MATHEMATICAL ASPECTS OF NASA MISSIONS

Teachers of mathematics, like most adults in today's world, can hardly fail to be aware of the rapid development of space science. We realize that the spectacular achievements of the space program have depended heavily on mathematics‹mathematics that is generally complex, advanced, and well beyond the level of most secondary school curricula. Even though this perception is valid, there are many significant aspects of space science that can be understood using only high school mathematics.

The exploration of space naturally uses the tools and techniques of astronomy. Astronomy in turn is gaining much new information as a result of sending scientific probes and satellites beyond Earth's atmosphere. Because astronomy has stimulated the growth of many of the concepts and methods of mathematics, the high school teacher will find here much that is familiar. However, in some instances the way mathematics is used to solve real-life problems is rather different from methods emphasized in school courses.

In this opening chapter, we shall examine several recent achievements of the National Aeronautics and Space Administration and identify mathematical ideas and questions that may be of interest to high school teachers and students. When appropriate, we will refer to a problem illustrating some aspect of the subject and worked elsewhere in the book.

THE SPACE SHUTTLE

The Space Shuttle (Fig. 1.1) is a true aerospace vehicle‹it takes off like a rocket, operates in orbit as a spacecraft, and lands like an airplane. To do this takes a complex configuration of three main elements: the Orbiter, a delta-winged spacecraft-aircraft, about the length of a twin-jet commercial airliner but much bulkier; a dirigible-like external tank, the only expendable element, secured to the Orbiter's belly and containing two million liters of propellant (Chapter 4, Problem 5); a pair of reusable solid rocket boosters, each longer and thicker than a railway tank car and attached to the sides of the external tank.

Each Space Shuttle is meant to be just one element in a total transportation system linking Earth with space. In addition to providing for continued scientific investigations by transporting such systems as the Spacelab and the Large Space Telescope, recently renamed the Edwin P. Hubble Space Telescope, into orbit (Chapter 3, Problem 4), the Space Shuttles are also expected to carry the building blocks for large solar-power space stations or huge antenna-bearing structures for improved communication systems (Chapter 4, Problems 9 and 10). Structures that would be too fragile to stand up under their own weight on Earth will be folded up in the Shuttle's cargo bay and assume their final shapes in the microgravity environment of space. The Shuttle will also be capable of carrying a work force of seven people and returning them home after the completion of their work.

One of the most basic mathematical problems raised by the launching and controlling of a Shuttle or any other spacecraft is that of describing its motion. This problem requires the ability to specify the position of the spacecraft's center of mass and its attitude (orientation) and to describe changes in both during flight. The specification of position and attitude can be accomplished by setting up suitable coordinate systems (Chapter 7, Problem 10). Instruments to determine a spacecraft's attitude are most effectively referenced to a spacecraft-based coordinate system, whereas ground control is best accomplished in terms of an Earth-based system. This dual-based system necessitates transformations between coordinate systems (Chapter 7, Problem 1, and Chapter 8, Problem 2). Describing a change of position and attitude requires an understanding of the measurement of time (Chapter 2, Problem 11). It is interesting to note here that our definition of a day on our rotating Earth must be redefined for a Space Shuttle Orbiter crew. For them the Sun might rise again and again every hour and a half!

THE PLANETARY PROBES

The launch of the two Voyager spacecraft in the summer of 1977 climaxed a series of fruitful missions of planetary exploration including the Mariner, Viking, and Pioneer series of probes to Mercury, Venus, Mars, Jupiter, and Saturn. All these missions sent back new information about the structure and composition of these planets and their associated moons. We focus in this book on some of the results of Voyager 1 and Voyager 2. These probes, which benefited from more highly developed instrumentation and computer capability than their predecessors, approached closer to Jupiter (Chapter 7, Problem 11) and Saturn than previous flights did. Stunning pictures resulted, showing the unanticipated presence of active volcanoes on Jupiter's moon Io (Chapter 10, Problem 6) and the fine structure of Saturn's rings.

Among the mathematical problems that arose in these missions were the following.

1. Transmitting spacecraft observations back to Earth (Chapter 5, Problems 2 and 3, and Chapter 8, Problem 1).

2. Determining the time of transmission of spacecraft observations (Chapter 3, Problem 5).

3. Calculating the rotation period for planets such as Saturn, which is not solid and has no outstanding observable features like Jupiter's Great Red Spot (Chapter 2, Problem 13).

SATELLITES

NASA began its formal existence in 1958 and by the end of 1979 had successfully launched more than 300 large and small satellites with missions as diverse as observing Earth's weather (Synchronous Meteorological Satellite [SMS] series) and resources (Landsat series), providing communication links for television signals (Applications Technology Satellite [ATS] series), and measuring solar radiation outside Earth's atmosphere (Orbiting Solar Observatory [OSO] series).

The design of these satellites and their experiments and the analysis of the data gathered involve a variety of mathematical questions. We shall consider some of the following examples.

1. The connection between the conic sections and the law of gravitation (See Appendix).

2. For elliptic orbits, the connection between the orbit parameters and the period of revolution (Chapter 9, Problem 11) and the determination of the exact position of a satellite in its orbit at a specified time (Chapter 9, Problems 19 and 20).

3. The geometry necessary to correct for distortions arising when flat pictures are made of a curved Earth (Chapter 7, Problems 7 and 9, and Chapter 10, Problem 2).

4. The need for logarithms to understand how radiation is absorbed by Earth's atmosphere (Chapter 6, Problem 3).

5. The mathematical analysis of the reflective properties of the conic sections needed to design an X-ray telescope (Chapter 9, Problems 21 and 22).

6. The judicious use of approximation (Chapter 3, Problem 8; Chapter 4, Problems 6 and 8; Chapter 7, Problem 6; Chapter 9, Problem 22).

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