A CONTRIBUTION TO THE THEORY

OF

ECLIPSING BINARIES

PROEFSCHRIFT

TER VERKRIJGING VAN DEN GRAAD VAN
DOCTOR IN DE WIS- EN NATUURKUNDE AAN DE
RIJKSUNIVERSITEIT TE UTRECHT, OP GEZAG
VAN DEN RECTOR-MAGNIFICUS Mr. J. C. NABER,
HOOGLEERAAR IN DE FACULTEIT DER
RECHTSGELEERDHEID, VOLGENS BESLUIT VAN
DEN SENAAT DER UNIVER TEIT TEGEN DE
BEDENKINGEN VAN DE FACl ^quot;^IT DER WIS- EN
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RECHERCHES
ASTRONOMIQUES

DE L\'OBSERVATOIRE

D\'UTRECH

IX

PART I

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A CONTRIBUTION TO THE THEORY

OF

ECLIPSING BINARIES

BY

J. THTLAAR

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C O N T E N T S.

Page

Introduction......................................................................................................1

Chapter I. The Determination of the Light-curves.

§ 1. Method of Observation and Reduction ........................................2

§ 2. On the Accuracy of the Photometric Magnitudes of the HP..nbsp;G
§ 3. Revised Reduction of the Grade-estimates to the Photometric

Scale......................................................................................................11

§ 4. A new Determination of the Magnitudes of the Comparison-starsnbsp;17

Chapter II. The Theory of Russell.

§ 5. On the physical Cause of the Light-variation..............................20

§ 6. Summary of Russell\'s Method:

A.nbsp;Notations......................................................................................21

B.nbsp;U-hypothesis................................................................................23

C.nbsp;D-hypothesis................................................................................32

Chapter III. Revised Method of Deriving the Elements of Eclipsing
Binaries.

§ 7. Objections to Russell\'s Method....................................................37

§ 8. The primary Light-variation shows a constant minimmu Lightnbsp;41

§ 9. The primary Light-variation shows no constant minimum Lightnbsp;40

§ 10. Ellipsoidal Stars..................................................................................49

§11. Determination of Theoretical Light-curves ..................................53

Chapter IV. Applications.

§ 12. General Remarks................................................................................57

§ 13. V 23 = SIV Cygni............................................................................02

§ 14. V 9 = Z Herculis............................................................................GO

§15. V V2 — RT Persci............................................................................73

§1G. F 48 = WZ Cygni............................................................................78

Appendlx. Tables ........................•............................................85

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INTRODUCTION.

During the past 18 years Prof. A. A. Nijland, Director of the Utrecht
Observatory, has observed a great number of variable stars, among which
some 70 binaries of the Algol (and p Lym^)-group. The determination of
the light-curves of these binaries has been examined and, in trying to de-
rive the best elements for some of these systems, I struck upon a modifi-
cation of Russell\'s well-known method. The revised method has been ap-
plied to the light-curves of four stars, with a view of extending this work
to the remaining stars in subsequent publications.

I am greatly indebted to Professor Nijland for his readiness in
putting the material necessary for this investigation at my disposal.

CHAPTER 1.

The Determination of the Light-curves.

§ 1. Method of Observation and Reduction.

All the observations have been made after a somewhat modified
Argelander method. The instruments used were the 10-inch refractor (apert-
ure a — 261 mm ; focal length / = 319 cm ; magnifying-power y = 94 as
a rule) and its 3-inch finder {a = 14. mm; / = 11.3 cm; v= 22). Some
of the brightest stars have been observed with a binocular.

In the following pages the words „stepquot; or „gradequot; (°) will be used
instead of „Stufequot;.

Nijland, however, did not follow the original method of Argelander,
but modified it into an interpolatio7i-m%ihod\'^). If for instance the brightness
of the variable v was estimated as being between that of the comparison-
stars a and b, the observer first estimated the ratio of the differences a—v
and v—b and then expressed these differences in steps. The observation
may be recorded as a 1 w 2 or aUv^h, or a2 v 4 b according to the estim-
ation in steps. These records are not identical; the interpolated magni-
tude of v, however, is in all cases the same, viz. v = a } {b—a).

The ratio of the differences remains the essential thing and for that
reason the magnitude of the variable must be found by interpolation. Not
infrequently, however, 1 or 3 comparison-stars have been used instead of
2 ; in these cases the magnitude of the variable was found by using the
derived step-value (p. 3).

1) A.N. 154, 413 (1901).

See also : Recherches Astronomiques de l\'Observatoire d\'Utrecht III, 14 (1908).

If photometric magnitudes could be considered as absolutely correct,
this interpolation would lead to Pickering\'s fractional method. Unfortunately,
however, photometric magnitudes are far from being correct, and they may,
and should, to a certain degree, be controlled and, if necessary, corrected by
the grade-estimates, so as to accord with the individual conception of the
observer. After a great number of observations, the intervals between the com-
parison-stars, given in steps, may furnish sufficient data to revise the photo-
metric magnitudes (see below).

This modified method is undoubtedly superior, not only to the old
Argelander method, which obliged the observers to stick to a constant
step-value (although this value is certainly not constant), but also to the
purely fractional method, which wholly ignores the step and assigns absolute
accuracy to instrumental photometry.

As far as possible the photometric magnitudes of the comparison-stars
were taken from the Harvard Photometry i) (HP), while magnitudes taken
from other sources have been reduced to it. Originally Nijland used the
following method of correcting the photometric magnitudes with the aid of
the step-scale (i. e. the list of step-differences with the faintest comparison-
star): The photometric magnitudes were plotted on squared-paper as abscissas
and the step-scale as ordinates, 5 mm, say, representing, O^l en 1 ° respect-
ively ; and a straight line was drawn passing as nearly as possible through
the points plotted. The magnitudes of the comparison-stars finally to be
adopted were now obtained by dropping perpendiculars on this line from the
plotted points.

It is clear that in this way equal weight is attributed to the photo-
metrically determined magnitudes and to the grade-estimates ; obviously
the estimated step-intervals are not used as they are given by the observa-
tional material, but have been modified so as to suit the photometric values.
The slope of the straight line gives the photometric value of 1 step for the
star in question.

As an example to illustrate this we choose V 23 = SIV Cygm\'^).

As contained in the „Annals of the Astronomical Observatory of Harvard Collegequot; (H.A )
•) For this notation of variable stars see A.N. 199, 215 (1914).

TABLE 1

The photometric magnitudes of the comparison-stars in the second
column have been taken from H.A. 74, except those of e and k whose mag-
nitudes were derived from the limits of vision for the two instruments used^)
(finder and telescope). The grade-estimates yield the scale given in the 3rd
column. In Fig. la these two values have been plotted for each comparison-
star ; through the points thus obtained a straight line has been drawn and
perpendiculars have been dropped on it.

^nbsp;B\'.\'b 9.0 io.Q ii.o dS.o

So

9\'ro io.onbsp;i\'t\'!h

^ho/ome/z\'ic mo^/i/Zoc^es

1) See on this subject: A.N. 205, 233 (1917).

In the fourth column we find the magnitudes as they have been adopted
for the comparison-stars after the adjustment described, whereas the 5th
column shows how the step-scale is modified. It follows from the slope of
the straight line, that 40 light-grades correspond to 4?56, so that for this
case 1° = 0?114.

As a matter of fact, however, the result obtained in this way is in
many cases far from satisfactory, either on account of the small number of
points used, or owing to the fact that the plotted points often strongly
deviate from a straight line.

The magnitudes of the comparison-stars once having been adopted,
the brightness of the variable is interpolated i). The corresponding phase is
taken from the elements at hand. As a rule the series of observations was
not closed until at least 250—400 estimates had been obtained. They were
put in order of phase and combined into normals of 12 estimates each.
These normals were then plotted and a smooth „light-curvequot; 2) was drawn
through the points obtained, satisfying as well as possible the following
conditions :

1nbsp;the algebraic sum of the residuals 5 should be zero : S 0 0 ;

2nbsp;positive and negative residuals should be equal in number ;

• 3°) the number of recurrences of sign should be equal to the number
of changes of sign ; _

4°) the mean error Sonbsp;should correspond to the mean error

Si found in computing the normals. The number of parameters |x
of the curve has been somewhat arbitrarily chosen, viz. :
IJL = 6 for curves without a stationary minimum ;
jx 8 „ „ with

Though Nijland himself has already derived light-curves for several
eclipsing binaries observed by him, these results cannot, for reasons ex-
plained in the following paragraphs, be regarded as final. We shall therefore

1)nbsp;See for the method of accounting for atmospheric extinction: Kcch. Astr. do I\'Dbs. d\'Utr. VIII,
Erste Abt., No. 14.

2)nbsp;In what follows the term „li^ht-curvequot; will be used for the curve giving the magnitude of the binary,
and the term „intcnsity-curvequot; for the curve giving the loss of light-intensity of the system.

here restrict ourselves to four systems (see chapter IV) whose light-curves
have been revised according to the principles of § 4.

The results are :

1 °) V 8 — Of05 ;

2°) number of positive residuals = 35,
„ negative „ =35,

„ zerosnbsp;rr: 14;nbsp;combined.)

3°) number of recurrences of sign = 27,
„ changes „ „ =39;

4°) £o = 0?031,
rn: 0\'?033.

It will interest the reader that for 17 Cepheids, treated in exactly
the same way, Nijland found the following figures :

1 °) S 0 = 0f05 ;

2°) number of positive residuals = 135,
negative „ =129,

zerosnbsp;=: 86; /nbsp;combined.)

3nbsp;°) number of recurrences of sign = 111,

„ „ changes „ „ = 153;

4°^ — 0®031 1

4nbsp;) eo - 0.U.51, ^^^^ ^^ ^^ Cepheids).

Si = 0?029. i

We may remark that, the mean error si of a normal being about 0?030,
the mean error s of a single observation comes out as 0quot;030Kl2 =0^10.

§ 2. On the accuracy of the Photometric Magnitudes of the HP.

We regret to state that this accuracy leaves rather much to be
desired. As to H.A. 14 and H.A. 24 Müller and Kempf already pointed
out 1), that the accuracy of the photometric determinations suffered from
the haste in which they were made. This is also proved by the large num-
ber of „discordant observationsquot; mentioned by Pickering. Moreover
Pickering was not quick in rejecting an estimate.

The magnitudes given in H.A. 14 are based on 3 observations, made

1) Publ. des Astrophys. Obs. zu Potsdam 9, 122 and 491 (1894).

on 3 different nights. Of these the mean was simply taken when the measure-
ments inter se did not differ more than one magnitude. Only when this
was not the case, four more observations were made and the mean of the
7 values was adopted as a final result. A measurement was only rejected
when it differed more than one magnitude from the mean of the other measure-
ments.

The magnitudes mentioned in H.A. 24 are the mean of 2 observations,
but here a Hmit of 0?6 has been adopted as the allowable difference between
the two observations 1). If these differed 0?6 or less the mean of the two
values was accepted as the observed magnitude of the star. If the difference
was greater, 3 more observations were made and the mean of those 5 observa-
tions was then accepted as the magnitude, unless one of the observations
differed by 0?6 or more from the mean of the other four ; in this case the
last mean was accepted.

The photometric observations mentioned in H.A. 44 and in H.A. 45
are more accurate, but here too great differences between the individual
measurements occur. In H.A. 44 2) at least 3 measurements have been taken
of each star, and in H.A. 45at least two. The residuals obtained by sub-
tracting the mean magnitude from the results of the individual measures
are-still often considerable. As a rule residuals greater than 0?65 have been
rejected, except in the case of a few stars of great Southern declination.

H.A. 70 and H.A. 74 give photometric magnitudes of fainter stars and
of special groups of stars. In H.A. 70^) 3 measures have usually been taken
of each star, 4 or sometimes 5 when the results of those 3 measures did
not agree and, besides, in the case of standard-stars. Here again discrepancies
of OTG and more between the individual maesurements are not exceptional.
In the case of the stars also occurring in H.A. 24 the difference has been
lt;given between the photometric magnitudes adopted in these two catalogues ;
among these differences values of 0®5 and more are by no means infrequent.
H.A. 74 contains various groups of stars, down to the 13th magnitude ; the

method of observation is substantially the same as in H.A. 70 ; every star
has been measured 1—6 times, but 3 times as a rule. In the case of stars also
occurring in H.A. 50 and H.A.54 (both being compilations of previous results),
the differences with the values occurring in these volumes have been com-
municated. These differences too amount often to 0?5 and more.

In H.A. 63 part II we find a list of magnitudes of comparison-stars
for 279 variables. These magnitudes, however, have not been determined
by accurate photometric methods. In the case of the brighter stars the
photometric magnitudes have been taken from H.A. 50 and H.A. 54 ; in
the case of the fainter stars from H.A. 74. Moreover the intervals between
the successive stars in the sequence were estimated in grades on 3 nights
and the means taken. The successive sums were then found, assuming the
light of the brightest star to be 0.0 grades. Points were next plotted with
the photometric magnitudes as abscissas and the estimates in steps as ordi-
nates. A smooth curve was drawn through these points and the magnitude
read from the curve. The difference between the photometric magnitude
and that derived from the means of the estimates has been given. This series
of differences contains large values of an irregular character. For instance
in the case oi V = XX SagUtarii (p. 174) : — 0\'f34 ; 0?60 ; — o?27 ;
0?22 ; — 0?29 ; 0?11 ; — 0\'^22.

Systematic differences between the two methods of observation are
sure to occur, but still from the above considerations it appears that the
accuracy of photometric determinations is far from unimpeachable.

In order to obtain a general idea of the accuracy of the Harvard cata-
logues we have computed the average mean errors s and So (of a single
measurement and of a catalogue-value respectively), applying the formulas
£2 = and £^0nbsp;to 10 pages arbitrarily chosen in H.A. 14, 24, 44

and 45 (each containing about 450 stars) and to 20 pages (about looo stars)
in H.A. 70. Table 2 gives the mean results in the 6th row, together with the
corresponding values for the Potsdam Photometric Durchmusterung (PD)\').

») Publ. des Astrophys. Obs. zu Potsdam 9, 491 (1894).

TABLE 2.

As we have seen the accuracy of the photometric magnitudes of the
HP leaves something to be desired; it is obvious that, with an average m.e. of
0*^105 of a catalogue-value, errors of 0T2 and even of 0?3 cannot be exceptional.
A simple calculation shows that e.g. in H.A. 24 with a total of 6700 we may
expect no fewer than 800 catalogue-values to have an error gt; 0?2 ; 400 to have
an error gt; 0?25 and 175 to have an error gt; 0?3. Of course, for stellar statistics
the photometric material collected at Harvard is invaluable, but in special cases,
as in our present investigation of the light-curves of eclipsing binaries the utmost
care should be taken when basing a light-curve upon the Harvard magnitudes
and we should certainly not follow the advice of the late Professor Pickering
to use the magnitudes given in H.A. 63. On the contrary, since the magnitudes
of the comparison-stars taken from the HP may sometimes differ to a large
extent from the true values, the magnitudes of the variable, if based on them
without further discussion, may be very inaccurate. The results to which
this may lead, will be clear from the following two examples :

1°). When the observations of V 10 = RR Delphini were plotted
the resulting light-curve was so far from smooth that it was wholly incom-
I)atible with any physical representation. As a matter of fact, both the descend-
ing and the ascending branches of the minimum showed a large hump. It
was only after new photometric magnitudes for the comparison-stars had
been determined at the Utrecht Observatory, one of them deviating consi-

derably from the value first applied, that a new reduction of the observations
led to a perfectly smooth light-curve.

2°) The first and most important part of Russell\'s method for
determining the orbital elements of the system is the derivation from the
light-curve of the ratio k of the radii of the two components. This value of
k must lie between 0 and 1. Every point of the light-curve may yield a
value of k. Now the values of k often proved to differ systematically from
each other according to the points chosen ; so for instance in the case of
V 2\'i = SW Cygni and F 25 = 51\' Cygni ; it also occurred that parts of
the light-curve did not produce any value of k at all. In connection with
what follows in §7, the possibility should be borne in mind, however, that
these discrepancies may also be wholly or partly due to the method employed.
When the magnitudes of the comparison-stars had been modified, however,
so as to bring them, into accordance with the step-scale (see §4) it was
surprising to see how well the theory of eclipses could be applied to the newly
obtained light-curve in cases where such an appHcation to the original curve
had been impossible.

We subjoin the results for V 2\'i = SW Cygni as an example :

TABLE 3.

The first three columns of table 3 are identical with the 1st, 2nd and
4th of table 1 (p. 4) ; consequently column H gives the magnitudes of the
comparison-stars after the adjustment described in § 1. Column H\' gives
the magnitudes modified according to the step-scale (see §4). The last column

gives the differences between the magnitudes of the comparison-stars taken
from H.A. 74 and those modified according to the step-scale.

The values of k found from various points of the light-curve by
Russell\'s method are given in table 4. This method, to be described in § 6,
starts either from the supposition of a uniform distribution of intensity on
both disks (U-hypothesis) or from the hypothesis of a complete darkening
towards the limb (D-hypothesis). The first two rows of the table were derived
from light-curve I, which was based on the magnitudes of the comparison-
stars mentioned in the third column of table 3 ; the last two rows were
derived from hght-curve H, based on the magnitudes in the fourth column
of that table. The notes of interrogation indicate that in these cases no values
of k are found between 0 and 1 (U-hyp.) or between 0.2 and 1 (D-hyp.).

TABLE 4.

§ 3. Revised Reduction of the Grade-estimates to the Photometric Scale.

The difficulties which have been mentioned in the foregoing para-
graph, may be due to the fact that in the derivation of magnitudes for the
comparison-stars equal weight has been assigned to the photometric values
of the HP and to the estimated light-steps. Therefore we decided to make
a revised reduction to the photometric scale, laying full weight upon the
observed steps, so as to bring the HP values into accordance with them.
However, before leaving the apparently firm ground of photometry in order
to tnist ourselves to a scale of steps, we should subject the latter to a closer
investigation. Then the following question at once presents itself: what is
the value of the step, expressed in magnitudes and what factors influence
this value ? As such we may consider :

the instruments used ;
the magnitudes of the stars observed ;
the number of steps estimated ;
the colour of the stars.

TABLE 5.
Step-value and Median Magnitude.

II

Ill

V

VI

1) Unless the reverse has beeti mentioned, the variables of the /i Lyrae-type are tacitly included.
When the stars of this latter type are separately treated, we shall indicate them for shortness\' sake
by the name of „Lyridsquot;.

From the very beginning it was plain that the step-value depends
on the stellar magnitude : it increases for fainter stars. As stated on p. 3 a
discussion of the comparison-stars of a variable may yield the value of 1 step
for any particular case. For this discussion echpsing binariesi) for which the
series of observations had been closed were available, and likewise 17 Ce-
pheids. The results have been collected in table 5.

The first column gives the name of the variable ; the stars marked
with an asterisk are Cepheids. The second column contains the median-
magnitude (mean of the maximum and minimum brightness). The third
column gives the step-value derived for each variable. These data have been
joined into six groups. Table 6 gives the mean median-magnitude and the
mean step-value for each group.

TABLE 6.

The values between brackets are obtained by excluding the Cepheids.
For both cases it is evident that there is a distinct increase of the step-value
with decreasing brightness. The last figure (group VI) is rather uncertain
on account of the scarcity of material in this group and of the difficulty of
estimating steps in an interval near the limit of vision.

In Fig. 2 the dots indicate the points we obtain if both Algols and
Cepheids are included, and the crosses the points we get if the Cepheids are
excluded. A glance at this Fig. 2 will show that the correlation may be con-
sidered to be linear.

Nijland obtained the same result in a different way. Hitherto it
was generally believed that the step-value increased with the number of

steps estimated^). Nijland himself found strong evidence for this variabihty
in a discussion of his observations of Algol In 1920, however, while stu-
dying many thousands of his estimates, to his great astonishment he came
to wholly different results, which may be summarized as follows :

O.öZ
OM
0.^0
0.09
o.o8
0.0^

-m ^
0.06

HTû b.O h.o 5.0 6.0 y.Ö S.o 9.0 äo.o ii.O iQ.O ib.o

1°) The step-value does not depend upon the estimated number
of steps 3); from this it follows that when a variable has been compared with
a comparison-star via another comparison-star as an intermediate, such
an estimate may be reduced with the same step-value as the directly esti-
mated intervals.

2°) If an interval has been estimated in two different instruments,
there is no perceptible influence of the instrument used ; i. e. the step-value
is the same for telescope, finder and binocular ; therefore it is unnecessary

gt;) Hägen : Uic veränderlichen Sterne, II, p. 200.

a.n. 154, 413 (I90I).

It is to be emphasized that this number never exceeded 8.

to treat the 3 instruments separately or to reduce them to each other i).

3°) For one and the same instrument the step-value depends on the
brightness of the stars included 2). For the study of this correlation two
sources of material were available, viz. :

a)nbsp;a comparatively small number of directly estimated step-intervals
of comparison-stars ;

b)nbsp;a large amount of indirect estimations, the variable itself being
the link between the comparison-stars. On account of what has been found
sub 1 an observation of this kind, say a m v n b, jaelds the step-interval
between a and b, viz. m n, as if a and b had been directly compared.

Both cases were treated separatel}^ and the second was given threefold
weight, in general, on account of the large number of facts included. The
results are given in table 7.

TABLE 7.

The last figure is very doubtful on account of the scarcity of the
material used. We have combined the results of table 7 as well as possible
with those of table G (Cepheids excluded) into 5 groups, represented in Fig. 2
by circlets. It is seen that the connection between step-value and magnitude
may be represented by a straight line.

Finally it seems worth while to consider whether the step-value
depends on the amplitude of the light-variation. To answer this question,
we liave with the aid of fig. 2, reduced the step-value following from the
discussion of the comparison-stars (p. 3) to the value, which would have been

1) The equivalence of two instruments docs, however, not hold for such stars as arc near the limit
of vision of the smaller instrument.

-) This statement, which might lead to a contradiction with that mentioned sub. 2°, will be sub-
mitted to a closer investigation later on.

obtained if the median-magnitude had been 8?0. Thus the step-values have
all been reduced to the same magnitude (table 8).

TABLE 8.

Star

Algol
k Tauri
0 Cephei

Ynbsp;Aquilae
p Lyrae
C Geminorum
0 Librae
RT Aurigae
S Sagittae
T Vulpeculae

Ynbsp;Sagittarii
R Canis Majoris
U Ophiuchi

U Cassiopeiae *
T Monocerotis *
X Cygninbsp;*

RZ Cassiopeiae
Z Herculis
RS Vulpeculae
TV Cassiopeiae
Z Vulpeculae

Ynbsp;Cygni
U Sagittae
U Cephei

U Coronae Borealis
TW Draconis
SZ Cygninbsp;*

RY Persei

The data of table 8 were grouped according to the amplitude, and
for each group the mean step-value was deduced. We thus get table 9, in
which the figures between brackets refer to the case of eclipsing binaries only.
This table does not show any influence of the amplitude on the step-

value. Indirectly it appears once more, that the step-value does not depend
upon the estimated number of steps.

TABLE 9.

Summarizing we find the following results from the preceding investi-
gation :

1 °) the step-value does not depend upon the estimated number of steps;

2°) the step-value increases with decreasing brightness.

§ 4. A new Determination of the Magnitudes of the Comparison-stars.

In order to connect the magnitudes of the comparison-stars as closely
as possible with the step-scale we proceeded as follows :

First the observed step-intervals in the sequence of comparison-stars
were reduced with the aid of Fig. 2 to the median magnitude of the variable.
By this reduction the change in the step-value during the process of the
light-variation is taken into account; obviously an appreciable effect is only
to be expected in light-curves of very great range, e. g. V 45 = WW Cygni.
Next this homogeneous step-scale and the photometric magnitudes taken
from the HP were plotted in the usual way and through the points obtained
a straight line was drawn. In a few cases, where the slope of the line could
not be determined with certainty from the points available, these being too
few in number, the „theoreticalquot; step-value derived from Fig. 2 was taken
into consideration. The difference with the process mentioned on p. 3 lies
in the fact that we did not, this time, draw perpendiculars on the line, but
lines parallel to the axis of abscissas, so that we leave the step-scale unalt-
ered. In other words we consider it to be free from errors. Are we justified
in doing so? It is true that in estimating steps, contrary to photometric

determinations, the remembrance of previous estimates in the same mterval
might influence the results, but apart from this the step-method in the hands
of an experienced observer need not be second to photometric determinations,
which are no more free from systematic errors even of a sometimes incon-
ceivable and complicate character i). In the HP according to p. 9 the mean
error s of a single measurment is about 0^18 and the mean error e« of a
catalogue-value aboutnbsp;The m.e. (s) of a grade-estimate is about 0?10

(see p. 6). In general, therefore, a grade-estimate is not less accurate than
a photometric determination. To this we may add :

1°) that the grade-estimates are usually very numerous (often 30—80
in the interval between 2 comparison-stars) ;

2°) the possibiUty of strong systematic personal differences in the
appreciation of brightness between the observer and the author
of the photometric catalogue, mostly due to the colour of the
stars.

Especially this latter consideration justifies keeping the step-scale
immodified. Except for the general course of the straight hne — its slope,
giving the step-value for each particular case — the photometric magnitudes
have no longer been taken into account. Only if the number of grade-esti-
mates in an interval is very small, and in some more special cases, we applied
little corrections to the step-value, so as to lead to a closer correspondence
with the individual photometric magnitudes.

As an example we once more take F 23 = SW Cygni.

The first four columns of table 10 are identical with those of table 1
(p. 4). Column 5 contains the observed step-intervals of the successive
comparison-stars; column 6 these intervals reduced to ll\'I\'O ; column 7 the
new step-scale. Column 8 gives the magnitudes of the comparison-stars,

1) See e.g. : Contr. from the Princeton Un. Obs. I : The Algol-system RT Pcrsei, by R. S. Dugan.
s) According to a statement on the same page these values arc much less for the PD, viz. 0\'.\'gt;084
and 0™059.

Dugan gives in the Contr. from the Princeton Un. Obs. 5, 29 (1920) for the probable error of
a single observation of full weight in the case of U Cephei, RT Persei, Z Draconis, RV Ophiuchi and
RZ Cassiopeiae an average of about 0™04.

TABLE 10.

obtained in the above way; see also Fig. (p.4) (the magnitude-scale is given
at the top). In this case a step-value of 0^109 is found for the median magni-
tude (11®0). The magnitudes of column 8 are those mentioned in the fourth
column of table 3 (p. 10), with which a new light-curve has been derived.
As already stated (p. 10) this curve gives much better values of k (see table4).

CHAPTER n.

The Theory of Russell.

§ 5. On the physical cause of the light-variation.

Two theories have been given which might explain the light-variation
of this group of variable stars :

1nbsp;The spot-hypothesis.

This explanation has first been treated by Zöllner and afterwards
fully by Bruns 2) and by Harting 3). Bruns found that it is always pos-
sible, in an infinite number of ways, to assume spots on the body of a star
located in such a way, that by its axial rotation the light-variation agrees
within arbitrarily chosen limits with the observations.

2nbsp;The eclipse-theory.

This explanation was originally given by Goodricke About a
century later E. C. Pickering \') based upon this explanation a theory which,
with various restrictions, enabled him to deduce the elements of the system
p Persei from the light-curve. His example was followed by Harting»),
Wilson «) and Blazko quot;), who expanded the theory and were able to drop
some of the restrictions. Blazko even went a little further by introducing
into his considerations the possibility of a diminution of light from the center
towards the limb of the star\'s disk, an idea which had been previously «)
suggested already.

1)nbsp;ZÖLLNER : Photometrische Untersuchungen IV, §§ 71—80. Leipzig 18G5.

2)nbsp;w. Bruns: Bemerkungen über den Lichtwechsel der Sterne vom Algoltypus. Akad. Berlin, 1881. S.48.

3)nbsp;JoH. Harting : Inaugural-Dissertation ; München 1889.

4)nbsp;John Goodricke: A Series of Observations on. and a Discovery of the Period of Algol. Phil.

Trans. 1783.

6)nbsp;E. C. Pickering : Dimensions of the fixed Stars, etc. Proceedings of the American Acad, of Arts

and Sciences ; Vol. XVI, 1881.

«) H. C. Wilson: Variable Stars of the Algol-type. Pop. Astr. 8; 113 (1900).

7)nbsp;S. Blazko : Annales de l\'Observatoire Astronomique de Moscou. 5 ; 70 (1911).

8)nbsp;Rödiger: Untersuchungen über das Doppelsternsystem Algol; Königsberg, 1902.

After the spectrographic observations of Vogel in the year 1889
it became at once evident that the true explanation of the phenomenon should
be sought in the direction of the eclipse-theory. Finally in the year 1912
H. N. Russell pubHshed an analytical method by which the orbital elements
may be deduced from the light-curve. A brief summary of this theory is
given in the next paragraph.

§ 6. Summary of Russell\'s method.

A. Notations.

Russell has considered two extreme cases, viz.:

I.nbsp;The star-disks are supposed to be „uniformlyquot; illuminated. This
case will be called the „U-hypothesisquot;.

II.nbsp;The light of both disks is gradually decreasing from center to
limb (where it has an intensity zero). This case will be called the
„D-hypothesis*

The cases actually observed will lie between these two extremes.

In explaining the theory we shall assume that the smaller star moves
in a circular orbit around the larger. The radius of the orbit is taken as the
unit of length and the total light of the system as the unit of light.

Russell uses the following notations:
P period of revolution in days ;
i inclination of the orbit, i.e. angle be-
tween hne of sight and normal to the
orbital plane. As a rule this angle lies
between 75° and 90°;
T time in days, measured from primary

mmmium

o

angle yt, mean anomaly of smaller
star in its orbit (see Fig. 3);
apparent distance of centers ; 1 Unit :
radius of larger star ;nbsp;) radius

radius of smaller star ;nbsp;) of orbit.

») a.n. 128, 289 (1890).

ratio of radii =

p ratio, in the case of circular star-disks, of Z—n (distance from the
center of the small disk to the circumference of the larger one) to kn
(radius of the small disk) ;
Pi, p2 densities of large and small star respectively ;
/i surface-brightness of larger star ;
/a surface-brightness of smaller star ;

Y the ratio ^; in the case of the D-hyp. y represents an average ;
Li light of larger star ;
Li light of smaller star ;
I light at any moment; hence 1 —I the

loss of light at that moment ;
\\ light at minimum ; hence 1—X the loss
of light at that minimum ;

N.B. The subscript 1 to the latter two quantities refers to the eclipse
of the small star by the large one ; the subscript 2 to the other
eclipse.

loss of light at any moment expres-
sed in terms of the loss at the mo-
ment of internal tangency (total or
annular eclipse) ; this does not imply
that internal tangency actually oc-
curs in the eclipse under considera-

tion.

It is to be remarked that a and a must be ^ 1, whereas aquot; may also
be gt; 1.

Obviously the units in which the quantities a, a\' and aquot; are expressed
have different values, according to the cases considered, as is shown
by the following table :

1) Properly speaking Russell defines a as that part of the surface of the disk of the smaller
star, which is at any moment eclipsed by the larger. Obviously the two definitions are equivalent

TABLE 11.

ao,aoanda^\'the values of a, a and aquot; at the middle of the eclipse.

Since we have supposed the orbits to be circular, the values of
«0 are in the U-hyp. the same for the two minima. In the D-hyp.,
however, this is not the case; here a^ and a^ are connected by
the relation (12) of p. 35.
o(o = 1 if the eclipse is total or annular and lt; 1 if it is partial;
Oo = 1 if the eclipse is total and lt;1 if it is partial;
oto gt; 1 if the eclipse is annular and lt; 1 if it is partial,
nnbsp;fraction of greatest loss of light in partial eclipses;

a;nbsp;coefficient of darkening ;

a, and a^ semi-major axes ;nbsp;\\

and hi semi-minor axes ;nbsp;I In the case of

£nbsp;excentricity of meridian-section; ( eUipsoidal stars.

znbsp;the quantity s® sin\'^ i ;nbsp;/

B. U-HYPOTHESIS.

Russell has first treated the problem in its simplest form^), sup-
posing the orbit to be circular and the disks to be uniformly illuminated
according to the cosine-law. In other words, he tried to solve the following
problem: Two spherical stars with uniformly illuminated disks, revolving in
circular orbits around their common center of gravity, eclipse each other ; how
to find, from considerations based upon the observed light-variation, the relative
dimensions and brightness of the two stars and the inclination of the orbit ?

») Sec for this Q-function p. 35.

*) IL N. Russell : On the determination of the orbital elements of eclipsing variables. Ap. J.
35, 315 (1912).

In this problem we may distinguish four cases :

1°) Both primary and secondary light-variation have been observed

and show a constant minimum-Hght.
2 The primary light-variation shows a constant minimum-light;
the secondary minimum cannot be exactly determined or is im-
perceptible ; at least the observations have not revealed it.
3°) Both primary and secondary light-variation have been observed,

but show no constant minimum light.
4°) Only the primary light-variation, showing no constant minimum
Hght, has been observed.

In the first and the second case we have to deal with total or annular
eclipses ; as a rule we cannot decide between these two before a value of
k has been found.

First of all we shall have to pass from the light-curve to the intensity-
curve (p. 5, note 2). If at any moment I represents the intensity, m the mag-
nitude, niQ the magnitude at the minimum, C and c two constants, we have :

L = C X 2.512^-quot;*

1 C X 2.512\'^-\'quot;o
L = 2.512\'«o-quot;»
whence : log. / rr= 0.4 (mo—m).

.nbsp;This formule enables us to con-

struct a table giving the loss of light corres-
ponding to an increase A w in stellar mag-
nitude (Table A at the end of this paper).

The part of the disk of the smaller
star, obscured at any moment, is the sum
of two circular segments (Fig. 4).

In A A Ml Ma:
cos pnbsp;and cos £

and further we readily find :

n r,»nbsp;—

or :

nl^--

These formulas are not given by Russell; we insert them because
we shall have to refer to them later on. It at once appears, that the ratio
a only depends on the ratios of the quantities ^i, r^ and o, for instance on
^ = ^ and = \\ ^ kp. Therefore we may write a = / J-).

For any given value of U we may invert this function and write

i =nbsp;or: 1 ^^ cpa).

Thus p becomes a function of k and a, which has been computed and put
in a tabular form by Russell(Table I at the end of this paper). For every
set of values of k and a this table gives a value of p and therefore of the func-
tion cp.

Further we derive from simple geometrical considerations :

d^ = cosHquot; sinH sin^O ..........................(la)

whence

cosH sinn\' sin={) = {\'f {k, «)=}..................(1^gt;)

An equation of this form may be deduced for any value of a—conse-
quently for any point of the light-curve. From several equations of the form
(16) we now must derive the three unknown quantities k, Vi and i. This
appears to be rather complicated; Russell gives the following method :

Let ai, «2, aa be any definite values of a and fti, »2, the correspon-
ding values of ft, which may be found from the light-curve. From the three
corresponding equations of form (16) we derive:

sin. -sin«,92 _ {r (A. a a a) (O^
^O.-li^^ - (A.nbsp;(A. «3)}^ - ^nbsp;«3).............(2)

The first member of this equation contains only known quantities.
Now choose once for all two fixed values for «3 and «3, viz. a^ = 0.6 and
ot3 = 0.9, corresponding to two fixed points a and b on the light-curve.
Thus lt;\\gt; becomes a function of k and ai only, and may be tabulated for suit-
-able intervals in these two arguments. If, for shortness\' sake, we put

A = sin»}), and B = sinquot; f), — sin« »3
equation (2) may be written :

sin«{), = A B^iKa,)................................(3)

gt;) Ap.J. 85, 333 (1912).

The value of k is now determined as follows : for several — say 12 —
values of a^, corresponding to as many points on the light-curve, we may
read off from that curve the corresponding values of which by means
of equation (3) furnish as many values of i. Then the d;-table gives a value
of k for each set of values \'h, a^. By taking a suitably weighed mean of
these values of k\'^), a theoretical light-curve can be obtained which passes
through the fixed points a and h, but will deviate more or less from the other
points of the observed curve. By slight changes in the assumed positions
of a and h (i. e. in the corresponding values of 8, or of - and therefore in
those of A and B), it is possible to obtain a computed curve which fits
the whole course of the observed curve as well as possible. The criterion
of this is that the upper (above a), the middle (between a and b) and the
lower (below h) parts of the observed curve will sensibly yield the same mean
value of kquot;^). The values of k found with the different values of a^, may of
course differ among each other to a certain extent.

This point and the possibilities which may arise from it will be more
fully treated in § 7.

When once k has been determined, we may with the aid of equation
(3) find values 0\' and bquot; for the moment of the beginning of eclipse («i o)
and that of the beginning of totahty (oti = 1). These computed values are
more accurate than those estimated from the observed curve\'). Since at these
phases of the eclipse o =ri rt = n (1 k) and oquot; = ri—rz = r^ (i —k)
respectively, we may derive from (Irt) :

The individual determinations of k are of very different weight. Between a and b (that is for
values of a^ between 0.6 and 0.9) »/\' changes very slowly with k. At the beginning and end of
the eclipse the stellar magnitude changes very slowly with the time, and hence, by (3) with n
The corresponding parts of the curve are therefore ill adapted to determine k. For the first approx
imation it is well to give the values of k derived from values of aj between 0.9o and 0 99 and
between 0.4 and 0.2 double weight (provided the corresponding parts of the curve are well fixed
by observation). The time of beginning or end of eclipse cannot be read with even approximate
accuracy from the observed curve and should not be used at all in finding k. The b \'
end of totality may sometimes be determined with fair precision, but docs not descrvc^\'as quot;muclquot;^
weight as the neighbouring points on the steep part of the curve. (Ap.J. 35, p. 322)

») If further refinement is desired, it can most easily been obtained by plotting the licht-
for two values of k and comparing with a plot of the observations. This will rarely be ncccssar ^ ^^^^

3) The latter values played an important part in the older treatment of the probleT^^^W
and others. Russell too has used them — though he gave them less weiphfnbsp;.gt; o ^

of k from various points of the light-curve.nbsp;quot;nbsp;dctermmation

(1 -\\-kf — Qos^i sinH sin^D\'
r^^ (l—k^) = cosHquot; sinHquot; sin^i}quot;
These equations finally give the elements r^ and i.
Moreover, when k has been found and the depth of the secondary
minimum is approximately known, it can be made out whether the ecHpse
at primary minimum is total or annular. In fact, we have :

= 1 —a Li; 4—1 — k^
or, Li Lz = I

OC = 1-4

consequently :

= 1-).. ^^....................................(4)

As has already been remarked on p. 22, the subscript 1 to the quant-
ities I and \\ refers to the eclipse of the smaller star by the larger ; the
subscript 2 to the other ecHpse. If at primary minimum the smaller star is
eclipsed (case Ei) we, therefore, have the equation. :

«0 = 1—Vnbsp;..................................(4rt)

and when at primary minimum the larger star is cclipsed (case Es) :

«0 =nbsp; ^..................................W

In the following chapters we have always used the equation connect-
ing the depths of the minima in the form (4 a) or (46).

If the principal eclipse is total or annular we have respectively :

Isec =nbsp;hr........................................(4c)

ornbsp;_

Xsec — j^a ........................................(4(i)

Since X must be less than 1, the first case is always possible. The second
case is only possible if k^ gt; 1—X/,r.

If, then, k has been found, (4c) and (4f/) give the depth of the secund-
ary minimum corresponding to each case, and now it can be decided whether
the principal eclipse is total or annular. For either (4r/) gives Kc gt; 1 and
then an annular eclipse should be excluded, or, if {Ad) gives l^cc lt; 1, the
depths according to each hypothesis are likely to differ so much») that the

J) _ » 1 _ (1 /(«) if the principal eclipse is total.

__ B j\', (l — V (1 A«)} if the principal cclipsc is annular.

The latter hypothesis therefore gives rise to the shallower secondary minimum.

choice is easily made — unless k should be near unity, but then the question
is of little importance.

If the primary minimum has a great depth, the eclipse at this mini-
mum can as a rule be directly said to be total, unless the value found for
k is large. In dubious cases equations (4c) and (4^^) must decide.

In the case of partial eclipses (cases 3° and 4°) the method just des-
cribed should be altered. A new unknown quantity is added, viz. a^, the
maximum obscuration (which is equal to unity in the case of a total or annu-
lar eclipse). Russell finds that in the case of partial ecHpses the problem
of deriving orbital elements solely from the hght-curve of the primary mini-
mum is indeterminate. For any value of k, comprised within wide limits,
it is possible to find an assumed percentage «o, and hence a set of elements,
such that the interval from the middle of echpse, at which any given mag-
nitude is reached, as calculated from any of these systems of elements, will
be the same within a fraction of 1 percent; in other words, such that nearly
the same light-curve will be found. If, however, the depth of the secondary
minimum has also been observed, Russell has succeeded in solving this
problem ; in this case the results may however be rather doubtful.

For various values « = ™ he finds the corresponding values of t
(and therefore of O) from the light-curve. In a way similar to that followed
in the case of a total echpse, he arrives at a function :

sinZ amp; (n) _ y {k, wap) — ygt; (k, ap) _

sin2^(J) — V\'(A,i«oj-V\'(AV«oquot;)nbsp;^ \'nbsp;.................(fi)

The first member contains only known quantities. The second is a
function of k, a^ and n, which may be tabulated for any convenient values
A table for y {k, «o, i) is constructed, and since it appears that the relations
between any pair of the -/-functions, corresponding to different fixed values
of n, is very nearly linear, we may write in general :

X {k, ao, n) = w, (n) -f ze^j. (n) y. a„, ...................... ^

and construct tables for the empirically determined functions w^ («) and
Wi {n). From (5) and (6) finally :

sin2 {} (n) = w, (n) sin^ {} w, (n) sin^ {) (J)
or, putting sin^}) = C and sin^ d (i) = D, this becomes

sin2 0 (n) = Cw., (n) Dw,in).........................(7)

If the value found for D = sin^ f) is preliminarily accepted, equation
(7) yields a value of C = sin^ [) for any value of n. If these values agree
sensibly for the upper, lower and middle parts of the light-curve, the mean
value of C is taken. If this agreement is not obtained — which is not infre-
quently the case — an attempt at improvement is made by changing the
value oi D a. little, which is again equivalent to slightly changing the point
on the light-curve (i.e. its value for t, or f)) corresponding to n = See
further § 7 on this subject.

This being accomplished we get from (5) :

iK «0, i) = §..................................--(8)

Unless the secondary minimum has been observed, we can proceed no
farther. But if we know the brightness at both minima (4) gives a second
relation between k and and values for k and «o may be derived by
means of the table for x (k,nbsp;The value finally adopted for x {k,

by means of (8) may, however, be very uncertain — the separate values
of C showing large discrepancies — so that the same may be the case for
the values of k and derived from it. Moreover the solution may occas-
ionally be indeterminate, and sometimes two solutions are possible, between
which it may be hard to decide.

Finally the elements and i are determined in the same way as in
the case of a total eclipse.

Russell has also considered the case of ellipsoidal stars (Lyrids)»).
The distance between the components of such a system is very small and
the period of their revolution most probably equal to that of their axial
rotation. Since in consequence of their strong mutual attraction the com-
ponents have an elongated shape, the longest axis lying in the line of their
centers, and since on the other hand the rotation gives them a polar flattening

m Ap.J. 86, 00—07 (1912).

(shortest axis perpendicular to the orbit plane) they will be as a rule ellipsoids
of three unequal axes. For the sake of simplification Russell assumes them
to be similar and similarly situated.

In the light-curve the ellipsoidal shape of the components is revealed
by the continuous light-change between the ecHpses, the brightness reaching
a maximum value midway. Strictly speaking this is the case with every
eclipsing binary. In most cases, however, the brightness between the eclipses
may be supposed to be practically constant if the distance between the com-
ponents is not very small. If, however, there is a distinct deviation from
a constant maximum light, we speak of a Lyrid (so-called after the best-
known representative, p Lyrae, of this group).

The ellipsoidal shape of the components, however, is not the only
cause of a continuous light-change between the minima. If the distance of
the components is so small, that they are nearly in contact, the light-curve
must also on this account present the same characteristics, a constant maxi-
mum light being either absent or of so short a duration that it can hardly
be observed.

The Lyrid-curves therefore differ from the Algol-curves :

1 °) on account of the ellipsoidal shape of the components ;

2°) on account of the very small distance of their centers the disks
being nearly in contact.

Either of these two circumstances will make the light-curve pass
continuously from one minimum to the other, without constant maximum light

The polar flattening cannot be determined from the light-curve but
it may be approximately estimated with the aid of plausible relations between
the three axes, based on Darwin\'s studies, when the elongation in the
equatorial plane has been found. Therefore in his investigation Russell
admits the case of two stars which by their mutual attraction have got the
shapes of similar prolate spheroids whose longer axes coincide with the line
joining their centers ; for if the dimensions of the ellipsoids in the direction
perpendicular to the orbital plane are modified in a constant ratio the ratio
of the eclipsed part to the surfaces of the two disks remain identical If L
and U represent the maximum values of the light of the two components

i. e. the values for D — 90°; 4 and 4 the amounts of Hght which would
reach us from each component if there were no eclipse; d^ and d^ the appar-
ent lengths of their major axes at that moment; a^ and a, the maximum
values of these axes, we have

Linbsp;Linbsp;ainbsp;^nbsp;\' gt;

whence, if / is the actual amount of light received by us at any time :
/ 4 (1—a) 4 = (1—sin^i C0S2 }))i {L, (1—«) Q

= {l—z C0S2 {L, (1—a) La}.......................(9)

When £ == 0 (and therefore 2 = 0) this reduces to the familiar
formula for spherical stars. The second factor of (9) is constant (= 1) when
there is no eclipse (a = 0) ; this means that 2 may be determined graphic-
ally from the light-curve outside of eclipses (near 90°) by plotting for various
points the values of (I—/\'\') against the corresponding values of cos^ {gt;. The
resulting points will lie on a straight line, whose slope gives the desired value
of z. When eclipse begins the plotted points fall above this straight line and
lie on an ascending curve. This method might seem to fail when the stars
are in actual contact because in that case the stars continually eclipse one
another more or less, except when U = 90°, so that the curve above described
jias no rectilinear portion. But the eclipsed surface is very Small, varying,
as can be easily shown, approximately as cos^ {). The tangent to the curve
determined by the plotted points at the point \'for which cos }) = o can be
drawn, and gives the value of ^ for this case.

Having found the light-curve may be „rectifiedquot;, removing all
apparent influence of the ellipsoidal shape of the components by subtracting
from the observed magnitudes the comimted variation due to the latter
cause. We then obtain a light-curve of the ordinary „Algolquot;-form, with con-
stant light between eclipses, which represents the variations in brightness
due to eclipse alone.

For the rest the solution runs parallel to that for spherical stars, only
slight changes being necessary in the formulas, since is replaced by

d,^ = a,2 (1—^ cos» a).

When the orbital elements have been determined, we may express
the density of each of the components in terms of the sun\'s density, putting

= 0.01344nbsp;; Pa = 0.01344 ^

The mass of the larger star is represented by my and that of the
smaller star by m{\\-y), the total mass of the system, expressed in terms
of the sun\'s mass, being m. The actual densities cannot be computed unless
the ratio of the masses of the two stars, and consequently y, is known. As
a rule this is not the case. But for a number of visual and spectroscopic bina-
ries the brighter star has proved to have nearly always the greater mass, which,
however, does not much exceed that of the smaller star (4 : 1 being the
maximum ratio that has hitherto been found).

Therefore in the above formula y is supposed to be i. e. the masses

of both components are supposed to be equal. At any rate the order of

density is found in this way, for which we find

0.00672

P = —^z-..............................(10).

This formula is Hkely to give too high a density for the faint star
and too low for the bright one, but in neither case the error is at all likely
to exceed 50 per cent of the computed values, or to be in the opposite

sense from that stated.

When the stars are ellipsoidal, our formula obviously becomes
^ ^72nbsp;^ ^ ^^^ ^ are the three axes of the ellipsoid.

rnbsp;P^abc \'nbsp;\'

Finally Russell has considered the effects of an eccentric orbit and of
reflexion. These effects are usually so small as to be detected only by the
most refined observations from the brightness between the principal eclip-
ses. We shall not discuss these possibilities 2), since we have not taken them
into account in the examples of chapter IV.

C. D-HYPOTHESIS.

The hypothesis of uniformly illuminated disks is most probably

incorrect, since most stars, like our sun, will show a decrease of brightness
towards the hmb. Russell assumes it to obey the following law :

1 = 1^ {\\—x XCOS\'S).
in which I is the surface brightness and 9 the angle between the line of sight
and the normal to the surface. The coefficient of darkening .t has for uni-
formly illuminated disks (U-hyp.) the value zero and for complete darkening
towards the hmb (D-hyp.) the value 1. Only these two extreme cases are
worked out; in the cases occurring in practice, a: is a fraction which is very
difficult to estimate, because we must then have a precise determination
of the whole hght-curve (including the non-eclipse portions as well as the
secondary minimum) of either a star with a conspicuous constant phase at
principal minimum (the shallower the better), or a star in which the sum of
the losses of hght at the two minima (after correction for ellipticity of the
components) is nearly equal to the whole hght of the system.

In the D-hyp. we shall no longer be able to distinguish by inspection
of the primary minimum between annular and partial echpses, because in
the former case a constant minimum light is absent. Moreover the relation (4)
no longer holds good on account of the dependence of the depth of the minima
upon the darkening-coefficient as well as upon k and the area obscured.

First the case of a total eclipse at primary minimum has to be
considered. By mechanical quadrature Russell determines values of a
for various values of p and k. From this tabulated material curves, repres-
ented by equations of the form 0.\' — f {k,p) were drawn for fixed values
of k. Having assigned definite values to k, it is possible to invert the
a\'-function just determined and read off from the curves as a function
of a and k. Thus he gets the /»-table for this hypothesis (Table II at the
end of this paper) and with the aid of this table he derives the other
tables, the method being for the rest the same as in the U-hyp.»)

From Russell\'s closer investigation of systems with total eclipses
it appears that, if darkening really exists, the solution made without regarding
it (U-hyp.) will lead to a density too low for the large (and usually faint)

M Ap.j. 36, 239 (1912). H. N. Russbll and H. Shapley : On darkening at the Jimb in eclipsing
variables. I.

component and too high for the smaller (and usually bright) one. The most
important effect is on the computed density of the smaller star, which on
the average will be about twice as great in the U-hyp. as in the D-hyp. These
densities are computed on the assumption that both components have equal
masses. As it is well known that the brighter component of a close double
star is regularly the more massive, this component will generally be more
dense, and the fainter less dense, than computed on the above assumption.
It is probable then, that the computed „uniformquot; densities very nearly
represent the real conditions — the unequal masses being compensated for
by a considerable degree of darkening toward the limb.

As to the partial eclipses i) it appears that besides equation (4), the
empirical linear relations between the \'/-functions which fortunately saved
the „uniformquot; partial eclipse problem from a long trial and error process,
Hkewise fail to hold for darkened stars.

The question whether an eclipse is annular or partial can arise only
when the larger of the two components is considerably the brighter, and
the smaller one is in front of it during the principal minimum, for other-
wise the minimum at which total echpse is possible will be deep enough to
make it certain whether or not it shows a constant phase. When this cannot
be definitely decided upon, the partial and the annular eclipses have to be
treated as one problem.

By using new tables, viz. :

1°) a table giving the value of p for various values of k and aquot; —
this latter quantity may now be greater than unity --;
2 °) a table for the function {k, aquot;) ;

3°) tables for the functions 7 {k, a«, n) for n = f, \\ and n = o-
and by using a „trial and errorquot; process, Russell has succeeded in giving
solutions for the following cases :

la) partial eclipse at primary minimum, with larger star in front •
\\b)

)gt; gt;gt;nbsp;gt;gt; )) smaller

2) annular eclipse at primary mmimum.

1) Ap. J. 36, 385 (1912). H. N. Russell and H. Shapley : On darkening at the limb in 1\' quot;
variables. II.

Sometimes, however, two solutions present themselves, between
which there is no choice.

The relation connecting the depths of the minima with k and ao is
now more complicated than in the U-hypothesis (equation (4)) for if the
stars are of unequal radii the intensity-curves of primary and secondary
minimum are no longer connected by the simple relations 1—/, = a L, and
1—4 = k\'^a.Li ; and one may show a constant minimum light when the
other does not. If (as stated on p. 23) denotes the fraction of the light of
the smaller star which is lost at the greatest phase of its ecHpse behind the
larger, Russell calls the fraction of the Hght of the larger star which is lost
at the corresponding phase during the other ecHpse Q {k, a^) and gives
a table for the new function Q {k, ctj,) (Table C at the end of this paper).
Further we have as before :

1—Xi =nbsp;I—X., = «0^1^(^,00),

whence, since L^ L^ = \\ :

lt; = ..............................(11)

(For uniform disks the (^-function reduces to /e^).

In the following chapters we will use this equation in the following forms:

= 1 —Ipr -Q ^-\'j (larger star in front at primary minimum;case F,). ..(11«)

or

ot; = 1 —Xs« 4-nbsp;(si^allcr star in front at primary minimum; case £quot;;).. .(116)

As has already been stated (p. 22), the loss of Hght when the large
star is eclipsed at primary minimum is expressed in the loss of light at the
moment of internal tangency as a unit. At the corresponding moment in
the other eclipse the smaller star would be totally hidden, and a\'„ = 1.
Thus, according to the above formulas, we have: 1—X^ = L^Q{k, 1)
and this is the unit to be used in measuring the maximum obscuration a\'«\'.
We have therefore the relation:

a^Qik, 1) = a\'oQik, ct;)..............................(12)

which gives the value of a\'o for each pair of values of k and a«.

quot;) Ap.J. 36, 394 (1912).

If we have to deal with ellipsoidal stars, the quantity Z is determined from: i)

I = 1 —Z cos2

nbsp;...........) sm^^

by plotting for various points on the non-eclipse portion of the light-curve
({} near 90°) the values of I—I against the corresponding values of cos^ft.

With the aid of the value found for Z the light-curve is „rectifiedquot;
by subtracting the change in stellar magnitude due to ellipticity from the
observed magnitudes. Next the quantity = s^ sin^z, appearing in the
formulas, is determined by

„_ 57 5 72

and the rectified curve treated in the way already described.

The rectification-factor Z is often (especially if the stars are nearly

spherical) almost equal to half the corresponding factor from the U-hyp. ;

consequently the factor to be used in the formulas of the D-hyp. is about

4- of the value of z in the U-hyp.
8

1) Ap.J. 36, 400 (1912).

CHAPTER m.

Revised Method of Deriving the Elements of Eclipsing Binaries.

§ 7. Objections to Russell\'s Method.

After new magnitudes for the comparison-stars had been obtained,
so as to bring them into closer agreement with the step-estimates (see § 4),
it was curious to see how much some of the most unmanageable light-curves
improved. But there remained some difficulties, which could not be due to
erroneous magnitudes of the comparison-stars. Their origin obviously lay
in the method employed and it soon became evident, that they chiefly arose
from the fact, that the elements — especially the fundamental ratio k —
are based on the choice of two fixed points {a and b).

1°) The determination of k from points of the light-curve near and
between the fixed points a and b appears to be very uncertain (see also p. 2G,
note 1), small variations in t bringing about large changes in k. If, for
instance, we change the values of t belonging to a = 0.50 ; 0.70 ; 0.80 ; 0.95
for the curve II (table 12) from 0^168« ; 0^140; 0^24 and0^087 into0^08 ;
0^140®; 0^124« and 0\'!086® respectively, the corresponding values of k:
0.20 ; 0.17 ; 0.00 ; 0.11 become 0.12 ; 0.00 ; ? ; 0.06. And if t = 0\'Jl24 for
the point belonging to a =0.80 is changed into t =0.123® the correspond-
ing value of k changes from 0.00 into 0.11.

Now, the part of the light-curve between the points corresponding
to a = 0.40 and a = 0.95 is that, which the observations can determine
with the greatest precision ; hence wc should prefer it in deriving reliable
.values of k. But for the reason mentioned above Russell is obliged to
attach less weight to the values determined from it.

Likewise slight changes in the values of t for the points a and b may
cause great alterations in the values of k especially for the above mentioned
points near and between a and b. Therefore slightly discrepant values of i
for the fixed points a and b may be responsible for the above mentioned

systematic deviations which the series of values for k often present (see p. 10).
These systematic deviations, however, may also be due to the fact that
usually the observed Hght-curve is less reliable between a = 0.00 and a = 0.40

and again between a = 0.98 and a = 1.00.

The choice and the position of the fixed points a and b have there-
fore great influence on the results.

2°) Generally : by slightly modifying the light-curve we may obtain
curves, which, though they still fulfil the conditions of § 1 (p. 5), may yield
in Russell\'s treatment of the problem values of k varying to a wide extent.

TABLE 12«.

0^260

TABLE 126.

(0.60)

0^16650^154
0.15

0.1545

0.20

0.168
0.20

0.168
0.25

0.1675
0.15

0.169
0.10

0.1695
0.10
0.99 1.00 k,

0t25

I

0.068 |0.058

0.13 I0.I8

0.0675 0.058
0.13 \'0.20

0.06^1(7.058
0.02 0.09
0^068 (U)58
0.09 |o.l4
0.067

0.058
0.22

0.058
0.17

Table 12« contains the coordinates of various light-curves of
= F23 Cygni. Curves I and II are the curves mentioned on p. 11, while
Ila, lib, lie, lid and 11^ have been obtained by slight modifications of
curve II. The curves Ila, 116, lie and lid represent the observations al-
most as well as curve II, whereas curve 11^ is not quite so satisfactory.

Table 126 gives the values of k, deduced after Russell\'s method,
for various values of a.

Finally for each curve the value of k has been given (last column of
Table 126), obtained by a modified method, to be discussed in § 8. The first
six curves practically yield the same value for k ; only curve lie, which
represents the observations less satisfactorily, gives a somewhat deviating
value. The original curve I, from which a value of k could not be derived
after Russell\'s method, now gives about the same value as the other curves.
It may be pointed out, however, that the theoretical curve, based on the
elements derived from curve I (^ = 0.25 ; n = 0.312 ; i = 77°42\') dif-
fers somewhat from this observed curve, as is shown in Table 13.

TABLE 13.

Since the differences 0—C usually are very small for the points from
a = 0.25 to a = 0.95 (from O\'JOO to O-^Ol), the above differences suggest
that, by some cause or other, curve I does not give the real course of light-
variation for this system.

At the same time we see here again that we ought to be very careful
in changing somewhat the value of x for points on the light-curve ; very
small changes having a great influence on the results.

3°) The chief objection to Russell\'s method lies in the fact, that,
in order to get a suitable ^-series from the quantities A and B (see p. 25),
Russell is obliged to shift the points a and 6 by modifying while the

values 0.6 and 0.9 for a are retained. The „correctionsquot; sometimes amount
to 0\'?003. Since this process alters the course of the curve, the neighbouring
points should also undergo a change ; but for these Russell retains the
original values of t. Anyway, the curve which had been previously drawn
so as to represent the observations as exactly as possible is changed, in order
to adapt it to the theory.

To this we may add that the introduction of slight modifications
in the quantities A and B often requires a rather long time and may be
worked out with success only when the observed curve is a very good one.
If this is not the case, even if there are but slight deviations from the real
light-variation, this process cannot be followed at all with good success, or
there is a great uncertainty in the value to be adopted for k and in the values
of the other elements.

4°) In the case of a partial eclipse similar objections arise. The series
of values for C, determined with the value of Z) sin^ {) as derived from
the light-curve, is usually very unsatisfactory ; the differences are as a rule
great and frequentl}^ show a systematic character. As in the case of total
eclipses they may of course be caused by a systematic error in the light-curve
These differences in C usually cannot be smoothed over satisfactorily by
changing the value of D. Here again the objection remains that the light-
curve, as it has originally been drawn through the normals, will be vitiated
by Russell\'s method, since the change in the value of D means an alteration
in the value of x for the point of the Hght-curve corresponding to n _^

At any rate the function {k, a,, = remains usually very
uncertain. Even more so the values of k and since they may be percep-
tibly changed by insignificant alterations in the value of y {k a This
again may result in a very unstable solution in the case of a partial eclipse
even if the value of y {k, a«, has been determined fairiy well

In practice it appears that in the case of a partial eclipse it is as a

rule impossible to follow the process of varying the value of D in order to

get a somewhat reliable series of values for C, and therefore to obtain values

of K n and z which give a theoretical curve agreeing with the observed
curve.

The question now arises whether the method can be modified so as
to use the whole light-curve for the determination of k and the other elements,
instead of two fixed points playing the leading part. We should also satisfy
the requirement that nothing shall be altered in the light-curve as it has
been drawn after careful considerations, according to the principles men-
tioned on p. 5. But then it is desirable to use only the steeper parts of the
curve, which as a rule have been determined with greater accuracy and
which yield more sharply determined values of k. An error e. g. in the mag-
nitude of a comparison-star may cause a fairly great error in t in the upper
part of the light-curve, and may consequently give greatly deviating
values of k. Moreover this part of the light-curve may be rather uncertain
in consequence of the fact that it contains a smaller number of observations
per unit of time. Obviously for the part of the light-curve very near the
minimum, similar objections hold good. These parts of the light-curve may
afterwards point out whether the observations yielded a discordant light-
curve. If the light-curve is a very reliable one, they also might give a decision
between the U- and the D-hypothesis.

These considerations have led us to suggest the following revised
method, in which Russell\'s /»-tables have been retained.

§ 8. The primary Light-variation shows a constant minimum Light («o = 1.00).

The following cases may now occur :

I,nbsp;U-hypothesis;

1nbsp;case U, : total eclipse at primary minimum ;

2nbsp;case U„: annular echpse at primary minimum ;

II.nbsp;D-hypothesis ;

case D,: total eclipse at primary minimum.

\'On page 25 wc have found for the apparent distance of the centers of the disks:
= cos\'i sin-J sin« ft and o^» = r^\'il kp)\\

•nbsp;cos^ i sin= i sinquot;- ^ t =

or, putting sin«nbsp;A:

(1—/I) sin«t -I- \'^pkr, -hnbsp;= 1.................(1)

If we take, say, m values of a, the corresponding values of \\—l are
given by a = ^ and then the corresponding values of - by the light-
curve; this enables us to compute the quantities A. If, as a first approxim-
ation, we choose for k a certain value h — for further particulars on this
choise see p. 43 — the p- tables I or II (for U- en D-hyp. respectively)
give the values of p for each value of a and the chosen value of k. We thus
get m equations of the form (1), which contain three unknown quantities,
viz. k, and i ] they may be combined into three groups.

Suppose m = 15 and a = 0,25 ; 0.30 ; 0.35 ;................o.95

respectively and let group I contain 6 equations (a = 0,25............0.50)j

group II 5 equations (a ==0.50............0.75) and group III 4 equations

(a rr: 0.80............0.95). Since the curve is steeper for increasing

values of a, this choice will make each group represent about equal lengths
of the light-curve. Moreover less weight is now given to the upper part of
the curve than to the lower, steeper part, which furnishes more reliable
values of t.

Taking the means of the equations of each group, we get :
{l—A\\) sinquot;J -f 2 /gt;! kr,^ k^r^^ = 1 1

(1—Z2) sinH -H 2 p^ kriquot; J} k^r,^ = ll..............(2)

(1—Z3) sinquot; i 2 p^ kr,^ p^\' = 1 )

from which k, r, and i must be solved. Subtracting the second equation
from the first and third respectively :

{Al—A[) sinquot; J -f 2 (A—A) kry\' nbsp;k\'^r,^ = 0 )

{A,—A,) sinquot; i 2 {p,-p,) kr{\' nbsp;k^r,^ = 0 1............(^a)

and regarding these equations as 2 linear homogeneous equations with
sinquot; i, kri^ and k^ri^ as unknown quantities, we find :

k = = _2 M-^2) ^z-p.) - i^A,) ip.-p,)

kr,^nbsp;\'{A-A,)nbsp;-nbsp;...........

The value of k thus found will be regarded as a second approxima-
tion, the first being K The new values of p will be taken from tables I or II
and then a third approximation will be obtained. The process is to be repeated

1) Taken from Russell: Ap.J. 35, 333 (1912) and Ap.J. 36, 243 (1012).

until no further change is found. As a rule it converges remarkably quickly,
but we may still shorten it by taking a value of k lying between k^ and k^.
Practice has taught that the exact value of k is not far from
It is remarkable that this way of proceeding, though nearly independent of
the choice of h, always leads to good results.

Substituting now the final value of k and the corresponding values
of p and p\\ in the first and the third equation of (2):

(1—Z;) sinH nbsp;=nbsp;•

(1_J3) sinH -^{l ip, k p,\' k^) == I \\

we get sin^ i and nS hence i and r^. It will be clear that the first and the
third equation have been chosen because here the coefficients of sin« i and
^i« are differing most widely. If required, the solution may be facilitated by
using a table which gives the coefficients of r-c for various values of k.

A first approximation of k may be found from the supposition that
the echpse is central. If tx is the semi-duration of the echpse and t^ that of
the constant minimum-hght we get:

-T \'2
Sin -p-

sin-Zi

hencenbsp;, ,

, _ tg :: gt;\'

for which we may also take the somewhat larger value k = . Approximate
values of t, and L may be taken from the observed light-curve. A graphical
construction will\'easily show that if the total eclipse is not central, A;
The expression for k, just found, is therefore an upper limit.

It is to be remarked that even with a fixed initial value of k, for
instance k = i the process described will, by the use of equation (3),

lead to a good result.

The process may be accelerated by a table, which gives for various

values of k the corresponding values of /)», p,; p^, pi\\ see for this

table at the bottom of tables I and II.

It remains to be decided whether in the U-hyp. the primary
minimum is due to a total or to an annular eclipse. This may be done with
the aid of the equations of p. 27 :

«0 - 1-V ......(case ........................(5«)

and

^nbsp; ^......(case E,)........................(56)

which enables us to see which of the two suppositions gives the closest
approximation of 7o to unity, and thus to make our choice.

By the above method the elements of the system have been derived
in two suppositions ; the U-hyp. and the D-hyp. respectively. These, how-
ever, represent two extrem.e cases, neither of which is Hkely to occur in
reality ; the actual cases lying as a rule between these two extremes. There-
fore the question remains how the degree of darkening towards the limb is
to be found.

In order to answer this question we might have recourse :

1 °) to that part of the hght-curve which as yet has not been used
in deriving the elements ;

2°) in the case of a total (annular) eclipse, to the depth of the second-
ary minimum, which as a rule will come out pretty differently according
to the hypothesis (U or D) which served as a base in the computation of the
elements.

Unfortunately our knowledge of the secondary minima is as yet very
small. Our purpose would be best secured by a system with a sharply
determined primary and a deep secondary minimum ; the computed depths
are then very different in both hypotheses and the observed depth may give
some information.

If for a system of this kind, e.g. Fo = Z Hcrculis, the light-curves
were determined with great accuracy for both minima, we might also in another
way expect to get some insight in the interesting question of darkening
toward the limb. For in this case the minimum at which the eclipse is annular
cannot have a constant minimum light.

As to the extreme parts of the light-curve, they are as a rule rather
inaccurately determined, while the theoretical curves based on the U- and
D-hyp. respectively do not differ much for the parts under consideration.

Summarizing we find, that only with the aid of light-curves enjoying
the highest accuracy along their whole course, it will be possible to make
a tolerably reliable estimate of the degree of darkening.

How then are we to interpolate between the U- and the D-hyp. for

intermediate degrees of darkening (e. g. 0.1 ; 0.2 ;........0.9) ? It is obvious

that the values of p can be interpolated directly from the tables I and II.
In the way described the value of k is now to be found; it appears that
this value may be interpolated directly between those for the U-hyp. and
for the D-hyp. Then we have to determine that degree of darkening, at which
the parts mentioned sub 1°) of the theoretical light-curve coincide as well
as possible with those of the observed curve.

It appears in practice that we might as well interpolate directly
between the coefficients of as they have been found in the U- and in the
D-hyp. But this method may be a little risky when the degree of darkening
toward the limb amounts to about 0.5 ; in this case it would be better to inter-
polate between the values p„ pu h\'gt; P^\'^ P^\'ynbsp;in the two hypo-
theses and to calculate the new coefficients of with the aid of the inter-
polated value of k.

There is still another case in which we can state something concern-
ing the darkening. It may occur in systems that gave rise to total eclipses
that sinquot; i comes out gt; 1. This case deserves special treatment, as it may lead
to an upper limit for the darkening, since it appears that the D-hyp. for
such systems gives a greater value for i than the U-hyp. We may distinguish

two possibilities :

1 tlie value sinquot; i gt; 1 presents itself in the U-hypothesis.

In this case the D-hyp. need not be considered. We are at once forced
to put sinquot;i\' = 1, so that, according to (2), we get three equations with
two unknown quantities, k and r, ; these equations lead again to the formula
(3) for k. Next, adding the three equations (2), in which sinquot; t = i, and
substituting thé value of k just found, we derive r,.

2°) The value sin«^\' gt; 1 only results in the D-hypothesis.

This is more likely to occur than the former case. Since, then, a dar-
kening 1.0 (D-hyp.) gives an imaginary value for i, whereas a darkening
0.0 (U-hyp.) gives a real solution, we may interpolate between the two
hypotheses (in the manner described on p. 45) so as to make sinHquot; equal
to unity. This degree of darkening (D\') will evidently be an upper hmit,
consistent with the observed light-curve.

In cases of high refinement of the latter, the possibihty subsists of
interpolating between the D\'- and the U-hypothesis, according to the method
explained above.

Finally r^ may be computed as sub 1°).

§ 9. The primary Light-variation shows no constant ntinimum-Light.

The following cases may now occur :
I. U-hypothesis ; the echpse at primary minimum may be :
1 partial, with larger star in front (case Upi) ;
2°) „ „ smaller,, „ „ (case Ups) ;

11. D-hypothesis ; the eclipse at primary minimum may be :
1 °) partial, with larger star in front (case Dpi) ;
2°) „nbsp;„ smaller „ „ „ (case Dps) ;

3°) annular (case DJ.

U-hypothesis.

According to p. 27 we have :

ao = l-lpr ........(case E,)......................(5«)

or

=nbsp;........(case EJ......................(56)

The depth of the secondary minimum, which was not made use of
in the case of total (annular) eclipses, is now supposed to be known.

From equations (5) limits for cto and k may be easily derived for both
cases (Upi and Ups). The upper limit for is 1.00 and the lower limit
is that which corresponds io k = 1.00. Conversely, the upper limit for k
is 1.00 and the lower limit is that which corresponds to «o = 1.00.

We shall consider 15 fractions w = ^ of the greatest obscuration

ao, viz. 71 = 0.25 ; 0.30 ;..........0.95. For a known or adopted value

of «0, these fractions give 15 values of a, which allow us to write down 15
equations of the form (1). Reducing these to a system of the form (2) we
again find for k the expression (3).

Here again we may start from some initial value of k, provided that
it lies between the limits just found. Equation {5a) or {5b) gives the correspond-
ing value of Oo and then formula (3) yields the value of k, following from
the light-curve. The initial and the final values of k will in general not agree.
With the new value of k we proceed in the same way, while this rather
rapidly converging process may be shortened as in the case of a total eclipse.
In working it out we shall soon be able to discriminate between the cases
Upi and Ups ; it appears that if we hit the right case in computing the value
of czo from (5), the resulting value of k will fall somewhere between the two
last approximations ; should it fall outside, then the other case is to be taken.

In practice the best way of proceeding is the following : Adopt,
between the limits found, a value of «„ and find with formula (3) the
corresponding value of k from the light-curve. The equations (5) give the
value of k for both cases Upi and Ups. The three values of k will in general
not agree. Repeat the process with another value of oo- It will soon be clear
whether we have to deal with Upi or with Ups; and only a small number
of repetitions will be wanted in order to get a value of for which the light-
curve and one of the equations (4) yield the same value of k.

In order to accelerate the process we have derived a number of tables

la......Ic from table I for various values of a„ (0.90......0.50). These

tables will give at once the values ofnbsp;p,; pc, J}, pi for different

values of k. For intermediate values of a« these quantities may be obtained
by interpolation.

The numerical results may be checked by computing y from equation
(96) or (106), § 11. The value thus obtained must agree with that yielded by the
depth of the minima. For in the U-hyp. the ratio of the losses of light at
both minima will be equal to 7. We may also compute the quantities
L, =nbsp;and I, =nbsp;(case Up,) or L, =nbsp;and Z, =

(case Ups) (p. 27) and examine whether the condition Li = 1 has
been satisfied.

Finally the elements n and i may be found from equations (4) in the
same way as for total eclipses, substituting the value found for k in the first
and the third equation of (2).

D-hypothesis.

The case Dpi, which we shall first consider, may be treated in the
same way as case Upi. Instead of (5) we must now use the relations of p. 35 :

lt; = nbsp;......El)......................(6«)

or

......(case Es)......................(66)

We may start with an initial value of «o, arbitrarily chosen, but
lying between the limits which may be derived from equation (6«) and are
the same as those given by equation (5 a) in the case Upi. Formula (3) gives
the corresponding value of k from the light-curve, after which Table C furn-
ishes Q {k, o\'o) and then (6«) a new value of a\'o. The latter gives with (3)
a second approximation of k etc.

In practice, as in the U-hyp., it is easier to proceed as follows: Adopt
a value of a\'«. Then equation (6a) gives Q (/e, a\'«) and Table C the corres-
ponding value of k. The light-curve gives, by means of (3), also a value of k.
These two values of k will in general not agree, but after some trials the
value of ot;, for which the required agreement is obtained, will be easily found.

Here again it seemed convenient to accelerate tbe process by deriving

from table II new tables lla----lie, for different values of (O.OO. o 50)

They give the values of p,, \'p„ p, ;nbsp;p,^ for any value of k. For inter-

mediate values of these quantities may be obtained by interpolation.

Finally we have to discuss the cases Dp« and Da. As has been said
on p. 22, we now take the loss of light at the moment of internal contact
as a unit; a\'o and Oo are then connected by the relation of p 35 •

\\)=o:.Q{k, lt;)...........................

Table IIP) gives the values of p for every pair of values of k and aquot;.
If the eclipse is central, put a\'o = I a: ; if it is annular, such values of
aquot; as exceed unity may be taken 1 0.2, 1 0.4 Jt;... .1 a;.

^^\'ith an arbitrarily chosen value of Oo we get the corresponding
value of k from {6b) and table C, and then equation (7) gives the correspond-
ing value of oo, from which, by the aid of the Hght-curve and formula
(3) a new value of k may be found. The two values of k will in general not
agree. The process must be repeated until the required agreement is
obtained.

Here again we have, in order to accelerate the process, derived from

table III new tables III«..III^ for different values of a\'o (0.90____0.50),

though the cases Dps and Da do not occur very frequently.

The numerical results may be checked by computing the quantities
= and i =nbsp;(case Dp,) or L, =nbsp;and = ^^

(case Dps) and examining whether the condition L^nbsp;= 1 has been

satisfied.

Summarizing, we may treat the D-hyp. in the following way : From
equations (6) we learn whether at the primary minimum the smaller or the
larger star is eclipsed. In the latter case, in order to know whether the eclipse
at primary minimum is partial or annular, we start from a\'o = 1.00 and proceed
as described for the cases Dpi and Da. A further criterion might be found
in the remark on p. 34. Except for this preliminary question, both cases
may be treated along the same lines. If in the case Da the observed light-
curve is of high refinement, the points of the theoretical light-curve, found
for aquot; = 1 0.2 X] etc. might again give some information about the
degree of darkening toward the limb.

Finally the elements r^ and i will be found by means of equations (4).

§ 10. Ellipsoidal stars.

In the case of a Lyrid the light-curve has to be rectified by the factor
2 = £2 sin«t (see p. 31).

quot; ^TlTkcn from Kussbll : Ap.J. 36, 39lt;) (1912).

Let at any moment d, and 4 be the apparent major axes of\'the two
stars and (h and a^ their maximum values, then :

d^^ =nbsp;cos^ft),

and we get :

^^ = cos^^- Sinn- sin^» = d,^ (1 kpY = a^ {\\-z cos^^) (1 kp)K

If we put again sin^n = A and cos^amp; =nbsp;^ = B, we

now obtain, instead of (1), equations of the form :

(1—^) sin^ S OaMl ^Pf = 1

ornbsp;1

(L _ A] sin^ i a,\'nbsp; p\'k\' a,\'= t-................(1*)

B B)

Taking again, say, 15 of these equations and combining them into
three groups, as before, we obtain three equations of the form :

sin^ (1 2 Mnbsp;=

i v- .nbsp;I ^nbsp;^ -nbsp;■nbsp;r

sinH- (1 2 M Pl^\') «i\' = ...................(2*)

0(3 sinH\' (1 2 p,k a,^ =

in which

1nbsp;. R — - - • etc

We may solve sin» i from the second equation and substitute this
value in the first and third equations :

(1 2 A ^ A\' - (1 2 A; Pi k^)} ai = a, [3, - a. [5,
{a. (1 2j,k nbsp;-«3(1 2Pik Pik^)] ai = a,

Dividing the first of these equations by the second and puttingnbsp;= o,

we get, after some reduction :

k —-quot;«le______„

^ ^ _ Pi («3 - «2 Piquot;) Qnbsp;quot;2 Pi - «1 Pi 4- («3 Pi - «J^) y ~~nbsp;)

If we start with a certain value of k and the corresponding values
of p and J\\ equation (3*) gives a better approximation; the process is repeat-
ed as above (p. 41), until the required agreement has been arrived at.

When k has been found, the other elements «i and i are determ-
ined b}\'\':

a, sinquot;^ (1 2 ^ nbsp;a^^ = j

assinquot;^ (1 2 nbsp;= \\..................

It is evident, that the method of p. 42 can no longer be applied to
the case of ellipsoidal stars, on account of the appearance of the quantities
B ; the second members of the equations (1*) being no longer equal to each
other, we cannot derive the peculiar equations (2 a) nor the simple formula (3).

It is obvious that the general Lyrid-problem, here considered, will
be reduced to the ordinary Algol-problem with a constant maximum light,
by putting B = 1 and consequently = 1.

We then have p = ^^ and a^ = 1—^^ etc. Substituting the
value of p in (3*), the numerator of the third term becomes zero and we
obtain for ^ :

, _ _ 2nbsp;Pi ^2—«2 Pz) Q

~~nbsp;«2nbsp; («3 -«2

___ Q («2—«3) («2^1— («2—«1) («3 Pi—lt;^2 Pz)

quot; quot; («2-«s) («2nbsp;Pi^) («2-lt;\'l) («3 ^-«2

_nbsp;__ o (ai—— («3—«2)

— — 2nbsp;- (^j-^J)

(Al-A^) ipi^-pi\') - (^3-^2) ipi--p2^)

Tliis is again the formula found on p. 42.

With a high degree of approximation a similar value of k may be
derived from equation (3*). Substituting the value of p, we get :

2 Onbsp;{»jPi—\'^iPi) (quot;2 Pl—»l t^i) (quot;3 /\'2—«2 fa) ,,

quot; (quot;anbsp;/\'»Hquot;« P^^-quot;\'

(n,—fli) (quot;1 /\'s—quot;3 (quot;3—quot;l) («2/1—«1 fia)______

(„Jnbsp;it) («2nbsp;Pi^) («2 /^l-«! k) («3 /\'2quot;-«2nbsp;~

or, after some reduction :

If p, = ,3., = p3 this equation reduces to the equation, which yields
the value of k given by (3) of p. 42.
If we put

a, -a, = Rr\', «3 -«2 pa = ; -«2) (^3 -^2) -(«3 - «2)nbsp;= R,,

in which the quantities R are constants, easily obtainable, equation (3**)
becomes :

fe2 4. onbsp;-Aaj^-M-^--_ —--^^ _--= = 0.(5*)

With a high degree of approximation we may write for this value :
=nbsp;In the same way : R, =

Bi Bsnbsp;^3

and ^3 =nbsp; nbsp; ^ .

Substituting these values in (5*), we finally find :

k^ 2

For an Algol-system B, = B, = B:, = I and then this equation
reduces to the equation, which yields the value of k given by (3) of p. 42.
Should it appear that the expression

I

B, {A,—A,) -fnbsp;4-

is very small — which may be examined before and always seems to be
the case — equation (6*) may be written in the same form as obtained
for stars with constant maximum light, viz.:

^2 i. 2 3nbsp;(Fi-P^ J

or :nbsp;_ _ _

k = —2nbsp;(Fl-Pi)

(Ai-A,) ip^^^) ~ B,nbsp;..............(7*)

In the same way equation (5*) may be approximated to
^ iPz^Pi) -

(Pi-Piquot;) - (fi-Pi) ...................^ \'

Since the quantities a and 3 are required when we want to get i and a

from equation (4*), the best way is to use (8*) instead of (7*).

The approximated values from (7*) and (8*) appear to be virtually

the same as those yielded by the rigorous equations (5*) and (6*).

§ 11. Determination of theoretical Light-curves.

The elements i, r^ k and 7 are supposed to be known. From the
relation :

cos» i sin» i sin» p
we get :

sm -p ^ — sin2» .........................(o;

Let /o = 1 be the maximum light of the system, I the light at a
certain moment and I the light at minimum ; let S and s be the surfaces
of the disks of the larger and of the smaller star respectively ; a the eclipsed
part of the disk of the smaller star at a certain moment and oo the eclipsed

part at minimum.

Then we have for the case Ei:
I _54-jy — «^y

i;- S^-sy

or, since ^ = k» and 4 = 1 :

/ = 1 -nbsp;.................^................(S)«)

so that :

V =nbsp;..................................W

And for the case Es:

/ _ s 5 y — « A» s
T,-quot; s s).

or

1=nbsp;.................................(iOa)

1 y
so that:

A«

Kpr — 1 — IAi

V =nbsp;.................................im

The formulae (96) and (106) appear to be the same as formulae (4«)
and (46) on p. 27. From (96) it follows: «0 = 1— hr nbsp;^^^^^ ^^

that case 7 = jquot;we get: cto = 1 — hr

Xnbsp;\'-«CX/nbsp;-nbsp;-

And in the same way from (IO6):nbsp; quot; T and, since

^nbsp;=nbsp;hec

\' 1 ~~ f\'pf

When T gt; 1 we have the cases Ut (Dt) or Upi (Dpi) at primary minimum.

When7lt;l„ „ » „ Ua (Da) or Ups (Dps) „
When fa cosi lt; r^ the eclipse is partial.

а)nbsp;Cases Ut and Ua-

Here a.^ is equal to unity.nbsp;^

In the case Ut (t gt; 1), equation (96) gives: Ipr =
For different values of oc Table I yields the corresponding values of p,
after which equation (8) gives the corresponding values of From equation
(9a) we derive the loss of light of the system, which is readily translated
into a difference of stellar magnitude by means of Table A. The light-curve

can now be plotted.nbsp;^^

In the case Ua (y lt; 1), equation (10amp;) gives: X^. = 1 —
For the rest the procedure is as described in case Ut, equation (9 a)

now being replaced by equation (10 a).

б)nbsp;Cases Upi and Ups.

In the case Upi (y gt; 1), the relation

32 = cosquot; i sinquot; i sinquot; 0
shows that at the middle of the eclipse 0 = UU, = cos i (Fig. 4 p. 24). At

that moment is, according to p. 24 :

--- ^ ................

where cos B - !2i .cosquot;- V ^nd cos s =

wnere COb P — 2 r^ cos 1nbsp;2 r^ cos »

Further, according to (96), the light at the middle of the eclipse is :
1 _ 1 _ y

npr — inbsp;1nbsp;;ij2 J, •

The maximum obscuration otp is found by means of (11). The values
of p, corresponding to the different fractions n of the maximum obscura-
tion «0, are easily derived from Table L E.g. for =0.80 and n =0.40
we enter that table for the argument a = 0.32. Equation (8) gives now the
corresponding values of x. The corresponding losses of light, expressed in
magnitudes, are derived by means of (9 a) and then the light-curve can be
plotted.

In the case Ups (y lt; 1) we have, according to (106): = l — ^ ^.

For the rest we proceed as in case Upi, equation (10a) taking the

place of (9 a).

c)nbsp;Cases Dt, Dpi; Dps and Da.

In these cases, instead of a («o), we use the quantities a (a\'o) or aquot;
(oto) i. e. the losses of light expressed in the loss of light at the moment
of internal contact (p. 22/23). In case Dt a\'o = 1. In the cases Dpi and Dps
we may first compute, by means of (11), the eclipsed part an of the smaller
star and then by the use of Tables I and II — respectively I and III —
the corresponding values of a\'o and a\'«. For the rest the procedure is the
same as in the U-hyp., provided that the values of p in (8) are derived
either from Table II (case Dt), or from Tables Ila----lie (case Dpi), or

from Tables Ilia----III^J (case Dps).

Since, finally, in case D» we have aquot; = 1 at the moment of internal
contact, we may j^roceed as in case Ua, provided that for p in (8) we use
the values from Table III. We now may also use values of aquot; gt; 1, until
(8) yields for t the value 0. Then a\'«\' has also been found.

d)nbsp;In the foregoing pages we have described how, in general, theo-
retical curves of echpsing binaries may be computed. As a rule, however,
the elements k, r, and i have been derived from an observed light-curve ;
we often want to compute the theoretical light-curve with these elements,
both to compare it with the observed curve and to check the computations.
In this case the process described above may be shortened, as now the
magnitudes corresponding to each a or n are available ; in fact they have

already been determined and used in the course of the computation of the
elements and we have only to compute the corresponding values of - by
means of (8). In other words : the quantity 7, which determines the amount
of hght at minimum, is not taken into consideration, since the range of light-
varation is given by the observed curve and is used for the derivation of
the elements.

In the case of ellipsoidal stars the values of x are computed by means
of the relation. :

cosH sinn\'sin2f) ^nbsp;cos» f)) (1 kpY,

whence

2.-r 1 — ai2 (1 4- kp)^
cos»0 = COS» -^T =nbsp;...................(12)

Between the points a (w) = 0.25 and a {n) = 0.95 the theoretical
light-curve will sensibly coincide with the observed curve, provided the latter
is tolerably accurately observed. This, however, will as a rule not occur
with the extreme parts. These differences and the consequences to which
they may lead, have been previously discussed (p. 44).

We may conclude this paragraph by drawing attention to a pecul-
iarity which we have met with while investigating the system 712 = RT
Persei (See § 15). The light-curve, as drawn through the normals, did not
show a constant minimum light; so far as it was used for the determination
of the elements, it led to an ecHpse, either just total (U-hyp.) or nearly so
(D-hyp.). But the theoretical Hght-curve began to systemetically deviate
from the observed curve after the point for which n = 0.90 • it showed a
stationary minimum of short duration. Since the plotted observations did
not contradict this eventuality, the lower part of the light-curve was slightly
altered and the elements of the system were again derived. They naturally
differed but very little from those found before and yielded a theoretical
light-curve which showed a perfect agreement with the observed curve.

CHAPTER IV.
Applications.

§ 12. General Remarks.

We shall now apply the preceding theory to a small number of ob-
served light-curves.

An explanation of each of the columns in the following Tables 14;
15 ; 16 ; 17, prefixed by the heading of the column, is first given :

a, or in the case of a partial eclipse n : the fraction of the greatest
loss of light;

I—I: the corresponding loss of light, computed from the loss of light
at mid-eclipse ;

m : the corresponding stellar magnitude, derived from the data in
the preceding column by means of Table A;

- . ^ -r; Degr. : the corresponding phase, in days, radians and

degrees respectively;nbsp;__ _

A : the quantities giving A^nbsp;which appear in the formulae

(3) and (4), from which the elements k, r^ and i are computed as explained

in Chapter III.

Finally the theoretical light-curve has been derived :

To : the corresponding phases of the theoretical light-curve ;

0—C: differences between the observed and the computed curve.

The preceding theory was based upon the light-ratio 2.512. All the
magnitudes of the comparison-stars used in the derivation of the light-curve
were either directly taken from the HP or reduced to the Harvard-scale.
Of late years, however, the exactness of this ratio has been doubted.
According to investigations of Nijland\') there exists a striking difference

~~ 1) A N 206, 233 (1017): ^bcr die Schgrciizc dcs Utrcchtcr Zchtu611crs und die photometrischcn
Skalcn von E. C. I\'ickkring und J. A. Parkhurst.
Sw also - Hcmel en Dampkring 14, 05 (1»10).

between the photometric magnitudes of faint stars as given by E. C.
Pickering and J. A. Paekhurst, though both assert having used the ratio
2 512. When Pickering is right, Parkhurst\'s ratio must - accordmg to
Nijland - be 2.06 ; on the other hand if Parkhurst has used the ratio

2 612 the Harvard scale is based on the number 2.94.

\' The question is most important since the results of many mvest,gâ-
tions in stellar astronomy depend on the exactness of Pickering\'s magni-
tudes of faint stars. Van der Bilt made some investigations with the polar-
izing photometer of the Utrecht Observatory, which invanably seemed
to point to the exactness of Parkhurst\'s scale. Later, at the suggestion
of Kapteyn and Seares, a wire-gauze screen was placed before the objec-
tive when the brighter of two stars was measured. The obscuration caused
by \'the screen was accurately known and apphed to the small difference
in brightness measured with the photometer ; the light of the faint stars
now agreed fairly well with Pickering\'s values. An extensive investigation
following both methods and a detailed knowledge, for a large number of
individual cases, about the way in which the magnitudes of famt stars have
been obtained at Harvard, seem necessary before a dehnitive judgment

can be obtainedi).

Since, therefore, a decision in this question is impossible for the

present we\' have decided to use both scales in deriving the intensity-curve

from the magnitude-curve. If we should have to adopt Parkhurst\'s scale,

we might keep the light-curves already obtained, mtroducmg now the ratio

2.05 (Table B).

In the following examples, therefore, the elements will be determined
according to four suppositions successively :
A Light-ratio 2.512.
a) U-hypothesis ;
h) D-hypothesis.
B Light-ratio 2.05.

a)nbsp;U-hypothesis ;

b)nbsp;D-hypothesis.

1) j. v. d. Bilt : Note on the photometric scales of Pickering and Parkhurst. 13. A. N. 30, 107 ( 1922).

If we had a well-determined light-curve with a deep and sharply
determined secondary minimum at our disposal, such a curve might give
some valuable information as to the controversy between the two light-
scales. For, according to formulae 5 (p. 46)

1 — /.i«c

Oo = 1--^-pr

or

oto = 1 — hec M----

The combined loss of light at the middle of the two minima has
unity for its upper limit; should the variable be a Lyrid, the light-curve
must first be „rectifiedquot;, which lessens the depths of both minima. According
to Tables A and. B the above limit will sooner be reached with Pickering\'s
ratio than with Parkhurst\'s.

If lt; 1, the combined loss of light is less than unity; the limit
1 will only be reached in a central eclipse of equal stars (k = 1). If,
moreover, the two components should have the same surface-brightness,
the loss of light at the middle of the two minima is I—Ipr = l—lsec = 0.5.
This value corresponds to a depth of 0?75 in the case of Pickering (Table
A), but of 0®96® (Table B) in the case of Parkhurst. If the eclipse is not
central, the two equal minima are shallower. Should we, therefore, meet
with a light-curve which shows a primary and a secondary minimum of
nearly the same depth which (if necessary after a correction for ellipsoidal
form of the components) exceed 0T75, this fact would support the vaHdity
of Parkhurst\'s scale. It should be borne in mind, however, that if the
minima are of equal depth, we ought to make sure first whether they are
really primary and secondary and not both primary, the secondary minimum
not being observable. For this purpose we may have recourse to spectroscopic
observations. In the first case the period is twice as long as in the second.
If the light-curve has one pronounced maximum midway between the minima
the second supposition must be rejected.

A light-curve fulfilling the crucial conditions has not presented itself
as yet. Various light-curves have been found to have equal or almost equal
minima — for instance V 4 = U Ophiuchi, Fl6 = RX Herculis; 719
= SS Carinae; F31 = TX Herculis; Fll = RS Scuti — but the am-

plitudes are always less than 0?75, with the possible exception of F 11 =
Saiti. It is true that so far as a rectification of the curve has been appHed,
this more or less uncertain process maybe as our judgment to a certain degree.

Fll = RS Scuti and Fl2 = RT Lacertae are systems, the accu-
rate determination of whose light-curves would be, in this connection, of
the highest interest. Shapley has tried to derive elements for these stars i),
based upon observations oi V 11 = RS Scuti by Ichinohe and of F 12
= RT Lacertae by Luizet and Enebo. But these observations are not
accurate enough, especially in the case of F 11 = RS Scuti, to yield a
rehable light-curve. For this star Shapley, from observations at maximum
light, decides upon an ellipsoidal form of the components. Though such a
form really appears to exist, it probably has not such a pronounced character
as Shapley presumes. Removing the influence of this eUipsoidal form on
the brightness, he obtains a „rectifiedquot; curve with minima 0T75 deep.
According to Pickering\'s scale this is at the very limit of possibility.

As to F 12 = RT Lacertae, observations of Luizet and Enebo
make it highly probable that the brightness remains constant between the
eclipses. The depths of the two minima are found to be I\'^.oe and 0?61, so
that according to Table A the loss of light at these minima reaches the va-
lues 0.6233 and 0.4298 respectively, their sum exceeding therefore unity.
The introduction of an ellipsoidal form of the components proved necessary
to allow of a solution of the problem ; though, as we pointed out before,
this form is hardly compatible with the light-curve at maximum»). In Park-
hurst\'s scale, however, the total loss of light is only 0f89.

Tables A and B show that the combined loss of light at both mi-
nima comes out greater in Pickering\'s scale than in Parkhurst\'s. The
quantity Oo, therefore, will sooner attain the value 1.00 in the first case than
in the second, so that k must lie between narrower limits (see p. 46) when
we use Pickering\'s ratio. Hence we should expect that the possibility of

1) Contr. from the Princeton Un. Obs. 3, (I9I5); 86, 88, 101, 104, 157, 171.

») Nijland\'s observations, A.N. 211, 357 (1920), however, showed a very pronounced Lyrid-char-
acter, the depths of the minima being l\'.quot;4 and l\'.quot;9, respectively. There remains, it is true, some
doubt as to the constancy of the brightest of the comparison-stars during the years 1910—192l\'; this,
however, cannot have affected the shape of the light-curve to any appreciable amount.

deriving elements, for the system would be greater when we use Parkhurst\'s
scale. Nevertheless the examples which will be treated below strongly point
to the superiority of Pickering\'s scale.

Parkhurst\'s scale sometimes yields no set of elements at all or no
satisfactory one. See V 9 — Z Herculis (§ 14) ; F 48 = WZ Cygni (§16)
and F 12 = RT Persei (§15). It should be borne in mind, however, that
a change in the light-curve or in the range of the light-variation resulting
from new observ^ations, may materially influence the results.

Such systems as present a total or annular ecHpse at primary mi-
nimum and an appreciable and reliable depth of the secondary minimum,
may be regarded as suitable to investigate the question. For the derivation
of the elements the depth of this minimum is not wanted and its theoretical
amount may be computed afterwards by means of the value found for k,
and compared with the observed value. From the formula for oto and the
Tables A and B it follows, that the computed depth is much greater when
based on Parkhurst\'s scale than on Pickering\'s. The observed depth,
therefore, may also be used as a criterion. Such variables are for instance

Vnbsp;9 = Z Herculis (§ 14) ; V 9 = Z Vulpeculae ; V 21 = TT Aurigae;

Vnbsp;G = W Crucis.

As to the following examples, we have always assumed the orbit to
be circular — the observations do not give any indication of ellipticity —
and likewise we have assumed the brightness between the eclipses to be
constant ; except for the Lyrid V 48 = WZ Cygni, where this is obviously
not the case.