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 1-7/8 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.[F]
[F] ‘Astrophysical Journal’, Vol. VII, January, 1898.
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 | ||
| Position (3), stars at apocenter Interval, 20 hours. | * | * | |
| Position (4), B eclipses 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 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.
Fig. 3. Light Curve of 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.
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.
Diameter of smaller star, 12,000,000 kilometers.
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.’
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.