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ASTRONOMIQUES

DE L\'OBSERVATOIRE

D\'UTRECHT

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ERRATA.

Page 10,nbsp;line 14, read quot;magnitudequot; for quot;magnitudesquot;.

,, 18,nbsp;last line, read quot;a. differencequot; for quot;differencequot;.

„ 24,nbsp;in the table omit quot;H.A. 47quot;.

„ 37,nbsp;in formula (4) read

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„ 38, line 12, readnbsp;for

„ 4(),nbsp;to the heading of table XIV add quot;in mms.quot;.

THE HARVARD MAP OF THE SKY

and

THE MILKY WAY

PROEFSCHRIFT

ter verkrijging van den graad van doctor
in de wis- en natuurkunde aan de rijks-
UNIVERSITEIT te utrecht, op gezag van den
RECTOR-magnificus dr. p. van romburgh,
HOOGLEERAAR in de faculteit der wis- en
natuurkunde, volgens besluit van den
senaat der universiteit tegen de
bedenkingen van de faculteit der wis- en
natuurkunde te verdedigen op maandag
15 october 1917 des namiddags te 4 uur door

ISIDORE HENRI NORT

GEBOREN TE GRONINGEN

drukkerij J. van boekhoven - utrecht

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RECHERCHES
ASTRONOMIQUES

DE L\'OBSERVATOIR

D\'UTRECHT

VIT

UTRECHT

j. van boekhoven

1917.

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THE HARVARD MAP OF THE SKY

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THE MILKY WAY

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CONTENTS.

Page

Introduction .. • .............................................................................................l

Part I. Star-counts on the Harvard Map of the Sky.

Chapter I. The Harvard Map of the Sky............................................5

Chapter H. The counts ............................................................................U

Chapter III. The determination of the limiting magnitude....................22

Chapter IV. Reduction of the plates........................................................30

Chapter V. Determination of the limiting magnitude by means of

overlapping fields; the reduced plates............................48

Chapter VI. The catalogue of star-density............................................79

Part IL The Milky Way.

\'Chapter VH. The star-density and the galactic latitude....................93

Chapter VHI. The position of the galactic plane......................................104

Chapter IX. The star-density and the galactic longitude......................117

Chapter X. The light of the Milky Way.....................................122

Chapter XL The system of stars down to 11?0......................................130

Summary ............................................................................................134

Appendix L Catalogue of star-density........................................................139

Appendix IL Bibliography.....................................................169

Plates.

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INTRODUCTION.

It is a well-known fact that star-counts by different authors and
the conclusions derived from them sometimes largely diverge. As to the
earlier counts, the circumstance that they are not based on a properly defined
photometric scale, may safely be considered to be the cause. However, there
exist two fairly recent counts, both of them based on a rigorously defined
magnitude-scale, and yet differing greatly from each other not only in
the number of stars of every magnitude, but also in the condensation
towards the Milky Way. I mean the counts by KapteynI) and by Chapman

and Melotte^).

Although Kapteyn admits that part of this great difference must be
ascribed to the fact that for the fainter stars his counts are based on
Parkhurst\'s visual scale and for the brighter on Pickering\'s photometric
scale and that according to an investigation by Nijland^) these two scales
diverge rather considerably, he is still of opinion that for the remaining
difference Chapman and Melotte must be held responsible. At any rate,
even after reducing Kapteyn\'s counts of the faintest stars to Pickering\'s
scale there remains a great difference between the results of these two investig-
ations which it would be desirable to explain. So when in June 1915 Prof.
Nijland proposed that I should attempt to elucidate this and some other
unexplained points in various star-counts, I readily accepted his suggestion
and at once started to collect all the literature on this subject I could find.
Since a fairly complete list of papers on star-counts was nowhere to be found
- only stroobanf gives a short list of twenty items in Tome XI, Fascicule II
of the „Annates de I\'Observatoire Royal de Belgique, Nouvelle sériequot; — this
collection took much time. In order to save other workers this trouble

C. Kapteyn, Publ. of the Astr. Lab. at Groningen, No. 18.

2) Chapman and Melotte, Memoirs R. A. S. 60, 145, 1914.

3 A A Nijland. Hemel en Dampkring, Sept. 1916. According to this investigation
the difference between the Harvard and the Parkhurst scales may be very satisfactorily
represented by the formula H-P=0\'^\'22 (H-9-33).

the bibliography I brought together has been added to this paper as
Appendix II.

Under A I have assembled the titles of all papers on star-counts I
have been able to find; a great part of these papers also contain theoretical
considerations on the distribution of the stars. This part of the list is probably
fairly complete up to December 1915; after this date completeness could
not be attained on account of the abnormal conditions resulting from the war.

Under B are gathered the titles of different papers in which the various
magnitude-scales are discusscd. Since the value of any star-count largely
depends on the magnitude-scale used and hence no investigation on the former
subject can ignore the latter, I thought it useful also to publish this part of
the bibliography. Here, however, I have not endeavoured to be exhaustive,
I simply mentioned under this heading the papers consulted by myself.

While collecting this literature I came across an article by M. Selga,
entitled: El mapa celeste de Harvardi). In this article Selga gives a short
summary of a paper by H. Henie on star-counts, carried out by the latter
on the Harvard Map of the Sky. Though Henie\'s counts do not include
fainter stars than those of the eleventh magnitude, so that this author does not
nearly go so far as Kapteyn or as Chapman and Melotte, still it seemed
to me from Selga\'s short paper that these counts might prove of great
value, not only because the number of fields counted is much greater than in
any other count, but especially because the material used for them is much
more homogeneous.

It is particularly on account of this latter circumstance that I studied
Henie\'s paper 2) in detail. The result is contained in the following pages.
So the original plan, namely to attempt to make Kapteyn\'s counts agree
with those by Chapman and Melotte, has not been carried out. However,
I hope to return to it within a reasonable time^).

M. Selga. Revista de la sociedad astronómica de Espana y América, Ano V, Num. 43.

2)nbsp;H. Henie. Lunds Universitets Arsskrift. N. F. Afd. 2. Bd 10. Nr. 1.

3)nbsp;When the present paper was nearly ready for the press, I received Nr. 27 of the
Publications of the Astronomical Laboratory at Groningen, containing an investigation by
Van Rhijn on the difference Kapteyn-Chapman and Melotte. After his elaborate explanation
the investigation planned by myself would be superfluous.

PART I.

Star-counts on the Harvard Map of the Sky.

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CHAPTER I.

the harvard map of the sky.

In Harvard Circular Nr. 71 Pickering gives a description of a photo-
graphic map of the whole sky which will henceforth for the sake of brevity be
called the Harvard Map. By means of two small anastigmatic lenses, one of them
mounted at Cambridge and the other at Arequipa, and both with an aperture
of one inch (2.5 cms.) and a focal length of about thirteen inches (33 cms.),
a large number of photographs were taken. Each plate, measuring eight by
ten inches (20 by 25 cms.), covers a region of over 30 degrees square. With
exposures of about one hour, stars as faint as the twelfth magnitude were,
in some cases, obtained.

From the plates obtained in this manner a set of 55 was selected,
covering together the whole sky and constituting the Harvard Map. Of
this original at the Harvard Observatory copies on glass have been made
by double contact printing and put at the disposal of astronomers. In table
I the current numbers of the plates are given, followed by the right ascension
and declination of the centre of each plate and its time of exposure. The
plates 1—^21, 24—^27 and 29 were taken at Cambridge, the others at Arequipa.

On the copy of the Harvard Map at Lund counts were carried out by
H. Henie, who gives the following description of this copyi): quot;As regards
quot;the images, these latter are generally very fine in a considerable part of the
quot;centre of the negative, even those of the faintest stars being visible. Towards
quot;the margins the images of the brighter stars are distorted and those of the

1) H. Henie. Lunds Universitets Arsskrift, N. F. Afd. 2. Bd. 10. Nr. 1, p. 4.

TABLE L

centres and times of exposure of the plates of the harvard map.

\'fainter ones difficult to discover.quot; The precise meaning of Henie\'s remark
(the italics are mine) I have failed to understand, since in my opinion it applies
not only to the marginal but also to the central images. On any photograph
of the sky the images of the stars will come out more faintly as the stars are
fainter themselves. This increasing faintness of the images is gradual and not
sudden, until finally the limit of visibility is reached. So on any star-photograph,
as well at the centre as near the margins, scarcely visible images will occur
and this will also have been the case\'on the copy of the Harvard Map used by
Henie. Only if he had stopped at a certain limit and so had not included
in his counts all stars down to the limit of visibility, his remark could be
understood, in which case it is to be regretted that he has not defined this
limit more sharply, especially in regard to the determination of the limiting
magnitude.

At any rate it appears from Henie\'s words that he was greatly pleased
with this copy of the Harvard Map and that he thought it very suitable for
carrying out counts. Now since the Utrecht Observatory also possesses a

P

copy of this Map, placed at my disposal through the kindness of Prof. Nijland,
it was not difficult to examine whether this copy also possesses the same
excellent qualities as the one used by Henie. I examined the plates with special
reference to the shape of the images and to the transparency of the plates.

As to the first point, forty-three out of the fifty-five plates show at the
centre and round it within a distance of about six centimetres perfectly
sharp circular images. On the remaining plates the images of the stars are
elongated, also at the centre, not always in the same direction. Near the
margins of the plates the images are always elongated and often strongly
distorted. Especially the deformation shown in fig. 1, /e, is very frequent.
On the 43 plates showing sharp round images at the centre, this deformation
of the marginal images is symmetrical with respect to the centre and is there-
fore only a consequence of the size of the field. On the 12 other plates the
deformation of the images seems to be the combined effect of the size of the
field and of the guiding-error. The different manners in which the deformation
of the images appears are schematically represented in fig. 1. The short lines
in the four quadrants show the principal direction of the deformation, not
its true size, this latter often varying as well in the four quadrants of a single
plate as from one plate to another.

d

J

In table II the appearance of the images on each of the 55 plates is
shown. The letters in the column lieaded quot;shape of the imagesquot; refer to the
different types represented in fig. 1. In the column headed quot;transparency
of the platesquot; the Roman figures indicate the four quadrants of every plate.
It has been assumed that the plates lie with the glass side upwards and that
the quadrants are numbered in the order shown in fig. l a.

TABLE II.

discussion of the plates of the harvard map.

Transparency of the plates.

I and IV fogged at the margins.

I and IV fogged at the margins; also II partly.

Ill partly fogged.

Ill fogged at the margins.

I and IV fogged at the margins.

I and IV slightly fogged at the margins.

nearly clear.

almost entirely slightly fogged,
quite clear.

I and IV slightly fogged.

I, II and III fogged at the margins.

nearly clear.

the margins of I and a great part of III and IV fogged,
nearly clear.

all quadrants partly fogged,
part of III badly fogged,
nearly clear.

fogged everywhere at the margins, III almost entirely
part of III slightly fogged.
Ill very slightly fogged,
nearly clear.

Ill and IV slightly fogged.

Table II. (continued).

By examining a single copy of the Harvard Map it cannot be settled,
of course, what part of the small deficiencies, enumerated in table II, originated

in the reproduction and what part is already present in the originals, but it
would seem to me that this latter part cannot be small. If this be really the
case, the small defects mentioned in table II will also for the greater part
be present in the plates counted by Henie, in which case his enthusiasm
about the Harvard Map would seem rather unjustified.

Doubtless these plates were not taken with the pre-conceived purpose
of carrying out counts on them; for other purposes they have proved eminently
serviceable to many astronomers. For star-counts, however, the plates of
the Harvard Map are only suitable in those portions where no deformation
of the images is observed. Obviously, if we want to determine the limiting
magnitude at various distances from the centre of the plates by a direct method
i.e. by estimating the diameters or the blackening of the images, their imperfect
circularity is a serious drawback. Evidently Henie has also felt this; in fact,
he determined the limiting magnitudes by an indirect method, namely by
basing the amount of its variation from centre to margin on a somewhat
arbitrary relation between the magnitudes and the numbers of the stars.

It would seem to the present author that this method of determining
the limiting magnitudes is essentially wrong. For it is exactly in order to arrive
at such a formula that, among other things, star-counts are undertaken and
to assume a priori the validity of such a formula- cannot but render the results
uncertain. Hence photographic plates, if they must serve for starcounts will
have to allow an exact determination of the limiting magnitudes by a direct
method and this condition is only partly fulfilled by the Harvard Map. Only the
central portion of the plates (a circular area of say six cms. radius) can be success-
fully used. Therefore it is important that Prof. Pickering should have placed
at the disposal of astronomers a Second Harvard Map, composed of 55 plates,
the centres of which appear near the corners of the plates of the first set. By
using these two sets combined and carrying out the counts only in the central
parts, we shall have a most suitable and probably very homogeneous material.

Very likely more reliable results might even be obtained by only
using the central parts of the only available set. Although this would
diminish the number of counted fields, the limiting magnitude might be
determined with much greater accuracy then when using the whole plate.

CHAPTER H.

the counts.

We shall now discuss the manner in which Henie carried out his counts.
He himself says on this point i):

\'Tn order to get the number of stars per square centimeter a most
quot;simple method is applied. The plates are negatives and consequently the images
quot;of the stars are black points on a transparent ground. The plates are therefore
quot;placed on a paper divided into squares of one millimeter, with thick lines for
quot;everv centimeter.

quot;The density may by this means be determined in any part of the
\'\'negative.

quot;Two rectangular axes were drawn on the paper, and the plate was
quot;placed, so that the approximative centre of the photograph coincided with
quot;the origin, and the outlines parallel to the axes.quot;

And later: quot;The counts are all made in good day-light under the same
quot;conditions through a lens magnifying about 3 times.

quot;As the density is not as a rule great the counts are easily made, the
quot;millimeter lines giving an excellent assistance.quot;

The film-surface of the plates measures 19 by 21 cms. If the plate
is laid in the above manner on the squared paper, there will consequently
be 18 x 20 entire fields of a square centimetre. Of these 360 Henie counted
in each quadrant 25 fields, the counted fields lying symmetrically with respect
to the centre, as shown in fig. 1 of Henie\'s paper.

In doing this Henie tacitly assumes that in order to get information
Henie. 1. c. p. 4.

on the apparent distribution of the stars it is not necessary to count the
whole sky, but that it is sufficient to count a number of regularly distributed
fields. Although it appears from the manner in which star-counts have been
and are still being made that this is the general view of astronomers, I have
not been able to discover that it has ever been tried to determine the amount
of the error which may in this way be introduced into the star-numbers.

I therefore made complete counts of four plates of the Utrecht copy
of the Harvard Map, covering a surface of 360 sq. cms. In order to obtain
results that would be comparable with Henie\'s, I followed in the main his
method of counting, using reflected light and working in quot;good day-lightquot;,
although this manner of counting certainly is not the easiest one nor the most
desirable. Without doubt transmitted artificial light is to be preferred Not
only the trouble of being continually obliged to count in the shadow of
one\'s head is then obviated, but, moreover, it is only in this way that it is
possible so to regulate the intensity of the light-source that there can be any
question of counting under the same conditions, which obviously cannot be
attained when counting by day-light in our climate.

In some other respects however I have departed from Henie\'s method.
So I dit not count on millimetre paper. Not only the greyish tint of this paper
makes an unsuitable background, but also the large number of lines is trouble-
some rather than being quot;an excellent assistancequot;. For one has to be very
careful not to count twice images of stars lying exactly on a line and for this
reason it is important to reduce the number of lines as much as possible.
I found that for plates outside the Milky Way lines at 1 cm. distance suffice
while for the Milky Way itself fields of 0.1 cm^. are not too large at all. The
necessary lines were drawn as fine as possible with Indian ink on white paper;
where images fell on the lines their grey colour contrasted clearly with the
blacker ink.

As to the optical means I had the choice between a BRiicKE-lens
magnifying five times, and an aplanatic magnifier by Zeiss, magnifying ten
times. The former was kindly put at my disposal by Dr. Moll, Lecturer in
the University of Utrecht, the latter by Prof. Lorentz, Director of the
Physical Laboratory of Teyler\'s Foundation at Haarlem. I wish to express

PLATE 2.

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Centre at

b = 56°
101°

PLATE 34.

9nbsp;13nbsp;10

9nbsp;9nbsp;5

ffnbsp;flnbsp;11

15nbsp;5nbsp;13

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19nbsp;21nbsp;12

7nbsp;11nbsp;14

17nbsp;20nbsp;12

20nbsp;10nbsp;15

8nbsp;17nbsp;11

18nbsp;11nbsp;12quot;
22nbsp;24nbsp;20
29nbsp;15nbsp;13
34nbsp;10nbsp;12
28nbsp;17nbsp;9
18nbsp;12nbsp;22
ISnbsp;18nbsp;15
17nbsp;13nbsp;14

13nbsp;11nbsp;13

14nbsp;8nbsp;18

PLATE 50.

-9 -8 -7 -0 -5 -4 -3 -2 -1

Centre at •

my sincere thanks to both these gentlemen. After a number of provisional
counts with both instruments the Zeiss magnifier was. chosen for the final
counts; although its field of view was a little Smaller than with Brücke\'s
instrument, it was superior to this latter in clearness and was moreover easier
to handle by its much smaller size.

When selecting the four plates to be counted, I was led by the consider-
ation that they should be representative, as far as was possible with such a
small number, of the different cases met with on the Harvard plates. So the
numbers 2, 6, 34 and 50 were chosen. The first two of these plates were taken
at Cambridge, the other two at Arequipa. The plates 6 and 50 cover regions
of small star-density, the other two parts of the Milky Way.

The results of my counts are schematically represented on pp. 13 and 14and
need no further explanation. It should only be noticed that the numbers for b
and ^ under each scheme indicate the galactic latitude and longitude of the centre
of the plate and that the itaUcised figures are from fields counted also by Henie.

It is now easy to ascertain how far the star-numbers which would have
been obtained for each of these plates by Henie\'s method agree with the
numbers obtained by complete counting. In plate 2 e.g. I counted:

in I 15972 stars
in II 17761 „
in III 11945 „
in IV 13261 „
total 58939 stars

With Henie\'s method we should have counted:
in I 4425 stars
in II 5050 „
in III 3424 „
in IV 3648 ..

total 16547 stars
which would yield for the whole plate by calculation

59569 stars

i.e. only 630 stars or 1.07 % more than the number actually found.

These results for plate 2, with the corresponding numbers for pi. 6,
34 and 50 have been collected in table III.

TABLE III.

IV Counted

In the first column of this table the number of the plate is given in the

next four the numbers found by complete counting in each of the four quadrants-

in the following four columns the numbers which would have been found

by Henie\'s method for each of the four quadrants; in the tenth column the

actually counted total for each plate; in the eleventh the total number which

would have been calculated by Henie\'s method. In the last column the percent

age is given by which the numbers based on Henie\'s method exceed the
actually counted numbers.

From table III it is seen at a glance that there is no question of a
systematic difference between the star-numbers obtained by either method For
although one of the Cambridge plates, when counted by Henie\'s method gives
more and the other fewer stars than when completely counted and although
the same holds for the Arequipa plates, still at Cambridge it is the Milky
Way plate that gives more stars, while at Arequipa this is the other way about

Also the absolute value of the error is very small. If we consider
the whole of the four plates, covering together an area of the sky of more than
3600 square degrees, the total of the stars found by Henie\'s method (in which
only about a quarter of the total surface is actually counted) would be 195408
while the complete count yields 197023 stars. So the error amounts only to
0.82 % of the exact number. This amount is certainly far exceeded by errors
resulting from other sources, viz. from the inaccuracy in determining the
limiting magnitude and from the uncertainty whether a spot on a plate is
caused by a star or by a dust-particle.

So we may take it as settled that in star-counts it is sufficient to
count a number of separate fields. However, a condition of paramount import-
ance is that these fields are spread as regularly as possible over the celestial
sphere.

When for the above-stated purpose I had completed my counts of
four plates of the Harvard Map, it seemed worth while to compare my
numbers with those found by Henie for these four plates. The results are
given in table IV. In the first column the number of the plate will be found,
in quot;the second that of the quadrant. The third column contains the numbers
of stars counted by Henie, the fourth the numbers found by myself. Of my
complete counts only those fields were, of course, retained which were also
counted by Henie. The last column states, in a percentage of Henie\'s
numbers, the difference H.—N.

TABLE IV.

comparison of counts.

This table reveals a very striking fact. In five out of sixteen cases Henie
counted fewer stars than I, in all other cases more. Bearing in mind the different
power of the optical apparatus used by Henie and myself, I — from personal
experience — should have been inchned to expect a very different ratio.
The absolute value of the difference between Henie\'s numbers and mine
varies from 1 % in the second quadrant of plate 6 to 50 % in the third quadrant
of plate 50, and in 11 of the 16 examined cases amounts to more than 25 %.
The cause of these enormous differences may of course lie either in the plates
themselves or in the counts or in both.

As to the first cause, the plates counted by Henie and myself are copies
of one and the same original and it is quite possible that these two copies
are not equivalent. One copy may be weaker than the other, either through
shorter exposure or, more probably, on account of less full development.
This would in itself not be serious. On the weaker copy one would count fewer
stars, to be sure, but on the other hand one would find a limiting magnitude
of less depth. This assertion is only true if the copies have been carefully
made, so as to bear the same ratio of density to the original all over the plate
If this were not the case the reduction of the plates would lead to very meagre
results. For in this reduction we start from the assumption, which probably
holds for the original so far as it is free from local fogging, that the limiting
magnitude is the same at equal distances from the centre. Now, if in taking
the copies, the various quadrants were unequally weakened, the above-stated
condition would no longer be fulfilled and the foundation on which the whole
reduction of the plates is based then becomes unreliable. How far this is true
can only be settled by one and the same observer counting different copies
under conditions as similar as possible. Since, as far as I know, only a single
copy of the Harvard Map is to be found in the Netherlands, namely at the
Utrecht Observatory, I requested Prof. Pickering to be so kind as to send me
another copy of plates 2, 6, 34 and 50. To my disappointment the circumstances
occasioned by the war made it impossible to forward these plates, so that
this part of the investigation had to be given up for the present.

As a second possible cause of the large deviation between Henie\'s
numbers and mine the counts themselves were mentioned. Not only difference

PLATE 6.

-9 -8 -7 -6 -5 -4 -3 -2 -1

-9 -8 -7 -6 -5 -4 -3 -2 -1

counted by
2 3 4 5 6

THE AUTHOR.
7 8 9