Chapter XIX: Short-Period Variables

The limit of length for “short periods” of stellar variation is conventionally fixed at thirty days; but it is seldom reached by objects of typical character. Rapid fluctuations are almost always accomplished with extreme exactitude both as to time and amount. To this rule there are very few exceptions, and a reason for it is not difficult to find. Variability of the kind in question is precise because it originates extrinsically. It might be called a “forced vibration” of change. Its course is, in some way, prescribed by the revolutions of a satellite. No more curious spectroscopic discovery has been made than that of the binary nature of short-period variables. By it a breach has been made in the wall of mystery surrounding stellar light-change. The breach has not yet been mounted, nor is it quite practicable; but by persevering efforts it can be gradually widened and levelled.

In dealing with variable stars we must proceed tentatively. The subject is so complex that no intelligible view of it can be gained all at once. A unifying principle is still lacking. We can only take things as they present themselves, noting differences, tracing partial analogies, and arranging into some fashion of order a multitude of heterogeneous examples. Thus stars fluctuating in short periods may be separated into three families. The first has δ Cephei, the second ζ Geminorum for its head, and their members may conveniently be designated Cepheid and Geminid variables. The third is represented profusely, but almost exclusively, in globular clusters. To begin with the Cepheids.

They are numerous and well known. Their changes are continuous and of moderate amplitude, but proceed unsymmetrically as regards time. The rise to maximum occupies on an average about a third of the period, or half the time allowed for the decline from it to minimum. This retardation is accompanied by an inherent tendency to a second maximum, sometimes barely indicated as a pause in descent, but in several cases giving rise to a pronounced “hump” on the downward slope of the light-curve. The variations of δ Cephei range from 3·7 to 4·9 magnitude in 5^d 8^h 47^m 39^s, of which 1^d 14^h 36^m suffice for the phases of increase. The spectrum is of the solar type, and does not change with the brightness. Oscillatory movements, however, of its constituent lines, detected by Bélopolsky in 1894,[619] betray the presence of an obscure companion revolving in the light-period. The ellipse described is so eccentric that the companion-bodies when at apastron are three times further apart than at their nearest approach; and its major axis deviates only by two degrees from a vertical plane passing through the earth. There is nothing indeed to show that it may not be highly inclined to the corresponding horizontal plane. The orbital level is undetermined, being evidently such as to exclude eclipses. This was unexpected, but it is certain. The criterion is simple. Radial velocity should vanish at minima if a transiting globe were concerned in their production; in point of fact, the epochs of least brightness precede the epochs of conjunction by a full day. The system of δ Cephei is not then an eclipsing system. The star’s variability must be otherwise accounted for. Nor would it be easy, without abusing the licence of hypothesis, to expound it as the result of occultations. The inducement to make the attempt is, at any rate, removed by the ascertainment of their non-occurrence. Nevertheless, the coincidence of periods assures us that orbital revolution, in one mode or another, prescribes the flow of change. But the ideas so far entertained on the subject scarcely bear examination. Obviously untenable, for instance, is Mr. Roberts’s view that the companion of δ Cephei is raised, by the heat received at periastron, from sensible obscurity to a nearly equal grade of lustre with its primary.[620] If this were so a double spectrum should be observed at quadratures. Again, the maximum should, on the hypothesis, fall short of twice the minimum brightness; actually, it exceeds it three times. Mr. Roberts himself adverts to these objections, but holds them not insuperable. Mr. Eddie’s suggestion[621] of luminous increase through intensified tidal action at periastral approach, is more plausible. Bodily strain due to mutual gravitation would, in so eccentric an orbit, gain twenty-seven-fold efficacy as the bodies moving in it come together from its farthest point; and the processes of internal circulation might possibly be sufficiently quickened by the disturbance to yield a largely augmented output of light. But commotions of the requisite violence could not subside with perfect regularity every five days, and they should inevitably be accompanied by gaseous outbursts spectroscopically evident. That they do not occur may be securely inferred from the one circumstance that no trace of emissive symptoms is met with in the light of this star. As to the scale of its system we are, moreover, completely ignorant, and are hence unable to estimate the absolute power over its members of tidal influences. The line-displacements of the bright star acquaint us merely with its rate of motion as projected upon the visual plane; they correspond to a mean orbital radius of 620,000 miles, the real path being perhaps six or eight times wider than that spectroscopically indicated, while the companion-ellipse traversed by the dark satellite may be of any imaginable size. A sapphire-blue star of the sixth magnitude forms with δ Cephei, which has a golden sheen, a combination resembling that so beautifully exhibited in β Cygni.

The variations of η Aquilæ are, in every respect, analogous to those of δ Cephei. They have the same range of 1·2 magnitudes, and a somewhat longer period of 7^d 4^h 14^m, which they divide, in about the same proportions, between a rapid increase and a leisurely decay of brightness. The abortive secondary maximum of δ Cephei, however, develops in η Aquilæ into a pronounced recovery of lustre. The light-curve shown in Fig. 22, from Dr. Schur’s delineation, renders the duplex phase conspicuous. The binary character of the star was detected by Bélopolsky in 1895,[622] and he computed its orbit from improved spectrographic data in 1897.[623] It proved to be of nearly the same _apparent_ size as that of δ Cephei, and the conditions of revolution were, in this case again, manifestly inconsistent with the occurrence of eclipses. An interval of two days was found to separate each minimum from the ensuing conjunction of the bright and dark spheres. Their conjunctions, accordingly, are not transits, since they are unaccompanied by any diminution of light. “Very remarkably,” Bélopolsky writes, “the same state of things is present in the variable star δ Cephei. Hence some other cause must be sought by which the variations in lustre and the spectral displacements of these two stars may be brought into harmony.”

FIG. 22.—Light-Curve of η Aquilæ (Schur).
]

Mr. W. H. Wright of the Lick Observatory renewed in 1899,[624] and substantially confirmed the Pulkowa investigations. His orbit, however, came out considerably more eccentric than that computed by his predecessor, and he located its major axis somewhat differently. But on the essential point for the theory of the star’s variability—the impossibility of eclipses—they agreed. Fig. 23 reproduces Mr. Wright’s drawing of the ellipse described by η Aquilæ round its invisible companion, supposed immovable at the focus (O). Its situation at minimum epochs (marked _Min._ in the figure) obviously corresponds to nearly the highest rate of speed in the line of sight, and to a comparatively wide visual separation of the coupled bodies. The periastron is at P; the points where the secondary phases take place are indicated respectively as _Min.__{2} and _Max.__{2}. They are unaccompanied by irregularities of movement. In fact, neither in this object nor in the analogous one decorating the crown of Cepheus is there any traceable connection, except their significant agreement in period, between the flow of spectroscopic velocity and the rise and fall of brightness. The spectra of these two variables are quite similar, only the lines of η Aquilæ are more diffuse. Both, too, are approaching the sun at uniform rates of about fourteen and nine miles a second respectively. Finally, both are subject to some slight disturbances in time.[625] Those of η Aquilæ, discussed by Dr. Lockyer,[626] have an amplitude of five hours, and are self-compensatory in a cycle of 400 light periods.

FIG. 23.—Orbit of η Aquilæ (Wright).
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The generalisation is a tolerably safe one that all Cepheid stars are binaries; but its establishment must be a work of time. Those of the shortest periods are the most promising for purposes of spectrographic inquiry, since they are likely to be in swift circulation, and hence to show conspicuous line-displacements. Negative results, however, as already said, may simply imply an unfavourable situation of the orbits; they do not necessarily indicate the solitary condition of the stars. A few examples of Cepheid variables are given below in the order of increasing length of period. Some brief description of the peculiarities of each follows.

┌──────────────┬──────────────────┬───────────┐
│ Name. │Limits of Change. │ Period. │
├──────────────┼──────────────────┼───────────┤
│ │ │ d. h. m.│
│ R Muscæ │6·6 to 7·4 mag. │ 0 21 10│
│ R Trianguli │6·6 „ 8·0 „ │ 3 9 20│
│ Australis │ │ │
│ST Cygni │6·6 „ 7·4 „ │ 3 20 10│
│ T Vulpeculæ │5·5 „ 6·5 „ │ 4 10 28│
│ Y Sagittarii │5·8 „ 6·6 „ │ 5 18 33│
│ U Sagittarii │7·0 „ 8·3 „ │ 6 17 46│
│ X Sagittarii │4·0 „ 6·0 „ │ 7 2 50│
│ W Sagittarii │4·8 „ 5·8 „ │ 7 14 16│
│ S Sagittæ │5·6 „ 6·4 „ │ 8 9 7│
│ X Cygni │6·4 „ 7·7 „ │ 16 9 15│
│ W Virginis │8·7 „ 10·4 „ │ 17 6 30│
│ T Monocerotis│5·8 „ 8·2 „ │ 27 0 18│
└──────────────┴──────────────────┴───────────┘

R Muscæ, being circumpolar at the Cape, can be observed to advantage only in southern latitudes. Once in twenty-one hours it doubles its light, and so emerges into naked-eye visibility, then sinks back again out of sight. The rise occupies just seven hours, or one-third of the period—the normal proportion for stars of this class. Evidence of rapid circulatory motion is pretty sure to be elicited by spectrographic means.

The phases of R Trianguli Australis were noticed by Gould in 1871, and have of late engaged the attention of Roberts. They show the peculiarity, surprising in a Cepheid star, of being uncertain in extent. The full measure of change is from 6·6 to 8·0 magnitude, but at certain maxima it mounts no higher than 6·8; at certain minima it descends no lower than 7·5 magnitude. It would be of interest to learn whether these oppositely incomplete phases occur together or disconnectedly. No methodical account of them has, we believe, been published. The increase of brightness in R Trianguli occupies little more than one quarter of the period.

The light-curve of ST Cygni is “humped” like that of δ Cephei. About forty-six hours after maximum the decline is stayed, then, a brief pause ended, resumes its course. The interval from minimum to maximum is 21½ hours, that from maximum to minimum seventy.[627] The variability of this star[628] was detected at Potsdam in 1896 by G. Müller and P. Kempf.

The spectrum of T Vulpeculæ is similar to that of δ Cephei.[629] Its changes, discovered by Sawyer in 1885, consist in a rise of one magnitude in 1^d 7^h, followed by subsidence in 4^d 3^h. The comparative brightness of the star brings its movements well within range of spectroscopic investigation.

The four stars in Sagittarius, which come next on our list, form a singular group, discovered by Schmidt in 1866.[630] They are included in a space of about ninety square degrees, and all vary after the fashion of δ Cephei in periods comprised between 5·8 and 7·6 days. Among them X Sagittarii, which attains once a week fourth-magnitude rank, is the most conspicuous. Its spectrum is of the solar type. One of its associates, U Sagittarii, distinguished by a strong orange tint, is the centre of a little cluster. Its fluctuations, upward in 2½, downward in 4¼ days, proceed with the regularity of clockwork.

S Sagittæ shows the retarded decrease and inflected curve distinctive of its class.[631] Yendell considered the maximum to be double (see Fig. 24). The change at minimum is unusually slow. In quality of light the star was found by Sir Norman Lockyer to be an exact match for δ Cephei.

X Cygni was discovered by Chandler in 1886. It rises without fail in 6^d 19^h to 6·4 magnitude, but shows an inconstant minimum brightness, sometimes descending to 7·7, at others stopping short at 7·2 magnitude. The period of 16^d 9¼^h is not known to vary. The star is quite colourless; information as to the character of its spectrum is not at present forthcoming.

W Virginis is uncertain both at maximum and minimum. Its highest rise is to 8·7, its lowest descent to 10·4 magnitude. But these limits are far from being always reached; phases deficient by quite half a magnitude either way are often observed, and seem to intervene casually. The period, on the other hand, of 17^d 6½^h is strictly conformed to. The time of increase, 8^d 5^h, is proportionately long for a Cepheid variable.

FIG. 24.—Light-Curve of S Sagittæ (Yendell).
]

That of T Monocerotis is, on the contrary, relatively short, although the entire cycle of twenty-seven days is the most protracted yet ascertained for a star of its class. And here again the amplitude of change is inconstant. Maxima occur as high as 5·8, as low as 6·4 magnitude, while the minimum brightness ranges between 7·4 and 8·3 magnitude. There are then spring and neap tides of stellar fluctuation; yet no clue can be found to the cause of their difference. The four Cepheid variables in which they appear—namely, R Trianguli, X Cygni, W Virginis, and T Monocerotis—have no other features visibly in common. Perhaps the spectroscope may reveal unlooked-for analogies connecting their physical qualities or their systematic relations; but its dicta have still to be pronounced.

About three dozen stars had been registered at the close of last century as variable on the model of δ Cephei. Most generalisations regarding them are liable to lose validity by future experience; but there are two properties in the absence of which they should be otherwise classified. In all, light-change progresses unceasingly; in all, it advances more rapidly in the direction of increase than of decrease. An inclination, more or less accentuated, to pause in descent is probably connected with this kind of dissymmetry. Among their other distinctive qualities the following may be provisionally enumerated:—(1) Their spectra are of the solar type. (2) They are binaries revolving in the periods of variation. (3) They are not eclipsing pairs; their orbits may be inclined at any angles to the visual plane. (4) Their fluctuations in lustre are unaccompanied by spectral change. (5) Being nearly devoid of proper motion, they are presumably at vast distances from the earth. They are then giant suns.

Future research will decide whether Cepheid stars are marked off from ordinary spectroscopic binaries by any peculiarities in their manner of circulation. Are their orbits, for instance, in all cases highly eccentric? Is there a fixed relation between the situation of the periastron and the point of lowest brightness? Do the major axes revolve? Above all, what differentiates short-period variables from revolving pairs constant in light? Why do δ Cephei and η Aquilæ show an incessantly changing lustre, while Polaris and Θ Ursæ shine steadily? The four appear to be of analogous constitution, and to suffer no diminution by eclipse. What form of influence, then, is it which acts so strikingly upon the two former objects, while leaving the two latter unaffected? Undoubtedly the close attendance of a satellite is instrumental to the production of the observed changes of luminosity; but other conditions also come into play—conditions that are absent in Polaris, in β Aurigæ, in α Virginis, in Θ Ursæ, in Θ Draconis. What is their nature? Here is the main issue as regards short-period variability.