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1996 | Buch

The Sheer Joy of Celestial Mechanics

verfasst von: Nathaniel Grossman

Verlag: Birkhäuser Boston

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Über dieses Buch

Dear Reader, Here is your book. Take it, run with it, pass it, punt it, enjoy all the many things that you can do with it, but-above all-read it. Like all textbooks, it was written to help you increase your knowledge; unlike all too many textbooks that you have bought, it will be fun to read. A preface usually tells of the author's reasons for writing the book and the author's goals for the reader, followed by a swarm of other important matters that must be attended to yet fit nowhere else in the book. I am fortunate in being able to include an insightful prepublication review that goes directly to my motivations and goals. (Look for it following this preface.) That leaves only those other important matters. In preparing the text, I consulted a number of books, chief of which included these: • S. Chandrasekhar, Ellipsoidal Figures of Equilibrium, Yale Uni­ versity Press, 1969. • J .M.A. Danby, Fundamentals of Celestial Mechanics, Macmil­ lan, 1962. Now available in a 2nd edition, 3rd printing, revised, corrected and enlarged, Willmann-Bell, 1992. • Y. Hagihara, Theories of Equilibrium Figures of a Rotating Ho­ mogeneous Fluid Mass, NASA, 1970. • R.A. Lyttleton, The Stability of Rotating Liquid Masses, C- ix x PREFACE bridge University Press, 1953. • C.B. Officer, Introduction to Theoretical Geophysics, Springer­ Verlag, 1974. • A.S. Ramsey, Newtonian Attraction, Cambridge University Press, 1949. • W.M. Smart, Celestial Mechanics, Longmans, Green, and Co, 1953.

Inhaltsverzeichnis

Frontmatter
Chapter I. Rotating Coordinates
Abstract
We want to look first at some geometric aspects of motion without paying any attention to the causes of motion or to the physical laws that might govern the motion.
Nathaniel Grossman
Chapter II. Central Forces
Abstract
A central force field is one whose action is always directed toward a fixed point. If that fixed point is taken to be the origin, and if i is taken to be a unit vector directed from the origin to the position r of a particle acted upon by the central force, then the force field can be written
$$F(r)=-mP(r)i,$$
(83)
where P is a scalar function and m is the mass of the particle. The equation of motion for a particle of constant mass can be written down using Newton’s Second Law; it is
$$\ddot{r}=-Pi.$$
(84)
Nathaniel Grossman
Chapter III. Orbits under the Inverse Square Law
Abstract
Kepler formulated his famous three laws of planetary motion during the first decades of the seventeenth century. It was an arduous task, proceeding strictly from observed data, the most accurate of which were contributed by Kepler’s mentor, Tycho Brahe.
Nathaniel Grossman
Chapter IV. Expansions for an Elliptic Orbit
Abstract
We have studied motion in an elliptic orbit under the inverse square law of attraction to a center. Among the formulas we derived in §111.3 and (196) were these:
$$r=\frac{a(1-e^2)}{1+e \cos f \prime},$$
(247)
$$\tan \frac{1}{2}f=\sqrt{\frac{1+e}{1-e}}tan\frac{1}{2}E,$$
(248)
$$r=a(1-e \cos E),$$
(249)
$$E-e \sin E=M.$$
(250)
Nathaniel Grossman
Chapter V. Gravitation and Closed Orbits
Abstract
The solar system appears to be stable: Calculations into the future using Newton’s Law of Universal Gravitation show that the Sun and the planets will behave in the future much as they have done in the past. The Earth will move in its yearly, very nearly elliptical, Keplerian orbit around the Sun, as will the other planets in their years. No planet will suddenly take off for deep space or fall into the Sun, nor will two of the (major) planets collide.
Nathaniel Grossman
Chapter VI. Dynamical Properties of Rigid Bodies
Abstract
Fix a set of cartesian axes in space. Suppose a finite system of particles is distributed in space, the ith particle having mass m i and position r i . The center of mass of the system is defined to be the point \(\bar{r}\), where
$$\bar{r}=\frac{\Sigma m_i r_i}{\Sigma m_i}$$
(425)
and the summation is over all of the particles. Being defined vectorially, the center of mass is independent of the orientation of the axes in space; it also is easy to see that it is independent of the choice of origin.
Nathaniel Grossman
Chapter VII. Gravitational Properties of Solids
Abstract
The potential energy at O of a particle of mass m at P is −Gm/r. The potential produced at O by a solid body with density function ρ is
$$V=-G\int \frac{\rho dv}{r}.$$
(463)
Nathaniel Grossman
Chapter VIII. Shape of a Self-Gravitating Fluid
Abstract
Consider a mass of liquid, incompressible but not necessarily homogeneous in density, that rotates around a fixed axis through its center of mass without any external force. Take the z-axis to be the rotation axis, and assume that the angular velocity is the constant ω The x- and y-axes will be fixed in the fluid mass with the origin at the center of gravity. Denote by p the pressure at a point (x, y, z) of the fluid; the pressure depends only upon the position. The force acting upon an element of volume dτ is
$$(X\,d\tau ,\,Y\,d\tau ,\,Z\,d\tau ,) = \left( {\frac{{\partial p}}{{\partial x}}d\tau ,\frac{{\partial p}}{{\partial y}}d\tau ,\frac{{\partial p}}{{\partial z}}d\tau } \right),$$
(554)
where (X, Y, Z) is the force per unit volume at (x, y, z). The equations of motion can be written in the rotating xyz-coordinate system by including a ‘fictitious’ centripetal force.
Nathaniel Grossman
Backmatter
Metadaten
Titel
The Sheer Joy of Celestial Mechanics
verfasst von
Nathaniel Grossman
Copyright-Jahr
1996
Verlag
Birkhäuser Boston
Electronic ISBN
978-1-4612-4090-7
Print ISBN
978-1-4612-8647-9
DOI
https://doi.org/10.1007/978-1-4612-4090-7