Chapter 1
Introduction

Spacecraft dynamics in this book means the study of motion of man-made objects in space, subject to both natural and artificially induced forces, which includes orbital mechanics and attitude dynamics.

Orbital mechanics is the science concerned with the trajectory motion of a spacecraft, whereas attitude dynamics is concerned with the orientation motion of a spacecraft.

This text book mainly focuses on trajectory motion, that is, the motion of mass center of a man-made spacecraft or a natural body. Natural body's motion is also within the scope of celestial mechanics. A basic introduction to attitude dynamics is given in this book.

Since dynamics are closely related to GNC (guidance, navigation and control), a brief discussion about its meaning is given here. Simply speaking, guidance answers the question where to go, navigation tells us where you are, and control is concerned with how to go. In this book, orbit and attitude determination belong to the category of navigation, while orbit maneuver, attitude stabilization and control are in the category of control. Dynamics form the foundation in analyzing a GNC system. Very little discussion is given to guidance in this book, and it can be simply regarded as a reference. Figure 1.1 shows the basic relations between these definitions using the classical feedback control diagram.

Figure 1.1 Basic relations between GNC and dynamics

Many sub-systems of a spacecraft are closely related to GNC, including attitude control, communication, power supply, thermal control, structure, propulsion subsystem, etc.

1.1 History

The scientists who have made great contributions to the development of orbital mechanics include Aristotle, Ptolemy, Copernicus, Tycho Brahe, Kepler, Galileo, Newton, Euler, Lagrange,....

1.1.1 Kepler's Laws

The Danish astronomer Tycho Brahe (1546-1601) gathered extremely accurate observational data on planetary motion. He developed and maintained detailed and precise records. But he didn't capitalize on observations, as he lacked thevision and mathematical skills. And he kept the Earth at the center of the planets.

Johannes Kepler (1571-1630), a gifted mathematician and astronomer, distilled Brahe's observation data to provide the first quantitative statements about orbital mechanics, which is known as Kepler's three Laws:

1. Law of ellipse: The orbits of the planets are ellipses with the Sun at one focus.
2. Law of equal area: The line joining a planet to the Sun sweeps out equal areas at equal times.
3. Law of harmony: The square of the orbital period (time to complete one orbit) is directly proportional to the cube of the average distance between the Sun and the planet.

Kepler's Laws are descriptive, which answer how the planets move around the Sun. But they can not explain why planetary motion satisfies these three laws. It was Isaac Newton (1642-1727), who established the mathematical foundation from which Kepler's Laws can be derived.

1.1.2 Newton's Laws

Newton is arguably the greatest physicist ever. His major discoveries and developments were differential and integral calculus, the gravitational laws, and also contributions in optics.

Isaac Newton (1642-1727) established the fundamentals of celestial mechanics based on the earlier work of Tycho and Kepler. Newton formulated the basic concepts of the Laws of Motion and Law of Gravitation around 1666, but it was not until 1687, at the urging of Edmund Halley, the Principia was published. It originally presented the three laws of motion.

1. Law of inertia: Every body continues in its state of rest or of uniform motion in a straight line unless it is compelled to change that state by forces impressed upon it.
2. Law of momentum: The rate of change of momentum is proportional to the force impressed and is in the same direction as that force. This law is also sometimes called the first principle.
3. Law of action and reaction: To every action there is always an opposed equal reaction.
4. Law of Universal Gravity: Any two bodies attract one another with a force proportional to the product of their masses and inversely proportional to the square of the distance between them.

Thus,

1.1

or in vector form:

1.2

where is universal gravitational constant, where is the force on and is a vector from to . This assumes a spherically symmetric point mass. See Figure 1.2.

Figure 1.2 Universal gravity

The subsequent major contributions in orbital mechanics are mainly in the sense of orbital determination. Newton needed three observations of position vectors; Halley mastered and refined Newton's methodology; Lambert used geometrical arguments; Lagrange developed mathematical basis for orbit calculations; Laplace developed new methodology; and Gauss summarized, simplified and completed the orbit determination work.

1.1.3 Space Missions

Some major figures in developing theory and practice in space technology should be mentioned. Konstantin Tsiolkovsky-Russia (1857-1935), who derived the fundamental equation of rocketry, is called the father of modern space technology. Robert Goddard-USA (1882-1945) built the laboratory experiments of rocket engine. Wernher Von Braun-Germany/ USA (1912-1977) brought the space technology to practical application, he designed the V2 rocket and Explore I satellite.

During the early space age, Sputnik was the first man-made object flying in space, having been launched in October 1957. Yuri Gagarin was the first man in space in April 1961. And Neil Armstrong was the first man to land on the Moon in July 1969. These are some of the milestones in human space exploration.

China's first satellite Dongfanghong-1 was successfully launched in 1970. In 2003, the Shenzhou-5 spaceship carried out manned spaceflight and Chang'e-1 in 2007 entered into lunar orbit. These have been the major breakthroughs in the history of Chinese space technology.

At the present time, space technology is beging used everywhere, including communications, weather forecasting, science exploration, navigation etc. See examples in Table 1.1 and some of the figures and data in Chapter 7.

Table 1.1 Classification and examples of missions

Mission Orbit/name (km) Period (hr) (deg) Communication Geostationary 42 158 24 0 0 Remote sensing Sun-synchronous 6500-7300 1.5 95 0 Navigation (GPS) Semi-synchronous 26 610 12 55 0 Comm/Intel Molniya 26 571 12 63.4 0.72 335 Space station ISS (ZARYA) 6770.6 1.5401 51.643 0.0008 Space station Tiangong 1 6731.5 1.5268 42.785 0.0008 Science HST 6968 1.617 28.5 0.001 Weather Fengyun 1D 7239.7 1.703 98.758 0.0016 Weather Fengyun 2F 42 166 23.936 2.196 0.0007 Navigation Beidou IGSO 2 42 167 23.937 54.625 0.0021 Navigation Beidou G3 42 167 23.937 1.51 0.0002

There are an extensive number of orbit types, for which a rough classification can be made. For a low Earth orbit (LEO), its height is normally less than 2000 km (altitude). Mid Earth Orbit (MEO)'s height is between 2000 and 30 000 km. Highly Elliptical Orbit (HEO) is an orbit with large eccentricity. Geosynchronous Orbit (GEO) (circular)'s period is the time it takes Earth to rotate once with respect to (wrt) stars, . Polar orbit's inclination is nearly 90 degrees. Molniya's orbit is a highly eccentric orbit with 12 hr period (developed by Soviet Union to optimize coverage of the Northern hemisphere).

In Table 1.1, ISS (International Space Station), Tiangong 1 are space stations whose orbits belong to low Earth orbit (LEO). And HST (Hubble Space Telescope) is also in low Earth orbit. Fengyun 2F is a geostationary orbit (GEO) whose inclination is near zero. Beidou 2 is a geosynchronous orbit whose inclination is not zero, but its period is still near 24 hrs. A GPS satellite belongs to a MEO satellite. Molniya is a high elliptical orbit designed by the former Soviet Union. Most Sun-synchronous satellites, such as Fengyun 1, are polar orbit satellites whose inclinations are near 90 deg.

An excellent source of information on specific spacecraft is the NASA JPL Missions, located on the World Wide Web at http://www.jpl.nasa.gov/missions/.

1.2 Coordinate...