Classical Orbital Elements, Kepler’s Equation & Barker’s Equation

Course Overview:

Satellite ground control stations, terrestrial facilities with antennas used for establishing line-of-sight communication with satellites, transmitting uplink signals, and receiving downlink signals, are essential for monitoring satellite health status and downloading valuable data. They are strategically located around the world to provide continuous connectivity with satellites as they orbit Earth. 

In determining the classical orbital elements of space objects, satellite ground control stations play crucial roles. In addition, they gather raw data from a satellite and feed it into orbit determination algorithms to calculate the classical orbital elements and define the satellite’s exact orbit. 

The concept of classical orbital elements, a set of six parameters that define an orbit around a central body, and known as Keplerian elements, is important for the prediction and understanding of the spacecraft and celestial body motion. In the years 1601 and 1619, Johannes Kepler built upon Tycho Brahe’s data to develop his three laws of planetary motion.

The Scientific Renaissance era resulted in Isaac Newton’s 1687 publication of his laws of motion and universal gravitation in Principia.

This course laid the foundation for the description and development of Keplerian orbital elements, which describe the shape, size, and orientation of an orbit, the importance of radar stations in determining spacecraft and celestial body orbital elements, and Kepler’s and Barker’s equations.

Course Modules:

This course is divided into seven modules: Module 1: Radar Stations & Classical Orbital Elements, Module 2: Time Since Periapsis & Special Case of Circular Orbit, Module 3: Kepler’s Equation for Elliptical Motion, Module 4: Baker’s Equation for Parabolic Motion, Module 5: Kepler’s Equation for Hyperbolic Motion, Module 6: Example questions, and Module 7: Exercises.

Course Modules Description:

a) Module 1: Radar Stations & Classical Orbital Elements:

• This module presents radar networks and stations, Classical Orbital Elements, and Determination of Orbital Elements from Position & Velocity Vectors and vice versa.
• Radar stations provide observational data important for determining celestial bodies and spacecraft orbital elements, helping to contribute to a better understanding of the space environment, and performing essential tasks such as spacecraft mission planning, ephemeris prediction, orbit determination, collision avoidance, and a host of others.

b) Module 2: Time Since Periapsis & Special Case of Circular Orbit:

• In this module, Time Since Periapsis & Its Applications, Development of Time Since Periapsis Equation, and Special Case of Circular Orbit are described.
• The time since periapsis is the elapsed time since an orbiting object last passed its periapsis, the point in its orbit where it is closest to the central body.

c) Module 3: Kepler’s Equation for Elliptical Motion:

• This module presents Elliptical Orbit Kepler’s Equation, Newton-Raphson Root Finding Method, and applications of Elliptical Kepler’s Equation.
• Kepler’s equation is important for determining spacecraft positions at any given time in their elliptical orbits around a central body like the Sun.

d) Module 4: Baker’s Equation for Parabolic Motion:

• In this module, the Barker’s equation and its importance are highlighted.
• Barker’s equation is a special form of Kepler’s equation for parabolic orbits that relates the time elapsed from periapsis passage to the position of the orbiting body.

e) Module 5: Kepler’s Equation for Hyperbolic Motion:

• This module presents Kepler’s Equation for Hyperbolic Trajectory, Newton’s Method for solving the equation, and its applications.

f) Module 6: Example questions:

• This module contains four example questions. The aim is to demonstrate how to apply the equations derived during the presentation.

g) Module 7: Exercises:

• For better comprehension of the developed concepts, exercises are provided.
• Through exercises, applications of the derived equations are explained in an easy-to-understand manner.

Learning Objectives:

In this module, you will learn about:

1. Understanding of the importance of radar in determining classical orbital elements of a planet or spacecraft.
2. Ability to determine orbital elements given position and velocity vectors, and vice versa.
3. Understanding of time since periapsis & special case of circular orbit.
4. Understanding of Kepler’s equation for elliptical motion and its derivation
5. Ability to be able to develop Barker’s equation and to understand its importance.
6. Development of Kepler’s equation for hyperbolic motion