its decline, a very little more slowly perhaps, recovering its normal brightness in a little less than five hours, the total duration of the light change being about 9h 45m. The curve of those changes, as determined by Dr. Scheiner from all available observations during the last two centuries, is shown in Fig. 4. Supposing the transit of the companion star to be central over the primary and the orbit to be circular, it is evident from Fig. 1 that the sum of the diameters of the two stars is equal to the line BB,, or the sum of their radii is one-half of that line. Representing the radius of A by a, that of B by b and that of the orbit by R, the duration of light variation by d and the total period of the variable by p, where r1, r1, v1 and v, are the radii vectores and the true anomalies of the points B, and B., and v, the true anomaly of the point B, which can be computed by the usual elliptic formula when the elements are known. Confining ourselves to the circular orbit, which is very nearly true in the case of Algol, and adopting Vogel's mean orbital velocity 26.3 miles per second and Chandler's period 24 20h 48m 55$ 247,735 68.814" we have = that is, the principal star is about one million miles from the center of gravity of the system. Assuming, from the loss of light at a minimum of about 1 magnitude, that the surfaces are in the ratio of 1.00 to 0.60, we should have for the ratio of the diameters of the stars 1 to 0.78; and for their volumes 1 to 0.475 or approximately 0.5. Again assuming, what may or may not be true, that their densities are equal, their masses will be in proportion to their volumes and their distances from the center of gravity will be as 1 to 2. So we may take in round numbers for the distance of the primary star from the center of gravity 1,040,000 miles, for the distance of the satellite from the center of gravity 1,040,000 0.475 = 2,190,000 miles, and for the distance of the center of the satellite from the center of the primary star 3,230,000 miles. The sum of their radii then becomes Dividing the sum in the ratio of 1 to 0.78 we have a = 781,000 Let us now see how these dimensions of the two stars will satisfy the light curve. In Fig. 5 let BE be the projection of a portion of the orbit of the satellite B near conjunction with A. If the transit were central, the distance AE would be zero; in general it is R sin i, where i is the inclination of the line of sight to the plane of the orbit. In the case of Algol the light is not stationary for any noticeable length of time at the minimum, so that the transit cannot be central. Let us assume that the inclination is just enough to cause the disc of B to wholly enter upon that of A, producing internal tangency at minimum; then Let p = the projection of AB and u = the angle BAE, then at any moment PNR cos2 u sin2 i + R' sin2 u (4) If t represent the time occupied by the satellite in passing from E to B The area from which the light is cut off at any moment is the sum of the two segments between the chord and the two arcs CC'. The segments are the differences between the corresponding sectors and triangles ACC' and BCC'. The areas of the sectors are in which , and 9, represent the angles BAC and ABC respectively. The areas of the triangles are From these we obtain for the area of the eclipsed portion of the If the loss of light be considered proportional to the area of the covered part of the disc and the satellite gives no appreciable light we may express the ratio of the brightness at any moment to the normal brightness by and the loss in magnitudes by = By the aid of these formulae I computed the values of 4m for Algol for the five hours before and after minimum, with the values a 781,000 miles and b = 609,000 miles. In the computation, however, I expressed a and b in terms of R by dividing each by 3,230,000. R then became unity in the formulae and a = 0.2415 and b = 0.1885. The values of 4m found agreed exactly with those computed in a similar manner by Dr. J. Wilsing and are represented by the lower dotted curve II in Fig. 4. It will be noted that this curve nowhere deviates more than 0.10 magnitude from the observed light curve. The small deviation is however systematic and larger than the errors of the observed curve. Dr. Vogel considered that the observations could FIG. 6. WILSING'S COMPUTED LIGHT-CURVE OF ALGOL. be better represented by supposing the two stars to be smaller in diameter but surrounded by extensive atmospheres, the absorption of which changes the character of the curve. He estimated the radii of the two stars as a 530,000 and b = 415,000 miles. The upper dotted curve I in Fig. 4 represents the light changes, which I have computed with these dimensions of the stars, neglecting the effect of their atmospheres. The duration of the light change would be only 6" 30" and there would be a stationary period of a half hour or more at the minimum. † Astronomische Nachrichten No. 2960. 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 SECONDAL EQUAL MINIMA FIG. 7. THEORETICAL LIGHT-CURVES. 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 5% 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. |