Hahn A.J. Basic Calculus of Planetary Orbits and Interplanetary Flight. The Missions of the Voyagers, Cassini, and Juno

Springer, 2020. — 387 p. — ISBN: 978-3-030-24867-3.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.Solutions to odd exercises are freely available;
Includes nearly 150 breathtaking color photos and extensive data from NASA’s Voyager, NEAR-Shoemaker, Cassini, and Juno missions
Highlights mathematics of the solar system and the role played by combining calculus and basic physical laws
Presents the basic working knowledge and insights into the applicability of polar calculus
Provides and in depth understanding of the uses trigonometric and hyperbolic functions and parametric equations in spacecraft navigation
Uniquely approaches basic calculus within the historical context of interplanetary flightFrom Kepler to Newton to a Picture of the Universe
Copernicus Moves the Sun to the Center
From Tycho to Kepler to Newton
The Conic Sections
Newton's Incisive Insights
Testing the Moon and Charting the Solar System
The Size and Scope of the Solar System
The Metric System of Units
Cavendish and the Gravitational Constants G and g.
The Sun
Galaxies and the Expanding Universe
Problems and Discussions
Exploring the Solar System
Rockets, Spacecraft, and the Hubble.
The Inner Planets.
The Outer Planets.
About Asteroids.
About Comets.
Trans-Neptunian Objects.
The Rocket Equation.
The Flight Path of Juno.
Problems and Discussions.
Calculus of Functions in Polar Coordinates
The Unit Circle and Trigonometry.
Polar Coordinates.
Polar Functions and their Graphs.
The Conic Sections in Polar Coordinates.
The Derivative of a Polar Function.
The Lengths of Polar Curves.
Areas in Polar Coordinates.
Spiral Galaxies and Equiangular Spirals.
Problems and Discussions.
Centripetal Force and Resulting Trajectories
A Basic Study of Forces.
The Mathematics of a Moving Point.
Centripetal Force in Cartesian Coordinates.
Going Polar.
From Conic Section to Inverse Square Law.
From Inverse Square Law to Conic Section.
Summary of Newton's Theory.
Gravity and Geometry.
Problems and Discussions.
Elliptical Orbits and their Precession
Setting the Stage
Determining Distance and Angle
Kepler's Equation
Determining Speed and Direction
Solving Kepler's Equation by Successive Approximations
Earth, Jupiter, and Halley
Orbital Questions and Definite Integrals
Perturbed Orbits and Precession
The Gravitational Force of one Planet on Another
Perihelion Precession for Mercury
The Relativistic Component of Precession
Problems and Discussions
Mathematics of Interplanetary Flight
NEAR-Shoemaker and Eros
Escape Velocity from Earth
Gravitational Sphere of Influence
Modifying an Orbit
Hyperbolic Functions
Moving along the Hyperbola
The Hyperbolic Kepler Equation
Solving the Hyperbolic Kepler Equation
Hohmann Transfer Orbits
Gravity Assist Flybys
The Voyagers and Cassini
The Cruise of Voyager1
The Cruise of Voyager2
The Cassini-Huygens Mission
Orbits and their Ephemerides
Problems and Discussions
References
Index