In the case of the masses of the two stars being nearly equal, these values will be reduced one half. GENERAL RESULTS. It will be sufficiently apparent that a definite determination of the density of any one of the stars discussed in this paper is an impossibility in the present state of our knowledge; we must content ourselves with being able to indicate a limit in one direction to the values sought. Precise values cannot be secured so long as the inclination of the orbit and the relative masses of the two components remain unknown. Gathering the results together we find for the densities. These are absolute maximum values of the density of the stars dealt with, assuming in the case of S Velorum a circular orbit, an assumption in keeping with the facts of variation. We may push the investigation one step further and deduce the mean average densities of all the stars, and in this case it will be sufficient to consider k and k, equal; their inequality will have very little effect on the average of all the values. The average density of the four systems dealt with in the foregoing paper is 0.13. That is, the average density of a close Algol variable is only one eighth that of the Sun. This result is sufficiently striking, but it is in no way new. The low density of close double stars has been indicated by more than one astronomer. Any value which the preceding investigation has does not depend upon its novelty but on its relation to the laws connecting stellar temperatures and densities. And it is as a contribution to any discussion which may arise. as to the validity or application of such laws that the investigation is offered. LOVEDALE, S. Africa April 1899. THE DENSITIES OF THE VARIABLE STARS OF THE ALGOL TYPE. By HENRY NORRIS RUSSELL. It is possible in the case of an Algol-star, assuming the eclipse theory of its variation, and a circular orbit, to deduce a limiting value for its mean density from its period and the duration of its light-variation, any uncertainty as to the result being due to the uncertainty in the ratio between these two quantities, which is, as a rule, not yet very accurately determined, Let be the radius of the bright component, r' that of the dark, and a the radius of the orbit. Let t be the period, and d the duration of the light-variation, that is, of the eclipse, and let m be the total mass of the system. The mean density p' of the system is the quotient of its mass by its total volume; that is When the eclipse begins or ends, the disks of the two stars are apparently tangent, and the distance of the projections of their centers upon a plane perpendicular to the line of sight is r+r'. Between the beginning and middle of the eclipse the stars move through an arc of their orbit equal to and the πα πα t relative displacement of their projections is a sin But this displacement is equal to the projection of the distance of their centers at first contact upon the line of relative motion. fore There the sign of equality holding only when the transit is central. We have also m == K 3 where k is a constant depending on the system of units used. Combining (1), (2), (3) and (4) we have (4) (5) where the sign of equality is to be used only in the case of a central eclipse of equal stars. This formula gives a superior limit for the mean density of the system independent of its dimensions. 3 K π It remains to determine the constant The period of a particle revolving close to the Earth's surface is 1.411 hours. That of two Earths revolving just out of contact would be twice this. The eclipse of one of these bodies by the other would occupy half the period. 3 K The sign of equality may in this case be used in formula (5) and, taking the Earth's density at 5.53, we obtain = 44.I where the unit of time is one hour, and the unit of density that of water. π The results of the application of the formula (5) to the seventeen known Algol-variables are given in the following table : The data are those of Chandler's Third Catalogue of Variable Stars, except as noted. It is possible that the gradual change of brightness near the beginning and end of the light oscillation is produced by some other cause than the interposition of an opaque body. If this is true, the tabulated values of d are greater than the true intervals between the contacts at eclipse, and the calculated limits of density are consequently too small. In the case of Algol, Vogel concludes that the real duration of eclipse is about 6% hours, and so finds a density about twice as great as the limit found in this article upon the assumption of an eclipse lasting 9.15 hours. Since it is improbable that the orbits of these systems are actually circular, the numerical values in the table are only rough approximations to the truth. An eclipse near periastron would occupy less time than if the orbit were circular, and the calculated limit of density would be too great. The reverse *Secondary minima. 2 DUNER, this JOURNAL, I, 285. 3 ROBERTS, A. J., 373. 4 Harvard Observatory Circular No. 5. 5 SAWYER, A. J., 450. "CERASKI, A. N., 3572. 7 A. N., 2947. would be the case at apastron, since the computed limit of density varies as the cube of the relative velocities of the stars. Since in an elliptical orbit the velocity is greater than in a circular orbit of the same period for more than half its circumference, and since the velocity at periastron is increased above the normal in a greater ratio than it is diminished at apastron, the mean value of the limit of density computed by formula (5) for a considerable number of stars would be increased by the effects of eccentricity above its true value. The stars Z Herculis, Y Cygni and R S Sagittarii have secondary minima. By taking the mean of the durations of the two minima—as has been done in the table— the eccentricity is nearly eliminated, and since Dunèr1 and Roberts have shown that in these systems the components are almost equal in size, and the transits almost central, the limits of density given above are probably close to the true values. Notwithstanding the causes of uncertainty, it is evident that the Algol-variables as a class are much less dense than the Sun, probably less than one fourth as dense. If these stars consist of a nucleus and an extensive luminous atmosphere, the nuclei may, of course, be much denser. PRINCETON UNIVERSITY, October 9, 1899. This JOURNAL, 1, 285. A. I., 265. 2 A. J., 373. |