Popular Science Monthly/Volume 57/October 1900/Chapters on the Stars IV

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CHAPTERS ON THE STARS.
By Prof. SIMON NEWCOMB, U. S. N.

VARIABLE STARS.

IT is a curious fact that the ancient astronomers, notwithstanding the care with which they observed the heavens, never noticed that any of the stars changed in brightness. The earliest record of such an observation dates from 1596, when the periodical disappearance of Omicron Ceti was noticed. After this, nearly two centuries elapsed before another case of variability in a star was recorded. During the first half of the nineteenth century Argelander so systematized the study of variable stars as to make it a new branch of astronomy. In recent years it has become of capital interest and importance through the application of the spectroscope.

Students who are interested in the subject will find the most complete information attainable in the catalogues of variable stars, published from time to time by Chandler in the 'Astronomical Journal.' His third catalogue, which appeared in 1896, comprises more than 300 stars whose variability has been well established, while there is always a long list of 'suspected variables'—whose cases are still to be tried. The number to be included in the established list is continually increasing at such a rate that it is impossible to state it with any approximation to exactness. The possibility of such a statement has been yet further curtailed by the recent discovery at the Harvard Observatory that certain clusters of stars contain an extraordinary proportion of variables. Altogether at the time of the latest publication, 509 such stars were found in twenty-three clusters. The total number of these objects in clusters, therefore, exceeds the number known in the rest of the sky. They will be described more fully in a subsequent chapter. For the present we are obliged to leave this rich field out of consideration and confine our study to the isolated variable stars which are found in every region of the heavens.

Variable stars are of several classes, which, however, run into each other by gradations so slight that a sharp separation cannot always be made between them. Yet there are distinguishing features, each of which marks so considerable a number of these stars as to show some radical difference in the causes on which the variations depend.

We have first to distinguish the two great classes of irregular and periodic stars. The irregular ones increase and diminish in so fitful a way that no law of their change can be laid down. To this class belong the so-called 'new stars', which, at various periods in history, have blazed out in the heavens, and then in a few weeks or months have again faded away. It is a remarkable fact that no star of the latter class has ever been known to blaze out more than once. This fact distinguishes new stars from other irregularly variable ones.

Periodic stars are those which go through a regular cycle of changes in a definite interval of time, so that, after a certain number of days, sometimes of hours, the star returns to the same brightness. But even in the case of periodic stars, it is found that the period is more or less variable, and, in special cases, the amount of the variation is such that it cannot always be said whether we should call a star periodic or irregular.

The periodic stars show wide differences, both in the length of the period and in the character of the changes they undergo. In most cases they rapidly increase in brightness during a few days or weeks, and then slowly fade away, to go through the same changes again at the end of the period. In other cases they blaze up or fade out, from time to time, like the revolving light of a lighthouse. Some stars are distinguished more especially by their maximum, or period of greatest brightness, while others are more sharply marked by minima, or periods of least brightness. In some cases there are two unequal minima in the course of a period.

Chandler's third catalogue of variable stars gives the periods of 280 of these objects, which seem to have been fairly well made out. A classification of these periods, as to their length, will be interesting. There are, of periods:

Less than 50 days 63 Stars.
Between 50 and 100 days 6 "
" 100 " 150 " 9 "
" 150 " 200 " 18 "
" 200 " 250 " 29 "
" 250 " 300 " 40 "
" 300 " 350 " 44 "
" 350 " 400 " 44 "
" 400 " 450 " 18 "
" 450 " 500 " 6 "
" 500 " 550 " 1 "
" 550 " 600 " 1 "
" 600 " 650 " 1 "

It will be seen from this that, leaving out the cases of very short period, the greater number of the periods fall between 300 and 400 days. From this value the number falls off in both directions. Only three periods exceed 500 days, and of these the longest is 610 days. We infer from this that there is something in the constitution of these stars, or in the causes on which their variation depends, which limits the period. This limitation establishes a well-marked distinction between the periodic stars and the irregular variables to be hereafter described.

Returning to the upper end of the scale, the contrast between the great number of stars less than fifty days, and the small number between fifty and one hundred, seems to show that we have here a sharp line of distinction between stars of long and those of short period. But, when we examine the matter in detail, the statistics of the periods do not enable us to draw any such line. About eight periods are less than one day, and the number of this class known to us is continually increasing. About forty are between one and ten days, and from this point upwards they are scattered with a fair approach to equality up to a period of one hundred days. There is, however, a possible distinction, which we shall develop presently.

The law of change in a variable star is represented to the eye by a curve in the following way. We draw a straight horizontal line A X to

PSM V57 D650 The law of change in a variable star.png
Fig. 1. The Law of Change in a Variable Star.

represent the time. A series of equidistant points, a, b, c, d, etc., on this will represent moments of time. One of the spaces, a, b, c, etc., may represent an hour, a day, or a month, according to the rapidity of change. We take a to represent the initial moment, and erect an ordinate aa', of such length as to represent the brightness of the star on some convenient scale at this moment. At the second moment, b, which may be an hour or a day later, we erect another ordinate bb’, representing the brightness at this moment. We continue this process as long as may be required. Then we draw a curve, represented by the dotted line, through the ends of all the ordinates. In the case of a periodic star it is only necessary to draw the curve through a single period, since its continuation will be a repetition of its form for any one period.

We readily see that if a star does not vary, all the ordinates will be of equal length, and the curve will be a horizontal straight line. Moreover, the curve will take this form through any portion of time during which the light of the star is constant.

There are three of the periodic stars plainly visible to the naked eye at maximum, of which the variations are so wide that they may easily be noticed by any one who looks for them at the right times, and knows how to find the stars. These stars are:

Omicron Ceti, called also Mira Ceti.
Beta Persei, or Algol.
Beta Lyrae.

It happens that each of these stars exemplifies a certain type or law of variations.

Omicron Ceti. On August 13, 1596, David Fabricius noticed a star in the constellation Cetus, which was not found in any catalogue. Bayer, in his 'Uranometria', of which the first edition was published in 1601, marked the star Omicron, but said nothing about the fact that it was visible only at certain times. Fabricius observed the star from time to time, until 1609, but he does not appear to have fully and accurately recognized its periodicity. But so extraordinary an object could not fail to command the attention of astronomers, and the fact was soon established that the star appeared at intervals of about eleven months, gradually fading out of sight after a few weeks of visibility. Observations of more or less accuracy having been made for more than two centuries, the following facts respecting it have been brought to light:

Its variations are somewhat irregular. Sometimes, when at its brightest, it rises nearly or quite to the second magnitude. This was the case in October, 1898, when it was about as bright as Alpha Ceti. At other times its maximum brightness scarcely exceeds the fifth magnitude. No law has yet been discovered by which it can be predicted whether it shall attain one degree of brightness or another at maximum.

Its minima are also variable. Sometimes it sinks only to the eighth magnitude; at other times to the ninth or lower. In either case it is invisible to the naked eye.

As with other stars of this kind, it brightens up more rapidly than it fades away. It takes a few weeks from the time it becomes visible to reach its greatest brightness, whatever that may be. It generally retains this brightness for two or three weeks, then fades away, gradually at first, afterward more rapidly. The whole time of visibility will, therefore, be two or three months. Of course, it can be seen with a telescope at any time.

The period also is variable in a somewhat irregular way. If we calculate when the star ought to be at its greatest brightness on the supposition that the intervals between the maxima ought to be equal, we shall find that sometimes the maximum will be thirty or forty days early, and at other times thirty or forty days late. These early or late maxima follow each other year after year, with a certain amount of regularity as regards the progression, though no definable law can be laid down to govern them. Thus, during the period from 1782 to 1800 it was from thirteen to twenty-four days late. In 1812 it was thirty-nine days late. From 1845 to 1856 it was on the average about a month too early. Several recent maxima, notably those from 1895 to 1898, again occurred late. Formulæ have been constructed to show these changes, but there is no certainty that they express the actual law of the case. Indeed, the probability seems to be that there is no invariable law that we can discover to govern it.

Argelander fixed the length of the period at 331.9 days. More recently, Chandler fixed it at 331.6 days. It would seem, therefore, to have been somewhat shorter in recent times. It was at its maximum toward the end of October, 1898. We may, therefore, expect that future maxima will occur in July, 1901; June, 1902; May, 1903; April, 1904, and so on, about a month earlier each year. During the few years following 1903 the maxima will probably not be visible, owing to the star being near conjunction with the sun at the times of their occurrence. The most plausible view seems to be that changes of a periodic character, involving the eruption of heated matter from the interior of the body to its surface, followed by the cooling of this matter by radiation, are going on in the star.

The star Algol, or Beta Persei, as it is commonly called in astronomical language, may, in northern latitudes, be seen on almost any night of the year. In the early summer we should probably see it only after midnight, in the northeast. In late winter it would be seen in the northwest. From August until January one can find it at some time in the evening by becoming acquainted with the constellations. It is nearly of the second magnitude. One might look at it a score of times without seeing that it varied in brilliancy. But at certain stated intervals, somewhat less than three days, it fades away to nearly the fourth magnitude for a few hours, and then slowly recovers its light. This fact was first discovered by Goodrick in 1783, since which time the variations have been carefully followed. The law of variation thus defined is expressed by a curve of the following form:

PSM V57 D652 Law or variation of a stab of the algol type.png
Fig. 2. Law or Variation of a Stab of the Algol Type.

The idea that what we see in the star is a partial eclipse caused by a dark body revolving round it, was naturally suggested even to the earliest observers. But it was impossible to test this theory until recent times. Careful observation showed changes in the period between the eclipses, which, although not conclusive against the theory, might have seemed to make it somewhat unlikely. The application of the spectroscope to the determination of radial motions, enabled Vogel, of Potsdam, in 1889, to set the question at rest. His method of reasoning and proceeding was this:

If the fading out which we see is really due to an eclipse by a dark body, that body must be nearly or quite as large as the star itself, else it could not cut off so much of its light. In this case, it is probably nearly as massive as the star itself, and therefore would affect the motion of the star. Both bodies would, in fact, revolve around their common center of gravity. Therefore, when after the dark body has passed in front of the star, it has made one-fourth of a revolution, which would require about seventeen hours, the star would be moving towards us. Again, seventeen hours before the eclipse, it ought to be moving away from us.

The measurement of six photographs of the spectrum, of which four were taken before the eclipses and two afterward, gives the following results:

Before eclipses: Velocity from the sun equals 39 km. per second.
After eclipses: "Velocity toward the sun equals 47 km. per second.

These results show that the hypothesis in question is a true one, and afforded the first conclusive evidence of a dark body revolving around a distant star. A study of the law of diminution and recovery of the light during the eclipse, combined with the preceding motions, enabled Vogel to make an approximate estimate of the size of the orbit and of the two bodies. The star itself is somewhat more than a million of miles in diameter; the dark companion a little less. The latter is about the size of our sun. Their distance apart is somewhat more than three millions of miles; the respective masses are about one-half and one-fourth that of the sun. These results, though numerically rather uncertain, are probably near enough to the truth to show us what an interesting system we here have to deal with. We can say with entire certainty that the size and mass of the dark body exceed those of any planet of our system, even Jupiter, several hundred fold.

The period of the star is also subject to variations of a somewhat singular character. These have been attributed by Chandler to a motion of the whole system around a third body, itself invisible. This theory is, however, still to be proved. Quite likely the planet which causes the eclipse is not the only one which revolves around this star. The latter may be the center of a system like our solar system, and the other planets may, by their action, cause changes in the motion of the body that produces the eclipses. The most singular feature of the change is that it seems to have taken place quite rapidly, about 1840. The motion was nearly uniform up to near this date; then it changed, and again remained nearly uniform until 1890. Since then no available observations have been published.

It is found that several other stars vary in the same way as Algol; that is to say, they are invariable in brightness during the greater part of the time, but fade away for a few days at regular intervals. This is a kind of variation which it is most difficult to discover, because it will be overlooked unless the observer happens to notice the star during the time when an eclipse is in progress, and is thoroughly aware of its previous brightness. One might observe a star of this kind very accurately a score of times, without hitting upon a moment when the partial eclipse was in progress. On the principle that like effects are due to like causes, we are justified in concluding that in the cases of all stars of this type, the eclipses are caused by the revolution of a dark body, now called 'Algol variables,' round the principal star.

A feature of all the Algol variables is the shortness of the periods. The longest period is less than five days, while three are less than one day. This is a result that we might expect from the nature of the ease. The nearer a dark planet is to the star, the more likely it will be to hide its light from an observer at a great distance. If, for example, the planet Jupiter were nearly as large as the sun, the chances would be hundreds to one against the plane of the orbit being so nearly in the line of a distant observer that the latter would ever see an eclipse of the sun by the planet. But if the planet were close to the sun, the chances might increase to one in ten, and yet farther to almost any extent, according to the nearness of the two bodies.

Still, we cannot set any definite limit to the period of stars of this type; all we can say is that, as the period we seek for increases, the number of stars varying in that period must diminish. This follows not only from the reason just given, but from the fact that the longer the interval that separates the partial eclipses of a star of the Algol type, the less likely they are to be detected.

 
STARS OF THE BETA LYRÆ TYPE.

The star Beta Lyræ shows variations quite different in their nature from those of Algol, yet having a certain analogy to them. Anyone who looks at the constellation Lyræ a few nights in succession and compares Beta with Gamma, a star of nearly the same brightness in its neighborhood, will see that while on some evenings the stars are of equal brightness, on others Beta will be fainter by perhaps an entire magnitude.

A careful examination of these variations shows us a very remarkable feature. On a preliminary study, the period will seem to be six and one-half days. But, comparing the alternate minima, we shall find them unequal. Hence the actual period is thirteen days. In this period there are two unequal minima, separated by equal maxima. That is to say, the partial eclipses at intervals of six and one-half days are not equal. At the alternate minima the star is half as bright again as at the intermediate minima.

It is impossible to explain such a change as this merely by the interposition of a dark body, and this for two reasons. Instead of remaining invariable between the minima, the variation is continuous during the whole period, like the rising and falling of a tide. Moreever, the inequality of the alternating minima is against the theory.

Pickering, however, found from the doubling of the spectral lines that there were two stars revolving round each other. Then Prof. G. W. Myers, of Indiana, worked out a very elaborate mathematical theory to explain the variations, which is not less remarkable for its ingenuity than for the curious nature of the system it brings to light. His conclusions are these:

Beta Lyræ consists of two bodies, gaseous in their nature, which revolve round each other, so near as to be almost touching. They are of unequal size. Both are self-luminous. By their mutual attraction they are drawn out into ellipsoids. The smaller body is somewhat darker than the other. When we see the two bodies laterally, they are at their brightest. As they revolve, however, we see them more and more end on, and thus the light diminishes. At a certain point one begins to cover the other and hide its light. Thus the combined light continues to diminish until the two bodies move across our line of sight. Then we have a minimum. At one minimum, however, the smaller and darker of the two bodies is projected upon the brighter one, and thus diminishes its light. At the other minimum, it is hiding behind the other, and therefore we see the light of the larger one alone.

This theory receives additional confirmation from the fact, shown by the spectroscope, that these stars are either wholly gaseous, or at least have self-luminous atmospheres. Some of Professor Myers's conclusions respecting the magnitudes are summarized as follows:

The larger body is about 0.4 as bright as the smaller.

The flattening of the ellipsoidal masses is about 0.17.

The distance of centers is about 1⅞ the semi-major axis of the larger star, or about 50,000,000 kilometers (say 30,000,000 miles).

The mass of the larger body is about twice that of the smaller, and 9½ times the mass of the sun.

The mean density of the system is a little less than that of air.[1]

It should be remarked that these numbers rest on spectroscopic results, which need further confirmation. They are, therefore, liable to be changed by subsequent investigation. What is most remarkable is that we have here to deal with a case to which we have no analogy in our solar system, and which we should never have suspected, had it not been for observations of this star.

The gap between the variable stars of the Algol type and those of the Beta Lyræ type is, at the present time, being filled by new discoveries in such a way as to make a sharp distinction of the two classes difficult. It is characteristic of the Algol type proper that the partial eclipses are due to the interposition of a dark planet revolving round the bright star. But suppose that we have two nearly equal stars, A and B, revolving round their common center of gravity in a plane passing near our system. Then, A will eclipse B, and, half a revolution later, B will eclipse A, and so on in alternation. But, when the stars are equal, we may have no way of deciding which is being eclipsed, and thus we shall have a star of the Algol type, so far as the law of variation is concerned, yet, as a matter of fact, belonging rather to the Beta Lyræ type. If the velocity in the line of sight could be measured, the question would be settled at once. But only the brightest stars can, so far, be thus measured, so that the spectroscope cannot help us in the majority of cases.

The most interesting case of this kind yet brought to light is that of Tau Cygni. The variability of this star, ordinarily of the fourth magnitude, was discovered by Chandler in December, 1886. The minima occurred at intervals of three days. But in the following summer he found an apparent period of 1 d. 12 h., the alternate minima being invisible because they occurred during daylight, or when the star was below the horizon. With this period the times of minima during the summer of 1888 were predicted.

It was then found that the times of the alternate minima, which, as we have just said, were the only ones visible during any one season, did not correspond to the prediction. The period seemed to have greatly changed. Afterward, it seemed to return to its old value. After puzzling changes of this sort, the tangle was at length unraveled by Dunér, of Lund, who showed that the alternate periods were unequal. The intervals between minima were one day nine hours, one day fifteen hours, one day nine hours, one day fifteen hours, and so on, indefinitely. This law once established, the cause of the anomaly became evident. Two bright stars revolve round their common center of gravity in a period of nearly three days. Each eclipses the other in alternation. The orbit is eccentric, and, in consequence, one-half of it is described in a less time than the other half. If we could distinguish the two stars by telescopic vision, and note their relative positions at the four cardinal points of their orbit, we should see the pair alternately single and double, as shown in the following diagrams:

A B
Position (1), stars at pericenter * *
Interval, 16 hours.
Position (2), A eclipses B *
Interval, 20 hours.
B A
Postion (3), stars at apocenter * *
Interval, 20 hours.
Position (4), B eclipses A *
Interval, 16 hours.
Position (1) is repeated * *

U Pegasi is a star which proved as perplexing as Tau Cygni. It was first supposed to be of the Algol type, with a period of about two days. Then it was found that a number of minima occurred during this period, and that the actual interval between them was only a few hours. The great difficulty in the case arises from the minuteness of the variation, which is but little more than half a magnitude between the extremes. The observations of Wendell, at the Harvard Observatory, with the polarizing photometer, enabled Pickering to reach a conclusion

PSM V57 D657 Light curve of the u pegasi.png

Fig. 3. Light Curve or U Pegasi, of the Beta Lyræ Type, from Observations by Wendell at the Harvard Observatory. Magnitude at Maximum, 9.32; at Principal Minimum, 9.90; at Secondary Minimum, 9.76. Period, 9 hours.

which, though it may still be open to some doubt, seems to be the most likely yet attainable. The star is of the Beta Lyræ type; its complete period is 8 hours 59 minutes 41 seconds, or 19 seconds less than nine hours; during this period it passes through two equal maxima, each of magnitude 9.3, and two unequal minima 9.76 and 9.9, alternately.

The difference of these minima, 0m. 14, is less than the errors which really ordinarily affect measures of a star's magnitude with the best photometers. Some skepticism has, therefore, been felt as to the reality of the difference which, if it does not exist, would reduce the periodic time below four and one-half hours, the shortest yet known. But Pickering maintains that, in observations of this kind upon a single star, the precision is such that the reality of the difference, small though it be, is beyond reasonable doubt.

Taking Pickering's law of change as a basis, Myers has represented the light-curve of U Pegasi on a theory similar to that which he constructed for Beta Lyræ. His conclusion is that, in the present case, the two bodies which form the visible star are in actual contact. A remarkable historic feature of the case is that Poincaré has recently investigated, by purely mathematical methods, the possible forms of revolving fluid masses in a condition of equilibrium, bringing out a number of such forms previously unknown. One of these, which he calls the apiodal form, consists of two bodies joined into one, and it is this which Myers finds for U Pegasi.

Quite similar to these two cases is that of Zeta Herculis. This star, ordinarily of the seventh magnitude, was found, at Potsdam, in 1894, to diminish by about one magnitude. Repeated observations elsewhere indicate a period of very nearly four days. Actually it is now found to be only ten minutes less than four days. The result was that during any one season of observation the minima occur at nearly the same hour every night or day. To an observer situated in such longitude that they occur during the day, they would, of course, be invisible.

Continued observations then showed a secondary minimum, occurring about half-way between the principal minima hitherto observed. It was then found that these secondary minima really occur between one and two hours earlier than the mid-moment, so that the one interval would be between forty-six and forty-seven hours and the other between forty-nine and fifty. The time which it takes the star to lose its light and regain it again is about ten hours. More recent observations, however, do not show this inequality, so that there is probably a rapid motion of the pericenter of the orbit.

It will be seen that this star combines the Algol and Beta Lyræ types. It is an Algol star in that its light remains constant between the eclipses. It is of the Beta Lyræ type in the alternate minima being unequal.

From a careful study, Seliger and Hartwig derived the following particulars respecting this system:

Diameter of principal star, 15,000,000 kilometers.
" smaller " 12,000,000 "
Mass of the larger star, 172 times sun's mass.
Mass of the smaller star, 94 times sun's mass.
Distance of centers, 45,000,000 kilometers.
Time of revolution, 3d. 23h. 49m. 32.7s.

It must be added that the data for these extraordinary numbers are rather slender and partly hypothetical.

Beta Lyræ is always of the same brightness at the same hour of its period, and Algol has always the same magnitude at minimum. It is true that the length of the period varies slowly in the case of these stars. But this may arise from the action of other invisible bodies revolving around the visible stars. This general uniformity is in accord with the theory which attributes the apparent variations to the various aspects in which we see one and the same system of revolving stars.

Another variable star showing some unique features is Eta Aquilæ. What gives it special interest is that spectroscopic observations of its radial motion show it to have a dark body revolving round it in a very eccentric orbit, and in the same time as the period of variation. It might therefore be supposed that we have here a star of the Algol or Beta Lyræ type. But such is not the case. There is nothing in the law of variation to suggest an eclipsing of the bright star, nor does it seem that the variations can readily be represented by the varying aspects of any revolving system.

The orbit of this star has been exhaustively investigated by Wright from Campbell's observations of the radial motion. The laws of change in the system are shown by the curves below, which are reproduced, in great part, from Wright's paper in the 'Astrophysical Journal.'

PSM V57 D659 Light-curve and radial velocity of eta aquilae.png
Fig. 4. Light-Curve and Radial Velocity of Eta Aquilæ.

The lower curve is the light-curve of the star during a period of 7.167 days. Starting from a maximum of 3.5 mag., it sinks, in the course of 5 days, to a minimum of 4.7m. It was found by Schwab that the diminution is not progressive, but that a secondary maximum of 3.8m. is reached at the end of the second day. After reaching the principal minimum it rises rapidly to the principal maximum in 2¼ days.

The upper curve shows the radial velocity of the star during the period of variation. It will be seen that the epoch of greatest negative velocity, which referred to the center of mass of the system, is 16.2 km, per second, occurs at the time of maximum brightness. The greatest positive velocity, 23.9 km., occurs during the sixth day of the period just after the time of minimum brightness.

Finally, the moments of inferior and superior conjunction of the dark body with the bright one are neither of them an epoch of minimum brightness, which takes place half-way between the two.

The most plausible conclusion we can draw is that the light of the star is affected by the action of the dark body during its revolution. But how the change may be produced we cannot yet say.

 
CLASSIFICATION OF VARIABLE STARS.

A classification of variable stars, based on the period of variation and the law of change, was proposed by Pickering. It does not, however, seem that a hard and fast line can yet be drawn between different types and classes of these bodies, one type running into another, as we have found in the case of the Algol and Beta Lyræ types. Yet the discovery of the cause of the variation in these types makes it likely that a division into two great classes, dependent on the cause of variation, is possible. We should then have:

(1) Stars, or systems, constituting to vision a single star, of which the apparent variability arises from the rotation of the system as a whole, or from the revolution of its components around each other.

(2) Stars of which the changes arise from other and as yet unknown causes.

The main feature of the stars of the first class is that we are under no necessity of supposing any actual change in the amount of light which they emit. Their apparent variations are purely the effect of perspective, arising from the various aspects which they present to us during their revolution round each other. If we could change our point of view so that the plane of the orbit of Algol's planet no longer passed near our system, Algol would no longer be a variable star. Under the same circumstances the apparent variations in a star of the Beta Lyræ type would cease to be noticeable, if they did not disappear entirely.

The stars of this class are also distinguished by the uniformity and regularity with which they go through their cycle of change.

The stars of the other class, which we may call the Omicron Ceti type, are different not only in respect to the length of the period, but in the character of the variation. There are certain general laws of variation and irregularities of brightness which stars of this class go through. Starting from the time of the minimum, the increase of light is at first very slow. It grows more and more rapid as the maximum is approached, in which time there may be as great an increase in two or three days as there formerly was in a month. The diminution of light is generally slower than the increase. The magnitude at corresponding times in different periods may be very different. Thus, as we have already remarked, Omicron Ceti is ten times as bright at some maxima as it is at others. The periods also, so far as they have been made out, vary more widely than those of stars of the other type.

The idea has sometimes been entertained that these variations of light are due to a revolution of the star on its axis. A very little consideration will, however, show that this explanation cannot be valid. However bright a star might be on one side, or however dark on the other, any one region of its surface would be visible to us half the time and a change of brightness from different degrees of brilliancy on different sides would be gradual and regular.

It is not impossible that the variability may be in some way connected with the action of a body revolving round the star. This seems to be the case with Eta Aquilæ. The radial motion of this object shows the existence of a dark body revolving round it in the same period as that of the star's variation.

From what has been said, it will be seen that, although a sharp line cannot be drawn, there seems to be some distinction between the stars of short and long periods. The number of stars which have been known to belong to the first class is quite small, only about fifteen, all told. On the other hand, there are still left some stars having a period less than ten days, which are otherwise not distinguishable from the Omicron Ceti type. It seems quite likely that the variations in the periods of these stars are, in some way, connected with the revolution of bright or dark bodies round them.

They also vary more widely than those of stars of the other two types. This might easily happen in the case of stars really variable through a cycle of changes going on in consequence of the action of interior causes.

The periodic stars of short period, which have not been recognized as of the Algol or Beta Lyræ type, form an interesting subject of study. Although the separation between them and the stars of long period is not sharp, it seems likely to have some element of reality in it. But no conclusions on the subject can be reached until the light-curves of a large number of them are carefully drawn; and this requires an amount of patient and accurate observation which cannot be carried out for years to come.

 
SUSPECTED VARIATIONS IN THE COLOR OF STARS.

The question whether certain stars vary in color without materially changing their brightness has sometimes been raised. This was at one time supposed to be the case with one of the stars of Ursa Major. This suspected variation has not, however, been confirmed, and it does not seem likely that any such changes take place in the color of stars not otherwise variable.

 
POSSIBLE SECULAR VARIATIONS IN THE BRILLIANCY OF STARS.

All the variations we have hitherto considered take place with such rapidity that they can be observed by comparisons embracing but a short interval of time—a few days or months at the outside. A somewhat different question of great importance is still left open. May not individual stars be subject to a secular variation of brilliancy, meaning by this term a change which would not be sensible in the course of only one generation of men, but admitting of being brought out by a comparison of the brightness of the stars at widely distant epochs? Is it certain that, in the case of stars which we do not recognize as variable, no change has taken place since the time of Hipparchus and Ptolemy? This question has been investigated by C. S. Pierce and others. The conclusion reached is that no real evidence of any change can be gathered. The discrepancies are no greater than might arise from errors of estimates.

There is, however, an analogous question which is of great interest and has been much discussed in recent times. In several ancient writings the color of Sirius is described as red. This fact would, at first sight, appear to afford very strong evidence that, within historic times, the color of the brightest star in the heavens has actually changed from red to a bluish white.

Two recent writers have examined the evidence on this subject most exhaustively and reached opposite conclusions. The first of these was Dr. T. J. J. See, who collated a great number of cases in which Sirius was mentioned by ancient writers as red or fiery, and thus concluded that the evidence was in favor of a red color in former times. Shortly afterwards, Schiaparelli examined the evidence with equal care and thoroughness and reached an opposite conclusion, showing that the terms used by the ancient authors, which might have indicated redness of color, were susceptible of other interpretations; they might mean fiery, blazing, etc., as well as red in color, and were therefore probably suggested by the extraordinary brightness of Sirius and the strangeness with which it twinkled when near the horizon. In this position a star not only twinkles, but changes its color rapidly. This change is not sensible in the case of a faint star, but if one watches Sirius when on the horizon, it will be seen that it not only changes in appearance, but seems to blaze forth in different colors.

It seems to the writer that this conclusion of Schiaparelli is the more likely of the two. From what we know of the constitution of the stars, a change in the color of one of these bodies in so short a period of time as that embraced by history is so improbable as to require much stronger proofs than any that can be adduced from ancient writers. In addition to the possible vagueness or errors of the original writers, we have to bear in mind the possible mistakes or misinterpretations of the copyists who reproduced the manuscripts.

 
THE PARALLAXES OF THE STARS.

It needs only the most elementary conceptions of space, direction and motion to see that, as the earth makes its vast swing from one extremity of its orbit to the other, the stars, being fixed, must have an apparent swing in the opposite direction. The seeming absence of such a swing was in all ages before our own one of the great stumbling blocks of astronomy. It was the base on which Ptolemy erected his proof that the earth was immovable in the center of the celestial sphere. It was felt by Copernicus to be a great difficulty in the reception of his system. It led Tycho Brahe to suggest a grotesque combination of the Ptolemaic and Copernican systems, in which the earth was the center of motion, round which the sun revolved, carrying the planets with it.

With every improvement in their instruments, astronomers sought to detect the annual swing of the stars. Each time that increased accuracy in observations failed to show it, the difficulty in the way of the Copernican system was heightened. How deep the feeling on the subject is shown by the enthusiastic title, Copernicus Triumphans, given by Horrebow to the paper in which, from observations by Roemer, he claimed to have detected the swing. But, alas, critical examination showed that the supposed inequality was produced by the varying effect of the warmth of the day and the cold of the night upon the rate of the clock used by the observer, and not by the motion of the earth.

Hooke, a contemporary of Newton, published an attempt to determine the parallax of the stars, under the title of "An Attempt to Prove the Motion of the Earth," but his work was as great a failure as that of his predecessors. Had it not been that the proofs of the Copernican system had accumulated until they became irresistible, these repeated attempts might have led men to think that perhaps, after all, Ptolemy and the ancients were somehow in the right.

The difficulty was magnified by the philosophic views of the period. It was supposed that Nature must economize in the use of space as a farmer would in the use of valuable land. The ancient astronomers correctly placed the sphere of the stars outside that of the planets, but did not suppose it far outside. That Nature would squander her resources by leaving a vacant space hundreds of thousands of times the extent of the solar system was supposed contrary to all probability. The actual infinity of space; the consideration that one had only to enlarge his conceptions a little to see spaces a thousand times the size of the solar system look as insignificant as the region of a few yards round a grain of sand, does not seem to have occurred to anyone.

Considerations drawn from photometry were also lost sight of, because that art was still undeveloped. Kepler saw that the sun might well be of the nature of a star; in fact, that the stars were probably suns. Had he and his contemporaries known that the light of the sun was more than ten thousand million times that of a bright star, they would have seen that it must be placed at one hundred thousand times its present distance to shine as a bright star. If, then, the stars are as bright as the sun, they must be one hundred thousand times as far away, and their annual parallax would then have been too small for detection with the instruments of the time. Such considerations as this would have removed the real difficulty.

The efforts to discover stellar parallax were, of course, still continued. Bradley, about 1740, made observations on γ Draconis, which passed the meridian near his zenith, with an instrument of an accuracy before unequalled. He thus detected an annual swing of 20" on each side of the mean. But this swing did not have the right phase to be due to the motion of the earth; the star appeared at one or the other extremity of its swing when it should have been at the middle point, and vice versa. What he saw was really the effect of aberration, depending on the ratio of the velocity of the earth in its orbit to the velocity of light. It proved the motion of the earth, but in a different way from what was expected. All that Bradley could prove was that the distances of the stars must be hundreds of thousands of times that of the sun.

An introductory remark on the use of the word parallax may preface a statement of the results of researches now to be considered.

In a general way, the change of apparent direction of an object arising from a change in the position of an observer is termed parallax. More especially, the parallax of a star is the difference of its direction as seen from the sun and from that point of the earth's orbit from which the apparent direction will be changed by the greatest amount. It is equal to the angle subtended by the radius of the earth's orbit, as seen from the star. The simplest conception of an arc of one second is reached by thinking of it as the angle subtended by a short line at a distance of two hundred and six thousand times its length. To say that a star has a parallax of 1" would therefore be the same thing as saying that it was at a distance of a little more than two hundred thousand times that of the earth from the sun. A parallax of one-half a second implies a distance twice as great; one of one-third, three times as great. A parallax of 0"20 implies a distance of more than a million times that of our unit of measure.

The first conclusive result as to the extreme minuteness of the parallax of the brighter stars was reached by Struve, at Dorpat, about 1830. In the high latitude of Dorpat the right ascension of a star can be determined with great precision, not only at the moment of its transit over the meridian, but also at transit over the meridian below the pole, which occurs twelve hours later. He, therefore, selected a large group of stars which could be observed twice daily in this way at certain times of the year, and made continuous observations on them through the year. It was not possible, by this method, to certainly detect the parallax of any one star. What was aimed at was to determine the limit of the average parallax of all the stars thus observed. The conclusion reached was that this limit could not exceed one-tenth of a second and that the average distance of the group could not, therefore, be much less than two million times the distance of the sun; if, perchance, some stars were nearer than this, others were more distant.

By a singular coincidence, success in detecting stellar parallax was reached by three independent investigators almost at the same time, observing three different stars.

To Bessel is commonly assigned the credit of having first actually determined the parallax of a star with such certainty as to place the result beyond question. The star having the most rapid proper motion on the celestial sphere, so far as known to Bessel, was 61 Cygni, which is, however, only of the fifth magnitude. This rapid motion indicated that it was probably among the stars nearest to us, much nearer, in fact, than the faint stars by which it is surrounded.

After several futile attempts, he undertook a series of measurements with a heliometer, the best in his power to make, in August, 1837, and continued them until October, 1838. The object was to determine, night after night, the position of 61 Cygni, relative to certain small stars in its neighborhood. Then he and hisassistant, Sluter, made a second series, which was continued until 1840. All these observations showed conclusively that the star had a parallax of about 0".35.

While Bessel was making these observations, Struve, at Dorpat, made a similar attempt upon Alpha Lyræ. This star, in the high northern latitude of Dorpat, could be accurately observed throughout almost the entire year. It is one of the brightest stars near the Pole and has a sensible proper motion. There was, therefore, reason to believe it among the nearest of the stars. The observations of Struve extended from 1835 to August, 1838, and were, therefore, almost simultaneous with the observations made by Bessel on 61 Cygni. He concluded that the parallax of Alpha Lyræ was about one-fourth of a second. Subsequent investigations have, however, made it probable that this result was about double the true value of the parallax.

The third successful attempt was made by Henderson, of England, astronomer at the Cape of Good Hope. He found from meridian observations that the star Alpha Centauri had a parallax of about 1". This is a double star of the first magnitude, which, being only 30° from the south celestial pole, never rises in our latitudes. Its nearness to us was indicated not only by its magnitude, but also by its considerable proper motion.

Although subsequent investigation has shown the parallax of this body to be less than that found by Henderson, it is, up to the time of writing, the nearest star whose distance has been ascertained. The extreme difficulty of detecting movements so slight as those we have described, when they take six months to go through their phases, will be obvious to the reader. He would be still more impressed with it when, looking through a powerful telescope at any star, he sees how it flickers in consequence of the continual motions going on in the air through which it is seen and how difficult it must be to fix any point of reference from which to measure the change of direction.

The latter is the capital difficulty in measuring the parallax. How shall we know that a star has changed its direction by a fraction of a second in the course of six months? There must be for this purpose some standard direction from which we can measure.

The most certain of these standard directions is that of the earth's axis of rotation. It is true that this direction varies in the course of the year, but the amount of the variation is known with great precision, so that it can be properly allowed for in the reduction of the observations. The angle between the direction of a star and that of the earth's axis, the latter direction being represented by the celestial pole, can be measured with our meridian instruments. It is, in fact, the north polar distance of the star, or the complement of its declination. If, therefore, the astronomer could measure the declination of a star with great precision throughout the entire year, he would be able to determine its parallax by a comparison of the measures. But it is found impossible in practice to make measures of so long an arc with the necessary precision. The uncertain and changing effect of the varying seasons and different temperatures of day and night upon the air and the instrument almost masks the parallax. After several attempts with the finest instruments, handled with the utmost skill, to determine stellar parallax from the declinations of the stars, the method has been practically abandoned.

The method now practiced is that of relative parallax. By this method the standard direction is that of a small star apparently alongside one whose parallax is to be measured, but, presumably, so much farther away that it may be regarded as having no parallax. In this assumption lies the weak point of the method. Can we be sure that the smaller stars are really without appreciable parallax? Until recent times it was generally supposed that the magnitude of the stars afforded the best index to their relative distances. If the stars were of the same intrinsic brilliancy, the amount of light received from them would, as already pointed out, have been inversely as the square of the distance. Although there was no reason to suppose that any such equality really existed, it would still remain true that, in the general average, the brighter stars must be nearer to us than the fainter ones. But when the proper motions of stars came to be investigated, it was found that the amount of this motion afforded a better index to the distance than the magnitude did.

The diversity of actual or linear motion is not so wide as that of absolute brilliancy. Stars have, therefore, in recent times, been selected for parallax very largely on account of their proper motion, without respect to their brightness. It is now considered quite safe to assume that the small stars without proper motion are so far away that their parallax is insensible.

Ever since the time of Bessel the experience of practical astronomers has tended toward the conclusion that the best instrument for delicate measurements like these is the heliometer. This is an equatorial telescope of which the object glass is divided along a diameter into two semicircles, which can slide along each other. Each half of the object glass forms a separate image of any star at which the telescope may be pointed. By sliding the two halves along each other, the images can be brought together or separated to any extent. If there are two stars in proximity, the image of one star made by one-half of the glass can be brought into coincidence with that of the other star made by the other half. The sliding of the two halves to bring about this coincidence affords a scale of measurement for the angular distance of the two stars.

The most noteworthy forward steps in improving the heliometer are due to the celebrated instrument-makers of Hamburg, the Messrs. Repsold, aided by the suggestions of Dr. David Gill, astronomer at the Cape of Good Hope. The latter, in connection with his coadjutor, Elkin, made an equally important step in the art of managing the instrument and hence in determining the parallax of stars. The best results yet attained are those of these two observers and of Peter, of Germany.

Yet more recently, Kapteyn, of Holland, has applied what has seemed to be the unpromising method of differences of right ascension observed with a meridian circle. This method has also been applied by Flint, at Madison, Wis. Through the skill of these observers, as well as that of Brünnow and Ball, in applying the equatorial telescope to the same purposes, the parallax of nearly 100 stars has been measured with some approach to precision.

A rival method to that of the heliometer has been discovered in the photographic telescope. The plan of this instrument, and its application to such purposes as this, are extremely simple. We point a telescope at a star and set the clock-work going, so that the telescope shall remain pointed as exactly as possible in the direction of the star. We place a sensitized plate in the focus and leave it long enough to form an image both of the particular star in view and of all the stars around it. The plate being developed, we have a permanent record of the relative positions of the stars which can be measured with a suitable instrument at the observer's leisure. The advantage of the method consists in the great number of stars which may be examined for parallax, and in the rapidity with which the work can be done.

The earliest photographs which have been utilized in this way are those made by Rutherfurd in New York during the years 1860 to 1875. The plates taken by him have been measured and discussed principally by Rees and Jacoby, of Columbia University. Before their work was done, however, Pritchard, of Oxford, applied the method and published results in the case of a number of stars.

One of the pressing wants of astronomy at the present time is a parallactic survey of the heavens for the purpose of discovering all the stars whose parallax exceeds some definable limit, say 0"1. Such a survey is possible by photography, and by that only. A commencement, which may serve as an example of one way of conducting the survey, has been made by Kapteyn on photographic negatives taken by Donner at Helsingfors.

These plates cover a square in the Milky Way about two degrees on the side, extending from 35° 50' in declination to 36° 50', and from 20h. lm. in R. A. to 20h. 10m. 24s. Three plates were used, on each of which the image of each star is formed twelve times. Three of the twelve impressions were made at the epoch of maximum parallactic displacement, six at the minimum six months later, and three at the following maximum. The parallaxes found on the plates can only be relative to the general mean of all the other stars, and must therefore be negative as often as positive. The following positive parallaxes, amounting to 0"1, came out with some consistency from the measures:

Star, B. D., 3972 Mag. 8.6 R. A. 20h., 2m. 0s. Dec. +35°.5 Par. +0".11
Star, B. D., 3883 Mag. 7.1 R. A. 20h., 2m. 3s. Dec. +36°.1 Par. +0".18
Star, B. D., 4003 Mag. 9.2 R. A. 20h., 4m. 58s. Dec. +35°.4 Par. +0".10
Star, B. D., 3959 Mag. 7.0 R. A. 20h., 9m. 14s. Dec. +36°.3 Par. +0".10
Against these are to be set negative parallaxes of -0".09, -0".08 and several a little smaller, which are certainly unreal.

The presumption in favor of the actuality of one or more of the above positive values, which is created by their excess over the negative values, is offset by the following considerations: The area of the entire sky is more than 40,000 square degrees, or 10,000 times the area covered by the Helsingfors plates. We cannot well suppose that there are 1,000 stars in the sky with a parallax of 0". 10, or more without violating all the probabilities of the case. The probabilities of the case are therefore against even one star with such a parallax being found on the plates. Yet the cases of these four stars are worthy of further examination, if any of them are found to have a sensible proper motion.

On an entirely different plan is a survey just concluded by Chase with the Yale heliometer. It includes such stars having an annual proper motion of 0".05 or more as had not already been measured for parallax. The results, in statistical form, are these:

2 stars have parallaxes between + 9".20 and + 0".25.
6 stars have parallaxes between + 0".15 and + 0".20.
11 stars have parallaxes between + 0".10 and + 0".15.
24 stars have parallaxes between + 0".05 and + 0'.10.
34 stars have parallaxes between + 0'.00 and + 0".05.
8 stars have parallaxes between - 0".05 and 0".00.
5 stars have parallaxes between - 0".10 and - 0".05.
2 stars have parallaxes between - 0".15 and - 0".10.
92, total number of stars.

It will be understood that the negative parallaxes found for fifteen of these stars are the result of errors of observation. Assuming that an equal number of the smaller positive values are due to the same cause, and subtracting these thirty stars from the total number, we shall have sixty-two stars left of which the parallax is real and generally amounts to 0".05, more or less. The two values approximating to 0".25 seem open to little doubt. We might say the same of the six next in the list. The first two belong to the stars 54 Piscium and Weisse, 17h., 322.

  1. 'Astrophysical Journal', Vol. VII, January, 1898.