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Vol. VIII. No. 3.

MARCH, 1900.

Whole No. 73.



The interesting articles referred to in the last number of POPULAR ASTRONOMY, on the Densities of the Algol Variables, by Messrs. Alexander Roberts and Henry Norris Russell*, led the writer to brush up his knowledge of this class of variable stars, and it was thought that some of the results of his study might be put in such form as to be interesting to the readers of the magazine. What is written is therefore presented with no claim to originality or a thorough study of the subject. We venture to hope, however, that it may be of use in putting clearly before the minds of perhaps some of the professional astronomers as well as of the average reader, the problem of interpreting the varying light of these peculiar stars.

These variables are regarded as exceedingly close double stars, whose light variation is caused by the eclipse of one star by the other. The planes of their orbits must be nearly parallel to the line of sight from the Earth to the star, so that each star in the course of a revolution will pass between the Earth and the other star, cutting off its light in part or wholly, just as the Moon eclipses the Sun. The parallel will be much better if we can imagine two suns close together, revolving about each other so swiftly that every few days or hours the one passes in front of the other. When they are separated, side by side, we receive the light from the whole hemisphere of each of them, and their combined light is at a maximum. When they are so nearly in line with the Earth that the disc of one appears to overlap that of the other, part of the light of the more distant star is cut off and their combined light is diminished. If the stars are equal in diameter and the one passes directly in front of the other, for a moment there is a total eclipse of the more distant star and the eye receives only the light of the nearer one. If the orbit be circular, under these conditions there are two equal minima of the star during each revolution, the intervals between them are equal and the periods of waning and increase of light are equal.

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Under the same conditions, if the two stars are of equal surface brightness, the greatest diminution of their total light will be 0.8 of a magnitude, for in case of a total eclipse of either by the other only half of their total light will be cut off. Since a star of a given magnitude on Argelander's scale is approximately 2.5 times as bright as one of the next lower magnitude, to divide its light by 2 would reduce its magnitude by 2/2.5


In case the two stars are of equal surface brightness but of unequal diameters, the light at either minimum, if the eclipse be central, is very nearly equal to that from the larger star and hence the diminution of light will be less than 0.8m. So when the components are of equal surface brightness, there can be no very marked minima. If the difference in size is very great the minima will be scarcely noticeable.

On the other hand, when one of the stars is darker than the other, the minima will be unequal, and may be so unequal that one will not be observable at all, while at the other the star may become nearly or quite invisible.

When the orbit of one star about the other is elliptic, as is usually the case, there are two minima which may be equal or unequal according to the inclination of the orbit and the relative brightness of the component stars: The intervals between the two minima of each revolution are generally unequal; the exception being when the major axis of the ellipse lies in the line of sight from the Earth to the star.

The figures, which we have prepared, will perhaps enable the reader to understand more clearly the statements already made, and prepare the way for those which wiil follow. In Fig. 1, Plate IV, we will suppose the circle A to represent a bright star, and B a smaller or darker star, both revolving in circular orbits around their common center of gravity C. Let EA be a portion of the line of sight from the Earth to the star. If this line of sight lie in the plane of the orbits AA'A" and BB'B', it will be clear that every time B passes the position BB, it must be projected in perspective against A, as shown in Fig. 2, Plate IV, obscuring a surface area of A whose projection is equal to the projection of the earthward surface of B. Also when A reaches A", B will be at B" and will be wholly obscured by A. Between those positions as at A'B' the stars will send their combined radiation earthward.

It will be easier to study the relative movements of the two stars if we transfer the center of movement to the center of the bright star and ascribe the whole change of position to the lesser star. The orbit of B about A will then be the larger circle B‚Â ̧Â ̧.

This circle represents truly the relative movement of B, for clearly AB, is equal both in length and direction to A'B', and A"B" has its counterpart in the line between AB, and AB. The angular movement of B is the same in both cases, but its linear velocity in the relative orbit is the sum of the real velocities of A and B.

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Using then the larger circle and regarding A as stationary, we may draw the parallel lines MN and M'N' tangent to the disc of A as the limits of a cylinder within which B must enter in order to eclipse A on the one hand or be eclipsed on the other. Supposing the eclipse to be central and taking no account of the atmospheres of the stars (a most important omission), as soon as B reaches the position B1, tangent to MN, the light begins to wane; slowly at first, then more rapidly, as the longer chords of B encroach upon A, then more slowly until at B,, the whole disc of B


overlaps that of A and the light is at a minimum. The minimum will continue, practically stationary, until B reaches B,, tangent to M'N', when the increase of light will begin, the order of change being the reverse of that of the waning, ending at B. From B to B the light will be constant, but from B, to B, a secondary minimum will occur, the light of B being cut off by A, waning from B to B, stationary from B, to B, and increasing from B to B. From B, to B, the light will remain constant again. Thus one will readily see that a variable of this class should have two minima and that generally the two minima should be unequal. Only in case the two stars are of equal surface brightness can the two minima be equal. In that case the brightness at both minima is practically that of the star A, for at the primary minimum B gives about as much light as it cuts off from A, while at the secondary minimum the light is wholly that of A.

The reader must understand that these stars are so distant from the Earth that they are mere points as seen through the most powerful telescopes and that the distance between the components is far below the limit of measurement with a micrometer. No one of them has ever been seen double in a telescope. Very little therefore can be determined with reference to their orbits. The fact of their being double was a mere inference from the phenomena of their light variations, until a few years ago. In 1888-9, Professor Vogel,* by a series of photographs of the spectrum of Algol proved that that star was moving away from the Earth before each minimum at the rate of 24.4 miles per second and was approaching at the rate of 28.5 miles per second after each minimum, proving conclusively that it was revolving about a center of gravity under the control of a force from a dark companion.

The other variables of this type are all fainter than Algol and their orbital motions have not, so far as I am aware, been determined spectroscopically, so that we must confine ourselves to what can be learned from their light changes and the similarity of these changes to those of the one known to be binary.

First, then, let us connect the light curve of Algol at minimum with the orbital velocity determined by Vogel. For about 59 hours Algol's light is practically constant, shining as a star of 2.3 magnitude. Nearly five hours before minimum it begins to decline, slowly at first then more rapidly, until at minimum its magnitude is about 3.4. It then increases in the reverse order of

* Astronomische Nachrichten, No. 2947.

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