Theory of Satellite Geodesy: Applications of Satellites to Geodesy

by William M. Kaula

Overview

The main purpose of this classic text is to demonstrate how Newtonian gravitational theory and Euclidean geometry can be used and developed in the earth's environment. The second is to collect and explain some of the mathematical techniques developed for measuring the earth by satellite.
Book chapters include discussions of the earth's gravitational field, with special emphasis on spherical harmonies and the potential of the ellipsoid; matrices and orbital geometry; elliptic motion, linear perturbations, resonance, and other aspects of satellite orbit dynamics; the geometry of satellite observations, including time and precise definition of coordinates, and observability conditions; and statistical implications and date analysis.
The completion of a first-year course in physics and a first-year course in calculus is assumed.

Product Details

ISBN-13:	9780486152219
Publisher:	Dover Publications
Publication date:	12/11/2012
Series:	Dover Earth Science
Sold by:	Barnes & Noble
Format:	eBook
Pages:	140
File size:	11 MB
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Theory of Satellite Geodesy

Applications of Satellites to Geodesy

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By William M. Kaula

Dover Publications, Inc.

Copyright © 2000 Gene Kaula
All rights reserved.
ISBN: 978-0-486-15221-9

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CHAPTER 1

THE EARTH'S GRAVITATIONAL FIELD

1.1. Potential Theory

The most pervasive fact of the familiar physical environment is the earth's gravitational attraction. According to Newton's universal law of gravitation, the force of attraction between two particles of masses m and M at a distance r from each other will be

F = k mM/r2, (1.1)

where k is the gravitational constant. If we combine Equation (1.1) with Newton's second law, which is

F = ma,

we obtain the acceleration of the particle of mass m with respect to the center of mass of the two particles,

a = kM/r2. (1.2)

This acceleration is the magnitude of a vector directed along the line between the two particles. A vector a equivalent to Equation (1.2) will be obtained by expressing the acceleration as the gradient of a scalar, called a potential. Thus

a = [nabla]V, (1.3)

where

V = kM/r. (1.4)

In (1.4), V is shown as a positive quantity, which is consistent with the sign convention of astronomy and geodesy. In physics V is conventionally taken to be negative.

For m negligibly small compared to M,Equations (1.3) and (1.4) are consistent with a coordinate system whose origin is at the center of mass of the particle of mass M. For the effect of several particles of masses Mi at distances ri, the combined acceleration can be expressed as the gradient of a potential, which is a sum of potentials Vi expressed by Equation (1.4). If these particles are conglomerated to form a continuous body of variable density ρ, this summation can be replaced by an integration over the volume of the body. Thus

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.5)

For a particular component ax of a derived from the point mass potential of Equation (1.4), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.6)

and for the second derivative, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.7)

Adding together the second derivatives for the other two coordinates, we get Laplace's equation,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.8)

We would get this same result for any element of mass ρ [dELTA]x dy dz in the potential of Equation (1.5) and hence for the summation thereof.

The coordinate system, rectangular or otherwise, that is most convenient in a physical problem usually depends on the geometry of the boundaries. The earth is rather round, which suggests spherical coordinates. Thus,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.9)

where r is the radial distance from the origin, φ is latitude, and λ is longitude measured eastward (that is, counterclockwise looking toward the origin from the positive end of the z -axis). The notation used in Equation (1.9) is consistent with geodetic practice; usually in mathematics there are used θ for colatitude and φ for longitude.

In order to convert the Laplace equation, (1.8), to spherical coordinates, we need partial derivatives of the spherical coordinates with respect to the rectangular coordinates. These can be obtained by differentiating Equation (1.9) with respect to r, φ, and λ in turn and then by solving the simultaneous differential equations for dr, dφ, δλ. Thus we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.10)

Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.11)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.12)

Carrying out these differentiations for

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

in turn and adding them together, we obtain for the Laplace Equation in spherical coordinates,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.13)

1.2 Spherical Harmonics

To express the variations of the potential V in the spherical coordinate system, it would be convenient if V had the form

V = R(r) Φφ) Λ(λ). (1.14)

By substituting Equation (1.14) in (1.13) and dividing by RΦΛ, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.15)

Because the first term of Equation (1.15) is the only term that is a function of r, it must be constant—say, as later turns out to be convenient—l (l + 1). Carrying out the differentiation and multiplying by R, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.16)

The form of Equation (1.16), in which R and each of its derivatives is multiplied by the equivalent power of r, suggests that R is of the form rk. Substituting this value in Equation (1.16) and solving the resulting equation for k, we get l and – l – 1 as the two admissible solutions, or

R = Arl + Br-l-1, (1.17)

where A and B are arbitrary constants. In the case of interest to us, a potential in free space vanishing at infinity, A must equal zero.

Substituting from Equation (1.16) into(1.15) and multiplying by cos2 φ, we obtain a separation of the Λ term. Thus

The last term of Equation (1.18), since it is the only one that is a function of ι, must be constant. On making this constant equal to—m2, we have

Λ = C cos mλ + S sin mλ, (1.19)

where C and S are arbitrary constants. Substituting – m2 for Λ term of Equation (1.18) and multiplying by Φ/cos2φ, we get an equation that is solely a function of φ. Thus

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.20)

or, substituting µ for sin φ,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.21)

The form of Equation (1.21) with m = 0, known as Legendre's equation, is solved by assuming that F is represented by a power series in µ (Apostol, 1962, pp. 359–364). In the case of m ≠ 0, the 1 – µ2 in the denominator makes the simple power series representation inconvenient, and the equation must be solved by cut-and-try. The try that succeeds is to assume that Φ has the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.22)

which leads to an equation for v. Therefore,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.23)

Assuming

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.24)

substituting in Equation (1.23), and requiring the coefficient of each power of µ to be separately zero, we obtain a recurrence relationship between alternate coefficients of the power series. From the coefficients of µk we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.25)

In order to obtain the maximum possible value of k, we set the numerator of Equation (1.25) equal to zero and solve for k. The result is

kmax = l - m. (1.26)

Hence m [??] l, and if v is to be represented by a finite power series in µ, the allowable powers will be l - m – 2t, where t is any non-negative integer [??](l –m)/2. Let us substitute (l -m – 2t) for k and the notation Tlmt for ak. Then, on taking advantage of the cancellation between some terms in the numerator to obtain a more compact expression, Equation (1.25) becomes

This solution still leaves Tlm0 to be defined. Because the whole expression is multiplied by the arbitrary constants BC and BS from Equations (1.17) and (1.19), Tιm0 is also arbitrary; any change in the value adopted for Tlm0 would merely result in an inversely proportionate change in BC and BS. As a result of the most common manner of derivation, Tlm0 is usually defined as (see, for example, Hobson, 1961, p. 91)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.28)

Applying Equation (1.27) successively to (1.28), we then get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.29)

The solution Φ of Equation (1.20) or (1.21) corresponding to a particular pair of subscripts l, m is called a Legendre associated function, Plm (sin φ) or Pim (µ); thus we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.30)

where k is the integer part of (l —m)/2.

The complete real solution of the Laplace equation, (1.13), from (1.17) (setting A = 0, B = 1), (1.19), and (1.30) is then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.31)

where the i subscript in the first term denotes the cos mι or sin mι term.

In addition to these real solutions, there are imaginary solutions that are not applicable to the potential problem in which we are interested.

An important property of the surface spherical harmonics Slmi is that they are orthogonal; namely,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.32)

for integration over the surface of a sphere. This property makes the spherical harmonics the natural means for general representation of a function over a spherical surface, analogous to Fourier series for a function in a rectilinear space. The integral of the square of Slmi is, for unit Clm or lm,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.33)

where the Kronecker delta δ0m is equal to 1for m = 0 and 0 for m ≠ 0. The factor (l + m )!/(l – m)! indicates that the magnitude of the functions (and hence of the coefficients) will vary greatly with the subscript m. In order to make coefficients more readily comparable in numerical work, it is generally convenient to use normalized functions, for example,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.34)

Spherical harmonics are most conveniently remembered in terms of their zeros. A surface harmonic Slmi will have (l – m) zeros in a distance p along a meridian and m zeros in the same distance along a parallel. Some examples are shown in Figure 1.

1.3 Potential of the Ellipsoid

The shape of the earth, as defined very closely by the geoid, or mean sea level (less meteorological effects), is determined not only by the gravitational potential V but also by the potential of rotation. These two potentials combine to make what is called the potential of gravity,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.35)

where ω is the rate of rotation.

It is an observed fact that the geoid is approximated within about 10-5 of the radius vector by an ellipsoid of revolution. The shape of this ellipsoid is conventionally expressed by the flattening f,

f = (a – b)/a, (1.36)

where a is the equatorial radius and b is the polar radius. The value of f is about 3.353 x 10–3. Hence, to explore the 10-5 departures of the earth's gravitational field from a reference potential V0 of an ellipsoid, it is necessary to carry the expression of V0 in terms of the mass M, radius a, rotation ω, and flattening f to terms of order f2. Furthermore, in order to connect results of systems affected by gravitation (satellite orbits) with those affected by gravity (gravimetry), it is desirable to connect the parameters of the ellipsoid convenient to the former (kM and C20) to those convenient to the latter (the equatorial acceleration of gravity ge and f). The algebra involved in making this connection is considerable, so we shall write out only the terms to O(f) in the solution outlined here.

The symmetries of an ellipsoid of revolution indicate that its radius vector can be expressed as a sum of even degree zonal harmonics. Thus, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.37)

where P20, P40 are defined by (1.29) and (1.30).

The customary manner of representing the potential of gravity of a reference figure, or "normal" potential, is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.38)

where

m = ω2a/ge, (1.39)

in which ge is the acceleration of gravity at the equator.

Expanding (1.37) by the binomial theorem gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.40)

when use has been made of

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.41)

Substituting (1.40) into (1.38), using (1.41), and neglecting P20P40 and P240 terms, we get a form for the potential U on the surface of the ellipsoid. Thus

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.42)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If the ellipsoidal surface is an equipotential, then U must be constant and equal to U0 thereon. The coefficients C2 and C4 must therefore be separately equal to zero, which yields two equations for kM, J2, and J4 in terms of the other parameters. A third equation is obtained by the condition that the negative of the radial derivative of U must be equal to ge at the equator. Thus,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.43)

The three equations then can be solved simultaneously for kM, J2, and J4. We have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In order to obtain r0, α2, and α4 in terms of the conventional parameters for an ellipsoid, a and f, we have the equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.44)

Converting from rectangular to spherical coordinates by Equation (1.9), using (1.36) to eliminate b, solving for r2, and then applying the binomial theorem to obtain r, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.45)

Integrating (1.45) from 0 to 1 with respect to sin φ, we get r0,

r0 = a]1 - f/3 + O(f2)],

and then α2 and α4,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Substitution in the equations for kM, J2, and J4 gives the final solution,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.46)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.47)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.48)

To summarize, for the purposes of celestial geodesy, we can consider the earth's gravity field as represented by a normal potential of an ellipsoid of revolution, Equation (1.38), plus small irregular variations expressed by a sum of spherical harmonics, as in (1.31).

CHAPTER 2

MATRICES AND ORBITAL GEOMETRY

2.1. General

The purpose of this chapter is to describe the geometry of an idealized situation: a vacuum with an earth rotating uniformly with respect to fixed inertial coordinates. This idealized situation will constitute both the reference frame for the development of close satellite orbit dynamics in Chapter 3, and for the description of observations of satellites and variations in the coordinate system in Chapter 4. As a preliminary to describing the geometry, as well as to some techniques of data analysis in Chapters 5 and 6, we summarize the rules of matrix algebra.

2.2. Matrix Notation

A vector x, or {x1, x2, x3}T, can be changed into another vector y by a linear transformation given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.1)

Equation (2.1) can be abbreviated as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.2)

or as

y = Ax.

A rectangular array A of numbers aij is called a matrix. Matrix algebra is the expression of algebraic operations on arrays of quantities, such as the transformation in (2.1), in compressed notation such as (2.3). In this text we shall be interested both in expressing transformations from one coordinate system to another and in the formation and solution of generalized least-squares problems. The principal rules of matrix algebra are:

The numbers aij that comprise a matrix are called elements.

(Continues...)

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Excerpted from Theory of Satellite Geodesy by William M. Kaula. Copyright © 2000 Gene Kaula. Excerpted by permission of Dover Publications, Inc..
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Table of Contents

TABLE OF SYMBOLS
TABLE OF NUMERICAL VALUES
1 THE EARTH'S GRAVITATIONAL FIELD
1.1 Potential Theory
1.2 Spherical Harmonics
1.3 Potential of the Ellipsoid
2 MATRICES AND ORBITAL GEOMETRY
2.1 General
2.2 Matrix Notation
2.3 Orbital Geometry
3 SATELLITE ORBIT DYNAMICS
3.1 Elliptic Motion
3.2 Perturbed Equations of Motion
3.3 Conversion of Spherical Harmonic Disturbing Function
3.4 Linear Perturbations
3.5 Nonlinear Perturbations
3.6 Resonance
3.7 Miscellaneous Effects
3.8 Summary
4 GEOMETRY OF SATELLITE OBSERVATIONS
4.1 General
4.2 Coordinate Transformations
4.3 Differential Relationships and Observation Equations
4.4 Observation Equations: Directional
4.5 Observation Equations: Range Rate and Range
4.6 Time and Precise Definition of Coordinates
4.7 Observability Conditions
5 STATISTICAL IMPLICATIONS
5.1 General
5.2 Time Series
5.3 Quadratic Sum Minimization
6 DATA ANALYSIS
6.1 Simultaneous Observations
6.2 Orbital Observations: Short-Term
6.3 Orbital Observations: Long-Term
INDEX