Numerical Simulation of Earth's Orbit Using Runge–Kutta 4th Order Method

Overview

This project focuses on numerical simulation of Earth’s orbit around the Sun using Newton’s Law of Gravitation and the Runge–Kutta 4th Order (RK4) method. By converting the two-body gravitational problem into a system of first-order differential equations, we compute Earth’s position and velocity step by step, starting from well-defined initial conditions.

The simulation produces a step-by-step numerical table for the first few days, helping students understand how position and velocity evolve with time. It then performs a full 365-day integration and visualizes the result as a 2D orbit plot, clearly showing Earth’s nearly circular path around the Sun. This project connects theory and practice, bridging classical mechanics, differential equations, and computational modeling.

Suitable For

• BS Mathematics – numerical methods, differential equations, dynamical systems

• BS Physics – classical mechanics, celestial mechanics, orbital dynamics

• BS Computer Science – scientific programming, simulation modeling, visualization

• BS Data Science – computational modeling, numerical integration, data visualization

Technologies Used

• Programming Language: Python

• Libraries: NumPy, Matplotlib

• Techniques: RK4 numerical integration, vectorized computation

• Visualization: Simulation tables, 2D trajectory plots, optional orbit animation

Features

• Uses realistic physical constants (G, M, AU) and a 1-day time step

• Sets up initial conditions for Earth at perihelion with tangential velocity

• Implements RK4 step-by-step integration for position and velocity

• Generates a simulation table showing the first 5 integration steps

• Produces a full-year orbital plot showing Earth’s nearly circular path

• Optionally creates an MP4 animation of the orbit

Deliverables

• Complete Python source code for RK4 simulation

• LaTeX-based documentation (theory, methodology, results, interpretation)

• Figures: orbit plot and simulation table

• Optional animation video showing Earth’s motion

Benefits for Students

• Hands-on experience with numerical integration of ODEs

• Clear understanding of how Newtonian gravity predicts orbital motion

• Practical skills in Python for scientific computing and visualization

• Excellent portfolio-ready capstone project combining math, physics, and coding