.
• Parameters:
  • t: Current time
  • S: State vector containing positions and velocities of the three planets
  • m1, m2, m3: Masses of the three planets

update

• Description: Updates the plot at each frame of the animation.
• Parameters:
  • frame: Current frame number
• Returns:
  • Updated plot elements (planet positions and velocities)

Code Structure

The code consists of four main sections:

1. Initialization: Initializes the variables and parameters for the simulation.
2. Numerical Integration: Solves the 3-Body Problem using numerical integration.
3. Visualization: Visualizes the trajectories of the three planets in a 3D plot.
4. Animation: Animates the visualization by updating the plot at each frame.

Example Use Cases

• Run the simulation with different initial conditions to see how the motion of the three bodies changes.
• Modify the interval parameter in the FuncAnimation function to change the speed of the animation.

Notes

• The solve_ivp function from SciPy is used for numerical integration, which provides a robust and efficient method for solving systems of ODEs.
• The FuncAnimation function from Matplotlib is used to create an animated plot by updating the visualization at each frame.

Installation

To install this project, run:

pip install numpy matplotlib scipy

Usage

Run the simulation using:

python 3_body_simulation.py

This will display a 3D animation of the three planets' trajectories. You can modify the initial conditions and time span by updating the m1, m2, m3 variables, initial_position_1, initial_position_2, initial_position_3, initial_velocity_1, initial_velocity_2, initial_velocity_3, t_s, and t_points variables in the code before running it.