PART VI APPENDICES

PART VI

APPENDICES

CONTENTS

PART VI

APPENDICES

Appendix

A Spherical Geometry 727

B Construction of Global Geometry Plots 737

C Matrix and Vector Algebra 744

D Quaternions 758

E Coordinate Transformations 760

F The Laplace Transform 767

G Spherical Harmonics 775

H Magnetic Field Models 779

Spacecraft Attitude Determination and Control Systems 787

J Time Measurement Systems 798

K Metric Conversion Factors 807

L Solar System Constants 814

M Fundamental Physical Constants 826

Index 830

APPENDIX A

SPHERICAL GEOMETRY

A.I Basic Equations A.2 Right and Quadrantal Spherical Triangles A.3 Oblique Spherical Triangles AA Differential Spherical Trigonometry A.S Haversines

James R. Wertz

Finding convenient reference material in spherical geometry is difficult. This appendix provides a compilation of the most useful equations for spacecraft work. A brief discussion of the basic concepts of spherical geometry is given in Section 2.3. The references at the end of this appendix contain further discussion and proofs of most of the results presented here.

A.I Basic Equations

Algebraic Formulas. Let Pi be a point on the unit sphere with coordinates (ai,oJ. The arc-length distance, ()(PI'P2), between PI and P2 is given by:

cos()( P I,P2) = cos()( P2,PI)

= sino lsin02 + cosOICOS02cos( a l - ( 2 ) (A-I)

The rotation angle, A(P p P2 ; P3), from PI to P2 about a third point, P3, is cumbersome to calculate and is most easily obtained from spherical triangles (Sections A.2 and A.3) if any of the triangle components are already known. To calculate directly from coordinates, obtain as intermediaries the arc-length distances ()( Pi' P), between the three pairs of points. Then

cos()( PI' P2) - cos ()( PI' P 3)COS()( P2 , P3)

cosA(PI,P2 ;P3)= . ()(P P)' ()(P P) SIn I' 3 SIn 2' 3

(A-2)

with the quadrant determined by inspection. The equation for a small circle of angular radius p and centered at (ao, 00) in

terms of the coordinates, (a,o), of the points on the small circle is, from Eq. (A-I),

COS P = sin 0 sin 00 + cos 0 cos oocos( a - ao) (A-3)

The arc length, {3, along the arc of a small circle of angular radius p between two points on the circle separated by the rotation angle, q, (q, measured at the center of the circle) is

{3= q,sinp (A-4)

The chord length, y, along the great circle chord of an arc of a small circle of

728 SPHERICAL GEOMETRY A.I

angular radius p is given by

cosy= 1-(I-cosCP)sin2p (A-5)

where cP is as defined above. The equation for a great circle with pole at (0:0>80) is, from Eq. (A-3) with

p=90°,

(A-6a)

The inclination, i, and azimuth of the ascending node (point crossing the equator from south to north when moving along the great circle toward increasing azimuth), </>0' of the great circle are

i=90° - 80

</>0=90° +0:0 (A-6b)

Therefore, the equation for the great circle in terms of inclination and ascending node is

tan8 = tan; sin( 0: -</>0) (A-6c)

The equation of a great circle through two arbitrary points is given below. Along a great circle, the arc length, the chord length, and the rotation angle, CP, are all equal, as shown by Eqs. (A-4) and (A-5) with p = 90° .

Finally, the direction of the cross product between two unit vectors associated with points PI and P2 on the unit sphere is the pole of the great circle passing through the two points. Find the intermediary, PI' from

(A-7a)

As shown iii Fig. A-I, PI is the azimuth of point PI relative to the ascending node

Fig. A-I. P is the pole of the great circle passing through PI and P2 and is also in the direction of the doss product PI xP2• PI is an intermediate variable used for computations.

A.l BASIC EQUATIONS 729

of the great circle through p. and P2• The coordinates, (ac,8c)' of the cross product p. XP2 are given by

sinfi.

tan8.

sin(a2 - a.) tan 82

(A-7b)

Combining Eqs. (A-7b) and (A-6a) gives the equation for a great circle through points PI and Pz:

(A-8)

Area Formulas. All areas are measured on the curved surface of the unit sphere. For a sphere of radius R, multiply each area formula by R 2. All arc lengths are in radians and all angular areas are in steradians (sr), where

I sr= solid angle enclosing an area equal to the square of the radius

_( ISO)2d 2 - - eg 'If'

The surface area of the sphere is

(A-9)

The area of a lune bounded by two great circles whose inclination is e radians is

(A-tO)

The area of a spherical triangle whose three rotation angles are e., e2, and e3 is

(A-ll)

The area of a spherical polygon of n sides, where e is the sum of its rotation angles in radians, is

Op = e - (n - 2)'If'

The area of a small circle of angular radius p is

Oc=2'1f'(I-cosp)

(A-12)

(A-l3)

The overlap area between two small circles of angular radii p and E, separated by a center-to-center distance, a, is

730 SPHERICAL GEOMETRY A.2

Qo = 27T - 2 cosparccos .. [ cos€ -cospcosa ]

smpsma

[ COSP-Cos€cosa ]

- 2cos€ arc cos .. sm€sma

[COS a - cos { cos P ]

-2arccos .. SlllfSIllP

(A-14)

Recall that area is measured on the curved surface.

A.2 Right and Quadrantal Spherical Triangles

Example of an Exact Right Spherical Triangle. For testing formulas, the isosceles right spherical triangle shown in Fig. A-2 is convenient. The sides and angles shown are exact values.

Fig. A-2. Example of an Exact Right Spherical Triangle. 'I'=arcsin{2j3 ""54.7356°",,,54°44'08".

Napier's Rules for Right Spherical Triangles. A right spherical triangle has five variable parts, as shown in Fig. A-3. If these components and their complements (complement of == 90° - <1» are arranged in a circle, as illustrated in Fig. A-3, then the following relationships hold between the five components in the circle:

The sine of any component equals the product of either 1. The tangents of the adjacent components, or 2. The cosines of the opposite components

For example,

sin A = tan <p tan(90° - <1» = cos(90° - A )cos(90° - 0 )

Quadrants for the solutions are determined as follows: 1. An oblique angle and the side opposite are in the same quadrant. 2. The hypotenuse (side 0) is less than 90 deg if and only if <p and A are in the

same quadrant and more than 90 deg if and only if <p and A are in different quadrants.

Note: Any two components in addition to the right angle completely determine the triangle, except that if the known components are an angle and its opposite side, then two distinct solutions are possible.

Fig. A-3. Diagram for Napier's Rules for Right Spherical Triangles. Note that the complements are used for the three components farthest from the right angle.

A.3 OBLIQUE SPHERICAL TRIANGLES 731

The following formulas can be derived from Napier's Rules for right spherical triangles:

sin A = tan cp cot cP = sinO sin A.

sin cp = tan A cot A. = sin 0 sin cP

cosO = cot cP cot A. = cOScpCOSA

cos A. = tancp cotO = cosAsin cP

cos cP = tanAcotO = cos cp sin A.

(A-IS)

(A-I6)

(A-l7)

(A-I8)

(A-19)

Napier's Rules are discussed in Section 2.3. Proof of these rules can be found in most spherical geometry texts, such as those of Brink (1942]; Palmer, et al., (1950]; or Smail [1952].

Napier's Rules for Quadrantal Spherical Triangles. A quadrantal spherical triangle is one having one side of 90 deg. If the five variable components of a quadrantal triangle are arranged in a circle, as shown in Fig. A-4, then Napier's Rules as quoted above apply to the relationships between the parts. (Note that the bottom component is minus the complement of e.) The rules for defining quadrants are modified as follows:

l. An oblique angle (other than e, the angle opposite the 90-deg side) and its opposite side are in the same quadrant.

2. Angle e (the angle opposite the 90-deg side) is more than 90 deg if and only if A and cp are in the same quadrant and less than 90 deg if and only if A and cp are in different quadrants.

90°

o

Fig. A-4. Diagram for Napier's Rules for Quadrantal Spherical Triangles. Note that complements are used in the three components farthest from the 90-deg side and the component opposite the 90-deg side is minus the complement of e.

The following formulas can be derived from Napier's Rules for quadrantal spherical triangles:

sin A. = tan cP cot cp == sin e sin A

sin cP = tan A. cotA = sin e sincp

cose= -cotcpcotA= -cosCPcosA.

COSA= - tan cpcote = cos A. sincp

coscp= - tan A. cote = cosCPsinA

A.3 Oblique Spherical Triangles

(A-20)

(A-21)

(A-22)

(A-23)

(A-24)

Three fundamental relationships-the law of sines, the law of cosines for angles, and the law of cosines for sides-hold for all spherical triangles. These may

732 SPHERICAL GEOMETRY A,3

be used to derive Napier's Rules (Section A,2) or may be derived from them. The components of a general spherical triangle as used throughout this section are shown in Fig. A-5.

Law of Sines.

Law of Cosines for Sides.

sin () sin A sin cp sin 8 = sin A = sin <P

COSA = cos()coscp + sin()sincpcosA

Similar relationships hold for each side.

Law of Cosines for Angles.

cosA = -cos8cos<p + sin 8sin <PCOSA

Similar relationships hold for each angle.

¢

e

Fig. A-5. Notation for Rules for Oblique Spherical Triangles

(A-25)

(A-26)

(A-27)

Half-Angle Formulas. A spherical triangle is fully specified by either three &ides or three angles. The remaining components are most conveniently expressed in terms of half angles. Specifically,

where

and

where

sin( (J - () )sin( (J - cp)

sin() sincp

cos(~ - 8)cos(~ - <P)

sin8sin <P

(A-28)

(A-29)

A.3 OBLIQUE SPHERICAL TRIANGLES 733

Similar relationships may be found for the other trigonometric functions of half angles in most spherical geometry texts.

General Solution of Oblique Spherical Triangles. Table A-I lists formulas for solving any oblique spherical triangle. In addition, in any spherical triangle, the

Table A-I. Formulas for Solving Oblique Spherical Triangles. See also Table A-2.

KNOWN TO FORMULA COMMENTS FIND

0=!(8+A+I/» 6. A,IP A sin !A= sin (0- 8) sin (0- tfij

2 sin 8 sin t/> 2

UNIQUE SOLUTION 1

e,A.«I> A cos! A= cos (E- Ell cos (E- «1» l:=!(EhA+«I>I

2 sin esin «I> 2

UNIQUE SOLUTION 1

I/> cos I/> = cos8 eos A + sin 8 sin A cos «I> UNIQUE SOLUTION 1

8,},,«I> A tan A-

sin r , tan «I> tan r , = tan A cos «I>

sin (8 - r , ' UNIQUE SOLUTION

e cos e = sin A sin «I> cos 8 - cosAcos«l> UNIQUE SOLUTION

8, A. «I>

A tan A= tan 8 sin r 2 tanr2 -tanAcos8 sin (<<1>+ r 2)

UNIQUE SOLUTION

I/> cos 8 sin r3 cot r3 = cos etan A sin (1/>+ r 3) =

cos A 2 VALID SOLUTIONS

8,}"a A sin A- sin asin A 2 VALID SOLUTIONS sin 8

tan r 4 = tan eeos A «I> sin (<<1>+ r 4) - sin r 4 tan A cot 8 2 VALID SOLUTIONS

cos esin r cot r5 - tan A cos 8 «I> sin (<<1>- r ) = 5

5 cos A 2 VALID SOLUTIONS

8,e,A A sin A = sin 8 sin A 2 VALID SOLUTIONS sine

tanrs = COl A tan 8

I/> sin (1/>- rS) - cot a tan Asin rS 2 VALID SOLUTIONS

'SECTION A-5 PRESENTS AN ALTERNATfVE FORMULA.

following rules are sufficient to determine the quadrant of any component: I. If one side (angle) differs from 90 deg by more than another side (angle), it

is in the same quadrant as its opposite angle (side). 2. Half the sum of any two sides is in the same quadrant as half the sum of the

opposite angles.

A.4 Differential Spherical Trigonometry

The development here follows that of Newcomb [1960], which contains a more extended discussion of the subject.

Differential Relations Between the Parts of a Spherical Triangle. In general, any part of a spherical triangle may be determined from three other parts. Thus, it is of interest to determine the error in any part produced by infinitesimal errors in the three given parts. This may be done by determining the partial derivatives relating any four parts of a spherical triangle from the following differentials, where the notation of Fig. A-5 is retained.

Given three angles and one side:

- sin A sin dO + d8 + cos cpdA + cos A d = 0

Given three sides and one angle:

- dO + cosdA + cosA dcp+ sincpsinA d8 = 0

Given two sides and the opposite angles:

cosOsin dO - coscpsin 8dcp+ sinOcosd - sincpcos8d8 =0

Given two sides, the included angle, and one opposite angle:

- sin dO + cosAsin 8dcp+ sinAd8 +cossinOdA = 0

(A-30)

(A-31)

(A-32)

(A-33)

As an example of the determination of partial derivatives, consider a triangle in which the three independent variables are the three sides. Then, from Eq. (A-31),

a8 I cosA cot A a;p A, II = - sincpsinA = - sincp

Infinitesimal Triangles. The simplest infinitesimal spherical triangle is one in which the entire triangle is small relative to the radius of the sphere. In this case, the spherical triangle may be treated as a plane triangle if the three rotation angles remain finite quantities. If one of the rotation angles is infinitesimal, the analysis presented below should be used.

Figure A-6 shows a spherical triangle in which two sides are of arbitrary, but nearly equal, length and the included rotation angle is infinitesimal. Then the

'" 06

Fig. A-6. Spherical Triangle With One Infinitesimal Angle

A.5 HAVERSINES 735

change in the angle by which the two sides intercept a great circle is given by

8<1>=<1>' -<1>= 180 0 -(A+<I»

=8eCOSA

The perpendicular separation, a, between the two long arcs is given by

a= 8esinA

(A-34)

(A-35)

If two angles are infinitesimal (such that the third angle is nearly 180 deg), the triangle may be divided into two triangles and treated as above.

A.S Haversines

A convenient computational tool for spherical trigonometry is the haversine, defined as

haversineO =havO =~ (1- cosO) (A-36)

The principal advantage of the haversine is that a given value of the function corresponds to only one angle over the range from 0 deg to 180 deg, in contrast to the sine function for which there is an ambiguity as to whether the angle corresponding to a given value of the sine falls in the range 0 deg to 90 deg or 90 deg to 180 deg. Given the notation of Fig. A-5, two fundamental haversine relations in any spherical triangle are as follows:

where

hav A = hav(O - </.» + sinO sin </.>hav A

sine a - 0 )sin( a-</.» havA= . Il .

smusm</.>

(A-37)

(A-38)

The first three formulas from Table A-I can be expressed in a simpler form to evaluate in terms of haversines, as shown in Table A-2. Most spherical geometry

Table A-2. Haversine Formulas for Oblique Spherical Triangles

KNOWN TO FORMULA COMMENTS FIND

8. A. </> A hav A = hav A - hav (8 - </»

sin 8 sin </J

8, J\,


texts (e.g., Brink [1942] or Smail [1952]) carry a further discussion of haversine formulas. The function is tabulated in Bowditch's American Practical Navigator [1966].

References

1. Bowditch, Nathaniel, American Practical Navigator. Washington, D.C.: USGPO,1966.

2. Brink, Raymond W., Spherical Trigonometry. New York: Appleton-CenturyCroft, Inc., 1942.

3. Newcomb, Simon, A Compendium of Spherical Astronomy. New York: Dover, 1960.

4. Palmer, C. I., C. W. Leigh, and S. H. Kimball, Plane and Spherical Trigonometry. New York: McGraw-Hill, Inc., 1950.

5. Smail, Lloyd Leroy, Trigonometry: Plane and Spherical, New York: McGrawHill, Inc., 1952.

APPENDIXB

CONSTRUCTION OF GLOBAL GEOMETRY PLOTS

James R. Wertz

Global geometry plots, as used :hroughout this book, are convenient for presenting results and for original work involving geometrical analysis on the celestial sphere. The main advantage of this type of plot is that the orientation of points on the surface is completely unambiguous, unlike projective drawings of vectors between three orthogonal axes. This appendix describes procedures for manually constructing these plots. Methods for obtaining computer-generated plots

Fig. 8-1. Equatorial Projection Grid Pattern

738 CONSTRUCTION OF GLOBAL GEOMETRY PLOTS Ap.B

are described in Section 20.3 (subroutines SPHGRD, SPHPLT, and SPHCNV). Interpretation and terminology for the underlying coordinate system is given in Section 2.1.

For most applications related to attitude geometry, the spacecraft is thought of as being at the center of the globe. Therefore, an arrow drawn on the globe's equator from right to left would be viewed by an observer on the spacecraft as going from left to right. This geometrical reversal is illustrated in Figs. 4-3 and 4-4, which show the Earth as a globe viewed from space and as viewed on the spacecraft-centered celestial sphere. Similarly, Figs. 11-25 and 11-26 show the orbit of the Earth about the spacecraft as viewed by the spacecraft. This spacecraftcentered geometry allows a rapid interpretation of spacecraft observations and attitude-related geometry.

In this book, we use four basic globe grids showing the unit sphere from the perspective of infinitely far away (i.e., half the area of the sphere is seen on each globe) as seen by observers at 30-deg latitude intervals from the equator to the pole. Tht" four grids are shown in Figs. B-1 through B-4 and are intended for

Fig. B-2. 30-Deg Inclination Grid Pattern

Ap.B CONSTRUCTION OF GLOBAL GEOMETRY PLOTS 739

reproduction by interested users. (For accurate construction, the globes must be reproduced to the same size as nearly as possible;"Therefore, reproductions of the different projections should be made at the same time on the same equipment.) Coordinate lines are at lO-deg intervals in latitude and longitude except within 10 deg of the poles, where the longitude intervals are 30 deg. The globe originals are handdrawn and are accurate to about I deg in the central regions and 2 deg near the perimeter.

The most important feature of the globes for the purpose of plot construction is that the geometry of figures constructed on the sphere does not depend on the underlying grid pattern. For example, if we take the globe from Fig. B-2, we may draw a small circle of 20-deg radius centered on the pole and an equilateral right spherical triangle between the pole and the equator as shown in Fig. B-5(a). (Any parallel of latitude is a small circle and the equator or any meridian of longitude is a great circle.) Having constructed the figure, we may rotate or tilt the underlying grid pattern without affecting the geometrical construction .. Thus, in Fig. B-5(b) the

Fig. B-3. 6O-Deg Inclination Grid Pattern


underlying coordinate grid has been rotated about 120 deg counterclockwise and the triangle/circle pattern has been left unchanged. Thus, in the new coordinate system (viewed by rotating the page 120 deg clockwise), we have constructed a small circle with a 20-deg angular radius centered at about 21 deg below the equator. By rotating the underlying grid an appropriate amount, we could center the small circle at any desired latitude.

Because of the symmetry of the underlying sphere, we may not only rotate the grid pattern, but also interchange the underlying grids among the four shown in Figs. B-1 through B-4. For example, Fig. B-5(c) shows the circle/triangle pattern with the underlying grid changed to the equatorial view and rotated somewhat counterclockwise. (Again, the grid may be rotated to any convenient angle.) The meridian lines on the grid pattern, along with the imaginary meridians between those that are shown, are great circles. Therefore, in Fig. B-5(c), the dashed line is a great circle passing through one vertex of the triangle and tangent to the small circle. Finally, in Fig. B-5(d), we have left the constructed figures unchanged and returned the underlying grid pattern to its original orientation from Fig. B-5(a).

Fig. B-4. Polar Projection Grid Pattern


(b)

(e) (d)

Fig. B-5. Construction of Global Geometry Plots. See text for explanation.

Thus, by using the grid pattern of Fig. B-1, we have constructed a great circle tangent to a small circle of 20-deg radius about the pole or, equivalently, at an inclination of 70 deg to the equator.

In practice, this construction is performed by first drawing the original figure of B-5(a), and then placing it on top of the equatorial view on a light table so that both grids can be seen. After rotating the grid patterns relative to each other until the desired orientation for the dashed curve is obtained, we can trace the dashed curve directly on the grid of Fig. B-5(a). This general procedure for drawing great and small circles by superposing grids on a light table (or window) and rotating them until the desired orientation is obtained has been used to construct nearly all the globe figures in this book. Note that whenever figures are constructed using superposed grids, the centers of the two grids must be on top of each other or, eqUivalently, the perimeters of the two grids must be superposed. This principle of grid


superposition can be applied to the construction of various figures, as discussed below.

Constructing Great Circles Through Two Points. Figure B-1 is the basic figure for constructing all great circles on the celestial sphere, because all possible great circles are represented by the meridian lines on the figure and the imaginary meridians between the ones drawn. To construct a great circle between any two points of any of the globes, place the globe with the two points on top of a copy of Fig. B-l. Keeping the perimeters of the two figures superposed, rotate the globes relative to each other until the two points lie over the same meridian on the underneath grid. This meridian is then the great circle defined by the two points. Note that this great circle is most precisely defined when the two points are neady 90 deg apart and is poorly defined if the two points are nearly 0 deg or 180 deg apart.

Measuring Arc Lengths. The grid pattern in Fig. B-1 can also be used to measure the arc-length separation between any two points on the sphere. The parallels of latitude (i.e., the horizontal straight lines in Fig. B-1) are separated by 10 deg of arc along each meridian. Therefore, to determine the arc length between two points, superpose the globe with the two points over a copy of Fig. B-1 and rotate it until the meridian forming the great circle between the two points is found. The arc length is then determined by using the parallels of latitude along the meridian as a scale. For example, consider the dashed great circle in Fig. B-5(c). Because the triangle is a right equilateral triangle, the distance between any vertex and the opposite side must be 90 deg. This may be confirmed by counting the parallels of latitude along the dashed great circle. Also, the diameter of the small circle in Figs. B-5(b) and (c) may be measured along the meridian passing through the center of the circle. In both subfigures, the measured angular diameter is 40 deg, as required. Note that arc length must be measured along a great circle; it cannot be measured along parallels of latitude or other small circles.

Constructing Great Circles From General Criteria. In general, any great circle is constructed by first finding two points on it and then drawing the great circle between these points. For example, to draw a great circle at a given inclination to the equator, first pick the intercept point on the equator. Measure along the equator to the right or the left (depending on the slope desired) 90 deg and then up from the equator (along a meridian) an angle equal to the inclination. This point and the intercept point on the equator define the great circle. This method could have been used to construct the dashed great circle in Fig. B-5(c) directly without considering the tangent to the small circle.

Figure B-6 illustrates the procedure for constructing a great circle through a given point, A, perpendicular to a given great circle, AA'. Locate the point A' along the given great circle 90 deg from A by the method described above. Measure along the meridian through A' 90deg in either direction to the point B. The great circle through A and B is perpendicular to AA I.

Constructing Small Circles. This construction has already been demonstrated by the example of Fig. B-S. The method described there may be used to construct


Fig. B-6. Construction of Great Circle AB Perpendicular to Great Circle AA'

small circles whose center is on the perimeter, 30 deg from the perimeter, 60 deg from the perimeter, or at the center. Small circles centered on the perimeter are straight lines on the plot and small circles centered in the middle of the plot are circles. The radius of the small circles constructed by this method is the colatitude (distance from the pole) of the parallel of latitude chosen on the underlying coordinate grid. For most purposes, one of these four sets of small circles is sufficient. They have been used for all of the constructions in this book. If it is necessary to construct a small circle at an intermediate arc distance from the perimeter, first construct a small circle of the desired radius at the desired latitude and as near the desired longitude as possible. Transform each point on this circle along a parallel of latitude a distance in longitude equal to the difference in longitude between the desired center and the constructed center. This procedure is illustrated in Fig. B-7.

Fig. B-7. Transforming a Small Circle in Longitude

APPENDIXC

MATRIX AND VECTOR ALGEBRA

C.I Definitions C.2 Matrix Algebra C.3 Trace, Determinant, and Rank C.4 Matrix Inverses and Solutions to Simultaneous Linear

Equations C.S Special Types of Square Matrices, Matrix Transforma-

tions C.6 Eigenvectors and Eigenvalues C.7 Functions of Matrices C.S Vector Calculus C.9 Vectors in Three Dimensions

C.I Definitions

F. L. Markley

A matrix is a rectangular array of scalar entries known as the elements of the matrix. In this book, the scalars are assumed to be real or complex numbers. If all the elements of a matrix are real numbers, the matrix is a real matrix. The matrix

All AI2 A ln

A21 A22 A2n =[ Aij] A= (C-l)

AmI Am2 Amn

has m rows and n columns, and is referred to as an m X n matrix or as a matrix of order m X n. The equation A = lAij] should be read as, "A is the matrix whose elements are Aij'" The first subscript labels the rows of the matrix and the second labels the columns.

Two matrices are equal if and only if they are of the same order and all of the corresponding elements are equal; i.e.,

A = B if and only if Aij= Bij; i= I, . .. ,m ;j= I, .. . ,n (C-2)

An n X n matrix is called a square matrix and is usually referred to as being of order n rather than n X n.

The transpose of a matrix is the matrix resulting from interchanging rows and columns. The transpose of A is denoted by AT, and its elements are given by

AT =[(AT)ij] =[ Aji] (C-3)

As an example, the transpose of the matrix in Eq. (C-l) is

C.I DEFINITIONS 745

It is clear that the transpose of an m X n matrix is an n X m matrix, and that the transpose of a square matrix is square. The transpose of the transpose of a matrix is equal to the original matrix:

(C-4)

The adjoint of a matrix, denoted by A t, is the matrix whose elements are the complex conjugates of the elements of the transpose of the given matrix, * i.e.,

The adjoint of the adjoint of a matrix is equal to the original matrix:

(At)t =A

The adjoint and the transpose of a real matrix are identical.

(C-5)

(C-6)

The main diagonal of a square matrix is the set of elements with row and column indices equal. A diagonal matrix is a square matrix with nonzero elements only on the main uiagonal, e.g.,

DII 0 0 0 D22 0

D= (C-7)

0 0 Dnn

The identity matrix of a given order is the diagonal matrix with all the elements on the main diagonal equal to unity. It is denoted by I, or by In to indicate the order explicitly. .

A matrix with only one column is a column matrix. An n X I column matrix can be identified with a vector in n-dimensional space, and we shall indicate such matrices by boldface letters, as used forvectors.t A matrix with only one row is a row matrix; its transpose is a column matrix, so we denote it as the transpose of a vector. The elements of a row or column matrix will be written with only one subscript; for example,

(C-8)

Bn

A set of m n X I vectors B('\ i = 1,2, ... , m, is linearly independent if and only if the only coefficients aj' i = 1,2, ... , m, satisfying the equation

m L ajB(j) = a.B(l)+ a2B(2) + ... +amB(m)=o (C-9)

.;=1

• The word adjoint is sometimes used for a different matrix in the literature. t Strictly speaking, a vector is an abstract mathematical object, and the column matrix is a concrete realization of it, the matrix elements being the components of the vector in some coordinate system.

746 MATRIX AND VECTOR ALGEBRA C.2

are aj = 0, i = 1,2, ... , m. There can never be more than n linearly independent n X I vectors.

C.2 Matrix Algebra

Multiplication of a matrix by a scalar is accomplished by multiplying each element of the matrix by the scalar, i.e.,

(C-lO)

Addition of two matrices is possible only if the matrices have the same order. The elements of the matrix sum are the sums of the corresponding elements of the matrix addends, i.e.,

A + B =[ Aij+ Bij]

Matrix subtraction follows from the above two rules by

(C-II)

(C-12)

Multiplication of two matrices is possible only if the number of columns of the matrix on the left side of the product is equal to the number of rows of the matrix on the right. If A is of order I X m and B is m X n, the product AB is the I X n matrix given by

(C-I3)

Matrix multiplication is associative

A(BC)=(AB)C (C-14)

and distributive over addition

A(B+ C)=AB+AC (C-15)

but is not commutative, in general,

AB =1= BA (C-16)

In fact, the products AB and BA are both defined and have the same order only if A and B are square matrices, and even in this case the products are not necessarily equal. For the square matrices A = U ~l and B = [~ ~l, for example, we have

AB=[19 43

22 ] =1= BA = [ 23 50 31

34 ] 46

If AB= BA, for two square matrices, A and B, we say that A and B commute. One interesting case is diagonal matrices, which always commute.

The adjoint (or transpose) of the product of two matrices is equal to the product of the adjoints (or transposes) of the two matrices taken in the opposite order:

(C-17)

C.2 MATRIX ALGEBRA 747

(C-18)

This result easily generalizes to products of more than two matrices. Multiplying any matrix by the identity matrix of the appropriate order, on the

left or the right, yields a product equal to the original matrix. Thus, if B is of order mXn,

(C-19)

The product of an n X m matrix and an m-dimensional vector (an m X I matrix) is an n-dimensional vector; thus,

(C-20)

A similar result holds if a row matrix is multiplied on the right by a matrix,

yT =XTAT = [ ~ Aji~l )=1

(C-21)

An important special case of the above is the multiplication of a I X n row matrix (on the left) by an n X I column matrix (on the right) which yields a scalar,

n

s=yTX= ~ Xj~ (C-22) j=1

For real vectors, this scalar is the inner product, or dot product, or scalar product of the vectors X and y. For vectors with complex components, it is more convenient to define the inner product by using the adjoint of the left-hand vector rather than the transpose. Thus, in general,

(C-23) ;=1

Note that, in general,

Y·X=(X·Y)* (C-24)

This definition reduces to the usual definition for real vectors, for which the inner product is independent of the order in which the vectors appear. Two vectors are orthogonal if their inner product is zero. The inner product of a vector with itself

n n

X·X= ~ X;*X;= ~ IXJ (C-25) ;=1 ;=1

is never negative and is zero if and only if all the elements of X are zero, i.e., if X = O. This product will be denoted by X2 and its positive square root by IXI or by X, ifno confusion results. The scalar IXI is called the norm or magnitude of the vector, X, and can be thought of as the length of the vector. Thus, with our definition of the inner product,

IXI =0 if and only if X=O (C-26)


which would not be true if we defined the inner product using the transpose rather than the adjoint, because the square of a complex number generally is not positive.

If we multiply an n X I row matrix (on the left) by a I X m matrix (on the right), we obtain an n X m matrix. This leads to the definition of the outer product of two vectors

(C-27)

If the vectors are real, the adjoint of Y is the transpose of Y, and the ijth element of the outer product is Xi lj.

Matrix division can be defined in terms of matrix inverses, which are discussed in Section C.4.

C.3 Trace, Determinant, and Rank

Two useful scalar quantities, the trace and the determinant, can be defined for any square matrix. The rank of a matrix is defined for any matrix.

The trace of an n X n matrix is the sum of the diagonal elements of the matrix

n

trA == ~ Aii i=1

(C-28)

The trace of a product of square matrices is unchanged by a' cyclic permutation of the order of the product

n n n

tr(ABC)= ~ ~ ~ AijBjkCki=tr(CAB) ;=1 j=1 k=1

However, tr(ABC)'" tr(ACB), in general. The determinant of an n X n matrix is the complex number defined by

(C-29)

(C-30)

where the set of numbers {PI,P2"" ,Pn} is a permutation, or rearrangement, of {l,2, .. "n}. Any permutation can be achieved by a sequence of pairwise interchanges. A permutation is uniquely even or odd if the number of interchanges required is even or odd, respectively. The exponent P in Eq. (C-30) is zero for even permutations and unity for odd ones. The sum is over all the n! distinct permutations of {l,2, ... ,n}. It is not difficult to show that Eq. (C-30) is equivalent to

n ~ ;+j

detA = ~ (-1) AijMij (C-31) j=1

for any fixed i= 1,2, ... ,n, where Mij is the minor of Aij' defined as the determinant of the (n - I) X (n - I) matrix formed by omitting the ith row and jth column from A. This form provides a convenient method for evaluating determinants by successive reduction to lower orders. For example,

123 456 =lxI561-2XI461+3XI451 789 89 79 78

=(5 x9-8 X 6)-2(4X9-7 X6)+3(4X 8 -7 X 5)=0 (C-32)

C.4 MATRIX INVERSES AND SOLUTIONS TO SIMULTANEOUS LINEAR EQUATIONS 749

The determinant of the product of two square matrices is equal to the product of the determinants

det(AB)= (detA)(detB)

The determinant of a scalar multiplied by an n X n matrix is given by

det( sA) = sndetA

The determinants of a matrix and of its transpose are equal:

detAT =detA

Thus, the determinant of the adjoint is

detA t = (detA)*

(C-33)

(C-34)

(C-35)

(C-36)

The determinant of a matrix with all zeros on one side of the main diagonal is equal to the product of the diagonal elem~nts.

The rank of a matrix is the order of the largest square array in that matrix, formed by deleting rows and columns, that has a nonvanishing determinant. Clearly, the rank of an m X n matrix cannot exceed the smaller of m and n. The matrices A, AT, At, AtA, and AAt all have the same rank.

C.4 Matrix Inverses and Solutions to Simultaneous Linear Equations

Let A be an m X n matrix of rank k. An n X m matrix B is a left inverse of A if BA = In. An n X m matrix C is a right inverse of A if A C = 1m. There are four possible cases: k is less than both m and n, k = m = n, k = m < n, and k = n < m. If k is less than both m and n, then no left or right inverse of A exists. * If k = m = n, then A is nonsingular and has a unique inverse, A -I, which is both a left and a right inverse:

A - IA = AA - 1 = 1 (k=m=n) (C-37)

A nonsingular matrix is a square matrix with nonzero determinant; all other matrices are singular. If k = m < n, then A has no left inverse but an infinity of right inverses, one of which is given by

(k=m<n) (C-38)

If k = n < m, then A has no right inverse but an infinity of left inverses, one of which is

(k=n<m) (C-39)

A L or A R is called the generalized inverse or pseudoinverse of A. Consider the set of m simultaneous linear equations in n unknowns;

X I'X2'···,Xn ;

• It is possible to define a pseudoinverse for a general matrix, which in this case is neither a left nor a right inverse. In the other three cases, the pseudoinverse is identical with A -I, A R, and A L, respectively. The results on solutions of simultaneous linear equations can be generalized with this definition [Wiberg, 1971).

AX=Y (C-40)

If k = m = n, then X = A - Iy is the unique solution to the set of equations. It folIows that a nonzero solution to A X = 0 is possible only if A is singular, i.e.,

AX=O for X:F 0, only if detA =0 (C-41)

If k = m < n, there are more unknowns than equations, so there are an infinite number of solutions. The solution with the smalIest norm, lXI, is

(C-42)

If k = n < m, there are more equations than unknowns; therefore, no solution exists, in general. However, the vector X that comes closest to a solution, in the sense of minimizing lAX - YI, is

(C-43)

Note that although AA L :F 1m , it is possible that AA Ly = Y for the particular Y in Eq. (C-40). In this case, Eq. (C-40) has a unique solution given by Eq. (C-43).

It is easy to see that if A is nonsingular, then A -I is nonsingular also and

(C-44)

Likewise, if A is nonsingular, then AT and A tare nonsingular and their inverses are given by

(ATfl =(A _I)T

(Atf'=(A-,)t

(C-45)

(C-46)

If two matrices, A and B, are nonsingular, their product is nonsingular also; and the inverse of the product is the product of the inverses, taken in the opposite order:

(C-47)

This result easily generalizes to products of more than two matrices. Various algorithms exist for calculating matrix inverses; several are described

by Carnahan, et ai., [1969] and by Forsythe and Moler [1967]. An example of a subroutine for this purpose is INVERT, described in Section 20.3.

C;S Special Types of Square Matrices, Matrix Transformations

A symmetric matrix is a square matrix that is equal toits transpose:

(C-48)

A skew-symmetric or antisymmetric matrix is equal to the negative of its transpose:

(C-49)

Clearly, a skew-symmetric matrix must have zeros on its main diagonal. An example of a skew symmetric matrix is 0 in Section 16.1. A Hermitian matrix is

C.S SPECIAL TYPES OF SQUARE MATRICES, MATRIX TRANSFORMATIONS 7S1

equal to its adjoint:

(C-SO)

A real symmetric matrix is a special case of a Hermitian matrix. An orthogonal matrix is a matrix whose transpose is equal to its inverse:

(C-51)

A unitary matrix is a matrix whose adjoint is equal to its inverse:

(C-S2)

A real orthogonal matrix is a special case of a unitary matrix. The product of two unitary (or orthogonal) matrices is unitary (or orthogonal). This result generalizes to products of more than two matrices. A similar result generally does not hold for Hermitian or symmetric matrices. A normal matrix is a matrix that commutes with its adjoint

AtA =AAt

Thus, both Hermitian matrices and unitary matrices are special cases of normal matrices.

By the rules for determinants of products and adjoints, it is easy to see that if A is unitary

IdetAl2 = 1 (C-S3)

Thus, detA IS a complex number with absolute value unity. Similarly, if A is orthogonal,

(detAf= I, detA = ± 1 (C-S4)

An orthogonal matrix with positive determinant is a proper orthogonal matrix; an orthogonal matrix is improper if its determinant is negative.

Let X be an n-dimensional vector and let A be an n X n matrix. Then A X is another n-dimensional vector and can be thought of as the transformation of X by A. If X and Yare two vectors, the inner product of AX and AY is

(AX)· (AY)= (AX)t(AY) = XtAtAY

If A is uni tary ,

(A X)· (A Y) = X . Y (C-55)

The dot product is unchanged if both vectors are transformed by the same unitary matrix. This result with X = Y shows that the norm of a vector is unchanged, too, so the unitary matrix can be thought of as performing a rotation of the vector in n-dimensional space, thereby preserving its length. If the vectors are real, the rotations correspond to proper real orthogonal matrices.

The transformations of a matrix are defined analogously to the transformations of a vector, but they involve mutliplying the matrix on both the left and right sides, rather than only on the left side. Several kinds of transformations are defined. If B is nonsingular, then

(C-56)


is a similarity transformation on A. We say that As is similar to A. A special case occurs if B is unitary. In this case we have a unitary transformation on A,

Au= BtAB

A second special case occurs if B is orthogonal, in which case

Ao=BTAB

defines an orthogonal transformation on A.

(C-57)

(C-58)

It follows directly from the invariance of the trace to cyclic permutations of the order of matrix products, Eq. (C-28), that

and

tr As = tr Au = tr Ao = tr A

Also, by the rules on determinants,

detAs = detA u = detAo = detA

It is easy to see that

(C-59)

(C-60)

(C-61)

(C-62)

Thus, Au is Hermitian (or unitary) if A is Hermitian (or unitary), and Ao is symmetric (or orthogonal) if A is symmetric (or orthogonal).

C.6 Eigenvectors and Eigenvalues

If A is an n X n matrix and if

AX=AX (C-63)

for some nonzero vector X and scalar A, we say that X is an eigenvector of A and that A is the corresponding eigenvalue. We can rewrite Eq. (C-63) as

(A -AI)X=O

so we see from Eq. (C-41) that A is an eigenvalue of A if and only if

det(A -}..l)=O

(C-64)

(C-65)

This is called the characteristic equation for A. It is an nth-order equation for A and has n roots, counting multiple roots according to their multiplicity.

Because the equation AX=AX is unchanged by multiplying both sides by a scalar s, it is clear that sX is an eigenvector of A if X is. This freedom can be used to normalize the eigenvectors, i.e., to choose the constant so that X·X = 1. From n eigenvectors of A, X('\ i= 1,2, ... ,n, we can construct the matrix

x}I) x}2) XP) x}n)

X~I) X~2) XP) x~n) (C-66) p==

X(I) n X (2) n X (3) n

x(n) n

C.6 EIGENVECTORS AND EIGENVALUES 753

Matrix multiplication and the eigenvalue equation (Eq. (C-63» give

AIXl l ) A2X F) A x(n) .~

n I .":~~;, ,

A X(l) A2X12) A x(n)

AP= I 2 n 2

=PA (C-67)

A X(I) I n A x (2) 2 n

A x(n) n n

where A is the diagonal matrix

AI 0 0

0 A2 0 A= (C-68)

0 0 An

The matrix P is nonsingular if and only if the n eigenvectors are linearly independent. In this case,

(C-69)

and we say that A is diagonalizable by the similarity transformation P -lAP. If A is a normal matrix, we can choose n eigenvectors that are orthonormal, or simultaneously orthogonal and normalized:

x(i). xv) = 8i == { 0 , I i"'} i=}

(C-70)

When the eigenvectors are orthonormal, P is a unitary matrix and A is diagonalizable by the unitary transformation A = ptAP. Any square matrix can be brought into Jordan canonical form [Hoffman and Kunze, 1961] by a similarity transformation

(C·71)

where the matrix J has the eigenvalues of A on the main diagonal and all zeros below the main diagonal. It follows from Eqs. (C-71), (C-59), and (C-60) that the trace of A is equal to the sum of its eigenvalues, and the determinant of A is equal to the product of its eigenvalues; i.e.,

n

trA=L Aj (C-72) i= I

(C-73)

Many algorithms exist for finding eigenvalues and eigenvectors of matrices, several of which are discussed by Carnahan, et af., [1969] and by Stewart [1973]. Using Eq. (C-61), we can see that the eigenvalues of Hermitian matrices are real numbers and the eigenvalues of unitary matrices are complex numbers with absolute value unity. Because the characteristic equation of a real matrix is a polynomial equation with real coefficients, the eigenvalues of a real matrix must either be real or must occur in complex conjugate pairs.


The case of a real orthogonal matrix deserves special attention. Because such a matrix is both real and unitary, the only possible eigenvalues are + 1, -1, and complex conjugate pairs with absolute value unity. It follows that the determinant of a real orthogonal matrix is (- l)m where m is the multiplicity of the root A= -I of the characteristic equation. A proper real orthogonal matrix must have an even number of roots at A= -I, and thus an even number for all A =1= 1, because complex roots occur in conjugate pairs. Thus, an n X n proper real orthogonal matrix with n odd must have at least one eigenvector with eigenvalue + 1. This is the basisof Euler's Theorem, discussed in Section 12.1.

It is also of interest to establish that the eigenvectors of a real symmetric matJix can be chosen to be real. The complex conjugate of the eigenvector equation, Eq. (C-63), is A X* = AX*, because both A and A are real. Thus, X* is an eigenvector of A with the same eigenvalue as X. Now, either X=X*, in which case the desired result is obtained, or X =1= X*. In the latter case, we can replace X and X* by the linear combinations X + X* and i(X - X*), which are real eigenvectors corresponding to the eigenvalue A. Thus, we can always find a real orthogonal matrix P to diagonalize a real symmetric matrix A by Eq. (C-69).

C. 7 Functions of Matrices

Let f(x) be any function of a variable x, for example, sinx or expx. We want to give a meaning to f(M), where M is a square matrix. If f(x)has a power series expansion about x = 0,

00

f(x)= L anxn (C-74) n=O

then we can formally (i.e., ignoring questions of convergence) define f(M) by

00

f(M)= L anMn (C-75) n=O

with the same coefficients an. It is clear that f(M) is a square matrix of the same order as M. If M is a diagonalizable matrix, then by Eq. (C-69),

M=PAP- I (C-76)

where P is the matrix of eigenvectors defined by Eq. (C-66), and A is the diagonal matrix of eigenvalues. Then,

Mn=(PAP-I)n=PNP- 1 (C-77)

and

f(AI) 0 0

f(M)=p(~oanAn )p-I=P 0 f(A2) 0

p- I (C-78)

0 0 f(An)

If M is a diagonalizable matrix, Eq. (C-78) gives an alternative definition of reM) that is valid when f(x) does not have a power series expansion, and agrees with Eq. (C-75) when a power series expansion exists.

c.s VECTOR CALCULUS 755

As an example, consider exp(!~t), where ~ is the 4x4 matrix introduced in Section 16.1,

0 W3 -W2 ~, I -W3 0 WI W2 ~=

W2 -WI 0 W3

-WI -W2 -W3 0

Matrix multiplication shows that ~2= -(wr+wi+wj)l= -w21, so it follows that

~2k = ( _ 1 )kw2k l

for all nonnegative k. Now,

00

=1 ~ k-O

where

~2k+1 =( _1)kw2k~

c=cos(iwt)

s=sin( iwt) i= 1,2,3

( 1 )2k+ I (_I)k 2(,)t

(2k+ I)!

(C-79)

This example shows that the matrix elements of f(M) are not the functions f of the matrix elements of M, in genera1.

C.8 Vector Calculus

Let cp be a scalar function of the n arguments X t,X2, ••• ,Xn• We consider the arguments to be the components of a column vector

X=[ X t,X2, ... ,Xn ]T

The n partial derivatives ofcp with respect to the elements of X are the components


of the gradient of cp, denoted by

:~ = [ :; , :; , ... , :; ] I 2 n

(C-80)

Note that acp / a x is considered a I x n row matrix. If we eliminate the function cp from Eq. (C-80), we obtain the gradient operator

a _[ a a a ] a x = ax ' ax , ... , ax I 2 n

(C-81)

The matrix product of the I X n gradient operator with an n X I vector Y yields a scalar, the divergence of Y, which we denote by

a nay; ax·Y=.~ ax.

1=1 1 (C-82)

The dot is used to emphasize the fact that the divergence is a scalar, although the usage is somewhat different from that in Eq. (C-23).

The mn partial derivatives of an m-dimensional vector Y with respect to X I'X2, ... ,Xn can be arranged in an m X n matrix denoted by

l.Y =[ ay; 1 ax aXj (C-83)

This is like an outer product of Y and a / a x; however, the analogy is not perfect because the gradient operator appears on the right in the matrix product sense and on the left in the operator sense.

C.9 Vectors in Three Dimensions

In this section, we only consider vectors with three real components. For three-component vectors, three products are defined: the dot product, the outer product, and the cross product. The cross product, or vector product, is a vector define.d by

r U2V3- U3 V2]

UxY= U3 VI -UI V3

UI V2 - U2 VI •

The following identities are often useful:

U·y= UIVI + U2V2+ U3V3= UVcosO

IUxYI= UVsinO

where 0(0,0, 180°) is the angle between U and Y. In addition,

UxY=-YXU

U·(UXY)=O

UI U2 U3

U·(YXW)=Y·(WXU)=W·(UxY)= VI V2 V3 WI W2 W3

(C-84)

(C-85a)

(C-85b)

(C-86)

(C-87)

(C-88)

C.9 VECTORS IN THREE DIMENSIONS

[u ·(VXW)J2=(UXV). [(VXW) X (WX U) J

= U2V2W2 - U2(V . Wi - V2(U . Wi - W2(U . V)2

+ 2(U . V)(V . W)(W . U)

UX(VXW)=V(U ·W)-W(U· V)

0= U x (V x W) + V x (W x U) + W x (U x V)

(UX V)-(WXX)= (U· W)(V ·X)-(U ·X)(V· W)

757

(C-89)

(C-90)

(C-91)

(C-92)

The following identity provides a means of writing the vector W in terms of U, V, and UXV, if UxV~O:

[(UXV)·(UX V)JW = [(VXU)·(VXW)JU + [(U XV)·(U XW)JV

+ [W·(UXV)]UxV

If A is a real orthogonal matrix,

(AU)X(AV)= ±A(UXV)

(C-93)

(C-94)

where the positive sign holds if A is proper, and the negative sign if A is improper. The tangent of the rotation angle from V to W about U (the angle of the

rotation in the positive sense about U that takes V X U into a vector parallel to WxU) is

IUIU·(VxW) tan e= --------

U2(V' W) - (U . V)(U . W) (C-95)

The quadrant of e is given by the fact that the numerator is a positive constant multiplied by sin e, and the denominator is the same positive constant multiplied by cose. If U, V, and Ware unit vectors, e is the same as the rotation angle on the celestial sphere defined in Appendix A. Equation (C-95) is derived in Section 7.3. (See Eqs. (7-57).)

References

1. Bellman, R. E., Introduction to Matrix Algebra. New York: McGraw-Hill, Inc., 1960.

2. Carnahan, B., H. A. Luther and J. O. Wilkes, Applied Numerical Methods. New York: John Wiley & Sons, Inc., 1969.

3. Forsythe, George E., and C. Moler, Computer Solution of Linear Algebraic Systems. Englewood Cliffs, NJ: Prentice-Hall, Inc., 1967.

4. Halmos, P. R., ed., Finite-Dimension Vector Spaces, Second Edition. Princeton, NJ: D. Van Nostrand Company, Inc., 1958.

5. Hoffman, Kenneth and Ray Kunze, Linear Algebra. Englewood Cliffs, NJ: Prentice-Hall, Inc., 1961.

6. Noble, Ben, Applied Linear Algebra. Englewood Cliffs, NJ: Prentice-Hall, Inc., 1.969.

7. Stewart, G. W., Introduction to Matrix Computations. New York: Academic Press, Inc., 1973.

8. Wilberg, Donald M., State Space and Linear Systems, Schaum's Outline Series, New York: McGraw-Hill, Inc., 1971.

APPENDIXD

QUATERNIONS

Lawrence Fallon, III

The quaternion representation of rigid body rotations leads to convenient kinematical expressions involving the Euler symmetric parameters (Sections 12.1 and 16.1). Some important properties of quaternions are summarized in this appendix· following the formulation of Hamilton [1866] and Whittaker [1961].

Let the four parameters (ql,q2,q3,q4) form the components of the quaternion, q, as follows:

q=q4+ iql + jq2+ kq3 (0-1)

where i, j, and k are the hyperimaginary numbers satisfying the conditions

i2=/=k2=-1

ij=-ji=k

jk= -kj=i

ki= -ik=j (0-2)

The conjugate or inverse of q is defined as

q*=q4- iql-jq2- kq3 (0-3)

The quantity, q4' is the real or scalar part of the quaternion and iq I + jq2 + kq3 is the imaginary or vector part.

A vector in three-dimensional space, U, having components UI' U2, U3 is expressed in quaternion notation as a quaternion with a scalar part of zero,

(0-4)

If the vector q corresponds to the 'lector part of q (i.e., q = iq I + jq2 + kq3)' then an alternative representation of q is

(0-5)

Quaternion multiplication is performed in the same manner as the multiplication of complex numbers or algebraic polynomials, except that the order of operations must be taken into account because Eq. (0-2) is not commutative. As an example, consider the product of two quatenlions

q" = qq' = (q4 + iql + jq2 + kq3)( q~ + iq. + jq2 + kq;)

Using Eq~ (0-2), this reduces to

" '( , , , + ') q =qq = -qlql-q2q2-q3q3 q4q4

+ i(qlq~+ q2q;- q3q;'+ q4qD

+ j( - qlq; + q2q~ + q3q. + q4q;')

+ k( q)q;' - q2q. + q3q~ + q4q;)

(0-6)

(0-7)

Ap.O QUATERNIONS

If q' = (q~,q'), then Eq. (0-7) can alternatively be expressed as

q" =qq' = (q4q~-q'q',q4q' + q~q+qxq')

The length or norm of q is defined as

Iql=yqq* =yq*q =Vq~+ qi+ q~+ ql

759

(0-8)

(0-9)

If a set of four Euler symmetric parameters corresponding to the rigid body rotation defined by the transformation matrix, A (Section 12.1), are the components of the quaternion, q, then q is a representation of the rigid body rotation. If q' corresponds to the rotation matrix A', then the rotation described by the product A' A is equivalent to the rotation described by qq'. (Note the inverse order of quaternion multiplication as compared with matrix multiplication.)

The transformation of a vector U, corresponding to multiplication by the matrix A,

U'=AU

is effected in quaternion algebra by the operation

U'=q*Uq

(0-10)

(0-11)

See Section 12.1 for additional properties of quaternions used to represent rigid body rotations.

For computational purposes, it is convenient to express quaternion multiplication in matrix form. Specifically, let the components of q form a four-vector as follows:

q- [!~ 1 (0-12)

This procedure is analogous to expressing the complex number c = a + ib in the form of the two-vector,

In matrix form, Eq. (0-7) then becomes

[ q;' I [ q~ qi' = -q; q;' qi q4' - qi

-qi qi q4

-q; (D-13 )

Given the quaternion components corresponding to two successive rotations, Eq. (D~13) conveniently gives the quaternion components corresponding to the total rotation.

References

1. Hamilton, Sir W. R., Elements of Quaternions. London: Longmans, Green and Co., 1866.

2. Whittaker, E. T., A Treatise on the Analytical Dynamics of Particles and Rigid Bodies. C~bridge: Cambridge University Press, 1961.

APPENDIXE

COORDINATE TRANSFORMA nONS

Gyanendra K. Tandon

E.l Cartesian, Spherical, and Cylindrical Coordinates E.2 Transformations Between Cartesian Coordinates E.3 Transformations Between Spherical Coordinates

E.. Cartesian, Spherical, and Cylindrical Coordinates

The components of a vector, r, in cartesian, spherical, and cylindrical coordinates are shown in Fig. E~ 1 and listed below.

Cartesian (x,y,z)

Spherical (r,O,cp) Cylindrical (p, cp, z)

z

O~ __________ -+ ______ ~,-+y

" I /"---, , I // ~ " p I //x

, I / , I /

-------------~'~/ v

Fig. E-!. Components of a Vector, r, in Cartesian (x,y,z), Spherical (r,IJ,IjI), and Cylindrical (p,IjI,z) Coordinates

The declination, 8, used in celestial coordinates is measured from the equatorial plane (x-y plane) and is related'to 0 by the equation

~ =900 - 0 (E-I)

The components in cartesian, spherical, and cylindrical coordinates are related by the following equations:

x = FsinOcoscp

y = rsinO sincp

=pcoscp =psincp

(E-2a)

(E-2b)

E.2 TRANSFORMATIONS BETWEEN CARTESIAN COORDINATES 761

z = rcos8 =z (E-2c)

r=(x2+ y2+ z2)1/2 2 2 1/2 =(p +z ) (E-2d)

8=arccos{ z/(x2+ y2+Z2)1/2} = arc tan(p/ z) 0" 8" 1800 (E-2e)

</>= arc tan(y / x) =</> 0" </> < 3600 (E-2f)

2 1/2 p=(x2+ y ) =rsin8 (E-2g)

The correct quadrant for </> in Eq. (E-2f) is obtained from the relative signs of x andy.

E.2 Transformations Between Cartesian Coordinates

If rand r' are the cartesian representations of a vector in two different cartesian coordinate systems, then they are related by

r'=Ar+a (E-3)

where a represents the translation of the origin of the unprime.d system in the primed system and the matrix A represents a rotation. For most attitude work, a=O.

The transformation matrix A (called the attitude matrix or direction cosine matrix in this book) can be obtained by forming the matrix product of matrices for successive rotations about the three coordinate axes as described in Section 12.1. The elements of matrix A are direction cosines of the primed axes in the unprimed system and satisfy the orthogonality condition. Because A is an orthogonal matrix, its inverse transformation matrix will be its transposed matrix; symbolically,

A -I =AT (E-4)

For many applications, the definition of the direction cosine matrix in terms of the orthogonal coordinate unit vectors in the two coordinate systems,

[ i'·x x'·y " 'J x ·z

A= y'·x y.'.y y'·z (E-5a) z'·x z'·y z' ~z

is useful for computations. As an example, let the primed coordinate system have its coordinate axes aligned with the spacecraft-to-Earth vector R, the component of V perpendicular to R, and the orbit normal vector RXV /IRXVI, where V is the spacecraft velocity:

x'=R/IRI y'=(RXV)XR/I(RXV)XRI

z'=(RX V)/IR X VI (E-5b)

Then, substituting Eq. (E-5b) into Eq. (E-5a) gives an expression for A which does not require the evaluation of trigonometric functions.

Euler's Theorem. Euler's theorem states that any finite rotation of a rigid body can be expressed as a rotation through some angle about some fixed axis. Therefore, the most general transformation matrix A is a rotation by some angle,

762 COORDINATE TRANSFORMATIONS E.2

~, about some fixed axis, e. The axis e is unaffected by the rotation and, therefore, must have the same components in both the primed and the unprimed systems. Denoting the components of e by e l , e2, and e3, the matrix A is

[ costfl+ ef(I -oostfl) e\e2(I -costfl)+ e)sintfl e\eil-oostfl)- e2Sintfl]

A = e\eil-costfl)-e)sintfl costfl+ei(I -costfl) e2e3(l-costfl)+e\sintfl

e\e)(l-costfl)+ e2sintfl e2e)(l-costfl)- e\sintfl costfl+erO -costfl)

(E-6)

In this case, the inverse transformation matrix may be obtained· by using Eq. (E.4) or by replacing ~ by -~, in Eq. (E·6), that is, a rotation by the same amount in the opposite direction about the axis e.

Euler Symmetric Parameters. The Euler symmetric parameters, q\ through q4' used to represent finite rotations are defined by the following equations:

,~ ql=e\sln'2

Clearly,

The transformation matrix A in terms of Euler symmetric parameters is

[ q: - qi - q~ + q~

A = 2( qlq2 - q3q4)

2( qlq3 + q2q4)

2( qlq2 + q3q4)

- q: + q~ - q~ + q~ 2(q2q3-q\q4)

(E·7a)

(E-7b)

(E-8)

The inverse transformation matrix in this case may be obtained by use of Eq. (E~4), or by replacing QI,q2' and q3 by - ql' - q2' and - q3' respect~vely, in Eq. (E-8) and leaving q4 unaltered.

The Euler symmetric parameters may be regarded as components of a quaternion, q, defined by

(E-9)

where i, j, and k are as defined in Appendix D. The multiplication rule for successive rotations represented by Euler symmetric parameters is given in Appendix D. The Euler symmetric parameters in terms of the 3-1-3 Euler angle rotation cp, 0, I/; (defined below) are as follows:

q\ = sin(O /2)cos«cp- 1/;)/2)

q2 = sin(O /2)sin«cp- 1/;)/2)

q3 =cos(O /2)sin«cp+ 1/;)/2)

q4 = cos(O /2)cos«cp+ 1/;)/2) (E-IO)

E.2 TRANSFORMATIONS BETWEEN CARTESIAN COORDINATES 763

Gibbs Vector. The Gibbs vector (components gl' g2' and g3) representation (see Section 12.1) for finite rotations is defined by

gl == qll q4 = el tan(~ 12)

g2 == q21 q4 = e2tan(<p 12)

g3 == q31 q4 = e3tan(<p 12) (E-ll)

The transformation matrix A in terms of the Gibbs vector representation is as follows:

2(gl g2+ g3)

1- gf+ gi- gj

2(g2g3- gl)

The inverse of A can be obtained in this case by the method of Eq. (E-4), or by replacing gj by - gj in Eq. (E-12).

Euler Angle Rotation. The Euler angle rotation (</>,(J,o/) is defined by successive rotations by angles </>, (J, and 0/, respectively, about coordinate axes i, j, k (Section 12.1). The i-j-k Euler angle rotation means that the first rotation by angle </> is about the i axis, the second rotation by angle (J is about the j axis, and the third rotation by angle 0/ is about the k axis. There are 12 distinct representations for the Euler angle rotation which divide equally into two types:

TYPE 1. In this case, the rotlltions take place successively about each of the three coordinate axes. This type has a singularity at (J = ± 90 deg, because for these values of (J, the </> and 0/ rotations have a similar effect.

TYPE 2. In this case, the first and third rotations take place about the same axis and the second rotation takes place about one of the other two axes. This type has a singularity at (J=O deg and 180 deg, because for these values of (J, the </> and 0/ rotations have a similar effect.

Table E-l gives the transformation matrix, A, for all of the 12 Euler angle representations. The 3-1-3 Euler angle representation is the one most commonly used in the literature. The Euler angles </>, (J, and 0/ can be easily obtained from the elements of matrix A. A typical example from each type is given below.

TYPE 1: 3-1-2 Euler Angle Rotation

</>= arc tan( - A211 A 22) 00;;;; </>< 3600

(J = arcsin(A 23) - 900 0;;;; (J 0;;;; 900

o/=arctan( - A \31 A 33) 00;;;; 0/<3600

(E-13a)

(E-13b)

(E-13c)

The correct quadrants for </> and 0/ are obtained from the relative signs of the elements of A in Eqs. (E-13a) and (E-13c), respectively.

TYPE 2: 3-1-3 Euler Angle Rotation

</>=arctan(A 311 - A 32)

(J = arc cos( A 33)

0/= arc tan(A 131 A 23)

00;;;; (J 0;;;; 1800

00;;;; 0/< 360 0

(E-14a)

(E-14b)

(E-I4c)

TY

PE

-1

E

UL

ER

AN

GL

E

RE

PR

ES

EN

TA

TIO

N

1·2

·3

1·3

·2

2·3

·'

2·'

·3

3-'·

2

3-2

-1

Tab

le

E-l

. T

he

Att

itude

M

atri

x,

A,

for

the

12

Pos

sibl

e E

uler

A

ngle

R

epre

sent

atio

ns

(S==

sine

, C

== co

sine

, I =

= x a

xis,

2 ==

Y ax

is,

3 ==

z ax

is)

TY

PE

-2

MA

TR

IX A

E

UL

ER

AN

GL

E

MA

TR

IX A

R

EP

RE

SE

NT

AT

ION

[~

C1/JS8~

+ S1/

JC~

-CI/J

S8C

4> +

SI/

J~

1 [ ·c

o S

8S¢

-SI/J

C8

-SI/

JS8~

+ C

I/JC4

> SI

/J58

C4>

+ CI/

J~

1·2

-1

SI/JS

O C.

I/JC

~ -S

I/JC

IJ~

SO

-C8~

COC~

CI/JS

O -S

I/JC

~ -CI/JC8~

[ ,,~

CwSO

C4>

+ S;J

J~

Co/ISO~ -So

/IC~

1 [ co

soc~

-58

CO

C4>

COS~

1·3

·1

-Co/

IS8

CwC8C~ -

S""

S¢

SI/JC

O So

/ISOC

4> -

Co/

IS¢

S1/J

58S¢

+ C

wC4>

So

/I58

-SwCOC~ -

Cw

S¢

[ ~~

58

-C8

S¢

1 [ C

wC

¢ -S

o/IC

8S¢

S>/IS

O

-CI/J

58C

4> +

SI/J

S¢

CI/J

C8

Co/IS

OS¢

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o/IC

¢ 2

-1·2

SO

S¢

co

So/I5

8C4>

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o/IS

¢ ":S

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-S

o/I5

8S¢

+ C

o/IC4

> So

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o/IC

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SO

[ "~.,

,,,.

SI

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-Co/

IS¢

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I/JS8

C4>

1 [

'''''H

'' C

""so

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JC~

+ C

o/I5

8S¢

CI/JC

O

SI/J

S¢ +

CI/J

SOC4

> 2

-3·2

-SOC~

co

C8S

¢ -5

8

COC~

So/I

COC~

+ C

o/IS

¢ So

/ISO

[ ''''-....

Co/

IS¢

+ S

o/ISO

C4>

-So/

IC8

1 [ ""

~,, .•

C>/I

S~ +

SwC

8C4>

-CO

S¢

COC4

> S8

3

·1·3

-S

.,C

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Cw

CB

S¢

-sw

s¢

+ C

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C4>

So/IC

4> +

CI/J

58S

¢ So

/I~

-CI/J

SOC

4>

CI/JC

O

SOS¢

-S8C~

[ ~~

CO

S¢

-so

1 [ ,,

,.,,~~.

Co/I

COS~

+ S

o/IC4

>

-Co/

IS¢

+ S

o/IS

OQ

p CI

/JC4>

+ S

o/I5

8S¢

SI/JC

O

3·2

·3

-So/

ICO

C4>

-C

o/IS

¢ -S

o/IC

OS~

+ C

o/IC4

>

SI/I

S¢

+ C

I/I58

C4>

-S

I/JC

~ +

CI/I

SOS¢

Co

/ICO

SOC~

SOS¢

-soc~

C"'~ +

So/

ICOC

~

-Sw

S¢

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wCO

C4>

SO~

C""

C8S

¢ +

So/

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-SwCOS~ +

Co/I

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IS¢

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C8C

4>

SOC~

-So/

IS¢

+ C

o/ICO

C4>

-C>/

ICO

S¢

-S>/

IC4>

SOS¢

-SI/

JC8S

~ +

Co/I

C~

Swso

1 C

wSO

C8

-C""

so

1 So

/ISO

·C8

1 1 1 1

-.l

0'1 +:-

(') o o ~ I:) Z

~ t'tl ~ (I) 2:l ~ ~ o z (I) tTl

N

E.3 TRANSFORMATIONS BETWEEN SPHERICAL COORDINATES 765

The correct quadrants for cp and t¥ are obtained from the relative signs of the elements of A in Eqs. (E-14a) and (E-14c), respectively.

Kinematic Equations of Motion. For convenience, the kinematic equations of motion (Section 16.1) for the 12 possible Euler angle representations are given in Table E-2. The kinematic equations of motion for other representations of the attitude matrix can be found in Section 16.1.

Table E-2. Kinematic Equations of Motion for the 12 Possible Euler Angle Representations (1:; x axis, 2:; y axis, 3:; z axis; WI' W2' w3 are components of the angular velocity along the body x, y, z axes.)

AXIS INDEX VALUES

SEQUENCE KINEMATIC EQUATIONS OF MOTION I J K

1·2 - 3 1 2 3 ; = (WI cos t/J - W J sin t/J ) sec 8 TYPE 1

2-3-1 2 3 1 8 = W J cos t/J + WI sin t/J

3- 1- 2 3 1 2 ~ = w K - (WI cos t/J - w J sin t/J ) tan 8

1-3-2 1 3 2 ; = (w I cos t/J + W J sin t/J ) sec 8

3 - 2 - 1 3 2 1 8 = W J cos t/J - WI sin t/J

2 - 1 - 3 2 1 3 $ = w K + (WI cos t/J + wJsin t/J) tan 8

1 - 2-1 1 2 3 ; = (wK cos t/J + W J sin t/J ) esc 8 TYPE 2

2-3-2 2 3 1 6 = W J cos t/J - w K sin t/J

3 -1 - 3 3 1 2 $ = wI - (wK cos t/J + w J sin t/J ) cot 8

1 - 3 - 1 1 3 2 ; = - (wK cos t/J - W J sin t/J ) esc 8

3-2-3 3 2 1 8 = w J cos t/J +wK sin t/J

2-1 -2 2 1 3 ~ =.wl + (wK eost/J - wJsin t/J )eot8

E.3 Transformations Between Spherical Coordinates

Figure E-2 illustrates the spherical coordinate system on a sphere of unit radius defined by the north pole, N, and the azimuthal reference direction, R, in the equatorial plane. The coordinates of point Pare (cp,O). A . new coordinate system is defined by the north pole, N', at (CPo' ( 0) in the old coordinate system. The new azimuthal reference is at an angle, cp~ relative to the·NN' great circle. The coordinates (cp',O') of P in the new system are given by:

cosO' = cos Oocos 0 + sin OoSin 0 cos(cp - CPo)

sin( cp' - cp~) = sin( cp - cpo)sin 0/ sin 0' (E-15)

766 COORDINATE TRANSFORMA nONS E.3

where 0 and 0' are both defined over the range 0 to 180 deg, and (<P - <Po) and W - <P~) are both in the range 0 to 180 deg or in the range 180 to 360 deg. Simplified forms of Eqs. (E-15) in two special cases are as follows:

Case I: <p=<p~=90°

cosO' = cos Oocos B + sin 0osinO sin <Po

cos <P' = - cos <Posin 0/ sin 0'

Fig. E-2. Transformation Between Spherical Coordinate Systems NR and N'R'

Case 2: <p=<p~=0

cos B' = cos Oocos 0 + sin Oosin 0 cos <Po

sin <p' = - sin <Posin 0 / sin 0 '

(E-16)

(E-17)

The most common spherical inertial coordinates for attitude analysis are the celestial coordinates (a,8) defined in Section 2.2. The right ascension, a, and the declination, 8, are related to <p and 0 by

a=<p

8=900 - 0 (E- 18)

APPENDIX F

THE LAPLACE TRANSFORM

Gerald M. Lerner

Laplace transformation is a technique used to relate time- and frequencydependent linear systems. A linear system is a collection of electronic components (e.g., resistors, capacitors, inductors) or physical components (e.g., masses, springs, oscillators) arranged so that the system output is a linear function of system input. The input and output of an electronic system are commonly voltages, whereas the input to an attitude control system is a sensed angular error and the output is a restoring torque. Most systems are linear only for a restricted range of input.

Laplace transformation is widely used to solve problems in electrical engineering or control theory (e.g., attitude control) that may be reduced to linear differential equations with constant coefficients. The Laplace transform of a real function, f( t), defined for real t > 0 is

e (f(t»== F(s)= (00 f(t)exp( - st)dt )0+

(F-l)

where 0+ indicates that the lower limit of the integral is evaluated by taking the limit as t~O from above. The argument of the Laplace transform, F(s), is complex,

s==(J+iw

where i==H. For most physical applications, t and w denote time and frequency, respectively, and (J is related to the decay time.

The inverse Laplace transform is

I jC+iOO e -1(F(s»==f(t)= -2 . F(s)exp(st)ds 7TI C-ioo

(F-2)

where the real constant C is chosen such that F(s) exists for all Re(s) > C, that is, to the right of any singularity.

Properties of the Laplace Transform and the Inverse Laplace Transform·. The Laplace transform and its inverse are linear operators, thus

e( af(t) + bg(t» = a e(f(t» + b e( g(t»

== aF(s) + bG(s)

e -1(aF(s)+ bG(s»= a e-I(F(s»+ be- I ( G(s»

= af(t) + bg(t)

where a and b are complex constants.

• For further details, see DiStefano, et 01., [I %7)

(F-3)

(F-4)

768 THE LAPLACE TRANSFORM Ap. F

The initial value theorem relates the initial value of J(t), J(O+), to the Laplace transform,

J(O+)= lim sF(s) s-.oo

(F-5)

and the final value theorem, which is widely used to determine the steady-state response of a system, relates the final value of f( t), f( 00), to the Laplace transform,*

J(oo) = limsF(s) s--->o

(F-6)

The Laplace and inverse Laplace transformations may be scaled in either the time domain (time scaling) by

E(J(t/a))=aF(as)

or the frequency domain (frequency scaling) by

E -l(F(as»= J(t/ a)/ a

The Laplace transform of the time-delayed function, J( t - to)' is

I.: (J(t - to» = exp( - sto)F(s)

(F-7)

(F-8)

(F-9)

where J(t - to) = 0 for t < to. The inverse Laplace transform of the frequency shifted function, F(s - so), is

(F-lO)

Laplace transforms of exponentially damped, modulated, and scaled functions are

I.: (exp( - at)J(t» = F(s+ a)

I.: (sinwtf(t» = [F(s - iw)- F(s + iw) J/2i E(coswtJ(t» = [F(s - iw)+ F(s + iw) J/2

E (tnf(t» = (- If dd;n F(s)

I.: (J(t)/t) = fooF(u)du s

(F-Ila)

(F-Ilb)

(F-Ilc)

(F-Ild)

(F-Ile)

The Laplace transform of the product of two functions may be expressed as the complex convolution integral,

1 f C + iOO E(J(t)g(t»= 2'lTi C-ioo F(w)G(s-w)dw (F-12)

Multiplying the Laplace transform of a function by s is analogous to differentiating the original function; thus,

• The final value theorem is valid provided that sF(s) is analytic on the imaginary axis and in the right half of the s-plane; i.e., it applies only to stable systems.

Ap.F THE LAPLACE TRANSFORM 769

(F-13)

Dividing the Laplace transform of a function by s is analogous to integrating the original function; thus,

e -1(F(s)j s)= {f(u)du (F-14)

The inverse Laplace transform of a product may be expressed as the convolution integral

e -1(F(s)G(s»= (I f(t)g(t-T)dT= (I g(t)f(t-T)dT Jo+ Jo+

(F-15)

which may be inverted to yield

F(s)G(s)= e (I f(t)g(t-T)dT= e (I g(t)f(t-T)dT Jo+ Jo+

(F-16)

A short list of Laplace transforms is given in Table F-I; detailed tables are given by Abramowitz and Stegun [1968], Korn and Korn [1968], Churchill [1958], and Erdelyi, et al., [1954].

Table F-1. Laplace Transforms

g(t) G(SI g(t) G(SI

dl SF(SI - 110+1 u It_.lt exp (-asl/s -d,

dl" 5" F(SI _.5"-1 110+1 , I/s2 -

"-2 dl I dl"-1 l d," - S ;;; ~+'" d,"-1 0+

," " 1/5"+1

,. na+lll S"+1 t

iflTldT exp (-atl 11 (5+al

F(SI/S

t" e • .p (-atl 11 (S+al0+1

," fltl (_I" dO F(S) sinwt wi (S2 +w21

dS" cos wt SI (S2 + w 2)

f(tl/, [ F (u)du exp (-at) sin wt wi ( (S+a)2 + w21

exp (-at) cos wt (S+a) I ( (S+0)2 + w21

flt/.) aF (as) (exp (-atl - exp (-b,) I I (a-bl 11 ( (S+a) (S+b) I

flt-'o) exp (-'oS) F(S) [a exp (-atl - b exp (-btl I I [b-al S/[ (S+a) (S+b) ,

sinh (atl a IIS2_a2 ) exp [t.o ) fltl F(S-So)

cosh (atl SI (S2_a2)

60 (t-a'" exp (-as)

NOTE: F(S) = I fit) exp (-stl d'; G(S) = 19h) exp (-stl dt

J o+ 10+ n DENOTES A POSITIVE INTEGER; a AND b DENOTE POSITIVE REAL NUMBERS.

"60 IS THE DIRAC DELTA FUNCTION.

t u IS HEAVISIDE·S UNIT STEP FUNCTION WHICH IS DEFINED BY u = 0 FOR, < a. u • 1 FOR t > a.

770 THE LAPLACE TRANSFORM Ap. F

Solution of Linear Differential Equations. Linear differential equations with constant coefficients may be solved by taking the Laplace transform of each term of the differential equation, thereby reducing a differential equation in t to an algebraic equation in s. The solution may then be transformed back to the time domain by taking the inverse Laplace transform. This procedure simplifies the analysis of the response of complex physical systems to frequency-dependent stimuli, such as the response of an onboard control system to periodic disturbance torques.

The solution to the linear differential equation

n d~ ;L ai(i? =x(t) 1=0

(F-17)

with an = 1 and forcing function x(t) is given by

(F-18)

n

where Xes)::::: E(x(t», r(s)::::: ~ ais i is the characteristic polynomial of Eq. (F-17) and i=O

are the initial conditions. Any physically reasonable forcing function, including impulses, steps, and

ramps, may be conveniently transformed (see Table F-I). The analysis of the algebraic transformed equation is generally much easier than the original differential equation. For example, the steady-state solution, f( 00), of a differential equation is obtained from the Laplace transform by using the final value theorem, Eq. (F-6).

The first term on the right-hand side of Eq. (F-lS) is the forced response of the system due to the forcing function and the second term is the free response of the system due to the initial conditions. The forced response, E -1(X(S)j ~(s», consists of two parts: transient and steady state.

Solving the differential equation (Eq. (F-17» is equivalent to finding the inverse Laplace transform of the algebraic functions of s in Eq. (F-I8). One technique involves expressing rational functions of the form

m n

R(s) = ~ his i j ~ ais i= N(s)n(s) (F-19) ;=0 ;=0

as a sum of partial fractions (n > m) using the fundamental theorem of algebra. The characteristic polynomial, r(s), may be factored as

r

Hs)= II (s+p;)mi (F-20) i= I

Ap.F THE LAPLACE TRANSFORM

where -Pi is the ith zero of r(s) with multiplicity mi and

The partial fraction expansion is

where

r mi C R(s)=bn + L L· ik k

;=1 k=1 (S+Pi)

C - 1 dm;-k [( + )miR()] ;k=(m;-k)! dsm;-k s Pi s S=-Pi

771

(F-21)

(F-22)

and bn =0 unless m = n. The coefficients Cit are the residues of R(s) at the poles - Pi- If no roots are repeated, Eq. (F-22) may be rewritten as

where

r. C R(s)=bn + L _il

;=1 s+p;

Cit =(s+ p;)R(s)ls= _PI

(F-23)

The zeros of r(s) may be determined using various numerical methods [DiStefano, et al., 1967].

The inverse Laplace transform of expressions in the form of Eq. (F-23) may be obtained directly from Table F-l. Other techniques for computing inverse Laplace transforms include series expansions and differential equations [Spiegel, 1965].

Example: Forced Harmonic Oscillator. The equation describing a I-degreeof-freedom gyroscope (Sections 6.5 and 7.S) is

d2tJ + D dO + KO = ~w(t) dt2 IG dt IG IG

(F-24)

where IG is the moment of inertia of the gyroscope about the output axis, D is the viscous damping coefficient about the output axis, K is the restoring spring constant about the output axis, L is the angular momentum of the rotor, and w(t) is the angular velocity about the input axis which is to be measured (see Fig. 6-45).

We assume that the input angular velocity is sinusoidal'" with amplitude, A, and frequency, we; i.e.,

(F-25)

The solution of Eq. (F-24) is given by (F-lS) as

(F-26)

°This is not as severe a restriction as it might seem because any physically reasonable w(t) may be expanded in a Fourier series. The result for a general w(t) is then obtained by linear superposition.

772

where

THE LAPLACE TRANSFORM

HS)=S2+ Ds/ fG+ K/ fG

X(s)= e(ALcos(,V/ fG )=ALs[ (S2+W;)IG]-1

OO=- 0lt=o

iJ = dO I 0- dt t=O

The characteristic polynomial, r(s), may be factored as

Hs)= (s+ PI)(S+ P2)

where

PI =-( D +1.y4KfG - D 2 ) I (2IG )

Ap.F

(F-27)

P2=-( D-l.y4KfG- D2 ) I (2IG) (F-28)

and we assume 4KfG-D 2>O. Substitution of Eq. (F-28) into Eq. (F-26) yields

O(t)=e- I[. ALs/fG j+e_I[DOo/fG+SOo+iJo] (F-29) (S2+w;)(S+ PI)(S+ P2) (s+ PI)(S+ P2)

The second term on the right-hand side of Eq. (F-29) is given in Table F-I as

(F-30)

The first term on the right-hand-side of Eq. (F-29) may be expanded in partial fractions as

e -I [ S ] = exp( - iWet) + exp( iWet) (S2+ w;)(s + PI)(S + P2) 2(pl - iWe)(P2 - iWe) 2(pl + iwe)(P2 + iWe)

1 ( Plexp( - Pit) P2exP( - P2t) } + ( ) 2 2 - 2 2 (F-31)

PI-P2 PI +We P2 +we

Equation (F-30) and the last term on the right-hand-side of Eq. (F-31) are the transient response of the system to the initial conditions and the forcing function. The transient response decays with a time constant and frequency given by

[Re(PI)r l = 'To=2IG/ D

Im(pl)=wl =V4KfG- D2 12IG (F-32)

Ap.F THE LAPLACE TRANSFORM 773

For t-H~J, the steady-state solution, the first two terms on the right-hand side of Eq. (F-31) dominate 8(t).These two terms may be rewritten as

(F-33)

where

DWe tanq,= 2

K-IGwe

Several features of gyroscope design are evident from these equations: I. The output response of the system to a constant or low-frequency input,

we~O, is linearly related to the input for t»'I"o; for example,

lim 8(t)=A (L/ K) "'e->O

2. The viscous damping constant, D, must be sufficiently high so that '1"0 is small compared with the gyro sampling period. However, if the damping is too high, the system output becomes frequency dependent and lags the input.

3. Systems with negligible damping, D~O, resonate at input frequencies near the characteristic frequency of the system,

wo=VK/IG

Integral Equations. Integral equations have the general form

y(t)= f(t)+ fU2k(u,t)y(u)du u\

(F-34)

where k(u,t) is the kernel of the equation. The limits of the integral may be either constants or functions of time. If u1 and u2 are constants, Eq. (F-34) is called a Fredholm equation, whereas if u1 is a constant and u2 = t,then Eq. (F-34) is called a Volterra equation of the second kind [Korn and Korn, 1968].

If the functional form of the kernel may be expressed as

k( u, t) = k( u - t) (F-35)

then the Volterra equation

y(t)= f(t) + fo1k(U- t)y(u)du (F-36)

may be solved by Laplace transform methods. Taking the Laplace transform of Eq. (F-36) and rearranging, we obtain

Y(s)= F(s)/(I- K(s» (F-37)

Ap.F

where Y(s) = e(y(t», F(s)= e(f(t», and K(s)= e(k(t», which may be solved for y(t) by taking the inverse Laplace transform.

774 THE LAPLACE TRANSFORM.

References

1. Abramowitz, Milton, and Irene A. Stegun, Handbook of Mathematical Functions. Washington, D.C., National Bureau of Standards, 1970.

2. Churchill, R. V., Operational Mathematics, Second Edition. New York: McGraw-Hill, Inc., 1958.

3. DiStefano, Joseph J., III, Allen R. Stubberud, and Ivan J. Williams, Feedback and Control Systems, Schaum's Outline Series. New York: McGraw Hill, Inc., 1967.

4. Erdelyi, A., et al., Tables of Integral Transforms. New York: McGraw Hill, Inc., 1954.

5. Korn, Granino A., and Theresa M. Korn, Mathematical Handbook for Scientists and Engineers. New York: McGraw Hill, Inc., 1968.

6. Spiegel, Murray R., Laplace Transform, Schaum's Outline Series. New York: Schaum Publishing Co., 1965.

APPENDlXG

SPHERICAL HARMONICS

John Aiello

Laplace's Equation, \l2U=0, can be written in the spherical coordinate system of Section 2.3 as:

a2u + I au + J.- a2u + cotO au + _1_ a2u =0 ar2 r ar r2 ao 2 r2 ao r2sin20 ell!>2 (G-l)

If a trial substitution of U(r,O,</» = R(r) YeO,</»~ is made, the following equations are obtained through a separation of variables:

d2R(r) dR(r) r2 __ +2r-d- -n(n+ I)R(r}=O (G-2)

dr2 r

a2y(o,</» ay(O,</» I a2y(O,</» --- +cotO ':'0 + -. -2 + n(n + I)Y(O,</»=o (G-3)

ao 2 v sm 0 a</>2

where n(n + 1) has been chosen as the separation constant. Solutions to Eq. (G-2) are of the form

R(r)=Arn+ Br-(n+l) (G-4)

Thus, solutions to Laplace's Equation (Eq. (G-I» are of the form

n =0,1,2, ... , (G-5)

These functions are referred to as solid spherical harmonics, and the YeO,</»~ are known as surface spherical harmonics. We wish to define U over a domain both interior and exterior to a spherical surface of radius r, and to have U continuous everywhere in the domain and to assume prescribed values Uo(O,</» on the surface. Under these conditions, Eq. (G-5) with B = 0 gives the form of U for the interior region of the sphere and with A = 0 represents its form in the exterior region.

To determine the surface spherical harmonics, the trial substitution

Y(O,</» = P(cosO)<fl(</» (G-6)

is made in Eq. (G-3). Multiplying by sin20/ p<fl and choosing a separation constant of m 2 yields

d2P(cosO) dP(cosO) [ m2 ] d0 2 +cotO dO + n(n+ 1)- sin20 P(cosO)=O

d2<fl(</» -- + m2<fl(</» =0

d</>2

The solutions to Eq. (G-8) are readily found to be

(G-7)

(G-8)

776 SPHERICAL HARMONICS Ap.O

11>( q,) = C cos mq, + S sin mq, (0-9)

in which m must be an integer, because 1I>(q,) is required to be a single valued function. Equatlon (0-7) can be rewritten substituting x=cosO as,

ddX[(I_x2)~~]+[n(n+I)- 1::2 ]P=O (0-10)

which is the generalized Legendre equation [Jackson, 1962]. For m = 0, the solutions to Eq. (0-10) are called Legendre polynomials and may be computed from either Rodrigues' formula

_ I (d)n 2 n Pn(x)- 2nn! dx (x -I) (0-11)

or from a recurrence relation convenient for computer use [Arfken, 1970].

Pn+ t(x)=2xPn(x) - Pn-t(x)- [xPn(x)- Pn-t(x) ]/(n + I) (G-12)

Rodrigues' formula can be verified by direct subs.titution into Eq. (0-10), and the recurrence relation can be verified by mathematical induction. When m i= 0, solutions to Eq. (0-10) are known as associated Legendre Junctions (of degree, n, and order, m), and may be computed by [Yevtushenko, et al., 1969]

(l-x2t/2 dn+m(x2_I)n Pnm(x) = rn! dxn+m (0-13)

or by [Heiskanen and Moritz, 1967]

P ( ) -2-n(l- 2)m/2.f;, (-l)k (2n-2k)! n-m-lk nm X - X k~O k!(n-k)!(n-m-2k)!x (0-14)

where / is either (n- m)/2 or (n-m-I)/2, whichever is an integer. Table 0-1 lists the associated Legendre functions up to degree and order 4 in terms of cosO [Fitzpatrick,1970].*

Table G-l. Explicit Forms of Associated Legendre Functions Through Degree n=4 and Order m=4

!';z 0 1 2 3 4

0 1

1 coso SIN 0

2 lcos2 8 -! 3 SIN 8 COS 0 3 SIN2 " 2 2 3 ~ (cos3 8 - ~ cos.) ~SIN 0 (COS2 8 - ;) 15 SIN2 0 COS 0 15 SIN3 0

4 ~(COS4._ !cos2.+~) ~ SIN 0 (cos3 8 - ~ COS 0) 10; SIN2 0 (cos20 - ;) 105 SIN3 tJ COS 8 106 SIN4 0 8 7 35

• Because Eq. (G-IO) is a homo'geneous equation in P, it does not define the normalizatio,:! o~ P. Equations (G-lI) and (G-13) define the conventional Neumann normalization, but other normahzatlOns are used (see Appendix H or Chapman and Bartels [1940D.

Ap.G SPHERICAL HARMONICS 777

Using Eq. (G-D), the functions Pnm can be shown to be orthogonal; that is,

f +l 2 (q+m)! Ppm (x)Pqm(x)dx = -2 -I ( _ ),8/ (G-15)

_I q+ q m.

where 8/ is the Kn;mecker delta. It is now possible towrite the complete solution to Laplace's equation as

00 n+ I n U(r,O,$)= L (7) L [CnmCosm$+ Snmsinm$ ]Pnm(cosO) (G-16)

n .. O m=O

describing the potential exterior to a spherical surface of radius a. Customarily, Eq. (G-16) is written in the form

00 n n+1 + L L (7) [Cnmcosm$+Snmsinm$]Pnm(COSO)

n=l m= I (G-17)

where In == Cno. Terms for which m = 0 are called zonal harmonics and the I n are zonal harmonic coefficients. Nonzero m terms are called tesseral harmonics or, for· the particular case of n = m, sectoral harmonics.

Visualizing the different harmonics geometrically makes the origin of the names clear. The zonal harmonics, for example, are polynomials incosO of degree n, with n zeros, meaning a sign change occurs n times on the sphere (0° ..: 0..: 180°), and the sign changes are independent of $. Figure G-l shows the "zones" (analogous to the temperate and tropical zones on the Earth) for the case of

Fig. G-!. Zones for P6(cos(J) Spherical Harmonics

778 SPHERICAL HARMONICS Ap.G

P6(COS(J). The tesseral and sectoral harmonics have n - m zeros for 0 0 < (J < 1800 ,

and 2m zeros for 00 ~ cp ~ 3600 • Figure G-2, the representation of P63(COS(J)cos3cp, illustrates the division of the sphere into alternating positive and negative tesserae. The word "tessera" is Latin for tiles, such as would be used in 'a mosaic. When n = m, the tesseral pattern reduces to the "sector" pattern in Fig. G-3.

Fig. G-2. P63(cos9)cos3</> Showing Alternating Positive and Negative Tesseral Harmonics

Fig. G-3. P66(cos9)cos6cp Showing Tesseral Pattern Reduced to Sectoral Pattern

For a more detailed discussion of spherical harmonics, see Hobson [1931].

References

I. Arfken, G., Mathematical Methods for Physicists. New York: Academic Press, Inc., 1970.

2. Chapman, Sydney, and Julius Bartels, Geomagnetism. Oxford: Clarendon Press, pp. 609-611, 1940.

3. Fitzpatrick, P. M., Principles of Celestial Mechanics. New York: Academic Press, Inc., 1970.

4. Hieskanen, W., and H. Moritz, Physical Geodesy. San Francisco: W. H. Freeman, 1967.

5. Hobson, E. W., The Theory of Spherical and EllipsOidal Harmonics. New York: Chelsea Publishing Co., 1931.

6. Jackson, John David, Classical Electrodynamics. New York: John Wiley & Sons, Inc., 1962.

7. Yevtushenko, G., et al., Motion of Artificial Satellites in the Earth's Gravitational Field, NASA, TTF-539, June 1969.

APPENDIX H

MAGNETIC FIELD MODELS

Michael Plett

Spherical Harmonic Model. This appendix presents some computational aspects of geomagnetic field models. A more qualitative description of the field characteristics is given in Section 5.1. As discussed there, the predominant portion of the Earth's magnetic field, B, can be represented as the gradient of a scalar potential function, V, i.e.,

B=-VV (H-I)

V can be conveniently represented by a series of spherical harmonics,

(H-2)

where a is the equatorial radius of the Earth (6371.2 km adopted for the International Geomagnetic Reference Field, IGRF); gnm and hnm are Gaussian coefficients (named in honor of Karl Gauss); and r, 0, and <p are the geocentric distance, coelevation, and East longitude from Greenwich which define any point in space.

The Gaussian coefficients are determined empirically by a least-squares fit to measurements of the field. A set of these coefficients constitutes a model of the field. The coefficients for the IGRF (Section 5.1; [Leaton, 1976]), are given in Table H-1. The first-order time derivatives of the coefficients, called the secular

Table H-l. IGRF Coefficients for Epoch 1975. Terms indicated by a dash (-) are undefined.

n m .(nT) hInT) g(nT/vr) h(nTlvr) n m g(nT) hInT) g(nTty,) hInT/v,)

1 0 -30186 -- 25.6 - 6 2 15 102 2.0 -0.1 1 1 -2036 5735 10.0 -10.2 6 3 -210 88 2.8 -0.2 2 0 -1898 - -24.9 -- 6 4 -1 -43 0.0 -1.3 2 1 2997 -2124 0.7 -3.0 6 5 -8 -9 0.9 0.7 2 2 1551 -37 4.3 -18.9 6 6 -114 -4 -0.1 1.7 3 0 1299 - -3.8 - 7 0 66 - 0.0 -3 1 -2144 -361 -10.4 6.9 7 1 -57 -88 0.0 -1.4

3 2 1296 249 -4.1 2.5 7 2 -7 -24 0.0 -0.1 3 3 805 -253 -4.2 -5.0 7 3 7 -4 0.6 0.3 4 0 951 - -0.2 - 7 4 -22 11 0.9 0.3 4 1 807 148 -2.0 5.0 7 5 -9 27 0.3 -0.7

4 2 462 -264 -3.9 0.8 7 6 11 -17 0.3 0.1 4 3 -393 37 -2.1 1.7 7 7 -8 -14 -0.5 0.8 4 4 235 -307 -3.1 -1.0 8 0 11 - 0.2 -5 0 -204 - 0.3 - 8 1 13 4 0.3 -0.2

5 1 388 39 -0.7 1.2 8 2 3 -15 0.0 -D." 5 2 275 142 1.1 2.3 8 3 -12 2 0.2 -0.2 6 3 -20 -141 -1.6 -2.0 8 4 -4 -19 -0.4 -0.3

5 4 -161 -99 -0.5 1.3 8 5 6 1 -0.3 0.4

5 5 -38 74 1.0 1.1 8 6 -2 18 0.6 -0.3 6 0 46 - 0.2 - 8 7 9 -6 -0.3 -0.6

6 1 57 -23 0.5 -0.5 8 8 1 -19 :"'0.1 0.3

780 MAGNETIC FIELD MODELS Ap.H

terms, are also given in Table H-1. With these coefficients and a definition of the associated Legendre functions, Pnm , it is possible to calculate the magnetic field at any point in space via Eqs. (H-I) and (H-2).

The coeffients of the IGRF assume that the Pnm are Schmidt normalized [Chapman and Bartels, 1940], i.e.,

(71 2 2(2-8~) )0 [P,:"(O)] sinOdO= 2n+I (H-3)

where the Kronecker delta, 8} = I if i = j and 0 otherwise. This normalization, which is nearly independent of m, is chosen so that the relative strength of terms of the same degree (n) but different order (m) can be gauged by simply comparing the respective Gaussian coefficients. For Schmidt normalization, the Pnm (0) have the form

m _{[ (2-8~)(n-m)! JI /2 (2n~ I)!!} . m

Pn (0)= ( + )1 ( _ )1 sm '(} n m. n m.

{ n-m (n-m)(n-m-I) n-m-

X cos 0 - 2(2n _ I) . cos 2fJ

+ cos 0-··· (n-m)(n-m-I)(n-m-2)(n-m-3) _ n-m-4 }

2 ·4(2n - I )(2n -- 3) (H-4)

where (2n-I)!!::::::1·3·5···(2n-I). The square root term in Eq. (H-4) is the only difference between the Schmidt normalization and the common Neumann normalization described in Appendix G. The computation time required for the field models can be significantly reduced by calculating the terms inEq. (H-4) recursively, i.e., expressing the nth term as a function of the (n - I)th term. The first step is to convert the coefficients in Table H-I from Schmidt to Gauss normalization, which saves about 7% in computation time [Trombka and Cain, 1974]. The Gauss functions, p n•m , are related to the Schmidt functions, Pnm , by

pm= S pn,m n n,m (H-5a)

where

S =[ (2-8~)(n-m)! JI/2 (2n-I)!! n,m- (n+m)! (n-m)! (H-5b)

The factors S are best combined with the Gaussian coefficients because n,m they are independent of r, 0, cf> and so must be calculated only once during a computer run. Thus, we define

gn,m:::::: Sn,mgnm

hn,m:::::: Sn,mhnm (H-6)

Using mathematical induction, it is possible to derive the following recursion relations for Sn.in:

Ap.H MAGNETIC FIELD MODELS

So,o= 1

s = S [2n-1 ] n,O n-I,O n n~1

( n - m + 1)( 8~ + I) m~ 1

n+m

The pn,m can be similarly obtained from the following recursion relations:

po,o=1

pn,n = sinOpn-I,n-1

pn,m = cosopn-I,m _ K n,mpn-2,m

where

n>1

n=1

781

(H-7)

(H-8)

(H-9)

Because the gradient in Eq. (H-I) will lead to partial derivatives of the pn,m, we need

apo,o --=0

ao

ap nn apn-In-I --' =(sinO) , +(coso)pn-I,n-I

ao ao n ~ 1 (H-IO)

a~;m = (cosO) ap~~I,m _ (sinO)pn-I,m _ Kn,m ap~~2'RI

Also note that

cos mq, = cos«m - 1 )q, + q,)

=cos«m - 1 )q,)cosq,- sinq,sin«m - 1 )q,) (H-II)

A similar recursion relation can be derived for sinmq,. The computational advantage of Eq. (H-II) is that it greatly reduces the number of times that sine and cosine functions must be calculated.

Given the coefficients gn,m and hn,m and recursion relations in Eqs. (H-7) ,through (H-ll), the field B is calculated from Eqs. (H-I) and (H-2). Specifically,

_ av k a n+2 n Br= ---a,:- = L (-;) (n+ I) L (gn,Rlcosmq,+hn,msinmq,)pn,m(O)

n=\ m=O

-I av k (a)n+2 n apn,RI(O) BO=-r--ao=- L -; L (gn,mcosmq,+hn,Rlsinmq,) ao

n= I m=O

. - 1 a V-I ~ (a)n + 2 . n B =-.---=-.- 4. - L m(-gn,msinmq,+hn,mcosmq,)pn,m(O) </> rsmO aq, smO n= \ r m=O

(H-12)

Here, Br is the radial component (outward positive) of the field, Bo is the coelevation compo'1ent (South positive), and B</> is the a:.imuthal component (East positive). (See Fig. 2-5, Section 2.3.) The magnetic field literature, however, normally refers to three components X, Y, Z, consisting of North, East, and nadir relative to an oblate Earth. These components are obtained from Eq. (H-12) by

X (" North") = - Bocosf. - Brsin£

Y (" East") = B </> (H-13) Z (" Vertical" inward positive) = Besinf. - Brcosf.

where f. == A - 0 < 0.2 0, A is the geodetic latitude, and 0 == 90° - 0 is the declination. The correction terms in sin € are of the order of 100 nT or less [Trombka and Cain, 1974].

The geocentric inertial components used in satellite work are

Bx = (Brcoso + Bosino)cosa - B</>sina

By = (Brcos 0 + Bosin 0 )sin a + B </>cos a

Bz = (Brsino - B(Jcoso) (H-14)

Note that B is still a function of longitude, cp, which is related to the right ascension, a, by:

(H-15)

where aG is the right ascension of the Greenwich meridian or the sidereal time at Greenwich (Appendix 1).

Dipole Model. Equations (H-6) through (H-14) are sufficient to generate efficient computer code. However, for analytic purposes, it is convenient to obtain a dipole model by expanding the field model to first degree (n = 1) and all orders (m=O, I). Eq. (H-2) then becomes

= ~(g?a3cosO+ g/a3coscpsinO+ h/a3sincpsinO) r

(H-16)

The cosO term is just the potential due to a dipole of strength g?a3 aligned with the polar axis. (See, for example, Jackson [1965].) Similarly, the sinO terms are dipoles aligned with the x and y axes. Relying on the principle of linear superposition, these three terms are just the Cartesian components of the dipole component of the Earth's magnetic field. From Table H-I, we find that for 1978,

g?= - 30109 nT

g/= -2006 nT

h/=5704 nT

Therefore, the total dipole strength is

a3Ho=a3 [ gr + gr + hf]I/2 =7.943 x 1015 Wb·m

(H-17)

(H-18)

Ap.H MAGNETIC FIELD MODELS 783

The coelevation of the dipole is

o,;,=arccos( !:)= 168.6° (H-19)

The East longitude of the dipole is

</>:n=arctan( :~l)= 109.3° (H-20)

Thus, the first-order terrestrial magnetic field is due to a dipole with northern magnetization pointed toward the southern hemisphere such that the northern end of any dipole free to rotate in the field points roughly toward the north celestial pole. The end of the Earth's dipole in the northern hemisphere is at 78.6° N, 289.3° E and is customarily referred to as the "North" magnetic pole. Frequently, dipole models in the literature use the coordinates of the North magnetic pole and compensate with a minus sign in the dipole equation.

The above calculations were performed for 1978 by adding the secular terms to the Gaussian coefficients of epoch 1975. The location of the dipole in 1975 can be similarly calculated and compared with the 1980 dipole. That comparison yields a 0.45% decrease in dipole strength between 1975 and 1980 and a O.071-deg drift northward and a O.056-deg (arc) drift westward for a total motion of 0.09-deg arc.

The dipole field in local tangent coordinates is given by

B, = 2 ( ~ r[ g?cosO + (gicos</> + hlsin</> )sinO]

B8 = ( ~ ) 3[ g?sinO - (gIcos</> + hlsin</> )cos 0]

Bep = (~f[ glsin</>- hlcos</>]

(H-21)

The field could be converted to geocentric inertial coordinates using Eq. (H-14), but the exercise is arduous and not particularly instructive. However, we may take advantage of the dipole nature of the dominant term in the field model to approximate the magnetic field of the Earth as due to a vector dipole, m, whose magnitude and direction are given by Eqs. (H-t8) through (H-20). Thus,

(H-22)

where R. is the position vector of the point at which the field is desired. Because this is a vector equation, the components of B may be evaluated in any convenient coordi'.late system. As an example, the field in geocentric inertial components can be obtained from the dipole unit vector,

[sinO';' cosam]

m= sinO';' sinam

cosO';' (H-23)

where aGO is the right ascension of the Greenwich meridian* at some reference time (aGo=98.8279° at Oh UT, December 31, 1979), daG/dt is the average rotation rate of the Earth (360.9856469 deg/day), t is the time since reference, and (0';',</>;") =(168.6°,109.3°) in 1978.

Then

(H-24)

where Rx, Ry , and Rz are the geocentric inertial direction cosines of R. The field components are

a3HO .' . I

Bx= 7(3 (m.R)Rx-smOmcosam]

a3H . By = R 30 (3 (m· R)Ry - sin 0';' sin am ] (H-25)

a3HO • Bz = 7 [3 (m·R)Rz -cosO';']

These equations are useful for analytic computations and for checking computer calculations. For example, if R is in the Earth's equatorial plane, then Rz =0 and

a3HO Bz = 7( -cos 168.6°) (H-26)

which is positive, i.e., north. Because the direction of the field line is customarily defined as that indicated by a compass needle, Eq. (H-22) is self-consistent.

For analytical work, the most useful coordinate system is the I, b, n orbit plane system (Section 2.2), in which R has the particularly simple representation

R. = R (V')COS v'

Rb = R ( v')sin v'

Rn=O

(H-27)

where v'is the true anomaly measured from the ascending node. Vectors are transformed into the [,b,n system from the geocentric inertial system by first rotating about the inertial z axis through n, the right ascension of the ascending node, followed by a rotation about the ascending node by the angle i, the orbital inclination. Using this transformation, the unit magnetic dipole is

m,= sinO';'cos(n- am)

mb = - sin O';'cos i sin(n - am) + cos 0';' sin i

mn = sin 0';' sin i sin (n- am) + cos 0';' cos i

(H-28)

• This technique of cOlfiiJuting aG is good to about 0.005° for I year on either side of the reference date. At times m~rll'distant from the reference date, a new aGO can be computed as described in Appendix J. Note that aGO is equal to the Greenwich sidereal time at the reference time of ()h UT, December 31, 1978.

Ap.H MAGNETIC FIELD MODELS 785

where Q is the ri5ht ascension of the ascending node and i is the inclination of the orbit.

Substituting Eqs. (H-27) and 01-28) into Eq. (H-22) yields the magnetic field in the I, b, n system. Although the equations are moderately complex, they can still be useful. Due to the simple form for R, especially for circular orbits, it is possible to analytically integrate the torque due to a spacecraft dipole moment as has been done for ITOS [Kikkawa, 1971].

A circular equatorial orbit is particularly simple because i = Q = 0 and, therefore,

m· R = sin():n (cos amcos v' + sinamsinv')

Substituting into Eq. (H-22) and simplifying yields

a 3H B{= ~sin():n[3cos(2v' - am) +cosam ]

2R

(H-29)

(H-30)

As in Eq. (H-26), the minus sign in the orbit normal component, Bn assures the northward direction of the field lines.

The torque resulting from a spacecraft magnetic dipole interacting with Bn is in the orbit plane, or, in this case, the equatorial plane. This torque causes precession around the orbit normal, or, for i = 0, right ascension motion. Torque out of the orbit plane is caused by the ascending node component B{ and the component Bb • For i = 0, out of plane is the same as declination motion. Thus, for an equatorial orbit, the ratio of declination motion to right ascension motion is at most on the order of

- - -04 1 B{,b 1_1 2 sin():n 1-Bn - cosO:n .

(H-31)

Consequently, for a satellite in an equatorial or low-inclination orbit, the right ascension is the easier to control.

Note that the declination terms BJ and Bb in Eq. (H-30) oscillate with a frequency of twice the orbital period. That is, the direction of the magnetic field in the orbit plane system rotates through 720 deg during the orbit.

Thus, B j and Bb change signs four times during the orbit. Declination motion then can be obtained in a certain direction by switching the polarity of the magnetic control coil four times or every quarter orbit. This is the basis for QOMAC control theory. If the satellite has a residual magnetic dipole, the Bn term will cause a secular drift in right ascension and the BJ , Bb terms will cause an oscillation in declination at twice the orbital period and a diurnal oscillation in declination due to the rotation of the Earth.


References

1. Leaton, B. R., "International Geomagnetic Reference Field 1975," Trans., A mer. Geophysical Union (Eff)S), Vol. 57, p. 120, 1976.

2. Chapman, Sydney and Julius Bartels, Geomagnetism. Oxford, Clarendon Press, 1940.

3. Trombka, B. T. and J. C. Cain, Computation of the IGRF I. Spherical Expansions, NASA X-922-74-303, GSFC, Aug. 1974.

4. Jackson, John David, CLassical Electrodynamics. New York: John Wiley & Sons, Inc., 1965.

5. Kikkawa, Shigetaka, Dynamic Attitude Analysis Study, Compo Sc. Corp., CSC 5023-10000-0ITR, Jan. 1971.

APPENDIX I

SPACECRAFT AlTITUDE DETERMINATION AND CONTROL SYSTEMS

Ashok K. Saxena

1.1 Spacecraft Listed by Stabilization Method 1.2 Spacecraft Listed by Attitude Determination Accuracy

Requirements 1.3 Spacecraft Listed by Type of Control Hardware 1.4 Spacecraft Listed by Type Of Attitude Sensors

This appendix summarizes spacecraft attitude systems and serves as a guide to mission specific attitude determination and control literature. The main table is an alphabetical listing of satellites by acronym with pertinent data as available. Subsequent sections list these spacecraft by stabilization method, accuracy requirements, control system, and sensor system. For example, if you are interested in gravity-gradient stabilization in low-Earth orbit, Section I.l lists DODGE, GEOS-3, and RAE-2 as gravity-gradient stabilized. The main table lists GEOS-3 as the only one of these in low-Earth orbit. Normal automated literature search procedures may then be used to obtain available literature citations for GEOS-3.

The material in this appendix has been collected from literature searches, the TRW Space Log, and Joseph and Plett [1974]. Design values are quoted for upcoming spacecraft, which are denoted by an asterisk after the spacecraft acronym. The superscript "b" is used for body and "w" is used for wheel.

1.1 Spacecraft Listed by Stabilization Method

Missions with mutiple phases (e.g., RAE-2) are listed in all appropriate categories.

Single Spin AEROS-I,2, ALOUETTE-I,2, ARIEL-III, ARYABHATA, ATS-3, CTS, DEA*, ESRO-IV, GOES-I, HEAO-I,C, HEOS-I, IMP-6,7,8, ISEE-I,B,C*, ISIS-I, II, ISS, IUE, LES-5,7, RAE-2, SIRIO, SM-3, SMS-I,2, SKYNET (U.K.I,2), SSS-I

Dual Spin ANS, ATS-6, DODGE, FLTSATCOM, HEAO-B*, IUE, NIMBUS-5,6, OAO-2,3, OGO-I, SAGE*, SEASAT*, SMM*, ST, SYMPHONIE

Momentum Wheels ANS, ATS-6, DODGE, FLTSTCOM, HEAO-B*, IUE, NIMBUS-5,6, OAO-2, 3, OGO-I, SAGE*, SEASAT*, SMM*, ST, SYMPHONIE

Gravity Gradient DODGE, GEOS-3, RAE-2

Magnetic AZUR-I, HCMM*

Gas Jets HEAO-I,C·

788 SPACECRAFT ATTITUDE SYSTEMS Ap. I

ORBIT PARAMETERS ACRONYM NAME PRINCIPAL MASS LAUNCH DATE

COUNTRY (KG I AND VEHICLE INeLlNA· PEAIGEE APOGEE' PERIOD nON HEIGHT H~~rT IDEGI IKMI (MINUTES)

AE·3 ATMOSPHERE USA 454 PLUS DEC. 16, 1973, 68.1 150 4,300 130 EXPLORER·3 PROPEL· BY DELTA LANT

AE·5 ATMOSPHERE USA 454 PLUS NOV. 19, 1975, 19 159 1,618 102.7 EXPLORER·S PROPEL· BY DELTA LANT

AEM GENERAL NAME FOR USA APPLICATIONS ex-PLORER MISSION; e.g., HCMM-. SAGE·

AEROS-' AERONOMY GERMANY 127 DEC. 16, 1972. 97.2 250 BOO 97.5 SATELLlTE-1 BY SCOUT SUN SVNrRONOUS

AEAOS-2 AERONOMY GERMANY 127 JUl Y 16, 1974, 96 .• 227 889 95.8 SATELlITE·2 BY SCOUT SUN SYNCHRONOUS

ALOUETTE-, CANADA 145.6 SEPT. 29,l962.BY THQR·AGENA 8

80.6 996 '.031 105.4

ALOUETTE·2 CANADA 146.1 NOV. 28, 1965.BY THQA·AGENA B

80.6 507 2,892 121.4

ANS NETHERLANDS NETHERLANDS 130 AUG. 30, 1974, BY 98.0 260 1,100 98.2 ASTRONOMICAL SCOUT SATELLITE

ARIEL·1I1 U.i<. MAY5,1967,BY 80 500 -500 96.6 SCOUT

AAYABI1ATA INDIA 360 APR. 19, 1975 50.4 S64 623 96.4

ATS·3 APPLICATIONS USA 361 NOV. 5, 1967, BY 0.4 35,772 1,436 TECHNOLOGY ATLAS- GEOSYN~HRONOUS SATELLlTE-3 AGENA 0

ATS·6 APPLICATIONS USA 1,360 MAY 30, 1974, 1.3 35,759 36.820 1,436 TECHNOLOGY BY TITAN·III C GEOSYNCHRONOUS SATELlITE-6

AZUR-1, GERMAN GERMANY 71.3 NOV. 8,1969, 102.9 239 1,955 121.8 ALSO RESEARCH BY SCOUT CALLED SATELLITE GRS-A

CTS COMMUNICATIONS CANADA JAN. 17. 1976, 0.1 35,786 1,436 TECHNOLOGY BY DELTA GEOSYNCHRONOUS SATELLITE

DE-A- DYNAMICS USA 2sa 1981. BY 90 275 23,918 417 EXPLORER·A DELTA

DE-O- DYNAMICS USA 305 1981,BY 90 276 1,200 100 EXPLOAER·B DELTA

DODGE OEPARTMEfoiT OF USA 195.2 JULY " 1967, 7.2 33,243 33,602 1,317 DEFENSE GRAVITY BY TITAN·III C eXPERIMENT

ERTS EARTH RESOURCES TECHNOLOGY SATEL· LITE RENAMEO LANDSAT

ESRO-IV EUROPEAN SPACE ESRO 120 NOV.21.1972, 91.06 254 1,143 99 RESEARCH BY SCOUT ORGANIZATION-IV

'PERIGEE AND APOGEE HEIGHT ARE MEASURED FROM THE SURFACE OF THE EARTH.

Ap. I SPACECRAFT ATTITUDE SYSTEMS 789

STABILIZATION ATTITUDE TECHNIQUE CONTROL HARDWARE POINTING ACCURACY ATTITUDE SENSORS MISSION OBJECTIVES

(SPIN RATE IN RPM)

DUAL SPIN MAGNETIC Z AXIS COil , ,0 BODY·MOUNTED IR HORI- ATMOSPHERE PROBE wb = 0.01-16 MAGNETIC SPIN COILS ZON SCANNERS WITH SOLAR ABSORPTION

Ww '" 214.4 MOMENTUM WHEEL WHEEL·MOUNTED IA EXPERIMENTS GAS JETS HORIZON SCANNERS

DIGITAL SUN SENSORS

DUAL SPIN MAGNETIC Z AXIS COIL ± ,0 BODY-MOUNTED IA HQRI· ATMOSPHERE PROBE wb '" 0.01-16 MAGNETIC SPIN COilS ZON SCANNERS

'" 214.4 MOMENTUM WHEEL WHEEL·MOUNTED IR

Ww GAS JETS HORIZON SCANNERS DIGITAL SUN SENSORS

SPIN wb ~ 10 MANUAL AND AUTO- ;!; 0.20 FOR SOLAR AS- TWO fA SENSORS AERONOMY, MEAS MATIe MAGNETIC PEeT ANGLE ANALOG FINE SUN SENSOR URES PHYSICAL COIL FOR PRECESSION DIGITAL SOLAR ASPECT PROPERTIES OF UPPER

SENSOR ATMOSPHERE IN COR· AUTOMATIC SPIN COR· ~ 3° FDA SOLAR AZI· THREE·AXIS MAGNE RELATION WITH EX RECTtON DURING DAY- MUTH TOMETEA TAE~E ULTRAVIOLET LIGHT RADIATION

SPIN wb ~ 10 MANUAL ,o\ND AUTO· .!. 0.2° FOR SOLAR AS· TWO IR SENSORS AERONOMY, MEAS MATIC MAGNETIC PECT ANGLE ANALOG FINE SUN SENSOR UAES PHYSICAL COl L FOR PRECESSION DIGITAL SOLAR ASPECT PROPERTIES OF UPPER

SENSOR ATMOSPHERE IN COR· AUTOMATIC SPIN COR ± 3° FOR SOLAR AZI- THREE·AXIS MAGNE RELATION WITH EX· RECTIQN DURING DA Y MUTH TOMETER TREME UL TRAVIQLET LIGHT RADIATION

SPIN wb - 1_43 MEASUREMENT OF EAATH'S IONOSPHE~E BY RADIO SOUNDING

SPIN wb = 2.25 IONOSPHERIC RESEARCH

MOMENTUM WHEELS THREE ORTHOGONAL ;!: 0.3° FOR SCAN AND STAR SENSOR ASTRONOMY ONBOARO COM- MOMENTUM WHEELS SLOW SCAN HORIZON SENSOR PUTER

X, Y, AND Z MAG- ~ 10 FOR STAR, OFFSET. SOLAR SENSOR NETIC COILS AND X·RAY MAGNETOMETEA

SPIN ~ 20 SOLAR ASPECT SENSORS MEASURE VERTICAL DISTRIBUTION OF MOLECULAR OXYGEN, MAP RADIO FREQUENCY NOISE IN GALAXY

SPIN wb = 10-90 NITROGEN JETS FOR ± 1° DIGITAL SOLAR SENSOR X·RAY ASTRONOMY SPIN TRIAXIAL FLUGATE SOLAR PHYSICS

MAGNETOMETER AERONOMY

SPIN wb = 100 HYDRAZINE JETS ! 1° STAR SENSOR COMMUNICATIONS V·SlIT SUN SENSOR METEOROLOGY SCANNING RADIOMETER

MOMENTUM WHEELS THREE MOMENTUM ± 0.1° IR EARTH SENSORS ERECT A LARGE AN-WHEELS DIGITAL SUN SENSORS TENNA STRUCTURE GAS JETS POLARIS TRACKER

INERTIAL REFERENCE PROVIDE GOOD TV GYROSCOPES SIGNAL TO LOW-COST INTERFEROMETER GROUND RECEIVERS

DEMONSTRATE USER-ORIENTED AP· PLICATION EXPERI-MENTS

PASSIVE MAGNETIC TWO FIXED MAGNETS ANGLE BETWEEN THE SUN SENSOR MEASUREMENT OF SUCH THAT THE RESULT GEOMAGNETIC FIELD RADIATION AND PAR-ING DIPOLE MOMENT VECTOR AND THE TWO·AXIS MAGNETOMETER TICLES IN THE POLAR POINTS IN THE DIREC· Z·AXIS WAS LESS ZONE AND IN THE TlON OF THE Z AXIS WITH THAN 120 VAN ALLEN BELT AN ACCURACY OF 8 MIN· UTES --

SPIN IN TRANSFER HYDRAZINE CATALYTIC ! 10 TRANSFER AND TWO NQNSPINNING EARTH COMMUNICATIONS AND DRIFT ORBITS THRUSTERS DRIFT ORBIT SENSORS wb = 60 TWO IR HORIZON SCANNERS

MOMENTUM WHEEL , 0,0 PITCH ANI NONSPINNING SUN SENSOR MOMENTUM WHEEL ROLL MISSION SPINNING DIGITAL SUN Ww = 3750 OABIT SENSOR

i 1.10 YAW

SPIN HYDRAZINE THRUSTERS ! 1° V·HORIZON SCANNERS INVESTIGATE ELECTRO· wb ~ 10 SUN SENSORS DYNAMIC EFFECTS IN

UPPER ATMOSPHERE

DUAL SPIN PITCH MOMENTUM WHEEL ! 10 HORIZON SCANNERS INVESTIGATE ELECTRO· wb:o 1 REV/OABIT MAGNETIC COILS SUN SENSORS DYNAMIC EFFECTS IN

HYDRAZINE THRUSTERS UPPEA ATMOSPHERE

GRAVITY-GRADIENT MOMENTUM WHEEL ! 16° SUN SENSORS DEMONSTRATE AUGMENTED BY TV CAMERAS GRAVITY-GRADIENT CONSTANT-SPEED STABILIZATION NEAR ROTOR AND MAG· GEOSYNCHRONOUS NETIC DIPOLE FOR ALTITUDE YAW

SPIN Wb = 65 MAGNETIC COilS ATTITUDE DETERMINA- IR HORIZON SCANNER STUDY NEAR-EARTH TION 1 20 DIGITAL SUN SENSOR MAGNETOSPHERE AND

IONOSPHERE ATTITUDE CONTROL i 50 TRIAXIAL FLUX GATE

MAGNETOMETEA


ORBIT PARAMETERS

ACRONYM NAME PRINCIPAL MASS LAUNCH DATE PERIGEE' APOGEE' CCUNTRV (KG} AND VEHICLE INeLlNA· PERIOD

TlON HEIGHT HEIGHT (MINUTES) IDEG) (KMI (KMI

eXPLORER GENERAL NAME GIV-EN TO SATELLITES; e.g .• EXPLORER-45 IS SAME AS SSS·1

FLTSATCQM FLEET SATELLITE USA 1.861 ATLAS- 1.436 COMMUNIt.;ATION CENTAUR GEOSYNCHRONOUS SYSTEM

GEOS-3 GfQDYNAMICS USA 340 APR. 9, 1975. 11. 83. 848 102 EXPERIMENTAL BY DELTA OCEAN SATEL-LlTE-3

GOES·' GEOSTATIONARY USA 295 OCT. 16, 1975. 1.0 35,770 35,796 1,435.9 OPERATIONAL BY DELTA GEOSYNCHRONOUS ENVIRONMENTAL SATELLITE·'

HCMM- HEAT CAPACITY USA -9' MAY 1978, 98 600 600 97 MAPPING MISSION BY SCOUT

HEAD·' HIGH ENERGY USA 3,150 AUG. 12. 1971, 22.75 .17 .3. 93 ASTRONOMY BY ATLAS· OBSERVATORY·' CENTAUR

HEAD-S- HIGH ENERGY USA 3,150 NOV. 1978, 22.75 - 435 -435 -9' ASTRONOMY BY ATLAS· OBSERVATORY ·B CENTAUR

HEAO·C· HIGH ENERGY USA 3,150 LATE 1979. 22.75 -- -- -95 ASTRONOMY BY ATLAS-OBSERVATORY ·C CENTAUR

HEOS-l HIGHLY ECCENTRIC ESRO 106 DEC. 5, 1968. 28.28 '2' 223,428 6,792 ORBIT SATELLITE BY DELTA

IMP-6 IN.TEAPLANETARY USA 286 MAR. 13, 1971. 28.8 148 122,146 5,628 MONITORING BY THOR·DELTA PLATFORM·6

IMP·7 INTERPLANETARY USA 376 SEPT. 23. 1972. 17.2 197,391 235.389 17,439 .MONITORING BY DELTA PLATFORM-7

IMP-8 INTERPLANETARY USA 371 OCT. 26. 1973, 28.2 196,041 235,754 17,362 MONITORING BY DELTA PlATFORM-8

ISEE·' INTERNATIONAL USA 31' OCT. 22, 1971, 28.4 288 134.000 2.29 SUN-EARTH BY DELTA DAYS EXPLORER-l

;

ISEE-S INTERNATONAL ESA 166 LAUNCHED 28.4 288 134,000 2.29 SUN-EARTH SIMULTANE- DAYS EXPLORER·S OUSlYWITH

ISEE·A ON THE SAME LAUNCH VEHICLE

ISEE·C· INTERNATIONAL USA - JULY 1978, IN PERI- APOHE- 1 YEAR SUN·EARTH BY DELTA EClIP· HELION LION AT EXPLORER-C TIC

PLANE2 AT 0.973 1~7 AU2

1515-1 INTERNATIONAL CANADA 238 JAN. 30, 1969. 88.43 .7. 3,522 128.3 SATELLITE FOR BY DELTA IONOSPHERIC STUDIES-l

ISIS-II INTERNATIONAL CANADA 284 APR. 1,1971. 88.16 1,356 1,423 113 SATELLITE FOR BY THOR-DELTA IONOSPHERIC STUDIES-2

ISS IONOSPHERE 135 - 70 -1,000 - r.ooo SOUNDING SATELLITE

ITOS-H '" IMPROVED TIROS USA 340 JULY 29. 1916, 102 1,507 1,522 116.2 NOAA·5 OPERATIONAL BY DELTA

SATELLlTE-8 = NA-TIONAL OCEANIC AND ATMOSPHERIC AD-MINISTRATION-S


Ap. I SPACECRAFT A TfITUDE SYSTEMS 791

STABILIZATION ATTITUOE TECHNIQUe CONTROL HARDWARE POINTING ACCURACY ATTlTUOE SENSORS MISSION OBJECTIVES

(SPIN RATE IN RPM'

MOMENTUM WHEELS THREE MOMENTUM" 1; 0.:20 TWO SPINNING EARTH WORLDWIDE COMMUNI· WHEELS AT 3000 t 300 SENSORS CATION SYSTEM FOR RPM, HYDRAZINE TWO SPINNING SUN SENSORS THE NAVY THRUSTERS ONE NONSPINNING EARTH SUPPORT AlA FORCE

SENSOR COMMANOANO TWO NONSPINNING SUN CONTROL SENSORS

GRAVITY GRADIENT' PITCH WHEEL ! 1.50 YAW THREE DIGITAL SUN SOLID EARTH PHYSICS DUAL SPIN REACTION BOOM :! 1.20 POINTING SENSORS AND OCEANOGRAPHY Ww = 2000 Z AXIS MAGNETIC COl L THREE-AXIS MAG-

NETOMETER

SPIN wb '" 90 HYDRAZINE JETS t 1.00 (TRANSFER ORBIT! FIVE fR HORIZON SCANNERS OPERATIONAL ! 0.10 (MISSION ORBIT) TWO OUAl-SLIT SUN METEOROLOGICAL

SENSORS SATELLITE

THREE·AXIS ONBOARD ONE MOMENTUM WHEEL ATTITUDE OETERMINA· ONE IR SCANNER MAP HEAT CAPACITY CONTROL. DUAL SPIN THREE MAGNETIC TION THREE DIGITAL SUN OF THE EARTH ""b" 1 REV/ORBIT COILS .t 0.70 IN ROLL.! 0.50 IN SENSORS

Ww = 1940 PITCH. :I: 20 IN YAW TRIAXIAL FLUX GATE ATTITUDE CONTROL MAGNETOMETER 1 10 IN ROLL AND PITCH 120 IN YAW

ONBOARD COMPUTER HYDRAZINE JETS , ,. COARSE SUN SENSORS ALL SKY X·AND SINGL~·AXIS SPIN, FINE SUN SENSORS GAMMA·RAY SURVEY TI1RE:E·AX1S TWO so x so FOV STABILIZED STAR TRACKERS

FOUR SOOF GYROSCOPES

ONBOARQ COMPUTER MOMENTUM WHEELS ONSOARD: 1 10 COARSE SUN SENSORS X·RAY TELESCOPE TO THREE·AXIS MAGNETIC COILS ARC MINUTE; FINE SUN SENSORS DETERMINE INTENSITY, STABILIZED HYDRAZINE JETS GROUND ATTITUDE THREE ;zo x ZO FOV SPECTRA, POSITION, AND

DETERMINATION: STAR TRACKERS TIME VARIATIONS OF 1 0.1 ARC MINUTE SIX SDOF GYROSCOPES X·RA Y SOURCES

ONBOARD COMPUTER HYDRAZINE JETS ATTITUDE CONTROL CO~RSE SUN SENSORS ALL SKY GAMMA·RAY SINGLE·AXIS SPIN· , ,. FINE SUN SENSORS SURVEY STABILIZED ATTITUDE DETERMINA· TWO so x so FOV

TION 10.050 STAR TRACKERS FOUR SDOF GYROSCOPES

SPINwti = 10 NITROGEN JETS TWO EARTH ALBEDO STUDY INTERPLANE· SENSORS TARY RADIATION. TWO SOLAR ASPECT SOLAR WIND, AND SENSORS MAGNETIC FIELDS SOLAR GATE SENSOR OUTSIDE THE

MAGNETOSPHERE

SPlNwb'" 5 FREON JETS ". OPTICAL HORIZON STUDY INTERPLANE· SCANNER TARY PARTICLES AND DIGITAL SUN SENSOR ELECTROMAGNETIC

RADIATION

SPIN w b " 46 FREON JETS , ,. OPTICAL TELES(X)PE STUDY SOLAR PLASMA, DIGITAL SUN SENSOR SOLAR WINO. SOLAR AND

. COSMIC RADIATION. ELECTROMAGNETIC FIELD VARIATIONS, EARTH'S MAGNETIC TRAIL

SPIN ""b .. 12-68 FREON JETS :l:O.SO OPTICAL TELESCOPE STUDY SOLAR PLASMA. DIGITAL SUN SENSOR SOLAR WIND, SOLAR

AND GALACTIC'COSMIC RAO,ATlON. ELECTRO· MAGNETIC FIELD VARI· ATIONS. AND INTER-PLANETARY MAGNETIC FIELD

SPINAwb '"' 20' COLO GAS JETS t.1o· PANORAMIC ATTITUDE ~DY MAGNETOSPHERE, SENSOR INTERPLANETARY SPACE SOLAR ASPECT SENSOR AND INTERACTIONS

BElWEEN THEM

SPINwb -20 COLD GAS JETS <"" V-SLiT COMBINED STUDY MAGNETOSPHERE. SUN/EARTH ALBEOO INTERPLANETARY SPACE SENSOR AND INTERACT'ONS

BETWeEN THEM

SPIN '"'b-20 HYORAZINE JETS < .0 PANORAMIC A"ITUDE STUDY MAGNETOSPHERE, SENSOR INTERPLANETARY SPACE SOLAR ASPECT SENSOR AND INTERAcrlONS

BETWEEN THEM

SPIN wb .. 2.938 MAGNETIC COIL WHEN MEASUREMENT OF COMMUTATEDWlTH EARTH'S IONOSPHERE SUN LINE OR GED- BY RADIO SOUNDING MAGNETIC FIELD VECTOR G'VES AT-TITUDE OR SPIN RATE CHANGE

SI"N'"'b- 2.9 MAGNETIC COIL FOR MEASUREMENT OF ATTtTUDE AND SPIN EARTH'S IONOSPttERE RAT~CHANGE BY RADIO SOUNDING

SPIN '"'b -'0 t. ,.SO IRSENSOR "!ORLD MAP OF ELEC-D'GITAL SUN SENSOR TRDN DENSITY OF THE MAGNETOMETER A~ERE

DUAL1IPIN MAGNEnC COILS SUFFICIENT TO MAIN- 'R HORIZON SENSORS METEOROLOG'CAL SATEL-"'b - • REVIORBIT TAIN CONSTRAINTS SOUUIASl'ECT SENSORS LITE .... -.40

792 SP ACECRAFT ATTITUDE SYSTEMS Ap. I

ACRONYM NAME PRINCIPAL MASS LAUNCH OATE COUNTRY !KG) AND VEHICLE

nOS-I", J" IMPAOVED TlROS USA DELTA OPERATIONAL SATELLlTE·I, J

tue' INTERNATIONAL USA 430 JAN. 26, 1978, ULTAAVIOLET BY AUG· EXPLOREA MENTED

THRUST DELTA

LANDSAT·' OAIGINAlLY USA 959 JULY 23, 1972, CALLED ERTS BY DELTA

LANDSAT-2 USA 959 JUNE 22, 1975, BY DELTA

LANDSAT·C· USA THIRD QUARTER 1977, BY DELTA

LANDSAT-O USA '" 1981

lE5-S liNCOLN EX- USA 102 JULY " 19!;57. PERIMENTAL BY TITAN-III C SATELUTE·5 (MULTIPLE PAY·

LOAD)

LES-7 LINCOLN EX- USA - 300 PERIMENTAL SATELLlTE·7

MAGSAr GEOMAGNETIC USA - 165 1979-1980 FIELD SATELLITE

MMS MUL TlMISSION MODULAR SPACE-CRAFT SERIES (e.g., SMM-)

NIMBUS-S USA 772 DEC. 1" 1972, BY DELTA

NIMBUS-6 USA 829 JUNE 12, 1975, BY DELTA

OAQ-2 ORBITING ASTRO- USA 1,998 DEC. 7, 1968, BY NOMICAL OBSER- ATLAS-VATOAY·2 CENTAUR

OAO·3 ORBITING ASTRO· USA 2,204 AUG. 21, 1972, NOMICAL OSSEA· BY ATLAS-VATORY·3, ALSO CENTAUR COPERNICUS

OGO-' ORBITING GEO- USA 487 SEPT. 4, 1964, PHYSICAL OBSERVA- BY ATLAS-TORY AGENA 8

050-7 ORBITING SOLAR USA 635 SEPT. 29, 1971, OBSERVATORY-7 BY DELTA

050-8 ORBITING SOLAA USA 1,090 JUNE 21,1975, OBSERVATORY-S BY DELTA

RAE-2 RADIO ASTRONOMY USA 205 JUNE 10, 1973, EXPLORER·2 (ORBIT) BY AUGMENTED

328 THRUST LONG-(LAUNCH) TANK DELTA

SAGe- STRATOSPHERIC USA -105 FEB. 1979 AEROSOL AND GAS· EXPERIMENT, ALSO KNOWN AS AEM·B-

SAS-2 SMALL ASTRONOMY USA 185 NOV. 15, 1972. SATELUTE-2 BY SCOUT


20RBIT ABOUT SUN·EARTH LAGRANGIAN p?INT L 1•

ORBIT PAAAMETERS

INeLlNA· PEAIGEE' APOGEE ' PEAIOD nON HEIGHT HEIGHT (MINUTES) loEG) {KM} (KMI

102 - 1,459 '" 1,466 115.2

28.7 25,706 45,876 1,436 ELLIPTICAL GEOSYNCHRONOUS

99 915 917 103.3

99 898 916 103.1

99 916 916 103.3

98 700 700 99

7.2 33,300 33,600 1,319

-0 NEAA GEOSTATIONARY

97 325 550 96

99.95 1,OB9 1,102 107.25

99_9 1,100 1,113 107.4

35 764 774 "100.3

35 738 750 99.7

31.1 260 150,000 64 HOURS

33 323 571 93.4

32.95 545.9 562.4 95.74

1,100 1,100 225 SELENQCENTAIC ORBIT

-so" -600 -600 - 97

2 443 632 96

Ap. I SPACECRAFT ATIITUDE SYSTEMS 793

STABILIZATION ATTITUDE TECHNIQUE CONTAOL HARDWARE POINTING ACCURACY ATTITUDE SENSORS MISSION OBJECTIVES (SPIN RATE IN RPMI

DUAL SPIN MAGNETIC COilS SUFFICIENT TO MAIN- IR HORIZON SENSORS METEOROLOGICAL SATEL· ~ • 1 REV/ORBIT TAIN CONSTRAINTS SOLAR ASPECT SENSORS LITES ww" '40

SPIN ABOUT AXIS HYDRAZINE JETS t ARC.sECOND PANORAMIC ATTITUDE UL TRAVIQLET SPECTRO-OF MINIMUM REACTION WHEELS SENSOR GRAPHY OF sye LLAR MOMENT OF IN· SPIN MOOE SUN SENSOR SOURCES ERTlA; THREE·AXIS FINE SUN SENSOR STABILIZED IN ANALOG SUN SENSOR MISSION NODE GYROSCOPES

FINE ERROR SENSOR

DUAL SPIN ATTITUDE CONTROL HORIZON SENSORS EARTH'S RESOURCES ± 0.70 ATTITUDE DETERMINA· TION ! 0.07°

DUAL SPIN ATTITUDE CONTROL HORIZON SENSORS EARTH'S RESOURCES t 0.70 ATTITUDE DETERMINA· TION to.07°

DUAL SPIN ATTITUDE CONTROL HORIZON SENSORS EARTH'S RESOURCES t 0.70 ATTITUDE DETERMINA-TION t 0.070

MMSPACKAGE ! 0.010 TWO STAR TRACKERS EARTH SURVEY ISEE SMM) GYROSCOPES

THEMATIC MAPPER FOR LANDMARK TRACKING

SPIN wb '" 10 TWO ORTHOGONAL ,,. EARTH SENSOR WITH EXPERIMENTAL SATEL· MAGNETIC COILS 3 FAN BEAM FIELD 4.TE

OF VIEW FOUR SOLAR GATE SENSORS

SPIN w w " 1100 GIMBALED FL VWHEEL to.l° IA EARTH SENSORS EXPERIMENTAL SATEL-PLASMA JETS LITE

DUAL SPIN MAGNETIC COILS ATTITUDE DETERMINA- TWO STAR CAMERAS ACCURATE MAPPING OF TION: 20 ARC-SEC FINE SUN SENSOR THE EARTH'S MAGNETIC

GYROSCOPES FIELD ATTITUDE TRANSFER STUDY OF THE EARTH'S SYSTEM CRUST, MANTLE, AND

CORE

MOMENTUM WHEELS FOUR MOMENTUM 01 0 HORIZON SENSORS METEOROLOGICAL WHEELS (TWO ON ROLL AXIS)

MOMENTUM WHEELS FOUR Mt')MENTUM , ,0 HORIZON SENSORS METEOROLOGICAL WHEELS (TWO ON ROLL AXIS)

MOMENTUM WHEELS THR EE MOMENTUM t 0.020 STAR TRACKER ULTRAVIOLET ASTRONOMY WHEELS RATE INTEGRATING

GYROSCOPES

MOMENTUM WHEELS THREE MOMEN'rUM t 0.1 ARC·SEC = FOUR GIMBALED ULTRAVIOLET ASTRONOMY WHEELS 3 x 10-5 DEG STAR TRACKERS

RATE·INTEGRATING GYROSCOPES

MOMENTUM WHEELS THREE MOMENTUM ! 20 TWO HORIZON GEOPHYSICAL WHEELS TRACKERS AIR JETS YAW AND SOLAR ARRAY

SUN SENSORS PITCH RATE GYROSCOPES

DUAL SPIN Z AXIS MAGNETIC COIL t 0.10 V-SLIT STAR SENSOR MEASURE SOLAR AND

""SAIL· 0 NITROGEN JETS V-SLIT SUN SENSOR COSMIC X·RAYS,

""w· 30 MAGNETOMETER GAMMA RAYS, ULTRA·

VIOLET RADIATION, AND OTHER ASPECTS OF SOLAR ACTIVITY

DUAL SPIN Z AXIS MAGNETIC COIL t 0_10 V-SUT STAR SENSOR SOLAR AND COSMIC w SA1L "0 NITROGEN JETS V-SLIT SUN SENSOR X-RAYS. MEASURE SOLAR

MAGNETOMETER ULTRAVIOLET LINE ""w,"6 PROF IlES, AERONOMY

SPIN FREON JETS FOR SPIN ATTITUDE DETERMINA- OPTICAL HORIZON SCAN- STUDY RF SOURCES THREE-AXIS AND PRECESSION TION: NER WITH VARIABLE GRAVITY GRADIENT :!:. 10 IN SPIN MOUNTING ANGLE

t 30 IN GRAVITY- DIGITAL SUN SENSOR GRADIENT CONTROL ATTITUDE CONTROL: ! 100 PITCH, ROLL t 2()0 YAW

MOMENTUM WHEELS TWO MOMENTUI!f1 ATTITUDE DETEAMINA- TWO IR SENSORS STUDY THE OZONE wb = 1 REV/ORBIT WHEELS TION: FIVE DIGITAL SUN LAYER

rHREE MAGNETIC t 0.50 PITCH SENSORS COILS t 0.70 ROLL MAGNETOMETERS

t 20 YAW ATTITUDE CONTROL: ! 10 IN PITCH, ROLL t 20 YAW

DUAL SPIN MAGNETIC COILS FOR t ,0 N-SlIT STAR SENSOR GAMMA·RAY ~ .. 0.5 TO 10 /SEC SPIN AND PRECESSION DIGITAL SUN SENSOR ASTRONOMY

ww • 2.000 MAGNETOMETERS

794 SPACECRAFT A TIITUDE SYSTEMS Ap. I

ORBIT PARAMETERS

ACRONYM NAME PRINCIPAL MASS LAUNCH DATE tNeLlNA· PERIGEE' APOGEE' COUNTRY {KG) AND VEHICLE PEFUOD

TION HEIGHT HEIGHT (MINUTES) toeG' (KM) (KM'

SAS-3 SMALL ASTRONOMY USA 189 MAY 7,1975. 2 .• 503 511 94.8 SATELLlTE·3 BY SCOUT

SEASAP USA 2.315 JUNE 1978, 108 790 790 100 BYATLA5-AGENA

SIRIO ITALIAN INDUSTRIAL ITALY 190 AUG, 25. 1917. 0.3 35,800 3S,BOO 1.436 OPERATIONS RE· BY DELTA GEOSYNrRONOUS SEARCH SATELLITE

SKYNET U.K. 129 NOV. 22, 1969, 2.4 34,100 36,680 1431.0 (UK-I' BY DELTA GEOSYNCHRONOUS

SM·3 SAN MARCO-J ITALY 362 APRIL 24, 1971, 3 136 449 93.B BYSCQUT

SM8-1.-2 SYNCHRONOUS USA 243 MAY 17,1974: 1B 35,785 35,788 1,436.1 METEOROLOGICAL FEB. 6. 1975, 0.4 35.482 36;103 1,436.5 SATELLITE." ·2 BY DELTA GEOSYNCHRONOUS

SMM* SOLAR MAXIMUM USA 2,087 1979. BY 28.5 -560 -560 -96 MISSION (FIRST DELTA OR 33 MUL TI-MISSION SATELLITE)

5SS-1 SMALL SCIENTIFIC USA 52 NOV. 15, 1971, 3 222 28,876 517 SATELlITE-1 BY5COUT

ST SPACE USA -1983, BY 2"" 500 500 95 TELESCOPE SHUTTLE

SYMPHONIE FRANCEI 221 DEC. 19, 1974, 0.5 35.017 35.852 1,418 GERMANY BY DELTA GEOSYNCHRONOUS

TIROS·IX USA 135 JAN. 22, 1965, 96.4 800 3,000 11. BY DELTA

1pERIGEE AND APOGEE HEIGHT ARE MEASURED fROM THE SURFACE OF THE EARTH.

Ap. I SPACECRAFT ATTITUDE SYSTEMS 795

STABILIZATION ATTITUDE TECHNIQUE POINTING ACCURACY ATTITUDe SENSORS MISSION OBJECTIVES

(SPIN RATE IN RPM) CONTROL HARDWARE

DUAL SPIN MAGNETIC COILS FOR ," TWO STAR CAMERAS X-RAY ASTRONOMY wb .. 0-5 REV/ORBIT SPIN AND PRECESSION N-5UT STAR SENSOR

IR HORIZON SCANNER SPINNING AND NONSPINNING SUN SENSOR MAGNETOMETER

DUAL SPIN MOMENTUM WHEEL ATTITUDE OETERMINA.· TWO IR HORIZON SENSORS OCEAN ~ .. 1 REV/ORBIT REACTION WHEEL TION: FOUR DIGITAL SUN PHYSICS

ww '" 2300 THREE MAGNETIC t: 0.20 IN PITCH. ROI,..L. SENSORS COILS AND YAW MAGNETOMETERS

ATTITUDE CONTROL: t: 0.50 IN PITCH. ROLL, AND YAW

SI!IN HYDRAllNE JETS , ,0 IR SLIT HORIZON SENSOR SUPER·HIGH·FREQUENCY wb,",90 IR HORIZON TELESCOPE COMMUNICATIONS

PLANAR AND V BEAM 12-18 GIGAHERTZ SUN SENSORS

SPIN GAS JETS EARTH HORIZON SENSOR COMMUNICATIONS DIGITAL SUN SENSOR

SPIN MAGNETIC COIL ,,0 SUN SENSOR OBTAIN DATA ON AT· MAGNETOMETER TRIAD MOSPHERIC DENSITY AND

MOLECULAR TEMPERA· TURE; DETERMINE NITROGEN CONCENTRA· TlON

SPIN HYDRAZINE JETS i: 1.00 ITRANSFER ORBIT) FIVE IR HORIZON SCAN· METEOROLOGY wb =90 !. 0.10 lMISSION ORBIT) NERS

TWO OVAL·SlIT SUN SENSORS

MOMENTUM WHEE LS FOUR MOMENTUM i: 0.1 0 ROLL THREE TWO·AXIS SOLAR ASTROPHYSICS ONBOARD COM· WHEELS ABOUT SUNLlNE; GYROSCOPES PUTER SIX MAGNETIC COILS !. 5ARC-$EC IN REDUNDANT FINE

PITCH AND YAW POINTING SUN SENSORS REDUNDANT TRIAXIAL MAGNETOMETERS TWO FIXED-HEAD STAR TRACKERS COARSE SUN SENSOR

SPIN MAGNETIC COIL AC- , ,0 STAR SCANNER STUDY MAGNETOSPHERE wb = 7 TIVATED AT PERIGEE OPTICAL HORIZON SENSOR

ONLY DIGITAL SUN SENSOR

ONBOARO COM- MOMENTUM WHEELS POINTING ACCURACY: TWO STAR TRACKERS MAJOR ASTRONOMICAL PUTER USING MAGNETIC TORQUING :to.1 ARC..sEC SUN SENSORS TI;.LESCOPE FINE ERROR STABILITY: FINE I[:RROR SENSOR SENSOR :t 0.01 ARC..sEC

MOMENTUM WHEELS MOMENTUM WHEELS :t 10 EARTH HORIZON SENSOR EXPERIMENTAL COM· GAS JETS DIGITAL SUN SENSOR MUNICATIONS SATELLITE

SPIN MAGNETIC COl LS , ,0 CONICAL SCAN METEOROLOGICAL HORIZON SENSORS


1.2 Spacecraft Listed by Attitude Determination Accuracy Requirements

.;;; l' .;;; 0.1° .;;; 0.25° .;;; 0.5°

ANS ATS-6 AEROS-I,2 HCMM* HEAO-B* CTS (mission) FLTSATCOM IMP-8 IUE GOES-I (mission) ISEE-C* LANDSAT-D* HEAO-I,C* SEASAT* MAGSAT* LANDSAT-I,2,C* OAO-3 LES-7 SAS-3 OAO-2 SMM* OSO-7,8 ST* SMS-I,-2 (mission)

.;;; 1° .;;; 1.5° ';;;ZO

AE-3,5 RAE-2 GEOS-3 ARIEL-III ARYABHATA SAS-2 ISS LES-5 ATS-3 SIRIO OGO-I CTS (acquisition) SM-3 DE-A*,B* SMS-I,-2 (transfer) GOES-I (transfer) SSS-I IMP-6,7 SYMPHONIE ISEE-I,B TIROS-IX NIMBUS-5,6

1.3 Spacecraft Listed by Type of Control Hardware

Magnetic

Jets

AE-3,-5, AEROS-I,2, ANS, AZUR-I (passive), DE-B*, ESRO-IV, GEOS-3 (acquisition), HCMM*, HEAO-B*, ISIS-I,ll, ITOS-8,I*,J*, LANDSAT-D*, LES-5, MAGSAT, OSO-7,8, SAS-2,3, SM-3, SSS-I, ST*, TIROS-IX

AE-3,5, ARYABHATA, ATS-3,6, CTS, DE-A*,-B*, FLTSATCOM, GOES-I, HEAO-I,B*,C*, HEOS-l, IMP-6,7,8, ISEE-I,B,C*, IUE, LES-7(plasma), OGO-I, OSO-7,8, RAE-2 (acquisition), SIRIO, SKYNET (U.K.I,2), SYMPHONIE

Momentum Wheel AE-3,5, ANS, ATS-6, GEOS-3 (mission), FLTSATCOM, HCMM*, HEAO-B*, IUE, LANDSAT-D*, MAGSAT*, NIMBUS-5,6, OAO-2,-3, OGO-I, SAGE*, SEASAT*, SMM*, ST*, SYMPHONIE

Gimbaled Flywheel LES-7

Reaction Boom GEOS-3

Ap. I SPACECRAFT ATIITUDE SYSTEMS

1.4 Spacecraft Listed by Type of Attitude Sensors

Star Sensors Star Scanner

ANS, ATS-3, OSO-7,8, SAS-2,3, SSS-I Fixed-Head Star Trackers

HEAO-I,B*,C*, LANDSAT-D*, MAGSAT*, SAS-3, SMM*, ST* Gimbaled Star Trackers

ATS-6: OAO-2,3

Horizon Scanners Optical

DODGE, IMP-6,7,8, ISEE-l,C*, IUE, RAE-2, SSS-l Infrared

797

AE-3,-5, AEROS-l,-2, ATS-6, CTS, ESRO-IV, FLTSATCOM, ISS, GOES-I, HCMM*, ITOS-8,1*,J*, LES-7, MAGSAT*, SAGE*, SAS-3, SEASAT*, SMS-l,-2, SIRIO

Sun Sensors Analog Sun Sensor

AEROS-l,2, ATS-3,6, GOES-I, IUE, SIRIO, SMM*, SMS-I,-2 One-Axis Digital Sun Sensors

AE-3,5, AEROS-I, ARYABHATA, ATS-6, CTS, ESRO-IV, IMP-8,9,1O, ISEE-l,C*, ISS, ITOS-8,I*,J*, IUE, RAE-2, SAS-2,3, SKYNET (U.K.I,2), SM-3, SSS-I, SYMPHONIE

Two-Axis Digital Sun Sensors ATS-6, CTS, GEOS-3, HCMM*, HEAO-l,B*,C*, IUE, MAGSAT*, RAE-2, SAGE*, SAS-3, SEASAT*, SMM*, ST*

Magnetometers AE-3-5, AEROS-I, ANS, ARYABHATA, GEOS-3, HCMM*, ISS, MAGSAT*, OAO-2,-3, OSO-7,8, SAGE*, SAS-2,3, SEASAT*, SM-3, SMM*

Gyroscopes ATS-6, HEAO-l,B*,C*, IUE, LANDSAT-D*, MAGSAT*, OAO-2,-3, SMM*

References

l. Joseph, M. and M. Plett, Sensor Standardization Study Task Report, Compo Sc. Corp., 3000-19300-01TN, May 1974.

2. TRW Systems Group, Public Relations Staff, TRW Space Log, Redondo Beach, CA (Annual Report).

APPENDIXJ

TIME MEASUREMENT SYSTEMS

Conrad R. Sturch

International Atomic Time, TAl, which is provided by atomic clocks, is the basis for the two time systems used in spacecraft time measurements. Ephemeris Time, ET, which is used in the preparation of ephemerides, is a uniform or "smoothly flowing" time and is related to TAl by

ET= TAI+32.18 sec

In contrast to ET, Coordinated Universal Time, UTC, uses the TAl second as the fundamental unit, but introduces I-second steps occasionally to make UTC follow the nonuniform rotation of the Earth. UTC is necessary for terrestrial navigation and surveying for which the rotational position of the Earth at a given instant is critical. It is this tIme which is broadcast internationally and is used for tagging spacecraft data and for all civil timekeeping. Finally, sidereal time. is a direct measure of the rotational orientation of the Earth relative to the "fixed" stars and, therefore, is used to estimate the position of a spacecraft relative to points on the Earth's surface. The characteristics of the various time systems are summarized in Table J-1.

Any periodic phenomenon may be used as a measure of time. The motion of the Earth, the Moon, and the Sun relative to the fixed statshas traditionally been used for this purpose. However, the need for increasingly accurate time measurements has resulted in the development of several .alternative time systems. The requirement for high accuracy comes from the cumulative effect of timing errors. An accuracy of I sec/day (I part in 105) would appear satisfactory for most scientific or technical purposes. However, an error of this magnitude in an ephemeris of the E~rth causes an error of 10,800 km in the position of the Earth

Table J-1. Time Systems

KIND OF DEFINED BY FUNDAMENTAL REGULARITY USE TIM!; UNIT

SIDEREAL EARTH'S ROTATION RELATIVE SIDEREAL DAY, IFtAEGUlAR ASTRONOMICAL OBSERVA-TO STARS 1 ROTATION OF EARTH TIONS; DETERMINING UT

AND ROTATIONAL ORIENTA-TION OF EARTH

SOLAR:

APPARENT EARTH'S ROTATION RELATIVE SUCCESSIVE TRANSITS IRREGULAR AND SUNDIALS TO TRUE SUN OF SUN ANNUAL VAAI·A-

TlONS

MEAN EARTH'S ROTATION RELATIVE MEAN SOLAR DAY IRREGULAR -TO FICTITIOUS MEAN SUN

UNIVERSAL UTO OBSERVED ur MEAN SOLAR DAY IRREGULAR STUDY OF EARTH'S WANDERING

POLE

un CORRECTED UTa MEAN SOLAR DAY IRREGULAR SHOWS SEASONAL VARIATION Of EARTH'S ROTATION

un CORRECTED un MEAN SOLAA DAY IRREGULAR BASIC ROTATION OF EAATH

iJTC- GMT ATOMIC SECOND AND LEAP MEAN SOLAA DAY UNIFORM EXCEPT CIVIL TIMEKEEPING; TEA· ·z SECONDS TO APPROXIMATE FOR LEAP SEC- RESTRIAL NAVIGATION AND

UT1 ONDS SURVEYING; BROADCAST TIME SIGNALS

EPHEMERIS, ET FRACTION OF TROPICAL YEAR EPHEMERIS SECOND UNIFORM EPHEMERIDES 1900

ATOMIC, TAl FREQUENCY OF 133 Ce RADIA· ATOMIC SECOND = UNIFORM BASIS Of ET AND UTC TION EPHEMERIS SECOND

Ap.J TIME MEASUREMENT SYSTEMS 799

after only I year, several orders of magnitude worse than what is acceptable for many unsophisticated measurements. Thus, the generation of accurate ephemerides requires a precise time measurement system.

The diurnal motion of celestial objects is the most obvious timekeeper. Until the Middle Ages "seasonal hours," one-twelfth of daylight or nightime periods, was used. Of course, this unit varies both with the season and with the observer's latitude. A more uniform unit of time is the apparent solar day, defined as the interval between two successive passages of the Sun across the observer's meridian. As discussed below, this interval varies throughout the year due to variations in the Earth's orbital speed and the inclination of the ecliptic. The Earth's orbital motion does not affect the sidereal day, the interval between two succesive meridian passages of a fixed star. However, irregularities in the rotation of the Earth cause both periodic and secular variations in the lengths of the sidereal and solar days.

The annual motions of celestial objects provide a measurement of time which is independent of the irregular variations in the rotation of the Earth. The tropical year, upon which our calendar is based, is defined as the interval of time from one vernal equinox to the next. The ephemeris second is defined as 1/31556925.9747 of the tropical year for 1900. Because of the precession of the equinoxes (Section 2.2.2), the tropical year is about 20 minutes shorter than the orbital period of the Earth relative to the fixed stars. This latter period is known as the sidereal year. Due to secular variations in the orbit and rate of precession of the Earth, the lengths of both types of year (in units of se' the ephemeris second) vary to first order according to the relations:

Tropical year = 31556925.9747 - .530 T

Sidereal year = 31558149.540+ .OIOT

where T is the time in units of Julian Centuries of 36525 days from 1900.0 [Newcomb, 1898].

The first satisfactory alternative to celestial observations for the measurement of time was the pendulum dock. The period of a pendulum is a function of the effective acceleration of gravity, which varies with geography and the position of the Sun and the Moon. The resonance frequency of quartz crystals has recently been employed in clocks; this frequency depends on the dimensions and cut of the crystal and its age, temperature, and ambient pressure. Atomic clocks are based on the frequency of microwave emission from certain atoms. An accuracy of 10- 14

(fractional standard deviation) may be achieved with atomic clocks; corresponding accuracies for quartz and pendulum clocks are 2 X 10- 13 and 10-6, respectively. For an extended discussion of time systems, see Woolard and Clemence [1966], the Explanatory Supplement to the Astronomical Ephemeris and the American Ephemeris and Nautical Almanac [H.M. Nautical Almanac Office, 1961] and Muller and Jappel [1977].

Solar Time. The celestial meridian is the great circle passing through the celestial poles and the observer's zenith. As shown in Fig. 1-1, the hour angle, HA, is the azimuthal orientation of an object measured westward from the celestial meridian. As the Earth rotates eastward, a celestial object appears to move westward and its HA increases with time. It takes 24 hours for an object to move completely around the celestial sphere or I hour to move 15 deg in HA; thus, I deg of HA corresponds to 4 minutes of time.

800 TIME MEASUREMENT SYSTEMS Ap.J

Fig. J-1. Definition of Hour Angle

The apparent solar time is equal to the local HA of the Sun, expressed in hours, plus 12 hours. Apparent solar time can be measured with a simple sundial constructed by driving a long nail perpendicularly through a flat piece of wood. If the nail is then pointed toward the celestial pole, the plane of the wood is parallel to the equatorial plane, and the shadow of the nail cast by the Sun onto the wood is a measure of the HA.

Due to the Earth's orbital motion, the Sun appears to move eastward along the ecliptic throughout the year. Because the Earth travels in an elliptical orbit, it moves faster when near the Sun and slower when it is more distant; therefore, the length of the solar day varies. Even if the Earth were in a circular orbit with a constant speed, the azimuthal component of the Sun's motion (parallel to the celestial equator) would vary due to the inclination of the ecliptic relative to the equator. To illustrate this, consider a satellite in a nearly polar orbit, as shown in Fig. J-2. The satellite changes azimuth slowly while near the equator and rapidly while near the poles. Although the variation in the length of the day due to the

Fig. J-2. Variation in Azimuthal Rate for a Satellite Moving Uniformly in its Orbit. A I.A2 ..... AS are azimuthal projections of the orbital points 1.2 .... 5 and are equally spaced in time.

Ap.J TIME MEASUREMENT SYSTEMS 801

eccentricity and inclination of the Earth's orbit is small, the cumulative variation reaches a maximum of 16 minutes in November.

To provide more uniform time than the real Sun, a fictitious mean Sun, which moves along the equator at a constant rate equal to the average annual rate of the Sun, has been introduced. Mean solar time is defined by the HA of the mean Sun. The difference between the mean and apparent solar times is called the equation of time.

Standard Time. Mean solar time is impractical for communication and transportation because it varies continuously with longitude. Therefore, the world has been divided into 24 time zones of approximately 15 deg each. Normally, these zones are centered on standard meridians which are multiples of 15 deg in longitude. The uniform time throughout each zone is referred to as Standard Time, and usually differs by an integral number of hours from the mean solar time at 0 deg longitude, or Universal Time, as discussed below. Table J-2 lists the standard meridiahs for time zones in the continental United States. The apparent solar time is converted to Standard Time by adding the equation of time for the date and subtracting the algebraic difference (expressed in units of time) between the observer's longitude and the standard meridian.

Table J-2. Standard Time

TIME ZONE STANDARD MERIDIAN UT MINUS STANDARD TIME UT MINUS DAYLIGHT TIME {DEG. EAST LONG.l (HOURS) (HOURS)

EASTERN 285 5 4

CENTRAL 270 6 5

MOUNTAIN 255 7 6

PACIFIC 240 8 7

Greenwich Mean Time, Universal Time. The O-deg longitude line is referred to as the Greenwich meridian because it is defined by the former site of the Royal Greenwich Observatory. Greenwich Mean Time, GMT, is the mean solar time at 0 deg longitude; that is, GMT is the HA of the mean Sun observed at Greenwich (called the GHA) in hours plus 12 hours, modulo 24. Greenwich Mean Time is also called Universal Time, UT, and, in spaceflight operations, Zulu, or Z.

Uncorrected UT or UTO (read"UT Zero") is found from observations of stars, as explained in the discussion of sidereal time below. UTO time as determined by different observatories is not the same, however, due to changes in the longitudes of the observatories caused by the wandering of the geographic pole. Therefore, UTO is corrected for this effect to give UTI, which is then a measure of the actual angular rotation of the Earth. The Earth's rotation is subject to periodic seasonal variations, apparently caused by changes in, for example, the amount of ice in the polar regions. When UTI is corrected by periodic terms representing these seasonal effects, the result is UT2. Even UT2 is not a uniform measure of time. Evidence from ancient eclipse records and other sources shows that the Earth's rate of rotation is slowing; also, unpredictable irregularities in the rotation rate are observed.

Before 1972, ihe broadcast time signals were kept within 0.1 sec of UT2. Since January 1, 1972, however, time services have broadcast Coordinated Universal

Time, UTe. A second of UTe is equal to a second of International Atomic Time, but UTe is kept within 0.90 sec of UTi by the introduction of i-sec steps, usually at the end of June and December.

Ephemeris Time. The irregularities in the Earth's rotation cannot be predicted; however, gravitational theories have been formulated for the orbital motions of the Earth, the Moon, and the planets. In particular, Simon Newcomb's Tables of the Sun, [1898], published at the end of the 19th Century, gives the position of the Sun for regular time intervals. These intervals define a uniform time called Ephemeris Time, ET. In theory, Ephemeris Time is determined from observations of the Sun. In practice, observations of the Moon are used because the Sun moves slowly and its position is difficult to observe. One method is to record the UT of a lunar occultation of a star; the tabulated value of ET for the observed lunar position, corrected for effects such as parallax, is noted and the difference

AT=ET- UT

is determined. A table of approximate A T values, both in the past and extrapolated into the future, is provided in The American Ephemeris and Nautical Almanac [U.S. Naval Observatory, 1973]. Ephemeris time at any instant is given by

ET=!:J.T+ UT

International Atomic Time. The cesium nuclide, 133 Ce, has a single outer electron with a spin vector that can be either parallel or antiparallel to that of the nucleus. The flip from one orientation to the other, a hyperfine transition, is accompanied by the absorption or emission of microwave radiation of a given frequency. In an atomic clock, the number of these transitions is maximized in a resonator by the introduction of microwave radiation from an oscillator tuned to the same frequency. The cycles of the oscillator are counted to give a unit of time. In 1967, the 13th General Conference on Weights and Measures established the Systeme I nternationale (SI) second as the duration of 9 192 631 770 periods of the radiation from the above transition in 133 Ceo This unit is the basis of International Atomic Time, TAl, and was chosen to make the SI second equal to the ephemeris second. The reference epoch for TAl is January 1, 1958, when <tOrnos TAl equaled Ohomos UT2. For most purposes, ephemeris time may be considered to be equal to TAl plus 32.18 sec, the value of !:J.T for January 1, 1958.

Sidereal Time. Sidereal time, ST, is based on the rotation of the Earth relative to the stars and is defined as the HA of the vernal equinox, 0('. The local sidereal time, LST, is defined as the local HA of 0(', LHA 0(', and the sidereal time at Greenwich, GST, is defined as the Greenwich HA of 0(', GHA 0('. Sidereal time may also be determined from the HA and right ascension, RA, of any star. The RA of a star is the azimuthal component of the star's position measured eastward from 0('

(see Section 2.2.2). From Fig. J-3 we see that

LST= LHA 0('= LHA * + RA *, modulo 24 (1-1)

where LHA* and RA* are the HA and RA (both converted to time) of the star. In the example in Fig. J-3, LHA * is 135 deg or 9 hours, RA * is 90 deg or 6 hours, and the LST is 15 hours. Similarly,

GST= GHA 0('= GHA * + RA*, modulo 24 (J-2)

Ap.J. TIME MEASUREMENT SYSTEMS S03

where GHA * is the GHA of the star (converted to time). In Fig. J-3, GHA * is 45 deg or 3 hours; thus, GST is 9 hours. Note that the sidereal time at Greenwich is equal to the right ascension of the Greenwich meridian. The difference between LST

GHAT

OBSERVER

LOCAL MERIDIAN

Fig. J-3. Sidereal Time. (View looking down on the Earth's North Pole.)

and GST (6 hours in this example) corresponds to the observer's East longitude (90 deg in this example). In general,

LST=GST+EL/15 (J-3)

where EL is the observer's East longitude in degrees. From the definition of mean solar time, it follows that GMT or UT equals the GHA of the fictitious mean Sun plus 12 hours, or

UT= 12 hours + GST- Ru

where ~ is the right ascension of the mean Sun. For a given UT of any calendar date,

GST= Ru -12 hours + UT

= 6h3Sffi45" .836 + 8640184".542 T + 0" .0929 T2 + UT (J-4)

where T is the number of Julian centuries of 36,525 days which have elapseq since noon (GMT) on January 0, 1900 [Newcomb, IS98]. The corresponding equation for GST expressed in degrees is

GST=99°.6910+36000°.7689 T+O°.OOO4 T2+ UT (J-5)

where UT is in degrees and T is in Julian centuries. Julian dates, or J D (Section 1.4), are convenient for determining T in Eqs. (J-4) and (J-5). The J D for Greenwich mean noon on January 0, 1900 (i.e., January 0.5, 1900), is 2415020.0. JD's for any date in the last quarter of this century may be obtained by adding the day number of the year to the JD for January 0.0 UT of that year listed in Table J-3. For example, to find the GST for 3h UT, July 4, 1976:

804 TIME MEASUREMENT SYSTEMS

Day number of July 4.125 (=3h UT July 4),1976 +JD for January 0.0,1976 =JD for July 4.125, 1976 -JD for January 0.5, 1900 = Tin days + 36,525 = T in Julian centuries 8640184.542 T + 0.0929 T2

+ first term Eq. (J-4) + UT GST

Ap.J

186.125 + 2442777.500 = 2442 963.625 - 2415020.000

27943.625 0.765054757

6610 214.340 sec = 76d12hlOm14~340

6 38 45.836 3 0 0.000

21h49m ~176

Due primarily to the varying distances of the Sun and the Moon, a small amplitude oscillation, known as astronomical nutation, is superimposed on the precession of the equinoxes. Sidereal time corrected for this effect is called mean sidereal time. GST in Eqs. (J-4) and (1-5) is mean sidereal time. The maximum difference between mean and apparent sidereal time is only about I sec.

Table J-3. Julian Date at the Beginning of Each Year From 1975 to 2000

YEAR JD FOR YEAR JD FOR YEAR JD FOR JAN 0.0 UT JAN 0.0 UT JAN 0.0 UT

2400000+ 2400000+ 2400000+

1975 42412.5 1984 45699.5 1993 48987.5 1976 42777.5 1985 46065.5 1994 49352.5 1977 43143.5 1986 46430.5 1995 49717.5 1978 43508.5 1987 46795.5 1996 50082.5 1979 43873.5 1988 47160.5 1997 50448.5 1980 44238.5 1989 47526.5 1998 50813.5 1981 446045 1990 47891.5 1999 51 178.5 1982 44969.5 1991 48256.5 2000 51543.5 1983 45334.5 1992 48621.5

Because of the orbital motion of the Earth, a solar day is longer than a sidereal day. As illustrated in Fig. J-4, the fixed stars are sufficiently far away that lines connecting one of them to the Earth are essentially parallel. Because the Earth's orbital period is approximately 360 days, angle A is approximately I deg. A sidereal day is defined as one complete rotation of the Earth, 360 deg, relative to the stars. The Earth has to rotate 360+A deg to complete a solar day. The ratio of the m-an solar day to the mean sidereal day is }.00273 79093; the mean sidereal day equals 23 hours, 56 min, 4.09054 sec of mean solar time and the mean solar day equals 24 hours, 3 min, 55.55536 sec of mean sidereal time [U.S. Naval Observatory, 1973]. Note that the "76 days" in the above example indicates the excess number of sidereal days, one for each year, that had occurred since the beginning of the century.

Sidereal time and mean solar time are affected proportionally by variations in the Earth's rotation. Although the irregular fluctuations in the Earth's rotation cannot be predicted, the general deceleration can be seen in Fig. J-5. The lengths of

Ap. J TIME MEASUREMENT SYSTEMS 805

Fig. J-4. Orbital Motion of the Earth for I Day (Exaggerated) as Viewed From North Ecliptic Pole

the two types of day in se are given approximately by: Sidereal day = 86164.09055 +0.0015 T

Mean solar day = 86400+0.0015T where T is in Julian centuries from 1900.0 [Allen, 1973). These terms are the dashed line in Fig. J-5.

1600 1700 1800

YEAR

1900 2000

Fig. J-5. Excess Length of the Day Compared With the Day Near 1900. Note the very irregular fluctuations about the mean slope of 1.5 ms/century [Morrison, 1973J.

Using Sidereal Time to Compute the Longitude of the Subsatellite Point. To determine the direction of geographic points on the Earth as seen from a spacecraft, it is necessary to know both the spacecraft ephemeris and the longitude of the subsatellite point. For any UT, Eg. (J-4) can be used to determine the Greenwich sidereal time, CST, which in turn can be used to determine the East longitude, ELspc' of the subsatellite point for any satellite for which the right ascension of its position in geocentric coordinates is known. From Eg. (J-3), we have

ELspc = RAspc - CST (in degrees)

where RAspc is the right ascension in degrees of the spacecraft at time CST. Because UTe is accurate to about 1 sec, the accuracy of the resulting longitude will be about 0.005 deg if the spacecraft ephemeris is known precisely.

References

1. Allen, C. W., Astrophysical Quantities, Third Edition. London: The Athlone Press, 1973.


2. H. M. Nautical Almanac Office, Explanatory Supplement to the Astronomical Ephemeris and the American Ephemeris and Nautical Almanac. London: Her Majesty's Stationery Office, 1961.

3. Morrison, L. V., "Rotation of the Earth from AD 1663-1972 and the Constancy of G," Nature, Vol. 241, p. 519-520, 1973.

4. Miiller, Edith A. and Arndst Jappel, Editors, International Astronomical Union, Proceedings of the Sixteenth General Assembly, Grenoble, 1976. Dordrecht, Holland: D. Reidel Publishing Co., 1977.

5. Newcomb, Simon, Astronomical Papers Prepared for the Use of the American Ephemeris and Nautical Almanac, Bureau of Equipment, U.S. Department of the Navy, Washington, DC, 1898.

6. U.S. Naval Observatory, The American Ephemeris and Nautical Almanac. Washington, DC: U.S. G. P.O., 1973.

7. Woolard, Edgar W., and Gerald M. Clemence, Spherical Astronomy. New York: Academic Press, 1966.

APPENDlXK

METRIC CONVERSION F'ACTORS

The metric system of units, officially known as the International System of Units, or SI, is used throughout this book, with the single exception that angular measurements are usually expressed in degrees rather than the SI unit of radians. By international agreement, the fundamental SI units of length, mass, and time are defined as follows (see, for example, NBS Special Publication 330 [NBS, 1974]):

The metre is the length equal to I 650 763.73 wavelengths in vacuum of the radiation corresponding to the transition between the levels 2PlO and 5d5 of the krypton-86 atom. The kilogram is the mass of the international prototype of the kilogram (a specific platinum-iridium cylinder stored at Sevres, France). The second is the duration of 9 192 631 770 periods of the radiation corresponding to the transition between two hyperfine levels of the ground state of the cesium-133 atom.

Additional base units in the SI system are the ampere for electric current, the kelvin for thermodynamic temperature, the mole for amount of substance, and the candela for luminous intensity. Mechtly [1973] provides an excellent summary of SI units for scientific and technical use.

The names of multiples and submultiples of SI units are formed by application of the following prefixes:

Factor by which unit is multiplied Prefix Symbol

1012 tera T 109 giga G 106 mega M 103 kilo k 102 hecto h 10 deka da 10- 1 deci d 10- 2 centi c 10- 3 milli m 10- 6 micro p. 10- 9 nano n 10- 12 pico p 10- 15 femto f 10- 18 atto a

For each quantity listed below, the SI unit and its abbreviation are given in parentheses. For convenience in computer use, most conversion factors are given to the greatest available accuracy. Note that some conversions are exact definitions and some (speed of light, astronomical unit) depend on the value of physical constants. All notes are on the last page of the list.

808 METRIC CONVERSION FACTORS Ap.K

To convert from To Multiply by Notes

Mass (kilogram, kg)

Atomic unit (electron) kg 9.109 6x 10- 31 (1) Atomic mass unit, amu kg 1.66053 X 10-27 (1) Ounce mass (avoirdupois) kg 2.834952312 5x 10- 2 E Pound mass, Ibm (avoirdupois) kg 4.535 923 7 X 10- I E Slug kg 1.459 390 294 X 101

Short ton (2000 pound) kg 9.071 847 4x IOZ E Metric ton kg LOx 103 E Solar mass kg 1.989 X 1030 (1)

Length (metre, m)

Angstrom m LOx 10- 10 E Micron m LOx 10- 6 E Mil (10- 3 inch) m 2.54x 10-5 E Inch m 2.54 X 10- 2 E Foot m 3.048 X 10- I E Statute mile (U.S.) m 1.609 344 X I ij3 E Nautical mile (U.S.) m 1.852 X 103 E Earth equatorial radius m 6.378 140x 106 (3) Vanguard unit m 6.378I66x106 (4) Solar radius m 6.959 9x 108 (1) Astronomical unit, AU m 1.495 978 70 X 1011 (5) Light year (tropical year) m 9.460 530 X 1015 (1)

Parsec (distance for which stellar parallax is I arc-sec.) m 3.085 678 X 1016 (1)

Time (second, s) (7)

Sidereal day, d* (ref. = cy» s 8.6I6409I8x104 (1)*

= 23h 56m 4.0918s (1)*

Ephemeris day, de S 8.64x 104 E Ephemeris day, de d* 1.002 737 89· (1)*

Vanguard unit s 8.068 1242 X 102 (4)

Keplerian period of a satellite in mm 1.658 669 x 1O- 4 x a3/ 2 (6) low-Earth orbit a inkm

Keplerian period of a satellite of de 3.652569 X IOZ X a3/ 2 (6) the Sun a in AU

Tropical year (ref. = CY» S 3.155692555 I X 107 (7)*

Tropical year (ref. =cy» de 3.652421938 8x IOZ (7)*

Sidereal year (ref. = fixed stars) se 3.15) 314954 8x 107 (7)*

Sidereal year (ref. = fixed stars) de 3.652 563 605 Ix 102 (7)*

• Epoch 1980.

Ap.K METRIC CONVERSION FACTORS 809


Calendar year (365 days) se 3.153 6x 107 E

Julian century d 3.652 5x 104 E Gregorian calendar century d 3.652425 X 104 E

Velocity (metre/second, m/s)

Foot/minute, ft/min m/s 5.08X 10- 3 E Inch/second, ips m/s 2.54x 10- 2 E

Kilometre/hour, km/hr m/s (3.6) - I = 0.277777 ... E

Foot/second, fps or ft/sec m/s 3.048 X 10- 1 E

Miles/hour, mph m/s 4.4704 X 10- 1 E Knot m/s 5.144 444 444 X 10- 1

Miles/minute m/s 2.682 24 X 101 E Miles/second m/s 1.609 344 X 103 E Astronomical unit/sidereal year m/s 4.740388 554 X 103

Vanguard unit m/s 7.905 389 X 103 (4) Velocity of light, c m/s 2.997 925 X 108 (I)

Acceleration (metre/second2, m/s2)

Gal (galileo) m/s2 LOx 10-2 E Inch/second2 m/s2 2.54 ><.10- 2 E Foot/second2 m/s2 3.048 X 10- 1 E Free fall (standard), g m/s2 9.80665 E Vanguard unit m/s2 9.798299 (4)

Force (Newton == kilogram· metre / second2, N == kg· m/ S2)

Dyne N LOx 10- 5 E Poundal N 1.38254954376 X 10- 1 E Ounce force (avoirdupois) N 2.780 1385 X 10- 1 (8) Pound force (avoirdupois),lbf==

slug· foot/ second2 N 4.448221 6152605 E

Pressure (Pascal == N ewton/ metre2 == kilogram· metre - I • second - 2,

Pa==N/m2 ==kg'm- I 's- 2)

Dyne / centimetre2 Pa LOx 10- 1 E Ibf/foot2 Pa 4.7880258 (8) Torr (00 C) Pa 1.333 22 X 102 (8) Centimetre of Mercury (OU C) Pa 1.33322 X IW (8) Inch of Mercury (320 F) Pa 3.386 389 X 103 (8) Ibf/inch2, psi Pa 6.894 757 2 X 103 (8) Bar Pa l.Ox 105 E Atmosphere Pa 1.013 25 X 105 E



Energy or Torque (Joule: Newton· metre: kilogram· metre2/second2, (2) J:N ·m:kg·m2/s2)

Electron volt, eV J 1.602 191 7x 10- 19 (8) Mass-energy of I amu J 1.49241 X 10- 10 (1) Erg: gram· cen timetres2 / second2

= pole· centimetre· oersted J LOx 10- 7

Ounce inch J 7.061 551 6x1O-3 Foot poundal J 4.214011 Ox 10-2

Foot Ibf = slug· foot2 / second2 J 1.3558179 Calorie (mean) J 4.190 02 British thermal unit, BTU (mean) J 1.055 87 x 103

Kilocalorie (mean) J 4.190 02 X I Q3

Kilowatt hour J 3.6 X 106

Ton equivalent of TNT J 420x 109

Power (Watt: Joule/second: kilogram· metre2/ second3, W: J / s: kg· m2/ S3)

E (8) (8) (8) (8)

(8)

(8)

E (8)

Foot Ibf/second W 1.3558179 (8) Horsepower (550 ft Ibf/s) W 7.4569987 X IOZ (8) Horsepower (electrical) W 7.46 X IOZ E Solarluminosity W 3.826 X 1026 (I)

Moment of Inertia (kilogram· metre2, kg· m2)

Gram·centimetre2 kg·m2 LOx 10-7 E Ibm·inch2 kg·m2 2.926 397x 10-4

Ibm·foot2 kg·m2 4.214011 X 10- 2

Slug·inch2 kg·m2 9.415 402 x 10-:-3

Inch ·lbf· seconds2 kg· m2 1.129 848 X 10- 1

Slug· foot2 = ft ·lbf· seconds2 kg· m2 1.355 818

Angular Measure (radian, rad). Degree (abbreviated deg) is the basic unit used in this book.

Degree

Radian

Solid Angle (steradian, sr)

Degree2, deg2

Steradian

rad

deg

sr

'7T/I80 E ~ 1.745 329 251 994 329 577 X 10 - 2

IW/'7T E ~5. 729 577 951 308 232 088 x 101

('7T/I80f E ~3.046 174 197867085 993 x 10-4

(I80/'7T)2 E ~3.282 806 350 OIl 743 794 x loJ

Ap.K METRIC CONVERSION FACTORS


Angular Velocity (radian/se~ond, rad/s). Degrees/second is the basic unit used in this book.

Degrees/second, deg/s

Revolutions/minute, rpm

Revolutions/second, rev /s

Revolutions/minute, rpm Radians/second, rad/s

Revolutions/second, rev /s

rad/s

rad/s

rad/s

deg/s deg/s

deg/s

77/180 E R:l1.745 329 251 994 329 577 X 10 - 2

77 130 E R:l1.047 197551 196597 746 x 10- 1

277 E R:l6.283 185307 179586477

6n E 180/77 E R:l5.729 577 951 308 232 088 x 101

3.6x102 E

Angular Momentum (kilogram· metre2 / second, kg· m2 / s) Gram· centimetre2 / second,

g·cm2/s kg·m2/s LOx 10- 7 E Ibm·inch2/second kg·m2 Is 2.926 397 x 10-4

Slug·inch2/second kg·m2/s 9.415 402 x 10-3

lbm·footl/second kg·m2/s 4.214 011 x 10-2

Inch· lbf· second kg·m2/s 1.129 848 x 10- 1

Slug·foot2/second =ft·lbf· second kg· m2 Is 1.355 818

811

Magnetic Flux (Weber == Volt· second == kilogram· metre2 • Ampere - I . second - 2, Wb == V . s ==kg·m2 • A -1' S-2)

Maxwell (EMU) Wb LOx 10- 8 E

B, Magnetic Induction (commonly called "magnetic field", (9) Telsa == Weber / metre2 == kilogram· Ampere - I . second- 2, T==Wb/m2=kg·A-I·s-~

Gamma (EMU) T LOx 10-9 E, (9) Gauss (EMU) T 1.0 X 10-4 E, (9)

H, Magnetic Field Strength (ampere tum/metre, Ajm) (9) Oersted (EMU) Aim (l 1477) X IQ3 E, (9)

R:l7.957747 154594766 788 x 101

Magnetic Moment (ampere·tum·metre2 ==Joule/Telsa, A·m2 ==J/T) Abampere'centimetre2 (EMU) A·m2 LOx 10- 3 E, (9) Ampere' centimetre2 (Practical) A· m2 1.0 X 10-4 E



Magnetic Dipole Moment (Weber' metre == kilogram· metre· Ampere - 1 • second - 2,

Wb.m==kg'm.A -1' S-2)

Pole· centimetre (EMU) Wb· m 4'7T X 10 - 10 E; (9) ~1.256 637 061435917 295 X 10- 9

Gauss·centimetre3 (Practical) Wb· m 1.0 X 10- 10 E

Temperature (Kelvin, K)

Celsius, C Fahrenheit, F Fahrenheit, F

Notes:

K K C

tK=tc +273.15 tK = (5 j9)(tF + 459.67) tc = (5 j9)(tF - 32.0)

E E E

E (Exact) indicates that the conversion given is exact by definition of the non-SI unit or that it is obtained from other exact conversions. (I) Values are those of Allen [1973]. (2) In common usage "Joule" is used for energy and "Newton-metre" for torque. (3) Value is that adopted by the International Astronomical Union in 1976

[Muller and Jappel, 1977]. Reported values of the equatorial radius of the Earth differ by about 20 m. It is therefore recommended that this unit not be used except in internal calculations, where it is given a single defined value.

(4) Values are those adopted in subroutine ROITAP, described in Section 20.3. Vanguard units should be avoided if possible because of differences in the definitions of the units involved. The Vanguard unit of length is equatorial radius of the Earth; the Vanguard unit of time is the time for an Earth satellite to move 1 radian if the semimajor axis is 1 Vanguard unit.

(5) Value is that adopted by the International Astronomical Union in 1976 [Muller and Jappel, 1977].

(6) Value is calculated from mass parameters adopted by the International Astronomical Union in 1976 [Muller and Jappel, 1977]. Actual period will differ due to various perturbation effects. (See Section 3.4.)

(7) For high-precision work, consult Appendix J on time measurement systems. The conversions for the length of the year are derived from values, given by Newcomb [1898], which define the unit of ephemeris time. The most convenient method for determining the time interval between events separated by several days or more is to use the Julian Date. See Section 1.4 and subroutine JD in Section 20.3 for a convenient algorithm for determining the Julian date and Appendix J for a table of Julian Dates.

(8) Values are those of Mechtly [1973]. (9) Care should be taken in transforming magnetic units, because the dimension

ality of magnetic quantities (B, H, etc.) depends on the system of units. Most of the conversions given here are between SI and EMU (electromagnetic). The following equations hold in both sets of units:

Ap.K

N=mXB=dXH B=p.H

METRIC CONVERSION FACTORS

m = I A for a current loop in a plane d= p.m

with the following definitions N=:torque B=:magnetic induction (commonly called "magnetic field") H=:magnetic field strength or magnetic intensity m=:magnetic moment I =:current in loop

813

A =:vector normal to the plane of the current loop (in the direction of the angular velocity vector of the current about the center of the loop) with magnitude equal to the area of the loop

d =:magnetic dipole moment p. =:magnetic permeability

The permeability of vacuum, P.o, has the following values, by definition: p.o=:1 (dimensionless) EMU P-o=:4'1T X 10- 7 Nj A2 SI

Therefore, in electromagnetic units in vacuum, magnetic induction and magnetic field strength are equivalent and the magnetic moment and magnetic dipole moment are equivalent. For practical purposes of magnetostatics, space is a vacuum but the spacecraft itself may have p. :j= p.o.

References


2. Mechtly, E. A., The International System of Units: Physical Constants and Conversion Factors, Second Revision, Washington, DC, NASA SP-7012, 1973.

3. MUller, Edith A. and Arndst Jappel, editors, International Astronomical Union, Proceedings of the Sixteenth General Assembly, Grenoble, 1976. Dordrecht, Holland: D. Reidel Publishing Co., 1977.

4. National Bureau of Standards, U.S. Department of Commerce, The International System of Units (SI) , NBS Special Publication 330 (1974 edition), 1974.

5. Newcomb, Simon, Astronomical Papers Prepared for the Use of the American Ephemeris and Nautical Almanac. Washington, DC: Bureau of Equipment, Navy Department, 1898.

APPENDIXL

SOLAR SYSTEM CONSTANTS

L.I Planets and Natural Satellites L.2 The Sun L.3 The Earth L.4 The Moon L.5 Potential Spacecraft Orbits

JamesR. Wertz

The mass, size, and gravitational parameters are those adopted by thelnternational Astronomical Union (IAU) in 1976 [Muller and Jappel, 1977]. The geocentric and geographical coordinate system conversions are based on the method adopted by the American Ephermeris and Nautical Almanac [H.M. Nautical Almanac Office, 1961] using the updated value for the Earth's flattening adopted by the IAU [Muller and Jappe!, 1977]. The properties of artificial satellites, both in orbit about solar system objects and in transfer orbits between objects, are calculated from the parameters given and are based on Keplerian orbits with no perturbative corrections. The properties of the Earth's upper atmosphere are from the 1972 COSPAR International Reference Atmosphere [1972], and the U.S. Standard Atmosphere [1976]. Additional constants are from Allen [1973], which is an excellent source of additional astronomical information. See Chapter 3 for definitions of orbital quantities and planetary magnitudes.

L.t Planets and Natural Satellites

Table L-l lists the orbital properties of the major planets. Because of orbital perturbations, the data here are not tabulated with the full precision normally used for ephemerides. If greater accuracy is needed, consult the American Ephemeris and Nautical Almanac for current osculating elements or Section 5.4 forepemerides for computer use. Quoted data are from Allen [1973].

Table L-2 lists the physical properties of the Moon and planets. Additional data on the Earth and Moon are given in Sections L.3 and L.4. Properties of the natural satellites of the planets are given in Table L-3.

Tab

le L

-I.

Pla

neta

ry O

rbits

SE

MIM

AJO

R A

XIS

S

IDE

RE

AL

PE

RIO

D

SY

NO

DIC

M

EA

N

ME

AN

PL

AN

ET

P

ER

IOD

Mg

~:~~

2 O

RB

ITA

L

EC

CE

NT

RIC

ITY

(106

KM

) (T

RO

PIC

AL

(D

AY

S)l

(D

AY

S)l

V

EL

OC

ITY

19

70

(AU

) Y

EA

RS

) (D

EG

) (K

MIS

EC

)

ME

RC

UR

Y

0.3

87

09

9

57

.9

0.2

40

85

8

7.9

69

11

5.88

4

.09

23

39

4

7.8

9

0.2

05

62

8

VE

NU

S

0.7

23

33

2

108.

2 0.

6152

1 22

4.70

1 58

3.92

1.

6021

31

35.0

3 0.

0067

B7

EA

RT

H

1.0

00

00

0

14

9.6

1

.00

00

4

36

5.2

56

0.

9B56

09

29.7

9 0

.01

67

22

MA

RS

1.

5236

91

22

7.9

1

.88

08

9

68

6.9

80

77

9.94

0

.52

40

33

24

.13

0.0

93

37

7

JUP

ITE

R

5.2

02

80

3

77

8.3

1

1.8

62

23

4

33

2.5

89

39

8.88

0.

0830

91

13.0

6 0

.04

84

5

SA

TU

RN

9

.53

88

4

14

27

.0

29

.45

77

10

759.

22

37

8.0

9

0.0

33

46

0

9.6

4

0.0

55

65

UR

AN

US

1

9.1

81

9

28

69

.6

84

.01

39

3

06

85

.4

36

9.6

6

0.0

11

73

2

6.81

0

.04

72

4

NE

PT

UN

E

30

.05

78

4

49

6.6

1

64

.79

3

60

18

9

36

7.4

9

0.0

05

9B

l 5

.43

0

.00

85

8

PL

UT

O

39

.44

5

90

0

24

7.7

9

04

65

36

6.73

0

.00

39

79

4

.74

0

.25

0

ME

AN

LO

NG

ITU

DE

P

LA

NE

TA

RY

D

AT

E O

F

LO

NG

ITU

DE

DF

A'

Z:f

INC

LIN

AT

ION

T

HE

AS

CE

ND

ING

A

n

OF

PE

RIH

EL

lON

,3

LO

NG

ITU

DE

, L

. P

ER

IHE

LIO

N

PL

AN

ET

T

O E

CL

IPT

IC.

i N

OD

E.

19

00

(D

EG

I 2

1,1

90

0

(DE

GI

JA

NU

AR

Y 0

.5.

PA

SS

AG

E 1

97

0

19

70

(D

EG

) C

EN

TU

RY

) C

EN

TU

RY

) 19

70

n (D

EG

) (D

EG

) ID

EG

I O

R E

AR

LIE

R

ME

RC

UR

Y

7.0

04

2

47

.14

58

+

1.1

85

3

75

.89

83

+

1.5

54

4

47

.98

25

D

EC

. 2

5.

1970

VE

NU

S

3.3

94

4

75

.77

97

+

0.8

99

7

130.

1527

+

1.3

91

7

26

5.4

14

4

MA

Y 2

1.

1970

EA

RT

H

--

-1

01

.21

97

+

1.7

16

7

99

.74

22

JA

N.

1,1

97

0

MA

RS

1

.85

00

4

8.7

86

3

+0.

7711

3

34

.21

83

+

1.8

40

6

12

.67

52

O

CT

. 2

1,1

96

9

JUP

ITE

R

1.3

04

7

99

.44

16

+

1.0

10

8

12

.72

08

+

1.6

10

6

20

3.4

19

7

SE

PT.

26

. 19

63

SA

TU

RN

2

.48

94

1

12

.78

88

+

0.8

72

8

91

.09

72

+

1.9

58

3

43

,00

55

S

EP

T.

8.1

94

4

UR

AN

US

0

.77

30

7

3.4

78

3

+0

.49

89

17

1.53

+

1.5

1

84

.29

02

M

AY

20

. 1

96

6

NE

PT

UN

E

1.7

72

7

130.

6811

+

1.0

98

3

46

.67

+

1.4

2

38

.92

33

S

EP

T.

2.1

87

6

PL

UT

O

17.1

7 1

09

.73

-

223

-18

1.65

4 O

CT

. 2

4.1

74

1

lON

E D

AY

: 8

6.4

00

SI

SE

CO

ND

S.

2M

EA

N D

AIL

Y M

OT

ION

IS

TH

E M

EA

N C

HA

NG

E I

N T

RU

E A

NO

MA

LY

AS

VIE

WE

D F

RO

M T

HE

SU

N.

3 TH

E L

ON

GIT

UD

E O

F P

ER

IHE

LIO

N, '

(;5,

IS M

EA

SU

RE

D F

RO

M T

HE

VE

RN

AL

EQ

UIN

OX

; T

HA

T I

S. '

;;;"

n

+ w

, W

HE

RE

n

IS M

EA

SU

RE

D A

LO

NG

TH

E E

CL

IPT

IC

FR

OM

TH

E V

ER

NA

L E

QU

INO

X E

AS

TW

AR

D T

O iH

E A

SC

EN

DIN

G N

OD

E A

ND

W

IS T

HE

AR

GU

ME

NT

OF

PE

RIH

EL

ION

ME

AS

UR

ED

FR

OM

TH

E A

SC

EN

DIN

G

NO

DE

AL

ON

G T

HE

OR

81

T I

N T

HE

DIR

EC

TIO

N O

F T

HE

PL

AN

ET

'S M

OT

ION

TO

PE

RIH

EL

ION

.

4A

T E

PO

CH

SE

PT

EM

BE

R 2

3.0

,19

60

.

r'

'tI ~ Z

tTl Vl > z '=' z ~ ~ t"'

CI:l ~ tTl

t"' t: ..., tTl

CI:l

00

V\

816

PLANET

THE MOON

MERCURY

VENUS

EARTH

I, MARS

JUPITER

I SATURN

URANUS

PLUTO

SOLAR SYSTEM CONSTANTS

Table L-2. Physical Properties of the Moon and Planets

EQUATORIAL RADIUS

(km)

'738.2

2.439

6.052

6378.'40

3397.2

71.398

60.000

25.400

24.300

2,500

0.00054

0.0

0.0

0.00335281

0.009

0.063

0.098

0.06

0.021

PLANETARY GRAVITATIONAL CONSTANT, GM 1

11012 m 3 /s2)

4.902786

22.03208

324.8588

403.5033

42.82829

126,712.0

37,934.0

5803.2

6871.3

40

MASS' 11024 kgl

0.073483

0.33022

4.8690

6.0477

0.64191

1,899.2

568.56

86.978

102.99

0.7

MEAN DENSITY (gm/cm3 )

3.341

5.4

5.2

5.518

3.95

1.34

0.70

1.58

2.30

INCLINATION OF EQUATOR

TO ORBIT IDEGI

668

< 28

- 23.44

23.98

3.08

26.73

97.92

28.80

L.l

MAJOR ATMOSPHERIC COMPONENTS tiN ORDER OF ABllNDANCEl

NONE

NONE

CO2 , N2 , O2 , H20

N2 , °2 , Ar, H20. CO2

CO2, Ar. CO. H20

H2, He, H20. CH4 , NH3

H2 , CH 4, NH35

H2,CH4

H2, CH 4

UNKNOWN L;EPTUNE

----~ ____ ~ ______ ~ ________ ~ ______ ~ __ ~L_ ____ _L __________ ~

VISUAL MEAN VISUAL SMALL SUB$OLAR DARK SIDE SIDEREAL.

A~~E~06 MAGNITUDE MAGNITUDE I~g~~~o ANGLE TEMPER- TEMPER- EQUATORIAL PLANET D!~~Au~~r7 OPPO~I~ION8 PHASE

A 8·V VARIATION. 11 ATURE ATURE ROTATION V 1',0) Vo A, (OK) 10 K) PERIOD

THE MOON 0.067 ~.23 -12.73 +0.91 ~.026 - '04 27.321661 DAYS

MERCURY 0.056 -0.36 _ 0.27 0.91 0.027 600 '00 59 DAYS

VENUS 0.72 ~-4.34 _ 4.22 7 0.79 0.013 240 240 244.3 DAYS'

EARTH 0.39 -3.9 - 0.2 - 295 280 23.93447 HOURS

MARS 0.16 -1.51 - 2.02 1.37 0.016 250 - 74.62294 HOURS

JUPITER 0.70 -9.25 - 2.6 0.8 0.014 '20 9 .. 8417 HOURS3

SATURN 0.75 _9.08 ~ 0.79 '.0 0.044 90 - '0.23 HOURS4

URANUS 0.90 -7.15 + 5.5 0.55 0.001 65 - 10.82 HOURS

NEPTUNE 0.82 ·-6.90 + 7.9 0.45 0.001 50 - '5.80 HOURS

PLUTO 0.145 -1.0 +14.9 0.79 - - - 6.40AYS

1MASSES AND GRAVITATIONAL CONSTANTS INCLUDE PLANET PLUS ATMOSPHERE PLUS SATELLITES (i.e., "EARTH" VALUE IS EARTH PLUS MOON); ACTUAL MASSES ARE LIMITED BY THE KNOWLEDGE OF THE UNIVERSAL GRAVITATIONAL CONSTANT, G ITAKEN AS 6.672 X 10- 11 m3 kg·- 1 5-2). THEREFORE, FOR PRECISION WORK, THE PLANETARY GRAVITATIONAL CONSTANT ~ G X IMplanet + Msat ) SHOULD BE USED. VALUES GIVEN ARE DERIVED FROM THOSE ADOPTED BY THE IAU [MULLER AND SAPPEL, 19771.

2RETROGRADE.

3ROTATION PERIOD 'y 9.928 HOURS AT HIGH LATITUDES.

4ROTATION PERIOO -10.63 HOURS AT HIGrl LATITUDES.

5TITAN, SATURN'S LARGEST SATELLITE, HAS AN ATMOSPHERE WHOSE MAJOR COMPONENT IS CH4.

6RATIO OF TOTAL REFLECTED LIGHT TO TOTAL INCIDENT LIGHT.

7V (1, 0) 15 THE VISUAL MAGNITUDE WHEN THE OBSERVER IS DIRECTLY BETWEEN THE SUN AND THE PLANET ANO THE PRODUCT OF THE SUN-PLANET DISTANCE (IN AU) AND OBSERVER-PLANET DISTANCE liN AU) IS 1. SEE SECTION 3.5 FOR FORMULAE FOR OTHER VIEWING

ANGLES AND DtST ANCES.

BAS VIEWED FROM THE EARTH; MAGNITUDES FOR MERCURY AND VENUS ARE AT GREATEST ELONGATION.

9WITH RING SYSTEM VIEWED EDGE·ON.

10BLUE MAGNITUDE MINUS VISUAL MAGNITUDE. SEE SECTION 5.6.

11 FOR SMALL PHASE ANGLES, t, IN DEGREES, THE VARIATION OF VISUAL MAGNITUDE WITH PHASE IS APPROXIMATELY V It I ~ V 10) +A,t. SEE SECTION 3.5. FOR SATURN, V DEPENDS STRONGLY ON THE ORIENTATION OF THE RING SYSTEM.

Tab

le L

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818 SOLAR SYSTEM CONSTANTS L.2

L.2 The Sun

Table L-4 lists the principal physical properties of the Sun. See Section 5.3 for properties of solar radiation and the solar wind.

Table L-4. Physical Properties of the Sun

PROPERTY VALUE

RADIUS OF THE PHOTOSPHERE (VISIBLE SURFACE) 6.9599 x 105 km

ANGULAR DIAMETER OF PHOTOSPHERE AT 1 AU 0.53313 Deg

MASS 1.989 x 1030 kg

MEAN DENSITY 1.409 gm/cm3

TOTAL RADIATION EMITTED 3.826 x 1026 J/sec

TOTAL RADIATON PER UNIT AREA AT 1 AU 1358 J s-l m-2

ESCAPE VELOCITY FROM THE SURFACE 617.7 km/sec

POLAR MAGNETIC FIELD AT SUNSPOT MINIMUM (1 t02) x 1O-4'T

APPARENT VISUAL MAGNITUDE AT 1 AU -26.74

ABSOLUTE VISUAL MAGNITUDE (MAGNITUDE AT +4.83 DISTANCE OF 10 PARSECS)

COLOR INDEX, B-V (SEE SECTION 5.6) +0.65

SPECTRAL TYPE G2 V

EFFECTIVE TEMPERATURE 5770° K

VELOCITY OF THE SUN RELATIVE TO NEARBY STARS' 15.4 km/sec TOWARD 0/ = 268 Deg, {j = +26 Deg

INCLINATION OF THE EQUATOR TO THE ECLIPTIC 7.25Deg

LONGITUDE OF THE ASCENDING NODE OF THE EQUATOR 75.48 Deg + 0.014 Deg/Year IN 1980 PLUS RATE OF CHANGE IN LONGITUDE

SIDEREAL ROTATION RATE OF THE SUNSPOT ZONE, AS A FUNCTION OF LATITUDE, L (ILl :S 40°)

(14.44° _3.0° SIN2 L) per Day

ADOPTED PERIOD OF SIDEREAL ROTATION (L = 17°) 25.38 Days

CORRESPONDING SYNODIC ROTATION PERIOD (RELATIVE 27.275 Days TO THE EARTH)

MEAN SUNSPOT PERIOD 11.04 Years

DATES OF FORMER MAXIMA 1957.9,1968.9

MEAN TIME FROM MAXIMUM TO SUBSEQUENT MINIMLJ:' 6.2 Years

"THE QUANTITY LISTED IS THE MODE OF THE VELOCITY DISTRIBUTION. THE MEAN OF THE DISTRIBUTION, WHICH IS MORE STRONGLY INFLUENCED BY HIGH VELOCITY STARS, IS 19.5 KM/SEC TOWARD .. s 2710

/; = +300. FOR FURTHER DETAILS, SEE MIHALAS [1968).

THE EARTH 819

L.3 The Earth

The principal physical properties of the Earth are listed in Table L-S. For general characteristics, see also Tables L-l through L-3. See Appendix J for a discussion of the length of day and year and the nonuniform rotation of the Earth. See Chapter 4 for an extended dis~ussion of Earth models.

Table L-S. Physical Properties of the Earth

PROPERTY

EQUATORIAL RADIUS, a

FLATTENING FACTOR (ELLIPTICITY) a;: '" I

POLAR RADIUSt,c

MEAN RADIUSt (.2 c) 113

ECCENTRICITy t (J a2~2)/a SURFACE AREA

VOLUME

ELLIPTICITY OF THE EOUATOR (aMAX - aMIN)/aMEAN

LONGITUDE OF MAXIMA, aMAX

RATIO OF THE MASS OF THE SUN TO THE MASS OF THE EARTH

GEOCENTRIC GRAVITATIONAL CONSTANT, GME '" "E

MASS OF THE EARTH*

MEAN DENSITY

GRAVITATIONAL FIELD CONSTANTS I~: J4

MEAN DISTANCE OF EARTH CENTER FROM EARTH·MOON BARYCENTER

AVERAGE LENGTHENING OF THE DAY (SEE FIGURE J·5)

GENERAL PRECESSION IN LONGITUDE (i.e., PRE'CESSION OF THE EQUINOXES) PER JULIAN CENTURY, AT EPOCH 2000

RATE OF CHANGE OF PRECESSION

OBLIQUITY OF THE ECLIPTIC, AT EPOCH 2000

RATE OF CHANGE OF THE OBLIQUITY (T IN JULIAN CENTURIES)

AMPLITUDE OF EARTH'S NUTATION

LENGTH OF SIDEREAL DAy,EPOCH 1980§

LENGTH OF SIDEREAL YEAR, EPOCH 1980§

LENGTH OF TROPICAL YEAR (REF =1') EPOCH 1980§

LENGTH OF ANOMALlSTIC YEAR (PERIHELION TO

PERIHELIOilll, EPOCH 19809

t BASED ON THE ADOPTED VALUES OF f AND a.

VALUE

6378.140 km

0.00335281 = 11298.257

6356.755 km

6371.00 km

0.0818192

5.10066 x 108 km2

1.08321 x 1012 km3

-5 - 1.6 x 10 (amax -amln ~ 100 m)

200 Wand 1600 E

332,946.0

3.986005 x 1014 m3 s-2

5.9742 x 1024 kg

5.515 gm/cm3

+1082.63 x 10-6 -2.54 x 10-6 -1.61 x 10-6

4671 km

0.0015 sec/Century

1.39697128 Deg/Century

+6.181 x 10-4 Deg/Centurl

23.4392911 Deg

(-1.30125 x 10-2 T -1.64 x 10-6 T2

+5.0 x 10-:-7T3) Deg

2.5586 x 10-3 Deg

86,164.0918sec: = 23 hr 56 min 4.0918 sec

3.1588149548 x 107 sec = 365.25636051 Days"

3.156925551 x 107 sec = 365.24219388 DaysD

3.1558433222 x 107 sec

= 365.25964377 Daysll

* ASSUMING G = 6.672 x 10-11 m3 kg- 1 s-2; THE'VALUE OF GME IS MORE ACCURATELY KNOWN.

§SEE APPENDIX J FOR FORMULAE AND DISCUSSION.

• ONE DAY'" 86,400 SI SECONDS.

Table L-6 summarizes the properties of the upper atmosphere of the Earth. The mean profiles between 25 and 500 km are from the COS PAR International Reference Atmosphere, CIRA 72 [1972]. Between 500 and 1000 km, the CIRA 72 profile for Too = I000K was used to indicate the densities to be expected. The maximum and minimum values of the density between 100 and 500 km were extracted from the explanatory material in CIRA 72 and indicate the variation in densities which can be obtained with the models. Sea level temperature and density are from the U.S. Standard Atmosphere [1976].

Geocentric and Geodetic Coordinates on the Earth. The geocentric latitude, </>', of a point, P, on the surface of the Earth is the angle at the Earth's center between P and the equatorial plane. The geodetic or geographic latitude, </>, is the angle between the normal to an arbitrarily defined reference ellipsoid (chosen as a close approximation to the oblate Earth) and the equatorial plane. Astronomical latitude and longitude are defined relative to the local vertical, or the normal to the equipotential surface of the Earth. Thus, astronomical latitude is defined as the angle between the local vertical and the Earth's equatorial plane. Maximum values

Table L-6. The Upper Atmosphere of the Earth

MEAN DENSITY (kg/m 3 ) SCALE ALTITUDE KINETIC HEIGHT

(KM) TEMPERATURE MEAN (KM)

(oK) MINIMUM MAXIMUM

0 288.2 1.225 x 10 +0 8.44

25 221.7 3.899 x 10-2 6.49

30 230.7 1.774 x 10-2 6.75

35 241.5 8.279 x 10-3 7.07

40 255.3 3.972 x 10-3 7.47

45 267.7 1.995 x 10-3 7.83

50 271.6 1.057 x 10-3 7.95

55 263.9 5.821 x 10-4 7.73

60 249.3 3.206 x 10--4 7.29

65 232.7 1.718 x 10-4 6.81

70 216.2 8.770 x 10-5 6.33

75 205.0 4.178 x 10-5 6.00

80 195.0 1.905 x 10-5 5.70

85 185.1 8.337 x 10-6 5 .. 41

90 183.8 3.396 x 10- 6 5.38

95 190.3 1.343 x 10-6 5.74

100 203.5 3.0 x 10-7 5.297 x 10- 7 7.4 x 10-7 6.15

110 265.5 6.0 x 10-8 9.661 x lp-8 3.0 x 10-7 8.06

120 334.5 1.0 x 10-8 2.438 x 10-8 6.0 x 10-8 11.6

130 445.4 4.5 x 10-9 8.484 x 10-9 1.6 x 10-8 16.1

140 549.0 2.0 x 10-9 3.845 x 10-9 6.0 x 10-9 20.6

150 635.2 1.2 x 10-9 2.070 x 10-9 3.5 x 10-9 24.6

160 703.1 6.5 x 10-1.0 1.244 x 10-9 2.0 x 10-9 26.3

180 781.2 2.4 x 10- 10 5.464 x 10- 10 9.0 x 10-10 33.2

200 859.3 1.0 x 10-10 2.789 x 10- 10 3.2xlO- 1O 38,5

250 940.2 4.0 x 10--11 7.248 x 10-11 1.6x 10-10 46.9

300 972.8 1.6 x 10- 11 2.418 x 10- 11 8.8 x 10- 11 52.5

350 986.5 2.0 x 10- 12 9.158 x 10- 12 6.0 x 10- 11 56.4

400 992.6 3.7 x 10- 13 3.725 x 10- 12 5.0 x 10-11 59.4

450 995.7 9.0 x 10- 14 1.585 x 10- 12 3.8 x 10- 11 62.2

500 997.3 1.3 x 10- 14 6.967 x 10-13 3.0xl0- 11 65.8

600 1000.0 1.454 x 10-13 79

700 1000.0 3.614 x 10-14 109

800 1000.0 1.170 x 10-14 164

900 1000.0 5.245 x 10-15 225

1000 1000.0 3.019 x 10- 15 268

L.3 THE EARTH

Table L-7. Coefficients for Determining Geocentric Rectangular Coordinates from Geodetic Coordinates on the Surface of the Earth. Based on f = 1/298,257.

'" s C (OEGI '" s c (OEG)

±o 0.993306 1.000000 ± 50 0.995262 1.001970 5 0.993331 1.000025 55 0.995544 1.002254

10 0.993406 1.000101 60 0.995809 1.002520 15 0.993528 1.000224 65 0.996048 1.002761 20 0.993695 1.000392 70 0.996255 1.002969 25 0.993900 1.000598 75 0.996422 1.003138 30 0.994138 1.000838 80 0.996546 1.003262 35 0.994401 1.001103 85 0.996622 1.003338 40 0.994682 1.001386 ±90 0.996647 1.003364

±45 0.994972 1.001678

821

of the deviation of the vertical, or the angle between the local vertical and the normal to a reference ellipsoid, are about I minute of arc. Maximum variations in the height between the reference ellipsoid and mean sea level (also called the geoid or equipotential surface) are about 100 m, as illustrated in Fig. 5-8.

The coordinate transformations given here are intended for use near the Earth's surface to correct for an observer's height above sea level and are valid only for altitudes much less than the radius of the Earth. For satellite altitudes, the coordinates will depend on the definition of the subsatellite point or the method by which geodetic coordinates are extended to high altitudes. For a discussion of geodetic coordinates at satellite altitudes, see Hedman [1970] or Hedgley [1976].

Geodetic and geocentric latitude are related by [H.M. Nautical Almanac Office, 1961]:

tancp = tancp' / (1- f)2;::::;: 1.006740tancp'

cp - cp' = (J + 1l)sin 2cp - (1 f2 +'1 f3)sin 4cp + t lsin 6cp

;::::;:0.19242 °sin 2cp - 0.000323 °sin 4cp

where f;::::;: I /298.257 is the flattening factor of the Earth as adopted by the lAO in 1976 [Muller and Jappel, 1977].

Let h be the height of P above the reference ellipsoid in metres; let Rffi be the equatorial radius of the Earth in metres; and let d be the distance from P to the center of the Earth in units of R(B. Then d and h are related by

h=Rffi[ d-(I-f)lVl-f(2-f)cos2cp' ]

d = : ffi + 1 - ( t ) f - ( (6 ) l + ( t ) f cos 2cp - ( 156 )f2COS 4cp + (9 (J3)

;::::;:(1.5679 x 1O-7)h +0.998327 + 0.001676cos2cp-0.OOOOO4cos4cp

To convert geographic or geodetic coordinates to geocentric rectangular coordinates in units of R(B' use the following:

dsincp' = (S + h x 1.5679 X 1O-7)sincp

dcoscp'=(C+hX 1.5679 X 1O-7)coscp

tancp' = (0.993305 + h X 1.1 X 1O-9)tancp


Table L-8. Physical and Orbital Properties of the Moon

PROPERTY VALUE

MEAN DISTANCE FROM EARTH 384401 ± 1 km

EXTREME RANGE 356400 to 406700 km

ECCENTRICITY OF ORBIT 0.0549

INCLINATION OF ORBIT TO ECLIPTIC 5.1453 Deg (OSCILLATING ± 0.15 DEG WITH PERIOD OF 173 DAYS)

SIDEREAL P.ERIOD (RELATIVE TO FIXED STARS) 27.32166140 + T x 1.6 x 10-7 WHERE T IS IN CENTURIES FROM 1900.0 Ephemeris Days

SYNODICAL MONTH (NEW MOON TO NEW MOON) 29.5305882 + T x 1.6 x 10-7 Ephemeris Days

TROPICAL MONTH (EQUINOX TO EQUINOX) 27.32158214 + T x 1.3 x 10-7 Ephemeris Days

ANOMALISTIC MONTH (PERIGEE TO PERIGEE) 27.5545505 - T x 4 x 10-7 Days

NODICAL MONTH (NODE TO NODE) 27.212220 Days

NUTATION PERIOD = PERIOD OF ROTATION OF THE NODE 18.61 Tropical Years (ROTROGRADE)

PERIOD OF ROTATION OF PERIGEE (DIRECT) 8.85 Years

OPTICAL LlBRATION IN LONGITUDE (SELENOCENTRIC ± 7.6 Deg DISPLACEMENT)

OPTICAL LlBRATIONIN LATITUDE (SELENOCENTRIC ± 6.7 Deg DISPLACEMENT)

SURFACE AREA NEVER VISIBLE FROM EARTH 41%

INCLINATION OF EQUATOR TO ECLIPTIC 1.542Deg TO ORBIT 6.68 Deg

RADII: a TOWARD EARTH, b ALONG ORBIT, c TOWARD POLE

MEAN RADIUS (b+cl/2 1738.2 km a-c 1.09 km a-b 0.31 km b-c 0.78 km

MEAN ANGULAR DIAMETER AT MEAN DISTANCE 0.5182 Deg FROM EARTH

RATIO OF MASS OF MOON TO MASS OF EARTH 0.01230002

MASS OF THE MOON" 7.3483 x 1022 kg

MEAN DENSITY 3.341 g em-3

SURFACE GRAVITY 162.2cm s-2

SURFACE ESCAPE VELOCITY 2.38 km/s

"ASSUMING G = 6.672 x 10-11 m 3 kg-1 S-2; MASS RATIOS ARE MORE ACCURATE.

L.S

where:

POTENTIAL SPACECRAFT ORBITS

c = [ COS2cp + (I - f)2sin2cp ] - 1/2

S =(1-f) 2 C

823

values of Sand C are given in Table L-7. In terms of Sand C, the distance to the center of the Earth for h =0 is

d 2= !(S2+ C 2) + t(C2 - S2)cos2CP

L.4 The Moon

The physical and orbital properties of the Moon are summarized in Table L-8. Additional general characteristics are given in Tables L-2 and L-3. To determine the Moon's visual magnitude, V(R,O, at any distance and phase, let R be the observer-Moon distance in AU and ~ be the phase angle at the Moon between the Sun and the observer. Then

V(R,~) =0.23 + SloglOR - 2.5logIOP(~)

where the phase law, pm, for the Moon is given in Table L-9 [Allen, 1973]. For additional details and a sample computation, see Section 3.5. Note that the visual magnitude of the Moon at opposition (i.e., full Moon) at the mean distance of the Moon.from the Earth is -12.73.

Table L-9. Phase Law and Visual Magnitude of the Moon

t p It I VIR. tl- VIR. 01 t

p It I V (R. tl - V IR. 01 IOEGI IOEGI

0 1.000 0.00 80 0.127 2.24

5 0.929 O.OS 90 0.089 2.63

10 0.809 0.23 100 0.061 3.04

20 0.625 0.51 110 0.041 3.48

30 0.483 0.79 120 0.027 3.93

40 0.377 1.06 130 0.017 4.44

50 0.288 1.35 140 0.009 5.07

00 0.225 1.62 150 0.004 5.9

70 0.172 1.91 160 0.001 7.5

LoS Potential Spacecraft Orbits

Table L-IO lists the transfer time and velocity required for Hohmann transfer orbits between the planets. The values cited are for minimum energy transfer orbits between the mean distances of the planets from the Sun. The upper number is the one-way transfer time in days; the lower number is the velocity change required to go from the orbital velocity of the planet of origin to the transfer orbit in km/sec. See Section 3.3 for relevant formulae. Finally, Table L-ll gives the velocity of escape, circular velocity, and synchronous altitude and velocity for potential artificial satellites of the Moon and planets.

Tab

le L

-IO

. H

ohm

ann

Tra

nsfe

r O

rbit

Pro

pert

ies.

(Se

e te

xt f

or e

xpla

nati

on.)

OR

IGIN

D

ES

TIN

AT

ION

M

ER

CU

RY

V

EN

US

E

AR

TH

M

AR

S

JUP

ITE

R

SA

TU

RN

U

RA

NU

S

ME

RC

UR

Y

-75

.6

105

171

853

2,02

0 5.

590

5.8

7.

5 8.

8 8.

2 6.

9 5.

4

VE

NU

S

75

.6

-14

6 21

7 93

1 2,

120

5,73

0 6.

7 2.

5 4.

8 6.

6 6.

0 5.

0

EA

RT

H

105

14

6

-2

59

99

7 2,

210

5,8

50

9

.6

2.7

2.6

5

.6

5.4

4.7

MA

RS

17

1 21

7 2

59

-

1,1

30

2,

380

6,08

0 12

.6

5.8

2.9

4.

3 4.

6 4.

2

JUP

ITE

R

853

931

99

7

1,13

0 -

3,65

0 7,

780

17.4

11

.4

8.8

5.9

1.5

2.4

SA

TU

RN

2,

020

2,12

0 2,

210

2,38

0 3

,65

0

-9

,94

0

18

.5

12.7

10

.3

7.6

1.8

1.3

UR

AN

US

5

,59

0

5,7

30

5

,85

0

6,08

0 7,

780

9,94

0 -

19.1

1

3.6

11

.3

8.7

3.3

1.5

NE

PT

UN

E

10,8

00

11,0

00

11,2

00

11,5

00

13,5

00

16,1

00

22,3

00

19.4

13

.9

11.7

9.

2 4.

0 2.

2 0.

7

PL

UT

O

16

,30

0

16,4

00

16,7

00

16,9

00

19,3

00

22,1

00

29,0

00

19.5

14

.1

11.8

9.

4 4.

3 2.

6 1.

1

TH

E M

OO

N

5.0

3.2

----

----

----

----_

.. --

----

----------------

,----

------

NE

PT

UN

E

10,8

00

4.6

11,0

00

4.3

11,2

00

4.1 i'-,~.

500

3.7

13,5

00

3.7

16,1

00

1.7

22,3

00

0.6

-

37,4

00

0.4

PL

UT

O

16,3

00

4.1

16,4

00

3.8

16,7

00

3.7

16,9

00

3.4

19

.30

0

2.5

22

,10

0

1.8

29

,00

0

0.9

37,4

00

0.3 -

j

00

N

.j:

:..

r.n o r > ::0

r.n -< r.n ..., m

3:

(') o Z

r.n ..., > Z ..., r.n

L'

N

L.5 POTENTIAL SPACECRAFT ORBITS 825

Table L-ll. Parameters for Potential Artificial Satellites of the Moon and Planets

VELOCITY IN SYNCHRONOUS ORBIT VELOCITY CIRCULAR ORBIT

PLANET OF ESCAPE AT THE ALTITUDE ABOVE VELOCITY (KM/SECI SURFACE SURFACE (KM/SECI

(KM/SECI (KMI

THE MOON 2.376 1.679 86,710 0.235

MERCURY 4.263 3.014 241,400 0.301

VENUS 10.346 7.316 1,536,000 0.459 1

EARTH 11.1BO 7.905 35,7B6 3.075

MARS 5.023 3.552 17,033 1.448

JUPITER 59.62 42.16 87,8202 28.22

SATURN 35.53 25.12 49,1502 18.63

URANUS 21.77 15.39 36,130 9.78

NEPTUNE 23.40 16.55 57,480 9.12

PLUTO 5.4 3.8 68,000 0.81

NOTE: THE VELOCITY OF ESCAPE FROM THE SOLAR SYSTEM AT THE DISTANCE OF THE EARTH'S ORBIT IS 29.7B5 KM/SEC.

1RETROGRADE.

2FOR EQUATORIAL ROTATION; THE PLANET'S ROTATION IS SLOWER AT HIGHER LATITUDES.

References

1. Allen, C. W., Astrophysical Quantities, Third Edition, London: The Athlone Press, 1973.

2. COSPAR, CaSPAR International Reference Atmosphere. Berlin: AkademieVerlag, 1972.

3. Hedgley, David R., Jr., An Exact Transformation from Geocentric to Geodetic Coordinates for Nonzero Altitudes, NASA TR R-458, Flight Research Center, March 1976.

4. Hedman, Edward L., Jr., "A High Accuracy Relationship Between Geocentric Cartesian Coordinates and Geodetic Latitude and Altitude," J. Spacecraft, Vol. 7, p. 993-995, 1970.

5. H.M. Nautical Almanac Office, Explanatory Supplement to the Astronomical Ephemeris and the American Ephemeris and Nautical Almanac. London: Her Majesty's Stationery Office, 1961,

6. Mihalas, Dimitri, Galactic Astronomy. San Francisco: H. W, Freeman and Company, 1968.

7. Muller, Edith A. and Arndst Jappel, Editors, International Astronomical Union, Proceedings of the Sixteenth General Assembly, Grenoble 1976, Dordrecht, Hoiland: D. Reidel Publishing Co., 1977.

8. Owen, Tobias, "Jovian Satellite Nomenclature," Icarus, Vol. 29, p, 159-163, 1976,

9. U.S. Standard Atmosphere, Washington, U.S.G.P.O" 1976.

APPENDIXM

FUNDAMENTAL PHYSICAL CONSTANTS

The physical constants are those compiled by Cohen and Taylor [1973a, 1973b] under the auspices of the CODATA (Committee on Data for Science and Technology of the International Council of Scientific Unions) Task Group on Fundamental Constants and officially adopted by CODATA. The astronomical constants are those compiled by Commission 4 of the International Astronomical Union and adopted at the 1976 IAU meeting in Grenoble [Muller and lappel, 1977]. The uncertainhes are the I a standard deviation expressed in parts per million (ppm). Additional constants are listed in Appendix K (Conversion Factors), Appendix L (Solar System Constants), Allen [1973], and Rossini [1974].

DIMENSIONLESS NUMBERS (Note I):

7T = 3.141 592 653 589793 238462643 ...

e=2.7IS 281828459045235360287 ...

PHYSICAL CONSTANTS: Uncer-tainty

Quantity Symbol Value Units (ppm)

Elementary Charge e 1.6021892 X 10- 19 C 2.9 Planck Constant h 6.626176 X 10- 34 l·s 5.4 Permeability of Vacuum flo 47TxlO- 7 N/A2 Exact

Fine Structure Constant a 7.2973506 X 10- 3 None 0.82

(floce 2/2h) Avogadro Constant NA 6.022045 X 1023 mol-I 5.1 Atomic Mass Unit u 1.6605655 X 10- 27 kg 5.1 Electron Rest Mass me 9. 109534 X 10- 31 kg 5.1 Proton Rest Mass mp 1.6726485 X 10- 27 kg 5.1

Neutron Rest Mass mn 1.6749543 X 10- 27 kg 5.1 Muon Rest Mass mp' 1.883566 X IO~ 28 kg 5.6

Electton Magnetic fle 9.284832 X 10-24 l/T 3.9 Moment

Proton Magnetic flp 1.4106171 X 10- 26 l/T 3.9 Moment

Muon Magnetic Moment flp. 4.490474 X 10-26 l/T 3.9

Rydberg Constant Roo 1.097373177 X 107 m- I 0.075 (fl5c3mee4/8h3)

Bohr Radius ao 5.2917706 X 10- 11 m 0.82 (a/47TR oo)

Classical Electron 'e 2.8179380 X 10- 15 m 2.5

Radius (0:3/47TRoo)

Ap.M FUNDAMENTAL PHYSICAL CONSTANTS 827

Uncer-tainty

Quantity Symbol Value Units (ppm)

Compton Wavelength of he 2.4263089 X 10- 12 m 1.6 the electron (h / mec)

Molecular volume of Vm 2.241383 x 10-2 m3/mol 31 ideal gas at S.T.P.·

Boltzman constant k 1.380662 X 10- 23 J/K 32

ASTRONOMICAL CONSTANTS:

Speed of Light in Vacuum C 2.99792458 X 108 m/s 0.004 Gaussian Gravitational k 1.720209895 X 10-2 rad/day Note (2)

Constant Earth Equatorial Radius Re 6.378140x 106 m 0.78

Earth Dynamical Form '2 1.08263 X 10- 3 None 9.2 Factor

Earth Flattening Factor j 3.35281 X 10- 3 None 6.0

I/j 2.98257 X IQ2 None 6.0 Earth Gravitation Constant GM e 3.986005 X 1014 m3 js2 0.75 Moon Gravitation Constant GMM 4.902794 X 1012 m3/s2 3.6 Sun Gravitation Constant GM0 1.32712438 X 1020 m3/s2 0.038

Gravitational Constant G 6.6720 X 10- 11 m3/(kg·s2) 615. (3) Mass of the Moon MM 7.3483 X 1022 kg. 615.(3)

Mass of the Sun M0 1.9891 X 1030 kg 615. (3) Mass of the Earth Me 5.9742 X 1024 kg 615. (3) Ratio of the Mass of the MM/ Me 1.230002 X 10- 2 None 3.6

Moon to the Mass of the Earth

Obliquity of the Ecliptic t: 23 0 26'21/1.448 at Epoch 2000 = 2.34392911 X 101 deg 1.2

General Precession in P 1.39697128 ( deg ) 30 Longitude per Julian century

Ephemeris Century, at Epoch 2000

Constant of Nutation, at N 2.55858 X 10- 3 deg 600 Epoch 2000

Astronomical Unit AU 1.49597870 X 10" m 0.013 Solar Parallax 7T0 2.442819 X 10-3 deg 0.80 Ratio of the mass of the. Sun to those of the planetary systems (planetary system masses include both atmosphere and satellites): Mercury 6.023600 X 106 580 Venus 4.085235 X 105 5.2 Earth + Moon 3.289005 X 105 1.5

828 FUNDAMENTAL PHYSICAL CONSTANTS

Quantity Symbol Value

Mars 3.098710 X 106

Jupiter 1.047355 X 103

Saturn 3.4985 X 103

Uranus 2.2869 X 104

Neptune 1.9314x 104

Pluto 3 X 106

Notes:

(1) Values from Abramowitz and Stegun [1970].

Units

Ap.M

Uncertainty (ppm)

26

24

430

9900

3900

(-0.3)X 106

+4

(2) The Gaussian gravitational constant has the given value by definition and serves to define the other astronomical constants.

(3) The gravitational constant enters theories of orbital dynamics only through the product GM. This product is well known for the various objects in the solar system. G itself, and consequently the masses in kilograms of the Sun and planets, is not as well known. Therefore, accurate analyses should use directly the product GM 0 and the ratios of the masses of the planets and the Sun.

EARTH SATELLITE PARAMETERS

ANGULAR REQUIRED ALTITUDE, h RADIUS PERIOD VELOCITY (km) OF THE EARTH (min) (km/secl

ENERGY (deg) (MJ/kg)

0 90.00 84.49 7.905 31.14

100 79.92 86.48 7.844 31.62

200 75.84 88.49 7.784 32.09

300 72.76 90.52 7.726 32.54

400 70.22 92.56 7.669 32.98

500 68.02 94.62 7.613 33.41

600 66.07 96.69 7.558 33.83

800 62.69 100.87 7.452 34.62

1,000 59.82 105.12 7.350 35.37

2,000 49.58 127.70 6.898 38.60

3,000 42.85 150.64 6.519 41.14

4,000 37.92 175.36 6.197 43.18

5,000 34.09 201.31 5.919 44.87

10,000 22.92 347.66 4.933 50.22

20,000 13.99 710.60 3.887 54.83

35,786 8.70 1436.07 3.075 57.66 (SYNCHRONOUS) (1 SIDEREAL

DAY)

00 0.0 00 0.0 62.39

Ap.M FUNDAMENTAL PHYSICAL CONSTANTS 829

In the above table, the angular radius, period, and energy required are valid for elliptical orbits of arbitrary eccentricity; however, the velocity is correct only for circular orbits. For non circular orbits, h should be interpreted as the instantaneous altitude when determining the angular radius of the Earth and as the mean altitude when determining the period and required energy. The mean altitude is hm =(P + A)/2, where P and A are the perigee and apogee altitudes, respectively.

References

1. Abramowitz, Milton and Irene A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. Washington, DC: National Bureau of Standards, 1970.


3. Cohen, E. Richard and B. N. Taylor, J. Phys. Chem. Ref Data, vol. 2, p. 663, 1973a

4. ---, CODATA Bulletin No. 11, Dec. 1973b. 5. Miiller, Edith A. and Arndst Jappel, Editors, International Astronomical Union,

Proceedings of the Sixteenth General Assembly, Grenoble 1976. Dordrecht, Holland: D. Reidel Publishing Co., 1977.

6. Rossini, Frederick D., Fundamental Measures and Constants for Science and Technology. Cleveland, Ohio: CRC Press, 1974.

INDEX

-A-Aberration 144 Abnormal termination in mission support

software, handling of 682---{)83 Absorption bands, in Earth's atmosphere 91-92 Absorption of radiation, torque due to 572 AC two-phase induction motot, torque

profile for 27a-27 I Acceleration, units and conversion factors 809 Accelerometen, as attitude determination

reference 17 Acquisition of signal (See In-triggering) Acquisition phase, of space mission 3-8 Active attitude control 18,503,506-509 ActuatOt (control system component) 502,589 Adams integraton 563-564 Adams-Bashforth integraton 563-564 Adams-Bashforth-Moulton Integraton 563-564 Adams-Moulton integratOt 563 Adaptive ruter, for state estimation 467 Adcole Corporation, Sun sensors 157, 161-166 Adjoint, of a matrix 745 ADL (See Attitude data link) Advanced tange instrumentation aitcraft

284,287 AE (Atmospheric Explorer)-

Application of block averaging to attitude solutions 371

Attitude determination accuracy 397 Attitude sensor bias evaluation 402 Attitude system of 788-789 Data collection for bias determination 475 Data curve fitting for 318 Data sample from AE-3 313 Earth-width data 233-234 Horizon sensors 177 Magnetic coil control system 509 Momentum wheel 202-203,601,602 Optimal magnetic matteuvers 640 Orbit generator accuracy 138 Pitch angle measurement 360 Shadow modeling for 573 Spin axis magnetic coils 205 Spin plane magnetic control 646, 64849 Stabilization method 503 Sun sensor 157 Telemetry data errors 311 Use of body-mounted horizon sensor 173 Use of carbon dioxide band horizon

sensor 92 Use of open loop control 663 Yaw inversion maneuver 651

AEM (Applications Explorer Mission)-(See also HCMM (AEM-A); SAGE

(AEM-B» Attitude system of 720,788-789

Sun sensor 157 Aerodynamic drag, effect on flexible

spacecraft 551 Effect on orbit 63---{)5

Aerodynamic stabilization 19 AerodynamictOtque 17,573-575 AEROS (German Aeronomy Satellite)-

Attitude acquisition Attitude system of Stabilization method Sun sensor

662 788-789

503 157 53 Agena (upper stage of launch vehicle)

Akademik Setgey Korolev (Soviet tracking ship) 290

83 816 571 79 83

Albedo Of the Moon and planets, table Torque due to Types of

Albedo seDSOt (See also Horizon sensor) Algebraic method of three-axis attitude

determination 421,424-426 Alignment, Math. model for

slit sensors 219-221 Math. model for two-axis sensors 221-223

Alouette (Canadian Ionospheric Research Satellite )-

Attitude system of 788-789 Altitude of Earth satellites, table of period and

Earth size vs. 828 Ambiguous attitude solutions, methods for

resolution of 371-373 American Ephemeris anti Nautical

Almanac Analog Sun seDSOn Analog-to-dlgital convenlon (A/D)

For magnetometers Analysis of variance

36,139-140 156-159 279,298

Angle of attack (in aerodynamic torque) ANGLED (subroutine)

250 429 574 690

Angstrom 808 Angular measure, units and conversion

factors 810 Angular momentum 516-521

Orbital 40 Storage in flywheel (See Momentum wheels) Units and conversion factors 811

Angular momentum axis, of spacecraft 487 Angular momentum control, with spin

plane magnetic coils 644, 646-649 Angular momentum sphere 499-501 Angular separation, between two points 727 Angular separation match, for star

identification 259, 262-263 Angular velocity, math. model for

gyroscope measurement of Units and conversion factors

267-268 811

Angular velocity vector, for Earth-referenced spacecraft 609

Annular eclipse 72,76 Anomalistic period, for Earth satellite 67 Anomalistic year 48 Anomaly (orbit parameter) 44-46 ANS (Netherlands Astronomy Satellite}--

Attitude system of 788-789 Antennas, Examples of tracking and

command 286-289 Telemetry and command 284-288

Antipoint, on celestial sphere 22 Antisolar point 22 Antisymmetric matrix 750 Aphelion 44 Apofocal distance 44 Apofocus 44 Apogee 44 Apogee height 44

Maneuvers to change 59 Apollo 53

Visual magnitude of during trip to Moon 79 Apollo 15 subsatellite, nutation of 495-497 Apolune 44 Apparent solar time 798, 800 Apsides 44

Rotation of (See Perigee, rotation of) Arc, of great circle Arc length, measurement of on global

23

geometry plot 742 Arc length distance, between two points 727 Arc length measurement 23,346-349

Notation for 23 Singularity conditions 406-407 Use in deterministic attitude solutions

364-365, 368-370 Arc length uncertainty, of single-axis

attitude solutions 372, 374-383 Arc minute 24 Arc second 24 Area on celestial sphere, formulas for 729-730 Areal velocity 46, 47 Argument or perigee 45-46 Argument of perihelion 49 Ariel (U.K. Satellite}--

Attitude system of Aryabhata (Indian satellite}-

Attitude system of Nutation damper

Ascending node Maneuver to change

Aspects of the planets (See Planetary configurations)

Asteroids, Lagrange point orbits Astrodynamics Astronauts, disturbance torques due to

788-789

788-789 628

44 59

55 487

motion of 576, 578-580 Astronomical constants 827 -828 Astronomical Ephemeris 36, 139 Astronomical latitude and longitude 820 Astronomical symbols 50 Astronomical Unit 41,808

INDEX 831

Atmosphere (unit of pressure) 809 Atmosphere of Earth (See Earth,

atmosphere of)

Atomic clock'S 799,802 ATS (Applications Technology Satellite}--

Attitude system of 788-789 Horizon sensors 175 Kalman filter state & covariance

propagation Momentum wheel Momentum wheel control system Nutation damping Orbit generator accuracy Polaris tracker Stabilization method Sun sensor Telemetry data errors

Attitude Definition Introduction to Introduction to analysis Parameterizations of -Table of relative advantages Transmission of results

465-467 202 508 628 138 189 503 157 311

I 1-21

343-361 410-420

412 292

Attitude acquisition (See also Acquisition phase) 502,661-667

Classifications of 662-663 CTS example 672 GEOS-J example 672--677

Attitude acquisition maneuvers 663-672 Attitude control 2, 588-678

Areas of current work 715 Closed loop and open loop 662-663 Example of 14-16 Example of function of 7 Functions of 2 Introduction to 502-509 Needed analytic projects 722-723 Table of methods 19,503 Transmission of commands 279, 292

Attitude control hardware (See also speCific item, e.g., Gas jets) 201-213

Earth-oriented spacecraft (table) 613 Introduction to 502-509 Listed by spacecraft & sensor type 796 Mathematical models 270-275

Attitude control strategy 16 Attitude control system 502

Attitude Data Link (ADL) 292 Attitude data simulators 709-712

Examples of structure 710-712 Functional requirements 709-710

Attitude determination (See also specific method, e.g., Earth width/ Sun angle) 2, 343-484

Areas of current work 715 Block averaging 370-373 Definitive vs. real time 681 Deterministic methods 362-435 -Advantages of 437-438 -Solution behavior 402-408

832

Deterministic vs. state estimation Example of Example of function of Functions of Hardware (See Attitude sensors)

436-438 10-14

7 2

INDEX

Introduction to analysis 343-361 Methods of 16-18 Mission support software 681-713 Needed analytic projects ,722-723 Proc. for elim. data anomalies 334-339 Reference sources, table of 17 Reference vector \0 -Attitude accuracy limits 393-397 Single axis 362-409 Spinning spacecraft 363-370 Star sensor methods 703-709 Three-axis 420-428 Uncertainties 345-346 -Expressions for 375-376,381-382,384

Attitude determination accuracy 397 Analytic solutions for 373-402 Direct calculation of 373, 429-435 Effect of Earth oblateness \05 Estimating systematic error 476 Estimating systematic uncertainty 402-408 Evolution of over time 714 For continuous measurements 376-382 For correlated uncertainties

382-383,431-432 For nonorthogonal measurements 377-382 For orthogonal measurements 376-377 For quantized measurements 374-376,431 For uncorrelated uncertainties

374-382,431-432 389-402 Geometrical limitations on

Identification of singularities -Table Limited by systematic error Sample computation (IUE) Single axis Single frame, summary table Spacecraft requirements, list of Three axis

Attitude disturbance torques (See Disturbance torques)

406-407 407 476

384-386 373-409

384 7%

429-434

Attitude dynamics 487-587 Flexible spacecraft 548-556 Introduction to 487,498-502 Mathematical models 521-523 Model of, in state vector formulation 440 -For attitude propagation 558-559

Attitude error ellipse (See Error ellipse) Attitude error parallelogram (See Error

parallelogram) Attitude extrapolation, for star sensor

attitude determination Attitude geometry (See also Global

geometry plots) As limitation on attItude accuracy For single-axis attitude solutions Unresolved analytic problems

705-706

22-35 389-402 362-402

723

Attitude kinematics 510-521 Approximate closed-form solutions

for 564-56,6 Equations fOf attitude propagatiori 558-559 lritroduction to 487-497 Model of, in state vector formulation 440

Attitude maneuver control 2, 502, 636-678 Attitude maneuver control program for

CTS spacecraft 700, 703 Attitude maneuver monitoring, in CTS

attitude system 702-703 Software for 700

Attitude matrix (See Direction cosine matrix) Attitude measurements (See also Arc

length measurement; Compound attitude measurement; Rotation angle measurement)

Equivalence of 344 Intersection of loci 344-346 Types of 346

Attitude measurement density 345-346 As limit on attitude

accuracy Expressions for For rOtation angles

392-393,394-396 384,386-388

352 Attitude measurement uncertainty (See

Attitude determination accuracy; Uncertainty)

Attitude motion (See also Attitude kinematics; Attitude Dynamics)

Math. model for gyro measurement of 267 Attitude perturbations, due to flexible

spacecraft Attitude prediction (attitude

548-556

propagation) 2, 558-587 Accuracy of for HEAO-I 708-709

Attitude sensor electronics 242-249 Attitude Sensor Unit (combined Earth/Sun

sensor) 155-20\, 178-179 Attitude sensors (See also item sensed,

e.g., Horizon sensors) \0 Distinction between hardware & math.

models Listed by spacecraft & sensor type Mathematical models Need for standardization Representative telemetry data errors Simple vs. complex designs Use in attitude control loop

217 797

217-270 718-721

311 716-718

502 Attitude stabilization (See also

Stabilization methods, e.g., magnetic stabilization) 2-3, 502, 588-635

Spacecraft listed by method 787 Table of methods 19

Attitude stabilization systems 604-625 AU (See Astronomical unit) Automatic control of spacecraft (See also

Onboard control; Inertial guidance) Automatic threshold adjust (for digital Sun

sensor) 163-165 Autumnal equinox 27

Symbol for Averaged attitude solution, estimating

reliability of Averaging of attitude solutlous Axial symmetry

50

Azimuth angle, attitude component Component of a spherical coordinate

system

373 370-373

524 360

25 Determination of, for spin-stabilized

spacecraft Azimuth biases

In horizon sensors Azur (German Research Satellite~

Attitude system of

-8-

366 239-242

235

788-789

Baffles, in fuel tanks 578 Ball-in-tube nutation damper 626, 627 Ballistic coefficient 64 Ballistic trajectory 52 Bang-bang control law 593

Example of 655~58 Bang-bang-plus-dead-zone control law 593 Bar (unit of pressure) 809 Barycenter 38 Batch estimator 437,439,448

For star data 704,707-709 Batch least-squares estimator 448-456

Advantages & disadvantages 456 Convergence 455-456 EX/l-mple of 454-455, 456-459

Batch mode of program operation 686 Bays, magnetic 123 BC/CD/CPD Number (star catalogs) 143 Bending stiffness, of spacecraft booms 548-549 Bessel, Friedrich Wilhelm 45 Betelgeuse (star), angular diameter of 167 Bias determination (See also Differential

correction; estimation theory) Application of scalar checking to Choice of observation models for Choice of state vector elements,

example Geometrical conditions for Need for Need for "simple" sensors for Operational procedures for

Bias momentum, dual-spin spacecraft Biases (See individual item; e.g.,

Magnetometer biases)

329-330 447

440-441 478-483 407-408 717-718 473-476

610

Effect on deterministic solution behavior Types of

404 477

295-298 Binary codes BipropeUant gas jet Block (of attitude solutions) Block averaging Block diagram, for control system Bode's law

206 370

370-373 588 49

INDEX 833

Body cone 491-492 Body-fixed coordinates (See Spacecraft-

fixed coordinates) Body-mounted horizon sensor 169, 173

Mathematical models 231-237 Body nutation rate 490, 525-526, 535 Bolometer, as energy detector for horizon

sensor 171, 178 Misalignment of in wheel-mounted

horizon sensor 236-237 Boud albedo 79 Bond number 578 Bonner Durcbmusterung (star catalog) 143 Boom deployment, deadbeat maneuver 669~71 Boresight (Sun sensor) 165 Brahe, Tycho 36 Branch, of root locus diagram 597 Brazilian anomaly in geomagnetic field 115 Breakaway point, of a root locus diagram 598 Brightness of planets and satellites 77-80 Brightness of spacecraft, sample calculation 79 Brouwer method, of general perturbations 137 Butterworth filter 324-327

-C-

Calendar time conversion subroutines Canopus (star), as attitude reference Cape Canaveral, Florida (See Eastern Test

Range) Cape Photographic Durchmusterung (star

catalog) Carbon dioxide, absorption bands

In Earth's atmosphere Use of absorption band for horizon

692 189

143 91-92

108

sensing 92 Cartesian coordinate transformations 761-765 Cartesian coordinates 760 Cartesian plot subroutines (graph, scale) 694 Catalog of Bright Stars 143, 146 Causal (in time-dependent linear systems) 243 Cayley-Klein parameters 414 Cayley, Arthur 416 Celestial coordinate systems (See

Spherical coordinate systems) Celestial coordinates (right ascension,

declination) 26-28 Transformation subroutines (RADECM,

VEC) 690 Celestial equator 26 Celestial mechanics I Celestial meridian 799 Celestial poles 26 C.elestial sphere 22-24

Plots of (See Global geometry plots) Center of mass, of a Keplerian orbit

(barycenter) Center of pressure CentralLUmt1beonn

38 573 431

834

Centroid of torque 274 Cbaracteristie equation, for a matrix 752 Cbaraeteristic frequency, of a linear system 773 Characteristic polynomial, of a differential

equation Charge coupled device star tracker Chebyshev polynomials (curve fitting) Chi-squared functioo Chord length, of a small circle Circular velocity Classical elements, of an orbit (See

Keplerian elements) CLIMSAT(climatology satellite)

Use of MMS spacecraft Clock angle (component of sensor

770 189

318-322 318

727-728 42

720

orientation) 422 Clocks 7':1':1

Atomic 799, 802 Ground based 299-302 SpaC'!Craft 298-299 Sundials 800

ClosecI-ioop control 604, 662 Closed"ioop control system 502, 588-600 Closed-loop poles, of root locus diagram 597 Closed-loop rate gyroscope 198-199 Closed-loop transfer function, in control

systems 595-596 CMG (See Control moment gyroscope) Coasting phase, of space mission 53 CORE, use of AEM spacecraft 720 Coding of data for transmission 295-298 Cold gas jets 206,209-210 Colored noise 269 Column matrix 745 Commands, transmission of 279,292 Committee on Space Research (COSPAR),

U.N. Committee 52 Commutation 293-294 Commutation angle, in magnetic control

system Commutation mode, in magnetic control

643

INDEX

system Commutator Commatater dwmel Compl!lll!llt uncertainty (of attitude) Compound attitude measurements

In single-axis attitude solutions Computer environment, Goddard Space

643 293

293-294 375

357-359 370

Flight Center Computer programs (See Software) Computers, onboard spacecraft COIle angle

682

210-213 23

CODe angle measurement (See Arc length measurement)

Cone intersections, analytic procedure for determining 364

368-370 691 691

364-370

Attitude solutions using Subroutine for (CONES8)

CONESS (subroutine) Use of in attitude solutions

Conk sections Coning (spacecraft attitude motion) Conjunction

Inferior vs. superior

38-40 489

49 49

Conservation of angular momentum, related to Kepler's Second Law 40

826--829 814-825 807-813

Constants, general Solar system Unit conversion factors

Consumables Control (See Attitude control)

18

Control bardware (See specifiC item, e.g., Gas jets)

Control law (See also Attitude control) 502,589 Implementation via onboard computer 210

Control moment gyroscopes 196,200-201 Nutation damping with 631

Control torque vs. disturbance torque 498 Control torques (See attitude control) ControDer, in control system Convergence, in a batch least-squares

estimator In a Kalman filter In estimation theory In integration procedures

Conversational software system Conversion, of telemetry data to

589

455-456 467-468

451 560 686

engineering units 304, 306--307 Conversion factors, for 51 (metric) units

807-813 Convolution integral Convolutional encoding Coordinate systems

243,768-769 282

24-31 Notation Parallax Table of inertial Transformations

Coordinated Universal Time (UTC)

xi 30 28

760-766

798, 801- 802 Attached to data 298

Copernicus spacecraft (OAO-3) (See OAO) Cordoba Durcbmusterung (Star Catalog) 143 Core catalog (star catalogs) 147 Cores, magnetic 205 Correction of invalid data 296--297, 307 Correlation, among measurement types 478

Among observations & noise in state estimation 444-445

In estimation theory 452-453 Limited attitude accuracy due to

390-391,394 Of a single measurement type at

different times 478-482 Of measurement uncertainties

374,378-379,382-383 Correlation angle 346,353-357

Expressions for 384, 388-389 Figure summarizing relationships 391 For a single measurement type at

different times 478-482

Correlation coefficient 378, 429, 452 COS (European Astronomy sateIlite}-

Slit horizon sensor ISun sensor 169,178-179,721

Cosine detector (Sun sensor) 156-0-159 Cosines, law of (in spherical triangles)

33-34, 731-732 COSPAR (See Committee on Space

Research) COSPAR International Reference

Atmosphere I \0 Covariance 429 Covariance analysis 429-434

In state estimation 452-453, 461, 465 Covariance matrix (See Error covariance

matrix) 430--434 CO2 (See Carbon dioxide; Infrared radiation) Crescent (illumination phase) 331 Crew, disturbance torques due to motion

of . 578-580 Critical angle prism (Sun sensor) 159-160 Critically damped control system 593 Cross product (See Vector product) CROSSP (subroutine) 691 Crosstalk, in magnetometers 250 CTS (Communications Technology SateJlite}-

ApplIcatIOn of block averagmg to attitude solutions 371

Attitude acquisition 661-663,666,669-670,672

Attitude data simulator 710--711 Attitude determination 10--12 -Accuracy of 397 -During attitude acquisition 422-424 Attitude software structure 698 Attitude support system 700--703 Attitude system of 788-789 Control system description 6 .. ' -613, 622-625 Correlation among measurement types

480-484 Data collection for bias determination 474 Determination of bias on Earth

angular radius 483 Determination of sensor mounting angle

bias Deterministic attitude subsystem Effect of flexibility on attitude dynamics

483 370 556 506 Gas jet control system

Horizon sensors Mission profile

169, 175-176

Modeling torque due to orbit maneuvers Multiple component software Nutation damping Spacecraft Spin rate change due to orbit maneuvers Stabilization method Sun sensor Telemetry data errors Use of body-mounted horizon sensor Use of carbon dioxide band horizon

4-8 582 716 630

6 582 503 157 311 173

INDEX

sensor Use of open-loop control

Curie point

835

92 663 115

Curve fitting Cusp Cusp region, geomagnetic field CylIndrical coordinates

317,318-322 90

120 760

-D-

D' Alembert, Jean 38 Damping (See also Nutation damping;

Libration damping) Of a control system 591-593 Of inertial control systems 658-659

Damping matrix (for flexible spacecraft) 553 Dark angle 88 Data-

Acquisition and transmission process 278-298

Correction if invalid 296-0-297, 307 Generation and handling of, onboard

spacecraft 278-283 Handling invalid data 315 Processing at receiving stations 303-304 Time tagging 298-304 Transmission from receiving station to

attitude computer Data adjuster, of CTS attitude system

292 701

Data anomalies, procedures for identifying 334-339

Data averaging, for single-frame attitude solutions 370--373

Data conversicu, in telemetry processor

Data dropout Data errors

. 304,306-0-307 310

310--311 Checking for in mission support

software 682-683 311 Table of representative examples

Data filters (See also State estimators) Data filters, Butterworth vs. least

squares quadratic vs. averaging Definition

325-327 437

317,322-327 Use for data smoothing Data flagging (See Flags) Data bandling, in telemetry processor 304-308 Data preparation subsystelll$, in attitude

software systems Data selection requiring attitude

information Data smoothing (See Smoothing) Data transmission (See Telemetry) Data validation (See Validation) Data weighting (See also Attitude

determination accuracy) DATE (subroutine) Date, conversion subroutines for Julian

697

334-339

370--373 692

836

dates (JD, DATE) Day, apparent.solar

Mean solar Sidereal

DC (subroutine) DE (Dynamics Explorer satellite}

692 799 805

799, 805 691

INDEX

Attitude system of 788-789 Effect of flexibility on attitude dynamics 556 Use of carbon dioxide band horiwn

sensor 92 Deadband, in attitude control Deadbeat boom deployment

607 669-671

64 28

Decay, of Earth satellite orbits Declination (See also Celestial coordinates) Deep Space Network, Jet Propulsion

Laboratory Definitive attitude determination

requirements Definitive orbit Delayed command system Delta functions Delta Launch Vehicle DELTIM (subroutine)

284

681 62

640 xii

3-5 693

Density of attitude loci (See Measurement density)

Density of Earth's atmosphere Descending nocie Design of mission support software Determinant, of a square matrix Deterministic attitude (See Attitude

107 44

681-713 748-749

determination, deterministic methods) Deterministic subsystem, in attitude

software systems 697 701 Of ITS attitude system

Development of mission support software 681-695

Deviation of the vertical Diagonal matrix Diagonalization, of a matrix Differential correction (See State

821 745

752-754

estimation; Data filters; Least squares) Differential correction subroutine (DC) Differential correction subsystem, of CTS

attitude system . Differential equations, solution using

691

702

Laplace transform 770-771 Differential spherical trigonometry 734--735 Diffuse reflection, torque due to 572 Digital codes 295-298 Digital processors (See Onboard computers) Digital sun sensor 156, 161-166

Spinning or one-axis, mathematical model

Two-axis, mathematical model Dihedral angle .

223-224 224--227

23 Dihedral angIe measurement (See Rotation

angle measurement) Dipole model, of geomagnetic field (See

Geomagnetic field, dipole model) Dipole moment, magnetic 204

Dirac delta function xii Direct match, for star identification

259,260-262 Direct orbit 53 Direction cosine matrix (attitude matrix)

Kinematic equations of motion for Parameterizations Summary of properties Table of as function of Euler angles

Directrix Discretization error, in integration

procedures Disturbance torques (See also SpecifiC

torques, e.g., Aerodynamic) As control system inputs Distinguished from control torques Due to engine misalignments Due to orbit maneuvers Environmental -Frequencies of Internal Mathematical models of Table of

411,424 512

410-420 761-762

764 40

560

502 589 498

580-583 580-583 566-576

318 576-580 558-587

17 558 Treatment of for attitude propagation.

Divergence-In Kalman filters Of a vector function

467-468 756

DODGE (Department of Defense Gravity Experiment Satellite}-

Attitude system of Dot product, of vectors Double stars, in star catalogs Downlink Draconic month Drag, on spacecraft orbits Drag coefficient Drift-

In gyroscopes Of spacecraft attitude

Drift rate ramp, iii gyroscopes

788-789 747

145-146 278

52 64

64, 574

200 498 200

DSCS (Defense Satellite Communications System}-

Shadow modeling for 573 Dual-flake horizon sensors 171 Dual-scanner single-axis attitude solutions 368 Dnal-spin spacecraft 202-203,601

Attitude acquisition via momentum transfer

List of Nutation of Pitch control Stabilization by

Dumbhell, rotation of Duty Cycle

Horizon sensor output Of reaction wheel command

Dynamo.: equations of motion For flexible spacecraft Torque free

667-669 787

536-539 617

503-505 487-488

172 270

521-523 552-555 524--529

Dynamics (See Attitude dynamics) Definition

-E---

e (Base of natural logarithms), value Earth (See also Nadir)

Albedo of Appearance from space -Due to oblateness Appearance of horizon at 14 to 16

microns As attitude determination reference

source Atmosphere of -Composition -Effect of during eclipses -Horizontal temperature gradients -Models -Structure -Table of properties -Variation in structure --Vertical temperature cross section Bias in sensed angular radius -Procedure to measure Dark angle Geocentric and geodetic coordinates

487

826

83 83-106 99-103

92-98

17 106--110

108 77

94-96 109-110 106-109

820 109 96

235 483

88

820--822 Geometrical distortion of surface as seen

from space Gravitational field models Horizon of (See Horizon) lllumination of as seen by nearby

87 123-129

spacecraft 334 Inertial rotational position of (GST) 802-805 - Irregulari ties in 805 Infrared appearance of 90--98 Infrared radiation from 82, 90--98 Magnetic field (See Geomagnetic field) Modeling procedures 82-110 Models of surface shape, table 98 Oblateness--Effect on orbit 65-69 -Modeling 98-106 -Term in gravitational potential 124 Orbit of (See also Ecliptic) 44,48-51 Path of conical scan on. 87, 178 Position of relative to Sun, Moon, and

planets 138-142 Properties of, table 819-821 Radiation balance, table 82 Radiation from 82-98 Shadow cone of 75 Shape of 99-103 Symbol for 50 Terminator -Identification -Modeling

331-334 86--90

INDEX 837

Thermodynamic equilibrium of Torque due to reflected sunlight from Visual appearance of

82 571

83-87 Earth-in (See In-triggering) Earth-Moon Lagrange points Earth-out (See Out-triggering) Earth referenced spacecraft

Table of characteristics

55

605, 608-625 612 828 Earth satellites, period vs. altitude table

Earth sensor (See also Horizon sensor) Combined Earth/Sun sensor 178-179

83,169 172, 3·p-349

348 105

336--339

Visible vs. infrared Earth-width measurement

Density of Error due to un modeled oblateness Pagoda effect in

Earth-width model, for horizon sensors 231,233-234

Earth-width/Sun angle single-axis attitude solutions 368

406--407 3,5

45-46 38,46

65 72, 75-77

75-77

Singularities in Eastern Test Range launch site Eccentric anomaly Eccentricity Echo I satellite Eclipse

Conditions for Of the Sun (See Solar eclipse)

Ecliptic As reference for solar system orbits Obliquity of Relative to celestial coordinates

Ecliptic coordinates Use in three-axis attitude system

Eddy current Iibration damper Eddy current nutation damper Eddy current nutation damping Eddy current rods, use of for Iibration

damping Eddy currents, torque due to Effective torque, for gas jet Eigenaxis inertial guidance maneuvers Eigenvalue Eigenvector Eigenvectors and Eigenvalues-

Interpret. of for attitude matrix Of moment-of-inertia tensor

Electric thrusters, for attitude control Electromagnetic units Electronic noise (gyroscopes), mathe

matical model Electronics modeling, attitude sensors ELEM (subroutine)

Basis of Elements--

Of an orbit (See also Orbit elements, Keplerian orbits)

Of a matrix Elevation-

44 48 48 27 28

425 632

626-627 614

633 575 274 661 752 752

411-412 519

19 811-813

269 242-249

692 60-62

46 744

Component in local tangent coordinates 30 Component in spherical coordinates 25

838

Ellipse Elliptical orbit, tahte of properties Ellipticity, of the hlirth Elongation, of a planet Encoder Energy-

38-40 47 99

49-50 280

DissipatiO!1, effect of on rotation 499-501 Required for spacecraft launch 54 Units and conversion factors 810

Energy ellipsoid 499-501 Discovery of 521

Energy optimal magnetic maneuver 642,648 Environmental torques (See Disturbance

torques) EPHEM file (See EphemeriS file) Ephemeris 36

Algebraic approx. for Sun, Moon, and planets 140-143

Spacecraft 133-,.-134 Sun, l\o1oon, and planets 138-142

Ephemeris file 133 Format 135 Subroutines for reading 693

Ephemeris subroutines-Analytic-for spacecraft, Sun, Moon,

pla.nets 692~93 Definitive (i.e., data set) 693 General-purpose Sun, Moon, spacecraft

(EPHEMX) Ephemeris TIme (El) EPHEMX (subroutine) Epoch (orbit parameter) Epoch, of celestial coordinates Equation of the center Equation of time

693 798,802

693 46 27

140 801

Equator (type of coordinate system) Equator, of a spherical coordinate system Equatorial e1ectrojet

28 24

123 692

48 EQUIN (subroutine) Equinoxes

Precession of -Sub. for updating coord. (EQUIN)

ERRS, use of AEM spacecraft Error-correcting codes Error covariance matrix

Analysis for attitude determination

27 692 720

296-297 452

378-379, 429-434 Error ellipse 376-381, 434 Error parallelogram 345-346,374-376 Error sigual, in control system 502-503 Errors in softwar~, avoidance of 682~83, 685 ERTS (Earth Resources Technology

Satellite) (See also Landsat)-Attitude system of 788-789 Horizon sensors 176-178

Escape velocity (See Velocity of escape) ESRO (European Space Research

Organization Satellites)-Attitude system of Nutation damping

788-789 627

EstImation theory (See State estimation)

INDEX

EstImation theory techniques Euler angles

Formulas

447-470 417-420 763-765

Kinematic equations of motion for

Euler axis, of rotation Euler rotation angle

513-514,765 413 413

Euler symmetric parameters (See also Quaternions) 414-416,583,758-759,762

Euler's equations 522; 558 General form for Earth-referenced spacecraft609~10

Mechanical integration via gyroscopes· 559 Solutions for torque-free motion 524-528

Euler's theorm 412, 487-488, 761~762 Basis of 754

EUVE, use of AEM spacecraft 720 Evaluation of state estimators 471-473 Even parity 295 Exosphere 106 Exospheric temperature 106 Explorer (general U.S. satellite name) 790 Explorer 1-

Instability of rotation Nutation

-F-

501 626

Fading memory (in Kalman filters) 463 Fall time (gas jet) 273 False sightings, in star sensors 192 Feedback control systems 502, 588~OO Filters (See Data filters; State estimators) Final value theorm 768

166 227-230

228

Fine Sun sensor Mathematical models Reticle pattern and photocell output

First point of aries (See Vernal equinox) Origin of term 27

Fixed-head star trackers 186, 189-190, 193-195 List of spacecraft using 797

Flags-Associated with telemetry data Internal vs. external Set by telemetry processor

Flake (component of horizon sensor) Flat spin Flattening, of the Earth Flexible spacecraft dynamics

Effects on equations of motion Flight path angle

313-315 315 307 171

501,576 99

18,548-556 552_555

61 Float torque derivative noise (gyroscopes),

mathematical model 269 Float torque noise (gyroscopes), mathe

matical model 269 FLTSATCOM (Fleet Satellite Communica

ti 'n System)-Attitude system of

Fluxgate magnetometer Mathematical model

790-791 182-184 249-254

Flyby trajectory Flywheel Focus, of an ellipse Force, Units and conversion factors Forced attitude solutions Forced response, of a linear system Forcing function, of differential equation Fourier series, use to solve linear

differential equations Fourier transform Frame, of data Frame synchronization signal Framing of telemetry data Fredholm equation Free response, of a linear system Freon, as gas jet fuel Fresnel reflection Friction modeling, for reaction wheels Fuel-

60 201 38

809 370 770 770

614 243 293 293 293 773 770 210 156 271

Budget for gas jets Loss due to engine misalignments Tanks, torques in Used for gas jets

207 582-583 577-57-8

207

-G-

Gain matrix, in sequential estimators 461 Gal (unit of acceleration) 809 Galactic coordinates 28 Gamma (unit of magnetic induction) 811 Gas jets-

Attitude control systems using 503, 506 Attitude maneuver analysis for 649-654 Disturbance torques due to propellant

slosh 577-578 Effects of thrusting on flexible space-

craft 551-552 Example of use in attitude control 14-16 Hardware description 206-210 List of spacecraft using 796 Math~matical models 272-275 Nutation damping with 630-631 Use for attitude control 19 Use for attitude stabilization 622-625

Gauss (unit of magnetic induction) 811 Gauss, Katl 113, 779 Gauss' Equation 45 Gauss-Newton least-squares procedure 448,455 Gaussian coefficients, Geomagnetic field 117

Table of 779 Gaussian measurement errors 430-431

Probabilities assQCiated with 381 Sample computation 434

GCI (See Geocentric Inertial Coordinates) Gemini program, use of horizon sensors

167, 168, 180 General perturbations, method of (orbit

analysis) 139 Geocentric coordinates, conversions with

geodetic 820-822

INDEX 839

Geocentric Inertial Coordinates 29 Geodetic coordinates, on the Earth's

surface Geoid Geoid height

Map of

820-822 98,99,125

99 125

Geomagnetic field (See also Magnetic, magnetometers, etc.) 113-123,779-786

Accuracy of models 118-119 Analytic approximations for 782-785 As attitude determination reference

source Dipole model -For Earth-referenced spacecraft -Rectangular components -Spherical components Diurnal variation General description Index of geomagnetic activity Magnetic storms Mathematical models Models Perturbations of Secular drift Solar perturbations Spherical harmonic model Subroutine for (MAGFLD)

Geometric albedo

17 782-785

613 784 783

122-123 113-120

122 121

779-785 117-123 120-123

113 120-123 779-182

693 79

Geometric method of three-axis attitude determination 421-424

Geometrical axes of spacecraft 487 Geometrical limitations of attitude

accuracy Applications

GeometryAttitude Effect of changes on information

389-402 397-402

22-35

content of measurements 478-482 Spherical (See Global geometry plots;

Spherical geometry) GEOS (Geodynamics Experimental

Ocean Satellite)Attitude acquisition Attitude system of Control system description Data records

662,666,672-677 790-791 612-617

304 Data sample Deadbeat boom deployment Fitting magnetometer data Libration damping Magnetic stabilization Momentum wheel Spacecraft Stabilization method

312,313,314 669 321 632 612

202-203 672

State vector for bias determination 503 441 157 Sun sensor

-Data correction Use of open-loop control

GESS (See Graphics Executive Support System)

GETHDR (subroutine)

330 663

693

840

Use of GETV (subroutine) Gibbous (illumination phase) Gibbs vector

Kinematic equations of motion for Gibbs, J. Willard Gimbal (gyroscope support)

133 693 331

416,763 512-513

416 196 196 Gimbal rotation axis (gyroscope)

Gimbaled star trackers 186, 187-189 List of spacecraft using

Global geometry plots Construction of Explanation of For attitude determination Spacecraft orbit on

797

737-743 22-26

397-399 398

Subroutines for (SPHGRD, SPHCNV, SPHPLn 694-695

Global Positioning System 8-10 GMT (See Universal Time) Goddard Space Flight Center

Attitude ,Data Link Computer environment Information Processing Division

292 682

292,299-303 Network Operations Control Center 284 Role in CTS mission 7 Role in receiving & relaying data 284 SCAMA (Switching, Conferencing, etc.) 291

Goddard Trajectory Determination System (GTDS)

subroutines for GOES (Geostationary Operational

Environmental Satellite)-Application of block averaging to

attitude solutions Attitude acquisition Attitude determination accuracy Attitude software structure Attitude system of Correlation among measurement

133-134 693

371 661 397 698

790-791

types 480-484 Data collection for bias determination 474 Determination of sensor mounting angle

bias 483 Fitting attitude solutions 322-323 Orbit generator accuracy 138 Spin rate change due to orbit maneuvers 582 Sun sensor analysis 717-718 Telemetry data errors 311 Use of carbon dioxide band horizon

sensor Use of open-loop control Use of body-mounted horizon sensor

Goodness-of-fit function GPS (See Global Positioning System) Gradient, of a scalar function Gradient operator Gradient searcb, method of differential

correction GRAPH (subroutine) GnpIIic software systems,

92 663 173 318

756 756

455 694

686--690

INDEX

Graphic Subroutine Package (GSP) 687 Graphic support systems \ 686--690 Graphics (See also Interactive graphics) Graphics Executive Support System

(GESS) Graphing subroutines Gravitation, Newton's development of

laws

688-690 694-695

Gravitational constant, accuracy of Gravitational constants, Earth, Moon, and

36-38 41

Sun 827 Gravitational field models 123-129 Gravitational potential 123-129 Gravity assist trajectory 60 Gravity-gradient attitude control 614-617 Gravity-gradient capture sequence 672-677 Gravity-gradient stabilization 19, 503, 505-506

Conditions for 611-612 List of spacecraft using 787

Gravity-gradient tensor 128-129 Gravity-gradient torqlJe 17

Effect on flexible spacecraft 549, 551 For dual-spin spacecraft, math.

model For Earth-referenced spacecraft Mathematical model

Gray Code--

568-570 609

566-570

Algorithm for conversion to binary 306-307 Conversion table 164 Output vs. Sun angle for sensors using 165 Reason for use 163-164 Reticle pattern for 164

Great circle 22 Construction of on global geometry plot 742 Equations for 728-729 Properties of 32

Greatest elongation 50 Greenwich Hour Angle 802-803 Greenwich Mean TIme (GMn (See also

Universal Time) 19,801 Greenwich meridian 80 I

Subroutine for right ascension of (RAGREN)

Greenwich Sidereal Time (GSn Gregorian calendar century GRO, use of MMS spacecraft

692 802-804

809 720

GSFC (See Goddard Space Flight Center) GTDS (See Goddard Trajectory

Determination System) Gyroscopes-

Accuracy of attitude propagation with for HEAO-I 708-709

As attitude determination reference 17 Attitude propagation with 564-566, 558-559 Biases 198,200 Effect of misalignmentS on slew

maneuvers 660 Hardware description 196-201 Mathematical models 266-270, 558, 559 Measurements from (rate and rate

integrating) 266-270

Modeling noise effects Solution of differential equation Spacecraft using

268-270 771-773

797 Gyrotorquer (See Control moment gyroscope)

-H-

Half-angle formulas, for spherical triangles 732 Hamming Code 296-297 Harmonic oscillator-

Equation for forced 614 Solution for forced 771-773

Haversines 735-736 Advantages over normal trig functions 735

HCMM (Heat Capacity Mapping Mission) (See also AEM)-

Attitude acquisition Attitude system Control system description Momentum wheel Nutation damping Scan wheels horizon sensor

662,672 720, 790-791

612-613,617-621 202 630

176-178 Use of carbon dioxide band horizon

sensor Yo-yo despin

HD number (star catalogs) Heading (orbit parameter) Heading angle (gas jet precession) HEAO (High Energy Astronomy

Observatory)Attitude acquisition Attitude system of Control system description Fixed-head star tracker Gyroscopes measurements Image dissector tube star sensor

92 667 143 61

652

662 790-791 605-608

190, 193-195 266-270

measurements 256-258 Inertial guidance maneuvers 655,661 Inertial reference assembly 197 Instrumental magnitude for star camera 258 Large data volume 308 Momentum wheel 202--203 On board computer 211,212-213 Star catalog for 147 Star tracker attitude determination

706,/08-709 Two-axis Sun sensor 158 Use of q method for attitude

determination 427 Heat pipe 627 Heat sink, use in horizon sensors 171 Height, used for distances measured from

the Earth's surface 43 Heliocentric coordinates 29 HelmholZ coil, for magnetometer testing 250 HEMITR (subroutine) 693 Henry Draper star catalog 143 HEOS (Highly Eccentric Orbit Satellite)

Attitude system of 790-791

INDEX

Nutation damping Hermitian matrix Hohmann transfer orbit (See also Plane

change orbit maneuvers) Between the planets, table

Horizon-Appearance of at 14 to 16 microns Definition dependent on sensor Identification of Of an oblate Earth

Horizon crossing vector (See also Earth width; Sun sensor/horizon sensor rotation angle)

Computation of For oblate Earth In single-axis attitude solutions

Horizon plane, for an oblate Earth Horizon sensors--

841

627 750

5&-59 824

92-98 167

331-334 99

238-239 103-105

370 101

Analysis of representative poor geometry for 397-399

Biases 234--237 Components of 169-172 Data validation 329 Example of use in attitude

determination Geometry of Hardware Lis t of spacecraft using Mathematical models Mathematical models of electronics Model of azimuth biases Optical system of Output Pagoda effect bias at small Earth

widths Path of scan on the Earth Representative output Representative spectral response Representative telemetry data errors Rotation angle (from Sun sensor)

models Slit horizon sensor/Sun sensor Use for single-axis attitude Use for three-axis attitude Visible vs. infrared

Horizon spheroid, for an oblate Earth Horizon/Sun rotation angle (See Sun

10-12 231

166-180 797

230-242 244--249 239-242

170 171-172

33&-339 87

172 170 311

237-242 178-179 362-409

426 83,169

101

sensor / horizon sensor rotation angle) Hot gas jets 206, 207-209 Hour angle 799 Hour angle of Greenwich Meridian,

subroutine for (RAGREN) Hour Angle, Greenwich Hour, measure of right ascension HR number (star catalogs) Huygens, Christian Hydrazine-

As gas jet fuel Thrust characteristics for

Hydrogen peroxide, as gas jet fuel Hyperbola

692 802-803

28 143 38

207-209 273-274

207 38-40

842

Hyperbolic anomaly 47 Hyperbolic orbit, table of properties 47 Hyperbolic velocity 42 Hypergolic fuel 53 Hysteresis, torque due to 575

-1-

Identity matrix IGRF (See International Geomagnetic

Reference Field)

745

Illumination during partial eclipse Illumination of planet, as seen by nearby

spacecraft

76

334 Illumination of sphere, as function of

phase, distance Image dissector, in star tracker

78-79,89 189

Image dissector tube star measurements, mathematical model 256-259

IMP (Interplanetary Monitoring Platform~ Attitude software structure 698 Attitude system of 790-791 Effect of flexibility on attitude dynamics 556 Telemetry data errors 311 Use of body-mounted horizon sensor 173 Use of convolutional encoding 282

Improper orthogonal matrix 751 Impulse (of force) 206 Impulse (of torque) 274 Impulse response function 242 Inclination, (orbit parameter) 44, 46

Maneuver to change 59 Of Earth satellite orbit 53

In-crossing (See In-triggering) Induction magnetometer Inertia wheel

181-184 201

Inertia, moment of (See Moment of inertia) Inertial coordinate systems 26-28

Table of 28 Inertial guidance 16 Inertial guidance maneuvers 655-661 Inertial nutation rate 490, 526, 535 Inertial reference assembly, for HEAO-1 197 Inertial spin rate 490 Inertially referenced spacecraft 605-608 Inferior conjunction 49-50 Inferior planet 49 Infinitesimal spherical triangles 734 Information Processing Division (IPD)

292, 299- 303 Infrared borizon sensors (See Horizon

sensors) Infrared radiation, from Earth Initial Valoe Theorm Injection

82,90-98 768

53 60-62

747 56-59

196

Determination of orbit elements from Inner product, of vectors In-plane orbit maneuvers Input axis (gyroscope) Instantaneous rotation axis, of

INDEX

spacecraft Instrumental star magnitudes

487-488 258

Integral equations, solution using Laplace transform 773-774

Integration methods 560-564 Choice of 564 Errors in 560 Subroutine (RUNGE) 692

Interactive graphics, use in mission support software 682, 686-690

Interactive software system 686 Interfaces, data set vs. core storage 699 Internal torques on spacecraft 576-580 International Astronomical Union, 1976

adopted astronomical constants 827-828 International Atomic Time (T Ai) 798, 802 International designation, of spacecraft 52 International Geomagnetic Reference Field 118

Coefficien ts of 779 Subroutine for (MAGFLD) 693

International System of Units 807-813 Prefixes 807

International Telecommunication Union 283 Interplanetary environment 130-132 Interplanetary rtight-

Sample calculation 58-59 Table of orbit properties 824--825

Interplanetary probe, distinguished from satellite

Inter-Range Instrumentation Group Intersecting cones, attitude solutions Intersection, of attitude measurement

52 282

12

loci Interstellar probe INTP (subroutine) In-triggering (Earth-in) Inverse Laplace transform Inverse, of a matrix INVERT (subroutine) Ion jets Ion thrnsters, for attitude control

345-346 52,60

693 II, 172,358

767 749-750

691 206

19 IPD (See Information Processing Division) IR (See Infrared) ISEE (International Sun Earth Explorer~


Attitude determination accuracy 397 Attitude software structure 698 Attitude system of 790-791 Earth and Moon coverage for ISEE-C 402 Effect of flexibility on attitude dynamics 556 Gas jet control system 210 Panoramic scanner 169, 173-175 Slit horizon sensor/Sun sensor

169, 178-179,721 Stabilization method Use of convolutional encoding

ISIS (International Satellite for Ionospheric Studies~

Attitude system of ISS (Ionosphere Sounding Satellite~

503,504 282

790-791

Attitude system of 7~791 ITOS (Improved Tiros Operational

Satellite>--"-Attitude system of 79~793 Momentum wheel 202,601,602

IUE (International Ultraviolet Explorer>--"Application of block averaging to

attitude solutions 371 Attitude acquisition 662 Attitude determination accuracy 397 Attitude software structure 698 Attitude system of 792.:..793 Computation of attitude determination

accuracy 384--386 Gas jet control system 207-209 Inertial guidance maneuvers 655, 659, 660 Momentum wheel 203 Onboard computer 210 Panoramic scanner 169, 173-175 Reaction wheels 270, 272, 601 Star tracker 190 Sun sensor 157, 166 Thruster characteristics 273 Use of convolutional encoding 282

-J-

Jacchia atmosphere models Jacobian elliptic functions JD (See Julian Day) JD (subroutine) JDS (See Julian Day for Space) Jet damping Jet Propulsion Laboratory, Deep Space

Network Jets (See Gas jets)

110 526-528

692

582

284

Jordan canonical form, for a square matrix 753 Joseph Algorithm (in sequential estimators) 461 JPL (See Jet Propulsion Laboratory) JPL ephemeris tapes 140

Subroutine for (RJPL T) 693 JSC (See Lyndon B. Johnson Space Center) Julian century 809 Julian Day 20

Conversion subroutines for (JD, DATE) 692 Table 804

Julian Day for Space 20 Julian period (basis of Julian Day) 20 Jupiter, effectiveness for gravity assist

trajectory Jz perturbations Jz term, in gravitational potential

-K-

60 67-69

126-127

INDEX

KaimanfUter For star data

448,462~9 708-709

Propagation of state & error covariance matrix 464--467

Kapustin Yar (Soviet launch site) Kepler, Johannes Kepler's equation

843

4 36-38

45 Numerical solutions of

Kepler's First Law Kepler's Laws Kepler's Second Law

46, 134, 140 37-40 37-42 37-40

Kepler's Third Law Keplerian orbit

Table of properties Keplerian orbit elements

As function of position & velocity, subroutine for

As function of injection conditions Table of

Kernal, of integral equation Kilogram, definition Kinematic equations of motion

Euler angle representations Torque free

Kinematics (See Attitude kinematics) Kinetic energy-

Of rotational motion Orbital (See also Vis viva equation)

Knudsen number Kosmonavt Vladimir Komarov (Soviet

tracking ship) Kronecker delta

-L-

37,41-42 35-37

47 46

692 60-62

46 773 807

511-514 765

529-531

517-519 38

108

290 xii

Lagrange, Joseph 55 Lagrange point orbits 55 Lagrange points 55 Lambert sphere 79

Reflected intensity from 85 Landmark tracking, potential use for

attitude determination 724 Landsat (See also ERTS; Earth Survey

Satellite) -Attitude system of 792-793 Horizon sensors 180 Landsat-D, use of MMS spacecraft 720 Momentum wheel 202

Laplace transform 767-774 Application to control systems 590 Example of use to solve linear differential

equations 615-616 Table 769

LaplaCe's equation, in spherical coordinates 775 Latitude, geocentric vs. geodetic vs.

astronomical 82~822 Latitude component, of a spherical

coordinate system 25 Launch

Added velocity required for prograde orbit 53

Energy required for 54 Launch phase, of space mission 3-7 Launch sites

844

Soviet U.S.

Launch vehicles Reignitable upper stage U.S.

Launch window, attitude accuracy constraints

Law of Cosines Law of Sines LBN coordinate system (orbit defined) Lead time constant, in control systems Least-squares estimator (See also Batch

3 2-3

53 3

399-401 33-34

33 28

591

least-squares estimator) Analytic basis Example of

437 447-470

454-455,456--459 322-327 691--692

Least-squares quadratic filter Least-squares subroutines Legendre functions, & Legendre

polynomials Legrange, Joseph Leibnitz, Goffried Length, units and conversion factors LES (Lincoln Experimental Satellite)-

776 45 38

808

Attitude system of 792-793 LFO (Landsat Follow-on Satellite)

Use of MMS spacecraft Libration Libration damping

Use of deadbeat maneuver for Libration points (three-body orbit) Lifetime, of Earth satellites Lift (aerodynamic) Light year Lighting conditions, on spacecraft Limit checking Limited stability, of a control system Line of apsides Line of nodes Linear independence, of a set of vectors Linear operators Linear system

720 625

631--633 669--671

55 64 63

808 71~80

314 594

44 44

745

Relationship to Laplace transformation

767 242 767

Lit horizon Local horizontal coordinates Local tangent coordinates Locator (horizon sensor signal

processing) Loc_

In defining attitude measurements Intersection of in attitude

85 30 30

94,172

344

measurements 344-346 Long Range Navigation-C

(LORAN-C) 299-300 Longitude component, of a spherical

coordinate system 25 Longitude of subsatellite point 805 Longitude of tbe ascending node 49 LORAN-C (See Long Range Navigation-C) Loss function (in estimation theory) 449-451 Loss of signal (See Out-triggering) Loxodrome (rhumb line) 653

INDEX

202 Lubrication, of spacecraft wheel bearings Lunar Orbiter (spacecraft), attitude

reference system i89 142-143

285 Lunar parallax Lyndon B. Johnson Space Center

-M-

MAGFLD (subroutine) 693 Magnetic attitude control (See also

Magnetic stabilization) 18-19,617.:..621 Hardware for 204--205 Maneuver strategy 639.:..642, 644--649 Maneuvers 636.:..649

Magnetic coil control systems 503,508-509 Magnetic coils 204--205 Magnetic dipole moment 204

Units and conversion factors 812,813 Magnetic disturbance torque 575-576 Magnetic equator 114 Magnetic field-

Interplanetary 13~132 Of the Earth (See GeomagnetiC field)

Magnetic field strength, units and conversion factors 81 f, 813

Magnetic nux, units and conversion factors 811 Magnetic induction field 251 Magnetic induction, units and conversion

factors 811,813 Magnetic materials- 205 Magnetic moment 204

Of a current distribution 252 Units and conversion fllctors 811,813

Magnetic nutation damping 614,629--630 Magnetic observatories 122 Magnetic permeability 813 Magnetic precession 636.:..649 Magnetic stabilization 671--672

List of spacecraft using 787 Magnetic storm 121 Magnetic systems of units 811 ~ 13 Magnetic torque 17 Magnetic torquing-

Continuous List of spacecraft using Quarter orbit (QUOMAC)

Magnetometers Bias determination Biases Data--Curve fitting for -Residual errors in -Validation of Example of use in attitude

determination List of spacecraft using Mathematical models On Apollo 15 subsatellite

Magnetopause

639 796 640

18.~184 329-330 251-254

321 328 328

13-14 797

249-254 495-496 106,120

Magnetosheath 120 Magnetosphere 106, 120 Magnetotail . 120 Magnitude (scale of brightness) 77

Instrumental 258 Moon and planets, table of 816-817,823 Of planets, satellites, and spacecraft 77-80 Sample calculation for spacecraft 79 Stellar 144--145

MAGSAT (Magnetic Satellite)-Attitude software structure Attitude system of Evaluation of horizon sensor data Fixed-head star tracker Need for accurate Sun sensor Scan wheels horizon sensor Star catalog for Use of carbon dioxide band horizon

699 792-793

471 193-195

166 176-178

147

sensor 92 Major frame 293-294 Major principal axis 500, 625-{;26 Maneuver control·(See Attitude maneuver

control) Marginal stability, of a control system Mariner (spacecraft), attitude reference

system Mariner 10 Marquardt's Algorithm Mars-

Hohmann transfer orbit to Oppositions of

Mask detector (Sun sensor) Mass expulsion system (See Gas jets) Mass expulsion torques Mass, units and conversion factors Master catalog (star catalogs) Master frame (telemetry) Master station (time signals) MATMPY (subroutine) Matrix Matrix algebra

Subroutines Matrix functions Matrix inversion

Subroutine (INVERT) Matrix multiplication subroutine

(MATMPy) Matrix notation Matrix Riccati equation Matrix transformations Mean (of a random variable) Mean angular motion

for Earth satellite Mean anomaly

Rate of change of Mean distance, in an elliptical orbit Mean free path, in atmosphere Mean of date coordinates Mean orbital elements Mean sea level (See also Geoid; geoid

height)

594

189 60

455-456

58-59 51

158

576-577 808 147

293-294 300 691 744

744--757 691

754-755 749-750

691

691 x

465 751-752

429 47,67

67 45-46

67 38

108-109 28 46

98,99

INDEX 845

Mean Solar Time Mean Sun

798,801,805

Measurement, as used in state estimation Measurement covariance matrix Measurement density (See Attitude

801 444 461

measurement density) Measurement uncertainty (See Attitude

determination accuracy; Uncertainty) Mercato~ Gerhardus 653

653 Mercator representation Mercury (planet), relativistic rotation

of perhelion Mercury program, use of horizon

sensors Meridian Mesopause Mesosphere Message vector (in a Hamming code) Metre, definition Metric conversion factors Micrometeorites, torque due to Minicomputers, use in attitude software

system Minor, of a matrix Minor frame

63

167,168 24

106 107 296 807

807-813 17

699 748 293

Minor frame counter (minor frame ID) 293-294

Minor principal axis 625-{i26 Misalignment, of rocket engine, torque

due to 58a-583 Mission Control Center 285 Mission mode, of space flight 661 Mission operations phase, of space mission 3-8 Mission orbit 53 Mission profile-

Future changes in 8-12 Representative 3-12

Mission support (See also Software) 681-713 Example of role of attitude determination

& control 3-8 Requirements during Space

Shuttle era 716-722,724 Software 681-713

MMS (Multimission Modular Spacecraft) (See also SMM-MMS-A)

MMS series spacecraftAttitude system Computer used on Momentum wheels Sun sensor

MOD coordinates (See Mean of date coordinates)

Modified Julian Day

718-720 210 203

166-167

Modulus, of Jacobian elliptic functions Molniya (Soviet communications satellite) Moment of inertia

21 526 290

Estimate of for Apollo 15 subsatellite Of spacecraft Transverse Units and conversion factors

Moment-of-inertia tensor

497 489 524 810

518-520

846

Momentum bias Momentum bias control system

Design of Momentum dumping

Need for Using gas jets

Momentum transfer maneuvers Momentum wbeels (See also Dual-spin

spacecraft; Reaction wheels)

201 600--603

203 602 203 650

667-669

As part of horizon sensor system 176-178 Attitude control systems using 503,507-508 Dynamic equations of motion for

spacecraft wi th Effect on spacecraft nutation Euler's equations for Hardware description List of spacecraft using Torque model Use in attitude acquisition sequence Use in attitude stabilization Use in inertial guidance maneuvers

Monopropellant gas jet Month-

Types of -Numerical values for

Moon-

522-523 536--539

559 201-203 787, 796

656 667-669 600--603 655-661

206

52 822

Analytic ephemeris subroutine (SMPOS) 693 Dark angle 88 Definitive ephemeris subroutines

(SUNRD, RJPL T) 693 Effect on acceleration of Earth satellite 127 Effect on Earth satellite orbit 63,70--71 Identification of from RAE-2 data 335 Lagrange points with Earth 55 Located horizon dependent on

temperature 168 Magnitude and phase law of, table 823 Multipurpose ephemeris subroutine

(EPHEMX) 693 Numerical values of different types 822 Orbit of 139, 141-143 Parallax of 142-143 Properties of 822-823 Properties of orbit 51-52 Shadow cone of 75 Types of 52

Moon-centered coordinates (See Seleno-centric coordinates)

Moving arc filter 322 Moving edge tracker (Earth sensor) 179-180 Multimission software 686, 721-722 Multiplexor 280 Multistep integrators 561

Nadir Nadir angle

-N-

Error due to unmodeled oblateness Nadir angle measurement

22,83 12, 23

105 344

INDEX

Density of Information content

Nadir angle/Sun angle measurement,

348 482

information content 480--482 Nadir angle/Sun angle single-axis attitude

solutions Singularities of

Nadir cone Use in attitude determination

Nadir vector

368 406-407

12 365

12 Nadir vector projection model, for horizon

sensors 232-233 Napier, John 34 Napier's Rules 34

For quadrantal spherical triangles 731 For right spherical triangles 730, 731

NASA Communications Network 291 NASA Monograph atmosphere model 110 NASA Standard Spacecraft Computers

(NSSC-I & NSSC-2) 210--212 NASCOM (See NASA Communications

Network) National Bureau of Standards, timekeeping

system 299 Natural frequency, of control system 591 NA VSTAR (See Global positioning

system) Near-real-time attitude determination

requirements Network Operations Control Center Neumann normalization, of Legendre

polynomials Neutral sheet, geomagnetic field Newcomb, Simon Newton, Issac Nimbus (meteorology satellite}--

Attitude system of Digital Sun sensor Horizon sensors Momentum and reaction wheels Momentum wheel State vector for bias determination Telemetry data errors Yaw reaction wheel

Nineteen Fifty (1950) Coordinates Node, of an orbit

Motion of (See Regression of nodes) Nodical month Noise, White vs. colored

681 284

776 121 802

36--38

792-793 157, 162 176--178

601 202

441-442 311 270

27 44

52 269

Noise correlation, as criteria for observa-tion model selection 444-445

Nongravitational forces, effect on orbit 63--65 Nonintersecting loci, use of in attitude

solutions Nonreturn to zero level pulse generation Nonreturn to zero mark pulse generation Nonspherical mass distribution, effect on

370 281 281

orbit 63, 65--69 Nonspinning Earth Sensor Assembly

169,175-176

Normal matrix 751 Notation-

For attitude angles 349 Use in this book x-xii

NSSC (See NASA Standard Spacecraft Computers)

Null (attitude geometry parameter) 350 Use of in evaluating correlation angles 391

Null (optical center line of a sensor) 188 Nutation-

Astronomical 804 Description from spacecraft frame

494,499-501 Effect on sensor da ta Effect on spin period measurement Effect on Sun angle data Equation for, in spacecraft frame Example of Apollo 15 subsatellite Measurement of Monitoring via Sun angle data Physical description

Nyquist criterion, for system stability

-0-

OAO (Orbiting Astronomical Observatory)-

Attitude system of Evolution of accuracy requirements Inertial guidance maneuvers Momentum wheel Onboard computer Reaction wheels Stabilization method Star trackers Sun sensor

Oblate spacecraft

534--548 544-548 535-536 522, 525 495-497 534--548 539-548 490-494

596

792-793 714

655, 660 202 210 601 503 188 157 491

Oblateness of Earth (See Earth, oblateness) Oblique spherical triangle-

Equations for components of Table of general solutions

Obliquity of the ecliptic Observability

In least-squares estimators Of state vector elements

731-734 733, 735

48

450 443

Observation (as used in state estimation) 443-444

Observation models-As used in state estimation Construction of Criteria for selecting Testing of

Observation model vector Observation residual vector Observation vector OCC (See Operations Control Center) Occultation Odd parity

444 443-447 443-447

473 439,449

449 439

Oersted (unit of magnetic field strength)

72-75 295 811

INDEX

OGO (Orbiting Geophysical Observa-tory)-

Attitude system of Deformation due to solar heating Horizon sensor Rubidium vapor magnetometer

847

792-793 550

168,180 184

On-off control law (See Bang bang control law)

Onooard computers 210--213 Interaction with ground-support

facilities 719-720 Use in attitude acquisition 662 Use of for HEAO attitude propagation 708

Onooard processing 8-9 One-step integrators 561 One's complement arithmetic 297 Open-loop control system 502, 589, 604, 663 Open-loop transfer function, in control

system Open loop zeros, of root locus diagram Operational procedures-

596--598 597

For identifying data anomalies For use of state estimators

Operations Control Center Role in CTS mission

334--336 473-476

285,292,299 7

Operations phase, of space mission (See Mission operations)

Opposition (planetary configuration) Of Mars, table of

Optical double stars Optical pumping

49 51

146 184

Optimal attitude determination methods (See also State estimation) 426--428

ORBGEN (subroutine) 141,692 Basis of 134--135

Orbit decay 64 Orbit defined coordinate systems 28-29

Use for three-axis attitude 425 Orbit determination I, 132 Orbit elements-

As function of injection conditions 6()'~{)2

Subroutine for determining from position & velocity 692

Orbit file format 133-134 Orbit generators-

Keplerian (ORBGEN) Numerical -Accuracy and applications of

Orbit maneuvers Torques due to

Orbit normal Orbit perturbations Orbit vs. trajectory Orbital motion of spacecraft, attitude

correction for

692 134-138 137-138

56--60 580--583

8 62-71

S3

365-366 Orbits (See also Ephemeris subroutines)

Apparent shape when viewed obliquely 73 Definitive spacecraft 133-134 Earth satellites 828 Example of types in typical mission 6--8 Lunar and planetary 138-142

848

Numbering of 53 Planetary 48-52 Potential artificial satellites of planets 825 Solar system 815,817,824-825 Subroutine for elements from position

and velocity (ELEM) Table of equations Two-body generator (ORBGEN)

Orthogonality, of vectors Orthogonal matrix Orthogonal transformation Osculating orbital elements OSO (Orbiting Solar Observatory)

Attitude system of Dual-spin stabilization Gas jet control system Gravity-gradient torque on Magnetic torquing on OSO-8 .Momentum wheel Nutation damping Orbit generator accuracy Spin axis magQetic coils Stabilization method Telemetry data errors V -slit star scanner

Out-of-plane orbit maneuvers Out-crossing (See Out-triggering) Out-triggering (Earth-out) Outer product, of vectors Output axis (gyroscope) 9verdamped control system Overshoot, in control systems Ozone, absorption bands

-P-

Pagoda effect Mathematical model

Panoramic Scanner

692 47

692 747 751 752 47

792-793 505 209 568 640 203 626 138 205 503 311

187, 190-192 56

II, 172,359 748 196 593 593

91-92

336-339 244-249

(attitude sensor) 169, 173-175 Data from 335 Mathematical models 231-242 Model of biases relative to Sun sensor 241

Parabola 38-40 Parabolic anomaly 47 Parabolic orbit, table of properties 47 Parabolic velocity 42 Parallax 30-31

Lunar 142-143 Parallel (component of a spherical

coordinate system) Parallel telemetry formats Parameter, of Jacobian elliptic functions Parity code Parking orbit Parsec (unit of distance) Partial derivatives-

24 279 526 295

5-6,53 808

Numerical vs. analytical evaluation 432-433 Procedure for testing correctness of 473

INDEX

Partial eclipse Passive attitude control Passive attitude stabilization Passive nutation damper PECE, integration method Pendulum nutation damper Penumbra Penumbral eclipse Periastron Pericyanthiane Perifocal distance Perifocus Perigee

Rotation of -Numerical formula for

Perigee height Maneuvers to change

Perihelion Perilune Period (o~bit parameter)

In an elliptical orbit

72,76 503-506

18-19 625

563":564 626

72 72 42 42 42 42 42 66

68-69 43 59 42 42

41 Of Earth or Sun satellite as function of

semi major axis 808 Of Earth satellite as function of altitude,

table 828 Of Earth satellite, numerical values 54, 828

Permalloy (use in magnetic coils) 205 Permeability of vacuum 813 Permendur (use in magnetic coils) 205 Perturbations-

In solar, lunar, and planetary ephemerides

Of orbits (See Orbit perturbations) Phase angle (See also Azimuth angle)

Of solar iIlumination Phase law-

Of Moon, table Of planets and satellites

Phase match, for star identification

139

78

823 78-79

PHASED (subroutine) Use in attitude computations

Photochemical reactions, in Earth's atmosphere

259,264-265 691 366

108 Photodiode, as energy detector for horizon

sensor 170, 178 Physical constants 826-829 Pi ('11'), value 826 Pioneer, nuclear power supply 156 Pioneer 10, 11 60 Pitch angle 360 Pitch axis 29 Pitch control, for dual-spin spacecraft 617 Pitch gain, in control systems 591 Plane change, orbit maneuvers 56, 59 PLANET (subroutine) 693

Analytic basis 14F-142 Planetary configurations (planetary aspects) 49

Symbols for 50 Planetary index, of geomagnetic activity 122 Planets-

Analytic ephemeris subroutine for (PLANET)

Definitive ephemeris subroutine (SUNRD)

Illumination of as seen by nearby

693

693

spacecraft 334 Magnitudes of 77-80 Orbits of 48-52, 138-142 -Table 815 Properties of 814-817 Spheres of influence on satellite orbits 71 Sym boIs for 50

Plant, in control system 589 Plesetsk (Soviet launch site) 4 PLOTOC (attitude data simulator) 711-712 Plots-

Computer generated (See Graphic support systems)

Of celestial sphere (See Global geometry pl&ls)

Plotting subroutines Plume (gas jet exhaust) Poinsot's construction, for rigid body

rotation Polar electrojet Polaris (pole star)

As attitude reference

694-695 208

530--531 123 27

189 Pole centimetre (urtit of magnetic dipole

moment) Pole, of a spherical coordinate system Poles, of control system transfer function POLYFT (subroutine) Polygon match, for star identification Polynomial fit, subroutine for (POL YFT) Poor geometry regions, for attitude

812 24

590 691 263 691

determination 389-402 Position-only control system 656-657 Position-plus-rate control system 590, 657-658

Use of by HCMM 617 Use of by HEAO-A 606-608

Postprocessing of attitude results vs. data preprocessing

Poundal (unit of force) Power, units and conversion factors Poynting-Robertson effect Poynting vector Preaveraging (to reduce data volume) Precession (attitude motion) Precession of the equinoxes

Subroutine for updating coordinate (EQUIN)

Precomplier, use in software systems Predicted vs. observed plots, for CTS

attitude system Predictor-corrector integrators Preprocessing of attitude data (See also

Smoothing; Validation; Telemetry processor)

Contrasted with postprocessing of results

318 809 810 64

156 317

14,498 27,48

692 684

702 563-564

310--334

318

INDEX 849

Effect on statistics 317 Pressure, units and conversion factors 809 Primary (one of two objects in an orbit) 38 Prime meridian 25 Principal axes 519

Discovery of 521 Of spacecraft 488-489 -Stability of rotation about 523

Principal moments of inertia 489,519 Printer plot subroutines (GRAPH, SCALE) 694 Probability density (of attitude) 375,376 Process (in state estimation) 438 Product of inertia 519 Prograde orbit 53 Programmable telemetry format 295 Programming (See Software) Programming standards Project Operations Control Center Project Scanner Prolate spacecraft Propagation (See Atlitude propagation) Propellant (See also Fuel)--

Disturbance torques due to slosh Proper motion Proper orthogonal matrix Proportional control system Proton precession magnetometer PSCTS, use of MMS spacecraft Pseudoevaluation (in numerical

integration) Psendoinverse, of a matrix Pseudoinverse state estimator Pulse amplitude modulation Pulse code modulation Pulse duration modulation Pyroelectric detectors, for Earth sensing

-Q-

684-686 285

92-96 491

577-578 144 751 590 184 720

564 749

468-469 281

280--281 281 171

Q metbod, of three-axis attitude determination

Quadrantal spherical triangle Equations for components of

Quadrature

421,426-428 34

731 50

Quality assurance-Of attitude solutions, need for Of software

Quality nag, on attitude data Quantb:ed measurements

Attitude determination accuracy Variance of

Quantum magnetometer Quarter orbit coupling Quarter Orbit Magnetic Attitude

Control (QUOMAC) Basis for

Quatemions (See also Euler symmetric parameters)

Algebra of

716 686 314 374

374-376 431

181, 184 601

640 785

758-759

850

Components of Kinematic equations of motion for Norm of Representation of attitude by Use in control laws Use for attitude propagation

758 511-512

759 414 605

INDEX

558-559, 564--566 Quicklook displays 305 QUOMAC (See Quarter Orbit MagnetiC

A ttitude Control)

-R-

RADECM (subroutine) Radiance profiles, of Earth in infrared Radiation balance, for the Earth Radiation pressure, torque due to Radiometric balance Earth sensor Radius of gyration, of gas jet propellant RAE (Radio Astronomy Explorer)-

Application of block averaging to

690 95 82

570--573 180 650

attitude data 371 Attitude acquisition 661,666 Attitude software structure 698 Attitude system of 792-793 Data from panoramic scanner 335 Deadbeat boom deployment 669 Effect of flexible booms 549, 555, 556 Gas jet control system 210 Gravity-gradient stabilization 505-506 High-altitude attitude magnetometers 181 Panoramic scanner 169, 173-174 Stabilization method 503 Sun sensor 157 Telemetry data errors 311 Thruster characteristics 273 Time-tagging of playback data 302-303 Use of convolutional encoding 282

RAGREN (subroutine) 692 Random error, in attitude measurements 402 Range and range rate, orbit measurements 132 Rank, of a matrix 749 Rate gyroscopes 196, 197-198 Rate-integrating gyroscopes 196, 199-200 Reaction wheels-

Hardware description Mathematical models Use for attitude control Use in attitude stabilization Use in inertial guidance maneuvers

Real-time attitude determin~tion requirements

Receiving stations Data processing at

Rectangular coordinate system,

201-203 270--272

19 603--604 655--661

681 291-292 303-304

advantages relative to spherical 25-26 RECUR (subroutine) 692

Application of on AE spacecraft data 318 Recursive estimator 437,439,448

Subroutine (RECUR) Recursive least-squares estimator

Advantages & disadvantages Reference meridian Reference orbit

692 459-462 461-462

25 62

Reference point, in spherical coordinate system 25

99 Reference spheroid, in Earth models Reference vectors (See Attitude determina

tion, Reference sources, see also individual reference vectors, e.g., Sun, mag. field)

Reflected binary code (Gray code) Reflection, specular vs. diffuse Regression of nodes

Numerical formula for Relativistic effects on orbit Residual, rms Residual editing Residual magnetic dipole Retrograde orbit Return-to-zero pulse generation Revolution vs. rotation Rhumb length Rhumb line Rhumb line attitude maneuver Rigid spaceeraft motion Right ascension (See also Celestial

coordinates) Related to time

295 571 66 68

37,63 453 320 252

53 281

53 654 653

652--654 523-524

28 802-803

Right ascension of the Greenwich Meridian (Greenwich Sidereal Time)

Subroutine for (RAGREN) 803 692

44,46 59

Right ascension of the ascending node Maneuver to change Motion of (See RegreSSion of nodes)

Right spherical triangle Equations for components Example of exact triangle

Rise time (gas jet) RJPLT (subroutine) Roberts atmosphere model Rocket engine misalignments, torque

due to Roll axis Roll, Pitch, Yaw coordinate system Root locus diagram

Use fo( selection of control gains Rosman STDN tracking station Rotating coordinate frames, rate of

change of vectors in Rotation-

34 730--731

730 273

140,693 110

580--583 29 29

596--600 622

284--289

514-515

Distinction from nutation and coning 489 Distinction from revolution 53

Rotation angle 23 Formula for 727 Formula for, in vector notation 757 Subroutines for (PHASED, VPHAZE) 691

Rotation angle measurement 23,346,349-352 Density 352 Notation for 23

Singularity conditions 406-407 Use in deterministic attitude solutions

364,365,369 Rotation angle models (Sun sensor/horizon

sensor) 237-242 Rotation axis of spaceaaft . 487-488 Rotational motion (See Attitude dynamics,

attituqe kinematics) ROUND (subroutine) 693 Round-off error, in integration procedures 560 Routh-Hurwitz criteria-

Example of use For nutation damper study For system stability

Row matrix Royal Greenwich Observatory ROITAP (subroutine)

Use of RPY coordinates (See Roll, Pitch, Yaw

coordinates) RVNGE (subroutine)

Analytic basis for Runge Kutta, integration method

Subroutine for (RUNGE)

S~band data transmission S-band telemetry subbands

619 626 596 745 20

693 134

692 562

561-562 692

282 282

SAGE (Stratospheric Aerosol Gas Experi-ment satellite) (See a/so AEM)-

Attitude system of 792-793 Scanwheels horizon sensor 176-178 Use of AEM spacecraft 720

San Marco Pladorm launch site 4 SAO number (star catalogs) 143 SAS (Small Astronomy Satellite)-

Analysis of dynamic motion Attitude acquisition Attitude system of Constant current source on SAS-3 Data records Disturbance torques Earth-width data Evaluation of horizon sensor data Fixed head star tracker Image dissec~or tube star sensor

538 661

792-795 206 304 580

233-234 471

193-195

measurements 256-258 Instrumental magnitude for star camera 258 Large data volume 308 Launch of SAS-I (Uhuru) 4 Momentum wheel 202-203

N-slit star scanner analysis 705, 706, 707 Nutation damping 627,629 Observation model for bias

determination Optimal magnetic maneuvers Programmable telemetry formats Scanwheels horizon sensor

445-446 640 295

176-178

INDEX

Spin axis magnetic coils Spin plane magnetic control Stabilization method Star catalog for Star identification for Star scanner Star tracker analysis State vector for bias determination Sun sensor Telemetry data errors Telemetry processor

Satelli~

851

205 646 503 147 263 187

706-707 440-441

157 311 305

Defined 38, 52 Distinguished from interplanetary probe 52 Local mean time of subsatellite point 68 Longitude of subsatellite point 805 Magnitude of (brightness measurement)

77-80 Names (See Spacecraft, names) Orbit of 52--62 -Lifetime 64 -Period around Earth or Sun vs.

semimajor axis 808 -':Period vs. altitude above Earth, table 828 -Potential artificial satellites of planets 825 -Properties 825 -Utility subroutines for 692--693

Satellite Automatic Tracking Antennas 284 Satellite Command Antenna 284, 289 Satellite tracldng stations (See Tracking

stations) Satellites, natural, table of properties Saturation limit, of momentum wheel Saturn V launch vehicle Scalar checking, for data validation Scalar product, of vectors SCALE (subroutine) Scale height, of atmosphere Scaliger, Joseph SCAMA (See Switching, Conferencing,

and Monitoring Arrangement)

817 508

3 328-334

747 694 108 20

Scanning mechanism, employed by horizon sensor 169

Scanwheels horizon sensor 169, 176-178 Sc;hmldt normalization, for spherical

harmonics· 780 Score (numerical measure of star

identification) 260 Scout launch vehicle 3 Search coil magnetometer 181 Search pattern (in fixed head star trackers) 189 SEASAT (ocean studies satellite)-

Anticipated horizon radiance variations 98 Attitude system of 794-795 Canted momentum wheels 602-603 Clock 299 Control system description 612--613,621--622 Effect of flexibility on attitude dynamics 556 Locator used on horizon sensor 172 Momentum wheel 202 Scanwheels horizon sensor 170, 176-178

852

Sun sensor -Coverage of Use of carbon dioxide band horizon

sensor Second (ephemeris) Second (SI unit)

Definition

157,166 402

92 799 802 807

INDEX

Secondary (one of two objects in an orbit) Sectoral harmonic coefficients,

38

explanation of Secular drift, geomagnetic field Secular terms, geomagnetic field model Selenocentric coordinates Self -correcting codes Semiconjugate axis, of hyperbola Semimajor axis Semiminor axis

777-778 113 779

29 296-297

40 38,46

38 40 Semitransverse axis, of hyperbola

Sensor electronics, mathematical models 242-249

Sensors (See A lIitude sensors; see also Object sensed, e.g., Horizon sensors)

Sequential estimator 437,448 For star data 704,707-709

Serial telemetry formats 279 Shadow bar Sun sensor 159 Shadow cone 72

Of Earth and Moon 75 Shadowing-

Effect on aerodynamic torque Effect on radiation torque

Short period variations, in orbital elements

Shuttle (See Space shullle) SI (metric) units

Prefixes Sidereal day Sidereal month Sidereal period

574 573

65

807-813 807

804-805 52 50

Distinguished from solar period for Earth satellite 55

Sidereal time 798, 802-805 Subroutine for (RAGREN) 692

Sidereal year 48 Sifting (to reduce data volume) 317 Signal conditioner 279 Similarity transformation 752 Simpson's Rule, for integration 561 Simulators, for attitude data 709-712 Simultaneous linear equations, solution of

749-750 Sines, law of (in spherical triangles)

Single-axis attitude Single-axis attitude determination

Accuracy Methods

Single-degree-of-freedom gyrnscope Single-spin spacecraft Singular matrix Singularities, in attitude solutions

33,731-732 343-346

373-409 362-409

196 503-504

749 403

Singularity conditions in attitude determination

Table of Sinter, Use in thermistor flake SIRIO (Italian experimental

communications satellite~

406-407 407 I7I


Attitude accuracy constraints on launch window 399-401

Attitude determination accuracy 397 Attitude software structure 698 Attitude system of 794--795 Correlation among measurement types 480 Slit horizon sensor/Sun sensor 178-179 Spin rate change due to orbit maneuvers 582 Sun sensor analysis 717-718 Use of body-mounted horizon sensor 173 Use of carbon dioxide band horizon

sensor Skew-symmetric matrix Skylab-

92 750

Attitude control system 197,201 Disturbancetotques due to crew motion 579 Spacecraft configuration 579

SKYMAP, star catalog 147 SKYNET (U .K. Communications

satellite}-- Attitude system of Skywave (in radio broadcasts) Slave station (time signals) Slew maneuvers Slit horizon/Sun sensor

794--795 301 300

601,655:...661 178-179

As possible standard coarse sensor Math. model for misalignment

Slit sensors, Analysis of alternative designs

Slit star sensor, mathematical model SLP ephemeris files

Subroutine for reading (SUNRD) Slug (unit of mass) SM (San Marco satellite)

721 219

717-718 254--256

140 693 808

Attitude system of 794--795 Small circle (spherical geometry) 22, 32

Area formulas 729-730 Construction of on global geometry plot

Equations for 739, 742-,743

727 Smithsonian Astrophysical Observatory,

star catalog 143-144, 146-147 SMM (Solar Maximum Mission) (See

alsoMMS~ Attitude acquisition 672 Attitude control law for 659 Attitude system 718~720, 794--795 Control system 608 Data collection for bias determination 475 Fine Sun sensor 166-167 Inertial reference assembly 187 Onboard computer 210

Smoothing, of attitude data and results 315-327

Application~ of Guidelines fdr

SMPOS (subroutine) Analytic basis

SMS (Synchronous Meteorological Satellite)-

Application of block averaging to

316 317-318

693 141-142

attitude solutions 371 Attitude acquisition 661 Attitude determination accuracy 397-399 Attitude software structure 698 Attitude solutions from 373 Attitude system of . 794--795 Behavior of single-frame solutions 403-405 Correlation among measurement types

478,480-482 Data collection for bias. determination 474 Earth-width data 233-234 Horizon sensor electronics

modeling Launch of Pagoda effect Sensor package characteristics State vector for bias determination Sun sensor analysis

244--249 5

336--339 721 440

717-718 311 173

Telemetry data errors Use of body-mounted horizon sensor Use of carbon dioxide band horizon

sensor Use of open-loop control View of Earth by

Snapshot, of star sensor data Snell's law Software-

92 663

84,91 706 223

Avoidance of errors in 682--683,685 Development of 681-713 Example of attitude support software

structure 700--703 For multimission support 686,721-722 General structure for attitude

support 696--700 Goddard Space Flight Center

environment 682 Safeguards for mission support 681--686 Standardization of 686,721-722 Systems, generaf structure for 696--700 Test procedures for state estimators- 471-473 Utility subroutines 69~95

Solar eclipse 72 Solar heating, effect on flexible spacecraft

Solar mass ratio, for planets Solar parallax Solar radiation-

Flux Pressure, effect on orbit -Effect on flexible spacecraft Stabilization Torque

Solar sail Solar System-

549,550 827--828

31

130 64--65

550-551 19

17,570-573 64

INDEX

Orbits Properties

Solar time Solar wind

Sectors Solid angle

Formulas Units and conversion factors

Solid spherical harmonics Sounding rocket

853

48-52 814--825

798, 799'-801 120, 129-'132

131-132 23

729-730 810 775

52 South Atlantic Anomaly (See Brazilian

Anomaly) Soviet Space Program

Launch sites Tracking and data acquisition

Space cone Space Mission, profile of

Future changes in Representative

'. Space navigation

3 3

290 491-492

8-12 3-12

I S~ce Precision Attitude Reference

-'System (SPARS) 708 3,8-9 Space shuttle

Effect on attitude determination and control 714,724

Orbit ephemeris 134 Payload mass as a function of altitude 9 Star tracker for 190 Thrust 53

Space TelescopeAttitude system of Onboard computer Pointing accuracy Reaction wheels Stability requirements

Spacecraft-Data generation and handling

onboard Effects of flexibility on dynamics Gyroscope measurement of angular

velocity Magnitude of when viewed from a

distance

794--795 211 714 604 604

278-283 548-556

267-268

Names and international designations Stabilization and control, methods of Stabilization, methods of (See also

79 52

18-19

Attitude stabilization) 3 Spacecraft attitude control (See Attitude

control) Spacecraft attitude determination and

control systems Spacecraft attitude dynamics (See

Attitude dynamics) Spacecraft attitude motlon

Example of Apollo 15 subsateIlite Introduction to

Spacecraft axes, alternative systems Spacecraft-centered celestial sphere Spacecraft-centered coordinates Spacecraft clocks Spacecraft ephemerides-

787-797

495-497 487-502 487-489

22-24 26--29

298-299

854 INDEX

Definitive subroutines 693 Two-body orbit generator (ORBGEN) 692

Spacecraft fixcd coordinates 26 Spacecraft orbits 52-62, 132-138

As function of injection conditions 60--62 MUltipurpose ephemeris subroutine

(EPHEMX) Spacecraft stability (See also A Ititude dy

namics; Disturbance torques;

693

Nutation; Flexible spacecraft ~vnamics) 523 Apollo IS subsatellite 495-497 With respect to hbration (GEOS

example) Spaceflight Tracking and Data Network

674

(STDN) 283-290 Time-tagging by 299-301

Spacelab, onboard computer 211 SPARS (See Space Precision Altitude

Reference System) Special perturbations, method of (orbit

analysis) 139 Specular reflection 84

Torque due to 572 SPHCNV (subroutine) 694 Sphere, illumination of as function of phase,

distance 78-79 89 Sphere of influence- '

F or spacecraft orbits 69-71 Table of for planets 69-71

Spherical coordinate systems 24--31, 760 Advantages relative to rectangular 25-26 Properties of 24--26 Transformations between 765-766

Spherical excess 32 Spherical geometry 31-35

Construction of global geometry plots 737-743

Equations for 727-736 Spherical harmonics 775-778

Expansion of gravitational potential in 124 Recursion relations 781 Representation of the geomagnetic

field Schmidt normalization for

Spherical plot subroutin\:s (SPHCNV,

779-782 780

SPHGRD, SPHPLT) 694-695 Spherical plots (See Global geometry plots) Spherical triangle 23

Infinitesimal 734 Notation for 33 Properties of 32-35

Spberical trigonometry 33-35 Differential 734--735 Example of 34--35 Table of general solutions 733, 735

SPHGRD (subroutine) 694 SPHPLT (subroutine) 694 Spin-axis drift,.during rocket engine firing 582 Spin-axis precession, magnetic 636-649 Spin rate-

Change during rocket engine firing 582

Control using gas jets 650 Effect of nutation on measurement

of 544--548 19, 503~504

3 787 172

231,234 467

Spin stabilization Spin-stabilized spacecraft

List of Split-to-index time, horizon sensors

Math. model (split-angle model) Square root filter SSS (Small Scientific Satellite)

Attitude system of 794-795 Effect of flexibility on attitude dynamics 556 Nutation damping 544,629 Telemetry data errors 311

ST (See Space Telescope) Stability-

Of control systems 594-600 Of rotation about a principal axis 523 Of spacecraft (See Spacecraft stability)

Stabilization (See A Ililude swbilization; Nutation damping; Libration damping)

Standard deviation 429 Standard notation

For attitude angles Standard symbols Standard Time Standardization-

Of attitude hardware Of attitude software

Star azimutb Star camera

x-xii 2~, 349

xi-xii, 50 801

718-721 686, 721-722

259 193

Star catalog, acquisition of by an attitude system

Star catalogs Star longitudes Star scanners

List of spacecraft using Star sensors-

704 143-150 149-150

186, 187, 190--192 797

Attitude determination methods for 703-709 Characteristics vs. accuracy require

ments Data selection and correction Example of use for nutation

monitoring Hardware List of spacecraft using Mathematical model of intensity

response Mathematical models Overview of attitude determination

with Representative telemetry data errors

190 704--705

538~539 184--195

797

258-259 254--259

703-709 311

Star trackersFixed-head Gimbaled List of spacecraft using

Stars-

186, 189-190, 193-195 186, 187-189

797

Angular diameter of 167 As attitude determination reference

source 17 Densities 145

Distribution, mathematical model of 263 Identification 706 Identification techniques 259-266 Instrumental magnitudes of 258 Magnitudes 144-145 Position modeling 143-150

State (in differential correction or "state" estimation) 439

436-483 437-438 407-408

State estimation Advantages relative to deterministic Need for Operational limits on accuracy Subroutine for (DC) Use in CTS attitude system

State estimators--Analytic basis Operational use of Prelaunch evaluation Subsystem in attitude software

476 691 702

447-470 471-484 471~473

system 697-698 Unresolv~d analytic problems 722-723

State noise covariance matrix 465 State plane trajectory 657

Examples of 658 For HEAO-I 607

State space 657 State transition matrix 450 State vector 436, 438-443

Choice of elements to be solved for 476-483 Construction of 438-443 Need for 407-408 Subroutine for updating (RECUR) 692

State vector elementsChoice of Observability of -Limitations

State weight matrix

439-443 443

476-483 449

STDN (See Spaceflight Tracking and Data NetwQrk)

Steady state (in Kalman filters) Steady-state error, in control systems Steady-state system response Steady-state trajectory

467 593 770 608

Steepest descent, method of (differential correction technique)

Steering law Stellar parallax Stepsize, in numerical integration STEREOSAT, use of AEM spacecraft Stictlon Stiffness matrix (for flexible spacecraft) STORMSAT-

Disturbance torques Use of MMS spacecraft

Stfapdown torque rebalanced· gyroscope Stratopause ~tratospbere SubcOmDlutated data Sub$atellite, Apollo 15, nutation of Subsatellite point (See a/so Nadir)

Alternative definitions Local mean time of

455 604

31 560 720 272 553

580 720 199 107 107

293-294 495-497

22 83 68

INDEX 855

Longitude of 805 Subsolar point 84 Summing point, in control system 589 Sun- (See also Solar)

Analytic ephemeris subroutines (SUN IX, SMPOS) 693

Approach by spacecraft 60 As attitude determination reference

source 17 Definitive ephemeris subroutines

(SUNRD, RJPLT) 693 Effect on acceleration of Earth satellite 127 Effect on geomagnetic activity 120-123 Energy flux from 130 Expression for mean motion 141 Interference with horizon sensors 169 Multipurpose ephemeris subroutine

(EPHEMX) Properties of Solar wind Symbol for

Sun angle measurement Density of For nutating spacecraft Information content

693 818

120, 129-132 50

11,23,344 347

539-548 480

Sun angle / nadir angle single-axis attitude solutions 368

406-407 12

364 156, 159-161

11,13 717-718

Singularities in Sun rime

Use in attitude determination SUD presence detector Sun sensor

Analysis of SIRIO vs. SMS design Calculation of coverage of celestial

sphere Calibration constants Combination Sun/Earth horizon

sensor Data validation Example of use in attitude control Example of use in attitude

34-35 230

178-179 329-330

14-16

determination 10-14 Field of view 225-226 Hardware 155-166 List of spacecraft by Sun sensor type 797 Mathematical models 218-230 Model of azimuth biases relative to

horizon sensor Nutation monitoring with Simple vs. complex Two axis. accuracy analysis Use for single-axis attitude Use for three-axis attitude

239~242

539-548 716-718 355-357 362-409

426 Sun .sensor /horizon sensor rotation angle

measurement 357'-359 Information content 480-482 Models 237~242 Use in attitude determination 364-365, 369

SUD shade (for star sensors) 186 Sun-synchronous orbit 68 Sundials 800

856

SUNRD (subroutine) SUNIX (subroutine)

Analytic basis Supercommutated data Superior conjunction Superior planet Surface spherical harmonics

140,693 693 141

293-294 49-50

49 775

Surveyor (spacecraft), attitude reference system

Switching, Conferencing, and Monitoring Arrangement (SCAMA)

Switching line (component of control system)

Symbols, astronomical Symmetric mas,s distribution, principal

axis of Symmetric matri,. Symmetric spacecraft SYMPHONIE (French/German

189

291

657 50

489 750 524

communications satellite)Attitude system of Horizon sensors

79.f-795 180 270 314 293

Synch speed, of reaction wheel Synchronization pattern, quality flag for Synchronization signal, role in telemetry Synchronization word, in NASCOM data

format Synchronous satellite

Of various planets. table Syndrome vector Synodic month Synodic period System gain-

291 55

825 296

52 50

Of a control system 590 Role in root locus diagram 597--600 Selection of for attitude control 622, 624

System mass matrix (for flexible space-craft) 553

Systematic error, in attitude measurements 402 Effect on deterministic solution behavior 404

Syzygy 49

-T-

Tachometers, for measuring wheel speed Tangent height Tangent plane coordinates (See Local

tangent coordinates) TCON20 (subroutine) TCON40 (subroutine) TDRSS (See Tracking and Data Relay

Satellite System) Tenon-

202 93

692 692

INDEX

Use in gas jets Use in momentum wheel bearings

Telemetry Generation and transmission of Time tagging

Telemetry anteODllS

206 202

293-298 278-298 298-304 28.f-288

Telemetry data errors (See also Data errors) 310-311

Table of representative examples 311 Telemetry formats 279, 293-295 Telemetry On-Line Processing System

(TELO PS) 292 Telemetry processor 3~308

In attitude software systems 696--697 Of CTS attitude system 701

Telemetry word 293 Tell-tale (data flag) 313-315 TELOPS (Telemetry On-Line Processing

System) 292 Temperature--

Of Earth's atmosphere Units and conversion factors

Tensor

107 812 519

Terminator Identification of

84,8Cr-90 331-334

Tesseral harmonic coefficients, explana-tion of 777-778

Testing-Of attitude software 686 Of state estimators 471-473

Thermal radiation (See Infrared radiation) Definition of 83

Thermistor, as energy detector for horizon sensor 171, 178

Thermopile, as energy detector for horizon sensor 171, 180

Thermosphere 107' Third-body interactions, effect on orbit

63,69-71 Three-axis attitude 343,359-361 Three axis attitude determination-

Accuracy 429-434 Example of least-squares estimator 456-459 Methods 410-434,420-428

Three-axis stabilized spacecraft 3 Thrust profile (of gasjet) 207,272-175 Tbrnsters (See Gas jets) Time--

Local mean time of subsatellite point 68 Measurement and broadcast facilities

299-302 Units and conversion factors 808-809

Time checking, of telemetry data 307-308 Time measurement systems 18-21,798--806

Conversion subroutines for 692 Table of 798

Time optimal magnetic maneuver 642, 648 Time tagging 278

Near-real-time data 302 Playback data 302 Representative telemetry data errors 311 Telemetry data 298-304

TlROS (meteorology satellite)-Attitude system of 79.f-795 First Use of Quarter Orbit Magnetic

Attitude Control 639

ntius-Bode law (See Bode's Law) TOD (See True of d4te) Torque (See also Disturbance torques;

Attitude control) Average of gas jet Due to magnetic moment Effect of, on spacecraft motion lntemal vs. external

, ,>Units and conversion factors Torque-free motion, of spacecraft Torque-free solutions, for attitude

motion Torr (unit of pressure) Total eclipse UtAAC, libration damping Trace, of a matrix

274, 813

498-502 521 810

487-497

524-531 809

Track pattern (in fixed-head star trackers) Tracking and Data Relay Satellite

72, 76 632 748 189

System (TDRSS) 8-10,287-290 Tracking stations 283-290

Location of 284-285 Timing systems 299-302

Trajectory, of gas jet precession 652 Trajectory, of spacecraft 53 Transfer elements, of a control system 588 Transfer function 244

Mathematical model of horizon sensor electronics 244-248

Of horizon sensor electronics 172 Use in control systems 589-590 Use of to evaluate stability 591-593

Transfer orbit (See also Hohmann transfer orbit) 5---{i, 53

Transfer time, in Hohmann transfer orbit

Between the planets (table) Transformations between coordinate

systems Transit

58-59 824

760--766 71-75

254 Transit time, in slit star scanner Transmission, of data and commands (See

also Telemetry) 278-292 Transpose, pf a,matrix 744 Transverse angular velocity 525 Transverse moment of inertia 524 Trapezoid model, of gas jet profiles 275 Trigonometry, spherical (See Spherical

trigonometry) Trojan asteroids 55 Tropical year 48 Tropopause 106 True anomaly 45-46 True of date coordinates 27-28 Truncation error, in integration procedures 560 Tumbling; of spacecraft due to crew

motion 579-580 Turbopause 108 Tum angle, of hyperbola 4O,60---{i1 Two-axis Sun sensor (See Sun sensor) Two-degree-of-freedom gyroscope 196

INDEX

Two tbousand (2000) coordinates Two's complement arithmetic Tyuratan (Soviet launch site)

UBV magnitudes UDU filter Uburu (See SAS)

-U-

857

27 297

4

144-145 467

Umbra 72 Uncertainty in attitudl~ measurements 345-346

Correlated 374, 378-379, 382-383 Due to systematic errors 402-408, 476 Expressions for 375-376,381-382,384 Uncorrelated 374-382

Uncertainty of state estimator soluti\,ns 476 Undershoot, in control systems 59,3 Unitary matrix 751 Unitary transformation 752 Units and converSion factors 807-813 Universal Time (UT)=GMT=Z

\9,798,801-802 Attached to data

Unpacking, of telemetry data UNVEC (subroutine) Uplink Upper stage rocket vehicles U.S. Coast Guard, time keeping system U.S. Naval Observatory, time keeping

system U.S. Standard Atmosphere UT (See Universal Time) UTC (See Universal Time; Coordinated

Universal Time) Utility subroutines

-V-

298 304-306

690 278

53 299

299-300 110

690---{i95

V-brusb, Sun sensor code 163 V-slit Sun sensor (V beam Sun Sensor) 161

Mathematical model 218-221 Validation-

Data flags and sensor identification Of attitude data

313-314 315-327

307,312-334 Of telemetry data Requiring attitude information

Vandenberg Air Force Base, Calif. (See Western Test Range)

Vanguard (tracking ship) Vanguard units Variance Variation of parameters, formulation of

334-339

284,287 808-809

429

attitude dynamics 531-534 VEC (subroutine) 690 Vector algebra 744-757

Subroutines 690---{i91 Vector calculus 755-756 Vector identities, in three dimensions 756-757

858

Vector magnetometer, mathematical model

Vector multlpUcation, inner and outer products

Vector notation Vector product (cross product)

'Equation for direction of Subroutine for (CROSSP)

Velocity-In a Keplerian orbit Units and conversion fact,)rs

Velocity of escape Planets and satellites, table

Vernal equinox Symbol for

VHF data transmission Vidicon, in star sensors Viking (Mars mission), launch dates

250-254

(47-748 x, xii

756 728 691

54,828 809 42

817,825 27,48

50 282-283

189

relative to oPPOSitiOO1 51, 57-58 Vis viva controversy 38 Vis viva equation 38, 42

Origin of 38 Viscous ring nutation damper 626, 627~29 Visibility of satellites and spacecraft 80 Visibility of spacecraft 71-80 Visible Ught sensor (See also Horizon sensor) 83 Visual magnitude, for spacecraft, sample

calculation Volterra equation VOP (See Variation of parameters) VPHAZE (subroutine)

-W-

Wallops Island, launch site Water, absorption bands Weighting (in estimation theory)

79 773

691

4 91-92

449-450

INDEX

Weighting data Weightlessness Western Test Range (launch site) WHECON, wheel control system

Active nutation damping with Wheel-mounted horizon sensor (See

also Scanwheels) Mathematical models Table of characteristics

White noise White Sands (launch site) World Warning Agency WWV time signals WWVH time signals

Yaw axis Year

Sidereal Tropical Types of

-Y-

Yo-yo despin maneuvers

-Z-

Z (time unit) (See Universal Time) Zenith Zenith angle Zero crossing magnetometer

370-373 41-42

3 613,622 63~31

169 234,236 176-178

269 4

52 299 299

29

799 799 48

663~9

Zeros, of control system transfer function Zonal luu1nonic coefficients-

22 85

250 590

Explanation of In gravitational potential

ZuIu;time i,

777 124,127

801

PART VI APPENDICES

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