Methods Of Orbit Determination For The Microcomputer

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Methods Of Orbit Determination For The Microcomputer

Boulet, 6.00" by 9.00", 564 pages, hardbound, published 1991, 2 Lbs. 9 Ozs. ship wt., $24.95.

This book describes how the principles of celestial mechanics may be applied to determine the orbits of planets, comets, and Earth satellites. More specifically, it shows how a dedicated novice can learn, by first-hand experience:

how orbital motion conforms to Newtonian physics
how a set of orbital elements can be translated into quantities which can be compared with observations, and
how a record of observed motion can be used to determine an orbit from scratch or improve a preliminary orbit.
Until recently, this exciting adventure with nature was beyond the reach of nearly all non-specialists. However the power of the microcomputer has swept away the drudgery of tedious calculations fraught with endless opportunities for careless error. With this book and a microcomputer the enthusiast may have the satisfaction of conquering problems which preoccupied astronomy for hundreds of years, and, in the process, gain a fresh appreciation for the genius and industry of the great mathematicians of the seventeenth, eighteenth, and nineteenth centuries.

This is a how-to-do-it book. Even though the derivations of many important relationships are described in some detail, the emphasis throughout is on practical applications. The reader need only accept the validity of the key equations and understand their symbology in order to use the computer programs to explore the power of the mathematical models. All the important principles have been reduced to complete computer programs written in simple BASIC that will execute directly on a Macintosh using Microsoft BASIC or (with the addition of a statement as line 1005 to reserve extra space in memory) an IBM-PC using BASICA or GWBASIC. For clarity, each program is preceded by an algorithm that describes the sequence of computation and ties it to the mathematics in the text. Further, the program is illustrated by at least one numerical example. Finally, the output from the examples is shown in the format produced by the computer routine. Magnetic media versions of the source code are also available for both IBM-PC and Macintosh computers—see last page in book for order form.

Who will find this book of value?

Amateur astronomers who want to determine the orbits of planets, comets or Earth satellite
Teachers and students of basic calculus, physics, astronomy or computer programming for a source of material that illuminates and expands upon subjects covered in the classroom.

Table of Contents

Preface
1. Fundamentals of Orbital Motion
   1.1. Introduction
   1.2. The Laws of Motion
      1.2.1. The Law of Inertia
      1.2.2. The Law of Acceleration
      1.2.3. The Law of Action and Reaction
   1.3. The Law of Gravitation
   1.4. Equations of Motion
      1.4.1. The Equation of Inertial Motion
      1.4.2. The Equation of Relative Motion
   1.5. Working Units and Constants
      1.5.1. The Heliocentric System
      1.5.2. The Geocentric System
   1.6. The Working Equation of Motion
   1.7. Numerical Example
2. Time and Position
   2.1. Introduction
   2.2. The Fundamental References
   2.3. The Empirical Frame of Reference
   2.4. Time Scales
      2.4.1. Universal Time
      2.4.2. Julian Date
      2.4.3. Sidereal Time
      2.4.4. Atomic Time
      2.4.5. Dynamical Time
   2.5. Coordinate Systems
      2.5.1. Celestial Equatorial Systems
      2.5.2. Terrestrial Equatorial Systems
      2.5.3. Celestial Ecliptic Systems
   2.6. Ecliptic-Equatorial Transformations
   2.7. The Fundamental Vector Triangle
   2.8. Reduction of Astronomical Coordinates
      2.8.1. Planetary Aberration
      2.8.2. The Instantaneous and Fixed Equator and Equinox
      2.8.3. Astrometric Positions
      2.8.4. Reductions for Aberration and Nutation
      2.8.5. Reductions for Precession
      2.8.6. Reductions for Geocentric Parallax
   2.9. Computer Programs
      2.9.1. Program LMST
      2.9.2. Program XYZ
      2.9.3. Program RAD
      2.9.4. Program CQTRAN
      2.9.5. Program ADAPP
      2.9.6. Program ADCES
      2.9.7. Program XYZCES
      2.9.8. Program ADLAX
   2.10. Numerical Examples
      2.10.1. Computing Local Mean Sidereal Time
      2.10.2. Converting Spherical to Rectangular Coordinates
      2.10.3. Converting Rectangular to Spherical Coordinates
      2.10.4. Converting Equatorial to Ecliptic Coordinates
      2.10.5. Reducing Apparent Place to Astrometric Place
      2.10.6. Reducing RA and DEC from Jxxxx.x to J2000.0
      2.10.7. Reducing Rectangular Coordinates from J2000.0 to Jxxxx.x
      2.10.8. Reducing Geocentric Place to Topocentric Place
3. The Two-Body Problem
   3.1. Introduction
   3.2. The Two-Body Equation of Motion
   3.3. The Orbital and Radial Rates
   3.4. The Laws of Two-Body Motion
      3.4.1. The Conic Section Law
      3.4.2. The Law of Areas
      3.4.3. The Harmonic Law
      3.4.4. The Vis-viva Law
   3.5. Two-Body Motion by Numerical Integration
      3.5.1. The f and g Series
      3.5.2. Taylor Series
      3.5.3. Runge-Kutta Five
      3.5.4. Numerical Error
   3.6. Computer Programs
      3.6.1. Program FANDG
      3.6.2. Program TAYLOR
      3.6.3. Program RUNGE
   3.7. Numerical Examples
      3.7.1. Two-Body Motion by f and g Series
      3.7.2. Two-Body Motion by Taylor Series
      3.7.3. Two-Body Motion by Runge-Kutta Five
4. Orbit Geometry
   4.1. Introduction
   4.2. General Relationships
      4.2.1. Angular Momentum and Angular Speed
      4.2.2. Radial Speed and True Anomaly
      4.2.3. True Anomaly and D
      4.2.4. Eccentricity, Semiparameter, and D
   4.3. Relationships between Geometry and Time
      4.3.1. Elliptic Formulation
      4.3.2. Hyperbolic Formulation
      4.3.3. Parabolic Formulation
   4.4. The Classical Elements from Position and Velocity
      4.4.1. Three Fundamental Vectors
      4.4.2. The Conic Parameters
      4.4.3. The Orientation Angles
      4.4.4. The Mean Anomaly
      4.4.5. The Time of Perifocal Passage
   4.5. Position and Velocity from the Classical Elements
      4.5.1. The Scalar Components of Elliptic Motion
      4.5.2. The Scalar Components of Hyperbolic Motion
      4.5.3. The Scalar Components of Parabolic Motion
      4.5.4. The Unit Vector Components of Motion
   4.6. Computer Programs
      4.6.1. Program CLASSEL
      4.6.2. Program POSVEL
   4.7. Numerical Examples
      4.7.1. Classical Elements for Mars
      4.7.2. Classical Elements for Comet X
      4.7.3. Classical Elements for Comet Y
      4.7.4. Classical Elements for GEOS
      4.7.5. Position and Velocity Elements for Pallas
      4.7.6. Position and Velocity Elements for Recon 1
      4.7.7. Position and Velocity Elements for Recon 2
      4.7.8. Position and Velocity Elements for Recon 3
5. Ephemeris Generation
   5.1. Introduction
   5.2. The Differenced Kepler Equations
      5.2.1. Elliptic Formulation
      5.2.2. Hyperbolic Formulation
      5.2.3. Parabolic Formulation
   5.3. The Closed f and g Expressions
      5.3.1. Elliptic Motion
      5.3.2. Hyperbolic Motion
      5.3.3. Parabolic Motion
   5.4. The Universal Formulation
      5.4.1. The Coefficients C, S, and U
      5.4.2. The Equations of Motion
   5.5. The Ephemeris
   5.6. Computer Programs
      5.6.1. Program SEARCH
      5.6.2. Program RADEC
   5.7. Numerical Examples
      5.7.1. Ephemeris for GEOS
      5.7.2. Ephemeris for Pallas
      5.7.3. Ephemeris for Comet X
      5.7.4. Right Ascension and Declination of Comet X
6. Special Perturbations
   6.1. Introduction
   6.2. Direct and Indirect Attractions
   6.3. The Method of Cowell
   6.4. The Method of Encke
   6.5. A Perturbed Ephemeris
   6.6. Computer Programs
      6.6.1. Program ATTRACT
      6.6.2. Program COWELL
      6.6.3. Program ENCKE
   6.7. Numerical Examples
      6.7.1. Solar and Planetary Attractions
      6.7.2. The Motion of Mars
      6.7.3. The Motion of Uranus
7. Applied Numerical Methods
   7.1. Introduction
   7.2. Finding the Root of an Equation
      7.2.1. The Bisection Method
      7.2.2. The Newton-Raphson Method
   7.3. Solving a System of Linear Equations
      7.3.1. Naive Gauss Elimination
      7.3.2. Partial Pivoting
   7.4. Polynomial Interpolation
   7.5. Polynomial Regression
   7.6. Multiple Linear Regression
   7.7. Numerical Differentiation
      7.7.1. The Interpolating Polynomial
      7.7.2. The Regression Polynomial
   7.8. Computer Programs
      7.8.1. Program PTERP
      7.8.2. Program PGRESS
      7.8.3. Program MGRESS
   7.9. Numerical Examples
      7.9.1. Polynomial Interpolation and Differentiation
      7.9.2. Polynomial Regression and Differentiation
      7.9.3. Multiple Linear Regression
8. Preliminary Orbit Data
   8.1. Introduction
   8.2. Principal Constraints
   8.3. The Topocentric Vector L
   8.4. The Topocentric Vector R
      8.4.1. Vector R for Geocentric Orbits
      8.4.2. Vector R for Heliocentric Orbits
   8.5. Computer Programs
      8.5.1. Program ADGRESS
      8.5.2. Program GEO
      8.5.3. Program HELO
   8.6. Numerical Examples
      8.6.1. Regression of Angular Data for Satellite GEOS
      8.6.2. Regression of Angular Data for Comet Rebek-Jewel
      8.6.3. Topocentric Vector to the Geocenter
      8.6.4. Topocentric Vector to the Heliocenter
9. The Method of Laplace
   9.1. Introduction
   9.2. Solution by Successive Differentiation
   9.3. The Scalar Equations for the Range and Rate
   9.4. The Scalar Equation for the Radial Distance
   9.5. The Scalar Equation of Lagrange
   9.6. The Vector Orbital Elements
   9.7. Program LAPLACE
   9.8. Numerical Examples
      9.8.1. The Orbit of Satellite GEOS
      9.8.2. The Orbit of Comet Rebek-Jewel
10. The Method of Gauss
   10.1. Introduction
   10.2. Solution by f and g Expressions
   10.3. The Scalar Equations for the Ranges
   10.4. The First Approximation
   10.5. The Scalar Equations Relating p and r at Epoch
   10.6. The Scalar Equation of Lagrange
   10.7. The Vector Orbital Elements
      10.7.1. Initial Position Vector
      10.7.2. Initial Velocity Vector
      10.7.3. Refinement of the Elements
   10.8. Program GAUSS
   10.9. Numerical Examples
      10.9.1. The Orbit of Pallas
      10.9.2. The Orbit of Comet Rebek-Jewel
11. The Method of Olbers
   11.1. Introduction
   11.2. Solution by Euler’s Equation
   11.3. The Scalar Equations for the Range
   11.4. The Vector Orbital Elements
      11.4.1. Three Radius Vectors
      11.4.2. The Velocity Vector
   11.5. Program OLBERS
   11.6. Numerical Example
      11.6.1. The Orbit of Comet Z
      11.6.2. The Orbit of Comet Rebek-Jewel
12. Orbit Improvement
   12.1. Introduction
   12.2. The Differential Equations of Condition
   12.3. Numerical Evaluation of the Partial Derivatives
   12.4. Comparing Observation with Theory
   12.5. Computer Programs
      12.5.1. Program IMPROVE
      12.5.2. Program CORRECT
   12.6. Numerical Examples
      12.6.1. Improved Orbit for GEOS
      12.6.2. Improved Orbit for Rebek-Jewel
      12.6.3. Improved Orbit for Pallas
Appendices
   A. Vectors
      A.1. Basic Vector Operations
      A.2. The Dot and Cross Products
   B. Elementary Calculus
      B.1 Differentiation
      B.2 Integration
   C. Astronomical Constants
      C.1 Constants Related to Units
      C.2 Masses of the Planets
Index