In the past few decades, Kessler syndrome (named after Donald J. Kessler) has become a point of concern in the field of Space Situational Awareness and the future of space missions. It refers to a scenario, where space debris in Earth's orbits collides and creates an exponential increase in space debris numbers leading to more collisions and more debris. In order to handle the resulting challenges like conjunction analysis, tracking, and probability of collisions, the State Transition Matrix (STM) and Tensors (STTs) of the orbit problem play a significant role. In addition, STM and STTs are ubiquitous in spaceflight dynamics, guidance, navigation, and control applications. Several methods exist in the literature for computing the STM and the STTs of the orbit problem; however, all these methods are either restricted by a simplified gravity model, computational accuracy or computational efficiency. In this dissertation, an adaptive Analytic Continuation is studied as a procedure for computing the STM and STTs of the perturbed Two-body problem. Analytic Continuation is a Taylor series based semi-analytic integration method that utilizes recursions of high-order time derivatives and the Leibniz rule to produce a solution with arbitrary accuracy at a fraction of the computational cost of finite difference methods. In this work, the method is used to compute the STM and the second order STT for the perturbed two-body problem. An adaptation technique is developed for keeping a balance between the number of higher order time derivatives and the time-step to achieve prescribed tolerances. Analytic Continuation is also adopted in a high-fidelity estimation framework (AC-EKF) to provide accurate orbit estimation results for a space-based space surveillance network of observers. Test cases on LEO, MEO, GTO and HEO show machine precision accuracy in the symplectic nature of the gravity perturbed STM and STT irrespective of the number of orbital revolutions. Gravity and atmospheric drag perturbed STM shows at least 3 times more accurate results when compared to finite difference methods in the initial error propagation of the trajectories in a span of 10 orbit periods. Furthermore, by incorporating second order STT, the error propagation results are improved by 2 - 4 orders of magnitude. Finally, results from AC-EKF show the utility of the method to accurately predict the error covariance in the absence of sensor coverage. As future work, Analytic Continuation will be expanded to compute arbitrarily high-order STTs with applications in orbit prediction and trajectory design.


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Graduation Date





Elgohary, Tarek


Doctor of Philosophy (Ph.D.)


College of Engineering and Computer Science


Mechanical and Aerospace Engineering

Degree Program

Mechanical Engineering


CFE0009841; DP0027782





Release Date

June 2023

Length of Campus-only Access


Access Status

Doctoral Dissertation (Open Access)