There are in the heavens many stars which shine with a variable
brilliancy. Amongst these there is a class which exhibits special
peculiarities; the members of this class are generally known as Algol
Variables, because the variability of the star Beta Persei or Algol was
the first of such cases to attract the attention of astronomers, and
because it is perhaps still the most remarkable of the whole class. But
the circumstances which led to this discovery were so extraordinary that
it seems worth while to pause a moment before entering on the subject.
John Goodricke, a deaf-mute, was born in 1764; he was grandson and heir
of Sir John Goodricke of Ribston Hall, Yorkshire. In November 1782,
he noted that the brilliancy of Algol waxed and waned (It is said that
Georg Palitzch, a farmer of Prohlis near Dresden, had about 1758 already
noted the variability of Algol with the naked eye. "Journ. Brit. Astron.
Assoc." Vol. XV. (1904-5), page 203.), and devoted himself to observing
it on every fine night from the 28th December 1782 to the 12th May 1783.
He communicated his observations to the Royal Society, and suggested
that the variation in brilliancy was due to periodic eclipses by a dark
companion star, a theory now universally accepted as correct. The
Royal Society recognised the importance of the discovery by awarding to
Goodricke, then only 19 years of age, their highest honour, the Copley
medal. His later observations of Beta Lyrae and of Delta Cephei were
almost as remarkable as those of Algol, but unfortunately a career of
such extraordinary promise was cut short by death, only a fortnight
after his election to the Royal Society. ("Dict. of National Biography";
article Goodricke (John). The article is by Miss Agnes Clerke. It is
strange that she did not then seem to be aware that he was a deaf-mute,
but she notes the fact in her "Problems of Astrophysics", page 337,
London, 1903.)
It was not until 1889 that Goodricke's theory was verified, when it
was proved by Vogel that the star was moving in an orbit, and in such
a manner that it was only possible to explain the rise and fall in the
luminosity by the partial eclipse of a bright star by a dark companion.
The whole mass of the system of Algol is found to be half as great again
as that of our sun, yet the two bodies complete their orbit in the short
period of 2d 20h 48m 55s. The light remains constant during each period,
except for 9h 20m when it exhibits a considerable fall in brightness
(Clerke, "Problems of Astrophysics" page 302 and chapter XVIII.); the
curve which represents the variation in the light is shown in a figure
titled "The light-curve and system of Beta Lyrae" (Fig. 7.).
The spectroscope has enabled astronomers to prove that many stars,
although apparently single, really consist of two stars circling around
one another (If a source of light is approaching with a great velocity
the waves of light are crowded together, and conversely they are spaced
out when the source is receding. Thus motion in the line of sight
virtually produces an infinitesimal change of colour. The position
of certain dark lines in the spectrum affords an exceedingly accurate
measurement of colour. Thus displacements of these spectral lines
enables us to measure the velocity of the source of light towards or
away from the observer.); they are known as spectroscopic binaries.
Campbell of the Lick Observatory believes that about one star in six is
a binary ("Astrophysical Journ." Vol. XIII. page 89, 1901. See also A.
Roberts, "Nature", Sept. 12, 1901, page 468.); thus there must be many
thousand such stars within the reach of our spectroscopes.
The orientation of the planes of the orbits of binary stars appears to
be quite arbitrary, and in general the star does not vary in brightness.
Amongst all such orbits there must be some whose planes pass nearly
through the sun, and in these cases the eclipse of one of the stars by
the other becomes inevitable, and in each circuit there will occur two
eclipses of unequal intensities.
It is easy to see that in the majority of such cases the two components
must move very close to one another.
The coincidence between the spectroscopic and the photometric evidence
permits us to feel complete confidence in the theory of eclipses. When
then we find a star with a light-curve of perfect regularity and with
a characteristics of that of Algol, we are justified in extending the
theory of eclipses to it, although it may be too faint to permit of
adequate spectroscopic examination. This extension of the theory secures
a considerable multiplication of the examples available for observation,
and some 30 have already been discovered.
Dr Alexander Roberts, of Lovedale in Cape Colony, truly remarks that
the study of Algol variables "brings us to the very threshold of the
question of stellar evolution." ("Proc. Roy. Soc. Edinburgh", XXIV. Part
II. (1902), page 73.) It is on this account that I propose to explain
in some detail the conclusion to which he and some other observers have
been led.
Although these variable stars are mere points of light, it has been
proved by means of the spectroscope that the law of gravitation holds
good in the remotest regions of stellar space, and further it seems now
to have become possible even to examine the shapes of stars by indirect
methods, and thus to begin the study of their evolution. The chain of
reasoning which I shall explain must of necessity be open to criticism,
yet the explanation of the facts by the theory is so perfect that it is
not easy to resist the conviction that we are travelling along the path
of truth.
The brightness of a star is specified by what is called its "magnitude."
The average brightness of all the stars which can just be seen with
the naked eye defines the sixth magnitude. A star which only gives
two-fifths as much light is said to be of the seventh magnitude; while
one which gives 2 1/2 times as much light is of the fifth magnitude,
and successive multiplications or divisions by 2 1/2 define the lower or
higher magnitudes. Negative magnitudes have clearly to be contemplated;
thus Sirius is of magnitude minus 1.4, and the sun is of magnitude minus
26.
The definition of magnitude is also extended to fractions; for example,
the lights given by two candles which are placed at 100 feet and 100
feet 6 inches from the observer differ in brightness by one-hundredth of
a magnitude.
A great deal of thought has been devoted to the measurement of the
brightness of stars, but I will only describe one of the methods used,
that of the great astronomer Argelander. In the neighbourhood of the
star under observation some half dozen standard stars are selected of
known invariable magnitudes, some being brighter and some fainter than
the star to be measured; so that these stars afford a visible scale of
brightness. Suppose we number them in order of increasing brightness
from 1 to 6; then the observer estimates that on a given night his star
falls between stars 2 and 3, on the next night, say between 3 and 4, and
then again perhaps it may return to between 2 and 3, and so forth. With
practice he learns to evaluate the brightness down to small fractions
of a magnitude, even a hundredth part of a magnitude is not quite
negligible.
For example, in observing the star RR Centauri five stars were in
general used for comparison by Dr Roberts, and in course of three months
he secured thereby 300 complete observations. When the period of the
cycle had been ascertained exactly, these 300 values were reduced to
mean values which appertained to certain mean places in the cycle, and
a mean light-curve was obtained in this way. Figures titled "Light curve
of RR Centauri" (Fig. 5) and "The light-curve and system of Beta Lyrae"
(Fig. 7) show examples of light curves.
I shall now follow out the results of the observation of RR Centauri
not only because it affords the easiest way of explaining these
investigations, but also because it is one of the stars which furnishes
the most striking results in connection with the object of this essay.
(See "Monthly notices R.A.S." Vol. 63, 1903, page 527.) This star has
a mean magnitude of about 7 1/2, and it is therefore invisible to
the naked eye. Its period of variability is 14h 32m 10s.76, the last
refinement of precision being of course only attained in the final
stages of reduction. Twenty-nine mean values of the magnitude were
determined, and they were nearly equally spaced over the whole cycle of
changes. The black dots in Fig. 5 exhibit the mean values determined by
Dr Roberts. The last three dots on the extreme right are merely the same
as the first three on the extreme left, and are repeated to show how
the next cycle would begin. The smooth dotted curve will be explained
hereafter, but, by reference to the scale of magnitudes on the margins
of the figure, it may be used to note that the dots might be brought
into a perfectly smooth curve by shifting some few of the dots by about
a hundredth of a magnitude.
This light-curve presents those characteristics which are due to
successive eclipses, but the exact form of the curve must depend on the
nature of the two mutually eclipsing stars. If we are to interpret the
curve with all possible completeness, it is necessary to make certain
assumptions as to the stars. It is assumed then that the stars are
equally bright all over their disks, and secondly that they are not
surrounded by an extensive absorptive atmosphere. This last appears to
me to be the most dangerous assumption involved in the whole theory.
Making these assumptions, however, it is found that if each of the
eclipsing stars were spherical it would not be possible to generate such
a curve with the closest accuracy. The two stars are certainly close
together, and it is obvious that in such a case the tidal forces
exercised by each on the other must be such as to elongate the figure
of each towards the other. Accordingly it is reasonable to adopt the
hypothesis that the system consists of a pair of elongated ellipsoids,
with their longest axes pointed towards one another. No supposition
is adopted a priori as to the ratio of the two masses, or as to their
relative size or brightness, and the orbit may have any degree of
eccentricity. These last are all to be determined from the nature of the
light-curve.
In the case of RR Centauri, however, Dr Roberts finds the conditions are
best satisfied by supposing the orbit to be circular, and the sizes and
masses of the components to be equal, while their luminosities are to
one another in the ratio of 4 to 3. As to their shapes he finds them
to be so much elongated that they overlap, as exhibited in his figure
titled "The shape of the star RR Centauri" (Fig. 6.). The dotted curve
shows a form of equilibrium of rotating liquid as computed by me some
years before, and it was added for the sake of comparison.
On turning back to Fig. 5 the reader will see in the smooth dotted curve
the light variation which would be exhibited by such a binary system as
this. The curve is the result of computation and it is impossible not to
be struck by the closeness of the coincidence with the series of black
dots which denote the observations.
It is virtually certain that RR Centauri is a case of an eclipsing
binary system, and that the two stars are close together. It is not
of course proved that the figures of the stars are ellipsoids, but
gravitation must deform them into a pair of elongated bodies, and, on
the assumptions that they are not enveloped in an absorptive atmosphere
and that they are ellipsoidal, their shapes must be as shown in the
figure.
This light-curve gives an excellent illustration of what we have reason
to believe to be a stage in the evolution of stars, when a single star
is proceeding to separate into a binary one.
As the star is faint, there is as yet no direct spectroscopic evidence
of orbital motion. Let us turn therefore to the case of another star,
namely V Puppis, in which such evidence does already exist. I give an
account of it, because it presents a peculiarly interesting confirmation
of the correctness of the theory.
In 1895 Pickering announced in the "Harvard Circular" No. 14 that the
spectroscopic observations at Arequipa proved V Puppis to be a double
star with a period of 3d 2h 46m. Now when Roberts discussed its
light-curve he found that the period was 1d 10h 54m 27s, and on account
of this serious discrepancy he effected the reduction only on the simple
assumption that the two stars were spherical, and thus obtained a fairly
good representation of the light-curve. It appeared that the orbit was
circular and that the two spheres were not quite in contact. Obviously
if the stars had been assumed to be ellipsoids they would have been
found to overlap, as was the case for RR Centauri. ("Astrophysical
Journ." Vol. XIII. (1901), page 177.) The matter rested thus for some
months until the spectroscopic evidence was re-examined by Miss Cannon
on behalf of Professor Pickering, and we find in the notes on page
177 of Vol. XXVIII. of the "Annals of the Harvard Observatory" the
following: "A.G.C. 10534. This star, which is the Algol variable V
Puppis, has been found to be a spectroscopic binary. The period 1d.454
(i.e. 1d 10h 54m) satisfies the observations of the changes in light,
and of the varying separation of the lines of the spectrum. The spectrum
has been examined on 61 plates, on 23 of which the lines are double."
Thus we have valuable evidence in confirmation of the correctness of the
conclusions drawn from the light-curve. In the circumstances, however,
I have not thought it worth while to reproduce Dr Roberts's provisional
figure.
I now turn to the conclusions drawn a few years previously by another
observer, where we shall find the component stars not quite in contact.
This is the star Beta Lyrae which was observed by Goodricke, Argelander,
Belopolsky, Schur, Markwick and by many others. The spectroscopic method
has been successfully applied in this case, and the component stars are
proved to move in an orbit about one another. In 1897, Mr. G.W. Myers
applied the theory of eclipses to the light-curve, on the hypothesis
that the stars are elongated ellipsoids, and he obtained the interesting
results exhibited in Fig. 7. ("Astrophysical Journ." Vol. VII. (1898),
page 1.)
The period of Beta Lyrae is relatively long, being 12d 21h 47m, the
orbit is sensibly eccentric, and the two spheroids are not so much
elongated as was the case with RR Centauri. The mass of the system is
enormous, one of the two stars being 10 times and the other 21 times as
heavy as our sun.
Further illustrations of this subject might be given, but enough has
been said to explain the nature of the conclusions which have been drawn
from this class of observation.
In my account of these remarkable systems the consideration of one very
important conclusion has been purposely deferred. Since the light-curve
is explicable by eclipses, it follows that the sizes of the two stars
are determinable relatively to the distance between them. The period of
their orbital motion is known, being identical with the complete period
of the variability of their light, and an easy application of Kepler's
law of periodic times enables us to compute the sum of the masses of the
two stars divided by the cube of the distance between their centres.
Now the sizes of the bodies being known, the mean density of the whole
system may be calculated. In every case that density has been found to
be much less than the sun's, and indeed the average of a number of mean
densities which have been determined only amounts to one-eighth of that
of the sun. In some cases the density is extremely small, and in no case
is it quite so great as half the solar density.
It would be absurd to suppose that these stars can be uniform in
density throughout, and from all that is known of celestial bodies it
is probable that they are gaseous in their external parts with great
condensation towards their centres. This conclusion is confirmed by
arguments drawn from the theory of rotating masses of liquid. (See J.H.
Jeans, "On the density of Algol variables", "Astrophysical Journ." Vol.
XXII. (1905), page 97.)
Although, as already explained, a good deal is known about the shapes
and the stability of figures consisting of homogeneous incompressible
liquid in rotation, yet comparatively little has hitherto been
discovered about the equilibrium of rotating gaseous stars. The figures
calculated for homogeneous liquid can obviously only be taken to afford
a general indication of the kind of figure which we might expect to find
in the stellar universe. Thus the dotted curve in Fig. 5, which exhibits
one of the figures which I calculated, has some interest when placed
alongside the figures of the stars in RR Centauri, as computed from the
observations, but it must not be accepted as the calculated form of such
a system. I have moreover proved more recently that such a figure of
homogeneous liquid is unstable. Notwithstanding this instability it does
not necessarily follow that the analogous figure for compressible fluid
is also unstable, as will be pointed out more fully hereafter.
Professor Jeans has discussed in a paper of great ability the difficult
problems offered by the conditions of equilibrium and of stability of a
spherical nebula. ("Phil. Trans. R.S." Vol. CXCIX. A (1902), page 1. See
also A. Roberts, "S. African Assoc. Adv. Sci." Vol. I. (1903), page 6.)
In a later paper ("Astrophysical Journ." Vol. XXII. (1905), page 97.),
in contrasting the conditions which must govern the fission of a star
into two parts when the star is gaseous and compressible with the
corresponding conditions in the case of incompressible liquid, he points
out that for a gaseous star (the agency which effects the separation
will no longer be rotation alone; gravitation also will tend towards
separation... From numerical results obtained in the various papers of
my own,... I have been led to the conclusion that a gravitational
instability of the kind described must be regarded as the primary agent
at work in the actual evolution of the universe, Laplace's rotation
playing only the secondary part of separating the primary and satellite
after the birth of the satellite.)
It is desirable to add a word in explanation of the expression
"gravitational instability" in this passage. It means that when the
concentration of a gaseous nebula (without rotation) has proceeded to
a certain stage, the arrangement in spherical layers of equal density
becomes unstable, and a form of bifurcation has been reached. For
further concentration concentric spherical layers become unstable, and
the new stable form involves a concentration about two centres. The
first sign of this change is that the spherical layers cease to be
quite concentric and then the layers of equal density begin to assume
a somewhat pear-shaped form analogous to that which we found to occur
under rotation for an incompressible liquid. Accordingly it appears that
while a sphere of liquid is stable a sphere of gas may become unstable.
Thus the conditions of stability are different in these two simple
cases, and it is likely that while certain forms of rotating liquid are
unstable the analogous forms for gas may be stable. This furnishes a
reason why it is worth while to consider the unstable forms of rotating
liquid.
There can I think be little doubt but that Jeans is right in looking to
gravitational instability as the primary cause of fission, but when we
consider that a binary system, with a mass larger than the sun's, is
found to rotate in a few hours, there seems reason to look to rotation
as a contributory cause scarcely less important than the primary one.
With the present extent of our knowledge it is only possible to
reconstruct the processes of the evolution of stars by means of
inferences drawn from several sources. We have first to rely on the
general principles of stability, according to which we are to look for a
series of families of forms, each terminating in an unstable form, which
itself becomes the starting-point of the next family of stable forms.
Secondly we have as a guide the analogy of the successive changes in
the evolution of ideal liquid stars; and thirdly we already possess some
slender knowledge as to the equilibrium of gaseous stars.
From these data it is possible to build up in outline the probable
history of binary stars. Originally the star must have been single, it
must have been widely diffused, and must have been endowed with a slow
rotation. In this condition the strata of equal density must have been
of the planetary form. As it cooled and contracted the symmetry round
the axis of rotation must have become unstable, through the effects of
gravitation, assisted perhaps by the increasing speed of rotation. (I
learn from Professor Jeans that he now (December 1908) believes that
he can prove that some small amount of rotation is necessary to induce
instability in the symmetrical arrangement.) The strata of equal
density must then become somewhat pear-shaped, and afterwards like an
hour-glass, with the constriction more pronounced in the internal than
in the external strata. The constrictions of the successive strata then
begin to rupture from the inside progressively outwards, and when at
length all are ruptured we have the twin stars portrayed by Roberts and
by others.
As we have seen, the study of the forms of equilibrium of rotating
liquid is almost complete, and Jeans has made a good beginning in the
investigation of the equilibrium of gaseous stars, but much more remains
to be discovered. The field for the mathematician is a wide one, and in
proportion as the very arduous exploration of that field is attained so
will our knowledge of the processes of cosmical evolution increase.
From the point of view of observation, improved methods in the use of
the spectroscope and increase of accuracy in photometry will certainly
lead to a great increase in our knowledge within the next few years.
Probably the observational advance will be more rapid than that of
theory, for we know how extraordinary has been the success attained
within the last few years, and the theory is one of extreme difficulty;
but the two ought to proceed together hand in hand. Human life is
too short to permit us to watch the leisurely procedure of cosmical
evolution, but the celestial museum contains so many exhibits that it
may become possible, by the aid of theory, to piece together bit by bit
the processes through which stars pass in the course of their evolution.
In the sketch which I have endeavoured to give of this fascinating
subject, I have led my reader to the very confines of our present
knowledge. It is not much more than a quarter of a century since this
class of observation has claimed the close attention of astronomers;
something considerable has been discovered already and there seems
scarcely a discernible limit to what will be known in this field a
century from now. Some of the results which I have set forth may then be
shown to be false, but it seems profoundly improbable that we are being
led astray by a Will-of-the-Wisp.