Intended for a one- or two-semester course, this text applies basic, one-variable calculus to analyze the motion both of planets in their orbits as well as interplanetary spacecraft in their trajectories. The remarkable spacecraft missions to the inner and outermost reaches of our solar system have been one of the greatest success stories of modern human history. Much of the underlying mathematical story is presented alongside the astonishing images and extensive data that NASA’s Voyager, NEAR-Shoemaker, Cassini, and Juno missions have sent back to us.
First and second year college students in mathematics, engineering, or science, and those seeking an enriching independent study, will experience the mathematical language and methods of single variable calculus within their application to relevant conceptual and strategic aspects of the navigation of a spacecraft. The reader is expected to have taken one or two semesters of the basic calculus of derivatives, integrals, and the role that limits play. Additional prerequisites include knowledge of coordinate plane geometry, basic trigonometry, functions and graphs, including trig, inverse, exponential, and log functions.
The discussions begin with the rich history of humanity’s efforts to understand the universe from the Greeks, to Newton and the Scientific Revolution, to Hubble and galaxies, to NASA and the space missions. The calculus of polar functions that plays a central mathematical role is presented in a self-contained way in complete detail. Each of the six chapters is followed by an extensive problem set that deals with and also expands on the concerns of the chapter. The instructor has the flexibility to engage them with greater or lesser intensity.
Author(s): Alexander J. Han
Edition: 1
Publisher: Springer
Year: 2020
Language: English
Pages: 386
Preface
Contents
1 From Kepler to Newton to a Picture of the Universe
1A Copernicus Moves the Sun to the Center
1B From Tycho to Kepler to Newton
1C The Conic Sections
1D Newton's Incisive Insights
1E Testing the Moon and Charting the Solar System
1F The Size and Scope of the Solar System
1G The Metric System of Units
1H Cavendish and the Gravitational Constants G and g.
1I The Sun
1J Galaxies and the Expanding Universe
1K Problems and Discussions
2 Exploring the Solar System
2A Rockets, Spacecraft, and the Hubble.
2B The Inner Planets.
2C The Outer Planets.
2D About Asteroids.
2E About Comets.
2F Trans-Neptunian Objects.
2G The Rocket Equation.
2H The Flight Path of Juno.
2I Problems and Discussions.
3 Calculus of Functions in Polar Coordinates
3A The Unit Circle and Trigonometry.
3B Polar Coordinates.
3C Polar Functions and their Graphs.
3D The Conic Sections in Polar Coordinates.
3E The Derivative of a Polar Function.
3F The Lengths of Polar Curves.
3G Areas in Polar Coordinates.
3H Spiral Galaxies and Equiangular Spirals.
3I Problems and Discussions.
4 Centripetal Force and Resulting Trajectories
4A A Basic Study of Forces.
4B The Mathematics of a Moving Point.
4C Centripetal Force in Cartesian Coordinates.
4D Going Polar.
4E From Conic Section to Inverse Square Law.
4F From Inverse Square Law to Conic Section.
4G Summary of Newton's Theory.
4H Gravity and Geometry.
4I Problems and Discussions.
5 Elliptical Orbits and their Precession
5A Setting the Stage
5B Determining Distance and Angle
5C Kepler's Equation
5D Determining Speed and Direction
5E Solving Kepler's Equation by Successive Approximations
5F Earth, Jupiter, and Halley
5G Orbital Questions and Definite Integrals
5H Perturbed Orbits and Precession
5I The Gravitational Force of one Planet on Another
5J Perihelion Precession for Mercury
5K The Relativistic Component of Precession
5L Problems and Discussions
6 Mathematics of Interplanetary Flight
6A NEAR-Shoemaker and Eros
6B Escape Velocity from Earth
6C Gravitational Sphere of Influence
6D Modifying an Orbit
6E Hyperbolic Functions
6F Moving along the Hyperbola
6G The Hyperbolic Kepler Equation
6H Solving the Hyperbolic Kepler Equation
6I Hohmann Transfer Orbits
6J Gravity Assist Flybys
6K The Voyagers and Cassini
6L The Cruise of Voyager1
6M The Cruise of Voyager2
6N The Cassini-Huygens Mission
6O Orbits and their Ephemerides
6P Problems and Discussions
References
Index
