Stellar motions in the Orion Nebula cluster

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Stellar motions in the Orion Nebula cluster
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Stars -- Motion in line of sight   ( lcsh )
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Thesis--University of Florida.
Bibliography:
Includes bibliographical references (leaves 77-79).
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by Frederick W. Fallon.
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Typescript.
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Vita.

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STELLAR MOTIONS IN
THE ORION NEBULA CLUSTER




















By

FREDERICK W. FALLON


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






UNIVERSITY OF FLORIDA
1975
















ACKNOWLEDGEMENTS


The following individuals deserve special thanks for

their roles in making the present investigation possible.

For the use of observational material. Dr. K. Aa.

Strand, for making available his measurements of the Yerkes

photography--the more deeply appreciated in light of his

own prior publication based on that material; W. D.

Cannell, for the free use of his measurements of the

McCormick series of 44 plates, forming the basis of his

own as yet unpublished paper; N. E. Wagmann, for the free

use of both early and recent plates from the Allegheny

collection; P. M. Routly of the USNO for permission to use

both the "SAMM" automatic and fringe-counting manual mea-

suring engines; A. Behall and J. Jostles of USNO for their

help and courtesy while I was using those instruments;

W. D. Heintz, for securing several recent epoch plates at

the Sproul refractor; and P. Ianna for permitting me to

inspect the McCormick plate collection and providing useful

information concerning it.

For assistance in observing: Special thanks are due

to M. DeGeorge, then Night Assistant at the USF Observatory,

for assisting in the tedious--and backbreaking--work of










collimation and polar alignment of the 26" Tinsley, as well

as in securing some of the photography. Graduate students

L. Twigg and B. Holman also assisted in the photography,

and in supporting work at the observatory generally.

R. Fox helped measure and reduce the photographic

photometry.

For discussions: Over the last several years Dr. H. K.

Eichhorn has been a focus of inspiration and stimulating

exchange of ideas. The modern era of rigorous astrometry,

to which the present work belongs, owes its inception

largely to his pioneering efforts. During the present work,

we have discussed some of the knottier and discouraging

problems that arose. I am grateful too for his critical

reading of the manuscript. The statements made herein,

however, except as otherwise credited, are the respon-

sibility of the author.

Mr. A. E. Rust has given much valuable advice on com-

puter problems, especially the optimum use of disk storage

which is a crucial part of the program with which the cal-

culations were done.

Finally, it is a great pleasure to thank Dr. S. Sofia

for pointing out the need for the investigation, and for

the guidance and encouragement he has given as advisor.

His insight into the basic fundamentals of the problem has

been invaluable, and discussions with him have always been

immensely fruitful.


iii

















TABLE OF CONTENTS


ACKNOWLEDGEMENTS . .. .

ABSTRACT . . .

SECTION
I. INTRODUCTION . .

Background . .
The Value of a New Astrometric Study
Scope of Present Investigation .

II. REFERENCE MATERIAL . .

III. THE PLATE MATERIAL . .

General Survey . .
The USF Photography . .
Coverage . .
Exposures . .
Measuring the Plates .
Preliminary Processing .

IV. FORMULATION OF THE PROBLEM .

Differential Plate Measurements .
Reference Proper Motions ......
Differential Overlap ....
Plate Overlap Method .


.. 1


. 22


V. THE COMPUTATIONS . .


Partition of the Plate Material
of the Proper Motion .
Behavior of the Solution .
The Computer Program ..


and


. .


Treatment


VI. RESULTS


Absolute Proper Motion of Trapezium
Velocity Dispersion .
Contraction . .


Cluster .


*
*


i I


. *












APPENDIX . .. .. 73

LIST OF REFERENCES . .. .. 77

BIOGRAPHICAL SKETCH. .. . 80










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



STELLAR MOTIONS IN THE
ORION NEBULA CLUSTER

By

Frederick W. Fallon

December, 1975

Chairman: Sabatino Sofia
Co-Chairman: Heinrich K. Eichhorn-von Wurmb
Major Department: Astronomy


Positions and proper motions for 113 stars in the Orion

Nebula are derived from measurements on 92 plates taken

with a variety of telescopes, with an average epoch difference

of 50 years. The positions and motions are measured with re-

spect to a set of some 40 reference stars; they are therefore

instrumentally absolute and unaffected by changes in plate

scale over time. In order to incorporate a sufficient num-

ber of reference positions into the solution and to obtain

the greatest possible accuracy from the data, the reductions

were carried out by the plate overlap method. A new version

of the method, particularly economical for the astrometry of

star clusters, was employed.

It is found that 1) the velocity dispersion of the stars

in the Trapezium Cluster (within 6' of the Trapezium) is less

than 2 km/sec, and 2) the Orion Nebula Cluster (radius 20')

vi










is contracting at a linear rate of (.'009/yr)/degree, cor-

responding to 10 km/sec at the 4 parsec radius of the

radio molecular cloud. These results imply that the Orion

Nebula is free from large turbulent motion, is gravitationally

contracting, and that star formation there has probably been

proceeding continuously since it began, with the greater

period of star formation yet to come. These conclusions

in turn are in agreement with the large (-105 solar masses)

mass now assigned to the Orion Nebula.


vii
















SECTION I
INTRODUCTION



Background

The Orion Nebula has been an object of special fascina-

tion to astronomers from its discovery three centuries ago

up to the present. It is now recognized as a complex in-

volving a ("the") trapezium, a star cluster, emission

nebulosity, infra-red sources and radio molecular clouds.

The coincidence of these components in a single, relatively

nearby (~500 pcs) object potentially makes it a key to

understanding a number of basic astrophysical processes.

At the same time, there are some perplexing problems which

stand in the way of arriving at a unified model of the

object.

Let us examine some of the special features of the

Orion Nebula and the questions relating to them.

1. The Trapezium itself is an interesting object, the

prototype of a small class of systems ("Trapezium system")

first recognized by Ambartsumian (1955). These systems are

intermediate between binaries and true clusters. Even when

a trapezium is considered as part of a surrounding cluster,

it is clear we are dealing with a distinct morphological

group since the trapezium always stands out as a small (4-8








2

members), central grouping at least five magnitudes brighter

than the next brighter cluster members, in contrast to ordi-

nary open clusters. Trapezia are known to be young because

they occur only among 0 and B stars. They would then be

disrupted within about 2 to 3.106 years, yet astrometric

studies fail to decide whether the Orion Trapezium has

positive or negative energy.

According to current thinking on the formation of

trapezia, they should consist exclusively of 0 and B stars,

yet the Orion Trapezium has two "comites" (O'E and 6'F)

both three magnitudes fainter than the faintest of the four

principal members.

2. The Trapezium is surrounded by a faint cluster, the

"Trapezium Cluster" (or "Orion Nebula Cluster"), discovered

only as late as 1931 (Trumpler, 1931; Baade & Minkowski,

1931). Its brightest members are 11-th mag. and it is

usually observed by the surrounding nebulosity. The clus-

ter shows well in the plate reproduced in Figure 1; on the

original plate the nebulosity was deliberately suppressed.

There are widely differing values given for the dia-

meter of this cluster. Baade and Minkowski judged its dia-

meter as 3' to 5', encompassing some 100 stars. Sharpless

(1966), referring to the Baade and Minkowski discovery,

misquotes the diameter as "10'"; actually, the 10' figure

appears in another context in another paper by the same

authors immediately preceding (in the journal) the cluster

















































FIGURE 1
THE ORION NEBULA SHOWING THE TRAPEZIUM CLUSTER

Two-fold enlargement of plate taken on 103a-G emulsion
with 3-69 filter, exposed for 12 min. at University of
South Florida Observatory. A circle 1' in radius has been
drawn around the star e'C. The faintest stars visible are
14-th magnitude.











discovery paper. Trumpler, on the other hand, though count-

ing the stars only in 4' diameter circle, actually gives

its diameter as 15' partly on morphological grounds.

Gradually, this larger value has crept into the literature.

Strand (1958) takes the diameter as 30'.

Turning to the evidence, one can see in Fig. 1 a

definite concentration of stars within a circle of 2'

diameter immediately surrounding the Trapezium. We shall

term this group the "Trapezium cluster." It is clearly

much smaller than a typical open cluster, and morphologi-

cally quite different; perhaps it might be better named the

"Extended Trapezium." A star count reveals a further con-

centration of stars in an area of about 5' radius, extending

north and also south-east toward the group 02. It is to

be noted that heavy absorption obscures the region south-

east of the Trapezium. In the absence of the dust we should

expect to see cluster members in that quadrant as well. (It

is to be expected too that many of the few stars visible in

that area, extending for at least 20', are foreground

objects.) Accordingly, we assign a diameter of 12' to this

grouping, calling it the "inner Orion Nebula Cluster."

Finally, in accord with current usage, we apply the name

"Orion Nebula Cluster" to the whole region of .30' in diameter

centered on the Trapezium, realizing that the stars in this

larger region may not comprise as well demarcated a group.









3. Separate theoretical arguments by Menon (1963, Kahn

and Menon 1961) and by Vandervoort (1963), based on the mass

motions and ionization of the inner region of the nebula,

both lead to an age of only 2*104 years (the "ultra-short"

age) for the Trapezium stars. This age refers to the time

since the stars have been capable of ionizing the hydrogen.

4. The proper motions of the cluster stars can in

principle yield the cluster's age, if the motions show a

radial expansion. Then the kinematic age is the reciprocal

of the expansion constant. (Strictly, this age refers only

to the time since greatest concentration of the cluster and

does not preclude earlier contraction and star formation.)

Some astrometric studies (Parenago 1953; Franz unpub-

lished, quoted by Sharpless 1966) have indeed confirmed the

ultra-short age of the Trapezium and this figure (104 yrs)

has become widely quoted in the literature. But other astro-

metric studies (Strand 1958, Duboshin et al 1971) result in

the much longer (though still very young) kinematic age of

2.105 years; while yet another (Meurers 1963) shows no evi-

dence of contraction, and still others (Cannell unpublished,

Vaerewyck 1972) report a slight contraction. Adding to the

confusion, some (Meurers, Strand) of these studies show a

rotation of the cluster. The compendious work of Parenago

(1954) has not yet been analyzed for evidence of cluster ex-

pansion. Parenago's material is extremely heterogeneous;

significantly, the stars for which he quotes the smallest

proper motion errors also have the smallest dispersions in

proper motions.











5. Color-magnitude diagrams for the Orion Nebula
1 2
Cluster (including 1 and 6 ) and surrounding regions as

given by several investigators, while disagreeing among

themselves, agree in pointing to a young age for the stars

in Orion. Some of the discrepancy can be attributed to the

different sizes of the area surveyed, while some is also

due to the differing reddening corrections applied. Walker

(1969) presents a particularly careful study. His photom-

etry shows the zero-age main sequence down to an intrinsic

(B-V) = -.07, corresponding to a contraction time of

"2 2/3 10 years. At fainter magnitudes, the scatter of

points above the main sequence would seem to suggest non-

coevality of star formation, but some of the scatter is no

doubt due to the patchy distribution of the absorption.

Significantly, no photometry has been carried out on

the stars of the Trapezium Cluster (except, of course, for

the Trapezium itself), due to the difficulty of observing

through the nebulosity. Its color-magnitude diagram might

well differ from that of the larger "Nebula" Cluster, and

be characteristic of an even younger age.

In this connection we should note that the unseen com-

panion in eclipsing binary BM (=8'B) Orionis, hitherto a

very puzzling object and "black hole" candidate, has re-

cently been explained by Popper (1975) on the basis of his

discovery of its spectrum. It is apparently a 1.8M

star in a state of pre-main sequence differential rotation,











2 magnitudes above the main sequence. The currently ac-

cepted contraction time for such an object is ~1.8 106

years.

6. Radio observations of the molecular cloud Orion A

have recently produced two surprises. First, the mass of

material (mostly in the form of H and not observed di-

rectly but deduced from the observed abundance of CO and

HCN) is enormously large: about 105 M (Liszt et al 1974),

or several hundred times that of the stellar mass in the

same volume. If the nebula really is expanding, it is puz-

zling why so large a fraction of the material failed to

form into stars.

Second, the widths of the CO and HCN lines are much

broader than the thermal Doppler widths, implying variable

mass motion along the line of sight. Fitting a non-LTE

model to the data, and assuming the motions to be large

scale, Gerola and Sofia (1975) are led to conclude a linear

contraction amounting to 12 km/sec at a distance of 1/20

from the center. This is in agreement with the large

gaseous mass of the cloud and apparent range of stellar

ages but not with the idea that the cluster is expanding.

One may indeed interpret the line profiles by an equal

expansion velocity, but then one finds a shell midway in

the cloud where the gravitational potential energy exceeds

the kinetic energy.











On the other hand, Zuckerman and Evans (1974) have

criticized the uniform contraction model, proposing instead

that the observed velocity spread is due to turbulent eddies

interspersed all along the line of sight through the cloud.

If this is so, then the cloud model of Gerola and Sofia is

invalid, and some entirely different model, as yet not

worked out, would be required.


The Value of a New Astrometric Study

The questions fundamentally at issue are these:

(1) Is the Orion Nebula cluster expanding, as is the Orion

Association as a whole, or is it still undergoing its ini-

tial contraction, with star formation still taking place?

(2) Are the motions in the nebula smoothly varying, or

dominated by turbulence? The answers to these questions

are important to our understanding of the process of star

formation in clusters.

A new study of the kinematics of the stars in the

Orion Nebula would form a valuable basis for answering these

questions. Thus far, the astrometric results have been

discordant and inconclusive. It is the purpose of this

investigation to derive proper motions of sufficiently high

accuracy to make a more conclusive kinematic study

possible.











Scope of Present Investigation

Four principal desiderata determined the scope of the

present study. (1) The proper motions should be referred to

reference stars exclusively and in no way depend on assump-

tions about the optical properties of the instruments with

which they are obtained. We call motions of this type

"instrumentally absolute." In particular, they are to be

completely independent of any assumption concerning the

variation of the telescope's plate scale with time. The

proper motions are, of course, relative in the astrometric

sense in that they are referred to the positions and mo-

tions of a system constituted by reference stars rather

than referred to the equator and equinox directly. Within

the accuracy of the determination, the positions and proper

motions derived here may differ from the best derivable

positions and motions systematically by an additive con-

stant, but not by a scale factor. (2) The coverage should

include enough reference stars to determine the plate scale

as accurately as possible. (3) As many stars of the

Trapezium cluster as practical should be included, while

equally faint stars farther out may be omitted. (4) The

formulation of the adjustment equations should be statisti-

cally fully rigorous; that is, the variance of the residuals

of all the (weighted) observations, taken together, is to be

minimized. This means, among other things, that the mathe-

matics used in the adjustment must take into account the











fact that a given star at a given epoch has one position

regardless of from whatever plate this position is derived.

Taking due account of this latter restraint is equivalent

to using the "overlap" method of Eichhorn (1963). It is

the imposition of the overlap condition that produces not

only higher internal accuracy than otherwise obtainable,

but also allows the reference stars to determine more

accurate estimates for the plate scales and hence the

expansion. This is so because there are not enough refer-

ence stars in the region, and therefore, the accuracy of

the plate scale which one would obtain if each plate were

reduced separately from all others is insufficient for our

purposes. By imposing the overlap condition, however, we

are able to make full use of plates on which as few as no

reference stars appear.

A fully rigorous solution, in the strict sense defined

above, would require that every single measurement be pre-

served separately throughout the reduction, i.e., no aver-

aging of individual settings of the measuring engine, or of

separate grating images, could be made. Since strict rigor

is laborious to enforce and really unnecessary when the

error distribution of such measurements is uncorrelated

with any of the parameters being sought, we have relaxed

the algorithm to allow the usual practices of averaging

direct and reverse measures, pairs of grating images, etc.,

in the interest of expediency.
















SECTION II
REFERENCE MATERIAL



By reference material we mean the positions and proper

motions of the set of reference stars with respect to which

the positions and motions of all the program stars are com-

puted. Since the role of the reference material is espe-

cially crucial in the present study, it deserves discussion

in some detail here.

As usable reference material we may only include posi-

tions whose errors are uncorrelated. In practice such posi-

tions are obtained from meridian observations, or photo-

graphic catalogues sufficiently global that our selection

of reference material is confined to a relatively small

region. Thus, heliometer or micrometer observations are

usable, even when confined to a small region of the sky; so

are photographic zone catalogues, if the zone is suffi-

ciently wide, even though such positions are "secondary."

On the other hand, we consider a catalogue based on photo-

graphs covering only the program region, e.g. the Zo-Se

Catalogue, unusable as reference material because the

errors are appreciably correlated through the plate con-

stant variances of the photographs.






















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The region from which the reference material is taken

is approximately bounded by 5 26 < a < 5 h42m, -6 < 6 <

-4V0 (orientation 1950). The actual limits were dictated

by the bounds of those Astrographic Catalogue plates which

happen to include the program region; these extend beyond

any of the other plate material.

Source Catalogues. We have taken our reference mate-

rial from the following sources:

1. FK4 and N30. The position of one of our reference

stars, i Ori, is listed in the FK4. The portions of three
1 2
additional stars are found in the N30: of e C, 2A, and of

BD-60 1255. The N30 positions and proper motions were

transformed to the FK4 system according to the tables of

Brosche et al. (1964). For i Ori, the mean of the trans-

formed position and that of the FK4 has been used.

2. Boss' General Catalogue (GC): For six of the

reference stars, positions are found in only the GC. The

positions and motions were taken directly from the SAOC and

are therefore on the FK4 system.

3. Yale Zone, Volumes 16, 17: The bulk of the refer-

ence stars is listed in the Yale Zone catalogues. Their

positions and motions on the FK4 system are taken from the

SAOC. When the Yale position is not listed in the SAOC, it

was taken from vol. 16 (Barney, 1945), or vol. 17 (Barney,

1945a), and transformed to the FK4 system by adding the

corrections (Yale to GC) given in the respective introduction










and the correction (GC to FK4) from Brosche, et al (1964).

Volumes 16 and 17 cover the zone -100 to -2, joining at

-6 with vol. 16 being the northern. These two halves had

been derived from one single belt of Yale plates and re-

duced as one zone; the proper motions, however, were ob-

tained from comparison with two different Strasbourg AGK1

zones which join at -60. A comparison of the proper motions

for the same star as it appears in both volumes for 24

stars in common, reveals a systematic difference between

the two in the sense P6 (vol. 16) = P6 (vol. 17) '012/yr.

We have therefore added the correction -'012/yr to all the

P6's from vol. 16. If this is not done, the systematic

difference is propagated through the plate constants and a

marked spurious contraction of the whole region in 6

results.

The epoch of this Yale zone is 1934. The AGK1 posi-

tions (epoch 1895) of these stars, used in the formation of

the proper motions listed in these Yale volumes, have such

large errors (a0.4), that the resultant error in the proper

motions is .'01/yr, corresponding to a positional error of

'.'45 at the epochs of both our earliest and latest plates

(1900, 1974). Accordingly, it is very desirable to improve

the proper motions of these reference stars with additional

reference positions of epoch well before or after 1934.

4. Meyermann's Heliometric Positions: B. Meyermann

(1903) measured the positions of 47 bright stars in the











vicinity of the Trapezium in 1903, using the G6ttingen heli-

ometer. This series is especially valuable because its

early epoch and the relatively high accuracy ('.'25 in each

coordinate) improves the accuracy of the proper motions

given in the Yale Catalogue from '.Ol/yr to .007/yr.

Thirty-seven of the Meyermann stars are included in the

present program; of these the data for 31 are in the Yale

Catalogue; for five, in the GC only. (The remaining star

can thus be used as a reference star only for plates taken

near the epoch 1903.)

Some 30 years later, in the course of a photographic

study of the same region, Meyermann discovered an error in

the scale of the heliometer; accordingly, in a subsequent

paper (Meyermann, 1938), he gave revised values for the

1903 positions, corrected for the scale error. At the same

time, he also transformed the positions to the system of

the FK3 by translation and rotation, using parameters de-

rived by a least-squares fit of the heliometer positions to

those in the PGC and the "Courvoisier"1 Catalogues. The

data for nine stars in all were available for this trans-

formation. The relations between the systems of the PGC

and Courvoisier catalogue to the system of the FK3 were

known and used by Meyermann.


Meyermann does not identify which of several
Courvoisier Catalogues is meant, but in any case he quotes
the actual positions.










In making use of these positions, we have first pre-

cessed the coordinates from orientation 1900 to 1950 using

Newcomb's constants, and transformed the PGC positions

quoted by Meyermann to the system of the GC (and eventually

to the FK4 system) for the same stars by a least-squares

adjustment and applied that transformation to all the

Meyermann stars. This amounts to a correction Aa = -S040,

arising from a systematic error in the PGC in the amount of

Aa = -!060.

5. Southern Reference System (SRS): This is the

reference catalogue currently being compiled at the U.S.

Naval Observatory from relative meridian circle observa-

tions and intended as a counterpart to the AGK3R in the

southern hemisphere. It contains the data for six stars

in the present program, all of these included also in the

Yale Catalogue. The average epoch is 1960, thus affording

a valuable improvement in the accuracy of their proper

motions.

Preliminary positions for these stars were made avail-

able prior to publication of the completed catalogue

through the courtesy of Dr. J. Schombert of the USNO.

Combination of reference sources. In the case of

stars included in more than one of the reference sources,

a weighted mean position, proper motion, and epoch has been

computed according to the rigorous expressions (eg.

Eichhorn, 1974 p. 113). An exception is made in the case











of the three FK4 and N30 stars, these positions not being

averaged with those from other sources. The standard devi-

ations characteristic of each source are listed in

Table 1.

Double Stars. The Aitken Double Star Catalogue (ADS)

was consulted for any entries lying within the range of our

program. None of our program stars shows any clear sign of

orbital motion from the observations listed in the ADS, so

no orbital corrections have been applied. Two uses were

made of the ADS material:

1. Doubles closer than 3" but wider than 1" were re-

jected from the reference list since they might appear

variously resolved on some plates and unresolved on others.

Some wider doubles (up to 5") were also rejected on the

shorter-focus astrographic plates for the same reason.

Doubles closer than 1" have all been retained since they

are presumably always unresolved. These criteria lead to

the rejection of the following reference stars: i Ori (on

AC plates), BD -4.1171, -4.1172, -4.1185, -4.1186 (on AC

plates), and -6.1255 (on AC plates).

2. Astrometric data for doubles were utilized in the

following special cases: (a) Wide pairs which occur as two

distinct reference stars, all of which happen to show zero

relative motion, were assigned equal reference proper motion

to both members, formed by the mean of the two catalogue

values. This amounted in some cases to a substantial























TABLE 1
SUMMARY OF REFERENCE CATALOGUE MATERIAL


Catalogue a a t
ca,s p e

FK4 ."05 '.004/yr 1912

N30 .05 .004/yr 1912

Yale (v. 16 or 17) .14 .010/yr 1934

GC .15 .010/yr 1900

Meyermann (H) .25 .007/yr 1903

S.R.S. .05 .007/yr 1965

Yale (v. 16 & 17) .10 .007/yr 1934

Yale + GC .10 .007/yr 1915

Yale + H .12 .007/yr 1924

Yale + S.R.S. .05 .003/yr 1960










departure from the original catalogue value, and always

yielded a better fit to the plate measurements. This pro-

cedure does not, of course, constrain the final proper

motions of the components to be equal. The following pairs
1 2 2 2
were treated this way: E C and 6 A; 0 A and 0 B; BD

-61233 and 1234. (b) For 1 E and 61F, the fainter com-

panions to the four principal Trapezium stars, the micro-

metric observations prior to 1920 are used in the deter-

mination of their proper motions, since these two stars

occur on no early epoch plates.

Parallaxes. Only two of our program stars have known

and measurable parallaxes: i Ori (n = '.'02) and 45 Ori (w =

'.'02). These values are too small to influence the overlap

solution, but are probably large enough to establish the

stars as foreground objects. Correction for the effects of

these parallaxes have been-applied to their reference

positions.

Magnitudes and colors. In general, we must antici-

pate that the plate reduction models may include terms in-

volving magnitude and color. Therefore, we need approxi-

mate values for them. For this purpose, they need to be

known with a relative accuracy of about 0.1. As a

criterion for membership in the cluster, however, we re-

quire more accurate values. As data we have (1) used the

photoelectric photometry of Johnson (1957) and Sharpless

(1962) when available, and (2) computed mV and mB for the







21


remaining stars from measurements on the iris photometer

of two pairs of blue and yellow plates, each pair taken

simultaneously especially for the purpose, and using the

Johnson and Sharpless stars as standards. The details and

final results of this photometry will be published else-

where; the colors given here should be considered

preliminary.
















SECTION III
THE PLATE MATERIAL



General Survey

In this sub-section the whole of the plate material

is summarized. The next two sub-sections deal with the USF

plates particularly and with the plate measurements carried

out in the present program.

The plate material (that is, the x,y coordinates of

the images of the program stars as measured on the photo-

graphic plates) used in this study is very heterogeneous.

This is so partly by necessity and partly by design. There

exists no one series of plates, taken at one telescope,

having at once all four of these properties: (a) epoch

difference of at least 30 years, (b) plate scale of at

least (say) 20u/", (c) inclusion of an adequate number of

reference stars, and (d) measurable images of the inner

cluster stars (i.e. stars within 5' of the Trapezium).

Thus, the Yerkes plates, for example, bear images of only

seven reference stars; the Allegheny plates are obscured

at the center by over-exposed nebulosity, while on the

McCormick plates the central region lacks all but the

brightest stars due to the use of a sector wheel. The

U.S.F. and Sproul plates avoid all these difficulties, but











have no early epoch counterparts. As for the possibility

of repeating the AC type "astrographic camera" photography

of this region with modern plates from the same telescopes,

this is not feasible in the present instance because the

San Fernando plates covering the region had an exception-

ally bright limiting magnitude (.llm), while the early

epoch "Zo-Se"plates (of excellent quality) cannot be re-

peated because the telescope with which they were taken has

disappeared. In any case, the focal length of the astro-

graphic cameras is woefully short--about one-third that of

a typical long focus parallaxx" instrument such as the

McCormick refractor.

It is regrettable that an objective grating was not

employed in any of the early epoch photography.

The plate material is summarized in Table 2. Explana-

tory notes and references are provided in the caption. A

list of the plate numbers of the individual plates used, in

the notation of their respective observatories, is given in

the Appendix (Table A-l). The information concerning the

circumstances of the USF photography, since it has not

appeared elsewhere, is also listed in the Appendix

(Table A-2).


The USF Photography


Telescope

A description of the astrometric reflector of the











TABLE 2
SUMMARY OF PLATE MATERIAL


Plate No. of
Scale b No. of No. of Ref.e f
Obs (I/") Epoch Plates Starsc Stars Msmt Source

S F 17 1893,1906 3 130 85 M 1
Z-S 17 1904-1916 5 M 2
Y 93 1905-1912 7 92 7 M 3
A 68 1921 6 70 15 M,A 4
McC 48 1924-1927 22 47 13 A 5
McC 48 1968 22 47 13 A 5
A 68 1970 5 70 15 A,M 4
Sp 53 1974 2 90 14 A 4
USF 50 1974 20 140 35 A,F 4

Name of observatory: S F = San Fernando, Spain; Zo-Se,
also "Zi-Ka-Wei," Shanghai, China; Y = Yerkes; McC = McCormick;
A = Allegheny; Sp = Sproul; USF = Observatory of University of
South Florida, Tampa.

Epoch: approximate mean value, or range.

CNo. of stars: number of program stars appearing on at
least 2 plates.

dNo. of ref. stars: number of reference stars with mea-
surable images on at least 2 plates. (The components of e1
and 82 are counted here as only two ref. stars.)

eMsmt: Type of measuring engine--M manual; A automatic
(impersonal); F fringe-counting (screw-independent).
f
Source: source of measurements as follows:
(1) Astrographic Catalogue, San Fernando Section, 4 (-5.
zone), p. 50 (plate 1496); and 5 (-6 zone), p. 50 (plates
1490, 3890).
(2) Chevalier (1933). The listing for plate "2" con-
tains many star misidentifications discovered and, where pos-
sible, rectified by this author. There are also several typo-
graphical errors in the measurements. This photography has
no relation to the Z6-Se equatorial zone, published shortly
before in the same series of publications. In general, all
of the photography and measurements carried out under the
supervision of Chevalier at Zo-Se were meticulously and in-
geniously done so as to yield very high accuracy.
(3) Strand (1958, 1972) for description; the measure-
ments on punch cards provided by Dr. Strand.











Table 2--continued

(4) Present investigation. Measurements made in 1973
and 1974, described below.
(5) Cannell (1970) for a description; the measurements
on punched cards provided by Mr. Cannell. Both the early
and late epoch plates were measured by him in 1970.



University of South Florida (USF) Observatory has not yet

appeared in the literature. The telescope is a 26-inch

Tinsley, f/15 "Schmidt-Cassegrain" with optics designed by

J. G. Baker. These are a full-aperture corrector plate

(not a "reflector-corrector"),an f/5 nearly spherical

CERVIT primary, and an approximately spherical secondary in

a Cassegrain configuration. This system has also been

termed a "Baker-Schmidt." The unobstructed field is 172.

Stellar images are sharp and round up to the edge of the

field, exhibiting no trace of coma.

As is the case with systems of the Cassegrain type,

image quality is critically sensitive to the collimation

of the elements. With the USF telescope, proper collima-

tion is achieved only with difficulty; moreover, the mirror

supports are such that collimation cannot be maintained for

more than a few weeks at a time. Consequently, the condi-

tions under which the plates used for this investigation

were obtained cannot with certainty be duplicated at a

later time, and such instrumental plate constants which

might otherwise have remained constant over time cannot be

assumed to have done so here.











The USF Observatory is located within an urban area

and suffers from a sky brightness that sets the limiting

magnitude at about 14m on yellow plates.


Coverage

The format of the USF plates is 5 x 7 inches. Cen-

tered at the Trapezium, this format encompasses only about

one dozen reference stars. In order to utilize more ref-

erence stars we must extend the field by overlapping adja-

cent plates. The overlap pattern chosen, shown in Figure 2,

was dictated by both the distribution of the reference

stars--which is far from uniform--and limitations of eco-

nomy. It is seen that there are very few reference stars

east and especially west of the cluster--a circular area

one full degree in diameter centered about 1/20 west of the

Trapezium contains but two reference stars. The distribu-

tion of plates is center-to-corner, with the Trapezium ap-

pearing on nearly all the plates; the region covered is

roughly co-extensive with the "Sword" of Orion, encompass-

ing a total of 39 reference stars. Sixteen of these plates

were used in this study.


Exposures

Photography of the Orion Trapezium cluster presents

several challenges. First, of course, is the nebulosity

itself. On the plate, the background light of the nebula

blots out the fainter stars imbedded in it, while











condensations of the nebula adjacent to stellar images can

shift the apparent photo-centers of those images. Second,

the magnitude range of the program stars is at least seven

magnitudes. Within the vicinity of the Trapezium espe-

cially, bright and faint stars are closely intermingled,

precluding the use of a sector wheel. An objective grating

can only be used with great care, lest the multitude of

images of the different stars impinge upon one another.

We have sought to circumvent these difficulties with

the following:

1. Choice of filter and emulsion. All exposures were

made on 103a-G emulsion, taken through a Corning 3-69 fil-

ter. The resultant combination isolates a 600 A-wide

region free of major nebular emission lines. The suit-

ability of this combination is revealed in Figure 1, in

which the nebulosity is clearly suppressed.

The filter can be expected to introduce some distor-

tion of the field. In order to counteract this, we posi-

tion the filter immediately in front of the plate, accord-

ing to standard practice, the spacing being 3mm. In addi-

tion, we have taken the unusual precaution of calibrating

the filter, as described below under "Measurement Process-

ing." To facilitate in applying this calibration, a ref-

erence mark is scribed on the filter so as to be sil-

houeted on plates exposed through it.











2. Objective grating. A coarse 26-bar objective

grating was constructed and employed in some of the expo-

sures. The direction of the bars was rotated until the

diffraction patterns of the Trapezium stars all avoided one

another--this proving a most delicate adjustment. The

grating constants were found by experiment to be 4.4, 4.7,

5.0, 6.3, respectively for the first-through 4th-order

images. (In retrospect, it seems probable that a more

evenly graded set of constants would have been more useful.)

3. Multiple exposures. As an alternative to the use

of the grating, some plates were taken with both a long and

a short exposure. An x,y translation in the focal plane of

less than 5 mm was found such that the images of one expo-

sure would not fall on any of those of the other. Exposure

times of 10min and 1min were usually made, separated in time

by only a few seconds while the plate holder was advanced

in the camera by the pre-set distance. Some of the plates

taken with the grating were also given two exposures.

Photographs were taken on either side of the pier,

usually at less than 30 min. from the meridian but occa-

sionally more. Exposure times were usually 9 minutes,

reaching 13m. Particulars are recorded in Table A-2.

All plates were kept horizontal during developing and

drying to reduce emulsion shifts. For the same reason,

washing time was reduced to a bare minimum even at the ex-

pense of incurring cosmetic blemishes, while drying time











was reduced by using a "drying box" with circulating warm

air.


Measuring the Plates


General Consideration

As noted above, measurements for some of the plates

used in the present study were already available. The

other plates were measured in the course of this investiga-

tion. The latter are 20 USF plates, 12 Allegheny plates,

and two Sproul plates.

Ideally, one of the automatic measuring engines, such

as the SAMM (Strand Automatic Measuring Machine) of the

U.S. Naval Observatory, would have been used for the mea-

surement of all plates because of both its impersonal cen-

tering action and much greater speed. In practice, we

found it is not always possible to use such a machine, for

the following reasons. (The limitations referred to here

apply specifically to the SAMM, but similar limitations

apply as well to the other automatic machines.) The effec-

tive scanning area perceived by the SAMM centering mechan-

ism is a circle of 250p diameter. On the SAMM it is impos-

sible to properly measure an image which extends beyond

this circle, or which is not isolated within the circle

which happens, for instance, when a part of another image

also falls within the circle; or when the background varies

measurably across the circle's area or is darker than










density .5. These limitations are often exceeded in the

Orion nebula photography, due to the crowding together of

grating images, the presence of the nebulosity itself, or

(in the case of some of the USF photography), the presence

of sky fogging. The large range of magnitudes encountered

in our program also has the result that, on plates taken

without a grating, the brighter stars (these include nearly

all the reference stars, which must be measured) produce

images with diameters larger than the critical 250p. This

phenomenon is especially at its worst on the early

Allegheny plates which were taken without the filter re-

quired to compensate for the difference in spectral regions

for which the chromatic aberration of the objective is cor-

rected and that to which the plates were sensitive.

The plates were inspected visually to judge the degree

to which these effects were present and each plate was

accordingly assigned to be measured either on the SAMM or

on a manual machine. Some plates were measured in both

modes--automatic for the smaller images and manual for all

the images. A brief account of the respective measuring

procedures follows.


Measuring Procedures

The automatic machine. The SAMM of the U.S. Naval

Observatory was made available through the courtesy of

Dr. P. M. Routly in 1973 and in 1974 and used for the











measurement of all these plates which could be measured on

it. This machine has been described elsewhere (Strand,

1971). Measurement sets were made in "direct" and "re-

verse" orientations and combined by a least-squares adjust-

ment (hereafter called a "rotation") solving for transla-

tion, rotation, and a magnitude term, just as is conven-

tionally done for manually made measurements. The resid-

uals (direct minus final) from these adjustments have dis-

persions of about 1.5 microns.

The calibration corrections given by the USNO for this

machine were applied to the measurements made in 1973. It

was found that the measurements made in 1974, after a series

of adjustments had been made to the SAMM, were better

fitted direct to reverse without the corrections, so the

corrections were not applied to them. At most, the cor-

rections amount to 2 microns.

The "fringe-counting" machine. Also at the USNO,

there is a conventional Mann two-screw measuring machine

which has been modified so that the stage travel is mea-

sured by Moire fringe-counters, as in many automatic ma-

chines, while the transport and centering remain unchanged

(hence this machine is called a "screw-independent" manual

machine). About half of the USF plates were measured on

this machine; the Allegheny plates are too large to fit on

its stage. The temperature at the machine could be mea-

sured but not held constant; therefore, measuring was










halted when the temperature varied by more than 15C. A

calibration of the machine that had been made at the USNO

shows that the corrections are smaller than one micron,

therefore, none were applied. Four settings were made on

each image in both 0" and 1800 orientations, and the two

orientations combined as described above. After the fit,

the rms error of the effective combined measurements is

1.75p.

Manual measuring machine. A conventional two-screw

Mann engine at USF was used to measure most of the Allegheny

places and a few of the USF plates. The temperature of the

machine remained within a 10C range during operation. Four

settings per image were made in each of four orientations:

0, 90", 180*, and 2700, thus giving two pairs of direct

and reverse measurement sets. Each direct-reverse pair was

"rotated" as above, then the two resulting combined sets

also rotated one into the other. The dispersion of the

residuals after adjustment confirms the value of measuring

four orientations: for each initial rotation the rms error

is about 2s microns, while for the final rotation, it is

1.5 microns. Thus the two-fold increase of information

contained in four as opposed to two orientations is com-

pletely effective in reducing the mean error of the final

combined measured coordinates.

Automatic and manual machines used jointly. The

plates measured on both the SAMM and manual machines were










treated as follows. At the SAMM, all measurable images

were measured, in direct and reverse positions, and the

measurement sets rotated and combined as for the other

plates. At the USF machine all the images were measured,

and the measurements were likewise rotated as above. Then

the combined "manual" set was rotated to fit the corres-

ponding "automatic" set, using as the transformation terms

translation, rotation, and scale. The rms error of this

adjustment was typically 2p. Finally, the transformation

parameters were applied to the remaining manual measure-

ments, producing a presumably homogeneous set of measure-

ments for all the images. We recognize that this procedure

of combining the measurements from two machines is not

fully rigorous. This is so because the adjustment para-

meters themselves contain some error, and this error, which

would not be present if the sets were not combined, infects

the manually-made measurements. The rigorous procedure

would be to segregate the two sets, and during the plate

reductions allow separate translation, rotation and scale

terms to be associated with each set. Since, however, at

least 20 stars common to both the SAMM and the manual mea-

surements, distributed uniformly over the plate, were

available in all cases for the adjustment, we feel no

errors greater than 1 micron have been introduced by hav-

ing adopted this procedure.











Preliminary Processing

It will suffice here to give just a brief outline of

the subsequent processing of the measurements. We have

taken into consideration the following:

Filter calibration. It should be anticipated that a

filter may shift the positions of the images as they appear

on the plate, since the filter may well depart enough from

optical flatness to introduce a measurable and non-uniform

distortion of the field. Such a distortion would not ap-

pear as a degradation in image quality, nor necessarily be

revealed at any step in the reductions. We control the

problem by "calibrating" the filters used at the USF Ob-

servatory. This is done by placing on the stage of the

measuring engine a grid and over it the filter in exactly

the same configuration as the photographic plate and filter

occur at the telescope, then measuring the coordinates of

the grid line intersections with and without the filter in

place. The shifts in the coordinates of the grid then pre-

sumably duplicate the distortion produced by the filter at

the telescope. The corrections are easily applied by mea-

suring the image of the filter reference mark on the plates

(see "USF Photography" above) at the same time the star

images are measured. The filter "Y" used for the present

USF photography proved to require no corrections even as

large as 1 micron.











No calibration has been made of the filters used for

the other plates in our study, i.e. the McCormick, later

Allegheny and Sproul plates. It is recognized that system-

atic errors of position and motion may remain by failing to

have done so, especially if different filters were used for

the early and late plates at the same telescope. In any

case, the filter used for the early McCormick plates was

broken shortly after the plates were taken.

Grating images. Where higher-order diffraction images

appear, the mean was taken of each pair of corresponding

images and subsequently treated as a single measurement for

that star, with its magnitude taken as the magnitude of the

star plus the appropriate grating constant. It can be

shown that second-order non-linearity of the plate scale

over a one-degree field introduces through this practice an

error of less than .Ol. The various orders were not

averaged together.

The dispersion of the differences between the central

image and/or means of the higher-order image pairs is 3.3p

for both X and Y for the USF plates and 4.1 for the Sproul

plates.

Multiple exposures. For plates with two (or three)

exposures, the average translation (AX,AY) between the

first and the second (and third) exposures was found. A

linear dependence of (AX,AY) on magnitude, as might arise

from a difference in magnitude term between exposures, was










allowed for. The values ( X, Y) were applied to all the

second (and third) exposure images, whether or not a first

exposure image was present, to form one homogeneous set of

measurements. If a small rotation between the exposures

also is present, that rotation will be incorporated into

the final meaned coordinates with no detrimental result;

however, a systematic rotation of the set of single expo-

sures with respect to the set of means of two exposures will

be introduced. This procedure is rigorously valid if only

the plate was moved between exposures. If the tangential

point was also shifted (as in re-centering the guide star)

an error is introduced amounting to 1~ over a 10 field only

when the shift exceeds 5 mm.

On two USF plates, the shorter exposures appeared to

be somewhat elongated. In these cases, the separate expo-

sures were treated as though they were separate plates.

Several McCormick plates also bore two exposures each; these

were all treated as separate plates.

Refraction and aberration. Since the apparent posi-

tions of the reference stars are altered by refraction and

aberration just as are the positions of the other program

stars, it is only the differential values of these effects,

across our 20 region, which need concern us. Moreover,

since our plate reduction models include linear and quad-

ratic terms with coefficients to be determined separately

for X and Y, it is only third-order differential refraction







37


and aberration which remain explicit to be allowed

for.

The largest zenith distance at which any of the plates

was taken is 45. At this value the maximum second-order

differential refraction is '.016; the third-order differen-

tial is less than "001. Third-order differential abbera-

tion is also less than "001. Consequently, corrections for

refraction and aberration have not been applied.
















SECTION IV
FORMULATION OF THE PROBLEM



The method used for the computation of positions and

motions in this study falls into that general category

termed "plate overlap" methods. In presenting our formula-

tion, we will develop it from the context of earlier me-

thods, beginning with the simplest, so that points of de-

parture are more clearly seen.


Differential Plate Measurements

The most direct computation of proper motion is simply

the comparison of the coordinates of the same star as it

appears on plates taken with the same telescope at differ-

ent epochs. The differences, after terms in arbitrary

translation rotation, and scale have been removed by a

least-squares adjustment, yield the motions directly. We

may write this formulation as follows. Let (x,y) be the

coordinates of a star on any plate, and xo, yo be initial

coordinates of the star, say the coordinates from one of

the plates chosen as a standard, at t = 0. For convenience

we can write x' ,y for the components of p parallel to the

axes of x y multiplied by the focal length. Then the

coordinates and motions for a star on the m-th plate are











related by observation equations involving a function f of
m
parameters (p ) and epochs tm for that plate:


fm(x,y,p) (P1+P2x+P3Y)m = x+(t +Ax) Ex -+r. (1)


For brevity, we let f stand for the vector of two equa-
m
tions, and p, x Ax, and r stand respectively for the

vectors (px,py), (XoYo), etc. The parameters (p ) are

found by a least-squares adjustment for each of the M

plates. The term in parentheses on the right side repre-

sents the "residual" for each star; it contains a random

error component and a component due to the proper motion.

The computation of the p from the M residuals can be done

rigorously by writing


ut = r (2)
m m

and solving for 1 by a least-squares adjustment. It is

more customary to combine the material in pairs of early

and late plates and compute the mean of the individual prop-

er motions, weighted according to the individual time

interval. It is to be noted that this will not yield

exactly the same value for p as will the least-squares re-

duction, unless the ranges of early and late epochs sepa-

rately are small compared to the whole range of epochs.

The great advantage of this method (apart from its

simplicity) is that as long as the plate centers are











approximately the same (a condition usually met in prac-

tice), the higher-order terms in the imaging properties of

the telescope do not appear. This means that the corres-

ponding errors in those terms do not appear in the errors

of the proper motions; and, since such plate constant

errors increase rapidly toward the edge of the plate

(Eichhorn and Williams, 1963), the proper motions as de-

rived by plate differences are equally accurate out to the

edge of the field, at least in comparison with proper mo-

tions derived through classical plate reduction. (Of

course, one may formulate the problem this way even when

the plates have been obtained with different telescopes,

but then the need for higher-order terms in the observation

equations vitiates the main advantage of the method.)

The disadvantage is that the terms in translation,

rotation, and scale absorb any net constant proper motion,

rotation or expansion of the stellar group itself. An at-

tempt is sometimes made to pin down these terms by imposing

external constraints: the change in plate scale, for exam-

ple, can be monitored by a field of known proper motions,

photographed at the epochs of the earliest and latest

plates. Again, the rotation might be monitored by trailing

some of the plates. The trouble with these devices is that

they do not really eliminate the problem. As long as any

factors which might give rise to rotation and scale terms,

such as differences in refraction, plate tilt, measuring










engine parameters, etc., remain unknown, adjustment para-

meters for them must still be included in the reduction.

An example of the application of this method is

Strand's (1958) study of the Trapezium cluster. As ex-

pected, the internal dispersion of proper motions is very

small (except as noted above), while his value for the

expansion has been subject to question.


Reference Proper Motions

The disadvantages mentioned in the preceding section

may be eliminated while retaining the advantages of a dif-

ferential method, by making use of reference proper motions

(not positions) when these are available. Then we have, in

the same notation as above, a set of observation equations

for the m-th plate,


fm(x,y pm) = (xo+ptm) + x (3)


where now pi is known for some of the stars. Again a least-

squares adjustment for the p's is made for each of the M

plates. Then, for each of the remaining stars on the m-th

plate,


Pm = (f ,yp)-Xo)/tm


(4)











In order to take full advantage of the material, one should

then treat these derived p's as known values in (3) and

iterate.

This procedure is still subject to the restriction

that all plates be taken with the same telescope and with

the same plate centers, and that there be a sufficient num-

ber of stars in the field with known proper motions to

determine the six adjustment constants.


Classical Plate Reduction

We can depart from the requirement that all plates be

taken with the same telescope with stable optical proper-

ties and with the same plate centers (termed together "pro-

jection conditions") by referring the measured coordinates

to a system of reference star positions instead of to one

another. Let the spherical coordinates of the stars pro-

jected onto a plane approximately tangent to the plate cen-

ter, the conventional "standard coordinates," be (E,n).

Then the observation equation relating these to the mea-

sured coordinates is, following the notation used above,


fm(S+pxtm, n+ ytm {p}) = x +Ax (5)


where f is in general some polynomial in &, n, and

magnitude. The f is now complicated by the appearance of

terms expressing tilt, coma, radial distortion, etc. (See,

for example, Konig, 1962 and Eichhorn, 1963 for general










discussions.) (Here, we have assumed that t and n vary

linearly with time; this is an approximation valid for the

small changes involved.)

These equations are solved for the {p}m by separate

least-squares adjustments for each plate. Then for each of

the non-reference stars we have, for each plate the vector

equation (where E now stands for the vector (5,n)),
-1
Cm = f (x)
m m


from which the proper motions are computed by solving for

each star the set of m equations


&0 + ntm = m


There are two advantages to the plate reduction method

over the differential methods. First, it does not require

that the plates have been taken with the same telescope and

plate centers. This in itself is the decisive factor when

the plate material is unavoidably heterogeneous, as in the

present program. Second, the plate reduction yields actual

star positions, in addition to the proper motions.

The disadvantage, as stated above, lies in the neces-

sity of including higher terms in the observation equation.

The presence of these terms necessarily increases the total

error due to all the parameters. Consequently, star posi-

tions which are computed from them contain large systematic

errors, especially toward the edge of the plate (Eichhorn











& Williams, 1963) and also in the spaces between widely

separated reference stars. This type of error ("parameter

error") is not revealed by the adjustment residuals of the

reference stars, but only by comparing the positions of

field stars obtained from several plates with an indepen-

dent position, or with each other. The more closely dupli-

cated are the circumstances of the plates, the less apparent

are these parameter errors.

Another weakness of the conventional plate reduction

procedure, implicit in equation (5), is the assumption that

only the plate measurements are subject to error. In fact,

however, the reference star positions are subject to error,

too. For example, a position error of '.3 (not at all

atypical for the Yale Catalogue) corresponds at the focal

plane of a typical parallaxx" telescope, to 15p--which is

far larger than the expected measuring error.

One might therefore be tempted to treat each plate as

a separate adjustment. Approximately, this can be accom-

plished by assigning the x and 5 relative weights according

to some assumed ox, a All that this accomplishes, how-

ever, is to divide each residual into a "star" component and

a "measurement" component in the ratio a /ax, effecting a

mainly spurious reduction in the measurement residuals.

Instead, it is much more important to enforce the condition

that the star position correction for a given star be a

single value, not a set of values for each plate on which










that star appears. It is the rigorous expression of this

condition which prompted the development of the plate over-

lap method treated below.

Among the proper motion investigations employing the

classical plate reduction method there are two of the Orion

Nebula cluster: Meurers and Sandmann (1963) and (for some

of the material) Parenago (1954).


Differential Overlap

We pause to remark here that the overlap condition can

be applied in a differential formulation as well.

In the notation used above, we have for the 1st star

on the firbt and m-th plates


fl(xl1,yl,{)1) t1 1 = xI + tl 1 + Ax (6a)

fm(x1,Ylml) + tmAl = + tm 1 + Ax

t Au = 0 + t AP1 (6b)


and for the i-th star on the m-th plate


f (xi,yi,{p} ) + t Ai = x + t i + Ax

t mAi = 0 + t Ami


Again we compute the {p}m and Aj for the reference stars by

a least-squares adjustment and by successive iterations, the

p's of the field stars. In forming the normal equations,

the conditions (6b) are weighted with respect to the










2 2
observations (6a) by a factor a (x)/2 (p) chosen a priori.

The matrix of the normal equations can be reduced to

dimension equal to the number of Ap, as described in the

section "plate overlap." The Ap will be distributed nor-

mally with mean zero and dispersion = o(i)/o(x) times the

dispersion in Ax.

Of all the formulations, this should give the most

accurate proper motions; but to be applicable, the plate

material must be homogeneous and there must also be an ade-

quate number of stars with known proper motion within the

field of view. These conditions are not satisfied in the

present program.

To the author's knowledge this formulation has never

before appeared in the literature.


Plate Overlap Method


The Observation Equations

We now write the observation equations in their most

general form. In order to illustrate the pattern more

clearly, we give here the equations for a reference star

(the first star) and a field star (the i-th star) as they

occur on the first and m-th plates.










fl(l+t1 l + F1 I+Flt lA1 =(xl+Ax1)F1 (7a)

f (i+tl,1P1p} + F1~Ai+tl11 =(xi+Axi)F (7b)

f (+ p + F AE +F t AV =(x +Ax)F(
m( t1 m x m m lmm1 1 1
fm(Ci+tm ix' p m + FmA i+F+M i= (x +Ax )Fm



+ A1 = O+A1 (7c)

tAp1 = O+tAu (7d)


where for brevity f(,{p)) stands for two functions, i.e.,

is a 1 x 2 matrix, and x,p,E,{p} stand for the vectors

(x,y), (Pxy ), ( (pP2...) respectively. Fm is the

reciprocal of the assumed focal length in the same units as

the x, of the telescope with which the m-th plate was

taken. (While the focal lengths are, in effect, among the

parameters being adjusted, only very approximate values are

needed for this role and they need not be revised in sub-

sequent iterations.) The C and p are in units of radians

and radians/yr., respectively. Carets denote the values

used in the initial approximation.

The tm are the plate epochs in years now measured from

the reference catalogue epoch, in the case of reference

stars, and from some arbitrary epoch in the case of field

stars. We have incorporated the approximation x(t) =
ax
--- t = Fi t, fy t into the terms involving p.
ac x y
In the right hand colum appear the observed quanti-

ties. In the equations (7c, 7d) representing the reference











positions and motions, the observed quantities are the de-

partures of the initial values of the reference position

or motions from the assumed values; these are equal to

zero. The A appearing in the right column symbolize the

errors of the observations; on the left side the A are

unknowns. The "Ap" of the field stars are, ordinarily,

the p themselves. Of course there must be at least two ob-

servations at sufficiently separated t for each field star

whose u is to be determined. If this condition is not met

for a particular star, we merely eliminate its p from the

equations.

In order to handle the entire set of equations by the

conventional least-squares algorithm, the equations are

scaled so that the scaled A are members of one normal dis-

tribution with a single standard deviation a Here this
u

standard deviation is measured in radians, hence the factor

F in the equations (7a). The proper motion equations (7c)

are each scaled by t, a time value such that to = o
u u
Finally, the non-uniformity of the a of the various cata-

logues and of the plate measurements is accounted for in

the usual manner by multiplying each equation by the appro-

priate value a /a(=/-w-).


The Normal Equations

The matrix of the resulting system of normal equations

is a single large array of dimension equal to the total










number of unknowns--rather than, as in the classical re-

duction, a set of matrices for each plate of dimension equal

only to the number of plate constants per plate. This total

number of unknowns is very large for the number of plates

and stars involved in a typical proper motion study; the

program which is the subject of this paper, involving 88

plates and 130 stars, leads to some 2,000 unknowns. Clearly

this is an unmanageably large dimension. We therefore have

taken steps to reduce the size of the largest matrix which

must be inverted.

First, we may always align the plates in the measuring

machine so that the instrumental x-axis is nearly parallel

to the F axis. Then


x = F(A + t a(.nt)= F(A + tV )
a( n,n t) 5 (Cnt) n


Thus the matrix E consists of elements of dimension 2 x 2

rather than 4 x 4.

Second, we assume the plate constants p appearing in

the equation f(C,n,{p)) = x to be independent of those in

f'(Cn,{p'}) = y. This means that the normal equations

involving the unknowns p, AC, Apx are decoupled from those

in p', An, Aly leaving two independent sets of mormals each

of half the rank of the original.

The system of normal equations may be represented

schematically as follows. Let pm represent submatrices due











only to coefficients of p (plate constants of m-th plate);

E represent the matrix due only to coefficients of star un-

knowns (lA,Ap): this is a block diagonal of 2 x 2 blocks;

C, the matrix of cross-terms, and V (i) the vector of plate

constant coefficients for the i-th star, on the m-th plate.

On the right side, the vector L consists of the homologous

vectors of the classical reduction normals, strung together,

while the elements of x are given by x = Wm Fm Xmi



0 0



0Q



C- L 1



In an earlier application of the overlap method, Googe

(1967) showed that the system of normals can be reduced in

dimension to only the total number of plate constants by

"folding," that is, by eliminating E and filling up the

entire block containing the P (In that application E was
m

a long diagonal matrix of single elements.) For the task

at hand, we instead eliminate the P, folding the matrix

into E and producing a solid block of dimension equal only

to n x n, n being the number of stars. Since n = 130 (as

opposed to about ten times that number of plate constants),










we are left with a matrix that can be easily handled and

inverted with present computer capacities.

These reduced matrices, S and T, can be formed in

place without the need of auxiliary matrices. They are

simply


S = E C 1 C
1 11C


T =E C 1 T
2 2 2 C2


where the subscripts merely refer to the homologous counter-

parts in the matrices of the original normal equations.

It can be seen that the matrix C is composed of strings

of vectors such that the i-th row of C is

V ,(i), V2,p(i), ... V ,p(i). Thus, the elements of S

(and similarly of T) can be written


M
S. = w.(6 ) I V (i)P V (j)
S1,i = (ij) m Vm(i) -Vm
m


with a similar equation for T .. The Kroenecker delta

denotes the original diagonal, Vm(i), is, again, the vector

of plate constant coefficients for the m-th plate and i-th

star, and wi are the star weights. For simplicity we have

ignored the proper motion components of S. Thus the entire

array can be formed from one plate at a time and built up

in layers. No more data need be stored at a time than the
-1
measurements for a single plate, the inverse P of the
m







52



matrix of normal equations generated by the single plate m,

and the arrays S and T with the updated sums as their ele-

ments. Only the upper halves of S and T need be maintained,

since they are symmetric. Another convenient feature of

the formulation is that the data generated by new plates

may be added as they become available without the necessity

of completely recomputing the already existing S and T.
















SECTION V
THE COMPUTATIONS



Partition of the Plate Material and Treatment
of the Proper Motion

The plate material naturally divides itself into three

groups: an early (,1900 group (A) of wide-field photo-

graphy; a later (Z1905-1927) group (B) of long-focus photo-

graphy; and the recent group (1968-1974) of long-focus

photography. We have taken advantage of this natural group-

ing to effect a certain measure of simplification. The

procedure adopted is (1) Reduction of the recent group in

a single overlap, without regard to the proper motion un-

knowns, using as initial values for the field stars the

positions calculated from plate constants obtained for a

subset of these plates reduced using reference stars only;

(2) Reduction of the early group as a single overlap, also

without regard to the unknown p; (3) Calculation of a

"secondary catalogue" of a, 6, a' 1 and means epochs for

the field stars, using the results from steps (1) and (2);

(4) Reduction of group (B) using this "catalogue" of field

stars as reference material of lower weight, in addition to

the original catalogue of positions and motions; (5) Cal-

culation of new proper motions of the field stars from the











solutions of steps (1) and (4); and (6) interaction of

steps (4) and (5), resulting in the final values of the

proper motions. This procedure is justified by the narrow-

ness of the range of epochs in groups (A) and (C), while

the 22-year spread of group (B) is adequately compensated

for by the use of the approximate proper motions for its

field stars.

The role played by the Astrographic Catalogue and

Zo-Se plates is in effect to provide starting values, well

determined with regard to scale and introduced systematic

errors, for the positions and motions of the field stars

appearing on the Yerkes and McCormick plates--these latter

plates containing few reference stars. The first-named

plates are well suited for this role because they cover a

wide area including many reference stars. (See Table 2.)

The relatively small scale of these plates is no great dis-

advantage here since high internal accuracy is not essen-

tial for the starting values. (We recall that these ini-

tial values are treated as reference material, not merely

as starting values.)

In contrast, the late-epoch photography (group (C)),

being overlapped, contains a sufficient number of reference

stars to allow a straightforward reduction without the need

for a "secondary" catalogue (but see below, "Stability of

the Solution").










Behavior of the Solution

We need not dwell on the routine mechanics of the

calculations. We are concerned here with properties of the

solution which are only revealed when the calculations are

actually made. To the extent that such properties are not

completely predictable in advance, we may say that, in

effect, they reflect the stability of the solution. It

ought to be remarked at this point that there has not pre-

viously appeared a plate overlap reduction of the general-

ity which obtains here, wherein the number of unknowns

greatly exceeds the number of reference stars as is typ-

ically the case in proper motion studies. The following

aspects of the behavior require special mention.


Iterations

A solution of the normal equation yields a set of star

position corrections, and plate constants obtained from the

initial positions. One therefore replaces the initial star

positions for the revised values and iterates the solution

until all the star position corrections are less than some

value. It was found that convergence is quite slow, re-

quiring about seven iterations for convergence to within

'.'02 for all the star positions. Convergence is more rapid

when fewer field stars are included as unknowns. If no

field stars are included, the solution convergences with

only two iterations.











Influence of Weighting

It was found that the solution is rather sensitive to

relative weights assigned to the several source catalogues.

If too high a weight is assigned to a star, the residuals

for that star from the various plates on which it appears

reveal a consistent error. Thus it was found that the

accuracy of positions and motions from the FK4 and N30 is

lower than the quoted mean errors for those catalogues.

If too low a weight is assigned to a star, there is

little effect if the catalogue value is very close to the

true value. If the initial position is incorrect by, say,

0?4, however, the reduction will over-correct its position.

Moreover, it can be argued intuitively that if a reference

star occurs in a relatively isolated region, the consequence

of assigning its position too low a weight is to allow its

position to adjust by too far, without the over-adjustment

being evident in the residuals. This is expected because

the plate constants are freer to absorb an error in a star

position if there are no surrounding stars to help con-

strain the values of the plate constants evaluated in that

region. Indeed, it was found by experiment that when the

position of one reference star, 15' from its nearest neigh-

bor, was allowed to vary over a range of 078, the residuals

for that star varied by no more than 'I! The implication

to be drawn from this experience is that it is preferable











to overestimate rather than underestimate the accuracy of

the catalogue positions.

The weight assigned to the plate measurements rela-

tive to the catalogue positions also were found to exert a

sensitive influence on the solution. We do not know in ad-

vance exactly the mean error to be assigned to the plate

measurements, especially those taken from the literature.

Within reasonable limits (4V to 8p), our choice of the mea-

surement mean errors is guided by the conjecture: given

several sets of observations each characterized by an inde-

pendent error distribution with some a, that set of a which

minimizes the variance of the adjustment made on the obser-

vations is the best obtainable estimate of the a.


Weighting of Field Stars

Since, by definition, the field stars have no a priori

positions, the initial values assigned to them have zero

weight; that is, the observation equations (7c)


i i = iA

corresponding to them reduce to


0 = 0.


It was found that, in practice, some non-zero weight

must in fact be assigned to the field star positions; other-

wise the solution is so ill-conditioned that successive











iterations rapidly diverge. This ill-conditioning worsens

as the number of field stars surpasses the number of refer-

ence stars, but improves with improvement in the initial

values.

We have therefore been compelled to assign a weight to

the initial values of the field star positions correspond-

ing to 1/10 that of the Yale positions at the same epoch.


Residuals

There is a residual for each star on each plate. The

residuals obtained from the final iterations have the fol-

lowing rms errors, calculated plate by plate, shown here

for each telescope. The values obtained from all the

plates except a few anomalous cases are included in the

ranges given.


TABLE 3
RMS ERRORS


Telescope rms

Yerkes .'09-'.'06

Allegheny .09-.07

McCormick .08-.06

USF .09-.04

San Fernando .40-.25

Zo-Se .35-.15










We also compute for each star the average of its resid-

uals for each plate. This mean residual, added to the

calculated position correction and the initial position,

represents the position for that star as calculated from

the final values of all the plate constants. It is this

value which is taken as the final calculated position of

the star at the corresponding mean epoch of the plates on

which it appears.


The Computer Program

The computer program with which these calculations

were carried out is written in Fortran IV. It is con-

tained in only about 800 statements and is conveniently

modular in structure. It utilizes two tracks of working

disk, but requires no tape storage. Taking advantage of

the cumulative character of the matrices of normal equa-

tions, the program accepts an indefinite number of plates,

with an arbitrary (up to the maximum value of n) number

of stars per plate, in arbirary order. The number n of

stars itself is completely arbitrary, within the limita-

tions of storage capacity. With core storage of 180K,

240K, and 300K, the corresponding maximum values of n are

80, 112, and 135. Were the proper motion explicitly in-

cluded, the corresponding values of n would be 1/2 of these

values. The catalogue of reference stars is treated as an

interchangeable data set. Copies of the program on cards

are available to interested persons on request.











The Plate Models

By plate model we mean the polynomial terms required

to transform the Standard Coordinates of the stars into

their (x,y) coordinates on the plate. Each telescope can

be expected, in general, to require its own model; finding

the proper model for a given telescope is a matter of

experiment.

For all plates, the terms whose coefficients are 1,
2 2
, r~, ,n (in the x-equation) and 1, E, n, n 2 n (in the

y-equation) are included, since these terms always arise

from the projection geometry. The plate models finally

adopted for each telescope include, in addition to the

above terms, the following.


TABLE 4
PLATE MODELS



Terms (and origin)
2 2 2 2
m xm, ym xr yr y ,x
Telescope (guiding) (coma) (radidist) (?)

USF X X

Allegheny X X

Yerkes X X X

McCormick X X

Zo-Se X X

San Fernando X X











No determination of terms involving color could be

made, even though such terms are undoubtedly present,

because the distribution of colors is too strongly cor-

related with magnitude and position. Most of the program

stars, while not members of the Trapezium cluster proper,

are members of the Orion aggregate; consequently, the bluer

stars are also brighter. Moreover, nearly all the red

stars are concentrated in the near vicinity of the

Trapezium. An attempt to model color terms in these cir-

cumstances could result in grave, undetectable systematic

errors.

It was found that the USF telescope shows neither sig-

nificant coma nor radial distortion; some plates show a

linear magnitude term in x or y. The Allegheny telescope

(Thaw refractor) also shows negligible coma, but does ex-

hibit some radial distortion; the earlier plates tend to

exhibit larger magnitude terms than do the recent plates.

The Zo-Se plates show surprisingly small higher-order

terms.

The McCormick plates show both linear magnitude terms

and large and variable coma, apparently amounting to about

lp/cm-mag. But the determination of the coma terms for

these plates is complicated by several factors: the stars

appearing on them, especially away from the center, have a

narrow range of magnitude, and they were taken without an

objective grating. Moreover there is a strong correlation







62


between the coma term and the magnitude term in declina-

tion. Rather than risk the introduction of unnecessary

parameter error, we finally pre-corrected the measurements

for the effect of coma, using for the purpose the coma

value found earlier by Eichhorn et al (1970) from photo-

graphy which was better suited than ours for its

determination.
















SECTION VI
RESULTS

The derived positions and proper motions, given for

orientation 1950, are listed in Table 5. The right ascen-

sions and declinations are given in decimals of degrees as

the best compromise between convenience of computation and

of identification. All the positions are given for epoch

1974.0, this being close to the actual epoch of the recent

plates. The weighted mean epoch of the recent plates is also

given for each star. The proper motion accuracies are cal-

culated from

02p = (V 2 + 02)/t ,
1 2
where ol and o2 are the mean errors of position at the mean

early and mean recent epochs. These in turn were calculated

simply from the average mean errors o of the n plates on

which the star appeared at each epoch according to


01,2 1,2 "

Absolute Proper Motion of the Trapezium

The average proper motion of the Trapezium Cluster is

a = '.010/yr, p. = 7001/yr

Part of this motion is due to secular parallax, which amounts

to (assuming a distance of 500 parsecs)

pa = +'001/yr, '6 = -.005/yr










The galactic rotation term in proper motion in the

direction of the Orion Nebula amounts to less than '0005/yr.


Velocity Dispersion


The most remarkable result emerging from the proper

motions is the very small value of the velocity dispersion.

This is given in Table 6 for various sub-groups of stars.

Here "o" denotes dispersion, not mean error.


Table 6


No. of
Group Radius Members a ("/yr) a ("/yr)



Trapezium 10" 4 .0010 .0009

81 and 92 2' 9 .0013 .0012

Trapezium Cluster 6' 43 .0015 .0014

Nebula Cluster 20' 93(a) .0036 .0026


(a): excludes stars assumed to be foreground objects.

The dispersion in proper motions for the inner members

of the cluster is so small as to call into question whether

any true motion is detectable at all. In any case it sets

a realistic upper limit to the true proper motion errors,

since random errors will act to increase the observed dis-

persion. It is believed that these are the smallest disper-

sions ever obtained for a group of stars.

Translating the proper motions into tangential velocities

at the assumed distance of 500 parsecs, we obtain a velocity










dispersion of 3 km/sec (in one coordinate) for the

Trapezium Cluster, 7 km/sec for the larger Nebula Clus-

ter. (It is worth recalling that the velocity dispersion

of the Pleiades, which is less than 1/3 the distance of

Orion, is about 1 km/sec.) We suspect that part of the

difference in the two figures is spurious, due to the increase

of error as one moves away from the region near the plate

centers and greatest concentration of stars. It is interes-

ting that Strand obtained an equally small proper motion dis-

persion for the larger Nebula Cluster, but a much larger

dispersion for the Trapezium and the Trapezium Cluster

('.008/yr., and .'004/yr., respectively). This is probably due

to the fact that on the Yerkes plates the nebula and the

brighter stars are overexposed.

Contraction

For each proper motion let the radial and tangential com-

ponents be p and y the radial distance R being reckoned
R T
from the Trapezium. Let the coordinates of the star measured

parallel to a and 6, with the Trapezium at the origin be A, D.

A linear expansion (or contraction) can be represented by a

coefficient k in units ("/year)/R(degrees). The expansion's

such that
contribution to R is an amount R, such that

Rk = u' u + Aip
R R R
Then in order to find k, we solve by least squares the

equations


Rik + (A/R)ila + (D/R)ia = (R)i + (R)i










where v., ia is the mean proper motion of the group. If

P yP are not known, a priori and if the distribution of

stars is non-uniform, they also must be treated as unknowns.

The value of k determined from our proper motion in the

Nebula Cluster, excluding the high proper motion star is

k = ('009 + .002/yr)/degree ;

that is, a contraction of the cluster amounting to '003/year

at its edge. This result was quite unexpected and prompted

a thorough search for sources of error that might have pro-

duced it as an artifact. As a weak check on the result we can

solve for contraction in a and in 6 separately (such "con-

tractions" having no physical significance considered

separately); it is found that the two coefficients are equal

within one standard deviation. It must be borne in mind that

the uncertainty of this result is greater than indicated by

its formal error. This is because the contraction coefficient

depends critically upon the proper motions of the stars at

the edge of the cluster (which in itself is indeed reflected

in the formal uncertainty of the result), while it is the posi-

tions and hence proper motions of precisely these stars which

are most afflicted (as pointed out above, p. 40) and to an

uncertain extent by the plate constants errors. In any event,

let us accept our result and examine its consequences.

Translated into linear velocity at the edge of the

Gerola-Sofia cloud, the R=4 parsecs, the contraction amounts

to 10 km/sec. This is in surprising agreement with the value

12 km/sec obtained by Gerola and Sofia for their model of the










cloud and lends strong support to their interpretation of the

cloud molecular line widths.

Quite independent of the contraction determination is

the evidence of the velocity dispersion in the Trapezium

Cluster. Since that result argues against the existence of

turbulent eddies with velocities much higher than 3 km/sec,

which seem to be required if the cloud is not contracting,

we have a double-barrelled support of the view that the Orion

Nebula is relatively quiescent, contracting under gravity,

and has not yet entered its most productive era of star

formation.

If the cluster is not expanding, the "expansion ages"

assigned to it in the post are invalid, and stars already

formed in the cluster cannot be assigned a single age on the

basis of kinematic considerations. The contraction removes

any discrepancy between the kinematic and other age determi-

nations, and allows for a continuous range of ages in the

member stars.












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APPENDIX:

DETAILED SUMMARY OF PLATE DATA










TABLE A-1
LIST OF PLATES USED IN THIS INVESTIGATION
(Numbers are in the system of the respective observatories)


Allegheny

15 572 (1919) 103 394 (1969)
15 573 103 512 (1970)
15 684 103 522 "
15 685 104 157 "
28 668 (1922) 104 295 (1971)
62 816 "

McCormick

17 148 (1924) 77 407 (all epochs 1968)
196 408
228 580
229 581
865 (1925) 624
866 625
966 650
967 651
18 905 698
906 699
922 726
923 727
19 480 (1926) 744
481 745
564 812
565 813
20 573 814
742 815
743 844
829 845
21 237 866
238 867

Yerkes

O 4 (1905) 0 18 (1906)
0 7 (1905) 0 30 (1907)
012 (1905) 0 52 (1908)
013 (1905) 0432 (1922)

z6-se


"Neb. d'Orion"a
1 (1902.913)


4 (1914.120)










TABLE A-l--continued


Zo-Se--continued

2 (1910.110) 5 (1916.069)
3 (1911.056)

San Fernando

1490 (1893)
1496 (1893)
3890 (1906)

University of South Florida

13b (all epochs 1974) 45
15 55
16b 57b
22 60
28b 63b
29b 65b
44

Sproul

T346b


aThe identification given in Chevalier (1933). The
times are given to three places to aid in identification.

Treated as two plates.


















DATA FOR


TABLE A-2
USF OBSERVATORY PLATES


Serial
No. Date Grating Exp. Center HA

13a 9 Mar 1974.189 7 min. 1 3 min.
2 + 5
15 9 Mar 1974.189 2 1 +156
16a 10 Mar 1974.192 12 1 + 66
2 + 67
22 14 Mar 1974.203 9 1 +103
28a 24 Mar 1974.230 2 9 1 +120
1.5 +125
29a 24 Mar 1974.230 9 1 +140
1.7 +146
44 17 Sep 1974.710 9 1 93
45 29 Sep 1974.743 8 1 35
55 17 Nov 1974.876 1.5 2 6
4.5 3
57a 17 Nov 1974.876 5 3 + 60
1 + 66
60 18 Nov 1974.879 1 2 12
6 7
63a 20 Nov 1974.885 2 9 3 4
1 + 1
65a 20 Nov 1974.885 2 4.5 1 + 78
1 + 81


Plate centers: 1 a
(Trapazium)


= 5 h. 48 min.,


= -5, 24'


2 30' N, 1 min. W of Trapezium
3 30' S, 1 min. E of Trapezium

Separate exposures reduced as separate plates.
















LIST OF REFERENCES


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Baade, W. and Minkowski, R. (1931). Astrophys. J. 86, 119.

Barney, I. (1945). Catalogue of the Positions and Proper
Motions of 8248 Stars Between dec's. -6' and
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Barney, I. (1945a). Catalogue of the Positions and Proper
Motions of 8108 Stars Between -2* and -6.
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Brosche, P., Nowacki, H., and Strobel, W. (1964). VerSff
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Cannell, W. D. (1970). M.A. thesis, University of
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79


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BIOGRAPHICAL SKETCH


Frederick W. Fallon was born and raised in Silver

Spring, Maryland, where he attended the public schools,

graduating from Wheaton High School. He took the A.B.

degree in Astronomy at Harvard in 1961. Subsequently, he

worked for seven years at the U.S. Army Map Service (now

Defense Mapping Agency) where he developed their program

for deriving geodetic positions from the photography of

artificial satellites. He also worked during summers as a

student trainee at the National Bureau of Standards at the

U.S. Naval Observatory.

Returning to graduate school, he attended the Univer-

sity of South Florida where in 1972 he took the M.A. degree

in Astronomy with a thesis on photometric evidence for an

atmosphere of la. This work earned an "Outstanding Grad-

uate Student Research Award" from the University of South

Florida chapter of the Sigma Xi.

He is currently on the faculty at the University of

South Florida, in Tampa, where he lives quietly. His spe-

cial areas of interest are astrometry and its application

to astrophysical problems, optics, and planetary atmos-

pheres. He is the author of some six papers in the profes-

sional literature on these topics. He is a member of the

American Astronomical Society and of Sigma Xi.

80











I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.




Sabatino Sofia, chairman
Professor of Astronomy


I certify that I have rea
opinion it conforms to accept
presentation and is fully ade wu
as a dissertation fpo.r-He degree,


and that in my
of s chol-a'rly "
pan-d quality,
io Philosophy.


/' (


Professor of Astronomy


I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.




Frank B. Wood
Professor of Astronomy











I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.




Stephen T. Gottesman
Associate Professor of Astronomy







I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.


This dissertation was submitted to the Graduate Faculty of
the Department of Astronomy in the College of Arts and
Sciences and to the Graduate Council, and was accepted as
partial fulfillment of the requirements for the degree of
Doctor of Philosophy.


December, 1975


Dea
NI.


.....""Oe -n ]








































UNIVERSITY OF FLORIDA
II I12620866 IIIII 20llll
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