Rigorous compilation of the Northern International Reference Stars

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Title:
Rigorous compilation of the Northern International Reference Stars
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xi, 133 leaves : ; 28 cm.
Language:
English
Creator:
Cole, Carl Stephen, 1954-
Publication Date:

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

Notes

Thesis:
Thesis (Ph.D.)--University of Florida, 1986.
Bibliography:
Bibliography: leaves 131-132.
Statement of Responsibility:
Carl Stephen Cole.
General Note:
Typescript.
General Note:
Vita.

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Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 000940976
oclc - 16656153
notis - AEQ2510
sobekcm - AA00004855_00001
System ID:
AA00004855:00001

Full Text












RIGOROUS COMPILATION OF THE
NORTHERN INTERNATIONAL REFERENCE STARS






BY


CARL STEPHEN COLE


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY



UNIVERSITY OF FLORIDA


1986















ACKNOWLEDGEMENTS


I would like to thank Drs. T. Corbin, J. Hughes and C.

Smith of the Astrometry Division of the U.S. Naval

Observatory and Dr. W. Warren the Astronomical Data Center

at the NASA Goddard Space Flight Center for providing those

data required to carry out this research. I would also like

to acknowledge the support which I received from the

Division of Sponsored Research at the University of Florida

in the form of a graduate research assistanceship.

Dr. Heinrich Eichhorn suggested the topic of this

research and has guided me through its execution. He has

also introduced me to the international community of

astrometrists and has gained their co-operation on my

behalf. I am very grateful for Dr. Eichhorn's advice and

for the knowledge which he has imparted to me.

Finally I owe many thanks to Leslie Gilbert for her

patience and the emotional support which she has given me

throughout my tenure as a graduate student.
















TABLE OF CONTENTS



PAGE

ACKNOWLEDGMENTS . .. ii

LIST OF TABLES . . ... v

LIST OF FIGURES . . ... ix

ABSTRACT . . .. x

CHAPTER

I. INTRODUCTION . . 1

The Establishment of an Inertial Reference
Frame . . 1
The International Reference Star Program 4

II. CATALOGUE COMPILATION . 7

The Estimation of Systematic Differences .. 7
Critique and Analysis of the Tabular
Procedure . 10
Description of Simultaneous Reduction 16

III. REDUCTION PROCEDURES . .... 20

Precession . . .. 20
Model Parameter Estimation . .22
Star Parameter Estimation . .. .25
Sequence of Iterations . ... .27

IV. RESULTS . . .. 110

Residuals Between the Two Versions of the
NIRS . . 110
Internal Errors . .118
Perth 70 Residuals . .. 122

V. CONCLUSIONS .. . . 129


iii









REFERENCES . .. .. .. .. 131

BIOGRAPHICAL SKETCH ....... ... .. 133

















LIST OF TABLES


TABLE

1. Succesive Corrections Computed with Biased
Estimates . . .

2. Smoothing Coefficients . .

3. Iteration Sequence . .

4. Independent Catalogues Used in the Compilation


of the NIRS


PAGE


. 15

. 25

. 30


. 32


Corrections Applied to the AGK3R .

Corrections Applied to the AGK2A .

Corrections Applied to the W20 .

Corrections Applied to the Albany 10

Corrections Applied to the Bonn 00

Corrections Applied to the Bord 50

Corrections Applied to the Sch .

Corrections Applied to the Bonn 25

Corrections Applied to the W2-50 .

Corrections Applied to the W3-50 .

Corrections Applied to the GCH 1-50

Corrections Applied to the Cape02 00

Corrections Applied to the Nice 10

Corrections Applied to the Nice 25

Corrections Applied to the GCH Z 10

Corrections Applied to the Paris 90


. 39

. 41

. 43

. 45

. 47

. 49

. 50

. 52

. 53

. 55

. 57

. 58

. 59

. 60

. 60

. 61


5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.









21.

22.

23.

24.

25.

26.

27.

28.

29.

30.

31.

32.

33.

34.

35.

36.

37.

38.

39.

40.

41.

42.

43.

44.

45.

46.


47. Corrections Applied to the Cin 00


Corrections Applied to the Paris 00

Corrections Applied to the GCH 2-25

Corrections Applied to the Berl 20 .

Corrections Applied to the GCH 00

Corrections Applied to the Toul3 00

Corrections Applied to the Cape 2-25

Corrections Applied to the Cape 3-25

Corrections Applied to the W 40 .

Corrections Applied to the W 00 .

Corrections Applied to the W ZOD 25

Corrections Applied to the Cape 1-50

Corrections Applied to the Bord 00

Corrections Applied to the ALB99 00

Corrections Applied to the Mun97 00i

Corrections Applied to the Mun97 OOii

Corrections Applied to the Kon 00

Corrections Applied to the Pulk99 00

Corrections Applied to the Madn 10 .

Corrections Applied to the Berg 1-25

Corrections Applied to the ABB-6 00

Corrections Applied to the Buch 50

Corrections Applied to the Bonn09 00

Corrections Applied to the ABB+20 00

Corrections Applied to the ABBO 00 .

Corrections Applied to the Lund44 50

Corrections Applied to the Stras 30


. 62

. 63

. 64

. 64

. 65

. 66

. 67

. 68

. 69

. 70

. 71

. 72

. 72

. 73

74

. 74

. 75

. 76

. 77

. 79

. 79

. 80

. 80

. 81

. 82

. 83

. 85









48. Corrections Applied to the PFKSZ .


. .. 87


49.

50.

51.

52.

53.

54.

55.

56.

57.

58.

59.

60.

61.

62.

63.

64.

65.

66.

67.

68.


Corrections Applied to the Lund42 50

Corrections Applied to the Cin 25 .

Corrections Applied to the Moscow40 50

Corrections Applied to the Tri 25 .

Corrections Applied to the Bruss 25

Corrections Applied to the Leid21 25

Corrections Applied to the Leid24 25

Corrections Applied to the Lund 25 .

Corrections Applied to the Leid27 25

Corrections Applied to the Berl Z 10

Corrections Applied to the Konl9 25

Corrections Applied to the Toul3 00-II

Corrections Applied to the Pulk 10 .

Corrections Applied to the Lick 17 .

Corrections Applied to the Lick 28 .

Corrections Applied to the Turin 10

Corrections Applied to the Bord 00-II

Corrections Applied to the Ottw28 25

Corrections Applied to the Ottw42 50

Corrections Applied to the Konl7 25


69. Mean Right Ascension Residuals, Corbin minus
Cole . .


. 114


70. Mean Declination Residuals, Corbin minus Cole

71. Mean Proper Motion in Right Ascension Residuals,
Corbin minus Cole . .

72. Mean Proper Motion in Declination Residuals,
Corbin minus Cole . .


115


116


117


vii


. 89

. 91

. 93

. 95

. 96

. 98

. 99

. 100

. 100

. 101

. 101

. 102

. 102

. 103

. 103

. 104

. 104

. 105

. 107

. 109








73. Mean RMS errors .. . 119

74. Position Residuals, Perth 70 minus NIRS 123

75. Mean Right Ascension Residuals, Perth 70 minus
NIRS .... ..... ... 127

76. Mean Declination Residuals, Perth 70 minus NIRS .128


viii















LIST OF FIGURES


FIGURE PAGE

1. Distribution of Right Ascension Residuals,
Corbin minus Cole . 111

2. Distribution of Declination Residuals, Corbin
minus Cole .. . 111

3. Distribution of Proper Motion in Right Ascension
Residuals, Corbin minus Cole . 112

4. Distribution of Proper Motion in Declination
Residuals, Corbin minus Cole. . 112

5. Distribution of Right Ascension RMS Errors 120

6. Distribution of Declination RMS Errors .. .120

7. Distribution of Proper Motion in Right Ascension
RMS Errors . . 121

8. Distribution of Proper Motion in Declination RMS
Errors .. . 121

9. Distribution of Right Ascension Residuals, Perth
70 minus NIRS .. . 125

10. Distribution of Declination Residuals, Perth 70
minus NIRS .. . 125















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



RIGOROUS COMPILATION OF THE
NORTHERN INTERNATIONAL REFERENCE STARS


By


Carl Stephen Cole


December 1986


Chairman: Heinrich Eichhorn
Major Department: Astronomy



The tabular method of determining the systematic

differences between two star catalogues is discussed. It is

noted that the tabular method is subjective in nature and

that the estimation of the model parameters does not use all

available model constraints. Furthermore, these estimates

are not least squares estimates nor are they unbiased.

The simultaneous estimation of both target parameters and

model parameters is applied to the compilation of a complete

star catalogue. By simultaneously using all available

constraints on all available data, more precise estimates

for the target parameters are obtained. The input material









of the Northern International Reference Stars (NIRS) is used

and the resulting catalogue is compared to the NIRS

catalogue compiled by Corbin using standard techniques. It

is shown that the new estimates of the star parameters have

smaller formal errors than estimates derived from the same

material but using conventional procedures. Both versions

of the NIRS are used to predict the star positions of the

later observed Perth 70: A Catalogue of Positions of 24900

Stars and these predicted positions are compared to the

actual observed positions. It is found that a simultaneous

reduction results in a slight but significant improvement in

the predicted positions.















CHAPTER I
INTRODUCTION



The Establishment of an Inertial Reference Frame

One of the goals of kinematic astronomy is the empirical

establishment of a frame of reference in which Newton's

first and third postulates of motion are valid. To those

involved in this endeavor, two important facts become

readily apparent. First, in all areas of science which

involve dynamics, such as lunar and planetary theory,

galactic dynamics, astronautics, among others, the

determination of an inertial reference frame to some

required accuracy is essential. Second, the complexities of

the determination of this inertial reference frame are often

completely ignored. The fact that the determination of an

inertial reference frame has been taken for granted is a

tribute to all the astronomers who have, over the centuries,

performed this indispensable service for their fellow

scientists. This work, however, is never finished. As

science progresses so does the precision of measurements

increase which in turn requires an increased accuracy of the

standard.

Due to the rotational and revolutionary motion of the

Earth, it has long been realized that stars can be used to








define, in practice, an inertial reference frame. The

sighting of stars has been used by sailors for centuries to

determine their latitude and longitude on the rotating Earth

and, more recently, star positions are being used to

navigate the Voyager spacecraft past Jupiter, Saturn, Uranus

and Neptune. In kinematic astronomy, the most frequently

used coordinates are not cartesian coordinates but spherical

polar coordinates. This is so because the position of an

object, which is defined by two angular coordinates (e.g.

right ascension and declination), can be accurately

determined to a fraction of a microradian while the radial

distance is known only to one or two significant figures if

at all. The position of a star, defined by these two

angular coordinates, fixes its place on the imaginary

"celestial sphere." Conversely, any two non-diametrically

opposed stars whose position and proper motion estimates

are given in a star catalogue, uniquely define the

coordinate system of that catalogue and provide the basis

for the establishment of an inertial reference frame.

A coordinate system (or simply a "system") in connection

with a star catalogue, is not necessarily inertial; rather,

estimates for the parameters needed to transform the system

to an inertial reference frame are assumed known. If

estimates for the distances and radial velocities of some of

the stars in a catalogue are known, this information can be

combined with the positions and proper motions in order to








estimate Oort's constants of galactic shear and galactic

rotation as well as the solar motion. Thus the kinematics

of our Milky Way galaxy are described and an inertial

reference frame is established.

A fundamental star catalogue contains the positions and

proper motions of at least several hundred stars and

therefore overdetermines, in a sense, the system which is

defined. In light of this, certain concepts related to a

catalogue's system need further clarification. Eichhorn

(1982) has given concise definitions to these concepts.

First, it must be borne in mind that the "star positions"

which are listed in a star catalogue are only estimates. If

the errors in these estimates are purely random, the system

is defined by any randomly selected subset of star positions

to the precision of the individual positions. However, if

there are systematic errors of star positions dependent on

which part of the sky is under consideration, or other

parameters charcterizing the stars such as brightness or

color, the system will be dependent on which subset of star

positions is used to define it.

It is often found that there exist differences in the

systems of catalogues which are functions of the sections of

the sky under consideration. In order to combine

independent star catalogues into a compilation catalogue,

these systematic differences must be modeled and the

parameters of these models must be estimated. In this way









it is possible to correct the systematic trends of each

independent catalogue in order to bring them all onto a

common "system."

This research investigates the techniques used to model

the systematic differences between the systems of star

catalogues, as well as the procedures used to estimate the

parameters of these models.



The International Reference Star Program

The International Reference Star program (IRS) is a

multinational effort whose execution has required more than

a quarter of a century. Its aim is to provide more than

40,000 accurate and precise star positions and proper

motions over the entire sky (Scott 1967, Scott and Schombert

1970, Smith 1979, Corbin 1985). Transit circle catalogues

from around the globe are being compiled into compilation

catalogues with a density of about one star per square

degree. With these catalogues, the fundamental system of

the Fourth Fundamental Catalogue (FK4) (Fricke and Kopff

1963) can be extended to fainter magnitudes such that over

300,000 star positions of all stars to the 9th magnitude can

be tightly related to the fundamental system.

The northern half of this program (NIRS) can be traced

back to the Catalog of Reference Stars for the Dritter

Katalog der Astronomischen Gesellschaft (AGK3R). One of the

aims of the NIRS was to provide proper motions for the AGK3R









stars. Unavoidably (and fortunately), this also led to

improved positions. The NIRS was compiled from observed

positions in 64 independent meridian catalogues whose mean

epochs date back as far as 1889.

The catalogue of Northern International Reference Stars

(NIRS) (Corbin 1974, 1977, 1982) contains positions and

proper motions of 20194 stars in the declination zone -5 to

+90 degrees of apparent visual magnitude 6.5 to 9.5. NIRS

was compiled from independent catalogues (ICs) which only

contain star positions measured at a given epoch. Corbin

constructed this catalogue in a two-step process. First,

the systematic differences between the star positions in

each of the ICs and the reference star positions of the FK4,

which is the target system, were calculated. Systematic

corrections were computed from these differences and applied

to all star positions in the ICs in order to bring them onto

the system of the FK4. Second, from the thus corrected and

weighted IC positions, a complete catalogue of appropriate

stars was constructed.

It must be noted that Corbin used no intercomparisons of

ICs to calculate systematic corrections. That is, when

estimating the parameters of the models of systematic

differences, the only constraints which were used were those

which minimized the systematic differences in star positions

between ICs and and the reference catalogue. The

constraints which require that the systematic differences in







6

star positions between independent catalogues also be

minimized were not used. In light of this fact, it is

apparent that better estimates of the parameters are

available with procedures which use all available

constraints on all available data (cf. Eichhorn and Cole

1985).















CHAPTER II
CATALOGUE COMPILATION



The Estimation of Systematic Differences

The difference between the position estimate of a star in

two catalogues originates from the random errors of the

observations from which the positions in each of the

catalogues were computed and the inconsistencies of the

systems defined by the star positions of the catalogues. In

computing corrections to bring a catalogue onto a system,

one seeks to minimize the differences between the defining

systems without changing the accidental errors in the

individual position estimates. In correcting for systematic

errors, the usual procedure is to model the source of the

error, guided by the geometry and, when indicated, the

physics of the actual measuring situation. In the case of

star catalogues this technique is impossible, in practice,

because there are too many small sources of systematic

errors which occur at different stages of the data reduction

process. Often, their presence is either unsuspected or

reasonably accurate models for them are difficult to

establish. Systematic errors can, for example, be

introduced by an inaccurate refraction correction. Likewise

it is difficult to determine the optical characteristics of








an instrument which was destroyed in the Second World War.

It is thus easier to lump errors from all sources together

and model them by some empirical interpolary function.

We see that the systems defined by star catalogues, since

they are only estimates, can only approximate the

unobtainable ideal target inertial reference frames.

Therefore the actual, true systematic errors of a catalogue

can never be rigorously and unambiguously found or even

defined. What can be defined and estimated are model

dependent systematic differences between the systems defined

by any two catalogues. Once systematic differences1 are

found, they can be applied to the positions in one catalogue

as systematic corrections in order to bring the two

catalogues ideally onto the same system. Regardless of the

functional form of the model for the systematic differences

between two catalogues, when two or more catalogues are

being combined, there are, in principle, two distinct

methods to compute these systematic differences.

The traditional methods utilize the comparisons of the

positions of only those stars common to each IC and the

reference system (the FK4 in this case) for the derivation

of the systematic corrections for that IC. Systematic

corrections are then determined from these individual

comparisons only. Since the star density in a typical IC is




1For the purposes of this research, the term "systematic
error" will be used to denote the systematic difference
between an IC and the FK4.








much higher than that of the reference system, systematic

properties of an IC have often been estimated from as few as

5 percent of the star positions in that IC.

Traditionally, position differences averaged over blocks

of the sky and smoothed with adjacent blocks are applied as

systematic corrections. It appears that several problems

arise with this traditional tabular method. First, the

smoothing coefficients are chosen on a subjective basis.

Second, not all of the available model constraints are used

in estimating the model parameters. This means that the

procedure which estimates the model parameters does not

constrain these model parameters to minimize some measure of

the systematic differences between all ICs but rather the

model parameters are constrained only to minimize a measure

of the systematic differences between each IC and the

fundamental system. Finally when estimating the parameters

in the tabular model, the estimates obtained are not least

squares estimates but estimates used only because, from a

computational standpoint, they are easily accessible. While

this results in an acceptable star catalogue, the process

involves a high level of subjective judgement which is

undesirable and uncommon in most scientific investigations.

More sophisticated models (Bien et al. 1978) define as the

measure for the systematic differences the sum of orthogonal

functions (Brosche 1966, Schwan 1977, 1985) and then use

individual positions in a least squares algorithm to

determine the parameters of these functions.







10

On the other hand, a simultaneous reduction sets up all

condition equations in closed form and solves for target

parameters (star positions and proper motions) and model

parameters (systematic errors) at the same time. The

advantage here is that all available constraints on all

available data are used to estimate both sets of parameters

and that all estimates are least squares estimates.



Critique and Analysis of the Tabular Procedure

In spite of the fact that the tabular method has been

producing very good results for over a century, it is poorly

defined and the underlying assumptions have never been

explictly stated. That is, the tabular method, with

smoothing included, has never been defined in terms of a

model, but only as a "cookbook recipe" as it were. Without

the benefit of a model and specifically, without the benefit

of the assumptions concerning the joint probability

distribution of the random quantities, it is impossible to

assign a meaning to terms such as the bias and variance of

the estimated parameters. One cannot speak of the bias of

an estimate unless one knows the expected value of that

estimate. One cannot know the expected value of an estimate

unless one knows the probability distribution of that

estimate. One cannot know the probability distribution of

an estimate unless there exists a model which specifies the

dependence of the estimate on the random quantities.









In the simplest case of a tabular method without

smoothing, the systematic error of an IC is considered a

fixed constant in each subjectively delineated domain. The

model for the observed error of the position of star p in

domain v, A is
11v

A =p +E U =1,2...n v=l,2...m (1)



where p is the fixed but unknown systematic error in domain

number v and E is an independent and normally distributed

random error with mean zero and constant variance o2 for all

y. Under these assumptions one invokes the principle of

least squares and minimizes the sum of all E2 In this way
Jiv
one obtains estimates p for p as the mean of all A
v V A .v
n A.

i=l n (2)


n of course, being the number of differences formed in the

2 2
vth domain. An unbiased estimate, s for a is
n


s = li-np (3)
v n -1

So far this is statistically sound, because the estimates of

the systematic errors are uncorrelated under the given

assumptions. The estimate of their (diagonal) covariance

matrix is

S2 = diag2/, S S/n) (4)
S2 = diag(S1/nl, S 2/ Sn2 n (4)








where m is the number of domains involved in the process.

Unfortunately, the model in (1) proves inadequate.

Experience has shown that the domains cannot at the same

time be chosen small enough to model with sufficient

accuracy the structure of the systematic differences and yet

large enough not to mask the random errors of the

observations. The accepted solution to this problem calls

for choosing smaller domains and then "smoothing" each

estimate, p with its immediate neighbors. It is during

this process of smoothing that the reference to a model and

its underlying assumptions is lost. However, this procedure

will still produce some kind of a numerical result.

If smoothing is involved, one can only infer a model

working backwards from the "recipe." It is implied that the

model for the observed error of the position of star n in

domain V is

m m
S j aj.p.+ jala.=l (5)
j=1 i] ]i 3y =1 3 a "


where the p.s are again fixed constants, E is again an

independent and normally distributed random error and the

a.s are subjectively chosen smoothing constants with

smoothing occurring over domains in the neighborhood of .

The method of least squares would yield the estimates, p ,

by minimizing the quantity

m n. m n. m
J2 J2
2 E e = E (A..- a (6)
j=1 i=1 3 j=l i=l 1 k= 1







13

In practice this is, however, not done; rather the pus are

obtained from equation (2)!

The assumptions of the model function and those of the

procedure for estimation of the model parameters thus

contradict each other. The model function (5) is predicated

on the assumption that the p s are correlated, thus giving
V
justification for the smoothing process, while the

estimation of the model parameters from (2) is based on the

assumption that the p s are independent.

Assuming the model (5), the estimates of the model

parameters from (2) are therefore biased. The bias of an

estimate p is the expected value of the estimate minus the

true value of the parameter or


m
= aj.p pV (7)
j=l

which is, in general, not equal to zero. One result of

these estimates being biased is that once systematic

differences are calculated, smoothed and applied as

systematic corrections, if systematic differences were again

calculated and smoothed using the same coefficients, the

resulting corrections would not be zero. That is, after a

catalogue is "corrected" using these biased estimates, if

systematic corrections were again calculated in the same

manner, the second set of corrections would be different

from zero.








As a demonstration of this phenomenon, the Catalogue

Meridien de 2024 Etoiles Reperes de la Zone +110 a +180

(Bord 50) was corrected to the system of the FK4 using the

tabular method with smoothing as described in chapter 3.

The upper part of table 1 lists the corrections to the

declinations of the Bord 50 in hundredths of arcseconds at

gridpoints separated by one hour in right ascension and five

degrees in declination. The individual positions of the

Bord 50 were corrected with this table using two-way linear

interpolation. The lower part of table 1 lists the

corrections to the declinations of the Bord 50 in hundredths

of arcseconds, computed with the same smoothed tabular

method, using the "corrected" Bord 50 and again only the FK4

as a reference system. Although the second set of

corrections is smaller than the first, the second set would

be identically zero if the first set had not been biased.

The concept of a covariance matrix of the model

parameters is very problematical in this contradictory

environment. Without an estimate of this covariance matrix,

the tabular method cannot be objectively compared to other

methods. Only a subjective comparison of the "goodness" of

results is available with the tabular method.

When one discuses and-inevitably-criticizes these

methods, one must remember that they were established at a

time when most calculations had to be performed with

logarithm tables and only later with mechanical desk



















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16
calculators. Often, the necessary computing effort was the

deciding factor as to whether a project was feasible or not.

With the advent of computers, it is now practical to

undertake enormous data reduction problems which require the

simultaneous estimation of tens of thousands of parameters

without taking any computational shortcuts which degrade,

even if ever so slightly, the results. It is now possible

to reduce the avoidable subjectiveness of a scientific

investigation so that its results can be judged on an

objective basis.

It must also be noted that there are many vital interests

in the results of astrometrists. For example, the time

standard provided by astrometrists is relied on to

syncronize global satellite communications and calibrate

radio navigation. Thus astrometrists must be absolutely

certain of their results. With this in mind one might argue

that caution should be exercised in the acceptance of any

new procedure. While this is a legitimate concern, because

of the importance of this work, catalogue astrometry does

deserve the full analytical treatment which is now

available. There is no excuse for not using a rigorous

reduction in the compilation of a complete star catalogue.



Description of Simultaneous Reduction

As mentioned above in regard to estimating the systematic

errors of a catalogue, there are two distinct methods to








compute these estimates. In the tabular procedure, one

estimates the systematic characteristics of a catalogue and

adjusts the catalogue in order to correct these systematic

trends. Once all the catalogues are on the same system,

estimates for the individual positions and proper motions

are calculated.

In a simultaneous reduction, systematic corrections

(model parameters) and positions and proper motions (target

parameters) are estimated simultaneously. In this way not

only are the systematic differences between the independent

source catalogues and the fundamental system of the FK4

minimized, but the systematic differences between all

catalogues are minimized as well. The incorporation of all

available information for the derivation of systematic

corrections yields a result more precise and accurate than

that achieved by traditional methods. The idea that all

parameters, used in the construction of a complete

catalogue, should be solved for in a single adjustment was

first suggested by Eichhorn (1974) and later described by

him in more detail (Eichhorn 1980).

Within the framework of this research I have employed

this principle to estimate simultaneously systematic

corrections for all ICs used to construct the NIRS on the

basis of exactly the same model and from exactly the same

raw material as those used by Corbin. Corbin's model

computes differences on a grid at each hour of right







18

ascension and each five degrees of declination and then uses

two-way linear interpolation to compute corrections to

individual IC positions. The smoothing coefficients and the

weights of the individual catalogues were also the same as

in the model used by Corbin. The only difference between

the the reductions was the method of computing systematic

differences.

Consider the linear matrix equation



Y = XB + ZP + E (8)


where Y is a vector of observations, B and P are the target

and model parameters respectively, X and Z are their

respective coefficient matricies and E is a vector of errors

with covariance matrix This equation could be

alternately and more compactly written


Y = [XZ] [] +E= A + F. (9)


A simultaneous reduction estimating both model and target

parameters would yield least squares estimates


^ T-1 T
[1] = = [A LA] A ZY. (10)
P

If model parameters are estimated first and target

parameters second, the least squares estimates are


^ = (zTz)-1 T (- ')
P= Z) Z E(Y-X6')


(11)


and









S(T -1 T T
S (X EX) X (Y-ZP)

where B' are preliminary estimates for In trying to

perform a simultaneous reduction of the NIRS using the same

model as Corbin, I had to decide how to handle two problems.

First there was the computational difficulty of inverting

the ATZA matrix in (10). Since I have approximately 40,000

star parameters and 20,000 model parameters, this means that

a 60,000 by 60,000 element matrix must be inverted.

Although there exist procedures (cf. Lawson and Hanson 1974)

which render a solution without the direct inversion of this

matrix, a problem of this size requires more virtual address

space than the Fortran application, which was at my

disposal, had available. This work was performed on a VAX

11/750. The second problem was that the model parameter

estimates that Corbin used were not least squares estimates

but the estimates described above. The obvious solution to

both of these problems was to iterate on a solution. An

iterative process converges toward the same solution as the

closed form (cf. Faddeeva 1959) but with less computational

difficulty and also allows for the use of traditional

tabular method of estimating model parameters.















CHAPTER III
REDUCTION PROCEDURES


In the previous chapter, I have criticized the tabular

procedure on three grounds: 1) the process is subjective in

nature, 2) not all available model constraints are used and

3) the estimates for the model parameters are biased. The

premise of this research is that by using all available

model constraints, a more precise and accurate result is

produced. Therefore, I have used the same subjective model

and the same biased parameter estimates as Corbin used for

my compilation of the NIRS. The differences in the two

versions of the NIRS are thus due only to the fact that I

have constrained the model parameters to minimize the

systematic differences in star positions between independent

catalogues as well as the systematic differences between

independent catalogues and the FK4. The aim of this

research was not to produce the best possible catalogue but

rather to show that a simultaneous reduction produces

superior results to those of the classical method.



Precession

The first step taken to recompile the NIRS was to process

all of the independent catalogue positions to the coordinate








system orientation of the FK4. The catalogues used in the

recompilation of the NIRS are given in table 4 at the end of

this chapter. The precession was carried out using Newcombs

constants of precession. Three angles, C z and e, were

computed for each catalogue epoch, these are (cf Eichhorn,

1974)


S= [(23402.253 + 139.75ti + 0.061t2)t
1 i f

+ (30.23 0.27t )tf + 18.0t 648000


z = G + [(79.27 + 0.66t.)t2 + 0.32t3] 6 0 (12)


) = [(20046.85 85.33t. 0.37t2)t
I f
2 3 ] 4T
(42.67 + 0.37ti)tf 41.8t3 648


where t. is the initial epoch of orientation relative to

1900.0 and t is the difference, final minus initial epoch

of orientation. Both t. and t are reckoned in Bessel
i f
millennia. The constant T/648,000 is necessary to convert

from arcseconds to radians. Next, the IC positions are

processed to 1950.0 with the above angles and the following

formulae

X = cos6 cos(a + -)

Y = cos6 sin(a +C -) cose + sin6 sine

Z = -cos6 sin(a + 7) sine + sin6 cose (13)

a50 = arctan (Y/X) + z + -
50 2









/ 2 2
50 = arctan (z/ x + y )

where X, Y, and Z are temporary cartesian coordinates, and

a, 6 and 50' 50 are the right ascension and declination at

the initial and final epoch respectively. A vector

resolution arc-tangent function was used to insure the

proper quadrant for a5
50


Model Parameter Estimation

After all IC positions had been processed, the next step

was to compute systematic corrections. Right ascensions are

used in the following discussion but an analogous procedure

was applied to declinations. It must also be noted that

right ascensions were first multiplied by the cosine of the

reference declination before differences were calculated.

First differences in positions were computed for each star

in each IC using the reference position and proper motion.

These differences were summed over blocks of one hour by one

degree centered on the hour and the degree. That is a block

that covered 13h would extend from 12h 30m to 13h 30m.

These sums are

h 0
oa+0.5 6+0.5 cos6
( ) = re [aref a(1950.0-T )-a] (14)
a-0h5 6-0.5 r

where Aa(a, 6) is the systematic difference, s2 is the

estimated variance of the reference star position at the

epoch of the IC position, aref and 6ref are the reference
ref ref







23

right ascension and declination, p. is the reference proper

motion and a is the IC position observed at epoch T,. The

estimated variance of an FK4 position was calculated using

the errors and central epoch given in the FK4 and the epoch

of the IC position. The variance for an NIRS position was

calculated in a like manner except that an additional

variance term was included to represent the error of the

system of the NIRS. That is, the reference system of the

FK4 is defined only in terms of the set of FK4 stars. If

that set of stars is altered, then the ideal reference

system, which the altered set approximates, is no longer

exactly that of the FK4. Even though the system of the NIRS

is an approximation to that of the FK4, they are not

identical. Therefore, the variance of an NIRS position

consists of two parts. The first is due to the error of the

star position within the system of the NIRS and the second

is due to the error of the system of the NIRS itself.

In order to estimate the variance due to the error of the

system of the NIRS, the Perth 70 and Corbin's version of the

NIRS were used. After matching 3324 stars between the Perth

70 and the NIRS, the one sigma dispersion of Perth 70

positions and the NIRS predictions of the Perth 70 were

found to be 0.22 arcseconds in right ascension and 0.30

arcseconds in declination. This dispersion is due to the

position errors within the catalogues as well as the errors

of the systems of the catalogues. Since the mean error ,of a






24

position is published in each catalogue, the errors of the

the systems of the catalogues can be estimated.


2 2 +2 2 2
S NIRS P70 + SNIRS SP70 (15)


The square of the dispersion, d2, is the sum of the squares

of the mean errors of a catalogue position, 2 and E2
NIRS P70
plus the squares of the estimated errors in the catalogue

systems, 2 and 2 The residual variance could be
SNIRS SP70
split equally between the systems of the two catalogues, but

I chose to have the ratio of system errors equal the ratio

of the mean position errors. The system error used for an

NIRS position was .064 arcsecond in right ascension and .070

arcsecond in declination.

Once tables of differences for each IC were calculated,

these differences were smoothed with adjacent differences

according to the following scheme:
+3h +60
E Z A.B.Aa(a + i, 6 + j)
h a 1 3
A'a(a6) = i=-3 j=-6 (16)
+h (16)
+3 +6
Z Z A.B. n(a + i, 6 + j)

i=-3h j=-6
where A. and B. are the smoothing coefficients in table 2

and n(a, 6) is the sum of the reciprocal variances for the

appropriate IC, hour and degree. The smoothing coefficients

used (regular or light) for each catalogue are listed in

table 4. After differences have been summed and smoothed,

they are then averaged over zones of five degrees:







0
+2 D A'a(a 6 + j)
Aa(a,) = =-2 (17)
+2
j=E23 Dj
where D is the denominator in equation (16) of the

associatedA'a(a, 6).



TABLE 2

Smoothing Coefficients



Regular Light
i A. j B. i A. j Bj
oh 4 00 10 Oh 8 00 10
Ih 2 10 8 1h 3 10 8
2h 1 20 8 2h 0 20 5
3 5 30 2
40 5 40 0
5 2 5 0
60 2 60 0




Once tables of systematic differences are computed for

each catalogue, they were applied to the IC positions as

systematic corrections using two-way linear interpolation.

Right ascensions were first multiplied by the cosine of the

declination, corrected and then divided by the cosine of the

declination.



Star Parameter Estimation

After correcting the systematic errors of the ICs, the

position and proper motion were calculated for each star









using a weighted least squares algorithm. Corbin (1982)

calculated weights for each IC used in the construction of

the NIRS using three different methods. Method A was based

on the deviations of an IC from the final NIRS compiled with

each catalogue receiving equal weight. Method B was

similarly based on deviations from a mean NIRS but this time

the mean NIRS was compiled without the particular IC whose

weight was being determined. Method C was based on the

deviations of an IC from the reference system which was used

to calculate its systematic corrections. Corbin then used

the arithmetic mean of these three methods in the final

compilation of the NIRS. I have used these same weights,

listed in table 4, in my compilation of the NIRS.1

The following algorithm was used to calculate star

parameters. First the central epoch, T, and position, P,

were calculated:



Sw.P.

C W.
i=l 1

(18)
Z w.T.
i=l
.-T-

E W.
i=l


e---------


1 For a discussion of selecting weights used in catalogue
compilation, see Khrutskaya 1980.









where T. is epoch and P. is the position of the star in
1 1
catalogue i and w. is product of the the weight associated
1
with catalogue i and the number of observations for that IC

position divided by the mean number of observations per IC

position for that IC. Next the T. were referenced to the
1
central epoch:



Ti' = Ti T (19)


and the proper motion was calculated:

1
L w.T. 'P.
i=l 1
= 1 (20)
^i2
SwiTi
i=l

Finally, estimates for the variance of the position and

proper motion, V and V., were calculated:


(P -
ZE wi(P. p Ti')
i=11 i 1i


(1 2) Z w.
i=1

1)2
il wi(Pi PTi


(21)


z2
(1 2) Z w.T.2
i=l 1 1


Sequence of Iterations

Now that the basic mechanics of the reduction have been

described, a discussion of the iteration sequence is in

order. Within an iteration the first step was to eliminate


V






Vii =







28

outliers (IC positions with large residuals), the second was

to determine and apply systematic corrections and the third

was to compile a new version of the NIRS.

Corbin used several criteria for selection of AGK3R stars

to use in the NIRS. Among these was the requirement that a

star with only two observations must have those two

observations separated by a minimum of 28 years. Because of

the convolution of these criteria with the selection of

outliers, I chose those stars which appeared in Corbin's

final NIRS catalogue to use in the compilation of my version

of the NIRS. This, however, did not eliminate the problem

of identifying the outliers.

In duplicating the model and method used by Corbin, I

chose the same criterion for rejecting outliers. Corbin

rejected an IC position if the absolute value of its

residual was 3.5 times the mean absolute residual for

positions in that catalogue. Corbin established residual

limits for each IC and I have used these same pre-set limits

in my compilation of the NIRS. Residual limits for each IC

are listed in table 4.

The problem here is that one must compute systematic

corrections before residuals can be analyzed. In using an

IC position with a large residual to calculate corrections,

the corrected system of a catalogue can be distorted such

that other positions, which would not normally be excluded,

now exceed the residual limit. I decided to take an







29
iterative approach to this problem. The largest outliers

were removed first such that the systems of the catalogues

were not influenced by them in the next iteration. Then the

residual limit was lowered and the next largest outliers

were removed. This process was repeated until the residual

limit was lowered to that of the individual catalogues. In

iterations one and two, no IC positions were removed. In

iteration three, only IC positions whose residual absolute

values were greater than five arcseconds were removed. The

iteration residual limit was lowered in succesive iterations

as given in table 3 until the residual limit for an IC was

the individual limit given in table 4.

For the first iteration, the only reference system was

the FK4. Only catalogues 1 through 10 had FK4 observations,

thus only these first ten catalogues were corrected. For

the second iteration, the reference system included the FK4

as well as 6317 NIRS positions and proper motions computed

in the first iteration, thus allowing all ICs to be

corrected. For the third and subsequent iterations, the

reference system included the FK4 and 20194 NIRS positions

and proper motions.

Only the first ten ICs were corrected in the first

iteration; thus for the first computation of the NIRS, only

positions from the first ten ICs were used. In addition,

only those stars with three or more IC positions were

compiled into the first version of 6317 NIRS positions and









TABLE 3

Iteration Sequence



Iteration Residuala Number of Stars
Limit Compiled into NIRS

1 none 6317
2 none 20194
3 5.0" 20194
4 2.0" 20194
5 1.8" 20194
6 1.6" 20194
7 1.4" 20194
8 1.2" 20194
9 1.0" 20194
10 0.8" 20194
11 0.6" 20194
12 0.4" 20194
13-25 b 20194

a. The residual must be greater in absolute value than
both the iteration residual limit and the individual
catalogue limit in table 4 in order for an IC position
to be rejected.
b. For the 13th through 25th iterations the individual
catalogue limits in table 4 were used.



proper motions. For the second and subsequent iterations,

all IC positions were used to calculate 20194 NIRS positions

and proper motions.

Tables 5 through 68 at the end of this chapter give the

corrections applied to the independent cataolgues for my

compilation of the NIRS. Each IC position was corrected

with values from this table using two-way linear

interpolation. The units are hundredths of arcseconds and

the right ascensions corrections have been multiplied by the

cosine of the declination. Furthermore, corrections whose







31
absolute values exceeded 99 hundredths of an arcsecond were

replaced by 99 hundredths with the appropriate sign.











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