Re-reducing the Southern Polar Zone of the Yale Photographic Star Catalogue


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Re-reducing the Southern Polar Zone of the Yale Photographic Star Catalogue
Physical Description:
xii, 172 leaves : ill. ; 29 cm.
Morrison, Jane E., 1965-
Publication Date:


Subjects / Keywords:
Stars -- Catalogs   ( lcsh )
Astrometry   ( lcsh )
Astronomy thesis, Ph. D
Dissertations, Academic -- Astronomy -- UF   ( lcsh )
bibliography   ( marcgt )
non-fiction   ( marcgt )


Thesis (Ph. D.)--University of Florida, 1995.
Includes bibliographical references (leaves 168-171).
Statement of Responsibility:
by Jane E. Morrison.
General Note:
General Note:

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University of Florida
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All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 002056280
oclc - 33803990
notis - AKP4291
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To Abby, Ben, Taylor and Atticus.


This project would never have been possible with the support and advice from

many members of the astronomical community, my family and friends. First of

all I would like thank my thesis advisor, Dr. Eichhor for his patience, advice,

insight and colorful conversations. By his own example I have developed a

passion for scientific research.

I would like to thank the members of my committee, Haywood Smith, John

Oliver, Ralph Selfridge, and Kwan-Yu Chen for their counsel and support.

I would like to thank my parents for their moral support and encouragement.

I would especially like to thank them for instilling in me the desire to always find

the best in every situation and for sharing with me their belief that I can achieve

any goal I set my mind to. Both of these qualities have enabled me to finish this

project while enjoying nearly every moment.

An extra special thanks goes to my friend and colleague, Ricky Smart, for

his enduring patience and unending support, without which this project may not

have been possible. I would also like to thank him for never a dull moment.

For fun, food and frolicking I would like to send hugs and kisses to Jaydeep

Mukherjee, Sumita Jayaraman, Caroline Simpson, Ron Drimmel, Chuck Higgins,

Leonard Garcia, Stephen Kortenkamp, Elaine Mahon and cast of thousands.

Thanks to Lenny and Croaker for ghostwriting my biographic sketch.


For providing the raw measurement of the Southern Polar Zone of the Yale

Photographic Star catalog and an early release of the International Reference Star

catalog, I would like to thank Tom Corbin and Sean Urban from the United

States Naval Observatory.

For providing background information concerning this project I would like to

thanks Dorrit Hoffleit, Fred Fallon, and Bill Van Altena.

The bulk of the computing has been carried out on the Northeast Regional

Data Centers IBM computer under their Research Computing Initiative program.

This computing time has saved me many hours.


ACKNOWLEDGMENTS .............................. iii
LIST OF TABLES ........ .............. ......... vii
LIST OF FIGURES ...... ........................ viii
ABSTRACT ........... ... .................. .....xi

1. INTRODUCTION ...... ......... ............... 1
Original Reduction ................ ........... 7
Why Do a Re-Reduction? ............................ 8

2. PHOTOGRAPHIC ASTROMETRY .................... 14
Standard Coordinates and their Relationship to the Equatorial
C oordinates . .. . 15
Determining the Equatorial Coordinates from the Measured
Coordinates ....... .. ............... ...... .. 17
The Relationship Between the Standard and Measured Coordinates 20
Some Commonly Used Models ......... ............. 28
4-Constant M odel ..... ....... ..... .......... 29
6-Constant M odel ....... ...... .. .......... 31
12-Constant M odel ........................... 32
Spherical Correction .. .......................... 34

3. LEAST SQUARES ............. ................. 45
Theory of Least Squares ................... ....... 45
Traditional (Linear Least Squares) .................. 47
Nonlinear Least Squares ........... ............... 49

4. METHOD OF PLATE REDUCTIONS .................... 56
Single Plate Reduction: Using Least Squares to Determine the Plate
Param eters. ........... ..................... 57
Final Positions from a Single Plate Reduction, .............. 68
Overlap Plate Reduction .......................... 69
Example of Overlap Reduction ................. ...... 83
Set up the equations of condition ................. 86
Determine J' ................................ 90
Determine a .............................. 93

Determine 3 ........

5. OBSERVATIONS .........
Observations for the Southern I
Measurement of Stars .....
Magnitude System .

Reference Catalog ........
External Catalog ........
Determining the Plate Model .
Single Plate Reduction .....
Final Positions from the Single
Overlapping Plate Reduction .
Conclusions ...........
Future Work ....... .



'olar Zone .


. ..
. .

. .. .. 94

. .. 96
. 96
. 97
. 10 1

. 102
. 102
. 104
. 107
. .. 108
. 114
. 119
. .. .. 126
...... 129

. 168

. 172



Yale Zones. ................................... 6

Refraction Information for plates 1-32 . ..... 43

Refraction Information for plates 33-64 . ..... 44

The Yale Plates ..... . ........... 98

Single Plate RMS Results ....... .............. 110

Overlap Plate RMS Results . ... 125

RMS Single Plate Comparisons: 15-constant model ... 127

RMS Overlap Plate Comparisons: 18-constant model ....... .128

Table 1:

Table 2:

Table 3:

Table 4:

Table 5:

Table 6:

Table 7:

Table 8:


Figure 1: The plates for the Southern Polar Zone of the Yale Catalogue .12

Figure 2: The Stars of the Southern Polar Yale Zone of the Yale
Photographic Star Catalogue . . 13

Figure 3: Gnomonic Projection . ..... ........ 15

Figure 4: Origin Shift ......... .... .... .. ......... 22

Figure 5: Rotation of x-y axis with respect to the -rq axis . 23

Figure 6: Nonperpendicularity of axes . . 26

Figure 7: 4-Constant Model: origin shift and rotation of axes .. 29

Figure 8: The effects of refraction on plate 1. . 39

Figure 9: Example of two overlapping plates . ... 84

Figure 10: Distribution of Magnitudes . ... 105

Figure 11: The different in positions found from the single plate adjustment
for plates I and 24 and plates 1 and 25. . ... 115

Figure 12: The different in positions found from the single plate adjustment
for plates 61 and 64 and plates 62 and 63 . .... 116

Figure 13: Magnitude Dependent Measuring Error, the units of the variances
are arcseconds squared ................... ...... 123

Figure 14: 6-Constant Single Plate Residuals . ..... 131

Figure 15: 6-Constant Single Plate Residuals . ..... 132


Figure 16: 6-Constant Single Plate Residuals .

Figure 17: 6-Constant Single Plate

Figure 18: 6-Constant Single Plate

Figure 19: 8-Constant Single Plate

Figure 20: 8-Constant Single Plate

Figure 21: 8-Constant Single Plate

Figure 22: 8-Constant Single Plate

Figure 23: 8-Constant Single Plate








Figure 24: 13-Constant Single Plate Residuals









13-Constant Single Plate

13-Constant Single Plate

13-Constant Single Plate

13-Constant Single Plate

15-Constant Single Plate

15-Constant Single Plate

15-Constant Single Plate

15-Constant Single Plate









Figure 33: 15-Constant Single Plate Residuals


..... . 134

. . 135

. . 136

. . . 137

. . . 138

. . . 139

. . . 140

. . 14 1

. . 142

. . 143

. . 144

. . 145

. . 146

. . 147

. 148

. . 149

. . 150

. . . 1 3 3

Figure 34: 20-Constant Single Plate Residuals

Figure 35: 20-Constant Single Plate Residuals ... .. 152

Figure 36: 20-Constant Single Plate Residuals . .... 153

Figure 37: 20-Constant Single Plate Residuals . .... 154

Figure 38: 20-Constant Single Plate Residuals . ..... 155

Figure 39: 13-Constant Overlap Residuals . ..... 156

Figure 40: 13-Constant Overlap Residuals . 157

Figure 41: 13-Constant Overlap Residuals . . ... 158

Figure 42: 13-Constant Overlap Residuals . ..... 159

Figure 43: 15-Constant Overlap Residuals . . ... 160

Figure 44: 15-Constant Overlap Residuals. . . 161

Figure 45: 15-Constant Overlap Residuals. . . 162

Figure 46: 15-Constant Overlap Residuals . . ... 163

Figure 47: 18-Constant Overlap Residuals. . . 164

Figure 48: 18-Constant Overlap Residuals . . 165

Figure 49: 18-Constant Overlap Plate Residuals . .... 166

Figure 50: 18-Constant Overlap Residuals . .....

. 151

Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy



Jane E. Morrison

August, 1995

Chairman: Heinrich Eichhorn
Major Department: Astronomy

We have re-reduced the Southern Polar Zone of the Yale Photographic Star

Catalogue (-70 to -90 declination) using an overlapping plate technique. This

region was photographed in 1955-56 with 64 overlapping plates and the reduction

of these plates (i.e. the determination of the equatorial coordinates of the star

images on the plates) was completed in 1971. Because of the scarcity of

observations from the Southern Hemisphere prior to the 1970's, this data set

is particularly valuable.

The plates where photographed so that they overlapped 50% in right ascension

and 50% in declination. Thus every region of the sky was covered by at least

2 plates and sometimes as many as 20 different plates. Originally these plates

were individually reduced. Thus after the reduction, some of the stars had many

as 20 different position estimates (i.e. right ascension and declination). Clearly

a star occupies one position at one time, but this can not be obtained without

mathematically enforcing this constraint on a multiple plate reduction process.

Instead, the separately computed values were averaged to give the "best" (in the

mathematical statistics sense) estimate for the stellar position.

Taking advantage of the high degree of overlap of these plates, we used a

more powerful reduction method for overlapping plates called the overlapping

plate technique. This method enforces the simple fact that, at a given instant

of time, a star is at only one position. When this constraint is enforced, all the

reduction parameters on all plates are obtained simultaneously by solving one large

system of equations. The application of these improved reduction procedures and

resources to the original measurements have yielded star positions which are a

20% improvement over those originally published. These new estimates of star

positions are capable of yielding improved proper motions, which are invaluable

data for many aspects of astronomy.


Astrometry, one of the oldest branches of astronomy, deals with the determi-

nation of locations and the positional changes of celestial bodies. The compilation

of star catalogs is the domain of astrometry devoted to the determination of ac-

curate and precise positions of celestial bodies. One of the most important tasks

of astrometry is to establish a reference frame and maintain it by determining the

motions of the reference stars which it represents. Usually these reference frames

are defined by a star catalog (i.e. FK4 (Fricke et al., 1963), FK5 (Fricke et al.

1988) and soon the HIPPARCOS catalog). The compilation of star catalogs has

been a continuing processes over the centuries. One of the first recorded star

catalogs was compiled by Hipparchus (128 B.C.) and contained 850 stars (Abell

et al., 1987). Throughout the centuries astronomers have devised and improved

upon the methods to compute accurate and precise positions for as many stars as

possible. Today the largest star catalog is the Guide Star Catalog (GSC, Lasker

et al., 1990) containing roughly 20 million objects with a limiting magnitude of

15m and positional standard errors from the plate centers to the edges which vary

from 0".5 to 1".1 in the northern celestial hemisphere and from 1".0 to 1."6 in the

southern celestial hemisphere (Taff et al., 1990). Containing fewer stars, but with

a much higher expected positional accuracy, the HIPPARCOS catalog is the first


star catalog to contain observations solely made in space. This catalog is being

produced by the European Space Agency from observations taken by the satel-

lite HIPPARCOS and will contain roughly 120,000 star positions with limiting

visual magnitudes between 7.7m and 8.7m depending upon the galactic latitude.

The individual rms-errors (root mean square-errors) are expected to be 1.3 milli

arcseconds (mas) per coordinate in position (Kovalevsky, 1995).

Star positions at only one epoch are of limited use, but they become of enor-

mous value when combined with several epochs and changes in the positions are

observed. From studying stellar positional changes several important astronomi-

cal parameters can be determined; probably the most important of these are stellar

parallaxes and proper motions. Trigonometric parallaxes are the only method for

determining stellar distances based only on geometry. Once the distances of stars

are known, then a host of other quantities can be found, for example, absolute

magnitudes and in double star systems, actual orbit semi-major axes and the sum

of the masses of the stars can be determined. Proper motions in combination with

radial velocities allow one to study the motion of stars in clusters, to identify stellar

associations, to determine the orbital motion of double and multiple star systems

and to study the motions within the Galaxy which leads to an understanding of

the evolution, structure and rotational curve of the Galaxy.

The considerable distances to stars reduce their proper motion to very small

amounts; for most stars these are considered negligible within the current proce-


dures of measuring precision. It is almost always an angle too small to measure

with precision in a single year. The majority of stellar proper motions can only

be determined after a cumulative effect (as a rule) over many decades produces a

measurable change in the stars' position. Most of the detectable proper motions

are for nearby stars. Though there are several hundred stars with proper motions

greater than 1."O per year (the largest motion being that of Barnard's Star, 10."25

per year), the mean proper motion of all naked-eye stars is less than 0."1 (Abell

et al., 1987). Therefore, in order to detect accurately proper motions we must

have extremely accurate and precise positions for the stars with preferably decades

having elapsed between observations. Star catalogs, in particular old star catalogs,

are extremely useful for the determination of proper motions for a large number of

stars. For example, stellar proper motions can be detected by comparing catalogs

containing the same stars with epochs several decades apart or by combining the

positions found in a single catalog with new observations of the same stars. Thus,

one way to improve upon the current proper motion estimates is to improve upon

the accuracy of the star positions found in the old star catalogs by using improved

reduction techniques and better reference material than was originally available.

This is one of the justifications for the work reported on in this paper.

For this project we have re-reduced (re-determined) the star positions found

in the Southern Polar Zone of the Yale Photographic Star Catalogue (-700 to

-900 declination, Lii 1971). Because of the scarcity of southern hemisphere

observatories in the first half of this century, there are few accurate old (by

this we mean 50 years or more) star catalogs containing southern declination star

positions. Thus, this region of the Yale Zone Catalogues is particularly interesting

for it contains some of the oldest accurate observations (epoch 1955-1956) taken

with photographic plates of the southern polar zone. In addition, the photographic

plates that were used to create this catalog were photographed so that they heavily

overlapped (50% in right ascension and 50% in declination); thus, many of the

same stars are on several plates. This heavily overlapped pattern is particularly

well suited to our reduction technique which has improved the accuracy of the

stellar positions and will lead to more accurate and precise proper motions when

calculated from the positions found in the improved catalog.

The Southern Polar Zone of the Yale Photographic Star Catalogue is one of

many zones which make up the Yale Zone Catalogues. In 1913 the director of the

Yale Observatory, Frank Schlesinger ("father of modern astrometry") initiated a

project to photograph the entire sky. He planned to re-observe by photography the

positions of the Astronomische Gesellschaft Catalogue (AGK series) stars and then

by comparing the older AGK positions with the new Yale determinations to detect

the proper motions of stars to the ninth magnitude. Ida Barney collaborated with

Schlesinger on this project and after Schlesinger's death in 1942, continued on her

own until 1959. Between 1925 and 1959 they (with very few other colleagues)

published some 20 volumes of accurate positions and attainable proper motions


for nearly 146.000 stars in the zone from -300 to +300, +500 to +600 and +850

to +900 (Hoffleit. 1962).

Dirk Brouwer, director of the Yale Observatory from 1941 to until his death

in January 1966, continued Schlesinger's project and extended it to more southern

declinations: -300 to -900. Early in 1960, Brouwer obtained a 2-year contract

from the U.S. Army Map Service to re-observe all the stars in the Cape Zone

Catalogues for the zones -300 to -500 and -600 to -90o, involving some 70,000

stars, in order to improve the positions and particularly the proper motions in

those zones. Dorrit Hoffleit was appointed to supervise the project, starting at

-300 and working southward (Hoffleit, 1983).

The southern catalogs, like the northern ones, were divided into zones (ex-

tended regions of the sky bounded by parallels of declination). Except for the

polar caps, each zone was covered by photographic plates centered at the same

declination, with the right ascensions of the centers spaced such that the vertical

center line of a plate almost coincided with the vertical edges of one of its neigh-

bors (i.e. overlap 50% in right ascension). This produced a double coverage of

the sky. The overlap in declination was at most only a few degrees and often

none. The Southern Polar Zone has a much stronger overlap pattern than the other

zones. This region consists of 4 belts of zones with a declination overlap of 50%

between the zones and an overlap of 50% in right ascension within the zones. The

Southern Polar zone was photographed according the scheme given in Table 1.

Table 1: Yale Zones

Number of Plates Center Declination Separation in R.A.
24 -75 1h 00m
20 -80 1 12
16 -85 1 30
4 -90 600

The strong overlap pattern is shown the polar plots in Figure 1. The first

plot shows the different declination zones the center representing the Southern

Celestial Pole (-900) and the other zones are represented by concentric circles,

moving outward from the pole: -850, -800, -750, -700 and -650. The first plot

is also only 3 plates, the next 4 plots are graphs for the plates in each zone and

the last plot shows all the plates for the region. The number of plates contained

is each zone given in Table 1.

The Yale Zone Catalogues have been of enormous importance to positional

astronomy and in particular the determination of stellar proper motions. Together

they contain hundreds of thousands of stars. Schlesinger had planned to eventually

cover the whole sky; however, this aim was not quite achieved, the zones +600

to +850, +300 to +500 and -500 to -600 are still lacking. The region we are

concerned with is the -700 to -900 zone. This was the last Yale zone to be

photographed (1955-1956) and has the largest star density and magnitude range

of all the Yale zones.


Original Reduction

The plate material for the Yale Southern Polar Zone Catalogue consists of

64 photographic plates with a sky coverage of 1 l x 110 each. These plates were

exposed between September 15, 1955 to August 7, 1956 at Sydney, Australia,

measured at the Yale Observatory between 1963 and 1968 and the reduction was

completed in 1971. The reduction of the plates (i.e. the determination of the

equatorial positions of the images on the plates) was completed in 1971 (Hoffleit,

1971). Figure 2 (top) shows the stars for this catalog on a polar plot. The bottom

graph in Figure 2 shows the references used to re-reduce the catalog.

In order to explain the reasoning behind our desire to reduce this catalog,

we will briefly outline the procedure for reducing plates. The details of this

are given in Chapter 4. The reduction of photographic plates requires the use

of a set of reference stars, whose stellar images are on the plate and whose

equatorial coordinates are known by previous investigation. In the traditional

method of reducing plates, estimates of certain plate parameters, which are

numerically different and characteristic for each plate, are determined from least

squares reductions of the reference stars' equatorial coordinates with respect to

the rectangular coordinate on the plate. These plate parameters are then used

to convert field stars images (i.e., nonreference stars) to equatorial coordinates.

This method of reducing each plate separately has become known as the "single

plate method."

The original reduction of the Southern Polar Zone was performed using a

single plate method by Phillip Li and Dorrit Hoffleit. The reference catalog used

was the Second Cape Catalog for 1950. The Yale plates were taken between 1955

and 1956, so the reference star positions taken from the Cape were corrected to

the epochs of the Yale plates by applying proper motions for intervals of three to

nine years. On the average there are 150 to 200 reference stars per plate.

After the reduction, the resulting positions of stars that had been recorded on

more than one plate were averaged together. The resulting catalog consists of

18,702 stars with their positions and proper motions on the FK3 system (Third

Fundamental Katalog, Astronomisches Rechen-Institut, 1957). "The tabulated

probable errors of the positions represent simply the internal consistency between

different Yale plates used for each star. They imply that the average probable

error of each coordinate determined from a single plate amounts to -0."33. This

is consistent with the average formal probable error of a plate solution, namely

+0."35" (Lii, 1971, p. 29). Note that the (usually quoted) standard errors are

about 1.5 times this amount or 0."53.

Why Do a Re-Reduction?

A major drawback of the single plate method is that it produces a different

position estimate (right ascension, declination) for each star on each plate, even if


that star also appears on more than one plate. Clearly the star can occupy only one

position at one time, but this cannot be obtained without mathematically enforcing

this constraint on a multiple plate process. Instead, the separately computed values

are averaged to give the "best" (in the mathematical statistics sense) estimate for

the stellar position.

As just mentioned, the most important constraint, which was never enforced,

is the simple fact that any one star, at a given instant of time, occupies only one

position. In the case of the Yale Southern Polar Zone, the plates were so strongly

overlapped that most stars occurred on 2 plates, some even on 20 plates. Thus after

performing the single plate reduction there were as many as 20 different position

estimates for some of these stars. Taking advantage of the rich overlap we have

re-reduced the plates using a more powerful reduction technique, the overlapping

plate technique (Eichhorn, 1960; for review Eichhorn, 1985). In this method the

critical constraint that a star can only have one position at one time is enforced.

Then all the reduction parameters on all plates are obtained simultaneously by

solving one large system of equations (i.e., by a block adjustment). Whereas the

single plate is limited to individual least square reductions of just the reference

stars on each plate, the overlapping plate method reduces all the plates at the

same time using all the stars (not just the reference stars). More accurate (because

the systematic errors on individual plates are diminished as a consequence of the

forced plate-to-plate agreement which must now occur) and thus also more precise

positions than those obtained from the single plate reduction are generated by the

overlap method. This method also has many capabilities that are not possible with

a single plate reduction, such as reducing systematic errors between neighboring

plates and the estimation of parameters common to certain data sets at varying


Though the theoretical formation of the overlap method was known when the

original reduction was done, it was not used. Originally it had been planned to

use an overlap reduction, but as Hoffleit explained in the preface of this catalog,

"the planned more elaborate reductions may be a long time forthcoming because

of a present shortage of both funds and personnel" (Hoffleit, 1971, p. 3). In fact,

when the computations for the Yale Catalogue were carried out, it would have

been very difficult to take advantage of the huge number of existing constraints,

because enforcing them requires a computing effort that, while almost modest by

contemporary standards, was totally prohibitive at the time. Many of the existing

geometric and physical constraints could not be enforced and were therefore

wasted. The solution using all the available information has become possible

only through the emergence of fast computers with large memory capacities.

In addition, we were able to use a more precise and accurate reference star

catalog, the International Reference Stars (IRS, Corbin, 1991) which was made

available to us by the US Naval Observatory (see Figure 2, bottom). Thus by

adjusting the originally measured position of the stars' images on the plates on


the basis of better reference star position estimates (which were not available

when the original reductions were carried out) and using a superior adjustment

algorithm, the early positions of these stars have been improved by roughly 20%.

The improvement in the positions found in this old star catalog will improve

the proper motion estimates calculated with them. For example the ARCS (Astro-

graphic Catalogue Reference Stars, Corbin and Urban, 1991) and PPM (Positions

and Proper Motion Catalog, Bastian et al., 1993) catalogs (both key reference

catalogs) used the Yale Southern Polar Zone Catalogue for their determination of

proper motions. Thus by improving the positions in the Yale Catalogue we can

improve other catalogs that use the positions from the Yale Catalogue. This is

especially important in the Southern Celestial Hemisphere, where there are few

old reference catalogs. The wholesale improvement of the precision and accuracy

of proper motion estimates is one of the keys to improving the precision and

accuracy of a large variety of astronomical data. This is one of the reasons why

the measurements in the Yale Catalogue, which is based on plates taken several

decades ago, are so important.

-75" Zone

Plates 1, 3 and 25

-80 Zone -85 Zone

-90' Zone -70 to -90 Zone
Figure 1: The plates for the Southern Polar Zone of the Yale Catalogue


Star Images for the Yale Southern Zone

IRS Reference Stars
Figure 2: The Stars of the Southern Polar Yale
Zone of the Yale Photographic Star Catalogue


One method of creating a star catalog (a list of right ascension and declination

of a group of stars) is to photograph a region of the sky with photographic plates.

Once the plates are developed, the positions of the star images on the plate in

a certain rectangular coordinate system (x,y) are determined with a measuring

machine. The positions of the measurements on the photographic plate can be

derived only with the use of a set of known positions of stars whose images

are among those recorded on the plate. Their equatorial coordinates are usually

found from existing star catalogs. We call such stars reference stars and all other

stars for which we do not have equatorial coordinates, field stars. The goal of

catalog compilation is to determine accurately the spherical coordinates (for our

purposes, equatorial coordinates) of all the stars imaged on the plate (field stars

as well as reference stars).

In the ideal case, one assumes that the optical system (telescope + plate

holder) is equivalent to a pinhole camera, in which is governed by the laws of

the gnomonic projection. A gnomonic projection is the projection of a spherical

surface onto a plane through a point. For descriptive purposes we will assume the

existence of the fictitious "celestial sphere" and we will further assume that the

center of the celestial sphere lies at the optical center of the objective system.

In this perfect case the portion of the "celestial sphere" where the telescope

is pointing is projected onto a plane through the focal point of the telescope.

The resulting rectangular coordinates projected onto the plane are known as

the standard coordinates ((,,I). The usefulness of photography for astrometric

positional work is connected with the fact that there exists a simple, rigorous and

parameter-free geometrical relationship between positions of objects on the sky

and their standard coordinates projected onto the plate.

Standard Coordinates and their Relationship to the Equatorial Coordinates

7 tangent plane

T(a0 ,8.)

celestial sphere

objective = center of celestial sphere

\ photographic plate

Figure 3: Gnomonic Projection


Imagine a plane which is tangential to the celestial sphere at the point ,o, 6,.

In the plane define a left handed coordinate system ((,r) (as seen from the center

of the sphere), such that its origin is at the tangential point. The positive tr axis

is tangential to the hour circle through co and points northward. The ( axis is

tangential to the parallel of declination at 6o and points east (in the direction of

increasing right ascension). This plane coordinate system is thus, by definition,

parallel to the focal plane of the telescope and therefore parallel to the plane of

the photographic plate being used to record the images. The tangential point can

also be described as the point in which a line normal to the plate coincides with

the optical axis of the telescope and penetrates the plate close to its geometrical

center. Thus star positions on the celestial sphere are projected in straight lines

through the center of the sphere (which is the also the focal point of the telescope)

and onto the tangent plate. This geometrical situation is illustrated in figure 3.

If we further assume that the radius of the sphere is equal to 1, the well known

relationships between a point on the celestial sphere (a, 6) and the gnomonic

projection ((, 7) of this point onto a plane which is tangent to the unit sphere

at (ao,6o) is given by
cos 6 sin (a ao)
sin 6 sin 60 + cos 6 cos 6o cos (a oo)
sin 6 cos 60 cos 6 sin 6, cos (a ao)
sin 6 sin 60 + cos 6 cos 6b cos (a ro).
These equations are rigorous and free of estimated parameters, and can be inverted

so that a and 6 can be found from ( and rf. Note that ( and 77 are in units of

the telescope's focal length.

However, in practice no real optical device produces images in the same way

as a pinhole camera. One needs to take into account such effects as coma, radial

distortion, rotation and tilt of the plate when photographed or measured. The

measured coordinates (x,y) are actually approximations to the standard coordinates


Determining the Equatorial Coordinates from the Measured Coordinates

The goal of a plate reduction is the derivation of the spherical equatorial

coordinates from the objects' measured positions on the plate. Usually this process

is achieved in two steps. First the measured coordinates are transformed to the

standard coordinates, then these are converted to equatorial coordinates. The

second step has already been dealt with (i.e., equation 1), so only the first remains.

The establishment of the relationship between the standard coordinates (,,r7)

and the measured plate coordinates (x,y) of the star's images is the fundamental

problem of determining the equatorial coordinates. In the ideal case (assuming a

pinhole camera), even though the precise position of the image of the tangential

point (0o, o) on the plate is not known, it is reasonable to assume that it is near

the geometric center of the plate. Plates are usually exposed in such a manner

that it is also reasonable to assume that the edges of the plate (i.e. the x and y

axes) are closely parallel to the ( and 77 axes, respectively. If the origin of the


measured coordinates is chosen to lie at the geometric center of the plate and

the axes are properly oriented, to first approximation the standard coordinates are

related to the measured coordinates by a factor of the telescope's focal length, s:

)= + +0(1). (2)

The unit of length on the plate in the (-r system is such that the standard

coordinates are equal to 1 at a distance of 1 focal length from the tangential

point. Each departure from the ideal imaging process, a gnomonic projection,

results in deviations from the perfect relationship in the equation

(x c S cos 6 sin (3)
y d sin p cos )\ 6 s

where c, d and 6 are all small quantities of the first order. The end result is a

mapping between the standard coordinates and the measured coordinates. Tradi-

tionally, this mapping is called a plate model. The plate parameters have tradition-

ally been called plate constants. We prefer the terminology "plate parameters"

which indicates that these are adjustment parameters to be estimated by some

statistical process.

In the traditional method of reducing plates one assumes there exists for each

plate the following relationships

f(xi,yi; mi, c; i, 6i; a, a2,,...,an) = 0


between the measured coordinates xi, yi of the image of the ith star, its magnitude

mi and color index ci, and its coordinates ai, Sii, on the one hand and a set of plate

parameters {ak ) on the other hand. If all the plates had been taken with the same

telescope, the plate parameters will belong to groups such that some elements vary

from plate to plate, others vary from plate to plate but are constrained, and finally

some are the same for all plates obtained on the same telescope. (Eichhorn, 1974).

After the functional form of the plate model is established, the plate constants

(or estimates thereof) are determined from a statistical adjustment (usually a least

squares adjustment), by fitting the measured coordinates to the corresponding

spherical coordinates of the reference stars (a, 6). In this case, equations (4) then

become the equations of condition for a least squares adjustment in which the xi,

yi, ai and 6i are regarded as observations and the at, a2, ..., a, as the unknown

parameters. The mi and ci are assumed to be known for every star. Of course,

there must be enough reference stars to make the system over-determined (i.e.,

the number of reference stars must be greater than the number of parameters). A

good rule of thumb is that there should be at least 3 times as many reference stars

as plate parameters. Once the plate parameters are determined the best estimates

of the standard coordinates for all the stars (field stars as well are reference stars)

can be obtained. Then, after routinely inverting the gnomonic projection (equation

1) the equatorial coordinates for all the stars are computed.


The Relationship Between the Standard and Measured Coordinates

Even under the most favorable conditions the assumption of a perfect

gnomonic projection can only approximate the actual circumstances. It is clear

that neither telescope nor measuring machine can be constructed with perfect ac-

curacy. First of all, there is no optical system in practical use which unites in a

single point all light rays that enter the front component of the objective from the

same direction. In reality there are many deviations from the ideal situation which

require small corrections to be added to equation (2). These departures from the

perfect gnomonic projection are "instrumental." Others, which rotate and distort

the coordinate system with respect to which the position estimates of the reference

stars are catalogued, may be termed astronomical effects and include refraction,

aberration, precession and nutation. We will deal separately with these two types

of "corrections." The "instrumental" corrections will be characterized by the plate

model and the "spherical" corrections will be discussed in the next section.

Below is a list of some of the more common departures from the ideal

gnomonic projection; this list is by no means exhaustive:

1. Shift in the origin of the x-y coordinate system with respect to the (-r7

coordinate system (c and d in equation 3

2. Rotation of the -rq axis with respect to the x-y axis.


3. The (-r; scale may be different from the x-y scale, due to a poorly known

focal length.

4. Tilt of the x-y focal plane with respect to the (-q focal plane, this will cause

an incorrect tangential point.

5. Different scales on each axis, due to different measuring screws or otherwise

introduced by the measuring device.

6. Lens aberrations (distortion, spherical aberration, coma, astigmatism, chro-

matic aberration) affect the position of the stellar centroids either by enlarging

them (thus making it more difficult to determine the center) or by an actual

shift in positions, both of which vary depending on the location on the plate.

These effects plus peculiarities of the process of photographic image forma-

tion render the positions of the star's images on the plate dependent on their

magnitudes and spectra expressed in terms of color indices. These conditions

lead to the inclusion of the mi and ci terms in the mapping between x-y and

(-7] coordinate systems.

7. Besides lens aberration "smeared out" or "blow up images can be produced

by guiding errors, developing errors (e.g. emulsion shifts), and bad seeing.

These can and do affect the position of the stellar centroids.

Each departure from the ideal imaging process results in deviations from the

relationship x=s<; y=srj. Given below are the relationships between the standard


coordinates and the measured coordinates to correct for many of the commonly

experienced errors.

1. Translation error: If the photographic plate is not centered correctly on

the measuring machine, there will be a systematic difference between the

measured coordinates and ideal coordinates.



Figure 4: Origin Shift

x s = c

y sl = d

4 -4-


2. Rotation: If the photographic plate is correctly centered, but rotated relative

to the measuring machine axes by an angle 0 (0 > 0 for counterclockwise).
x = s( cos 6 + / sin 0)
v = .( rcos 6 ( sin o).


Figure 5: Rotation of x-y axis with respect to the &-7 axis

3. Imprecise focal length: An incorrect focal length can cause differences in

the scales between the ,rj system and x,y system. This can be corrected by

making small changes in ( and r:
x s =
y sq = er.


4. Center error (tangent point error): For wide fields ( say over 50 x 50) the

extraneous methods for the determination of no and 60 yield approximate

values of these equations and thus they must be corrected differentially. If

the actual position of the tangential point is (oo+dco), (6o+db0) instead of

oo,60, values of jq will be incorrect if calculated only from to,So. It can be

shown to first order in doo and d6o (Eichhorn, 1971) that the centering error

can be corrected (to first order) by
x s, = (cos Sodo),2 + d6,r
y srs =(cos bodoao)( + d6orl2
provided that the plate was oriented during measurement so that the x and (

axis as well as the y and q axes are nearly parallel and (, r1) are in radians

If dao and dbo turn out to be large, it is recommended that the calculation be

repeated with the newly calculated ,r7 now referred to the tangential point ao+dao

and 60+d60. This may be repeated until further repetitions result in no change in

the assumed ao, 6o (Eichhorn, 1974).

5. Tilt: If the photographic plate is not perpendicular to the optical axis of

the telescope but titled by an angle w relative to the focal plane, the errors

introduced will be
x s = (pg2 + q@) tanw
y sr = (p + qri2) tanw
where p and q are constants (Taff 1981). One can see the terms for the

corrections for zero point error and tilt have the same form. This is because if

we have a situation where there is a tangential point error this is the equivalent

of having an imaginary plate whose true tangent point coincides with our

assumed one, but is inclined to the actual plate. Since these two effects can

be modeled in a similar manner, they are sometimes combined under that

same name. If construction and proper alignment of the telescope has been

carefully executed such second order errors may be neglected. However, in

the case of large plates (over 50 x 50), it is important that these higher order

terms be included (K6nig, 1962).

6. Radial distortion: If the image process produces a net radial distortion, it can

be modeled. Radial distortion originates at the intersection point of the optical

axis with the focal plane and its origin will, therefore, also be very close to the

tangential point. This aberration is radially symmetric and usually modeled

in the form
x S= E a(k + 772)1
-1 (10)
y s''= 1 Rk(W + q2)k
where R is a constant (Eichhorn 1974). Unless the radial distortion is very

strong, and for the field sizes used on photographic catalog astrometry, only

the first terms of equations (9) are necessary for appropriate modeling of

radial distortion. Thus with n = 1,
x s=R~((2 + rl)
y si==Rr(r2 + r2).


7. Nonperpendicularity of axes: If the axes of the measuring machine are not

perpendicular to each other (in other words x is not perpendicular to y), let

be the angle between the q and y axes (McNally, 1975)
x s=qr tan u'
v sq=(1 sec ,)r.

y rl
A 4


Figure 6: Nonperpendicularity of axes

8. Coma: The effective focal length of the instrument may depend on the

apparent magnitude m of the object that is being imaged. This is coma;

it is radial and accounted for by
x s=tmx
y srt=tmy.


Experience shows that the coefficient s is rather sensitive to the rate of change of

the ambient temperature during the exposure. Thus it can not be regarded as a

constant for any one objective, although the values obtained in practice will have

a tendency to cluster around a certain value (see Eichhorn & Gatewood, 1967).

9. Decentering distortion: If all the components of the objective are not prop-

erly aligned, the resulting imperfection is known as decentering distortion.

Brown (1966) found that the appropriate corrections to the measured x and

y coordinates are

=1 (14)
y-s,=={2[P,,7+P&,r]+P,( 2 +i2)} 1+ Pk+2(&2 +2,)k
again, under the conditions imposed by catalog photographic astrometry, the

terms with Pk+2 may usually all be assumed to equal zero, this results in a

second order effect
x s&=2[P,& +P297] + P, (2 +r2)
y sq=2[Plq+P2,2] + P, (2+2).
One can see by comparison, the first brackets are exactly the same structure as

those that model the correction to the tangential point error (number 4) and tilt

(number 5). If the model includes the provisions for correcting the tangential

point (or tilt) anyway, the above decentering terms in the brackets can be may

"lumped" with them. Then by just adding the Pi (&2 + q72); P.(2( + q2)terms

one can account for the corrections due to decentering distortion.


As just indicated the sources of errors are not independent of each other

and there is much crossover in effects but they can be modeled in a first order

(sometimes second order) approximation that will more than suffice considering

their magnitude and the magnitude of the measuring errors. It is worth noting

that the coefficients which describe the distortions are very insensitive to changes

in the ambient temperature and may thus be regarded as more or less constant

for any one objective. This statement is not true for coma, which is, in effect, a

dependence of the effective focal length (scale) on magnitude.

Some Commonly Used Models

The difficulty in establishing the relationship between the standard and mea-

sured coordinate system is in determining which terms to include in the model.

Eichhom (1985) writes the relationship (2) in the form

(x) = s(/ +Ea (16)

where E, the model matrix, is a function of (, r7, m and c, and a is the vector

of plate parameters. The form of the model matrix is determined by probing

possible deviations from the gnomonic projection, first adding appropriate terms

to a very basic model, and then investigating the adjustment residuals produced by

additional terms. When significant terms are found they are added to the model.

This trial and error process continues until one is satisfied that one has found the


"best" possible model which approximates the "true" system. We will now look

at some of the models which have been employed in the past.

4-Constant Model

The first model we will look at is the simplest and therefore the most basic

model; it corrects for errors in the measurements. We will assume the tangential

(ao 60) to be known or determinable by extraneous methods and thus not subject

to the adjustment. The scale and orientation of the standard coordinates are fixed

by their definition. However, the system of the measured coordinates is dependent

on the measuring machine and the accidental orientation of the photographic plate

on it. Consequently, the two systems may not agree because the origins of each

system might be different and the axes of one system may be rotated with respect

to the ones of the other. This situation is demonstrated in figure 7.



S--- -- -- -


Figure 7: 4-Constant Model: origin shift and rotation of axes

x = s( cos o + sin o) + c
y = s( coso sin o) + d.
where o is the angle of rotation, s the focal length and c and d the difference in
the origins. Combining these effects results in the following model. If we let a

= s coso and b = s sino, then the equation becomes,
x =a + br + c
y = a7l b + d.
Frequently, equation (18) is written in the form

(x) = 5 r( ) + ( qrl r, l 0 a
=) s ( 0 )( (19)

In this case plate model, E, is the 2x4 matrix and the vector of plate constants

a is (a, b, c, and d). There are four plate constants, so this model is called the
4- constant model.

Usually, the plates are measured in such a way that the x-axis is almost

parallel to the I-axis which will (if x and y are strictly rectangular) render the
y-axis almost parallel to the qr-axis. This will make b very small compared to

a. So, when O 0,
a = pcos o cos 0 m 1,
thus, a p
b= psin sin s 0,
therefore, b 0.

Thus the term a is the correction to the assumed focal length. If the x-y and -rq
axes are tilted with respect to each other, this term may have a small effect due
to this rotation, but the dominant effect is the correction to the focal length.

The inversion of the above equation is

(' = 1 (a -b f (x c (22)
q )- + b b a bd

or after introducing certain quantities

[()=(B AB)()+(C) (23)

where A, B, C, D make up the matrix of plate constants. Thus for the 4-constant
model a rigorous equation exists for and 77 to be solved in terms of the plate
constants and x and y.

6-Constant Model

The 4-constant model does not correct for different scales in the x and y
measurements. Correcting for this effect, equation (17) becomes

(x ( cos sinm (') + (c ( (24)
Y ) (- sin cos 0 ) d e ()

or after multiplying out
x = s cos Of + s sin rl + c a(
y = -s sin O+ + s cos Or + d er.


After substituting the following values, a = cos o e cos o b =sin o,

f = -sino, c= and d= d


(X+ 0 0 0 i (26)
1. a & e: correction to assumed focal length.

2. b & f: correction for rotation of x-y system with respect to (-r system. This

also corrects for nonperpendicularity of the x and y axes. These should be

approximately equal and opposite in sign.

3. c & d: differences in origin of the x-y system and (-r1 system.

For the 6-constant model there exists an equation analogous to equation (23)

for the 4-constant model. In other words ( and r can be rigorously solved for

from the plate constants and x and y. This is not true for the other models we

will be using. In those cases, an initial approximation is used and the solution

then iterated until a set tolerance is met.

12-Constant Model

So far our models have only dealt with terms linear in the standard coordinates

( and r7. However, there are many "second order" effects that need to be

considered, especially when dealing with large plates like the Yale plates (11

x 110). A standard model for the conversion of the measured coordinates to the


standard coordinates considers corrections for 1) rotation, 2) translational error,

3) tangential point correction, 4) radial distortion, 5) tilt and 6) coma ; it is the

12-constant model:


0 0 0 q 1 (q q2 0 m mo qm q1(2 + r12) p



1. a & e: correction to assumed focal length (different in x and y).

2. b & f: correction for rotation of x-y system with -q system. This also

corrects for nonperpendicularity of the x and y axes

3. c & d: differences in origin of the x-y system and (-77 system

4. p & q: Tangential point correction and tilt

5. i & j: magnitude "equations"

6. g: coma

7. h: radial distortion


All terms of -a are at least one order of magnitude smaller than x, y or (, rl and

s is, of course, an approximation for the focal length. This clearly shows that the

rigorous formulas give (, r/ not in radians, but rather in units of the focal length;

they are indeed dimensionless quantities.

This model is very similar to one we chose to best represent the spherical

coordinate system. The actual model and method of determining the model for

the Yale plates is given in Chapter 6.

Spherical Correction

We stated earlier that we split the deviations from the perfect gnomonic

projection into instrumental and astronomical effects. The instrumental effects

are corrected by the plate model. The astronomical deviations arise because

the image formed on the photographic plate represents the refracted apparent

topocentric positions of the objects. The standard coordinates are computed from

the coordinates listed in star catalogs (e.g. mean positions). To account for the

difference between the refracted topocentric apparent positions (x, y) and the mean

positions (, rt), corrections for refraction precession, and for extreme accuracy,

diurnal aberration, should be applied. We will not consider diurnal aberration,

since its effect is smaller than our measuring error. The effects of precession are

accounted for by correcting the reference catalog and influence the results only

slightly through refraction.


Atmospheric refraction causes light from a celestial body to be bent as it passes

through the Earth's atmosphere. Assuming a simple model for the atmosphere as

a plane atmosphere and for small zenith distances, total refraction in the zenith

distance can be calculated by

z -z = Az = 3 tan z (28)

where z and zo are the topocentric apparent (i.e., unaffected by refraction) and

observed refracted zenith (i.e., effected by refraction) distance of the object,

respectively. 3 is the constant of refraction and depends weakly on the temperature

and pressure at the observing station (usually this value is about 1 minute of arc).

Since Az > 0, astronomical refraction raises celestial objects toward the zenith.

Astronomical refraction ideally does not affect the azimuth.

For large zenith distances, allowance has to be made for the variation of 3

within the field of the plate; in practice, it suffices to put 3 = 3o + 3' tan2 z,

where 3' and 3o are refraction constants and can be found in refraction tables

(KOnig, 1933). Thus equation (28) becomes

Az = /o tan z + /' tan3 z. (29)

We have neglected higher order terms for zenith distance less than 700.

We will split the effects of atmospheric refraction into those of "absolute"

and "differential." By "absolute" we mean the amount of refraction affecting the


center of plate or more precisely the tangential point. We can think of refraction

as shifting all the positions on the plate by this constant amount plus a differential

amount which varies across the plate. For example in Figure 8 (top) we have

plotted the effects of refraction for plate 1. As one can see the positions are

shifted mostly in the y direction (an average of about 49"). In Figure 8 (bottom)

we have shown the effect across the plate by subtracting the tangential point

refraction. Note that the scales for the two graphs are not the same; in the bottom

left hand corner of each graph we have plotted the scale: 50" for the top graph

and 5" for the bottom one. We will apply a correction for differential refraction,

and since our plate model already has a linear term in it we will allow the linear

effect of absolute refraction to be taken care of by our plate model.

It has been a frequent practice to account for differential refraction by adding

higher order terms to the model. This is usually achieved by adopting a full

second and third order polynomial expansion in the coordinates. There are two

major disadvantages to using this approach.

Firstly, if the number of parameters is large an unfavorable error propagation

of the resulting star positions in the plate field due to the random errors in

the x-y measurements and the reference star positions is introduced (Eichhorn

and Williams, 1963); in particular when the number of reference stars per plate

is small. Secondly, all effects are mixed up in the model and therefore it is

impossible to separate different contributions and evaluate the physical meaning

of the plate parameters in detail.

As we mentioned above we will pre-correct the x,y measurements for the

differential atmospheric refraction. These values are used as observations in the

plate reduction. In this way, the number of parameters in the model can be kept

to a minium which will give us a better chance to disentangle the complicated

origins of many of the plate parameters (Zacharias et al., 1992).

The usual approach in plate reduction calculations is to transform the equa-

torial system to the horizon system and make refraction corrections in a simple

way in the horizon system. In this manner the effect of refraction is added to

the reference positions. However, this leads to the situation where the same star

imaged on different plates has different positions due to refraction. From Chapter

1 we stated that the power of the reduction technique that we use, the overlap-

ping plate method (Chapter 4), is that it enforces the constraint that a single star

photographed on many plates must have only one position in the sky. Therefore

correcting the stars' equatorial coordinates for refraction leads to a different posi-

tion depending on what plate it was imaged, thereby invalidating our constraint.

We have chosen, instead, to pre-correct the measured star images so that they

represent the position on a plate if the plate had been observed outside of the

atmosphere. We have used the equations developed by K6nig (1933) for calcu-

lating the corrections for refraction, Ax and Ay, which are to be applied to the

apparent topocentric coordinates, x and y. After calculating the zenith distance of


the plates at the time of exposure and then allowing for temperature and pressure

we can determine 30 from the refraction Tables 7-13 in K6nig (1933) and let 3'

= -0.082". Table 2 and Table 3 contain all the necessary information to determine

refraction with the aid of Konig's refraction tables. In addition, the latitude at the

Sydney observatory is -330 51' 41."1, the longitude is -10h 4m 49.s06, and we

have assumed an altitude at sea level and a barometer reading of 750 millibars.

Refraction Correction: Absolute

0.037 -0.034

3.50318 YRMS:

Refraction Correction: Relative

0 % %( v 41 11 Hi

It, ,

.. .. C. -. *- -
~N- \ -

<-* I '*;. ^< *

* / ,,, ",', ,, '' ,,, :

,^ '^ ;n^'.

n 1i n



3.10016 YRMS:

Figure 8: The effects of refraction on plate 1

-0.108 i











i i I r i i I i I

The tangential point is taken to be the refracted ao, o0. The rigorous mathe-

matical relations between the refracted positions and the topocentric positions are

somewhat complicated. For practical application, only development into series

can be considered; it is therefore necessary to state which terms must be retained

and which can be neglected.

In order to obtain the simple expressions for the coefficients of the series,

Kbnig introduces the following auxiliary quantities

k = tan z sin y

k2 = tan z cos X
k3 = 1 + k

k4 = 1 + k2

where y is the (refracted) parallactic angle at the tangent point (taken to be the

refracted tangential point) and ki and k2 have a simple geometrical meaning;

they are the tangential coordinates of the zenith in the plane of the plate. The

corrections for the measured values are (to the third order):

Ax = [3k3 +20/'(1+ kl + k )k ]x

[3k(k2 + tan bo) + 23'(1 + + k + )kk2]y

+ 2[3 + 20'(1 + k + k2)]klk2y klk3x2 (31)

23k k2xy /3kk4y2 + /k 2x3 + 30klk2k3X2y

+ ,(kak4 + 2k k2)xy2 + /kik2k4y3

Ay = [3k(k2 + tan o) + 23'(1 + k2 + k )kik2]x

+ [3k3 + 23'(1 + k2 + k )k ]y

+ [3 + 23'(1 + kf + k:)] (k2 k )y (32)

3k2kax2 23kik xy 3kk4y2 + 3kik2k3x3

+ 3(k3k4 + 2k2 k)x2y + 33kik2k4xy2 + 3k y3.
XR = 3lkk3x2 23kfk2xy 3klk4y2 + 3k x3
+33klk2k3x2y + 3(k3k4 + 2k k )xy2 + 3klk2k4y3
YR 3k2kax2 20kik xy 3k2k4y2 + 3klk2k3x3
+i3(k3k4 + 2kI k)x2y + 3/3klk2k4xy2 + 3k y3
also define
A = 3ka + 2'(1 + k + k2)kl

B = [Oki(k2 + tan) + 2/'(1 + k2 + k )kik2]
C = 2[3+20'(1 +k k) + k i]k2y

D = [ + 2/'(1 + ki + k2)] (k2 k).

Substituting these variables into equation we can rewrite equations (31) and (32) as
Ax = Ax + By + C + XR
Ay = -Bx + Ay + D + YR.

Written in this form the refraction terms which cause a change in the scale (A),

rotation (B), linear refraction terms (C and D), residual refraction (XRand YR) cab

be separated. Generally, for small fields and for zenith distance less than 70 the


terms A and B can be neglected. However, this is not so for plates taken at very

high declinations. We have included the terms for change of scale and residual

refraction, but not rotation. Rotation was not included because the rotation caused

by refraction cannot be separated from that caused by the position of the plates

in the measuring machine.

Table 2: Refraction Information for plates 1-32

1 0.0 0.108 10 15
2 1.0 1.042 11 11
3 2.0 2.058 12 5
4 3.0 2.75 11 20
5 4.0 4.342 1 6
6 5.0 5.408 1 13
7 6.0 6.158 1 8
8 7.0 7.092 1 8
9 8.0 8.092 1 18
10 9.0 9.108 1 18
11 10.0 10.063 2 7
12 11.0 11.258 4 12
13 12.0 11.958 4 11
14 13.0 13.083 7 3
15 14.0 14.008 4 10
16 15.0 15.233 6 12
17 16.0 16.250 6 12
18 17.0 17.150 6 5
19 18.0 18.150 6 5
20 19.0 18.883 6 5
21 20.0 20.238 9 19
22 21.0 21.175 10 13
23 22.0 22.208 10 15
24 23.0 23.275 10 15
25 0.0 0.096 11 20
26 1.2 1.413 10 13
27 2.4 2.496 11 20
28 3.6 3.713 11 20
29 4.8 4.629 1 13
30 6.0 6.213 1 9
31 7.2 7.046 1 13
32 8.4 8.579 1 18

68 0.108 41.15 48.25
70 0.042 41.14 48.05
70 0.058 41.14 48.06
70 23.975 41.14 48.05
70 0.342 41.21 48.18
75 0.408 41.26 47.78
75 0.158 41.16 47.63
75 0.092 41.14 47.61
77 0.092 41.14 47.43
77 0.108 41.15 47.43
75 0.063 41.14 47.60
65 0.258 41.18 48.58
65 23.958 41.14 48.51
60 0.083 41.14 48.99
65 0.008 41.14 48.51
60 0.233 41.17 49.04
60 0.250 41.18 49.05
60 0.150 41.15 49.00
60 0.150 41.15 49.00
60 23.883 41.15 48.99
60 0.237 41.18 49.04
68 0.175 41.16 48.27
68 0.208 41.17 48.28
68 0.275 41.19 48.32
70 0.096 46.14 57.22
68 0.212 46.16 57.47
70 0.096 46.14 57.22
70 0.113 46.14 57.23
75 23.829 46.15 56.70
75 0.213 46.16 56.71
75 23.846 46.15 56.70
77 0.179 46.15 56.49

Table 3: Refraction Information for plates 33-64


33 9.6 9.717
34 10.8 10.967
35 12.0 12.258
36 13.2 13.404
37 14.4 14.729
38 15.6 15.792
39 16.8 16.971
40 18.0 18.088
41 19.2 19.433
42 20.4 20.567
43 21.6 21.746
44 22.8 22.875
45 0.0 0.088
46 1.5 1.583
47 3.0 3.250
48 4.5 4.550
49 6.0 5.900
50 7.5 7.854
51 9.0 9.325
52 10.5 10.575
53 12.0 12.308
54 13.5 13.783
55 15.0 15.183
56 16.5 16.750
57 18.0 18.000
58 19.5 19.771
59 21.0 21.154
60 22.5 22.550
61 0.0 0.133
62 6.0 6.350
63 12.0 12.083
64 18.0 18.263

77 0.117 46.14 56.48
65 0.167 46.15 57.79
65 0.258 46.17 57.82
65 0.204 46.16 57.80
60 0.329 46.18 58.41
60 0.192 46.15 58.35
65 0.171 46.15 57.79
60 0.088 46.14 58.33
60 0.233 46.16 58.36
65 0.167 46.15 57.78
68 0.146 46.15 57.45
68 0.075 46.14157.44
68 0.088 51.14 68.46
70 0.083 51.14 68.20
70 0.250 51.15 68.23
70 0.050 51.14 68.20
75 23.90 51.14 67.56
75 0.354 51.16 67.61
75 0.325 51.16 67.61
75 0.075 51.14 68.90
65 0.308 51.16 68.90
65 0.283 51.15 68.89
65 0.183 51.15 68.87
60 0.250 51.15 69.55
60 0.000 51.14 69.52
60 0.271 51.15 69.55
60 0.154 51.14 69.53
68 0.050 51.14 68.46
68 0.133 56.14 82.15
77 0.350 56.14 80.76
65 0.083 56.14 82.62
60 0.262 56.14 83.42


Since we will use the method of least squares to determine our plate parame-

ters for both the single plate overlapping plate reduction techniques, we will first

give an outline of this data analysis method.

Theory of Least Squares

Consider a set of n observations represented in vector form by xo, and assume

these observations are unbiased (i.e. without systematic error). The vector of

true values is represented by x and v = x xo is the vector of errors in the

observations. From the assumptions that the observations are unbiased we can

assume that the errors will have a Gaussian distribution (multivariate normal

distribution) and we will regard their covariance matrix, 0r, as being known. Thus

we can express the probability density distribution function of the errors as the

multivariate gaussian

k(v) = Ce-vT V. (37)

We further assume that the observations satisfy a set of equations of condition

F(xo + v, a) = 0 (38)


where a is a vector of q parameters. Some of the components of the vector

equations F=O may not explicitly contain any of the components of x; such

equations would be condition equations involving parameters only. In this analysis

we will distinguish these from the general condition equations by calling them

parameter constraints.

If the q parameters are not mutually independent, the relations existing be-

tween them must be included among the parameter constraint equations. The

observations must be functionally independent, that is, not one of the n observa-

tions can be derived from any or all of the remaining (n-1) observations. There

must be at least n observations; in other words, n>q. Otherwise there will be a

deficiency in the model. If q = n then such a circumstance will lead to a unique

solution. When n > q, redundancy is said to exist and adjustment is needed in

order to obtain a the best set of estimates for the model variables. Adjustment is

meaningful only in those cases in which the data available exceed the minimum

necessary for a unique determination. Since the data points are usually obtained

from observations, which contain errors, redundant data are usually inconsistent

in the sense that each sufficient subset yields different results from another subset.

No unique result (no one vector a) will satisfy all the equations, but will give rise

to (at least some non-zero) residuals, v.

Making the basic assumption, called the principle of maximum likelihood, we

assume that the set of measurements which we obtain is actually the most probable


set of measurements. Thus the best estimates of the coefficients a are the ones that

maximize the probability of obtaining the particular set of measurements which

we actually obtained. Clearly, the way to maximize 0 is to minimize the value of

the exponent. The principle of maximum likelihood thus leads to the conclusion

that we should minimize vTa-v.

The principle of least squares consists in finding estimates of v and a

which minimize the quadratic form vT0r-lv while simultaneously satisfying

F(xo + v. a) = 0. Thus if the random variables to which the observations re-

fer are normally distributed, the least squares method will give identical results

to those from the maximum likelihood method.

Traditional (Linear Least Squares)

Traditional least squares is when we assume that the observation errors are

of the same precision and uncorrelated and that exactly one observation occurs in

each equation of condition. Since the observations are uncorrelated the covariance

matrix a becomes diagonal. Furthermore, if the observations are of the same

precision the covariance matrix will be of the form o = ooI, where I is the

identity matrix. The minimum principle of least squares becomes :

1n (39)
vT-1V = V 2
v 1 v =- v -- minimum.

Thus in this case the minimum of v2 = vTv and vTa-lv coincide. This
last case is the oldest and the most classical, and possibly the one that gave rise

to the name "least squares" since in this case we seek the "least" of the sum of

the squares of the residuals.

In traditional least squares the condition equations F(x,a) = 0 usually have

the form

Aa + Xo = -v. (40)

where A = -. In this case the "adjustment unknowns" or adjustment

parameters", a, occur linearly in the condition equation, and each equation

contains exactly one observation. The "normal equations" for finding the estimates

of a are easily found for this case.Multiplying the above equation by its transpose


(aTAT + x)(Aa + Xo) = (-vT)(-v)
aTATAa + 2aTATxo + x0xo = vT
for matrix A and vectors a and x. Also, note that vT v is not a vector or a matrix,

but a number and we may therefore transpose any of the terms without knowing

its value. To find the minimum of vT v, we differentiate with respect to each

component of a and set the derivative equal to zero
(vTv) a(aTATAa + 2aTATxo + xoxo)
Oa 9a (42)
=2(aTATA+x A) =0


aTATA = -xTA. (43)

After transposing and rearranging, we get the traditional least squares solution

in its simplest form:

a = -(ATA)-fATxo (44)

the solution of the normal equations.

Nonlinear Least Squares

The conditional and the constraint equations involved in an adjustment prob-

lem can, in general, be nonlinear. However, least squares adjustments are gener-

ally performed with linear functions, since it is rather difficult and often imprac-

tical, at least at present, to seek a least squares solution of nonlinear equations.

Consequently, whenever the equations in the model are originally nonlinear, some

means of linearization must be performed on the equations. Series expansions, a

Taylor's series in particular, are often used for this purpose, where only the zero

and first-order terms are used and all other higher-order terms are neglected.

When applying a series expansion, an initial set of approximate values for the

unknowns in the equations must be chosen. The choice of those initial approx-

imations is an important aspect of solving the problem at hand. Unfortunately

there is no concrete and unique way of choosing approximations that can be ap-

plied to all adjustment problems. In all cases an attempt should be made to obtain


the closest approximations that can be obtained by using relatively simple and

uninvolved techniques.

We will now use the more generalized least squares treatment to drop some

of the restrictions found in the traditional least squares method. In the following


1. Observations may be correlated.

2. More than one observation may appear in an equation of condition.

3. The equations of condition are assumed nonlinear in both parameters and


4. Nonlinear constraints among the parameters are allowed.

As before the vector Xo is the vector of observations, x is the vector of true

values x=xo + v where v is the vector of corrections. In order to find estimates

of the vector of parameters a and x in the generalized least squares we will use the

method of Laplace multipliers. At the solution the numerical value of vTcr-v =

S. These conditions are the same as S* = vTo-lv 2FT(a, xo + v)A, where

A is a Lagrange multiplier. This means that the values of S and S* reach their

minimum at the same values of a and v.

As before, a has q components, v has n components and F has p components.

Thus there are only p+n-q free components. Differentiating S* with respect to


the components of a and v gives

-= -lv 2XTA
OV (45)
-- = -2ATA

ofi af_ X
_v av2 *
Xpxn (46)
S x=xo,a=ao (46)

Of af af

OF 3aal 5a,
Apxq = (47)
/a 2 x= xo=xa=oaa=ao
-da1 ea2 iaq x=xo,a=ao

The following equations must, therefore, be rigorously satisfied at the solution
v = XTA n equations

ATA = 0 q equations (48)

F = 0 p equations.
This means that A and X must be evaluated at the solution for a and for x. When

one considers that A has p unknowns, then one can see that there are exactly the

same number of unknowns as there are equations.

The equations are in general nonlinear and therefore must be solved by

successive approximations. A general rigorous and non-iterative algorithm for the

solution exists only for the case that the elements of x and a occur linearly in the

function F. When F is nonlinear in the elements of either x or a, or both, equations

which are practically equivalent to the F and which are linear in the pertinent

variables can be derived, e.g., in the following way: assume that an approximate

solution of the above equations has been obtained somehow, linearize these

equations about the approximate solution and by iterations determine corrections

to it. Start by taking an approximation ao for a and using the observations xo

as approximations for x. If we write a = ao + a, with a being small corrections

to ao, the equation F=0 can be developed as a Taylor series

,F(xo, ao) 0F(xo, ao)
F(x,a) = F( + (x Xo) + (a ao) + 0(2)= 0
xr Ba
F(x, a) = F(xo, ao) + Xov + Aoa + 0(2)= 0
where Xo and Ao are evaluated at x0 and ao; and also, let F(xo, ao) = F0.

From equation (48) we have v = aXTA; so, by substituting into the above

equations we get

Fo + XooXoTA + Aoa = 0. (50)

Combining this with ATA = 0 yields the Generalized Normal equations

Xo 7XT Ao[ A] [Fo] [01
AT 0 0 0 (51)

Note that A is needed for computing the correction v. If XaXT is nonsingular,

we can eliminate A from the equation by multiplying the equation (50) by

AT(XoXT) -1 yielding
A (Xo X0) Fo + AOA + Ao(XoTX)-1Aoa = 0. (52)

Since, ATA = 0, rearranging gives us
AT(XoaXor)-IAa = -AT(XoCaXT)-Fo
a = -[AT(XoaXT)-I A]-'AT(XoaX;)-^Fo.

Pre-multiplying equation (50) by (Xoo'XT)- and solving for A gives

(XoaXo)-1Fo + A + (XoXoX)-Aa = 0
A = -(Xoc'Xo)-1Aa (XoaXT)-'Fo.
A = -W(Aa + Fo) (55)

where W = (XoaXT)-1. W is frequently called the weight matrix.

Since we know ATA = 0, we can pre-multiply equation (55) by AT, and get

AT = ATWAa ATWFo = 0. (56)

Solving for a gives

a = [ATWA] ATWFo (57)

It can be shown (Brown 1955; Jefferys 1980, 1981) that (ATWA)-1 is the

covariance matrix of a, so that the square roots of its diagonal terms are the
standard deviations of the corresponding terms of a. Note that XoXT is not
always nonsingular; for instance, when F=O contains equations in which none of


the components of x occur explicitly i.e., parameter constraints. This, however,

produces in general no essential singularities.

It is now possible to solve for v. From v = oXTA we get

v = X T[WA[ArWA]-ATW W}Fo. (58)

anew = ao + a
Xnew = Xo + V
define the improved solution. If the improved solution is insufficiently accurate,

the process should be repeated until satisfactory convergence is attained.

The natural starting point for this scheme is to set xo= x as the initial

approximation, so that v = 0, and to use a vector a of initial approximations

for a for ao. Obtaining a suitable a is sometimes a difficult question, it must be

close enough to the solution that the process given above converges.

Comparison with Traditional Least Squares

It is informative to compare this general method with the results from the

traditional least squares method. In the traditional case we are dealing with the a)

noncorrelated observations of equal precision (i.e. ar = aoloI = OTolm) of which

b) exactly one observation occurs in each equation of condition, and that c) the

equation of condition is solved with respect to it, we have X = --I and therefore

(XoXT)-' = ~Im, so that in this very specialized case a = -(ATA) ATFo


Also, the observation errors reduce using this simplification and

X = -Im

a = (o7Im (60)

W = Im

v = (XT[WA[ATWA]-lATW W}Fo

v = aolm(-Im)[-ImA(AT ImA)-AT Im -Im]Fo (61)
70o 00 070 (0"
= [Im A(ATA) AT Fo.

This formula is well known from classical least-squares theory and evidently

independent of a0.


We will now use the theory of least squares for the determination of the

plate parameters. Recall that from chapter 2 the determination of the equatorial

coordinates is a two-step process. First the reference stars' equatorial coordinates

are converted to standard coordinates. Second, after the functional form of a

plate model is established between the measured coordinates and the standard

coordinates a least squares reduction is performed on the measured and standard

coordinates of the reference stars to determine the plate parameters. Usually a

great deal of analysis is involved in determining the functional form of the plate

model. First a simple 4 or 6-constant model is used and approximate values

are obtained for some of the plate parameters. A least squares reduction using

only the reference stars on the plate (single plate reduction) is done to find the

best values for the parameters. After the reduction, the residuals from the actual

measurements (x, y) and those calculated using these plate parameters and the

cataloged equatorial coordinates are studied. From the residual plots it is decided

which new terms, if any, need to be added to the plate model. Once the "best"

plate model has been established, the plate parameters corresponding to this model

are used to determine the equatorial coordinates of all the stars imaged on the

plate (not just the reference stars). After position estimates for all the stars have



been determined an overlap solution can be performed. This is because in an

overlap solution all the images of stars that occur on at least two plates or are

reference stars are considered in the normal equations.

Single Plate Reduction: Using Least Squares to Determine the Plate Parameters.

We will first reduce each plate separately. In the next section we will develop

a multiple plate reduction technique. We will use the formulation developed by

Eichhom (1985) to set up the equation for a single reduction and also an overlap


In a least squares adjustment we regard the measured relative coordinates

of the images and the position estimates from the reference star catalogue as

observables. These two types of observables give rise to two types of condition

equations. Consider data from only one plate on which there are n stars, m of

which are reference stars. Without restricting generality, we may assume that the

measured image coordinates of the vyth star, x/, yp are related to their standard

coordinates ip, t77 by the equation

=( s(} + Z=a. (62)

,, is the model matrix, which depends on 7, rt, and possibly other data pertaining

to the star in question, such as magnitude or color index, a is the vector of the

plate parameters, which constitutes a subset of the adjustment unknowns and is

a vector of dimension p. In what follows, we shall keep the model matrix E

general and hold to the assumption that Ea is at least one order of magnitude

smaller than s( and sj.

Using this relation we establish the first set of condition equations

(yv (,i ,a

F =\ y2(a( 662))
(2 ) X) o. (63)

(Xm ( m(Om, m m)\
\ mm (m(am, 6m) /

In our approach, the observables are xu and yp and we regard the adjustment

parameters to be not only the plate constants a, but also the spherical coordinates

(aCp,6p) of the stars. ( and i7 are used only as auxiliary quantities. We consider

the tangential point, ao and bo, as being known. As stated in Chapter 2 if the

assumed tangential point is not the actual one, then this is can be accounted for

by an appropriate form of E and additional components of a.

The second type of observations (estimates of the reference star's spherical

coordinates from an existing catalogue) states that the actual spherical coordinates

(c 6/) are equal to those estimated in the catalogue plus corrections. To take

care of convergence of the meridians, we introduce the factor cosb in the right

ascension equations yielding
/(Oc ao) cos \,
61c 61
(a2C 0 ) cos 6,
G(2mx) = 6= 0 (64)

(Omc Om) COS 6m
\ m m /
where aic...bmc are the catalogue positions and are the adjustment un-

knowns which occur both in F=0 and G =0. We will now combine the 4m

condition equations F and G into one set H.

HT = (FT GT). (65)

Each equation, regardless of whether it belongs to F=0 or G-0 contains exactly

one observable, therefore

( X = I4m. (66)

We assume that the measurements of all the coordinates are free from bias

and not correlated. The variance (in units of squares of the units in which xy

and yp were measured) of x; is vp and yp is Op. The covariance matrix of the

rectangular coordinate measurements is

Orx = diag(/iv, &,2, v2, 02,..., Vm,
and in most cases, it will suffice to write

Ox = mI'2m (68)

where arn is the common variance of all coordinate measurements and 12m is, of
course, the identity matrix of order 2m.

Also let's assume that all the reference star coordinate's are uncorrelated,
so that the covariance matrix a,, of the catalogues reference star coordinate
estimates acos6 and 6 will be

Fyav = diag(pl, 7,..., pn, OIm). (69)

The factors cos2 6 are introduced in order to make vp/v the dispersion of
cutcos(y6), that is the dispersion on the great circle. Not only will this simplify
some of the formulas we need in our calculation, but it compensates for the
convergence of the meridians.

Thus, the observations are

XO = (xx,yX2 z,Y2,...,Xm, Ym; OYlcCOS 61, 1l,...QamcCOS bm, m) (70)

and their covariance matrix is

a = diag(vl, 0l, V2, 12, ...Vm, ;m ; pI, a 2, 2, -...P,, &m) (71)

o 0
xv = (V (72)
0 )
(0 :

The vector a of adjustment parameters consists of the corrections to the

reference star positions and the initial plate parameters. As approximations for

the position we choose their cataloged values which renders Go = 0. As initial

approximations to the vector of the plate parameters we choose 0. Thus we have
Fo = F(a,,,,0) = d

Ho = d (73)

The vector of the adjustment parameters a = with
3T = (cos 6dda1, db6, ..., cos 6,dam, d6m). (74)

From our least squares analysis in Chapter 3, the corrections to the parameters

a are given by

a = -[ATWA]-'ATWHo (75)

where W = (XcrXT) Since all the equations have only one observation,

X = I and our weight matrix becomes W = r-1Im so we can write

a = -[ATor -A]-'ATcrHo. (76)

In order to solve the normal equations, we have to evaluate explicitly the

terms of the matrix A and since a = ( we can rewrite the above equation as

A A() = -A oT-1 (). (77)

A is the Jacobian matrix of the condition equations with respect to the adjustment

parameters, so we can generally write,
OH O(F ( ) G) (78)
A- 8a)- 0(3.a) G) (dG)) )
If Icl << 1 and if the initial parameter vector is a = 0 then, the values we

obtain for a from equation (77) will be the actual parameter values rather than

the corrections.

Let's take a closer look at what these partial derivatives equal:

3d, (04 cos6b,6 b) (79)

where B = ( ) 0 F I When v does not equal = 0.

Therefore (a) is block diagonal, consisting of m blocks of dimension 2 x 2 on

the main diagonal; these blocks are -sBy. The factor matrix is required because

we regard cos6 do, and not do as the correction, thus

d( = ( o) ) cos da + -. (80)
\ \ / cos 6 \d6
The other elements of A are

a= = -S (81)

OG 9G &9G G

I 9G 2G 1G

a = 0. (83)


A=-(B ) (84)

We will substitute this into equation (77) and examine the remaining terms.

A -T o ;7
sBlT 1 (85)

( s nT o ,x I o ra 1

T -1 0
SSBnTox'ln BToxl (86)
SaBT -1BS 1 01- (
-, .sZTr B ST--1 }
It is informative to look at the dimensions of A and the elements that make up
A. For m reference stars and p plate parameters we have

* B: block diagonal (2m x 2m), whose elementary matrix is a (2 x 2).
* 1 : block diagonal (2m x 2m), whose elementary matrix is a (2 x 2 ).

* -1 : block diagonal (2m x 2m), whose elementary matrix is a (2 x 2).

* : (2m x p)

Therefore the dimensions of the 2 x 2 blocks of A are:

* (1,1) position : (2m x 2m)
* (1,2) position : (2m x p)

* (2,1) position : (m x 2p)

* (2.2) position : (m x m)

After substituting in equation (86) into equation (77) we get

(.2BTcx IB + a sB r -' J\ sBTx1 1 d
s )\1B ) = S1O ) )-1 O 0 (87)

and expanding this gives two equation vectors

(s2BTo'r B + 3a') + sBTo -a = sB Tad (88)

-T xlB + T 1a= OTi ld. (89)

Solving the first for 3 gives

3 = (s2BTo-B + a 1) sB T l(d Za). (90)

Substituting in 3 to the second equation and letting

Y =s2-l 1B(s2BTr lB + -1)-BT x,- (91)

we (after some algebra) get the following

.T(y a~ )d = T(Y-_rl).a. (92)

Let J = o Y or

J= ('-s 2 1B (s2B Trx B+ or-- BTr-1 (93)


using the inversion lemma (Henderson and Searle, 1981) we get

J= (x + sBoB') ) (94)

Substituting this equation for J into equation (92) yielding

ETJEa = WTJd. (95)

We can solve this equation for a:

a = (ETJ) 'TJd. (96)

Note that J is also block diagonal, consisting of 2 x 2 blocks on the main

diagonal. We can therefore write more explicitly

a 6= ( VT) J .. (97)
V=l V=l
Next, we have to find explicit expression for the terms of J. These depend on ax,

oa and the terms of B. The terms of B are essentially the terms of ( ) Since

we have good approximations of the values of (, j which are approximately (x/s)

and (y/s) respectively, we will determine J from these quantities. Following the

procedure suggested by Eichhorn (1985) we will define
R2 = 1 + 2 + 72

T2 = 2 + (cos 60 r sin 60)2

S = R2 sin 60/T
U = ,R(sin bo + cos bo)/T

V = R2(cos 60 7 sin o)/T

W = (2Rcos 6o/T + V/R

and note that

R SU + VW. (99)

B can now be written as

B = (100)

From our earlier discussion we wrote J = (ox + s2BaaBT)- However with the

formulas above the elements of B can be easily calculated without the explicit

knowledge of the star's spherical coordinates. If we further introduce the variables,
QS = -pUW + OSV

Y = pU + ci V (101)

Z = pW2 + oS2

and introduce

(P 0 (S2 )0
(= )aS= (S 2& (102)

the weight matrix J can be written in the form

J= (v+Z -Q (103)
-Q 0 + Y)-

Note that for any nonsingular matrix D the following is true

l{d2 -d12
D- 2= -d dl (104)
A -dai d ).


Therefore the determinant of J is A = vo + Zo + Yv + R'ap and we get the

weight matrix belonging to the vth star as

J,= I 1 Ov +=Y, Q, (105)
V",vO + Yvv + ZVov + R6ap v Qv V + Z

We now need to explore more fully the computation for the corrections to

the star positions. We now assume that we have calculated the vector of plate

parameters, a, from equation (97), and henceforth we consider that the vector

d Ea is known. As mentioned before, we have m different independent sets

of equations for calculating the corrections3T = (cos 6Sda, d6b) for each star.

From equation (90) we have the formula for the corrections to the star parameters,

expanding this equations out (and substituting &a'1 = s2a1) we have

1 T T--
3= B B + a ) sB (d a)
1 [( 1 0 U 10 W _-S -1
S 1 S V 1 V
(W U U)(-(1 0td
V 1 0)(d Ea)


Remember that 3 is different for each star and that the symbols W, U, V, S

above are simple numbers and not matrices. To calculate the inverse of the 2 x

2 matrix in this equation, we need its determinant. After some algebra we get

the determinant of 3 to be

S= p(v + Yv + Z + R6p). (107)


Except for the factor. I, this is the reciprocal of the determinant of J. After

multiplying the equation for 3 (equation 106) by its determinant we get

.s( o 4 Zo + Y "' + R6pa)
3( [-S: Ioor 0 V

Note that 3 is in radians. Its is (if one investigates more closely ) a sophisticated

weighted average of the catalogue positions and that which follows from the plates.

Final Positions from a Single Plate Reduction,

Once we obtain, by an adjustment procedure, estimates for the plate constants,

we use them with an inverse of the plate model to determine estimates for the

standard coordinates for all the measured images (reference and non-reference).

Then using the inverse gnomonic projection, the equatorial coordinates are de-

termined. One of the drawback of the single plate solution is that if the star

occurred on more than one plate, the estimates for the same star's coordinates

will, of course, be different depending on which plate they originate from. Thus,

the single plate solution does not use the simple fact that a star can at one time oc-

cupy only one position. Since all the plates are reduced individually, a star that is

found on n plates will have n different values for its coordinates. Clearly the star

only has one position at one time, but this cannot be obtained without mathemat-

ically enforcing this constraint on a multiple plate reduction process. Instead, the


separately computed values are averaged to give the "best" (in the mathematical

statistics sense) estimate for the stellar position. In addition, the random errors are

reduced by a factor of 1 but this does not reduce the systematic errors. Another

drawback of the single plate solution is that when using only the reference stars

in the reduction we would more than likely be extrapolating the magnitude terms

to the field stars which are typically fainter than the reference stars.

It was considerations of this kind which led to the method of overlapping

plates, in its full form first published by Eichhorn (1960). The method of

overlapping plates avoids the multiplicity of several "best" star positions and

increases considerably the accuracy of at least some of the plate constants, thus

reducing the systematic errors (i.e., increasing the accuracy) of the final positions.

However, its implementation requires an arithmetic effort exceeding that which

is necessary for traditional plate reductions by a factor of at least 2 orders of

magnitude. This task has become possible only within the last few years, because

cheap, powerful computers have become easily available, removing the label of

"major job" from a complete overlap solution.

Overlap Plate Reduction

The basic principle of the overlapping plate method is that all the star

coordinates, not just the reference stars, are considered together with the plate

parameters, as adjustment parameters (unknowns). Since the overlap solution


regards the coordinates of each star as adjustment parameters as long as images of

the star were measured on more than one plate, the number of parameters increases

in proportion to the ratio of the field stars to the reference stars. Obviously -

since the number of unknowns has increased dramatically the analytical and

computational problem is exponentially more complex and time consuming than

the single plate reduction. In addition, the adjustments pertaining to individual

plates also lose their mutual independence because any two plates which have a

star in common can now no longer be reduced individually.

Another unavoidable defect of the single plate reduction is a large parameter

variance (Eichhorn and Williams, 1963). Even if the geometry of the imaging sys-

tem was known precisely and the errors of the measuring machine were modeled

exactly, there will still exist unavoidable random errors in the x-y measurements

and reference star positions. This leads to an unfavorable error propagation of

the resulting star positions, in particular when the number of parameters is large

and the number of reference stars per plate is small. The accidental errors of

these plate parameter estimates thus reappear as systematic errors in the positions

that were calculated with them. Such systematic differences even showing up

as functions of model terms are unavoidable (and their magnitudes can even be

estimated) for those stars (field stars) which were not involved in the derivation of

the single-plate parameter estimates. An overlap solution enforces the constraint

that a star can only have one position at one time, producing more accurate plate


parameters, thus inevitably leading to smaller systematic errors in the star posi-

tions. Also, since the positions themselves are regarded as adjustment unknowns,

there can not be any systematic residuals between plates (except those introduced

by model deficiencies). In addition, when there is a strong overlap between plates,

fewer reference stars are needed per square degree to obtain the same accuracy

since, in effect, the total area covered by the overlapping plates is treated as one

large plate.

We assume that there are altogether m stars involved in the adjustment,

including both reference and field stars found on more than one plate. Each

star number: yp = 1, 2,..., m. The numbers t are assigned to the stars in some

organized fashion. There are n plates and each plate is assigned a number v =

1, 2,..., n. On the vth plate there are my stars. For example, (xyv, ypv) are the

measured rectangular coordinates of the images of the lth star on the vth plate.

mr of the m stars are reference stars and the reference stars are assigned a number

Pri, PTr2, ..., Prm, -

The stars used in a overlap reduction are either field stars which occur on at

least two plates or reference stars. The estimates for the field stars' equatorial

coordinates are found from the single plate solution. When the star appears on

more than one plate the average of the single plate solution is used. Thus the

averaged ay, 6b of the field stars are used to find the standard coordinates /vy,

rJuv on all plates on which the star appears. However, for the reference stars


the catalogued values ac/, b6c are used as initial approximations to these stars'

spherical coordinates.

For the establishment of the condition equations, it is important to fix the

order of the equations and the order of the adjustment parameters. The condition

equations are arranged in the following order. Plate by plate we establish the

F-type equations for each star on that plate which either occurs on at least on

other plate or as a reference star in order of increasing star number. These are

followed by the G-type equations which are produced by the estimates of the

spherical coordinates of the reference stars, also in the order of increasing star

numbers. The order of the observations is the same as the order of the equations

which they generate. We chose the order of the parameters as follows: star by

star, in numerical order, corrections to the coordinates, i.e. cos 6Sdca, and d6,

with p = 1, 2,... m are followed by the frame parameters al, ..., an.

The equations of condition are identical to those in the single plate solution.

As before the total vector of conditions equations H=0 consists of "plate equa-

tions", F=0 and "catalogue equations", G=0. First let's discuss the F=0 equations.

Plate by plate we establish the F-type equations for each star on that plate which

either occurs on at least one other plate or as a reference star. The observations

are xpand y,,, the measured coordinates of the image of the /th star on the Vth

plate. Exactly one observation occurs in each equation of condition, as before.


The frame equations for the v plate are

F, = (v v -a,v. (109)

There are my such equations for each plate, for a total of 2my equations. (uP

and rytv are the standard coordinates of the yth star with respect to the assumed

tangential point of the vth plate. S,- is the model matrix for the /th star on the

v th plate and av is the vector of plate parameters on the vth plate. Expanding

the above equation, we get

x/A S-4 \ y a
YI V s 7 av
(XIVL/ S^i.f "
F. = ,vY s% a (110)

ACvm -- SAVmV V-- vav
-, v s ) IsnI ^

At first glance the above notation looks rather confusing. However, x,,, Y is the

first star on the v th plate. Since all the stars are given a particular number

designation numbered 1, 2...., m. There has to be a way to keep up with the

running star number p and the star number on each plate, which we will refer to

as vk. The number of stars on the vth plate run from p to pu,, These subscripts

are useful in designating the total running star number and the star number on

a particular plate. See the example at end of this chapter on how to assign the

running star number.

The entire vector F is

F= = 0. (111)

Each vector F; has the dimension 2mv x 1 and the dimension of the total vector

is E (2m, x 1).

Now on to the reference star condition equations. There are a total of mr

reference stars, numbered Pri, 1r2,, --,rm,. For each reference star we have an

estimated value of its position (acy, 6cp ) from the catalog. The G-0 equations.

look like this:

(ac',. C, ,) cos S 6r + Erl
ScLrl 6rL, + Elrl
(ac,-r2 ac2,2) cos 6 r2 + Er2
G = bcr2 6r2 + Lr2 .(112)

(aC/rmr arm,r cos 6/im, + ,rm,
6Crmr 6 rMr, + E,,rmr

Since only a small fraction of all stars are reference stars, it follows that

Pfk and pLrk+1 of two neighboring reference stars normally do not have the

same number. That means that the subscript r is just to keep track of reference

stars, but just because a star has a subscript r it does not mean that it is given a dif-

ferent number from that of a field star. G does not consist of a separate equations

for each plate, as they do for the equations F, it has a dimension of 2mr x 1.


As before, the total vector H consists of the plate equations F and the catalog

equations G.

H= =0. (113)
Likewise the vector of residuals is

H = (dT,0) (114)

The covariance matrix is similar to that in a single plate setup, except that ox is

broken down into blocks belonging to the various frames. Again we are assuming

no correlation between observations, so the covariance matrix is diagonal. It

consists of V v, lUpv, the variances of Xjv and ypv, and the pp ,ap which are

cos261P times the variance of ap and the variance of 6y, respectively, for those

Cp which belong to a reference star

a = diag(o, 7x2, ,... Ox.; 7a) = (x 0 (115)

The dimensions of each a,, : 2m, x 2m,, so the dimension of the total matrix
ax : E (2m, x 2m,), while the dimension of ea : 2m, x 2m,.
As with the single plate each equation contains only one observation so,

X () = I, where x = (xl, yl, X2, Y2, ..., Xm, Ym; alc Cos 6, 1, ...amc COS 6m, m),

so XaXT = ) = x 0
(0 0a
The mathematical development of an overlap solution is very similar to the

single plate method discussed in the last section. The major difference is the


increase in the observations (all the stars), which enormously increase, the size

and structure of the matrices. We will present the mathematical development

similar to the single plate development, noting the changing the structure of the


As before, the correction to the parameters is given by

a= AT,-1A 'A T-Ho


where a = (3) and 3 is the vector of corrections to the spherical coordinates

(a,, cos 6,6, 6b). The total vector of plate parameters for all the plates is a, which

is made up of the individual vectors for each plate

/a, \



Our primary concern is the establishment of the matrix A f(

= (OH = (F, G) ( ) (a
A = =(,a) (9F (

Similar to the single plate reduction we have for each star


or the overlap case:

.) (118)
Ga ) /


OFpAV, O(a ( cos k.6) -sB k
0, a (a',, Cos 61, k, 1,) vk)


where each BLk is a 2 x 2 matrix. For each plate we get


S= o B 0 i 0) (120)
0 ... 0 B ... ... ... ... 0O

0 0 ... 0 B ... ... ... 020)

where the dimension of each By matrix is (2mv x 2m). Thus each line pair

contains exactly one Bp,, matrix (of dimension 2 x 2) but only in those column

pairs which belong to a star whose image was measured on the vth plate. In other

words, the rows correspond to the stars on plate v, while the columns are for all

the stars, therefore B ,, is not the null matrix only when the pth star is on the

Vth plate. The entire matrix B for all the plates is written as


sB= (121)

and this matrix is of dimension E 2m, x 2m.
Furthermore, we have

S= -=, (122)

where ,, is of dimension 2my x ly, ly being the number of components of the

vector a on the vth plate. In E,, each star with number p that occurs on the vth


frame generates a 2 x l; matrix 2,,. Thus

.= (123)

and the total matrix E is of the form

OF ,

n n
and has dimension E 2mE, x Z l,.
v=l v=-
0G,, 0G_, OG,
G-- = G OG= -K. (125)
d3 cos 6,,VOv, O6v,,

All nonzero elements of the matrix K are 1. However, since not all stars are

reference stars, K is not the identity matrix. K has the dimension of 2mrx 2m. If

the pthstar is also a reference star then the row pair corresponding to this reference

star and the column pair for this pth star will contain the 12 identity matrix.

Since G is independent of the plate parameters, we have (as in the single

plate result)

= 0. (126)

Combining all these results (as in the single plate reduction), the matrix A can

be written as

A =- o (127)

and has dimension 2 i m, + 1m, x 2m + E 1,.

Now that we have looked at the various matrices that make up the matrix A,

let's look at the structure of the normal equations. The structure of the normal

equations looks the same as in the single plate solution:

(AT A) = -ATa-ol (128)

However, these equations are much more complicated than in the single plate

case. In detail

AT- sBTox-1 KTO-a1 B 2
A T A= a
a A=-1 0 \K 0
x 0 \ (129)
fsBr B + KT--K s2Bo-- .'
S .T.x- lB / )
,T -=Tx

which can also be written as

\ s.< erBn 0 n ,Bn /

Q :B a 1 0n
'A= (1 T O1 1BT, i 1B) ... (

with L = KT rK + s2BTc,' B.

Now let's take a closer look at structure of the matrices that make up equation

(130). The example given later on will help to clarify these matrices, so the

reader may want to refer to this example. Because ax is a diagonal matrix,


multiplication by this matrix leaves the structure of the matrix by which it is

multiplied unchanged.

We noted from the discussion above that the By (2mv x 2m) are very sparse

matrices and have a structure similar to that of K. Each row pair contains exactly

one non-zero 2 x 2 matrix, however not every column pair has one of these

blocks, (i.e. some of the column pairs contain all zero elements). Nonzero

elements occur in a column pair of By only if the corresponding star occurs on

the plate corresponding to the row for that plate. Compact blocks with nonzero

elements in BTv will interact with compact blocks in By, wherever the pth star's

image was measured on the vth plate. The result of the interaction will result in

a 2 x 2 block in the Pth row pair and column pair of L. The structure of

BT aB is thus similar to that of BT, except that the vth column pair would be

"replaced" by a matrix of the width Iv. The matrix KTo, aK will be a diagonal

matrix, however some of the elements on the diagonal may be zero. The result of

adding this matrix to s2BTc, 'B is that the diagonal blocks (pp, a ) will add to

the diagonal blocks in the pth double row and column. L is thus a block-diagonal.

Since the dimensions of the matrices BTv are 2m x 2my and those of =,V

are 2my x ly, the product sBf'aV-1, is a sparse matrix of dimensions 2mxly,

whose nonzero 2 x 2 elements sB a ,-,1VS occur only on those of its line pairs

whose numbers correspond to a star which occurs on the vth plate.


7 -xL is block-diagonal, the individual blocks being ,, the matri-

ces of the product sums of each plate's model matrix with the covariance matrix.

Finally, consider the right hand side of the normal equations (equation 128),
Arl-1H. The resulting vector has the dimension (2rn + F l,) x 1 which is a
vector of the order of twice the number of stars plus the total number of plate

A0TdsB T -K T OW -1 0 ) \ /d
A7H= =T
S0 (131)
-sB T
= T )7x 1d

Consider the 2 x I vector: s E B wildly,. Nonzero contributions to these
sums are made only when images of the uth star were measured on the vth plate.

Substituting equation (130) and (131) in the normal equations (128) yields
( L sBTar'1 ( lBT
gS -1B T (a) = ( BT) d. (132)

The first row gives L3 + sBTx l-a= sBTxld and the second row gives

sTxlBi3 + I Txl'Sa = Scrd. As before, the solution to the normal equa-

tions is done in steps. First we eliminate the star parameters, 3, and solve for the

plate parameters a. Solving the first equation for 0 gives :

3 = sL- 1BTr (d a). (133)

Substituting 3 into the second row of the normal equations yields

('1 xI s 'x1BL-1B T )(d Sa) = 0. (134)
(a / x


J' = x s2a1BL-BT-I1 (135)

T (J')(d Ea) = 0

TYJ'd = eTJ'Sa (136)

a = (TJ' ) TJId.

The inversion lemma cannot be used to simplify J' as it was used to simplify

J because the 2m x 2m matrix KTr-1K term in L is a diagonal matrix with

nonzero elements on the diagonal only in those double blocks whose ordinal

numbers correspond to reference stars, thus it is singular. Unlike the matrix J', J

is not block-diagonal. Since we have established above that L is block-diagonal

with blocks of dimension 2 x 2, the structure of J' is the same as that of BBT.

Thus the resulting T J'S is "banded-bordered" (De Vegt and Ebner, 1972) and

thus there are simplified routines to invert it.

Now that we have determined (i.e., estimated) the plate parameters, we can

use them to find the star parameters by equation (133). In terms of the individual


n n
= s ,.,,,,.L1 T-1

SI = s 2 + s B T B B lT o-(dr ,,a,). (137)
V=1 V=1
v=l v=l

The symbol bvy is of Kronecker type, it equals zero if the pth star is not a reference

star, and it equals 1 for reference stars. The 2 x 1 vector B/ = (da, cos 6P, d6)T


is a sub-matrix of the previously encountered 2my x 1 vector 3/ which stands

for the pairs (da cos 6, d6) of all the stars which occur on the vth frame. Thus if

a star occurs on n plates, there will be n different 3's after the reduction, whereas

in the case 3p, all the star parameters for one star are calculated after all the

plate parameters have been calculated. This substantially amounts to taking a

weighted mean for all the contributions of all frames on which images of the star

were measured.

Example of Overlap Reduction.

Although the solution of the normal equations in the overlap case is very

similar tot that of the single plate case, the interaction of the stars imaged on

more than one plate complicates the reduction. It is therefore extremely useful to

illustrate the method with an example.

We will use a very simple example. Consider the case where we have 8 stars

imaged on 2 plates, 6 stars on the first plate and 6 stars on the second. Let three

of the stars be reference stars.


plate 1

5 3
4 1

plate 2

S : field star

reference star

Figure 9: Example of two overlapping plates

One can see that each plate has a star in the corer to which no number has

been assigned. Each of these two stars occurs on only one plate, and none is a

reference star, thus in the overlap reduction they are ignored. After the reduction

has been performed, the plate parameters are used to calculate the equatorial

coordinates of these isolated stars.

Using our notation established earlier we have



m = 3

m = 5

m2 = 5

Indexing for the stars on plate I is PLV,

the first star on plate 1, pi, 1 = 11 = 1.1

the second star on plate 2, p1, = 221 = 2, 1

the third star on plate 1, P 1, = 331 = 3, 1

the forth star on plate 1, i1 = 441 = 4, 1

the fifth star on plate 1, /1il1 = 551 = 5, 1

or t r, = 2

or /r2 = 4

Indexing for the stars on plate 2 is

* Star 1: the first star on plate 2, P2'22 = 112 = 1,2

* Star 3 : the second star on the second plate, P222 = 322 = 3, 2

* Star 4 : the third star on the second plate, p2,2 = 432 = 4.2 P12 = 4

* Star 5 : the forth star on the second plate, 2, 2 = 542 = 5, 2

* Star 6 : the fifth star on the second plate25,2 = 652 = 6, 2 or 11r3 = 6

We have four prime goals:

1. Set up the equations of condition.

2. Use these equations to find J'.

3. Use J', to determine a.

4. Use a, to find 3.

* Star 1

* Star 2

* Star 3

* Star 4

* Star 5

Set up the equations of condition

The equations F have the form

F ( ) = S, Ev,a, 0 (138)

and the equations G become

G ((CArk- 6rk) COS /irk =0, (139)
(bcMk -rk)

where k goes from 1 to mr, the total number of reference stars. The index ac r

is the pth number of the k'h reference star. Therefore in our example:

* Pr1 = 2

* Pr, = 4

* lrT3 = 6

It is important to fix the order of the equations. In the last section we stated

that the equations were ordered plate by plate and within a plate, in order of

increasing star number. However, from a programming point of view we found

it simpler to order the equations star by star. In other words we group all the

stars together that occur on the separate plates. This is equivalent to ordering the

equations plate by plate (the row and columns of the matrices are interchanged).

Therefore the equations are ordered star by star in order of increasing star number.

The total vector H becomes

I I ,- ,I a
Yii Stil
(xii s) -\

Y12 s712 /
S -i _2a -
Y12 srh 2
Xz2 SJ21
Y21 Sqz21
/X31 -31 d
31 S731 / d12
(3'2 S<32 \ d32
Y32 S32 32a
(41 41 d32
S- d41
H4= S42 42a d42 (140)
Y42 S42 d,-
X5 S51 d5
Y51 S7751 ad,
(x2 s52 d-52
S52 s<522
Y.52 S752 52a2
(X62 Sf62 \ ,
=62a2 \ 0 /
\(62 S7762 )
(aC2 a2) cos 6,
(bC2 62)
(ac4 04) cos S4
(b4 64)
(6c6 66) cos 66
(6c6 -6 6)

The (total) covariance matrix is a = ) where the covariance matrix of
0 0a

the measurements is

0r11 0 0
0 (7,, 0
0 0 0',
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
o0 00
o0 00
o0 o00

Each sub-matrix in -x is a 2

0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0
(731 0 0 0 0 0 0
o (732 0 0 0 0 0
0 0 o41 0 0 0 0
o 0 0 (742 0 0 0
0 0 0 0 (751 0 0
0 0 0 0 0 0o6, 0
0 0 0 0 0 0 62,

x 2 diagonal matrix of the form a,, = P(P



n n
The matrix ox then has dimensions E 2mg, x E 2m,. The covariance matrix
v=l v=1
of the reference stars has the form

(a = 0



where the diagonal elements are 2 x 2 block matrices of the form

S(a cos2 6S
-a = 0


so the dimension of 0a is 2mr x 2mr.

By has dimension 2my x 2m, so in this example the total matrix B is a

01 =

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