Comparative Analysis of Numerical Methods for Earth Orbit Calculations in Celestial Mechanics

Abstract

Numerical methods are indispensable for simulating celestial mechanics, specifically Earth's orbit around the Sun. These calculations are vital for understanding orbital dynamics, influencing satellite trajectories, seasonal patterns, and climate prediction. By leveraging numerical methods, researchers can analyse complex gravitational interactions and validate fundamental principles like Kepler's Laws. However, many numerical methods suffer from accuracy limitations, leading to errors in long-term orbital calculations, which can affect practical applications like satellite operations and space missions. This study compares Euler’s method and the Fourth Order Runge-Kutta (RK4) method to validate Kepler's First Law, which posits that planets follow elliptical trajectories. MATLAB simulations were employed to implement these methods, and a convergence test was conducted to evaluate their accuracy. The findings reveal that while Euler’s method offers simplicity, it accumulates significant numerical errors, leading to deviations in trajectory calculations. Conversely, RK4 method demonstrates superior precision and efficiency, accurately simulating Earth’s elliptical orbit with minimal error. These results underscore the critical role of higher-order numerical methods in modelling orbital dynamics, offering valuable insights for applications in celestial mechanics and astrodynamics.