Smithsonian

Contributions to Astrophysics

VOLUME 3, NUMBER 9

GEODETIC USES

OF

ARTIFICIAL SATELLITES

by GEORGE VEIS

SMITHSONIAN INSTITUTION

Washington, D.C.

1960
Publications of the Astrophysical Observatory

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Tables, and World Weather Records.

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the Editorial and Publications Division, Smithsonian Institution, Washington, D.C.

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Contents

Page

Introduction 95

The coordinate systems 96

Elliptical coordinate systems 96

Terrestrial rectangular coordinates 97

Sidereal rectangular coordinates .. 97

Geodetic rectangular coordinates 100

Accuracy of the transformations 106

Determination of directions 108

The observations 110

Measurement of distances 110

Optical measurement of directions Ill

Electronic measurement of directions 112

Correlations between the observed quantities 112

Keduction of observations to the station 115

Correction for aberration 115

Correction for optical refraction 117

Correction for electronic refraction 122

Observations to objects of known positions 122

Use of angles in space 122

Use of directions and distances 123

Accuracy of determined positions 126

Observations to objects of unknown positions 129

Unconditioned equations of observation 129

Introduction of conditions 132

Connection of geodetic systems 133

Observations to orbiting objects 135

Quasi-simultaneous observations 136

Use of a first-approximation orbit 136

Use of a second-approximation orbit 139

Use of a third-approximation orbit 143

Orbit of unknown mean distance .. . 145

Applications of the various methods 145

The orbit: ephemerides and visibility 145

The observations 146

Method for a known orbit 147

Method of orbital interpolation 148

Methods for simultaneous observations 150

in
Page

The use of the moon 152

The orbit of the moon: libration 152

The observations 153

Determination of positions and expected accuracy 155

Acknowledgments 157

Symbols used 157

References 159

IV
Geodetic Uses of Artificial Satellites

By GEORGE VEIS2

Introduction

The geodetic methods heretofore applied to the

data provided by the artificial satellites have

been based on a dynamic approach. That is,

the perturbations in the motion of a satellite

have been used to determine the gravitational

field of the earth, and thus to obtain information on the shape of the geoid (Jacchia, 1958b;

O'Keefe, 1958; O'Keefe and Eckels, 1958;

Kozai, 1959).

This paper presents another way of using

the satellites in geodesy. The methods depend

on a geometric approach, which we may say is

related to mathematical geodesy in the same

way that the dynamic approach is related to

physical geodesy. The geometric method consists in performing a triangulation in space and

determining the positions of a certain number

of observing stations whose positions are unknown.

The science of geodesy today covers a wide

field, but its main purpose is to determine the

size and shape of the earth as a whole, or of

large areas on its surface. This problem, at

least at first sight, may seem to be a purely

geometric one. But as soon as geodesists

began using the direction of the vertical as a

reference direction, the problem became dynamic as well. Selecting the vertical as the

direction of reference is fully justified for the

survey of a small area—say, a few square

kilometers—since we can assume that the

directions of the vertical are parallel to themselves over that area. But to relate them over

large areas at different points on the earth we

need to know the shape of the geoid, since the

vertical is defined as being perpendicular to

the geoid.

Since an approximation for the geoid is an

ellipsoid of revolution, the geoid over a large

area can be replaced by a reference ellipsoid.

We therefore use the vertical as if it were the

normal to the ellipsoid, and thus establish the

relation between the reference directions. This

is the method of "classical" geodesy.

The angle between the normal and the vertical (deflection of the vertical) can attain

several seconds of arc, however, and over very

large areas (of the order of continents) the

errors accumulate so that the methods of classical geodesy are not adequate.

With the methods of physical geodesy

(Stokes, 1849;VeningMeinesz, 1928) we try to

find the geoid as it actually is. To do this we

theoretically need measurements of gravity

over the entire earth. Even if these measurements do not cover the whole world, we can

nevertheless obtain an accuracy to within a

few seconds. We can then relate the reference

directions over any large area, provided we

have measurements of gravity.

But even if we have related the reference

directions at the different points, another

problem remains to be solved: the effect of

refraction.

The different lines of the geometric figures

that we establish on the earth's surface are

defined by rays of light. Because of atmospheric refraction the light rays are bent, and

this bending occurs almost completely on a

vertical plane. Although we correct for the

effect of refraction, the proper amount of correction is very uncertain, and the errors in

vertical angles (for long lines) can attain several

seconds. It is important to note, however,

that such errors diminish as the light rays move

away from the earth's surface.

• This paper is based on a dissertation, "Geodetic Applications of Observations to the Moon, Artificial Satellites, and Rockets," presented to the

Institute of Geodesy, Photogrammetry, and Cartography, of Ohio State University.

* Consultant Geodesist, Optical Satellite Tracking Program, Actrophytical Observatory, Smithsonian Institution; and Refearch Associate,

Harvard College Observatory.

05
96 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

These difficulties have led to a duality in the

methods of geodesy. Determinations of positions are made separately, for the planimetry

and for the elevations. The planimetry is

based on a more or less geometric procedure,

triangulation, while the vertical control is

based on a physical method, precise levelling.

The methods presented here depend on observations of orbiting objects such as artificial

satellites. These methods have the advantage

that they use a unique system of reference

direction which is not affected by the earth's

gravitational field (or the geoid), and that

the errors from refraction are reduced to a

minimum. The mathematical tools are fairly

simple, consisting mainly of analytical geometry

in three dimensions.

In principle, the method involves our performing a three-dimensional space triangulation, with the observed objects (satellites) and

the observing stations on the earth as the vertices. On this space-triangulation net we

compute the positions of the stations. Since

the observed object will usually be at a high

altitude, errors from refraction are reduced

to a minimum.

An important feature of the methods presented here is that instead of measuring angles

to obtain directions, we determine them directly, in relation to a reference system defined

by the stars. The relative positions of the

stars, corrected for proper motion, aberration,

and parallax, will be the same for all observers,

everywhere on the earth. Therefore if we connect a system of reference to the stars, we can

then define the system by the equator and the

ecliptic of a certain epoch.

For practical reasons, however, we may also

use a system of reference that is not fixed with

respect to the stars but is moving; i.e., we

use a system defined by the instantaneous

equator and ecliptic. This motion (precession

and nutation) actually will not affect the accuracy of the definition of our reference

system. The accuracy depends only on the

consistency of the positions of the stars, and

the accuracy of their proper motions.

Finally, this system will be connected to a

reference system on the earth. Since the two

systems will have a relative motion, we must

have continuous astronomic observations from

fixed observatories, to connect the two systems.

Although the methods presented here were

devised specifically for use with the artificial

satellites, the same methods can be applied,

with some modifications, to such targets as

the moon, rockets, and even to terrestrial

objects. Because these methods have such

general applications, the presentation that

follows will commonly use the word "object"

rather than "satellite."

The coordinate systems

The several systems of coordinates and their

transformations present various problems.

Elliptical coordinate systems.—The system

commonly used in geodesy is that of ellipsoidal

coordinates. A point Q is defined by projecting

it along the normal to a reference ellipsoid.

The projection Q' on the ellipsoid is defined by

the latitude 4> and longitude X. The distance

QQ' defines the height H of the point Q above

the ellipsoid.

Some question exists whether the curved

vertical should be used instead of the normal.

For points near the surface of the earth, however, the differences are extremely small. For

a height of 10 km, for example, the differences

are 4 cm in <j>, 0 in X, 10~4 cm in H.

In an ideal system of coordinates, the reference ellipsoid would be centered at the center

of the mass of the earth and the 3-axis would

coincide with the mean axis of rotation. This

is called the terrestrial ellipsoid (Heiskanen

and Vening Meinesz, 1958). Since the center

of mass of the earth is not known, this ideal

system of terrestrial coordinates unfortunately

cannot be realized. It is therefore necessary

to substitute either the astronomic or the

geodetic coordinate system.

The astronomic coordinates fa and \A are

defined as the angles which the vertical makes

with the equator and the meridian of Greenwich.

These coordinates are related to the geoid and

can be obtained with an accuracy of the order of

0*1 by astronomic observations. They differ

by several seconds from the terrestrial coordinates on account of the irregularities of the

gravitational field. Gravimetrically corrected

for the deflection of the vertical, the astronomic
GEODETIC USES OF ARTIFICIAL SATELLITES 97

coordinates can come much closer to the terrestrial.

Terrestrial rectangular coordinates.—The geodetic coordinates 4>o and Xo are defined in the

same way as the terrestrial coordinates <f> and

X, but they refer to a particular computation

ellipsoid which may not be centered at the

center of gravity of the earth, and may not be

oriented appropriately. These coordinates are

computed from geodetic surveys on a reference

or computation ellipsoid, which is oriented at

the origin of the geodetic system (Bomford,

1952).

For points referred to the same geodetic

system, the relative positions will be correct

within the accuracy of the survey. They will

not be correct if the points do not refer to the

same system.

The elliptical coordinates (0, X, H) are not

convenient for points far from the earth's surface. For such points, it is much more convenient to use a system of rectangular coordinates.

The particular rectangular coordinate system X, presented here, has its origin at the

center of gravity of the earth (or of the terrestrial ellipsoid) and is oriented so that the

3-axis is directed toward the mean north pole,

as defined by the International Latitude Service, and the 1-3 plane is parallel to the mean

meridian of Greenwich (the meridian instrument at Greenwich does not lie in the 1-3

plane), as defined by the Bureau International

de l'Heure (Stoyko, 1955).

This coordinate system is fixed with respect

to the earth's surface, and the coordinates of

any point on the earth are fixed and do not

change with time, if we assume no movements

of the crust.

A point is defined in this system by a vector

X* (X1, X3, X3). This system is related to the

ideal system of the geographic coordinates by

the following formulas of transformation:

X

l= (N+H) cos 4> cos X=p cos /3 cos X

X

t= (JV-j-H) cos <f> sin X=p cos 0 sin X

Z*=[(l— e*)N+H] sin <f>=p sio 0 (1

where AT=radius of curvature in prime vertical,

e=eccentricity of the ellipsoid, p=radius vector, and /3=geocentric latitude.

The first set of equations is more convenient,

since tables (Perrier and Hasse, 1935; Army

Map Service, 1944) may be obtained for the

values of N.

Sidereal rectangular coordinates.—For points

that do not rotate with the earth, it is convenient to have a coordinate system that does not

rotate with the earth. This is the sidereal system. We can reach this, however, only by

means of an intermediate system, the instantaneous terrestrial coordinate system Y.

The axis of rotation of the earth is not fixed

with respect to its surface. The motion of the

true pole is studied by the International Latitude Service, which gives the coordinates of the

instantaneous pole with respect to a mean pole

(the same pole used to define the ^-axis).

If we know the coordinates of the apparent

(instantaneous) pole we can define another system Y of coordinates which uses this instantaneous axis as the y-axis. The I^J^-plane

contains the point where the mean meridian of Greenwich intersects the equator, which

is also the J^-axis, since the instantaneous

zero meridian is thus defined by the Bureau

International de l'Heure (B.I.H.) for the time

UTl. (According to the decision of the International Astronomical Union at the 1955 meeting in Dublin, UTO is the observed time; UTl

is the observed time corrected for the motion of

the pole; and UT2 is UTl corrected for the

seasonal variations of the rotation of the earth.)

Let x and y (fig. 1) be the angular (spherical)

coordinates of the instantaneous pole P with

respect to the mean pole P. We have taken

X

1

 as the primary axis, but since x and y are less

than 1 second of arc, the result is practically the

same.

The transformation from the X to the Y system is given by the expression

in which Man , is the matrix:

cosX'Y1

cosX'Y2 cosX'F2
SMITHSONIAN CONTBIBUTIONS TO ASTROPHYSICS

X

3

VOL.*

FIGURE 1.—Terrestrial and sidereal coordinate systems.

From the spherical triangles in figure 1 we

obtain the formulas given in equation (2) and,

consequently, the relation in equation (2a).

In the sidereal coordinate system Z, the

Z*-*xis will coincide with the F'-axis, and the

Z*-axis will be directed toward the apparent

vernal equinox T. The transformation from

the Y to the Z system will be

with

r

cos t —sin t (T

sin t cos t 0

Kfi 0 1J

t being the angle PZ1 or

Z

2

'cos t — sin t (T

sin t cos t 0 Y

2

Y

3

(3)

cosX

2

Y

1=sinx siny COBX3

Y

1=—sinx cosy

cosJ£2F

3

=—cosxsiny cosX3F

3

=cosx cosy.

r

cos x -f sin x sin y —sin x cos y"

0 cos y -f-sin y

j3in x —sin y cos x -f-cos x cos y..

(2)

(2a)
GEODETIC USES OP ARTIFICIAL SATELLITES 99

From the definition, the angle t is clearly

the same as the Greenwich apparent sidereal

time as defined by the B.I.H. (Stoyko, 1955).

We can go directly from the X- to the Zcoordinates, since

(4)

or as in equation (4a).

The sidereal time t must be considered as an

angle in space and not as elapsed time.

To find t we must use UTl rather than UT2

since UT2 is corrected for seasonal variation

in the speed of rotation of the earth and thus

is not a true measure of the angle between

F

1

 and Z1. This correction (periodic) has an

amplitude of about 0!03 or 0!5.

We have now obtained a system of coordinates (Z) that is not affected by the rotation of

the earth. This system is not fixed in space,

however, since the axis of rotation of the earth,

which defines our system, is moving under the

influence of precession and nutation. There

are two methods that could be applied here:

1. Keep the Z system and reduce all systems to the

apparent positions. This reduction can be made easily

by using the classical methods of reduction to apparent

positions (Besselian Star Numbers or Independent Star

Numbers). The necessary tables can be found in the

various Ephemerides.

2. Use another system of coordinates, W, that will be

defined as being the mean sidereal system at a certain

fixed epoch To. This system will be defined as follows:

The W1 axis will be directed toward the mean equinox ^p

of To and the W* axis will be directed toward the mean

pole P of To.

The relation between the Z and the W systems

will be obtained in two steps through a system

which will be defined as being the W system but

at epoch To; i.e., the epoch in which the Z system is defined. We will call this system Z. This

step is necessary since the motion of the polar

axis is given in two parts, as precession and as

nutation.

Let K and « be the angle between the line of

intersection of the mean equator at To and the

mean equator at T with the W and Z2 axes respectively (fig. 2). Let v be the angle between

the planes of the two mean equators (same

figure). The transformation from the W to Z

system will be given by the formula:

or by the rigorous expression of equation (5).

The values of K, G> and v are given as (Danjon,

1952):

K=[23042T53+139!73(r0-1.900) +

0T06(T0-1.900)2] (T-T0)+[Z0:23-

0!27 (To-1.900)1 (r-T0 )

2

+

is'.oo(T-Toy

«=[23042T53+139T73 (To-1.900) +

OTO6(77o-1.9OO)2] (T-T0)+[l09'.50+

0T39(7T

o-1.900)l (T-To )

2

+

18T32(7T-r0)

3

v= [20046T85-85T33 (To-1.900) —

0r37(T-1.900)2] (r-r o ) + [-42!67-

0T37(r-1.900)] (r-T 0 )

2

-

4ir8O(2T-7To)8

where To and T are expressed in units of a

thousand tropical years.

Z

2

r

cos t cos x cos t sin x sin y —sin t cos y —cos t sin x cos y —sin t sin y""

sin t cos x sin t sin x sin y +cos t cos y —sin t sin x cos y +cos t sin y

sm x —cos x sin y cos x cos y

'Z1'

Z

2

r—sin K sin «+cos K COS W COS V —cos K sin «—sin K COS U COS V —cos w sin v^

sin K cos w+cos K sin u cos v cos K COS W—sin K sin u cos v —sin « sin v

„, cos K sm v

512692—60 2

-sm K sm v COSP

X

2

W

2

(4a)

(5)
100 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

FIGURE 2.—Instantaneous and mean sidereal coordinate systems.

If we denote by An, Av, Ae, respectively, the

nutation in right ascension, declination, and

obliquity, we get

or with sufficient approximation, omitting

terms of the order 10~9,

zr

z?

z?

=

"i

AM

Av

-An

l

Ae

-Av^

—Ae z*(5a)

The values of An, Av, Ae, can be obtained

from an almanac for the epoch T. Combining

the two rotations we get

and

where

We have now achieved a system of rectangular coordinates (W) centered at the center of

gravity of the earth. This system is not fixed

in space, but—what is more important—it has

no rotation. We have also established the

formulas for making the transformations from

one system to another. All these systems are

geocentric, which means that we should know

the position of the center of gravity of the earth.

Geodetic rectangular coordinates.—Just as we

were obliged to use a system of geodetic coordinates as substitutes for the geographic coordinates, so we are required to use a system of

"geodetic rectangular coordinates" as a substitute for the terrestrial geocentric coordinates.

The geodetic rectangular coordinate system

X is defined by the geodetic coordinates

<po, \J, Ha given to our triangulation points

and by the parameters of the computation

ellipsoid used. If we compute the rectangular

coordinates X* from equation (1) by using

ipo, Xo, Ha, we obtain exactly our geodetic

rectangular coordinates. For each different

geodetic system, we obtain a different system

of rectangular coordinates.
GEODETIC USES OF ARTIFICIAL SATELLITES 101

FIGURE 3.—Geodetic and terrestrial coordinate systems.

A, \A> HA be the coordinates of the

origin of a geodetic system obtained by astronomic methods and leveling, and let £, % f be

the absolute deflections, so that

17= (X^—X) cos <p

In general, to the origin Qo of the system we

assign the astronomic coordinates q>Ay \A, HA.

But to those coordinates corresponds another

point QA on the terrestrial (geocentric) ellipsoid. Since the point Q is the same physical

point on the earth, we shall have a displacement

of the system of coordinates. This displacement will be a parallel translation since at the

origin of the system we make the theoretical

normal at QA correspond with the observed

vertical at Qo> These two lines are parallel

(fig. 3), while the direction of the X* axis is

identical with the direction of the pole (which

is observed directly). We assume here that

there are no errors in the determination of the

azimuth.

If we express £ and 17 in linear units by using

the radii p+H and N+H, the total translation

will be (€*+i?2+fs)*. In the X system, the

coordinates of the new origin will be X*c and

the transformation is given by matrices (6) and

(7).

It should be noted that the same formulas

could be obtained by differentiating equation

(1), and that equations (6) and (7) apply only

for small values of £, 17, f.

x%

XL

=

'sin v* cos Xo

sin «po sin Xo

. —COS <po

sin Xo

—cos Xo

0

-co s w, cos V

—cos <po sin Xo

—sin <po j

V

J

and

"sin <po cos Xo sin <po sin Xo —

sin Xo —cos Xo 0

,,—cos?tcosXo — cosvofflnXo —sin

(6)

(7)
10

2 SMITHSONIA

N CONTRIBUTIONS TO ASTROPHYSICS TOL.

8

The parallelism between the geocentric ter-

restrial system (terrestrial ellipsoid) and the

geodetic system (computation ellipsoid) is cor-

rect not only if we use the astronomic values

of ipy X, and azimuth A, but also if we apply

corrections to the astronomic coordinates for

deflection of the vertical, provided the astro-

nomic azimuth is also corrected for the Laplace

equation. In this case £, 17, f are the residuals

from the absolute deflections.

So if x* are the coordinates in the geodetic

system we have the equation,

X*

c

. (8

)

Until now we have assumed that the astro-

nomic coordinates at the origin of the system

were correct. An error in <pA, \A or HA will

have no effect on the parallelism between the

two systems (at least to the first order), since

it will introduce only a different displacement.

However, an error in the azimuth, which is

related to the longitude, does affect this

parallelism.

Obviously an error dA in the azimuth at the

initial point of the system will result in a rota-

tion of the system of coordinates around the

vertical, by the amount dA (in this case dA

is the error in the azimuth plus the effect of the

error in X). We will compute the effect of this

rotation.

First we notice that since the axis around

which the system rotates does not pass from

the origin, we will have also a translation. The

rotation matrix can be obtained as a product

of five rotations, expressed by the matrix (8a).

Since the angle dA will be very small (of the

order of 1"), we find the matrix MA for the

rotation as follows:

We know that a vector r* after an infinitesimal

rotation dA* will become rA=r* + dA* X r*.

If we consider a matrix M whose columns

are the vectors r[, r\ . . . r*

h th

e matrix MA

of the same vectors after the rotation will be

MA=M+dA*XM. (9

)

The cross product is simply the matrix whose

columns are the vectors of the cross product

dA* X r*j. If M ha

s as columns the three unit vectors

of the Xi—Xj} system (the X system after the

03 00

a•a0981os- •i^>

o

1—1 oo

9>a

•53

o9

-\

i—i

o

J

sin vo

oo

-cos 1\

.a

•3 o
NO. 9 GEODETIC USES OP ARTIFICIAL SATELLITES 103

parallel displacement), it is clear that MA will

be the matrix of rotation so that

(X'-Xti-x^MU^ix'-xl), (10)

xl being the coordinates of the initial point of

the geodetic system.

But then M=I (the unit matrix) so that

equation (9) gives the matrix

1 -dA3 dA2

dA3 1 -dA1

•dA2 dA1 1

(11)

In our case if dA is the magnitude of the

error in A, then

dAi=dA(—cos 0o cos Xo, —cos <fo sin Xo, —sin fo)

and we obtain the matrix in equation (12).

This matrix is not orthonormal since the

second and higher order terms of dA have been

disregarded. However, since dA is expected

to be less than 1", which is about 5X10~8, the

discrepancies will arise in the 11th decimal.

If we then introduce the values of M*u)f

to equation (10), we get:

X

1

 = XI + xl + [sin ntf-xl)-

cos <po sin Xo(a^—Xo)] dA,

cos ¥* cos XoCz3—xl)] dA,

X

3

 = XI + 3? +[cos ft sin \i{xx-x\)-

cos ft cos Xotx2—xl)] dA.

(13)

The formulas are correct to at least 10"10.

The computation ellipsoid (and thus the

geodetic system), as defined by the origin of

the system, will be parallel to the terrestrial

ellipsoid and the terrestrial system, if we assume

no error in the astronomic azimuth. Since the

triangulation is calculated on the same computation ellipsoid, there is apparently no reason

for any tilt between the computation and the

terrestrial ellipsoid.

Indeed, that would be the case if the coordinates of the different points were computed

with a consistent geometric method. However, the measurements are made with reference

to the geoid, while the computations are made

on the ellipsoid. Furthermore, the heights are

measured by leveling and are measured from

the geoid. This means that there is a kind of

distortion in the triangulation net.

If the computation ellipsoid is very near the

geoid over the area of the triangulation, the

distortion will be minimum. It is possible,

however, that the computation ellipsoid may

be tilted with respect to the mean geoid over

the area. This would happen if, on account

of local attraction at the origin of the system,

the geoid were very rough and thus the true

vertical were tilted from a mean vertical by

an amount d% (and drj) (fig. 4).

Since heights of points are measured from

the geoid, we must tilt the computation ellipsoid

so that the computed position Q[ is on the

true point Qt. This means a tilt around point

Qo by d£ (and drf). This tilt will bring an

inconsistency in that the computed direction

of the pole (axis z

8

) will not be the one that

has been .observed.

We must notice, however, that the angle dl

(and drj) will be zero or very near zero, if we

have corrected the heights to those corresponding to the computation ellipsoid, e.g., by making

astronomic leveling; or if we have corrected

for the deflection of the vertical and hence for

the local attraction by, e.g., a gravity survey

around the initial point of the system.

Let d£ and <Zij be the magnitude of the tilt.

The vectors of rotation dp rfij' will be

<££'=<2g(sm Xo, —cos Xo, 0),

drit=dn(—sin ft, cos Xo, —sin ft, sin Xo, cos ft),

with positive direction to the west for

to the north for drj*.

and

1 +dA sin <po —dA cos ft sin

—dA sintpo 1 +dA cos

te dA cos ft sin Xo — dA cos ft cos Xo 1

(12)
104 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

Normal

Mean

vertical

True

vertical

We can compute the rotation matrices M(£) and

M(v) with the help of equation (11), replacing

dA* with d? and dnf.

FIGURE 4.—Tilt of the computation ellipsoid.

X* - XI - i$)=M\»i(x'-xl)

We then obtain equations (14) and (15).

Thus if we consider only the effect of d£ or

dt\ we will have

The total rotation from the effect of dA,

£, and di\, will be

1

0

jl£ cos

0

1

sin

—d£ cos

—d% sin

1

1 — di\ cos ^o

dq cos ^ 1

jitl sin ^b sin Xo —dy sin ^D COS

—di\ sin ^ sin Xo^

dti sin ^> cos

(14)

(15)
GEODETIC USES OF ARTIFICIAL SATELLITES 105

(since we have eliminated second order terms

the order of multiplication does not affect the

result), or as in equation (16).

If we include all three errors of orientation,

dA, dZ, dy, then equation (10) will be written as

(17)

or

(x'-xo>). (17a)

Grouping the terms, if g* is the vector (dA,

dl-, dti) we get

or

X

1

 = x

1

 -{- ^ c +[sin <po(x2—xl) —

cos <po sin ^(x 3 — X<J)]GL4 —

[cos Xotx3—Xo)] <2£—[cos ^(x2—Xo) +

sin <po sin ^(x3—x§)] dtj, (18a)

X

2

 = x

2

 -f- X% -\-[—sin ^(x1—a-J) +

cos <po cos X0(x3—X3,)] dA —

[sin XoOc3—xl)] tf£+[cos ^o(xx—xj) +

sin <po cos X0(x3—x§)] dij, (18b)

*\. — Js ~\~ ^\.^~p"^COS ip§ Sill A0^X """XQ^ TM~"

COS ^ ) COS Ao^if/ ~~~iCQ^ J (Ijfx ~y~

[cos Xo(xl—xj) + sin X0(x2—x§)] rfj +

[sin #0 sin Xo(x*—xl) —

sin #> cos Xo(x2—x§)] ^17. (18c)

Let us now consider the effect of an error in

scale. Since triangulation schemes are scaled

by geodetic base lines, nonconnected triangulations may be at different scales.

Then, again eliminating second-order terms,

we should add « (x*—xj) to equation (18). With

Ax' used for (xi—xj1), the final equations of

transformation will become

X* = x

l

 -\- Xc + G)g} + eAx', (19)

or

X

1

 = x

1

 -f Xh + (sin poAx2 —

cos <po sin XoAx^d-A — (cos XoAx*)d{ —

(cos *?oAx2 + sin <p0 sin XoAx3)^ + «Axl,

(19a)

A = X -f- .A£ -j- v— Sin ^oAX -|-

cos <po cos \)Ax3)dA — (sin XoAx*)rff -f-

(cos ^oAx1 + sin <p0 cos XoAx3)^ + eAx2,

(19b)

(19c)

cos <p0 cos XoAx2)^ + (cos XoAx

sin XoAx2)^ + (sin ^>0 sin

sin $9o cos \>Ax2)dr) + eAx3;

or also,

X< = x* + XJ. + ( - /)Ax' + e • Ax*. (20)

Finally, we have another group of systems of

rectangular coordinates, which are parallel to

the geocentric but have their origin at a particular station Q. Such a system will be called

"topocentric" and will be designated by a prime

accent. If we refer to terrestrial coordinates,

the transformation is given by the relation

Similarly, in geodetic coordinates we have

-dA sin <po-\-dr) cos <p

dA cos <po sin Ao+^£ cos \o-\-drj sin <po sin

dA sin ipo—

dA cos ipo sin

cos <pa

1

—dq sin <po cos Xo

—dA cos v>o sin Xo—d£ cosXo—dq sin ^ sinXo

dA cos v3o sin Xo—d£ sin Xo-f cfij sin v?o cos Xo |« (16)

1

•}
106 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

Accuracy of the transformations.—The transformation from one coordinate system to another requires that we use a number of parameters that are derived from observations and

theory. We shall try to estimate the expected

accuracy of the transformed coordinates, using

the accuracy with which the parameters are

known.

In making transformations from the X system to the Y, from the expression

we get the equation

dY<=dM\ ,X>- (21)

Taking the differentials of the elements of the

matrix Mum), we obtain equation (21a). Since

the values of x and y are small, we can eliminate

the second-order terms so that

0 0 -dx"*\

0 0 dy

jix —dy 0 >

and therefore

dY*

ro

o

0

0

dy

-dx''

dy

0 «,

X* (22)

It is not easy to estimate the accuracy with

which the coordinates x and y of the pole are

given. We accept ±0?02 as a measure of the

accuracy (Melchior, 1954, p. 36). With this

value for dx and dy we have, for points on the

surface of the earth, errors (expected) of about

0.40 meters for the transformation to the Y

system.

When we perform the transformation from

the Y (or X) to the Z system, we have

oJ'. (23)

We evaluate:

r

 —sin tdt —cos t

cos tdt —sin tdt

0 0

and therefore

CdZl~\ f—sin tdt —cos tdt 0

dZ2cos tdt —sin tdt 0

0 0 0.

F

2

LF3J

(24)

This formula is correct if Y* has no errors.

But from equation (21) we see that Y* has the

error dY* so that equation (23) must be written

or, if we replace dY* from equation (21), as

Since F'~X<,

Or we have the matrix shown in (24a).

"—sin xdx sin y cos xdx-\- cosy sin xdx sin x cos ydy—cosy cos xdx

0 — sin ydy cos ydy • (21a)

.cos xdx — cos x cos ydy-\- sin x sin ydx — cos x sin ydy—sin x cos ydxj

Therefore

cos

sin

, 0

t

t

—sin

cos

0

t

t

or

0

i

0

0

Jx

0 — dx*

0 dy

—dy 0j

= 0

jlx

0 —cos

0 —sin

—dy

tdx— sin

tdx +cos

0

r—sin <d£ —cos 2cft —cos &fa —sin ^y"

cos tdt —sin <</< —sin fcfc -f-cos tdy

dx —dy 0

(24a)
GEODETIC USES OF ARTIFICIAL SATELLITES 107

Or, finally we have the form shown in equation

(25).

Note that we could obtain the same result by

taking the differentials of the elements of the

matrix M\XZ)} from equation (4) and eliminating

the terms of second order.

The study of the accuracy of the angle t is

very complicated. The apparent sidereal time

of Greenwich is the hour angle of the apparent

equinox T • It is related to the rotation of the

earth and can be determined directly by astronomic observations at Greenwich.

In practice, however, we use Universal Time

from which we find the apparent sidereal time

at Greenwich by using the special tables in the

ephemerides. This transformation is based on

Newcomb's tables. Universal Time itself is

obtained in the opposite manner, i.e., from the

apparent sidereal time as determined from

astronomic observations which have been transformed to Universal Time (mean time) with the

same tables.

At first glance we might think that since the

apparent sidereal time is determined directly

(the use of the mean time as an intermediate

stage has no effect) it is given with the accuracy of the observations themselves. This

is not true, however, mainly because the observed values of the sidereal time have been

smoothed to make them agree more closely with

the time given by the clocks, to obtain a time as

nearly uniform as possible. This smoothing

eliminates the effect of changes in the rotation

of the earth, at least those of short period.

It is obvious that the effect of the secular

deceleration of the earth, as well as of the effect

of the term B of fluctuation (Jones, 1939), is

zero, since both have an extremely slow effect

and UT is not corrected, for the moment, for

this effect. However, corrections for the seasonal variations are applied according to the

resolutions of the International Astronomical

Union at the Dublin assembly. These corrections (ATs) are published in advance in the

Bulletin Horaire. Since we do not require

a uniform time, but the true angle of the point

T, we must not use UT2 and convert it to the

sidereal time but, instead, we must use UTl

which is not corrected for ATs. Since the

radio time signals refer to UT2, we must apply

the correction — ATs to obtain UTl.

In analyzing the errors, we find that they

result from errors in the astronomic observations made at the observatories of the B.I.H.,

and from the very short period irregularities

of the rotation of the earth. Since we smooth

our results we do not know how much of the

error derives from each of these two categories.

We assume that the B.I.H. can provide the

sidereal time with an accuracy better than

± Of007. How accurately we can determine

the time of an observation is another question.

Expressing the estimated errors in radians,

we have

dx=dy= ±0.1 X10"6

dt = ±0.5X10-'

so that equation (25) becomes equation (25a).

dZ2

dZ2

Or

dZ2

r—sin tdt —cos tdt —cos tdx —sin tdy'

cos tdt —sin tdt —sin tdx +cos tdy

dx —dy 0 _

=F0.5 sin < 3=0.5 cos * =F0.1(cos £+sin t)

±0.5 cos £ =F0.5 sin t ±0.1 (cos t—sin t)

L ±0.1 T0.1 0

f—Xl sin *—X2 cos t —X3 cos * — X3 sin *

X

1

 cos t—X2 sin < X3 sin * Xs cos *

0 X1 X2 s

'X1'

X

2

(25)

X

2

dx

.dyJ

(25a)

(26)
108 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

For a point at 0=40°, X=45°, H=0 and

for a sidereal time <=3h, we get a total uncertainty in position of about 2.6 meters.

Equation (25) may also be written in the

form of equation (26), or

dZl=Mt

CdV

dx

If V{t, x, y) is the variance of t, x, y, then the

variance V{Z*} will be

(27)

Disregarding the errors in x and y, we obtain

equation (28) where <rt is the standard error in

t, of the order of 0.5 X1O~6.

In transforming from the Z to the W system

or vice versa, one should expect to introduce

errors arising from errors in the values of the

astronomical constants (precession, etc.). This

source of error is more apparent than real,

however, since such errors will be eliminated

when we determine from observations the

position of the mean pole and the sidereal

time.

Determination of directions.—Since the stars

lie at so great a distance from the earth, the

direction of a star is the same from the center

of the earth as from any station on its surface.

The directions of the stars are given in the

various catalogs or ephemerides in the W or

Z system, as right ascension (R.A.) and declination. Therefore we know the direction

of the line connecting the center of the earth

with the star; this direction is the same as

that of the line connecting any station on the

earth with the star.

In determining directions, if we use the Z

system of coordinates we must use the true

positions of the stars for the time of observation,

i.e., referred to the equinox of the epoch. This

direction (defined with a and 8) is the true and

not the apparent. To get the apparent direction

we must correct for aberration as well, or use the

apparent positions of the stars.

If we use the W system of coordinates, we

must use the mean positions of the stars for the

epoch TQ. The mean positions must be cort

o

\

o

I

co *

© .

e. O

1 k

a

•a

i 8

"** on

•i I

•a
GEODETIC USES OF ARTIFICIAL SATELLITES 109

rected only for the proper motions of the stars,

and if we need the apparent directions we must

correct also for aberration.

The fact that a star has the same (known)

direction from all stations on the earth indicates a

method by which we can determine the direction

of an object from a station in the sidereal

system.

Let Q be the station and S be the object

(fig. 5). Also let the object be in front of a

FIGURE 5.—Directions in the sidereal coordinate system.

fictitious star whose topocentric position is

(a',8'). It is obvious that the direction of the

line^Q/S will be given by a' and 8f.

If the direction cosines of QS in the Z system

are

Z^cos a1

Z

2

=cos a2

Z*=cos a9,

we have the following simple relations between

Z<>nd a'fi'i

Z^cos 8' cos a'

and

Z

2

=cos 8' sin a'

Z»=sin 8'

sin 5'=Z8

tan a'=ji*

(29)

(30)

Using equation (3) we can find the direction

cosines m* in the Y system by making Zl a unit

vector. We then obtain

m

1

=cos 8' cos (a'—t),

TO2=COS 8' sin (a'—t),

TO8=sin 8'.

When we introduce 8=a—t= — (t—a)=GHA,

we obtain

TO1=COS 8' cos 8',

m*=cos 8' sin 8*, (31)

TO8=sin 8',

and

,_m 2

TO1

sin 8'=TO8.

(32)

Similarly, we get the direction cosines n* in

the X system by using equation (2). We obtain

n

1

=+sin x sin S'+cos x cos 8' cos 8',

n*— —cos x sin y sin 5'+sin x sin y cos 5' cos 6'+

cos y cos 5' sin 6', (33)

n

8

= -f-sin tf sin y cos 8'—sin x cos y cos 8' cos 0'+

cos x cos y sin 8'',

and

n

2

 cos y+n3 sin y ^

n

1

 cos z-f sin x n2 sin y—sin x ns cos y

sin 8'=nl sin x—cos x n2 sin y+cos x n8 cos y.

(34)

To obtain directions in the X system, we get

the direction cosines q* in the geodetic system X

by using equation (17) in the form

(35)

or

q

1=nl

 + (—n2 sin «* -f n

3

 cos <po sin \t)dA-\-

(n3 cos Xo)d£+ (n2 cos ^ +

2

= n 2+ (n1 sin «% — n8 cos «* sin

(n8 sin Xo)d$+ (—n1 cos <po—

n

3

 sin W) cos \))drif

(36a)

(36b)
110 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

g

s

=n3 + (—n

l

 cos ipo sin Xo + n

2

 cos <po sin

(—n

1

 cos Xo — n2 sin Xo)</£+

(n1 sin <po sin Xo — n

2

 sin ^ sin Xo)rfi7. (36c)

The big advantage of using the sidereal system of coordinates lies in the fact that we can

get the direction of an object in this system

directly from a photographic method suggested by V&isala (1946).

The object is photographed with a metric

camera, with the stars as background. Using

the methods of photographic astrometry we

find the topocentric apparent R.A. a' and declination 5' of the object. These values of a'

and 5' then define the direction of the line

Q-S. By using formulas (29) we can find the

direction of the line between station and object

with respect to the Z system of coordinates.

If we wish we can determine the direction in

the Y, X, or X system by using formulas (31),

(33), or (36), provided we know the parameters

that relate these systems with the Z system, i.e.,

t, x, y, dA, d£, and drj. The effect of these parameters is small but we must know the time of

the observation in order to determine t to better

than 10 msec to have positional errors smaller

than 5 meters. However, as another alternative

we can fix the camera in the X (or the X) system,

determine the orientation of the camera, using

the stars in the same X system, and then determine the direction of the object, knowing the

orientation of the camera. In this case the time

of the observation is not needed.

A source of error is the possible error in the

positions of the stars used as reference, as given

in the various star catalogs.

Since the FK3 catalog (Dritter Fundamentalkatalog, 1937, 1938) is used for the definition

of time, it must also be used for computing

the apparent places of stars.

When we say the sidereal time is the hour

angle of point T, we must make it clear that

the point T itself is not observed. The stars

are observed; and knowing the R.A. of the

stars we can find the hour angle of T. In

other words, our Z system of coordinates is

defined by the coordinates of a number of

stars from one catalog and by the constants

of precession, nutation, and aberration as well

as by the proper motions that we have assumed.

We might more correctly say that the point

whose hour angle gives the sidereal time is not

the intersection of the planes of the equator

and the ecliptic, but the mean of the points

defined as being — at from the stars.

So if we change the catalog from which we

take our data, we change also our system of

reference and our Z system. Corrections exist

which enable one to shift from one catalog to

another, and the same corrections should be

applied to change from the Z system defined

by the one catalog to the system defined by

another.

Unfortunately the FK3 catalog that should

be used contains only 1,500 stars, which are

not always enough for our purpose. If we

use a camera with long focus and a small field

of view it is very probable that we will not

observe a large enough number of stars included in FK3. For this reason we will be

obliged to use additional stars that do not

belong to FK3, and must reduce their positions

to the FK3 system.

It is very difficult to say what the degree of

accuracy is within the FK3 system. The internal errors are mainly accidental. Some

systematic errors exist, resulting from errors

in the astronomical constants, but these will

be eliminated, since the Z system is defined

with the same constants (the same errors with

opposite sign are also made in the determination of the time). This is also true for the

errors Aaa that are partly eliminated in the

same way.

The observations

Although for orbiting objects the dynamic

quantities (e.g., velocities) can be measured,

this discussion will deal only with the measurable geometric quantities, i.e., distances and

angles or directions.

Measurement of distances.—The only possible

way to measure directly the distance to the

object seems to be to use electromagnetic

waves, or radar. The electromagnetic waves

are reflected from the object and we measure

the time needed for the waves to travel the

double distance. Knowing the velocity of the

electromagnetic waves, we can find the distance.

Better results can be obtained if the object

carries a receiver-transmitter (transponder) to
GEODETIC USES OP ARTIFICIAL SATELLITES 111

send back the signal. Both the pulse system

(Shoran, Hiran) (Canada, 1955) and the phase

system (Raydist) (Comstock and Hastings,

1952) can be used.

The accuracy of the distances thus measured

depends on two things: the accuracy with

which we can measure the small time interval

the wave needs to travel; and the accuracy

with which we know the velocity of the wave

in the medium traveled. The distance r will

be given by the equation

r=~ (37)

where Vm is the mean velocity along the path

and T the time interval.

Differentiating, we obtain:

or

1

 v A MdV- (38)

The value of Vn is about 0.3 X109 m/sec,

and if T can be measured within ±0.02 msec,

the distances will be given within ± 3 meters.

An error dVm causes an error proportional

to the distance. Vm is computed from the

value of c (velocity in vacuum) by applying

a correction for the index of refraction. The

latest value of c, accepted by the International

Association of Geodesy at Toronto in 1957,

gives c=299,792.5±0.4 km/sec, which gives a

value of dc/c= 1.3 XI 0~fl. However, the value

of dVm/Vm will be higher than that and probably of the order of 10~8

, because of uncertainties in the index of refraction. Furthermore,

the curvature of the path will cause additional

errors.

Finally, we must consider possible errors in

the timing of the observations if the object is

moving (compare with p. 112, where this problem is treated for the measurement of directions). Since time is a parameter, we will

include the effect of an error in timing in the

errors of the observed quantities, by using

time as the independent variable.

If f is the change of the observed distance

per unit of time and o? the variance of timing,

the variance of r will be

and this value should be added to the variance

<J? as given from an analysis of the accuracy

of distance measurements to fixed points; thus

(39)

The value of f can be easily obtained if we

have continuous observations.

Optical measurement of directions.—Measurements by the optical method can be referred

either to a system of local coordinates (for

example, azimuth and zenith distance) or to a

system of universal coordinates, such as apparent right ascension and declination (see

p. 108 ff.).

Directions may be obtained from photographs

taken with a calibrated camera. With the

methods of terrestrial photogrammetry we can

use a camera with known elements of exterior

orientation. From the picture we can get the

direction of a photographed object analytically

by measuring the photographic plates with a

comparator or directly by using a photogoniometer.

The directions so obtained will refer to

the horizontal system h* of the station and

thus will be affected by the deflection of the

vertical. We can get directions in a general

system if we photograph the object with the

stars as background. With the help of the

stars we will find the apparent right ascension

and declination, i.e., the directions in the

sidereal system Z of coordinates. Those directions can be transformed to any other system

with the equations given on page 108 ff. The

determination of apparent positions with the

help of photography is a common method in

astronomy; a summary of the techniques can

be found in Smart (1956, chapter 12).

For geodetic applications, the observed

object will be moving with respect to the

stars, and the timing of the exposure is highly

important. Since the stars and (usually) the

object itself will not be very bright, the exposure

will vary from a fraction of a second to some

seconds, depending on the camera. On the

other hand, we must know the timing of the

exposure to within about 1 millisecond, depending on the apparent velocity of the moving

object.
112 SMITHSONIAN CONTBIBTrnONS TO ASTROPHYSICS VOL. 3

Various methods can be used.

a. The camera is equatorially mounted and follows

the stare; thus the image of the object will be a trail.

With the help of a special shutter, the trail will be

interrupted and the timing of the interruptions will be

taken.

b. The camera follows the stars, and the relative

motion of the object is compensated with a rotating

plate. The time is taken when the rotating plate is in

the normal position; this method is used by Markowitz

(1954).

c.The camera is kept fixed so that both the stars and

the object produce trails as images. The trails will

be interrupted for timing with a special shutter.

d. The camera follows the stars while the object

(invisible) produce flashes at known times and preferably at equal intervals. Then the plate will contain

point images of the stare as well as point images of the

flashes. This method has been proposed by Vais&la

(1946) and by Atkinson (in 1957).

Methods (a) and (c) can always be used,

provided the trail of the object is bright enough

to be photographed. Method (b) requires

that the relative motion of the object be known.

But even if the motion is not completely

known, the method will be a help in photographing fainter objects, since the trail will be

much smaller. Method (d) applies only to a

specially instrumented object.

Figure 6 gives a schematic version of plates

taken with the different methods.

! /

! / !

1

 / ,

!

i

i

FIGURE 6.—Schematic version of moving objects photographed

with four methods.

Alternative methods exist. Instead of using

a rotating plate to integrate the light of the

object and thus increase the effectiveness of the

optics, the camera can follow the object. This

principle is used in the Baker-Nunn satellite

tracking camera (Henize, 1957). Also, for a

flashing object the camera may be kept fixed;

the stars will then produce trails and the object

will give point images. If a shutter is used to

interrupt the trail (of the object or the stars), a

correction must be applied either to the positions or to the time, to allow for the fact that

the shutter has a finite speed.

Electronic measurement of directions.—Electronic methods using the principle of interference have also been developed to determine

the direction of an object which has a transmitter. The Minitrack system (Easton, 1957)

is perhaps the best of this type. Such methods

give the directions with respect to a local

system of antennas, specially arranged and

fixed on the ground. By careful and repeated

calibrations, however, they can give directions

in a general system such as the terrestrial or

even the sidereal system.

The accuracy of these electronic methods, at

present, is low. Minitrack can provide directions accurate to within one or two minutes of

arc, which is low for geodetic applications.

Electronic direction methods probably cannot

be used, but observations made with electronic

systems could give valuable information regarding the variation of the orbital elements,

and as such they can be used as described on

page 140.

Since conventional radar will give at best

an accuracy of ±0°l, its use is out of the

question. A possible method using electronic

optical observations is discussed by Merrill

(1956).

Correlations between the observed quantities.—

If the measurements on the photographic plate

were made with equal accuracy to every direction and there were no errors of timing, the

values of a' cos 8' and 5' would be determined

with the same weight. However, the position

of the image of the object on the plate is not

likely to be determined with the same accuracy

in the direction of the apparent motion as in

the perpendicular direction. This is more

obvious in the case of the methods (a) and (c)
HO. • GEODETIC USES OF ARTIFICIAL SATELLITES 113

on page 112, because the image is in the form of

an interrupted line.

Also, we must introduce the effect of errors

in timing. Since time is a parameter, we will

introduce the error of the timing in the positions observed, but such an error will affect the

accuracy only in the direction of the orbit.

We introduce a coordinate system x and $ on

the plate oriented along the orbit (fig. 7) which

c>, we will have

V

\

AS'

f\

V Aa*

X

t

COS 5 '

FIGURE 7.—Orientation of the coordinate systems x and

along the orbit.

makes an angle <p with the axis of R.A. We

assume that the standard error in the direction of x is A: times the standard error a in the

perpendicular direction f. The value of k

will depend on the image and on the standard

error of timing. It can be evaluated as follows.

If because of the shape of the image alone the

standard error in the direction of the orbit is p

times the standard error in the perpendicular

direction, and the standard error in timing is

or

(40)

where x is the change of x per unit of time (x

can be evaluated from the plate directly).

If the image of the object is a point, p will be

1. Otherwise, it must be determined experimentally, or taken as 1.

The axes x and 4> being the primary axes of

the error ellipse, we have

(41)

and since

\Aa' cos i'J \cos <p —sin <p/ \w \w

therefore

or as shown in equations (42) and (43).

We have found the variance of a' cos h' and &'

as well as the variance of r for the effect of the

errors in timing. To complete this analysis of

the correlations, we must find the variance of a'

cos 5', 5', and r as a whole, since all three elements

are correlated.

We begin with the fact that if we have a number of reasons to introduce errors to a vector

with the partial variances Vt, V9, . . ., the total

variance of the vector will be the sum of the

partial variances. Let Aa' cos 6', A5', and Ar

be a system of coordinates having its origin at

the object, and parallel to the directions of the

and 8inceP=F~1,

cos2 (p (1c2 — 1)sin tpcos $>]

tp k* cos2 ^>-f sin* <p J

r . , I . , /. i\ .

cos2 *>+p sin2 ip ( 1 —p j sin <p cos <p

( 1 —TJ ) sin <p cos <p sin2 9+15 cos2 *p

(42)

(43)
114 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS VOL. 8

apparent R.A., declination, and radius vector

(fig. 8).

Ar

FIGURE 8.—Orientation of the velocity vector.

Let <r be the standard error in the measurement of the directions 8' and a' cos 8' (assumed

to be equal) and at the standard error of the

distance measurements. If the object were not

moving, the variance of &', a' cos 8', r would be

8' ^ Co2 0 0

Vi- a' cos 8'

r

0 a3 0

0 0 <r*

(44)

This represents an ellipsoid of revolution.

Further, let w be the relative velocity of the

object with respect to the observer, the direction

of the velocity being defined in our coordinate

system with the angles <p and i (fig. 8).

An error in the timing will introduce new

errors in the position. As has already been

pointed out, the errors in timing are introduced

in the observations, since time is kept as a

parameter. The errors thus introduced will be

waT in the direction of the velocity. They will

represent an error ellipsoid that has become a

segment of a straight line.

If we introduce the system of coordinates x,

\p, w as in figure 8 (the axis x to the direction of

to), the variance in x, $, « will be

0 0

0 0 0

0 OJ

but equation (45a) shows that the variance

expressed in 5', a' cos 8', r will be

5'

(45)

V2 a' cos 5' E'.

Or, the variance V* will be that shown in equation (46), and the total variance will be

or that of equation (47).

AS'

Aa' cos 5'

Ar .

sin <p cos i co8<p —Bin % sin <p

r r r

cos (p cos i —sin <p —sin i cos <p

r

sin i 0

r

cos i

=E (45a)

~2 sin2 <p cos 21

1 . .. . 1 . . . ."

~2 sm <p cos* % sm <p - sin <p cos i sin %

1 , . . 1 , , .

-5 cos tp cos1* t sm <p -j- cosz ^ cosz i - cos <p cos i sin i

1 . .. . 1 .. . . . .

- sm <p cos % sin % - cos <p cos i sin i smz i

(46)

r

r ~i sir

. gW2 . ,

% <XJI —jr sin ^ cos ip cos

^ 2

rs sin 9 cos £ sin

T -j cos <p sin <p cos2 i a

2

 + <rr -j cos2 ^> cos2 i <r| -g- cos ^> cos £ sin i

^r - j sin ^> cos i sin i o-| - j cos <p cos ^ sin i

r" r*

sin2

(47)
GEODETIC U8ES OF ARTIFICIAL SATELLITES 115

Introducing the rate of change 6', it' cos &',

and f, we obtain the variance as written in

equation (48).

The weight matrix is obtained as

P=V~l.

We may make the following comments on

these methods:

Equations (39) and (42) are special cases of the general equation (47).

If the standard errors in R.A. and declination are

not equal, this fact can easily be taken into account.

From equation (47) we have eliminated the effect of

the image (coefficient p), but this effect can also be

taken into account.

Similarly, we could also introduce the effect of errors

from refraction, since they will also introduce more

correlations. However, this effect will not be significant

from a practical point of view.

Reduction of observations to the station.—

Sometimes the instruments used to measure

directions or distances are not centered at the

station but are placed eccentrically (in general

not far from the station). Since different instruments are needed to measure directions

and distances, it will be impossible to have

both instruments at the same point. We will,

therefore, have to reduce the observations made

at the eccentric station. The reduction will be

given by a differential formula (see p. 114 ff.).

If we call &'„ a',, rt the values observed at the

eccentric station with coordinates X\, and i',

a', r are the observations that would have been

made from the true station with coordinates

X£, then we will have the formula shown in

equation (49).

Correction for aberration.—Usually the object

will be moving with respect to the observer and

its velocity will be rather high (the velocity of

the artificial satellites is of the order of 8 km/

sec). This relative motion will introduce an

aberration effect which will apparently displace

the ray of light.

We could correct for the effect of aberration

by the method used in astronomy for observations of planets and comets (planetary aberration), and apply a correction to the time of

the observations instead of correcting the

observed directions. However, if we wish to

keep the time of the observation unconnected,

we must apply the corrections to the observed

quantities &' and a!. In this case, let Q and S

be the positions of the station and the object

(fig. 9); let V© be the velocity of the earth with

respect to the sun, VI be the velocity of the

observer with respect to the earth, V* be the

velocity of the object with respect to the

earth, and V& be its total velocity, VQ+V.

The velocity of the object with respect to the

observer will then be

The total correction for the aberration will be

e (fig. 9) and will have the value

tocos i (50)

w cos i being the projection of w on the plane

perpendicular to QS, and c the velocity of light.

It will lie on the plane defined by w* and Q.

We introduce the system of coordinates

Aa' cos 6', A5', Ar to the point S, the same

1

 ^'(a'co s $') o\h'r

rid'(a'COS8') o*+o*T(a' COS «')» a\(a' cosa')r

o\h'r <rr(af cos 6')r

(48)

6'

J

sin 6',

rt

COS

0

r, cos 8', r, cos 6i

„ CO8 0J cos 8', sin 9i cos 8't sin 6 ^ l»^«—•"t t

(49)
116

SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

Apparent direction^

of object and star

vi

FIGURE 9.—Correction for aberration.

system given on page 114 ff. If wm, wt, w, are the

components of w* in the three new axes, then

wa=w cos i cos 4>

wt—w cos i sin <f>

wr=w sin t.

Then the corrections e to a', 8', r will be

v>a 1

In the case of photographic determination of

the directions, it will be easy to find the values

of wa and Wt bv measuring on the plate the

change a and 8' of a' and h' with time. We

have then

ccos5

(52)

c cos 8

.,=*:.

(51)

(See footnote.1)

'Some explanation most be given for this formula. It may seem

ridiculous to apply a correction for the velocity of light to the measurement of distances. Actually, no measurement can be made instantaneously, and e should be the. . . velocity of the used yardstick! Since

the distances will be measured with electromagnetic waves we must use

the velocity of light. Notice, also, that if for the time of the observation

we take the mean time between the emission and the reception of the

signal, the correction will be zero.
GEODETIC USES OF ARTIFICIAL SATELLITES 117

When the directions are obtained by the use

of stars, as described on page 111, we will get

the apparent direction of the object if the positions of the stars are also apparent. We then

apply the corrections given by equation (51)

to find the true position (or true direction) of

the object. It will be more convenient, however, and will avoid many computations if

we use the true rather than the apparent

positions of the stars. In this case (fig. 9) we

must add (or subtract the negative of) the

effect of the aberration of the stars, i.e., the

effect of the motion of the observer with respect to the sun (annual and diurnal aberration).

This means that we must use V*3 instead of

to*. Thus we compute the position of the

object by using the true positions of the stars;

we then apply the correction for the annual

aberration for the position of the object as

well as the diurnal aberration, and finally

apply the correction for the aberration of the

object according to equation (51).

Since the diurnal aberration is less than

0?3, it will in most cases be negligible. The

annual aberration can be computed by Bessel's

method:

Cc+Dd in R.A.

Cc'-\-Ddr in declination.

We could further simplify the computations

if instead of using the true position of the stars

we use the mean position for a given epoch,

but the positions of the stars should be corrected

for their proper motions.

The position of the object, thus determined,

and corrected for the annual aberration as well

as for the aberration of its own velocity, will

be the mean position for the given epoch,

and thus will refer to the W system of coordinates. We can then compute the true position

of the object by using known methods of reduction, such as Bessel's method, if we want our

directions to refer to the Z system rather than

the W.

Correction for optical refraction.—Although

the observed objects will be at rather high

altitudes, the optical rays will have to pass

through the atmosphere and thus undergo a

curvature. The existing formulas for the corrections for astronomic and geodetic refraction

cannot be applied in our case, because they

assume that the observed object is either at

infinity or at a rather small distance and low

altitude.

We will therefore try to find a formula that

can be applied for the intermediate case. We

will assume that the density of the atmosphere

p as a function of altitude H is given by the

relation,

log P=Kt H + K2. (53)

The observed values of p at different altitudes H (Whipple, 1954) fit accurately enough

to those of equation (53) for heights up to 80

to 100 km. The results from the artificial

satellites for the density at heights of 200 to

600 km show an important deviation from the

values of equation (53) (Whitney, 1959), but

this does not affect our solution (see p. 118).

The coefficients Kx and Ka will be determined

experimentally. Or we can write

P=e

1 "~\ I -K-2 where a=-r4> O=-T-7 M M (M=log10«).

But if n is the index of refraction, (n— l)=Bp

according to Gladstone's law, and so

or

(n-l)=kettB, (54)

where it=.Be6=7io— 1, if iu> is n for H=0.

We assume that the index of refraction as a

function of the altitude H is given by equation

(54), k and a being coefficients whose values will

be determined experimentally.

Let Q (fig. 10) be the station and S the

observed object. The ray of light will follow

the curved line SMQ, and the observed zenith

distance will be Zo, the angle between the

vertical and the tangent to the curve at Q.

Also let 6 be the angle between the tangent at

any point M with the tangent at Q.

Assuming no curvature of the earth (this

assumption will hold for Z<45°), we have the

relation

=—tanZ—»

n

(55)
118 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS TOL.8

Vertical

FIGURE 10.—Correction for refraction, when curvature of the

earth is not considered.

which, since Z will be constant, will give by

integration

= tan ZoQn nQ—In n). (56)

Since n is very near 1, we can expand the

equation so that

e= tanZ0 [(w0 -l)-(w-l)]. (57)

If we introduce a system of coordinates $ and

il so that | is the tangent at Q we have

^ = 6 = tan Z0[(n*-1) - (n-1)] (58)

or with the help of equation (54)

But since the angle e is small, we can replace

£ with r (r is the distance QS); thus H=r cos Z.

Then equation (58) is written

^ = & tan Zo—k tan Zo^C08 z», (59)

or

, = C\k tan Z0-

Integrating, we obtain

Thus we obtain an equation that expresses the

line followed by the ray of light:

»-*«•» ^HIf R is the correction, we have tan i?= |

since R is small, and if we replace $ with r we

get an equation which gives the correction for

the refraction R as a function of the zenith

distance Zo and distance r:

This is a general formula and can be applied

for any r provided Z<45°. We can find that

for r= oo (we have a<C0), the correction is

k tan Zo, which is the well-known astronomic

refraction.

For the constants, we take k=(n0—1) =

292X10"8 or 60T25 for the normal case (0°,

760 mm Hg), while for Kt of equation (53)

we have estimated from the work of Whipple

(1954) that Kx=— 0.06 km"1, which brings

a=-0.1385 km"1.

Then for the normal case we get

For {=0, i7=0; therefore

#=60r25tanZo -435r0 ^ ^

(62)

where r is expressed in km.

From equation (57) we see that for an observation made at a height between 100 km

and infinity, the angle e will be

cioo= tan Z^oo—1),

and since (nl00— l)£^0.5X10-9=0!0001,

«ioo=0?0001 tan Z,

which proves that even for values of Z=80°,

the angle e^ will be less than 07001. This

means that we need not consider the atmosphere above a height of 100 km, since the errors

introduced by this neglect will be very much

smaller than the errors of our measurements.

As follows from equation (60), the curve of

the ray of light has an asymptote (fig. 11),

given by the expression,

acosZo

(63)
H0.» GEODETIC USES OF ARTIFICIAL SATELLITES 119

FIGURE 11.—Asymptote of a refracted ray of light.

The curve will fast approach the asymptote.

With the assumed constants, the distance between the curve and the asymptote will be

6==0.0021tanZoe_0.1386eOBZot

cosZo

For positions of the object not far from Q the

elimination of the curvature of the earth (and

thus the curvature of the equal n layers) will

not have any important effect. This will be

true for points within Z^45° and #=10 0 km.

In such a case, confusing H with r cos Z will not

have important effects. For points beyond this

limit, we cannot replace H with r cos Z. But

provided Z<45°, the curve will almost have

completed its shape when it reaches the height

of 100 km. In such an extreme case, the value

of b will be about 1 mm.

Actually, in this case, we could use the equation of the asymptote, and from it compute the

correction,

k tan Z0H

page 111 ff. the correction should be AR=€m—R8

or AR=Ba—Rs- We then obtain

(65)

COS

a cos Zor

We prefer to leave the formula for the correction as given in equation (61) since for points

at distances farther than our limit, the term

(1— e

atoaZ

o

r

) will differ from unity by only

10~9. Thus equation (61) holds also for every

value of r, provided Z<45°.

If the directions were obtained photographically with reference to stars, as described on

A nomogram (fig. 22) is given on page 158 for

this correction, which is always negative.

This correction will be in the zenith distance.

The correction in a' and 6' will be given by the

formulas

Ca=—AJ? sec h' sin g'

C,=— Aflcosg'

where g' is the apparent parallactic angle of the

object.

We may want also to consider the effect of

the earth's curvature. Since layers of equal

n have a curvature following the curvature of

the earth, the value of Z in equation (55) will

not be constant, and equation (56) will no

longer be its integral. The angle Z (fig. 12)

will be

and since the height of the atmosphere has

been taken as 100 km, « will be a small angle

so that

«=—tanZo,

p

where p is the radius of curvature of the earth
120 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

S

FIGURE 12.—Correction for refraction, including the effect of the earth's curvature.

Then But €=-£ so that

. „ . „ fftan

tan Z=tan Zo jp cos*

(66)

ii=k tan Zo \d£ — Tc tan Zo [ef3[ -y

Thus equation (55) will be

, , rwdn. tan Zo

d€=—tan Zo—r

n p cos2 Zo n

Integrating, we obtain

(67)

ap cos

2

 Zo

p cos* n

Also

and since

p cos

2

 Zo Jo

dH cos Z'

tan2 Zo

0J

(68)

cos Z=cos ZO(1H H),

H. dH cos Zo p cos Zo

So equation (68) is written as in equation (68a).

n=> cos

ktan3 Zp

p

2

 cos3 Zo

f

p cos

3 ^

ap cos3 Zo

(

J p cos3 Zo J Ha"dH2 2 i

ap2 cos2 Zo

fgg "^g+ Artan

2

Z

° fdf. (68a)

J ap cos2 Zo J
GEODETIC USES OF ARTIFICIAL SATELLITES 121

Integrating and eliminating second order

(

H*\

terms containing —j- j , we obtain

, y /k tan Zo , k tan Zo

p cos Zo

k tan Zo

_kJ^nZo\ a

a?p cos3 Z o / '

ap cos Zo

The constant of integration is found from the

condition that at the origin 17=0. We get

finally equation (69) and thus, since R=% we

obtain equation (70).

Introducing the values of the constants, we

get equation (71) which gives the correction for

the refraction for values of Z up to perhaps 80°.

The last term usually can be neglected.

(All formulas given in this section are based

on normal conditions for the value of k. If the

temperature and pressure at the station are

different, k should be changed. A nomogram

(fig. 23) is given on page 159 for this correction. The height of the station Ho will not

change the results except for extremely high

altitudes. In such a case the integration must

begin from Ho.)

It can be seen that for Z0<C45°, equation (71)

gives practically the same correction as equation (62).

Formulas (69) to (71) will be correct for

Zo<75° (or perhaps 80°) and #<10 0 km

since we have expanded Z in series and eliminated second order terms. For i7>100 km,

as we have shown, the path of the ray will,

practically, already have taken its shape and

will follow the asymptote. The equation of

the asymptote will be

17 = (k tan Zo+— tan Zo sec2 Z^) ( + \ ap /

- tan Zo sec Zo |~1 +— (2 sec2 Zo+tan2 Zo)T (72)

a L °P J

Thus the correction will be:

i? =it tan Zo + — tan Zo sec* Zo+

ap

— tan Zo sec Zofl +— (2 sec2 Zo + tan\Zo)T (73)

or [_ ap • J

But as with equation (61), we leave the

formula for the correction as given in equation

(71) since the additional terms will bring practically no change.

Thus we can check the fact that for points

at infinity we get

or

fi=60!25 tanZo—0T0682 tanZ0sec*Zo

#=60ri8 tanZ0 -0!0682 tansZ0,

which is exactly the general formula of astronomic refraction.

v=(ktfinZo)S+[— tanZosec2Zolf + -tanZoBecZori + — (2 sec* Zo + tan* Z0)"l (1 -

\_ap j a [_ ap j

— tan Zo sec Z0(sec* Zo+ tan1 ZJHf*. (69)

R=k tan Zo+— tan Zo sec* Zo + — tan Zo sec Zo [~1 +— (2 sec1 Zo + tan* Zo)l (1 — ap at [_ ap J

apr

tan Zo sec Z0(sec2 Zo + tan2Z0)HfB.

= 60T25 tanZo—0T0682 tan Zo sec* Zo—43570 sec Zo[l—0.00113(2 sec2

 Zo+

tan* Zo)] (1 -e-0-

W!Mwr)-0r0682 tan Zo sec Zo(sec2

 Zo + tan* Zo) *j-e-°'13%6a.

(70)

(71)
122 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

When the directions are obtained photographically with the stars as reference (p. Ill),

the correction will be that given in equation

(74). The last term generally is negligible.

Also, since the term 0.00113 (tan2Z0+2 sec2Z0)

is about 0.04 for Z0=75°, we could say that in

most cases equation (65) will be sufficient.

Correction for electronic refraction.—The refraction of electromagnetic waves in the earth's

atmosphere is a very complicated problem. It

depends on the wave length and also on the

dielectric condition of the air. Different ionized

layers exist at altitudes that vary with place

and time, where reflections and refractions occur.

This greatly complicates the problem. However, the shorter the wave length the less is the

influence of those layers. The directions thus

obtained with radio methods, as described on

page 112, must be corrected.

The distances obtained with radio methods

need two corrections, the first for the curvature

of the path followed by the electromagnetic

wave, and the second for the change of the

value of the velocity along the path.

These corrections have been studied (Jacobsen, 1951; Williams, 1951) for low altitudes for

applications in electronic geodesy (Shoran, etc.),

but it is very doubtful whether they can be

applied also in the case of high altitudes. With

the use of radio methods, however, the distances

will probably be more accurately measured than

the directions.

Observations to objects of known positions

Several methods exist for making observations

to objects of known positions.

Use of angles in space.—Let us consider first

the case in which we have observed angles to

three known positions. Let Su S2, St (fig. 13)

be the known positions of the object, and Q

the unknown station where the angles /3<12>,

/3UM, pm) have been observed.

FIGURE 13.—Determining the position of an object by use of

three angles in space.

In general there is a solution giving the

position of Q, provided Si, Sa, S3 are not

colinear. The problem is that of resection in

space, and Q will lie on the intersection of three

tores. The direct solution gives a system of

high order and is not applicable. This problem

is of great importance in photogrammetry. Of

the many solutions suggested, the best, which

has also the advantage of being easily generalized in a case of more than three known

points S, seems to be one in which we compu te

corrections dXl to an approximate position X*

of the point Q. This principle is very much

used ia geodesy.

Briefly, the solution is obtained as follows:

Let (jj_be the approximate position of Q.

Using Q and Si, S2, St, we compute the angles

is"2*, ^23), j? 3 " . __

If Q is near Q and the corrections to Q are

dX* we obtain the equation:

5 C ^1 2 >

 dX*=coa /3(12>-cos J3<12) (75)

and two more equations with /3(23) and /3(S1).

[l-0.00113(tan2 Z0 + 2 sec2 Z0)l (1 -«-«•»«")-

Zo+ sec2

(74)
W0.» GEODETIC USES OF ARTIFICIAL SATELLITES 123

The coefficients of dX* ere of the form

d cos /3(>

w

, r 1 cos

cosa*Us,—mrj

where a* is the direction angle of the line SQ.

The solution of this system of three equations

with three unknowns gives the correction

dX* to the approximate values ~X*.

The method can be generalized for the case

in which we have observed more than three

angles /3(iW. The solution will be obtained by

the method of least squares. We must notice,

however, that if equation (75) is applicable in

the case of three known points, it is not applicable if we must make an adjustment, since

cos p

ut) is not the measured quantity. In

this case we must use equations with /3<iw

—jS*'*' since the angles /? are assumed to be

measured directly.

We must replace

dcos/3 m U U 5j8 d cos ft. d/3

X

OT dX dcos/3*

We then get equations of the form

(76)

where

A (W cos a) 1 cos /3"*H

S, QS, J

cosaj |~ 1 cos/3a*H

sin/3«*>L<2S, QSk J

The accuracy of this method of determining

the position of the station depends on the

accuracy of the observations and on the net

configuration (the positions of the object are

assumed to be errorless).

An analysis to estimate the accuracy could

be made, but since this method will not be of

much use we shall not attempt it. The reader

may refer, however, to a similar study for photogrammetric applications (Doyle, 1957). We

may note, also, that if the different positions

of the object lie near a straight line (as happens

with an artificial satellite) the solution is

very weak.

Use of directions and distances.—Let ZQ

(fig. 14) be the coordinates of the station

Q, Z*s the coordinates of the object S, r* the

vector QS, and I* the direction cosines (unit

vector) in the Z system of the vector r

1

 corresponding to a! and 8'.

z'

FIGURE 14.—Relation between the positions of the station and

the object.

The equation relating the positions of Q and S is

or

S=ZJQ-\-I r. K*O

If the coordinates are referred to the terrestrial system, we will have

A.8=j\.Q-x~n r. \io)

Again, since we almost always know the approximate positions, we will develop formulas

relating the corrections to the approximate coordinates Z* or X* and use the method of variation of coordinates.

Differentiating equation (77) we get

where I* is a function of a' and 6'. Differentiating equation (29) we get

dll=—(cos a' sin i')dir—(sin a' cos 6')daf,

dl*= — (sin a' sin 6')d8' + (cos a' cos h')da!, (80)

dl* = (cos 8')d8'.

Then equation (79) can be written in the

form shown in equations (81).

512692—60 8
124 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS VOL. 8

(dZ\—dZ1

Q) = — (cos a' sin V)r d8f—

(sin a')r cos V da'-\- (cos a' cos V)dr,

(dZ2

a-dZl) = - (sin a' sin 5')r d8' + (81)

(cos a')7 cos S' 6^'+ (sin a' cos 8~')dr,

(dZ%-dZl) = + (coB I')r d8'+

0+ (sin 8')dr.

Inverting the system (81) (the matrix is

orthonormal) we get

or

( I ) (<*Z|

(82)

dr=(cos a' cos 3') (dZ^

(sin a' cos 5') (dZI-

(Bint')(dZl-dZZ).

We will write this system

(82a)

In the Y system, assuming that t is known, we

replace a' with 0'=a'—t, and Z* with Y* and

find that

(83)

(sin ?' cos ? ) (dY%-dY%) +

LdrJ

(83a)

Equations (83) or (83a) can also be used for

the terrestrial (X) and geodetic (x) systems,

since the omitted terms will be of the order of

dXXlO'* (for dX=1000 meters we will omit

terms of 1 millimeter).

Hence we get

da'

with

(84)

cos 0' sin 5' sin 0' sin 8' cos 8'

r

sin0'

7 cos 5'

r

COS0'

7 cos 3'

cos 0' cos 5' sin 0' cos J'

and d8'=8'—5', da'=a'—a', dr=r—r, which

are the observed values minus those computed from the approximate coordinates.

The computed values of 5', a', (or ?'), 7 can

be obtained from the rectangular coordinates

of the object, with the help of equations (30),

(32), (34), or (35), depending on the system in

which the coordinates are given. If the position

of the object is given in spherical coordinates

a, S, B, in the Z system, we can find the apparent a', 1', (or 0'), 7 as follows (fig. 15):

(tana'—a)=tan (fi'—0)=

t 7/ cos (9'- tan o =

FA sin 0—F^cosfl

= =-= '

(85)

cos (0—0) cos 5

where

A=B cos 8—Tl cos 0—¥% sin 0.
NO. 9 GEODETIC USES OF ARTIFICIAL SATELLITES 125

If from one station we have observed a', b'

and r of an object of known position, we can

find the position of the station with the help

of equation (84).

If we have observed from the same station

many positions j of the object and we have all

a'*, bf), r*, we adjust by the method of least

squares for the coordinates of the station by

using the appropriate weights and correlations,

if any.

Y

1

FIGURE IS.—Topocentric and geocentric directions.

It may happen that we do not observe both

directions and distances. If we observe only

directions, i.e., a'* and 8'*, for every observation

we will have two equations, the first two of

equation (84). If we have measured only the

distances r*, we will have for every observation

one equation, the last of equation (84).

We can also get a solution if we have

observed only declinations but we do not get a

complete solution if we have observed only

R.A., since a is independent of Xs.

The same equation (84) can be applied both

to problems of resection (i.e., from known

positions of the object compute the position of

the station) and to problems of intersection

(i.e., from known positions of stations compute

the position of the object). In the latter case,

since the object will be moving, we must

make the observations simultaneously from

all stations.

If we do not want the rectangular coordinates

of the station (or the object) but instead want

the ecliptic coordinates <p, X, H (or 5, a, R),

we must replace dX*Q with dtp, d\, dH (or dX*8

with dS, da, dB). We therefore introduce a

system of coordinates with origin at the approximate point Q. The 3-axds is directed toward

the normal (zenith), the 2-axis is directed

toward north on the horizontal plane, and the

1-axis toward east.

We will call this system horizontal and we

will use the symbol h. We find that the relation

between this and the terrestrial system is given

by the formula of equation (86). Or,

"X* and ip, X being the coordinates of Q. By

using differential displacements, and since

dkl=N cos <p d\, dh?=pd<p, dh*=dH, we obtain

the matrix (87) or (88). Or,

*N cos <pd)C

pd<p

dH

dX*

'"—sin X cos X 0

—cos X sin ip —sin X sin ip cos <p

„. cos X cos ip sin X cos <p sin ip.

—sin X cos X 0

—cos X sin tp —sin X sin ip

w cos X cos tp sin X cos ip

*—iVsin X cos tp —p cos X sin ip cos X cos <p

JVcosXcos? —p sin X sin ip sin X cos ip

0 pcos p sin??

(86)

dX* (87)

dip

[dHj

(88)
126 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

dtp (88a)

If the height of the station is not negligible,

N and p should be replaced with N-\-H, and

P+H.

Substituting in equation (84) we get

jir ^

— KjjlJL a— d<p

[dHj

(84a)

Similarly, we get the matrix in equation (89).

Or,

dX*=Si da

KdRj

and equation (84) becomes

(89a)

da' da (84b)

Accuracy of the determined positions.—After

the solution of the matrix equation (84), or

after the adjustment of the same equation, in

the case of superfluous observations, the accuracy with which the positions are determined

depends on three things: the accuracy of the

measured quantities, the accuracy of the given

positions, and the net configuration.

Since the problems of resection and intersection are the same (in both cases we have measured the same directions and distances), we

shall study the case in which we determine the

position of one station Q from a number of

known positions j (j = 1, 2,. . . s) of the object

(or objects), having measured all or some of

the elements a', &', r.

Both the accuracy of the observations and

the net configuration will affect the accuracy of

the determined positions. On the assumption

that the positions Xi

Sj are given and correct, the

solution for dX& (or X*) will be given by the

system

(90)

Here A\ is a matrix consisting of matrices

— 6< arranged in columns, and I' is a vector

consisting of vectors (ddf, da', dr) arranged in

a column. If we have observed all three elements at every position j , we have g=3j.

The solution by least squares will be given

by the equation

or

NdX=A'-P-l

dX=N-1A'-PJ, (91)

where N—A'-P-A and P is the weight matrix,

or P= V~l, V being the variance of the measured

quantities I9.

Furthermore, the variance V {XQ} which is

the same as the variance of dXq will be

V{XQ}=N- (92)

1=(A'-P'A)-1

.

If the values I' are not correlated and have

the same variance <r2

h P= -a J and

(92a)

We see from equation (92a) that the variance

of XQ (or in other words the standard error of

XQ) depends on <rj=the standard error of the

observed quantities. There is no doubt that

the smaller the value of <r%, the more accurate

point Q will be. But V{XQ} also depends on

(A' -A) ~

x

 (or in the more general case on N~x) and

(A''A)'1 is a function of the geometry of the

net, as can easily be proved.

dX2

—R cos 0 sin 5 — R sin 0 cos 8 cos 0 cos 5'

—Rsin 0sin 5 R cos 0 cos 8 sin0 cos 5

L R cos 5 0 sin 5

da

\JRJ

(89)
GEODETIC USES OF ARTIFICIAL SATELLITES 127

Finally, to compute the error ellipsoid for

the point Q, we try to find a system of axes

such that no correlation exists between them,

i.e., so that the variance matrix will be written

as

"a1 0 0

0 a2 0

.0 0 a\

Let T be the matrix of the rotation. T is a

3X3 orthonormal matrix, and its elements can

be expressed as a function of three quantities

(e.g., 0, Xand^L).

If V0{XQ} is the variance of XQ after the

rotation T, we will have

V 0 0

0

10

(93)

Solving equation (93) we can find both a* and

T; a* is obtained from the solution of the

discriminating cubic:

\V{XQ} — a'/|=0 , (94)

and T is obtained from the solution of the

system:

t'tti=\ (95)

where tt are the column vectors of T (or

principal directions).

Although we must have the matrix A to be

able to find the variance XQ, every observation

will define completely the position of the

station if all three elements a', 8', and r have

been measured. Or, every observed quantity

will define one locus which in the case of small

corrections dXQ, will be a plane. To get higher

accuracy for XQ, these planes must intersect in

right angles if possible.

If we measure only distances r and the given

positions S lie on a straight line, there is no

solution (the planes will intersect in one line).

The closer the positions S are to a straight line,

the weaker the solution will be. The solution

will be weak also if the given positions and the

station lie on the same plane.

Furthermore, the last part of equation (84)

(if we assume the positions of the object as

given and correct) can be written:

(96)

dr__

r

cos 0 cos 5

r

sin

dX1-

0 cos 5dX2

 =— dX3.

r r

Therefore the system (84) can be written as

shown in equation (84c), the matrix being

orthonormal.

As we said previously, dh' and da' cos &' will

in general have the same weight. If drfr also

has the same weight,* the variance of XQ as

obtained from equation (84c) will be

V{XQ)=a*r.% (97)

a being the standard error of the weight unit.

This proves that the accuracy in the determination of the station Q is independent of the

net configuration and the error ellipsoid is a

sphere (the error, however, will be proportional

to the distance).

The same is true if we have more than one

observed position of the object (provided we

have observations of the same weight). In

this case we see that the matrix A is composed

• in most of the cases the errors In distance measurements are supposed

to be proportional to the distance. In this case the weight of drfr will

be the same for all distances. In the electronic methods of distance

measurements, the errors are sometimes supposed to be independent of

the distance. In this case irlr will not have the same weight for all rfs.

'd8'

dot' cos ti

dr

cos 0' sin h'

sin?'

sin 0' sin 5'

-cos ~0'

-cos b'

0

—cos 0' cos b' —sin 8' cos 8' —sin 8'

(84c)
128 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS VOL.«

of orthonormal matrices with a factor 1/r*, so

we can easily prove that

o

In most cases <r*=<ra eoiC=cr. Then Pi=-j>

•Af"/-

or

Introducing this expression to equation (92a),

we obtain

V{XS\-X?P-I (98)

where P* is the harmonic mean of (r*)*. Equation (98) is of the same type as equation (97)

and thus the same conclusions can be obtained.

We can therefore state the following general

rules concerning the accuracy of our determinations of position:

a) When we have observed only distances

(having the same weight), we obtain the best

result if the positions of the observed object

can be grouped by threes so that the lines

connecting them with the station are mutually

perpendicular.

b) When we have observed 8' and a' (and

8' and a' cos 8' have the same weight), we

obtain the best result if the positions of the

observed object can be grouped by twos, so

that the lines connecting them with the station

are perpendicular (the distances being assumed

to be almost equal).

c) When we have observed all three elements

(i.e., r, 8', a'), the net configuration will not

affect the accuracy, provided we have the same

weights.

d) If we have to choose between measuring

distances and directions, we must compare the

weights; that is, compare

with P2*

where a refers to standard errors.

If Pi>Pa it will be more accurate to measure

direction, and if Pi<P2, it will be more accurate

to measure distances.

and so we compare a with — -\/2, and if

=V5.

the measurement of directions is

more advantageous, while if — -J2 the opposite is true.

As an example, if 8' and a' cos h' can be

measured with an accuracy of ±3 " or about

1.45X10"8, the measurement of distance will

be better only il the distances can be measured

with an accuracy better than 10~8

csd0~8.

Let us now consider the effect of errors in

the given points (in this case the object points).

Let 8X3 be the errors in the position of the

object. Then the system (90) will be, with

the help of system (84),

AidXQ=l<+&f8Xs, (99)

where $ is a gX<Z matrix having in its diagonal

the matrices —9 and the other elements zero.

If all the elements have not been observed, the

matrices 0 will not be square, and thus the

matrix $ will not be square either.

This means that we introduce the errors 8XQ

in XQ which arise from the errors 8XS in the

given positions Xs:

A-SXQ=$-8XS. (100)

We want to find the variance V{XQ} of 8XQ

(or of XQ), knowing the variance V{XS} of 8Xa

(or of Xs). From equation (100) we have

or

K (101)

If V{Xs}=<r%'I, equation (101) will be simplified to

(102)
HO. 9 GEODETIC USES OF ARTIFICIAL SATELLITES 129

Considering that the matrices A and $ are

made from the submatrices —6^, we have the

matrices in (102a). Therefore

(103)

Or, the variance of XQ from the standard

error aa in the position of the given points is

independent of the net configuration, and the

standard error of XQ is equal to the standard

error of Xs divided by <y[s, the result to be

expected.

Observations to objects of unknown

positions

We shall now discuss the computation of position for the case in which we do not know the

position of the object, but have observed it

from stations of both known and unknown

positions. Since the object is moving, the observations must be made simultaneously.

For one solution, we could first compute the

positions of the object by using observations

from the known stations, and eventually make

an adjustment if we have superfluous observations. Then from the already computed positions of the object, we could find (or adjust for)

the positions of the unknown stations by using

the observations made from the unknown stations to the object.

It will be more advantageous, however, if we

adjust the space net as a whole, since we will

get higher accuracy.

Unconditioned equations of observation.—Let

us suppose we have n known stations Qk, m

unknown stations Qu and s unknown positions

of the object Sj. Let us also suppose that we

have observed the object from both known and

unknown stations, but have not necessarily observed all its positions nor all of the elements

«', «', r.

For every element observed, we will have an

observation equation relating the position of

the station and the object that will be of the

form of one of equations (84).

We will call Qit the matrix of equation (84)

connecting the j position of the object with the

known station k, and Qju the matrix connecting

the j position of the object with the unknown

station u.

The matrix Qju or 6,* may be the complete

matrix of equation (84) or it may be incomplete, that is, it may contain only certain rows

of that matrix, depending on how many and

what elements have been observed from the

station. We will then have a system of observation equations of the form

A>dX=l, (104)

where A is a matrix consisting of submatrices

G# and Qju, dX a vector consisting of the vectors dX^m and dX*Si (corrections to the approximate coordinates XQ% and Xst) and I a vector

consisting of the vectors (db', da', dr) obtained

from the observations at both known and unknown stations. Needless to say, these vec-

- G x 0 ... ^

0 -G 2 . . .

r-e

0

l o .. q

- e ; ... =

Gx€

0

)i 0 ...

GJGJ . . .

or

(102a)

and

-er

- e , =«/.
130 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS YOU*

tore need not have all three components; they

will depend on how many and what elements

we have observed.

The matrix A will be of the general form

shown in equation (104a).

We notice that in each of the three partitions

(the partition with all zeros is excluded), at

every row, there is only one submatrix 9, and

in the two upper partitions, for every element

—Qi% in the left partition, there is a corresponding element QJn on the right, in the same row.

An example of a matrix A appears in equation

(104b).

The solution by the method of least squares

for dX will be given by the system of normal

equations

N-dX=A'Pl, (105)

where N=A''P-A, and P = the weight matrix

(if the observations are of different weight).

From the general form of the matrix A we can

see that the matrix N of normal equations will

be of the form shown on page 131. Since the

subscripts in parentheses are constant numbers,

it is understood that there is no summation

with respect to those indices. The matrices P

are the corresponding weight matrices.

The matrix N (equation 106) is symmetric,

as would be expected. To every row and every

column there corresponds one point Qu and St

in sequence. Then every element of the matrix

is formed by summing all the terms of the form

0'P9 where 6 is the matrix that relates the two

points that correspond to the column and row

of that element. Since no observations have

been made between the stations or between the

objects, the corresponding elements will be zero.

It may happen, however, that we also have

observations between the unknown stations.

In this case the elements of the matrices that

correspond to those stations will not be zero.

These observations in practice would be mainly

distance measurements by Shoran or any other

electronic method, since no intervisibility will

exist for observations of directions, because of

the long distance between the stations. Such

distance measurements between the stations by

any electronic method will be very desirable

because they will strengthen our net.

Submatrix

with elements

—GJU

Submatrix

with elements

0.. .

Submatrix

with elements

Observations from

unknown stations

Observations from

known stations

(104a)

Observations from

unknown stations

(104b)

Observations from

known stations
GEODETIC USES OF ARTIFICIAL SATELLITES 131

If we observed all three elements from

every station, and they have the same weight,

the matrix N will be much simplified. The

expression [Qj»PQju] will be (?/?*) I, where v is

the total number of observed positions and

f* is the harmonic mean of the squares of the

corresponding distances, the standard error

being unity. In this case the matrix JV will be

as shown in equation (107).

0 '(/1)(«1)P0 (/I) («

N=

' (it) («ai)

(106)

« [G0. O ••• •

[-40 [-40 - [-*

(107)
132 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS VOL. »

The variance V{dX} of the unknowns will be

given by the formula: N~l=(A''P'A)~1. In

practice, the weights are given in terms of an

appropriate weight unit, in order to have simple

numbers. Therefore we have

(108)

o* being the variance of the weight unit.

In all our derivations and formulas we have

used the terrestrial coordinate system X. The

same formulas hold for any other system provided we change the matrices 6 as indicated on

page 124 ff.

In case we have geodetic rectangular coordinates x*, equations (104) to (108) are

exactly the same, since the matrices 0 will be

the same.

The only restriction is that the coordinates x*

of the known stations must be referred to the

same system and of course the positions of the

unknown stations and of the object will refer

to the same system.

Also it must be mentioned again that the

values of a' and 5' will be obtained from equations (32), (34), or (35), depending on the

system of coordinates used; e.g., if we use the

X system we will first find the approximate

direction cosines n* with the approximate

coordinates 3 * and then with the help of equation (34), we find 6 and 5.

If we use the X system we have to go through

the direction cosines q

1

 of equations (35) and

(36), provided we know the values of dA, d£,

dti. Since in most cases those values will be

unknown, the directions in the X system will be

the same as in the X.

We could simplify the computations if all

observations were made within a short interval

(some days), during which the instantaneous

pole will be practically fixed. In this case we

could make all the computations in the Y system

and afterward make the transformation of the

adjusted coordinates.

Introduction of conditions.—There may be

certain conditions that must be fulfilled between

the unknown stations—for example, the distance

between the unknown stations must be kept

fixed. In making our adjustment we must then

impose a number of condition equations between the values dX*.

Let the equation

CdX=k (109)

be the condition equations in matrix notation,

while

A-dX=l (HO)

is, as previously, the observation equation.

The solution by least squares will be given

by the system

dX\ (A'Pl\

where N=Af'P'At the same as in equation (105)

and X are the auxiliary unknowns (correlates)

equal in number to the conditions.

From the solution of equation (111) we get

the unknowns dX, while their variance is given

by the formula,

(112)

cr being the standard error of the weight unit.

In most cases, the conditions will relate the

coordinates of the unknown stations Qu. It is

more probable that there are fixed lengths or

directions between the unknown stations.

There may also, however, be conditions between

the coordinates of the object (e.g., we may

know the distance between two positions of an

object if we know its velocity and the elapsed

time).

The conditions will be mainly of two kinds:

imposed length and imposed direction.

Let us first consider the conditions of imposed

length. Let Lo be the known and fixed distance

between any two points Qt and Q2. Let Z be

the computed distance between the approximate

positions Q and <&. We have

(113)

where AX^Xi-Xl

Differentiating, we get

L dL=&X1(dX\-dX\)+&X2(dXl-dX*l)

AX*(dXl-dX\)
GEODETIC USES OP ARTIFICIAL SATELLITES 133

or

A y*

(dXi-dX\)-\-~- (dX\-dX\) +

A V 3

or, further, by using n*, the direction cosines of

the line QiQ2, we obtain

dL=n1(dX1

2-dX\)+n*(dXl-dX{)+

n*(dXl-dX\).

If dL—L0—Z, dX* is the correction to the

approximate coordinates Xf, and so the condition equation is written

n

x

dX\-nldX\ +n*dX\-n*dX\+nldX\-

n*dX\=Lo-L. (114)

For every imposed fixed length, we have one

condition equation of the form (114) which

could also be written as

ntdXl-n'dX^Lo-L. (114a)

If we take the approximate positions 7$ so

that Lo—Z we have

By comparing equation (114) with the last

of equation (84), we see that they are the same.

We can write the conditions also in the form

shown in equation (116).

We must be sure that we use the direct

distance between the two points (also called

chord distance), and not the distance along the

terrestrial ellipsoid Qength of the geodesic).

Next let us consider conditions imposed on

the directions. A line connecting two unknown

stations may have a known direction that has

to be kept fixed in the adjustment. One direction in space has two freedoms, and so for every

imposed direction we will have two condition

equations.

Instead of using the direction cosines of the

imposed direction, we will find it more convenient to use their equivalent elements, i.e.,

the angles 0' and 8'. Let d'o and 8'0 be the

imposed values and 0' and 5' the computed

values from the approximate coordinates. It

will not be necessary to derive the condition

equations, since we know beforehand that they

will be the same as equations (84a) and (84b).

We can therefore write the condition equations

by using L for the distance between the two

points Qi and Q2, as shown in equations (117)

and (118).

(117)

cos 8

If the imposed directions of 0'o and 8'0 have

been obtained with the help of astronomic observations, as they will be in general, they refer

to the Y system. If we are using a different

system, the computed 6' and j ' must be obtained

by using the T* coordinates of the approximate

points.

Regarding the condition equation, we must

make the following remark: if we have measured

the distance, e.g., by Shoran, between two of

the unknown stations, we must not use this

measurement as a condition equation, but rather

as an observation equation. We can correctly

use it as a condition equation only if we want

the distance to be kept fixed after the adjustment or, perhaps, if the accuracy of the measurement between the stations is much higher than

that of the measurement to the object.

The connection of geodetic systems.—One

important problem that often arises in geodesy

is that of connecting two geodetic systems; that

is, to find the relative positions of the two computation ellipsoids or, equivalently, to find the

relative positions of the two X systems of coordinates. The connection can be made by com-

(cos 0'cos 8')(dX\-dXX) + (sin 0'cos 8')(dX*a-dX\) + (sin 8')(dX\-dXf) ==U~ (116)
134 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS TOL.»

paring the coordinates of common points of

the two systems.

If we have observed the positions of an object

from stations belonging to both systems, we

will be able to make the connection. For that

we use the points belonging to the one system

as known points, and we find the coordinates of

the points belonging to the other system (as if

they were unknown), but we introduce the

necessary condition equations so that the net

will not change as a whole. We then compare

the coordinates computed in this way with the

coordinates of the same points as given in the

second system.

Instead of comparing the coordinates of

points of the one system, as computed from

the other system, with those originally given,

we could compare the coordinates of the object

as computed from the two systems. This will

be much more simple, since we will have to compute positions of an object from known stations

and will have only groups of normal equations

with three unknowns.

The relation between the two systems will be

given by one translation (3 unknowns), one

rotation (3 unknowns), and perhaps one scale

factor (1 unknown).

If the two geodetic systems have no orientation errors dA, d£ and drj, they will be parallel,

since both will be parallel to the terrestrial system (see p. 109). Connecting the two systems will

then require only translation, and no rotation.

Further, if the two surveys are scaled correctly

(or are scaled far better than we can detect), we

have only three unknowns. Then only one

position of an object, observed and computed

from both systems, will give us the solution.

More positions will help in eliminating the accidental errors since we can make an adjustment.

From equation (8) we have the expression

x[=Xi—Xci for the coordinates of one point

in the first system, and x\==X*—Xi

c% for the

coordinates of the same point in the second system; therefore

i—*J—-Ac,—•

A

-ov—-A-c (119)

Thus the relative position X^., (which is the

same as the relative position of the center of the

computation ellipsoid of the second system with

respect to the first), is simply the difference of

coordinates of the same point computed in the

two systems. If we have more points we will

take the mean.

If we want to express this translation in terms

of the deflections £, tj, f, we will use equation

(7). The £ and 17 so obtained will be expressed

in length units, but we can express them in

angular units by using the radii of curvature.

We must remember, however, that those deflections are not absolute but relative, i.e., with

respect to the computation ellipsoid of the first

datum.

If in equation (7) <po, \> are the coordinates

of the origin of the second datum, the deflections

will correspond to that point. If we want the

deflections at any other point (^, X), we have to

replace <po with <p and Xo with X.

The residuals of X^^, obtained from the different positions of the object, should be within

the accuracy of the computed positions. If the

residuals are bigger, this indicates that the systems also have orientation errors dA, d£, and drj,

and/or are at different scales.

It is not possible to find the absolute errors

in orientation of the two systems, but we can

find the relative orientation error of, e.g., the

second system with respect to the first.

If we take the first system as reference, we

can write equation (119) as follows:

(120)

where X^^ are the coordinates of the origin of

the second system with respect to the first,

g2~i the orientation errors at the origin of the

second system with respect to the first system

(computation ellipsoid), and 68-i is the scale

difference of the second system with respect

to the first.

In equation (120) we have seven unknowns,—

three for the translation, three for the rotation,

and one for the scale. In most cases the scaling

of the two systems will be so much more correct

than can be obtained with this method that

we may disregard the last term of equation

(120).

Every observed and computable position of

the object will provide the three equations of

(120).

If we disregard the scale difference, two positions of the object can give the solution for the
GEODETIC USES OF ARTIFICIAL SATELLITES 135

six unknowns. If we have more positions, we

can make an adjustment by the method of least

squares, and eventually introduce the scale difference. After the adjustment, we could express ^ 2 _ , in terms of the deflections.

If the two geodetic systems are connected by

Shoran, the connection is not complete because,

although we have the horizontal positions referred to the same ellipsoid, the heights refer to

the geoid, and it is not possible (except with a

gravity method) to connect the geoid of the

two areas (cf. Bomford, 1952, p. 367).

If such connections exist, we will introduce

them to our solution either as condition equations, if the measured distances should be preserved, or as observation equations if the

Shoran distance should also be corrected.

First, we assume no errors in orientation.

Let L be the distance between the connected

points. If Xcsl=Xc, we have

(121)

and so the distance

(122)

This equation is not linear with respect to

the unknown xc and thus can not be used.

We will use, therefore, an expansion into the

Taylor series:

LdL= (zj-zf

or

or

t

c=dL = L-L, (123)

For the value of Xc we can take the mean

value obtained before we introduced the condition equation.* L is the observed distance

and L is the distance computed with the approximate value xc, and can be obtained from

equation (122) by replacing x& with Xc.

* The term xc'can also be neglected completely. Since x& b expected

to be smaller than 600 meters, while L to bigger than 300 kilometers,

XcfL will be of the order of 10~»; therefore the omitted terms amount to

It is very important to introduce the correct

weight to our equation, if equation (123) is an

observation equation.

In general, where we have also the unknowns

dA, d£, dq, and e, if the vector (dAa-u d&-i, dih-i)

is denoted by g\-\, then, through the use of

equation (120), equation (121) will become:

Therefore,

(124)

(125)

Differentiating and eliminating the secondorder terms, we obtain

or:

i [xl-xl+xi

c)'dxtc+j- [(*4-

[(J (126)

Equation (126) is the observation equation

(or condition equation), the same as (123) but

it contains all seven unknowns. We will have

as many equations of this form as there are distances measured between the two systems to be

connected.

Observations to orbiting objects

If the observed object is orbiting and the orbit

is assumed to be known, we could treat the

problem as one involving an object of known

position (p. 122 ff.). If the orbit is completely

unknown (as for a rocket's orbit in the earth's

atmosphere), we could treat the problem as one

involving an object of unknown position (p. 129

ff.), by making simultaneous observations. In

the latter case, however, we would make continuous observations of the object and, having

also the time of the observations, we can interpolate for fictitious simultaneous observations.

We will call this method that of quasi-simultaneous observations.

But very frequently we will meet the problem

in which we know the theory of the orbit (more

or less), but we do not know the values of the
136 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS TO1.1

parameters that define it. It is then possible,

from observations from known stations, to find

those parameters and eventually to find the positions of unknown stations. Since the object that

we can use in this way will be an artificial satellite, we will limit ourselves to elliptic (or quasielliptic) orbits.

Quasi-simultaneous observations.—We assume

that we have continuous observations of the

object from both known and unknown stations

and for the same period of time. The result

of those observations can be either a continuous

recording against time or individual values at,

preferably, equal and short intervals of time.

If we have a recording, we can easily find the

value that corresponds to a given time Tt. We

will then take the times Tu T% . . . and find

the values that would have been observed from

the different stations at those times. We then

apply the method devised for simultaneous

observations.6

If the data are given as individual observations, we must make a numerical interpolation;

we may use a linear interpolation or second and

third differences, our choice depending on how

smooth the values run. Any interpolation

formula can be used, although if the observations are not at equal intervals, we must use

Newton's or Lagrange's formulas for unequal

intervals.

We could also extrapolate the orbit beyond

the observed values. This is dangerous, however, unless we are very sure that the values

will continue to run smoothly. Even then, the

extrapolation should not be made for an argument far outside the two extreme observed

values.

The accuracy that we can get depends on

the accuracy of the timing, as well as on the

factors mentioned earlier (p. 129 ff.). If we

want the position of the object to be accurate

within ± hX and the velocity of the object is

V, the timing accuracy of the observations at

all stations should be better than within ST,

ay

where ST<~

The interpolation itself will not introduce

errors, provided that the observed values run

• It should be pointed out that since the object is moving, to be really

simultaneous the observations should refer to the time that the signal

(light pulse for optical observations) left the object, and not to the time

that the signal was received by the observer.

smoothly, that there is no Tn«.iriTniiTn or minimum or inflection point, and that the interpolated interval is small. On the contrary,

since every fictitious observation will be

obtained from a number of true observations,

this method will tend to reduce the effect of

accidental errors. On the other hand, we must

consider the correlation between the observed

values (see p. 115).

Use of a first-approximation orbit.—Let us

assume the object has an elliptical orbit

around the earth, and conforms to Kepler's

laws. Unquestionably the orbit of an object

not too far from the earth's surface will be much

more complicated, but if mean elements are

used, the elliptical orbit is very nearly correct

for a small interval of time, and can be used as

a first approximation.

We will use the following elements to define

the orbit at epoch To:

mean anomalistic motion n

eccentricity e

R.A. of ascending node Q

inclination i

argument of perigee «

mean anomaly Mo

We will use 6" to designate the vector consisting of these elements, in the same sequence;

i.e., 6-=(n, e, Q, i, to, To), (u = 1, 2, . . . 6).

Instead of the mean motion, we could use

the period P or the semimajor axis a, as we

have the relations,

r*Jp> (127)

n

2

a

3=k*en. (128)

Here k\ is the geocentric gravitational constant

with the value (Herrick, Baker, and Hilton,

1958):

k\= 1.4350087 Mm3/min2, (129)

where me=mass of earth, m,=mass of orbiting

object, and Am=an augmentation of the masses

for the effect of the perturbations.

For an object of negligible mass orbiting

around the earth with an orbit of small eccentricity, we have with sufficient approximation
GEODETIC USES OF ARTIFICIAL SATELLITES 137

the relation (see also equation (129b), p. 144):

-|sin 8

(129a)

where a e=th e earth's equatorial radius, J=

the coefficient of second harmonic of the earth's

ellipsoid with the value J=1.624X10- 3 (Jacchia, 1958a).

When these six elements are given, the orbit

is defined and we can find the position of the

object at any time T. The positions can be

computed in either rectangular or polar coordinates. We shall use the polar coordinates a,

6, and R as more convenient. (Note that the

values a and 5 are not the apparent but the

mean coordinates, referring to the equinox and

the obliquity of the epoch for which the elements

of the orbit are given.) They are expressed as

follows:

a=G+ar c tan [cos i tan («+»)], (130)

sin 5=sin i sin (&»+»), (131)

R=a(l — ecosE). (132)

Here the auxiliary elements o=true anomaly,

E= eccentric anomaly, Af=mean anomaly, are

given by the formulae

n

f

 (i33)

M=E — e sin E, (134)

M=M0+n(T-T0), (135)

and a is given by equation (128) .

7

The classical methods of Laplace and Gauss

may be used for determining the elements of

the orbit from three observations (Moulton,

1958), although the orbits of the earth's nearby

satellites can be determined also with other

more simple and less accurate methods. In

all those methods, however, we must assume

that we know the positions of the stations with

respect to the center of gravity of the earth,

since the equations of the orbit are given with

respect to that point.

• The element M% can be replaced by Tt, which is both the epoch and

the time of a certain perigee crossing. The expressions for this latter

can are given in Veis (19S8).

By differentiating equations (90) and (92) to

(97), and using l=w+v, we obtain the following:

ba_ tan I sin i

di ~~ 1+tan2 I cos2 i

ba cos i

=

bu cos2 /(1+tan21 cos2 i) ba

bR

cos i

cos2 Z(l+tan8 Zoos2*)

= 1— e cos E

bR w —— = — a cos E

de

bR

oh, a e sin

dM

bM

bs

bi~

bs

ba

bs

bv

dp

bE~

bv

be"

bM

bE

be"

ba

bn

bR

bn

cos i sin I

cos 6

sin i cos I

cos 5

sin i cos I

cos 5

sine

"sin 2?

sin v

1—ecosE

sin 2?

1— ecosE

2a

s

"3^

2o(l—ecosE)

3 n
138

But

SMITHSONIAN CONTBIBTniONS TO ASTROPHYSICS

\bvbEbMbn

/dado, da do

de

da do

/babv bE bM\,

n

/da do bE dM

V^^Md M

bRbEb

TOL.I

da\ .

(137)

bR,bRbE

(138 )

When we introduce the values of the partial

derivatives, we obtain equations (139), (140),

and (141).

We now write equations (139) to (141) in a

matrix form:

da =u*

de

dQ

di

dco

(142)

Or, if we introduce the vector dbn=(dn, de,

dQ, di, do, dM0), we obtain

'di

da

.dRj

=U*db\ (143)

Equation (143) relates the differentials of

the position of the object with the differentials

of the parameters of the orbit bm.

If we introduce equation (143) into equation

(84b), we obtain

dec

jir ^

=Qr

t-Si-U*'db«-ert-dXiQ. (144)

This equation will help us to find the orbit

from observations from known stations, with

fsin i cos I sin o (T— To)

Lcos a sin E(l—e cos E)

s t sin L I ,. , I sin t cos

i*icosl*nv / 1 + 1

cos 8 \1—e2 1—ec

sin t cos Z sin o

cos a sin E(l—e cosE) (139)

da= cost sin V(T—TQ)

Lcos2 Z(l+tan21 cos2 i) sin^U—e cos

cos i sino / 1 . 1

Lcos2 Z(l+tan2 Z cos21) Vj— ^ l —

cost "J

cos2Z(l+tan2Zcos2t)J

Hr

r ae

, , TO , F tan Z sin i "1 ,. ,

d e + +L""l+tan2

 Z cos2 i\ +

cosisinv

cos2 Z(l + tan2 Z cos21)

2a(l—ecoaE) , oesinff

o«sin.E

Z(l+tan21 cos21) sin E{\-e cos £ )

I"

—acos

L

ag sin2 g

1—ecos£

\ J

(141)
GEODETIC USES OF ARTIFICIAL SATELLITES 139

the method of variation of coordinates. In this

case dX£=0 and thus

/dS>\

Ida' \=e-S'U'db't. (145)

\dr /

From an approximate orbit £" we find the

elements of the matrices 0, S, U, as well as the

computed quantities J/ 3 / r. If we have

measured 5', a', r for two positions of the

object8, we can solve for dbu. If we have

observed only the directions, we need three

positions. If we have more observations, we

will adjust by the method of least squares.

We can also introduce as unknowns the correction dX£ to the coordinates of the stations

and solve simultaneously for both dbu and dX^,

using equation (144).

If the stations from which the observations

have been made belong to the same geodetic

system and there is no orientation error, the

value dX& will be the same for all the stations;

therefore we can introduce only one dX* which

will be identical with X£. In this case the

observation equations will be

The variance of bm (or dbu), Xl

c, and g' will be

\dr/

Q-S-U-db'-Q-Xh, (146)

with a total of 6+3=9 unknowns.

If we introduce also the orientation errors

g

s=(dA, d£, drf), the observation equations

will be:9

1

. (147)

If we have more than the minimum required

number of observations, we will adjust by the

method of least squares (as described on p. 122

ff. and p. 129 ff). Whenever this method of

adjustment is used, it may be necessary to

iterate the solution, since the coefficients of the

observation equations depend on the approximate orbit.

[(e -s-u, -e , -e-G)'P(es-u, -e , -e-6)]-1,

where (0 'S-U, — 0, — 0 -0) is a matrix con mating

of the submatrices Q-S-U, —0 and —Q*Q.

The variance V{dbu, X£, g

}

) can be found

only after the observations have been made,

since it depends on the configuration of the

net. We can, however, notice that the elements of the orbit will be more accurately determined if the positions of the object are not

concentrated in only one part of the orbit.

For the accuracy of the positions of the stations

(or of Xc and g

$

, which is the same), see page

126 ff.

Use of a second-approximation orbit.—To the

elliptic orbit (considered on p. 136 ff.) we

will introduce the secular perturbations caused

by the oblateness of the earth (second harmonic) and by the air drag.

The oblateness of the earth introduces a

rotation of the line of nodes on the equatorial

plane (regression of SI) and a rotation of the

line of apsides on the plane of the orbit. The

two motions will be uniform. They depend

both on the orbit (n, e, i) and on the flattening

of the earth (or the constant </).

Various formulas have been developed (Brouwer, 1946; Spitzer, 1950; Davis, Whipple,

and Zirker, 1956; King-Hele and Gilmore, 1957)

for the values of those motions that will be

denoted by Q and u (the dot denotes derivatives with respect to time). Cunningham

(1957) gives the simple formulas

Q = —nj(2S) cos t,

\P /

* ct and K must not refer to apparent positions bat to those

corresponding to the epoch for which we want to compute the elements

of the orbit.

* Compare with equation (130).

where p=a (1-e*), the parameter of the ellipse.

The effect of the air drag is to change the

shape and dimension of the orbit on its plane,

i.e., to change both a (and thus n) and e. Formulas for d and e are given by Davis, Whipple,

and Zirker (1956) for different assumptions.

A very interesting result is that the perigee g

changes very little.

512692—60 5
140 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS VOL. t

Assuming a distribution for the densities,

we can integrate dq/da numerically and obtain

the relation q—j(a) (Jacchia, 1958a).

A secular variation in the inclination will

arise from the rotation of the atmosphere,

(Sterne, private communication; Merson, KingHele, and Plimmer, 1959;. The inclination will

diminish,

The secular variations of the elements can

be assumed to be linear over a short period of

time, but not in general. The most important

effect will be the pseudo-periodic variation in

the densities at high altitudes, which has been

correlated with variations of the flux of solar

radiation (Jacchia, 1959a,b). These variations

(at altitudes of 700 km the densities may vary

by as much as 100 percent) introduce considerable variation in the acceleration (Veis, 1959).

The existence of a third harmonic term in the

earth's potential will also introduce a periodic

variation in the orbital elements (O'Keefe and

Eckels, 1958).

If we take only the secular part, the orbit

could be expressed in the general form of a

polynomial,

(135) we will integrate

ndT.

where h is the vector (n, e, Q, i, «, Jiio), b the

vector (n, e, 0, i, », Sin), etc. The number of

terms will depend on the maximum value of

(T—To). Additional (e.g., exponential or trigonometric) terms can be added if needed.

To find the position of the orbiting object

at time T, we will use equations (130) to (134),

using for e, Q, i, <•>, the instantaneous values

corresponding to time T, and for equation

r.

One method that could be applied to determine the orbit (second approximation),

similar to the one on page 138 ff., is the following: With the new definition for the orbit,

equation (143) will be written as in (143a),

where db^, db', dSw . . • are the corrections

to the approximate values 6*, 6f, $"

(db0, db, db . . . may not be of the same dimensions; actually M=n so that b does not

contain M).

Then equation (144) will be written as in

(144a).

Assuming the stations known (i.e., dX^—0),

we solve for db^, db' .. . as on page 138.

We have applied this method for orbit determinations at the Smithsonian Astrophysics!

Observatory (Veis, unpublished) and obtained

very satisfactory results.

A second method would be to find the mean

elements from observations within a short

interval of time by using, if necessary, approximate values for the variations of the elements

to make the reductions. Provided the time

interval is short, errors in the values of b,

J . . . will not have much effect. Given the

mean elements for different epochs Tit T2, . . .,

we can find the value of b as a function of time.

We now consider the simultaneous determination of the orbit and the positions of the

stations. If we try to solve equation (144a)

for both the orbit and the positions of the

stations vn will have a large number of un-

'd8

da ==f/7* T7*(T—T~} T7*(T—T~\* db' (143a)

da'

'dbJTl

db' (144a)
GEODETIC USES OF ARTIFICIAL SATELLITES 141

knowns, and thus the solution will be weak.

Therefore we solve separately for the orbit at

epoch and the positions of the stations, and

separately for the variation of the elements of

the orbit.

The larger the part of the orbit we use and

the longer the interval of our observations, the

more accurate will be the elements of the orbit,

as well as their variations. But on the other

hand, the smaller the part of the orbit we use

and the shorter the period during which the

observations are made, the more likely it is

that the computed positions of the orbiting

object will be correct. For this reason we will

divide the observations into two groups.

In the first group, the observations will be

made over a long period of time to determine

the variation of the elements b, $, . . . If the

observations are made from enough stations

(which need not belong to those we use for

geodetic purposes; on the contrary, they

preferably should be spread all over the world),

and for a rather long period of time (many

revolutions;, they can provide us with accurate

enough values for b, f>, . . ., although the

observations and the positions of the stations

may not be so accurate.

The second group of observations will be

made as described on page 136 ff. Our purpose

will be to find the elements of the mean orbit

at epoch and the corrections in the positions

of the stations. For this we will use an approximate mean orbit 6" for an arbitrary epoch To

(e.g., the middle of the observations). We will

then compute the positions of the orbiting object

by using T>$+bw(T— To) as orbit, where b' is

the vector obtained from the first group of

observations. In most cases, b" will be constant

in the interval (T— To). If not, we must also

include terms of 5", b*, etc.

If we assume there are no errors in the value

of 6f, the discrepancies will be due only to

d&", ie., the correction to the mean orbit of

epoch To. Thus the observation equations for

stations of the same system will be similar to

equation (147):

the matrix U being evaluated for the value of

Separating the unknowns as described above

may not always be the most efficient method.

For each situation we must decide which separation will give the strongest solution. For

example, the mean anomaly n will often be

more accurately determined from the first group

of observations than from the second (it should

be remembered that n is actually the secular

variation of M).

With the help of equation (146)—where Xc

will be replaced by dX^, which is the correction

to the coordinates of the station—we could

use stations not belonging to the same system, but such a method would be less accurate

because of the large number of unknowns. In

such a case we must also have distance measurements to the object.

From stations of the same geodetic system

we will not usually be able to observe the

object over a large part of its orbit, and thus

the orbit will not be well determined. This

inaccuracy will, in turn, bring large errors in

the determination of X*c and g

1

.

There are two possible methods for dealing

with the problem.

1) The first method uses two groups of stations, belonging to two different geodetic systems, from which the object can be observed

at two different parts of its orbit during the

same revolution (fig. 16).

The observation equations will be two groups

of the form of equation (148), the one referring

da' ]= e-S-U>dbl-QXh-QGg1, (148) FIGURE 16.—Connection of two geodetic system* by fir»t

method.
142

SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS YOL. 1

to observations from stations of the first system

and the other to observations from stations of

the second system; or,

(

d

£) = 9SUdb»-9X<->

/dl'\

[da'] = OS- U- dbl-Q-Xh,-<

\dr)

(149)

This gives a total of 18 unknowns, or 12

unknowns if the geodetic systems have no

orientation errors.

If we have independent unknown stations

between the two systems, we could compute

their positions by observing the object while

it moves from one geodetic system to the other.

For this we will compute the position of the

object for the times of observation, using the

orbit 6j+t#J+&f (T—To)+. . . , as obtained

from the adjustment of equation (149), and

continue according to the procedure described

on page 122 ff.

Or, we could introduce the observations from

the unknown stations to the system (149) and

adjust as a whole. This would give a more

rigorous solution, but would result in a system

with more unknowns. Similarly, if we have

more than two geodetic systems along the orbit,

we will have to introduce for each one an

additional group of equations, of the form (149).

2) The second method is to use a small part

of the orbit, which is the same for the observations from both geodetic systems. This can

be done since different areas will be under the

same part of the orbit during each revolution

(fig. 17) because of the rotation of the earth.

The orbit determined in this manner may not

be very accurate but, on the other hand, the

poorly determined elements willl not greatly

affect the determination of the positions. The

observation equations will be the same as those

of equation (149).

Errors in the orbit and the timing will have

certain effects. From equations (143) and (89a)

we can see that an error Sdu in the orbit will

introduce an error in the position of the object,

N revolution

N +l revolution

FIGURE 17.—Connection of two geodetic systems by second

method.

given by the expression

(150)

To find the effect of an error 5Tin the timing

of the observations, we must differentiate

equations (129) to (134) with respect to T so

that we can find the effect of dT in a, 8, R.

The result will be

/da\

I dd )=u>

\dR/

dT

where u* is a vector, the last row of the matrix

U multiplied by n.

So the introduced errors in the position of

the object will be

dX^Siu^T. (151)

If, furthermore, we have an error sb* in the

value of bv (we again use the index u to indicate

the complete vector b, 5 . . . with zero elements if they do not exist), we will introduce

an additional error db*(T— To) to bu, or an

error in the position of the object,

6Xl

a=(T-T0)SU6i>«. (152)

From equations (150), (151), and (152) we

see that the variance of Xl

s

 will be

V{Xt

s}=S>U'V{bl}-U'-S'+S-u-V{T}.u'-S'+

(T-T0)S-U'V{b«}>U'>S'(T-T0).
GEODETIC USES OP ARTIFICIAL SATELLITES 143

The variance of XQ, then, can be found with

the help of equation (101).

The effect of an error bT in bu (T— To) will be

of the second order and thus may be neglected.

The accuracy with which we can expect to

determine the orbit and the positions of the

unknown station (or the elements Xc and g*)

depends not only on the accuracy of the observations and the net configuration (see p. 139^,

but also on the accuracy of the elements 6"

(since they are determined separately) and on

the correctness of our theory of the orbit.

An error 8b* in 6" in the determination of

the orbit with the method described on p. 140 ff.

will introduce an error to the value of

8bu

0=8b»E{(T-T0)}, (153)

where E{ (T— To)} is the mean value of (T— To)

of the different observations used. If we select

To as the mean time of the observations,

E{(T— To)} will be zero; thus we will not

introduce errors in bo (provided 6*=0). However, an error 8bn

 will introduce errors 8X3 in

the positions given by equation (152).

We must keep 8XS as low as possible, and

this can be done by keeping 8b" low (by making

many observations from many stations well

spread along the orbit and over a long period

of time), and by keeping short the interval of

time used for the geodetic connections (using

no more than a few revolutions, if possible).

If we desire an accuracy such that the errors

in position do not exceed /x', then 8b", and

T— To should satisfy the conditions,

SU 8om (T— To) </i*. (154)

Also the time must be measured with an

accuracy such that from equation (151) we

obtain

(155)

Finally, some errors result from the fact that

our theory for the orbit is not complete. Without a complete theory it is not possible to find

these errors. We can, however, consider the

following remarks:

1. The second-approximation orbit should not be

very far from the true orbit, and if we limit ourselves to

the use of a small part, it is more likely to approach

the true orbit. Also, the farther from the earth's

surface the orbit, the smaller the effect of the perturbations from the earth's gravitational field and

from air drag.

2. In our second-approximation orbit we have

assumed that the one focus of the instantaneous elliptic

orbit is always at the center of gravity of the earth.

This assumption may not be correct for the ellipse

that we fit to the true orbit. In such a case, the

values Xc will not be correct, since they will correspond

not to the true center of gravity, but to an assumed

center of gravity, at the focus of the fitted ellipse.

This can be checked by computing Xc for the same

geodetic system with different orbits. However,

although the Xc values may not be correct, the relative

values between systems connected with observations

on the same part of the orbit will be much more nearly

correct, since both will refer to the same assumed center of

gravity.

3. There are also periodic perturbations that we

have neglected, which introduce a periodic variation

in the position of the object from the elliptic orbit.

The effect of such errors is reduced by the fact that

we fit the elliptical orbit to the observed positions.

There will be, however, a displacement of the ellipse.

For best results, therefore, we should use only a

small part of the orbit. Thus the'method given on

page 142 is more advantageous. The results will also

be improved if we make the same connection, using

different orbits.

4. Perturbations caused by the earth's gravitational

anomalies are expected to be insignificant for the small

interval of time covered by our observations.

5. For the computation of the orbit we have assumed that the constant fc|/i in equation (128) is

known and correct. Actually, an uncertainty exists

of a little more than 10~§ Mm'/min* (Herrick, Baker,

and Hilton, 1958); this includes the uncertainty of J.

Thus the value of a computed from n may be wrong

by about 80 meters, which will introduce an error

of the same magnitude in the geocentric positions.

To eliminate this error we can introduce the constant

k\n as an additional unknown, or, what amounts to

the same thing, define the orbit with both a and n.

Thus, the orbit will be defined with seven elements;

therefore bM and £/£ will have different meanings.

The use of a third-approximation orbit.—

Until now we have not considered any periodic

variations in the orbital elements. However,

such periodic variations exist because of the

non-sphericity of the earth's gravitational field,

and because of the variation of air densities

with altitude. These variations may be of

long period, « or 2« (some months), or of short

period, v or 2v (of the order of 1 hour).

Pseudo-periodic variations (mainly in n) also

exist, due to pseudo-periodic variations in the

densities, of a geophysical or solar character.
144 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS VOL. *

If we use the methods already described we

need not worry about the long period variations,

if we limit ourselves (as we do) to observations

covering only some days. (The periodic variation of the orbit due to the third harmonic

(Kozai, 1959) can easily be taken into account.)

The situation is different with the short period

variations, however; they can be reduced to

some extent only if our observations are very

well spread in true anomaly over the orbit.

At the bottom of this page are expressions

for the short period variations (or perturbations), reduced to four elements. They were

derived by Kozai (unpublished).

Associated with these expressions is the following value for /*:

w (*)'(

(129b)

These expressions show that the discrepancy

from an orbit without short period perturbations can amount to 1 to 2 kilometers, and thus

cannot be neglected. Since these variations

can be computed from their theoretical values

with sufficient accuracy (the effect of an error

in J is completely negligible), we will correct

the orbital elements for short period perturbations to obtain a third-approximation orbit.

Note that it is not an oscillating orbit, and we

shall therefore call it a mean orbit. When we

use it to compute the position of the object at

a certain time, we should add the effect of the

short period perturbations to the elements as

given from the previous expressions.

Aside from the effect from the nonspherical

potential,10 a short period variation occurs in

the orbital elements because the air densities,

and thus the air drag, vary with altitude. Indeed, the assumption of a secular variation in

n cannot be correct since the greater part of

the effect of the air resistance during one revolution will occur within a very small region

around perigee. However, the amplitude of

this periodic variation cannot be more than

the total variation per revolution. In any case,

the orbits that will be of any use for geodetic

applications will have small values of h; therefore this periodic effect can be neglected (the

displacement along the orbit per revolution will

be equal to n/w2).

In applying the third-approximation orbit we

will use the methods described previously, with

one exception. In computing the position of

the object at the time of observation, T, in

order to determine the computed values d',

a', r, we will not use the approximate orbit

t=bo+&(T-To)+ ... . Instead, we will use

the orbit b-{~5b, where 8b are the short period

variations of the orbit.

» The abort period effect of the other terms In the expansion of the

gravity potential is very small.

«Q=—

sin i /co s 2(i>+«)+« cos (o+2w) + | ecos(3»+2w) V

^ \ cos t -[ (v—M)+e sin v~ sin 2(t>+»)—^ e sin (v+2a)— i e sin (3v+ 2«) \ •

yK1"!8in2 0 [-^

^y [(2-§sin'i §sin'i) (o-M+ - e sin - § sin2

-5Lvw)sin v+

•J 1— VT—i5)sin 2 o V— f n"~R sin2ijesin (v-\-2u)—

\2~l 2

 Sm 2

 V S i n 2(

t;+<l>)~~6 ( 1—smS

 ^e s m (3tH-2«)|-
GEODETIC USES OF ARTIFICIAL SATELLITES 145

An orbit of unknown mean distance.—We shall

now consider the case in which we know the

orbit except for the mean parallax (or mean

distance, which corresponds to an unknown

value for kl). (The moon's orbit is an example;

after the perfection of Brown's (1899a, 1899b,

1901, 1908) theory, it is expected that the only

doubtful value will be the mean parallax, cr0-)

Since the orbit will be based on the unknown

mean distance, there will be an unknown scale

factor (1+K) by which the given distances Rt

should be multiplied to give the correct distance R. Thus

or

B=0.+K)R,

dR=R—Rt=KRt,

where K is supposed to be a small quantity.

But a difference dR in R will introduce a

difference in X% given by equation (89); or

dX%=cos 6 cos 8 dR= (Rt cos 6 cos 8)K—S1K,

dX%=sm B cos 8 dR= (Rt sin d cos 5)K=S2K,

d3T|=sin 8 dR = (Rt sin 8) K =S>K.

(156)

Introducing equation (156) to equation (84)

we get

/d8'\

ida'MsU-edXl.

These are the observation equations for the

computation of both dX*Q and #c. They are the

same as equation (84b) with d8=da=0,

Applications of the various methods

Any of the three methods described earlier can

be used to obtain geodetic information from

the artificial satellites. The choice of method

will depend on the kind of satellite, its orbit,

shape, instrumentation, etc.

If we know the orbit of the satellite with a

good degree of accuracy (of the same order as

that with which we want to determine the positions of the stations), we can apply the method

described in pages 122 to 126. If we do not know

the orbit accurately enough, but we do know

that the variation of the elements is more or

less smooth, we can use the method of orbital

interpolation as described in pages 135 to 144.

This more flexible method can be used with

any spherical satellite of rather high specific

gravity and a perigee height of more than 500

km.

If we do not know the orbit at all, we can

use the satellite by making simultaneous observations and using the method described in

pages 129 to 133. Simultaneity of observations

can be ideally achieved if the satellite is specially

instrumented to send flashes of very short duration; such an object we shall call a flashing

satellite.

The orbit: ephemerides and visibility.—Random variations occur in the orbital acceleration,11

 because of variations (of geophysical and

solar origin) in the densities. Such random

variations make it almost impossible to determine an orbit that could be extrapolated for a

long period of time and give a sufficiently accurate position for the satellite. The acceleration varies from 10 per cent (perigee heights of

170 km) to 100 per cent (perigee heights of

700 km) of its value. These variations are

sufficient to displace a satellite by several kilometers along the orbit (or some seconds in time)

in one day, even for a perigee height of 700 km.

However, such variations in acceleration do not

seem to be of a sudden character.

For these reasons, the orbits and ephemerides

prepared by the various agencies (e.g., the

Smithsonian Astrophysics! Observatory) are

revised at least once every week. Although

the extrapolation of an orbit may be in error

by several kilometers, the a posteriori orbit

determination is expected to be much more

exact. Preliminary results of orbit determinations made with the method described on page

140 without correction for the short period perturbations and with observations of moderate

accuracy, gave residuals in the position of the

satellite of the order of 1 km. The final orbits

from accurate observations are expected to be

accurate to within ± 100 meters, or better.

Visibility is an important factor. Unless we

measure only distances, or use electronic meth-

" By orbital acceleration we mean the variation of the period P or of

the mean motion m, per revolution or per day. In the notation of p. 140

the acceleration will be i.
146 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS TOL.J

ods for determining the directions, the satellites

must be visible to be observed or photographed.

During the day a satellite is rarely visible.

At night it will be visible if it is illuminated,

as would be possible if the satellite carried a

light source, either continuous or flashing.

O'Keefe (1956) has suggested the use of retrodirective reflectors illuminated from the ground

by searchlights.

The simplest solution would be to let the sun

illuminate the satellite. Indeed the satellite

can be out of the earth's shadow and thus

illuminated by the sun, while an observer

underneath it is in the shadow (fig. 18). The

Point where the satellite

enters the earth's shadow

Area of visibility

FIGURE 18.—Area of satellite visibility.

satellite will enter into the shadow of the earth

when at an angular distance D from the line

of the terminator given by the formula

cos D f

where p is the radius of the earth. The effect

of refraction has been excluded.

The observer, on the other hand, should be

in darkness, the degree depending on the

apparent magnitude of the satellite. If the

satellite is rather bright, the nautical twilight

(12° depression of the sun) must have ceased

for the observer, but if the satellite is faint,

the astronomical twilight (18° depression)

should be considered. This means that the

observer should be farther than 12° (or 18°)

from the terminator.

In addition, the satellite must be above the

horizon of the observer, preferably more than

15° above, to reduce the errors from refraction.

Or, the angular distance d between the satellite

and the observer should be

sin (75°-d)=

psin 105°

P+H '

Figure 18 shows the form of the visibility

area on the earth, which can be found easily

with a graphical solution if we use a polar

stereographic projection (fig. 19), and assume

the earth to be a sphere. A polar stereographic

map of the world may be combined with this

graphical solution to give the visibility areas on

the same map. This method is now in use for

visibility predictions at the Smithsonian Astrophysical Observatory (Schilling, 1958).

The observations.—Both optical (photographic) and electronic methods can be used

for observing the satellites. For the moment,

however, only photographic methods seem to

be useful for geodetic purposes, since present

electronic methods do not yet give the needed

accuracy.

Since the apparent angular velocity is fairly

important and satellites are not always very

bright, the Smithsonian Astrophysical Observatory constructed a special camera, the BakerNunn Satellite Tracking camera, designed by

J. G. Baker and J. Nunn under the direction of

F. L. Whipple (Henize, 1957). The instrument is a Super-Schmidt F/l camera with

FIGURE 19.—Area of satellite visibility on a stereographic

projection.
GEODETIC USES OP ARTIFICIAL SATELLITES 147

focal length 50 cm and a field of view 5° X 30°,

and is expected to photograph satellites fainter

than 10th magnitude.

The focal field is spherical, and a cinemascope

film stretched on a focal spherical surface is

used for the emulsion support. At a scale of

406" per mm, an accuracy of ±2 " is expected

in the determination of the directions (Henize,

1958).

In addition, a time unit, controlled by a

crystal clock, gives the time of the middle of

the exposure to O'.OOOl (Davis, 1958). However, the accuracy of the timing is not expected

to be higher than ±08.001. This error in

timing will introduce errors in the direction of

the motion, in view of the fact that the satellites have rather rapid motion. The variance

of d' and a' cos 5' can be computed from

equation (42).

The Baker-Nunn camera was developed

especially to photograph faint satellites. If

* they are not particularly faint we can use astrographic cameras or even long focus refractors,

provided we use a special shutter to interrupt

the trail and get the timing. For a flashing

satellite no shutter is needed.

Markowitz (1959b) has developed a dual-rate

camera on the same principle as that of his

moon camera (Markowitz, 1954). Ballistic

cameras (e.g., of the type made by Wild

Heerbrugg Ltd., Switzerland) also can be used

for bright satellites. These cameras, developed

for missile tracking, can give directions with

an accuracy between 3 " and 5" when properly

used. For a flashing satellite they can be

used with or without a shutter.

The direction of a satellite can also be determined electronically (see p. 112); there is

then no problem of visibility but, on the other

hand, the satellite must carry a transmitter.

Electronic methods do not as yet give sufficient

accuracy and can be used only as described on

page 140 ff.

No direct distance measurements to the

satellites have yet been made, but are expected

in the near future. Johns (1958) discusses

this possibility and expects an accuracy of

±30 meters in the measured distances, a rather

optimistic estimate for long distances. If we

also include the effect of errors in timing

(p. Ill), the accuracy will be still less.

The observed directions (or distances) must

be corrected for aberration (p. 115 ff.) and

refraction (p. 117 ff.).

Method for a known orbit.—The method

based on the assumption of a known orbit is

perhaps the simplest in regard to the geodetic

computations, but it is the most difficult to

apply because of the difficulty in obtaining an

accurate orbit. The theory of this method is

discussed on pages 122 to 128.

If we know the orbit, to determine the position of a station we have only to observe the

satellite (or satellites) from the unknown station with an appropriate instrument (e.g.,

photograph the object with a Baker-Nunn or

ballistic camera against the star background).

We observe the satellite at a minimum of two

positions, and record the times of the observations.

The reduction of the observations, after the

appropriate corrections have been applied, will

furnish the observed elements (say a' and 5')

and their variances. We obtain the solution

as described on page 122 ff. The corrections to

the approximate coordinates will be given by

equation (91) and the variance by equation

(92).

If we have a number of stations all belonging

to the same geodetic system, we introduce as

unknowns the coordinates X*c of the origin of

the system and, perhaps, the rotation g* and

the scale factor «.

The observation equations using equations

(20) and (84) in this case will be:

'dS")

da'

,dr J

=-er

ix

t

e-eTtG

t

ig

t

-er

t(Xi

-x$)e. (157)

It is important to know the system of reference in which the orbit is defined and the

system in which the observations were made,

in order to apply the correct expression and to

reduce the observed and computed 5', a', r to

the same system, as explained on page 124.

Since this method requires that we know the

position of the satellite to a high degree of

accuracy, we shall use it only for satellites

whose orbits can be determined accurately.

Since, as we have seen, random variations in
148

SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

the accelerations displace the satellite considerably along the orbit, we must use an orbit

derived from a rather large number of observations made around the same time that the

observations from the unknown station Cor

stations) were made. For a spherical satellite

of a high specific gravity and a perigee height

of more than 700 km, and preferably of 1000

km, and using good observations, we can get

an orbit accurate to ±100 meters or better,

with the method described on page 143. The

larger part of this uncertainty will be of an

accidental character.

This orbit, and thus the unknown stations as

well, will refer to a system of reference that may

not be geocentric. Indeed, the origin of this

system will be a kind of weighted mean of the

origins of the different systems to which the

stations, used for the determination of the

orbit, were referred. Some complications will

also arise from the fact that the true orbit does

refer to the center of gravity of the earth.

If we use many stations belonging to different

geodetic systems all over the world to determine

the orbit, we can expect that a large part of the

errors in the different systems will be compensated; thus the origin of the mean system in

which the orbit will be given will be close

enough to the center of our ideal terrestrial

ellipsoid. New positions for the stations could

be determined with this orbit and an iteration

could be performed to improve the positions of

the stations and thus also of the orbit.

Another approach is to compute the orbit by

using only stations that belong to one and the

same system. The positions of the unknown

stations thus determined will refer to that

same system.

The larger part of the uncertainty in the

position of the satellite will be accidental in

character. Therefore if we use a certain number of observations to determine the position

of the station, and those observations are distributed in every direction, we will considerably

improve the final result as given by equation

(84). The same is also true for the effect of

errors in the observations. Combining equation (98), which will hold for a good distribution

even if we do not measure distances, with

equation (103), then for the standard error of

the position of a station aQ we obtain

(158)

o

If we adopt the rather conservative values

«s=±100 meters, <r=±4*=±2X10-6, r=3

megameters, and «=25, we get <rQ= ±23 meters.

The effect of the systematic errors may increase the error to about ±30 to 40 meters,

this value being always rather conservative.

The velocity of the satellite being about 7-8

km/sec, a systematic error of 1 millisecond in

the time (which is rather optimistic) will introduce an error of 7-8 meters in the position.

Since this error will be in the direction of the

satellite's orbit, the final result can be improved

by using different passes of the satellite so that

the projections of the orbits on the earth will

intersect at different angles.

Finally we have errors arising from errors in

the theoretical values for the perturbations of

short period (p. 143) which will introduce

equivalent errors in the positions.

Since the theory and the values of the constants are good to a few parts in a thousand,

and since the amplitude of those perturbations

is less than a few kilometers, the errors thus

introduced will be less than ± 10 meters. The

use of different orbits will reduce the errors

still further.

Method of orbital interpolation.—With this

method we use the observations to the

satellite to determine simultaneously the positions of the stations and the elements of the

orbit, as described on pages 129 to 133. This

method is very similar to that used for a known

orbit. In a final analysis, the only difference is

whether or not we solve separately for the orbit

and for the positions of the stations. The

remarks made for the previous method are thus

valid also for this one. The choice of method

will depend on the magnitude of the variations

of the orbital elements.

This method is best fitted for connecting

geodetic systems as described on page 140 ff.

The satellite then will be observed during one

or two passes over the two systems and the

solution will be given by equation (149). The
GEODETIC USES OP ARTIFICIAL SATELLITES 149

variation of the elements will be determined

(see p. 140) from observations from other stations over a longer period of time. It may also

prove better to use these auxiliary observations

to determine the mean motion n (which is

actually the variation of M). Since the observations that will be used for the connection

cover a rather short period of time, they may

introduce an important uncertainty in n. In

this case equation (149) will not include dn as

unknown and the vector 6" will be (e, 12, i, «, Mo).

If any other element of the orbit is assumed to

be known for any reason, we can eliminate it as

unknown from equation (149).

To apply this method, the satellite must be

visible during a single revolution over both

geodetic systems, and for as long an arc as

possible. This condition is not unrealizable,

and the higher the position of the satellite, the

easier it will be. Such a favorable situation will

have a duration of several days and will occur

every few months. As an example, we give

in figure 20 a favorable condition for a connecFIGURE 20.—Possible connection of the North American and

European geodetic systems. Dash-dot line, satellite orbit;

solid line, visible part of orbit; broken line, area of visibility.

tion between the European and the American

geodetic systems. The satellite is assumed to

have an inclination of 60° and a height of 500

km. The sun has a declination of +10°, and

the difference between 12 and ao is 6h. If we

use the second method given on page 142, the

visibility conditions are much more favorable.

The accuracy with which the positions of the

stations and the orbital elements will be computed is discussed from a general point of view

on pages 139,140, and 143. For best results, it is

essential that we have a good net configuration

as described on page 126. If we observe only

directions, this will not be very difficult to

accomplish. If we measure only distances, it

will be rather difficult, and it will be impossible

for independent stations. At one station, it is

impossible to get mutual perpendicular directions to the satellite during the same revolution.

A deviation from the ideal conditions for the

net configuration will appreciably reduce the

accuracy, and the errors may be as much as

five times as large.

The kind of orbit is also very important for

the accuracy. The smaller the eccentricity,

the less is the effect of the position of the perigee. The higher the perigee of the satellite,

the more undisturbed the orbit will be. But

since the accuracy of the directions is ±2 " or

10~6, high orbits will bring large linear error,

and should be avoided unless we use long focus

cameras. A height of 1,000 to 2,000 km may

be a good value.

The final accuracy will depend on the accuracy with which the orbit at epoch could be

determined, and the accuracy with which the

variation of the orbital elements and the

theoretical corrections to the orbit (short

period terms) will be given.

The value of b can be determined from

continuous observations over a long period of

time. They need not be of very high accuracy,

but should be made from a rather large number

of stations.

An accuracy of ± 7"/day can be expected for

12, provided the change of 12 is smooth (the

accuracy will depend on the inclination). It

follows from equation (152) that the linear

accuracy of the satellite's position will be ± 10

meters for a duration (T—To) of 1 hour and a

satellite height of 700 km. This error will be

in a direction perpendicular to the polar axis.

If the timing at the auxiliary stations is taken

to±l*, the change of the period can be determined to about ±0\025/day. It follows that
150 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS VOL.S

the error in the position of the satellite will be

±7 meters for a value of (T-To) equal to one

hour. This error will be in the direction of the

orbit.

With the help of equation (152), we find that

the main term in the error of the position, for

an error 8d> in 6> and S'e in e will be, respectively,

and

a(T—T0)6'e.

It follows that for a duration (T-To) of one

hour and a height of 700 km, to get an accuracy

of 7 meters in the position of the satellite, we

must have

&i><±50" per day,

8e<±0.000024 per day,

the first value being for «=0.1.

A very small eccentricity will increase the

accuracy of the determination of both e and a.

Finally, the accuracy will increase if the inclination of the orbit is near 63?4 where u is

zero.

The values of b (and b, etc.) can also be computed from their theoretical expressions, provided the constants are known to a good degree

of accuracy. Q and o» are proportional to J.

For the moment, J is known to about 1 part in

a thousand, and since those variations are

about 5° per day they will be given to about

±0?005. The theoretical determination of h

and e will be more difficult, because the uncertainties of the densities are rather important.

For the errors arising from possible errors in the

short period terms in the perturbations and

systematic errors in the timing, we refer to the

method used when the orbit is assumed to be

known (p. 147).

The accuracy with which the geodetic connections will be made can be computed only

after we have made the observations, because

of the very strong correlations between the

different elements. A rough estimate can be

obtained by making some simplifications such

as the following.

Let <r be the (linear) standard error of our

observations, i.e., the standard error of the

determination of the satellite's position with

respect to the geodetic system, the latter assumed to be self-consistent.

If we use a part of the orbit of length Li to

determine the orbit, and we extrapolate it to a

distance L, the errors will be c(L/Li), and since

the orbit is determined from two independent

parts (the parts observed from the two geodetic

systems) with, say, lengths L\ and L%, we will

have as errors o(LjijL\-\-L\).

If we have v observations, considering that

we have m unknowns, we get as errors

The errors in the determination of the elements

for the connection of the two systems will be

of the same order as those in the determination

of positions.

The value of a will be of the order of ±20

meters.

If the lengths Lx and Z* are equal and the

distance between the two systems is 2Lt, considering that m=12, we get as errors

in meters.

If we have a total of 30 observations, we will

get an accuracy of about ±7 meters.

This estimate will be valid if we assume the

orbital theory to be correct. Including the accidental and systematic errors from other

sources, the total error will increase, but in all

probability it will stay below ±30 meters. By

performing the same geodetic connection with

different orbits we should be able to improve

our results still more.

Methods for simultaneous observations.—This

method does not demand any knowledge of the

orbit, but requires only that we make simultaneous observations from two systems. Since

the problem is purely geometrical there is no

reason to restrict the use of the method to

satellites alone. Any visible object

 M at a certain height can serve our purpose.

There are no dynamic parameters entering

directly or indirectly into the solution. This

fact makes the method a very accurate one

«* V&isalft (1046) and Atkinson (unpublished) have proposed the use of

a flashing rocket in a similar way. For further Information on the

application of the method to rockets, see Veis (1958).
GEODETIC USES OP ARTIFICIAL SATELLITES 151

which may prove to give the best results for

geodetic connections or for tieing individual

stations (mainly islands) into a geodetic system.

The only limitation is the height at which the

satellite must be if we are to observe it simultaneously from the two geodetic systems. At

an altitude of 1000 km we can connect systems

that will be 15° apart, by maintaining a strong

net configuration and the elevation angles above

25° to 30° (to keep the uncertainties from refraction as small as possible). If we want to

connect geodetic systems that are apart by 30°

and 50°, as will be the case for intercontinental

ties, the satellites should have altitudes of the

order of 5000 km to give strong solutions.

The observations may be reduced to simultaneity either by using the method of quasisimultaneous observations described on page 136,

or by synchronizing the shutters of the different

cameras so that they open at the same time.

In this case, however, the observations will not,

strictly speaking, be simultaneous, because they

will not refer to the same time that the light

pulse left the satellite (aberrational effect). A

reduction could easily be made to correct for

this error since we will know approximately

the distances and the velocities. The correction will be given by equation (52), which in

this case will be written:

Ar

CCOBS

e =8'—,

where Ar is the difference in the topocentric

distances from the two stations to the satellite.

Simultaneous observations are best obtained

when the satellite emits flashes. The techniques

for a flashing satellite are discussed in greater

detail by Whitney and Veis (1958).

The timing of the simultaneous observations

will be needed only if we use the star background to determine the direction (the usual

procedure). The timing will be used to relate

the sidereal and the terrestrial coordinate systems; thus it does not need to be of very great

accuracy since an error of 10 milliseconds will

introduce an error in the positions of only 5

meters at the equator. The timing can be secured with an accuracy better than ±10 milliseconds for a flashing satellite, by various methods (Whitney and Veis, 1958). The methods

described in pages 129 to 133 can be used to

compute the connections of the geodetic systems, or to determine the position of the independent stations.

To achieve a strong solution it is very important to obtain a good net configuration. We

should try to arrange the observed positions of

the satellite so that the lines from the known

stations to the satellite, as well as the lines

from the satellite to the unknown stations,

intersect as nearly as possible at right angles.

The final accuracy with which the unknown

stations or the elements of the geodetic connection (i.e., Xci-i, g

1

, e) will be obtained can be

determined only after the final solution, since

it depends on the net configuration. We can

estimate this accuracy, however, by assuming

a perfect net configuration; we can then apply

equation (158) by replacing a\ with

in accordance with equation (98), where rj is

the harmonic mean of the squares of the distances r from the n unknown stations.

Equation (158) then becomes

0

 sn 8

where f\ is the harmonic mean of the squares

of the distances r from the unknown station.

Assuming that <r=±2"=±10-», f«=ft=3

megameters, «=5, n=5, we get

meters.

The effect of systematic errors and an imperfect net configuration will increase the errors

to about 20 to 30 meters. By repeating the

observations we may get a final accuracy of

±15 meters.

This value corresponds to a relative accuracy

of 10~6 for a distance of 1,500 km between the

stations to be connected. The existing triangulations, which are used as a sort of base line,

are expected to be accurate to only 10~" and

may further reduce the final accuracy, especially

for the connection of distant stations.
152 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS rout

The use of the moon

The moon being a natural satellite of the earth,

there is no reason that the methods described

for artificial satellites could not apply for the

moon as well, although it is by no means the

best object for observation, and is somewhat

farther from the earth than is desirable for our

purposes.

The advantage of using the moon is that, for

the present, it is the only terrestrial satellite

for which we have a good orbital theory. In

the future, when satellites will be launched to

heights of some earth radii, they may be

expected to replace the moon completely, and

advantageously, for geodetic purposes.

Only methods that involve direct observations to the moon are discussed here. For

methods involving indirect observations (solar

eclipses and occultations of stars), see Lambert

(1949), Kukkamaki and Hirvonen (1954),

Kaiaja (1955), O'Eeefe and Anderson (1953),

and Army Map Service (1954).

The orbit of the moon: libration.—The presently accepted theory for the moon's orbit is

that of Brown (1899a, 1899b, 1901, 1908), who

formulated special tables (Brown, 1919) for the

preparation of an ephemeris of the moon. The

National Ephemerides give the position of the

moon (computed with Brown's tables) for every

l

h

 interval, to 0*01 in a, and OTl in 5. With

the improvement of electronic computers, the

U.S. Naval Observatory undertook to prepare

improved ephemerides of the moon, in which

the positions were given to 0*001 in R.A., OfOl

in declination, and 0TO01 in parallax, and

Brown's theory was used directly without the

help of his tables.

In the new ephemeris the empirical (periodic)

term of Brown's theory has been removed; the

mean longitude has been corrected by an amount

-8T72-26T74 T-11T22 T2,

to bring accordance with Newcomb's tables of

the Sun, Mercury, and Venus; and a small

correction has been added to the apparent

longitudes to correct for aberration.

The Improved Lunar Ephemeris for 1952-59

(U.S. Naval Observatory and Greenwich Royal

Observatory, 1954) contains the most accurate

material yet available for the position of the

moon. The value for the mean horizontal

parallax, BJQ, may be wrong, however, and for

this reason it may be better to introduce an

unknown scale factor (1 -f K) . Thus the distance

to the moon will be

Rsin 0o

ae .

The ephemeris (as well as the theory) gives

the position of the moon as a function of a

time that must be absolutely uniform (sometimes called Newtonian). Such a time is, unfortunately, unrealizable for the moment (although the so-called atomic clocks may be able

to furnish it). We use, instead, "Ephemeris

time" (ET) which is very close to uniform and,

as a matter of fact, is computed from observations of the moon. The second of ephemeris

time is defined by the I.A.U. as 1:31 556 925.975

of the tropical year 1900.

ET is obtained from UT2 as:

where

ET=UT2+AT,

A!F=+24"349+72?318r+29i.950T2+

1.82144 B.

Tis in centuries from 1900.0.

The value of B, determined by direct observations of the moon (Jones, 1939), is B—l^

— C, where l^=observed longitude of the

moon, and (from Brown's tables)

C= — 10T71 sin (140°. 0 T— 240?7) +

5!22r2+12:96 T + 4T65.

Using the Improved Lunar Ephemeris we have:

AT= 1.82144 (lt-lB)

where lK is the longitude of the moon given by

the new ephemeris.

Ephemeris time can also be computed directly

with the observed a and 5 by inverse interpolation in the ephemeris. The value obtained

from R.A. will be more accurate than the one

obtained from the declination (the accuracy will

depend on the rate of change of those elements).

But to obtain ET we must have the geocentric

observed values of a and S, which means that

we have to know the position of the station,
GEODETIC USES OF ARTIFICIAL SATELLITES 153

to make the appropriate reductions given by

equation (85). We can eliminate the effect of

the error in the position of the station by making

two observations of the moon at positions differing in hour angle by 12h

. In this case, as can be

seen from the second of equation (83), the errors

made in the two reductions will be equal and

of opposite sign. Thus the mean value of ET

computed from those two observations will not

be affected by errors in the position of the

station.

To make observations of the moon at positions differing by 12b

 in hour angle, we must

observe at both moonrise and moonset from the

same station.

The errors arising from the fact that the

R.A. of the moon does not change linearly with

ET will be smaller than 0*001.

The orbital theory gives the position of the

center of gravity of the moon, but this is an

inaccessible point for observations. Instead, a

point called "center of figure" is observed.

By that we mean the center of a circle that fits

best the apparent limb of the moon.

But the moon does not always show the same

limb to the observer because of the libration.

The problem then is how to bring the centers

of the best fitted circles to the same reference

datum, and also, since very often we observe

several points on the limb, to bring the topography of the moon to the same reference datum.

Provided the topography of the moon is

"accidental" the best fitted circles will have

the same radius and the same center. But the

topography of the moon has also a systematic

character. Yakovkin (1954) finds different

radii for different values of libration in latitude

(libration in radius), and different radii have

been found for the east and west limbs.

The foremost completed material is that of

Hayn (1914) and Weimer (1952). A new and

more rigorous study, expected to be completed

in the near future, has been undertaken by the

U.S. Naval Observatory under the direction of

Watts (Watts and Adams, 1950). Its purpose

is to give the heights of different points of the

profile of the moon expressed in seconds of arc

with reference to a unique datum, and for

different values of the libration.

The datum must finally be connected to the

center of gravity of the moon. This connection

has not been yet definitely established. However, if the center of figure is used as the datum

it probably will not be far from the center of

gravity.

The observations.—Markowitz (1954) has

devised an ingenious method for measuring the

apparent coordinates of the moon, using a

special dual-rate moon camera.

The camera, which can be mounted on almost

any telescope,13 takes pictures of the stars for

about 20 seconds, the plate following their

apparent motion. If at the same time the

image of the moon were formed on the photographic plate, the image would not be sharp

because of the moon's relative displacement

with respect to the stars. To compensate,

Markowitz uses a parallel plane plate rotating

around an axis that is perpendicular to the

direction of the apparent relative orbit, and at

an angular velocity such that the image of the

moon is kept fixed with respect to the images

of the stars. This plate is at the same time a

dark filter, which reduces the brightness of the

moon.

At the instant at which the parallel plate is

perpendicular to the optical axis, and thus the

image of the moon is in its true position with

respect to the stars, an electric contact affects

a chronograph and records the time corresponding to this fictitious instant of the exposure.

The plates are measured on a specially constructed comparator, and the readings are

punched directly on IBM cards. About 10

stars near the moon are selected (from the

Yale Catalogue) as reference stars, and their

positions are measured twice. The position of

the center of the moon is determined as follows:

We find an approximate center 7) of the moon

(fig. 21). Using this point as origin, we measure

the polar coordinates of a number of points on

the limb. According to practice at the U.S.

Naval Observatory, 30 points are taken at every

6° along the bright limb.

Using the measured polar coordinates (0, p),

we adjust by the method of least squares for

the coordinates (z, y) of the center of the best

fitted circle, as well as for its radius R.

The observation equations will be

M For the I.O.Y., special telescopes were made with an aperture of

12" and a focal distance oflSO".
154 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

FIGURE 21.—Determination of the position of the center of

the moon.

and thus the coefficient matrix for the normal

equations is:

f[sbD*6]

N=

[sin e cos 0] [sin

[sin 0 cos 0] [cos* 0] [cos 0]

JanO] [cos0]

(159)

where v is the number of measured points.

For the method used by the U.S. Naval Observatory, we thus have with sufficient approximation:

and thus

fl5 0 201

0 15 0

,20 0 30j

(160)

r+o.6

0

L-0.4

0 -0.41

+0.0667 0

0 +0.3J

(161)

if we assume that the standard error of measurement of p is unity.

In general,

N=

and

- 0 — 2

 u 3

0 \ 0

J °

— 0

p

o 2

p

1 2 — « 0

p

(162)

(163)

As can be seen from equations (161) or (163),

the determination of the center O will not be

made with the same accuracy in every direction.

The error in the y direction (line of terminator)

will be one-third the error in the x direction.

As a result, the determination of a' and h'

will in general not be of the same weight, and

there will be an additional correlation.

If a is the position angle of the terminator

(fig. 21), we find that the variance for the

determination of S'o and a'o cos 5'0 of the center

will be that given in equation (164), where o> is

the standard error of p.

This variance should be combined with the

one given by the errors of timing, equation (46).

The motion of the moon is about 0T5 per second

of time, so even if the error of timing is 0"01,

the introduced errors in the positions will be

only 0T005. This means that equation (46)

can be disregarded and the variance will be

given by equation (164).

To evaluate <r, of equation (164), we have to

consider both the errors in the measurements

and the errors resulting from the topography.

If the standard error of measuring the distance

cosSi/J "\ -

18sin2« + 2

16 sin w cos w

— 16sinwcosa>

(164)
GEODETIC USES OF ARTIFICIAL SATELLITES 155

p is o- and the variance of the topography is

<*\ov will have the equation,

(165)

If there is no correction for the topography

of the moon, atov will be equal to the RMS of

the topography, which is of the order of 0T7.

This value will drop to 0T2 with the use of the

existing maps of the moon and perhaps to less

than Of 1 after the work of Watts is completed.

The value of <r, on the other hand, is expected

(Markowitz, 1959a) to be of the order of 1/* on

the plate. With a focal distance of 180", this

corresponds to about 0T04. This means that

the errors are due mainly to insufficient knowledge of the moon's topography.

As an example, we give the following values

of <rj,, and oati assuming the position angle of

the terminator «=0 and the declination 8'=Q

(in this case there is no correlation):

Without correction for the topography:

With correction for the topography from

existing maps:

With correction for the topography from

Watts' maps (expected):

The ratio between the weights in all cases

is 9:1.

There is no doubt that if the moon could

be observed around all the limb, the results

would be much better. In such a case there

would be no correlation between 5' and a' cos 6',

and the standard error of one coordinate would

be

Determination of positions and expected accuracy.—The position of a station from which

we have observed the moon with a moon camera

will be obtained with the same method given

for a satellite of known orbit (p. 147). Since

the value of the moon's mean parallax is doubtful, it would be valuable to introduce it as an

additional unknown, as described on page 143 ff.

We introduce an unknown scale factor (l+«)

and we determine the value of K.

But in this case, since we have not measured

distances but only directions, it will be impossible to solve for K. This follows immediately

from a geometric consideration. Thus, the

computed positions of the stations will be given

within the unknown scale factor (1+*). If

we have two stations of known distance apart,

the value of K can be obtained by comparing

the computed with the known distance.

This means that to get a complete solution

we must have stations geodetically connected.

The introduction of the conditions will be made

as on page 132 ff.

The accuracy obtainable depends on the

accuracy of our observations, on the correctness

of the lunar theory, and on the net configuration. A general analysis is given in page 126 ff.

We will here introduce some numerical values

to get a better idea.

For simplicity, we will assume that the moon

is always at the same distance r and that the

observed values of a' and 5' have no correlation.

But we will assume that the weight of a' cos &'

is % the weight of £'. (This is the case in

which the position angle of the terminator is

zero.) The normal matrix will then be that

shown in equation (166), on the following page.

If we have a great number s of observations,

we can replace the elements of the matrix with

their mean (expected) values, considering that

e takes the values X— 9O°<0'<X+9O°, i

—30°<5<+30°, and assuming that the values

are randomly distributed.

We find approximately,

ro.5 o o

0.5 0

0 0.9 J

or

2 o o

0 2 0 (167)

I 0 1.1 J

Therefore the variance of X' is

r

2 0 0

8

Lo o

a being the standard error of the weight unit.

0 2 0

512692—60 6
1 5 6 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

*o The standard errors <r* in X* will be

Ms

O* = i

<> If we make no correction for the topography

a °g we have (see p. 154) a = 0T18, and if we have

•^ *2. 100 observations we will get the following errors

' in the coordinates:

^=±4 8 meters,

o*=±48 meters,

u*=±36 meters.

i

With the additional corrections for the topography from existing maps and from Watt's

„ maps (p. 155) the errors will be respectively:

§ a

x= ± 13 meters,

•b <r*= ±13 meters,

a

y <rs= ± 10 meters

• and

•£ (^=±8 meters,

o*= ±8 meters,

ff*= ± 6 meters.

This estimated accuracy looks very promising. Unfortunately, this is not the true accuracy obtainable, since we have neglected the

systematic errors. Many efforts have been

^ made to eliminate those errors occurring in the

*» camera and the comparator (Markowitz, 1959a).

§ Both accidental and systematic errors in the

K> positions of the stars used as reference introduce

.9 another source of error.

«. If the standard error of one coordinate of a

o star is ±0Tl and we use 10 reference stars, the

*j* computed coordinates of the moon will have an

/ additional error of about ±0!06. This will

increase the errors a* by some meters.

Only after the final adjustment has been

made will we be able to determine the accuracy

of our result. However, we may expect that the

coordinates of the stations will be computed

with an accuracy of tbe order of 30 to 50 meters.
GEODETIC USES OF ARTIFICIAL SATELLITES 157

A final source of error that we must consider is

the correctness of the ephemeris of the moon.

The coordinates of the moon in the Improved

Lunar Ephemeris are given to within 0T01 in 5,

0*001 in a, and 0T001 in vs. From this rounding

we thus have errors approximately equal to ± 5

meters in 8, ± 7 meters in a, and ±30 meters in

a. With a great number of observations, however, as mentioned on page 128, the errors so

introduced will be very small.

What is important is the correctness of the

theory of the moon from which the ephemerides

were prepared, since it will introduce systematic

errors. As has already been pointed out, the

mean parallax of the moon, GJO, is very doubtful.

For this reason it will be advisable to use the

mean distance

 M of the moon as an additional

unknown. We will then get much better results, considering that the mean distance may

be wrong by several kilometers.

It is not easy to estimate the correctness of

the theory of the moon and how it will affect

our solutions. On the other hand, by analyzing

the residuals we should be able to check and

probably correct the orbit of the moon. The

same holds also for the relative position of the

center of figure with respect to the center of

gravity.

Acknowledgments

The author acknowledges his indebtedness for

help received during the preparation of this

paper, especially to Profs. W. A. Heiskanen,

R. A. Hirvonen, and F. J. Doyle of the Institute

of Geodesy, Photogrammetry and Cartography,

Ohio State University, and to Drs. L. G.

Jacchia, C. A. Whitney, Y. Kozai, and Profs.

J. A. Hynek and F. L. Whipple of the Smithsonian Astrophysical Observatory.

Symbols used

Following is a list of symbols used and their

meanings:

u This is affected not only by errors of the mean PM»IIM-, bat also by

the earth's equatorial radios.

a, semimajor axis

A, azimuth

b

u

, vector or orbital

elements

e, eccentricity

E, eccentric anomaly

H, terrestrial height

Ha, geodetic height

HA, astronomic height

«/, coefficient of second harmonic

of the earth's

potential

k\, geocentric gravitational constant

I*, direction cosines

in the sidereal

system

L, geometric distance between

the two stations

TO', direction cosines

in the instantaneous terrestrial system

M, mean anomaly

n', direction cosines

in the terrestrial system

n, mean anomalistic

motion

N{, matrix of normal

equations

P{, weight matrix

q{, direction cosines

in the geodetic

system

q, perigee distance

r, topocentric distance

R, geocentric distance (on pp.

118 to 122,

R=correction

for refraction)

/, Greenwich sidereal time

T, time as independent variable

v, true anomaly

V, variance

W, coordinates in

mean sidereal

system

z', coordinates in

geodetic system

X', coordinates in terrestrial system

Y*f coordinates in instantaneous terrestrial system

Z', coordinates in apparent sidereal

system

Z, zenith distance

a, right ascension,

geocentric

a', rignt ascension,

topocentric

T, equinox

6, decb'nation, geocentric

8', declination, topocentric

r, HA-H

% deflection of the

vertical in

prime vertical

6, a—t

X, terrestrial longitude

Xo, geodetic longitude

\A, astronomic longitude

£, deflection of the

vertical in

meridian

p, radius vector,

geocentric

<r, standard deviation

r, small time interval

u, argument of

perigee

Q, right ascension of

the node
158 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS THUS

H or

rcosZo

in km

1000-

900-

800^

700-

600-

500:

400:

300--

200--

100--

9 0 : :

80 —

70- :

60--

5 0 -

4 0 :;

30 '-}

20- -

Zo

--80°

-70 °

--60°

--50°

-40 °

--30°

--20°

-- 10°

• 2°

AR

(always negative)

-10"

1

 9'

- 8'

• T

- 6'

- 5"

- 4'

- 2'

-- 1"

:

 - 0".9

:

 - 0'.8

:

 - O'J

- - 0r.6

— 0'.5

- - 0'.2

FIGURE 22.—Nomogram giving correction for refraction.
GEODETIC USES OF ARTIFICIAL SATELLITES 159

Temperature

F° C°

8 0 -

6 0 -

4 0 -

2 0 -

-+30

-+20

--+10

Pressure

inches mm.

- 0

--1 0

3 0 -

2 8-

-760

-720

-680

26--

- 660

Corrector factor

--1.10

--1.08

- - 1.06

-- 1.04

- - 1.02

-- 1.00

- - 0.98

- - 0.96

- - 0.94

- - 0.92

- - 0.90

--0.88

FIGURE 23.—Nomognm for correcting refraction for temperature and pressure.

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