Dr. Wilsing by using Vogel's radii of the stars and assuming that the atmospheres surrounding the stars extended to the limits which in our first computation we gave to the stars themselves, i. e. to 781,000 miles and 609,000 miles from the respective centers, was able to represent the observed light curve in all parts within 0.02 magnitude, the coefficient of absorption which he deduced being only one-fortieth of that of the Earth's atmosphere. In Fig. 6 I have platted his final values, in which he made a small correction for the eccentricity of the orbit, which he found to be 0.011, together with the observed light curve. The dotted line here coincides so closely with the smooth line that it was difficult to draw them so as to show both. No more complete representation of observation by theory could be asked for.

In the first part of his investigation Dr. Wilsing takes up the question of the possibility of two such bodies existing in so close proximity and shows that the deformation of one of the bodies by the attraction of the other amounts to only oneninetieth part, which is much less than the flattening of Jupiter by its own rotation. He also shows that if the satellite shines by reflected light only, its brightness cannot be more than of that of the primary star and therefore it could have no influence upon the light curve. That its light is very feeble is evident from the fact that Algol has no secondary minimum. Dr. Plassman* thinks he has observed slight changes in the light of Algol be

Astronomy and Astrophysics Vol XI, p. 419.

tween minima. These changes are very slight and must be explained in some other way than by the eclipse theory, for he finds two secondary minima, neither of them anywhere midway between the primary minima.

In Fig. 7 I have drawn some theoretical curves, which, though having no basis in actual observation, may assist in comprehending the problem. The dotted curve was computed on the hypothesis of two stars, without atmospheres, equal in diameter and intensity of surface luminosity, revolving about each other, or rather their common center of gravity, at the distance of the components of Algol. The curve is the same for both minima. The two smooth line curves were computed on the same hypothesis as above, with the exception that surface intensity of one of the stars is twice that of the other. The transits are supposed to be central in each case and the diameters 1,060,000 miles. The sharp points of the curves at minima can be rounded off either by assuming atmospheres or by supposing the

0.0

-0.1

0° 30 60 90 120 150 180 210 240 .270 300 330 360°

.2

.3

.4

.5

.6

-7

-0.8

FIG. 8. LIGHT-CURVE OF EQUAL STARS IN TANGENCY, AND OF ẞ Lyræ. eclipses to be only partial. The symmetry of the curve before and after minimum may be destroyed either by eccentricity of the orbits of the stars or by elongation of the bodies themselves, provided that elongation be not exactly parallel to the line joining their centers. In Fig. 8 the smooth curve represents the light variation, during an entire revolution, of two stars of equal intensity of surface illumination and equal diameters, revolving tangent to each other. This is an impossible case, for the mutual attractions of the two bodies would destroy their spherical figure and cause the contiguous portions to merge, forming figures perhaps like the ellipsoid of Jacobi, the dumb-bell or the apioid of Poincaré, figures which have been shown to be of at least temporary stability of equilibrium*. The curve bears so

* See POPULAR ASTRONOMY, Vol. III, pp. 489-519.

much resemblence however to the curve of B Lyrae, which I have drawn as the dotted line in the same Figure, that it would seem that by proper assumptions as to the forms, distance, intensity and atmospheres of the two components, the variations of that star and all of its class might be represented on the eclipse theory. In this class of variables the light change is not confined to a relatively short portion of the period, near minimum, but is continuous throughout the whole period. This would be the result with the Algol variables if the components should be near enough together for their atmospheres to blend.

Dr. Chandler has shown that the period of Algol is variable, having decreased, with several fluctuations, up and down, from 2d 20h 48m 58.0 to 2 20h 48m 51.1 during the last century. He attempts to account for this change by the attraction of a second dark body in the system, and is inclined from his study of the problem to regard the mass of this second companion as greater than that of the bright star. We should thus have the anomaly of a bright satellite revolving about a dark primary body, which is certainly a very interesting cosmological problem, much as we may doubt its possibility. From an investigation of the proper motion of Algol, Dr. Chandler thinks he has found evidence of this orbital motion of the bright star, and that its period is about 131 years and the semimajor axis of the orbit 1".33. There are also slight indications of disturbances by still other hypothetical dark companions, in other words perhaps of a complex planetary system akin to the solar system. M. Tisserand, however, denies the necessity of assuming the dark primary body and shows that the observed change of period can be accounted for by the attraction of a smaller dark body, if one or both of the close companions are flattened by rotation. His theory also requires the presence of the third member, and possibly more, of the system.

Similar irregularities in the length of period are found in the case of several others of the Algol-type variables, and suggest the same explanation, so that we may, perhaps, naturally conclude that motions like those shown to exist in the case of Algol, and furnishing evidence of complex planetary systems, somewhat similar to that of the Sun, are not rare exceptions but possibly the rule in the stellar universe.

There is another irregularity in the light changes of certain of these stars, however, which seems more difficult of explanation

Astronomical Journal Vols. VII, p. 180; IX, p. 121. See also POPULAR ASTRONOMY Vol. V, p. 302.

and almost fatal to the satellite theory. That is well shown in the light curve of U Ophiuchi in Fig. 9; a distortion of the curve shortly after minimum, due to something which checks the recovery of light for a time. This retardation is so slight that, if it were noticed in one star or by one observer alone, it might be regarded as due to some fault in the observing. But it seems to be characteristic of several of the Algol stars and Chandler speaks of noticing the same tendency in Algol. It must, therefore, be taken into account in any theory which is to explain the changes of these peculiar stars.

LIGHT CURVE OF U OPHIUCHI.

Chandler

Sawyer

423

FIG. 9.

In order to obtain any idea of the absolute masses of the bodies, we must know their actual distances or parallaxes and the elements of their orbits. Rough approximations to their maximum densities, however, as compared with the Sun, may be obtained even when these are lacking, from the data of the light variations. Assuming the law of gravitation to be the same in other systems as in our own, when two bodies have satellites, the ratio of their masses is determined by the proportion*

3 rå

M+m: M1 + m1 ::

t2

t12

(10)

in which M and M, are the masses of the primary bodies, m and m, the masses of the satellites, r and r, the distances of the satellites from their respective primaries, and t and b their periodic times. From this

Young's General Astronomy. p. 342.

Adopting the mass of the Sun as unity and neglecting that of the Earth, since it is insignificant in comparison, expressing the time in units of a year, and taking the Earth's mean distance from the Sun as 93,000,000 miles, we have

Since the density of a body is its mass divided by its volume, the density of a star and that of the Sun will be in the proportion of their masses divided by the cubes of their diameters, or

where da is the density of the star, d, that of the Sun, a and s their diameters and Ma and M their respective masses. In units of the Sun's mass and density

If 8 represent the average density of the two components of a binary, whose diameters are a and b

Substituting the second member of equation (12) for (Ma + Mb) and 866,400 miles for s, we obtain

Evaluating (16) for Algol, taking Vogel's dimensions of the system, we get

that is, the average density of the components of Algol is only one quarter of that of the Sun.

If we should include the atmospheres in the volumes of the stars, which is what we must do when we have no other data than the light curve, in the case of Algol a3 + b3 would be (1,562,0003 +1,218,0003) and we should obtain d = 0.078.

In formula (16) let a = pr, and b = qr. Then