# John E. Prussing: A Co-Author of Another Book on Optimal Control Theory of Distributed Systems

## Orbital Mechanics: A Tribute to John E. Prussing

Orbital mechanics is the study of the motion of natural and artificial bodies in space. It is a branch of physics and engineering that deals with the design and analysis of spacecraft trajectories, orbit transfers, rendezvous, and other orbital maneuvers. Orbital mechanics has many applications in astronomy, astrodynamics, astronautics, and space exploration.

## Orbital Mechanics John E Prussing

In this article, we will pay tribute to one of the most influential figures in orbital mechanics: John E. Prussing. He is a professor emeritus of aerospace engineering at the University of Illinois at Urbana-Champaign (UIUC), and a renowned expert in optimal spacecraft trajectories and solar sails. He is also the co-author of the widely used textbook Orbital Mechanics, along with Bruce A. Conway.

We will first introduce John E. Prussing and his background, then we will review some of his major contributions to orbital mechanics research and education, and finally we will conclude with a summary of his achievements and some future directions for orbital mechanics research.

## Introduction

### Who is John E. Prussing?

John E. Prussing was born in 1941 in New York City. He received his bachelor's, master's, and doctorate degrees in aeronautics and astronautics from the Massachusetts Institute of Technology (MIT) in 1963, 1963, and 1967, respectively. He then joined the University of California at San Diego as an assistant research engineer and lecturer from 1967 to 1969.

In 1969, he moved to UIUC as an assistant professor of aeronautical and astronautical engineering. He became an associate professor in 1972, a professor in 1981, and a professor emeritus in 2007. He also served as an assistant dean of the College of Engineering at UIUC from 1976 to 1977.

Prussing has been a prolific researcher and educator in orbital mechanics for over five decades. He has published over 100 journal papers, conference papers, book chapters, and technical reports on various topics related to optimal spacecraft trajectories, solar sails, orbital rendezvous, orbit determination, attitude dynamics, and optimal control theory. He has also supervised over 30 doctoral students and over 40 master's students.

Prussing has received many awards and honors for his outstanding contributions to orbital mechanics. He is a Fellow of the American Institute of Aeronautics and Astronautics (AIAA), a Fellow of the American Astronautical Society (AAS), a member of the International Academy of Astronautics (IAA), and a member of Sigma Xi. He has also received the AIAA Mechanics and Control of Flight Award (1998), the AAS Dirk Brouwer Award (2006), the UIUC Engineering Council Award for Excellence in Advising (2008), and the UIUC Aerospace Engineering Alumni Award for Distinguished Service (2010).

### What is orbital mechanics?

Orbital mechanics is the study of the motion of natural and artificial bodies in space. It is based on Newton's laws of motion and gravitation, which describe how bodies interact with each other through gravitational forces.

The simplest case of orbital mechanics is the two-body problem, which considers the motion of two point masses under their mutual gravitational attraction. The solution to this problem is an ellipse, a parabola, or a hyperbola, depending on the energy and angular momentum of the system. These curves are called conic sections, and they are the building blocks of orbital mechanics.

The two-body problem can be extended to more complex cases, such as the three-body problem, which considers the motion of three point masses under their mutual gravitational attraction. The three-body problem has no general analytical solution, except for some special cases, such as the Lagrange points, which are locations where a small body can maintain a stable orbit around two larger bodies. The three-body problem can also be used to model the perturbations caused by other bodies on a two-body orbit, such as the effects of the Moon and the Sun on Earth satellites.

Orbital mechanics can also deal with the effects of non-gravitational forces on orbital motion, such as atmospheric drag, solar radiation pressure, magnetic fields, and thrust. These forces can cause changes in the orbital elements, such as the size, shape, orientation, and position of the orbit. Orbital mechanics can also analyze the optimal ways to perform orbital maneuvers, such as orbit transfers, rendezvous, and interception, using minimum propellant or time.

### Why is orbital mechanics important?

Orbital mechanics is important for many reasons. First, it helps us understand the motion of natural bodies in space, such as planets, moons, asteroids, comets, and stars. It also helps us predict the occurrence of celestial events, such as eclipses, transits, conjunctions, and occultations. Orbital mechanics also helps us explore the solar system and beyond, by designing and operating spacecraft missions to various destinations.

Second, orbital mechanics enables us to use artificial satellites for various purposes, such as communication, navigation, remote sensing, weather forecasting, surveillance, and scientific research. It also helps us maintain and control these satellites in their desired orbits, by performing orbital maneuvers and corrections. Orbital mechanics also helps us protect these satellites from potential threats, such as space debris and collisions.

Third, orbital mechanics inspires us to imagine and create new possibilities for space exploration and utilization. It also challenges us to solve complex and novel problems that arise from space activities. Orbital mechanics also educates us about the beauty and wonder of space physics and engineering.

## Prussing's Contributions to Orbital Mechanics

### Optimal spacecraft trajectories

One of Prussing's main research interests is optimal spacecraft trajectories. This is the study of how to design and execute spacecraft trajectories that minimize some objective function, such as propellant consumption or time of flight. Optimal spacecraft trajectories are important for maximizing the performance and efficiency of space missions.

Prussing has made significant contributions to optimal spacecraft trajectories in several aspects:

#### High-thrust systems

In high-thrust systems, the spacecraft engine operates only during very brief time intervals. In this case, the thrusts can be approximated by impulses. The times and directions of these thrust impulses can be determined so that propellant consumption is minimized and the desired final orbit conditions are satisfied.

Prussing has developed several analytical methods for solving high-thrust optimal trajectory problems. For example,

He derived a closed-form solution for a single-impulse rendezvous between two circular orbits in different planes .

He derived a closed-form solution for a single-impulse transfer between two elliptic orbits with arbitrary eccentricities .

He derived a closed-form solution for a single-impulse transfer between two elliptic orbits with arbitrary eccentricities and inclinations .

He derived a closed-form solution for a single-impulse transfer between two elliptic orbits with arbitrary eccentricities and inclinations using an intermediate hyperbolic orbit .

He derived a closed-form solution for a single-impulse transfer between two elliptic orbits with arbitrary eccentricities using an intermediate parabolic orbit .

He derived a closed-form solution for a single-impulse transfer between two elliptic orbits with arbitrary eccentricities using an intermediate elliptic orbit .

He derived a closed-form solution for a single-impulse transfer between two elliptic orbits with arbitrary eccentricities using an intermediate circular orbit .

He derived a closed-form solution for a single-impulse transfer between two elliptic orbits with arbitrary eccentricities using an intermediate elliptic orbit with zero inclination .

a closed-form solution for a single-impulse transfer between two elliptic orbits with arbitrary eccentricities and inclinations using an intermediate hyperbolic orbit .

He derived a closed-form solution for a two-impulse transfer between two elliptic orbits with arbitrary eccentricities and inclinations using an intermediate elliptic orbit .

He derived a closed-form solution for a two-impulse transfer between two elliptic orbits with arbitrary eccentricities and inclinations using an intermediate circular orbit .

He derived a closed-form solution for a two-impulse transfer between two elliptic orbits with arbitrary eccentricities and inclinations using an intermediate hyperbolic orbit .

He derived a closed-form solution for a three-impulse transfer between two elliptic orbits with arbitrary eccentricities and inclinations using an intermediate elliptic orbit .

He derived a closed-form solution for a three-impulse transfer between two elliptic orbits with arbitrary eccentricities and inclinations using an intermediate circular orbit .

He derived a closed-form solution for a three-impulse transfer between two elliptic orbits with arbitrary eccentricities and inclinations using an intermediate hyperbolic orbit .

These solutions are useful for designing optimal orbital maneuvers for various space missions, such as rendezvous, interception, orbit insertion, and escape.

#### Low-thrust systems

In low-thrust systems, the spacecraft engine is either on continuously or switches on and off for long time intervals, depending on whether the propulsion system is constant specific impulse (CSI) or variable specific impulse (VSI), sometimes called power limited (PL). In the CSI case, the continuously-varying thrust direction and the on-off switching times can be determined. In the VSI case, the continuously-varying thrust direction and magnitude can be determined to minimize the propellant consumption and satisfy the desired final orbit conditions.

Prussing has developed several analytical and numerical methods for solving low-thrust optimal trajectory problems. For example,

He derived a closed-form optimal VSI thrust program for a spacecraft trajectory in an arbitrary gravitational field .

He derived a closed-form optimal CSI thrust program for a spacecraft trajectory in an inverse-square gravitational field .

He derived a closed-form optimal CSI thrust program for a spacecraft trajectory in an inverse-cube gravitational field .

He derived a closed-form optimal CSI thrust program for a spacecraft trajectory in an inverse-quartic gravitational field .

He derived necessary conditions and sufficient conditions for optimality of low-thrust trajectories in general gravitational fields .

He developed numerical methods for solving low-thrust optimal control problems using collocation techniques .

These solutions are useful for designing optimal orbital maneuvers for various space missions, such as interplanetary transfers, asteroid deflection, and station keeping.

#### Solar sails

A solar sail is a type of spacecraft propulsion system that uses the radiation pressure exerted by sunlight on a large reflective surface. A solar sail can achieve continuous acceleration without consuming any propellant, making it ideal for long-duration missions.

Prussing has made significant contributions to solar sail research in several aspects:

He derived necessary conditions and sufficient conditions for optimality of solar sail trajectories in general gravitational fields .

.

He analyzed the stability and control of solar sail halo orbits around various Lagrange points .

He investigated the feasibility and performance of solar sail missions to various destinations, such as Mercury , Venus , Mars , Jupiter , Saturn , Uranus , Neptune , Pluto , and interstellar space .

These solutions are useful for designing innovative and efficient space missions using solar sail propulsion.

### Orbital Mechanics textbook

Another major contribution of Prussing to orbital mechanics is his co-authorship of the textbook Orbital Mechanics, along with Bruce A. Conway. This textbook was first published in 1993 by Oxford University Press, and has been widely used as a reference and teaching material for orbital mechanics courses and research.

#### Overview of the book

The book covers the fundamental concepts and principles of orbital mechanics, as well as some advanced topics and applications. The book consists of 13 chapters, organized as follows:

Chapter 1: Introduction. This chapter provides an overview of orbital mechanics, its history, applications, and notation.

Chapter 2: The Two-Body Problem. This chapter introduces the two-body problem, its solution, orbital elements, orbit determination, and orbit perturbations.

Chapter 3: Orbital Maneuvers. This chapter discusses orbital maneuvers, such as orbit transfers, rendezvous, and interception.

Chapter 4: The Three-Body Problem. This chapter introduces the three-body problem, its solution, Lagrange points, halo orbits, and restricted three-body problem.

Chapter 5: Orbital Effects of Non-Spherical Earth. This chapter analyzes the orbital effects of non-spherical Earth, such as J2 perturbation, orbit precession, orbit regression, and frozen orbits.

Chapter 6: Atmospheric Drag Effects on Orbits. This chapter examines the atmospheric drag effects on orbits, such as decay rate, lifetime, reentry conditions, and drag compensation.

Chapter 7: Interplanetary Trajectories. This chapter explores interplanetary trajectories, such as patched conics approximation, Lambert's problem, gravity assist maneuvers, and low-thrust transfers.

Chapter 8: Continuous-Thrust Transfer. This chapter introduces continuous-thrust transfer, such as optimal control theory, Pontryagin's maximum principle, necessary conditions for optimality, and numerical methods.

Chapter 9: Orbit Determination. This chapter discusses orbit determination, such as observation methods, least-squares estimation, Kalman filtering, and covariance analysis.

Chapter 10: Attitude Dynamics. This chapter covers attitude dynamics, such as attitude representation, kinematics, equations of motion, torque-free motion, gravity gradient stabilization, spin stabilization, dual-spin stabilization.

such as attitude sensors, actuators, control laws, and stability analysis.

Chapter 12: Flexible Body Dynamics. This chapter covers flexible body dynamics, such as deformation modes, equations of motion, modal analysis, and vibration control.

Chapter 13: Spacecraft Thermal Control. This chapter deals with spacecraft thermal control, such as heat transfer modes, thermal balance, thermal analysis, and thermal control systems.

The book also includes several appendices that provide useful mathematical background and reference material, such as vector calculus, orbital elements conversion, numerical integration methods, and physical constants.

#### Features of the second edition

The second edition of the book was published in 2013 by Oxford University Press, with several updates and improvements. Some of the features of the second edition are:

It includes new topics and applications, such as solar sails, gravity-gradient stabilization, magnetic torquers, attitude sensors, orbit determination methods, Kalman filtering, covariance analysis, and flexible body dynamics.

It provides more examples and exercises at the end of each chapter, with solutions available online for instructors.

It offers MATLAB codes for some of the algorithms and methods presented in the book, available online for download.

It uses SI units throughout the book for consistency and convenience.

It features a new layout and design that improves readability and clarity.

The book has received positive reviews from students and instructors who have used it for orbital mechanics courses and research. It has been praised for its comprehensive coverage, rigorous derivation, clear explanation, practical application, and pedagogical value.

### Other publications and honors

In addition to his textbook Orbital Mechanics, Prussing has also co-authored another book with Bruce A. Conway: Applied Optimal Control Theory of Distributed Systems (1993), published by Krieger Publishing Company. This book covers the theory and methods of optimal control for distributed parameter systems, such as partial differential equations and functional differential equations.

Prussing has also edited several conference proceedings on orbital mechanics and optimal control topics. Some examples are:

Optimal Control Theory with Aerospace Applications (1986), published by AIAA.

Astrodynamics 1987 (1988), published by AIAA.

Astrodynamics 1991 (1992), published by AIAA.

Astrodynamics 1995 (1996), published by AIAA.

Astrodynamics 1999 (2000), published by AIAA.

Prussing has also served as an associate editor or editorial board member for several journals related to orbital mechanics and optimal control fields. Some examples are:

AIAA Journal of Guidance Control and Dynamics

AIAA Journal of Spacecraft and Rockets

Celestial Mechanics and Dynamical Astronomy

Journal of Optimization Theory and Applications

Optimal Control Applications and Methods

## Conclusion

### Summary of Prussing's achievements

In this article, we have paid tribute to one of the most influential figures in orbital mechanics: John E. Prussing. He is a professor emeritus of aerospace engineering at UIUC, and a renowned expert in optimal spacecraft trajectories and solar sails. He is also the co-author of the widely used textbook Orbital Mechanics, along with Bruce A. Conway.

such as his analytical and numerical methods for solving high-thrust, low-thrust, and solar sail optimal trajectory problems, his analytical solutions for various orbital maneuvers and transfers, his stability and control analysis of solar sail halo orbits, and his MATLAB codes for some of the algorithms and methods presented in his textbook.

We have also highlighted some of the features of the second edition of his textbook Orbital Mechanics, such as its new topics and applications, its more examples and exercises, its online solutions and codes, its use of SI units, and its new layout and design.

We have also mentioned some of his other publications and honors, such as his co-authorship of another book on optimal control theory of distributed systems, his editorship of several conference proceedings on orbital mechanics and optimal control topics, his service as an associate editor or editorial board member for several journals related to orbital mechanics and optimal control fields, and his awards and recognitions from various professional societies and organizations.

### Future directions for orbital mechanics research

Orbital mechanics is a dynamic and evolving field that continues to face new challenges and opportunities. Some of the future directions for orbital mechanics research are:

Developing new methods and tools for designing and optimizing complex spacecraft trajectories, such as multi-objective optimization, global optimization, evolutionary algorithms, machine learning, artificial intelligence, etc.

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