Gauss Method for Preliminary Orbit Determination

This repository contains MATLAB code for implementing the Gauss method to determine the preliminary orbit of a celestial object based on tracking station observations. All equations are taken from Orbital Mechanics for Engineering Students, 2nd Edition by Howard D. Curtis. The provided script includes the following features:

• Tracking Station Position Calculation: Determines the inertial position vector of the tracking station at multiple observation times considering Earth's flattening factor.
• Directional Cosine Vectors: Computes unit vectors in the direction of the slant range vector.
• Scalar Quantities Computation: Calculates cross products and dot products essential for further orbit determination steps.
• Polynomial Coefficients and Roots Calculation: Forms and solves an eighth-order polynomial to find the range at the second observation.
• Initial Position and Velocity Vectors: Derives the position and velocity vectors at the second observation time.
• Iterative Refinement: Applies an iterative algorithm for improved accuracy based on a tolerance level.
• Orbital Elements Calculation: Computes specific angular momentum, inclination, ascending node, eccentricity, argument of perigee, true anomaly, and semi-major axis.
• Orbit Plotting: Visualizes the calculated orbit in a 3D space along with the equatorial plane and Earth's position.

Key Concepts

• Gauss Method: An analytical technique for determining the orbit of an object from three observations, following Algorithm 5.5 from Orbital Mechanics for Engineering Students, 2nd Edition
• Lagrange Coefficients: Used to relate the position and velocity vectors at different times.
• Newton-Raphson Method: An iterative numerical technique used for finding successively better approximations to the roots (or zeroes) of a real-valued function.
• Universal Kepler Equation: A generalized form of Kepler's equation that applies to all types of orbits (elliptical, parabolic, hyperbolic) and is used to compute the position of an orbiting body over time.
• Stumpff Functions: Special functions used in solving Kepler's equation for orbital motion.
• Iterative Improvement: Enhances the accuracy of the determined orbit by iterating on the calculated ranges, following Algorithm 5.6 from Orbital Mechanics for Engineering Students, 2nd Edition
• Orbital Elements Calculation: Extracts orbital elements from the state vector using Algorithm 4.2 from Orbital Mechanics for Engineering Students, 2nd Edition

Getting Started

Clone the repository and run the MATLAB script to execute the orbit determination process. Ensure MATLAB is installed on your system.

git clone https://github.com/ShreyasP36/GaussIOD.git
cd GaussIOD

Required

To use this script, you need the following data:

• Right Ascension at three different time points.
• Declination at three different time points.
• Local Sidereal Time at three different time points.
• Altitude of the tracking station.
• Latitude of the tracking station.

Usage

Open the gauss_orbit_determination.m script in MATLAB and assign data to the input variables as per your needs. Execute the script to display intermediate results and plot the determined orbit.

Requirements

• MATLAB
• Basic understanding of orbital mechanics and celestial navigation

Repository Structure

• gauss_orbit_determination.m: Main MATLAB script for orbit determination.
• README.md: Description and usage instructions.

Example Output

Below is an example plot of the determined orbit:

License

This project is licensed under the MIT License - see the LICENSE file for details.

Citation

• Howard D. Curtis, Orbital Mechanics for Engineering Students, Third Edition, 2010