Satellite Orbit Determination

Course contents

1 Dynamics

• Introduction to dynamics

• Planetary Gravity field

• Tides and the three-body problem

• Hill radius and Roche limit

• Relation to planetary sciences and astrodynamics

• Solving Equations of motion

• Reformulate orbit problem as a system of ordinary differential equations

• Efficiency and accuracy of numerical integration methods

• Implementation of numerical integration methods

2 Observations techniques and reference systems

• Observation techniques

• Laser, Doppler, optical and VLBI observations

• Refraction, Electromagnetism, radio- and optical technology

• Tropospheric and ionospheric refraction

• Relativity and the definition of time,

• Classification of time systems (UTC, TAI, etc)

• Light-time effect

• Quality of clocks (Allan Variance behavior of clocks)

• Coordinate reference systems

• Local and global coordinate systems

• Definition of geoid and reference ellipsoid, height systems

• Precession and nutation, polar motion, polar wander.

• Newton or Einstein, consequences for reference systems

3 Statistics

• Random variables, probability density functions, moments, hypothesis testing

• Least squares minimization

• Unconstrained linear parameter estimation,

• Data weighting

• Nonlinear parameter estimation.

• Rank deficient equation systems

• Singular value decomposition and Empirical Orthogonal functions

• Penrose Moore inverse

• Compatibility conditions

• General and homogeneous solutions

• Constrained linear parameter estimation

• Mechanization of parameter estimation algorithms

• Choice of algorithms

4 Orbit determination

• Perturbation analysis and variational problems

• State transition matrix for initial state vector problems

• Partial derivatives for dynamical parameters

• Parameter estimation

• Identification of parameters

• Batch least squares

• Sequential least squares: The Kalman filter, theory and implementation

5 Applications

• Global Navigation Satellite Systems:

• Technology and terminology,

• Various data processing strategies and available software

• Modeling deformation of the solid Earth,

• The Earth's gravity field and thermospheric density

• Satellite laser ranging and Doppler tracking via DORIS

• Technology and terminology, results and applications

• Observing changes in the cryosphere with satellites

• Hydrology and Oceanography observed with satellites

6 Individual assignments in quarter 2

• The assignments will be announced during the quarter and typically take the following form

• One test-your-knowledge exercise related to dynamics, observation systems or reference systems

• A graded assignment related to GNSS applied to orbit determination

• A graded assignment related to batch or sequential least squares methods

7 Numerical Practical in quarter 3

• Exercises with the Ghost software, typical examples are to solve:

• Initial value problems (state vector estimation)

• Problems with parameters in a dynamic model (drag parameter estimation)

• Problems with time bias parameters

The final grade will come together from:

1. Written examination resulting in a grade

2. Nominally two individual assignments resulting in a weighted average grade

3. Numerical practical resulting in "pass" or "fail".

The grading rules are as follows:

1. The numerical practical result should be: "pass"

2. The written examination result should be at least a 5,0

3. In order to participate in the numerical practical you must have passed the individual assignments with at least a weighted average of 5,0

4. No claims can be made with regard to individual assignment results or numerical practical results obtained before the previous academic year. For ae4-872 in 2025-2026 it means that results from 2024-2025 are still accepted, however, we do not accept results from 2023-2024 or earlier academic years. (Academic years run from September to August)

Due to the master curriculum refresh it is expected that ae4-872 orbit determination will run for the last time in academic year 2025-2026. This means that all course elements (exams, assignments and numerical practical) should be completed by the students in academic year 2025-2026.