Mechanics of the Solar System

J.A Evans

Description

This book develops methods of computing astronomical phenomena from basic ideas. The position of a celestial body is defined by a vector, with components referred to a system of coordinate axes. The relations between various systems in regular use by astronomers are described. In cases where two systems differ in spatial orientation, they are related by a rotation matrix. These matrices are discussed in considerable detail in the mathematical notes. Other topics discussed include: Kepler's Laws and the dynamics of planetary motion, Precession and Nutation, transits of Venus and Mercury, Lagrange points. While no previous knowledge of Astronomy is necessary, it is assumed that the reader is familiar with elementary algebra, trigonometry and calculus.

Sujets

• Sciences et techniques
• Astronomy

Informations

• Publié par: Brown Dog Books
• Date de parution: 12 mars 2021
• Nombre de lectures: 0
• EAN13: 9781839522352
• Langue: English
• Poids de l'ouvrage: 2 Mo

Extrait

MECHANICS of the SOLAR SYSTEM
For Dorothy

First published 2021
Copyright © J.A. Evans 2021
The right of J.A. Evans to be identified as the author of this work has been asserted in accordance with the Copyright, Designs & Patents Act 1988.
All rights reserved. No part of this book may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, electrostatic, magnetic tape, mechanical, photocopying, recording or otherwise, without the written permission of the copyright holder.
Published under licence by Brown Dog Books and The Self-Publishing Partnership, 7 Green Park Station, Bath BA1 1JB
www.selfpublishingpartnership.co.uk

ISBN printed book: 978-1-83952-234-5 ISBN e-book: 978-1-83952-235-2
Cover design by Kevin Rylands Internal design by Mac Style
Printed and bound in the UK
This book is printed on FSC certified paper
Contents
Introduction
Chapter 1 Time and Space in Astronomy
1.1 Time
1.2 Universal Time
1.3 The solar year
1.4 The modern calendar
1.5 Julian Day numbers
1.6 Sidereal time
1.7 Flamsteed and the RGO
1.8 The celestial sphere
1.9 Coordinate systems
1.10 Ecliptic and equatorial coordinates
1.11 Transformation of coordinates between the ecliptic and equatorial systems
1.12 Equatorial coordinates of the Sun
1.13 Parallax
1.14 Angular separation
1.15 Celestial line-ups
1.16 Terrestrial coordinates
1.17 Directions of cardinal points in terms of a & A
1.18 Rising and setting
1.19 Stellar transits across a local meridian
1.20 Exercises
Chapter 2 Planetary Motion
2.1 Ancient and modern conceptions of the Cosmos
2.2 Kepler and Newton
2.3 Properties of a planetary orbit
2.4 The mean anomaly
2.5 Orientation of the orbit
2.6 Determination of the ecliptic coordinates of a planet
2.7 Non-periodic orbits
2.8 Exercises
Chapter 3 The General Kepler Problem
3.1 Introduction
3.2 Determination of the anomaly
3.3 An approximation for ϕ in a periodic orbit
3.4 Conclusion
3.5 Examples
3.6 Exercises
Appendix
Chapter 4 Calculation of the Position and Velocity of a Planet
4.1 Calculation from orbital elements
4.2 Comets
4.3 Calculation from VSOP
4.4 Geocentric position and velocity vectors
4.5 Time derivatives of geocentric coordinates
4.6 Exercises
Chapter 5 Geocentric Observation
5.1 Relative motion
5.2 A schematic model
5.3 Geocentric orbits
5.4 Special events
5.5 Realistic calculations
5.6 Sample results for two planets
5.7 Planetocentric data
(i) Planetocentric declination of the Sun and Earth
(ii) Longitude of the Planet’s central meridian
(iii) The angles P & Q
5.8 Exercises
Chapter 6 Corrections to the Apparent Position of a Planet
6.1 Method of calculation
6.2 ΔT
6.3 Aberration
6.4 Nutation
6.5 Conclusion
6.6 Stellar aberration
6.7 Exercises
Chapter 7 Precession
7.1 The Earth as a very large top
7.2 Low accuracy calculation of the effects of precession 121
7.3 The precession matrix
7.4 Equatorial coordinates
7.5 The Eulerian angles method
7.6 Instantaneous rate of precession
7.7 Reduction of orbital elements to a standard epoch
7.8 Exercises
Chapter 8 Solar Transits of the Inner Planets
8.1 Rare events
8.2 Anatomy of a transit
8.3 Transits in history
8.4 Method of calculation
8.5 Transit systematics
Mercury
Venus
8.6 Topocentric effects and visibility of transits
8.7 Determining the size of the Solar System
8.8 Exercises
Appendix
Chapter 9 Seven Assorted Topics
9.1 Lagrange and the restricted three-body problem
9.1.1 Introduction
9.1.2 Lagrangian mechanics
9.1.3 The restricted three-body problem
9.1.4 Equilibrium points
9.1.5 Stability
9.1.6 The normal modes at L1, L2 & L3
9.1.7 The points L4 and L5
9.1.8 Discussion
Appendices A, B
9.2 Orbital elements from the position and velocity vectors of the planet
9.2.1 Calculation of the orbital elements
9.2.2 Time dependence of the orbital elements
9.3 The anomalous motion of Mercury
9.3.1 Perihelion precession
9.3.2 New theory of gravity: GR
9.3.3 Application to Mercury
9.4 MOID
9.4.1 Oumuamua
9.5 Simple theory of the EOT
9.5.1 Qualitative argument
9.5.2 Lowest order calculation
9.6 Apparent and absolute visual magnitude
9.6.1 Stellar parallax
9.6.2 Stellar visual magnitude
9.6.3 Visual magnitude of a planet
9.7 Celestial navigation
9.7.1 Lunar distance
9.7.2 Stellar altitudes
Chapter 10 Mathematical Notes
10.1 Vectors
10.2 Matrices
10.3 Rotations of the coordinate axes
10.4 Non-commutation of finite rotations – experimental proof
10.5 The G-matrices
10.6 More about vector products
10.7 Angular velocity
10.8 General forms for finite rotations
References
Solutions to the exercises
Introduction
Mathematics and astronomy have been closely linked since antiquity, and mathematics has sometimes signposted major conceptual developments in astronomy. In the 16th century it was Copernicus’ discovery of serious inaccuracies in the predictions of Ptolemy’s algorithms which led him to his radical revision of the structure of the cosmos. In the next century Kepler used Brahe’s data on planetary positions to show that the planets move on elliptical, rather than circular, orbits. Later in the same century Newton demonstrated mathematically that, according to his laws of motion and gravity, a single planet would move in an elliptical orbit around a single star. Newton’s work forged the first direct link between astronomy and physics. Kepler’s empirical laws of planetary motion are all derivable from Newtonian mechanics applied to a single orbiting planet.
This book presents numerous demonstrations of how mathematics can be used to compute planetary phenomena. While no prior knowledge of astronomy is required, a familiarity with basic algebra, trigonometry and calculus is assumed. Any serious work in this subject, such as the computation of planetary ephemerides, demands a familiarity with computer programming in a high level language such as Fortran. The text relies heavily on vector analysis and matrix algebra, and these areas of mathematics are treated in some detail in Chapter 10 . For some applications it is useful to treat vectors as matrices with a single row or column, creating a distinction between ‘row’ and ‘column’ vectors, which are written horizontally and vertically respectively and enclosed in square brackets. In other applications this distinction is unimportant and all vectors are written horizontally in round brackets. It is not anticipated that this will lead to any confusion. Equations are numbered independently within each chapter, so that they can be referenced in the text. In cases where it is necessary to refer to equations in other chapters, the chapter number is placed in square brackets together with the equation number; e.g. [8(6.3)] refers to equation ( 6.3 ) in Chapter 8 .
The main tool for calculating planetary positions and velocities is VSOP, as described in Chapter 4 . The VSOP 87 data files can be accessed via the link ftp://ftp.imecce.fr/pub/ephem/planets/vsop87 .
In addition to orbital elements, there are five other versions of the theory, designated A–E as indicated below.
A. heliocentric rectangular coordinates ref. to equinox and ecliptic of J2000.
B. heliocentric spherical coordinates ref. to equinox and ecliptic of J2000.
C. heliocentric rectangular coordinates ref. to equinox and ecliptic of date.
D. heliocentic spherical coordinates ref. to equinox and ecliptic of date.
E. barycentric rectangular coordinates ref. to J2000.
The data given in the appendix of Reference 1 is a subset of that in version D.
The site also lists a Fortran source code which uses VSOP87 to calculate planetary positions.
Chapter 1
Time and Space in Astronomy
Astronomy is the science which studies the distribution and movements of stars, planets, comets etc. in the cosmos. Before going on, in later sections, to describe how the locations of celestial objects in space are recorded, we begin by discussing the measurement of time.
1.1 Time
Any regular periodic phenomenon can be used to measure time. As most human activity is carried out in daylight, and the most important in early civilisations, viz. agriculture, depends critically on the seasons, the Sun is a fairly obvious choice of timekeeper. However, many primitive cultures used the Moon and even hybrid combinations of Sun and Moon to measure long periods of time. This sometimes resulted in a chaotic situation with the calendar seriously out of step with the seasons. In a successful calendar, the equinoxes and solstices should occur on or very close to the same dates each year.
Julius Caesar found it necessary to reform the Roman calendar, which, as his biographer Suetonius wrote: ‘the pontifices had allowed to fall into such disorder, by intercalating days or months as it suited them, that the harvest and vintage festivals no longer corresponded with the appropriate seasons’. In order to achieve his reform he added 80 days to 46 BCE making it, with 445 days, the longest year in history. His new “Julian calendar” was inaugurated on the designated first day of 45 BCE January 1. The Julian calendar assumed that the mean tropical year (i.e. from vernal equinox to vernal equinox) contains exactly 365¼ days, and introduced the familiar leap-years of 366 days so that each four-year period contained 365 × 3 + 366 = 1461 days. The Julian calendar was in general use throughout the Christian world until the late 16th century and until much later in some countries.
1.2 Universal Time
The time used for everyday life in general, and astronomical observations in particular, must depend ultimately on the Sun, i.e. on the Earth’s diurnal rotation. This need is met by Universal Time (UT) which is based on observations of the transits of stars across the local meridians of observatories across the globe. These