Jan¿ 1906. Dr. Böberts, Algol Variable Sta/R. 123 on a Method Of

Jan¿ 1906. Dr. Böberts, Algol Variable Sta/r. 123 On a Method of Determining the Absolute Dimensions of an Algol Variable Star. By Alex. W. Roberts, D.Sc. I. Introduction. The present paper deals with an extension of the problem of Algol variation in the direction of a determination of the absolute dimensions of a close binary star. Although the more definite and more accurate consideration of the dimensions of such systems falls more properly within the area of spectroscopic research, yet, theoretically at least, the light-curve of any eclipse variable exhibits data which when properly discussed yield a determination of the absolute size of the system. The theory that underlies this important determination is the simple one that light takes an appreciable interval of time to traverse the orbit of a binary star. A moment’s reflexion will make it evident that this circumstance must make itself manifest as an acceleration in the apparent occurrence of both the primary and secondary maximum phases. The time of passing the primary and secondary maxima will, however, remain unchanged ; that is, the approach and recession of the component stars relative to the Earth as they revolve round one another will be translated, owing to the measurable velocity of light, into a corresponding hastening and retardation of the successive phenomena of eclipse. It will be clear, therefore, that if we had the means of ascer- taining the light-curve of an Algol variable with perfect precision and completeness, and if all the phenomena of eclipse were capable of geometrical explanation and exposition, then, at all times, the light-curve of the system under consideration would provide sufficient data for a definite determination of the absolute dimensions—size, mass, and density of the eclipsing stars. It will require little acquaintance with the many perplexities of variable star curves to assure one that, while in theory the problem of determining the dimensions of a binary star from an examination of its light changes has the merit of simplicity, in actual practice the solution is one beset with difficulties and obscured by uncertainties. (1) It is not possible to determine the light-curve of a close binary star with the accuracy and refinement necessary for a numerical solution of the problem, except in cases where the variable completes its full period in a few hours. In the latter circumstance the intervals between the four cardinal phases of variation, principal minimum, principal maximum, secondary minimum, secondary maximum, can be determined to within a minute of time. This is a quantity sufficiently refined to indicate, at least, a major limit to the size of the star. (2) If all the phenomena of variation are to be included in a L © Royal Astronomical Society • Provided by the NASA Astrophysics Data System 124 Dr. Boberts, On Determining the Absolute lxvi. 3, consideration of the problem, and any system of exclusion must weaken the weight of the solution, then we must travel beyond the simple facts of eclipse to a region of physical causes and 1906MNRAS..66..123R conditions to find a complete explanation of their origin and an interpretation of their character. We must, for instance, take into account the ever-changing figure of the component stars, due to eccentricity of orbit, as well as their rhythmic fluctuations in brightness, consequent on alterations in surface-pressure and tension. (3) A number of untenable assumptions have always to to be made regarding the form and surface brightness of the component stars. Notwithstanding these hindrances to a hopeful dealing with the question of the dimensions of an Algol variable, it seemed to me worth while instituting an inquiry as to the extent and nature of the uncertainty which must surround a solution so conditioned as that already indicated in the preceding paragraphs. I accord- ingly, early in 1905, worked out a graphical solution of the problem, taking the Lovedale observations of Algol variables as the materials of my investigation. To my surprise , the results obtained were much more definite than I had expected. They were, indeed, sufficiently final in character to convince me that in certain cases—i.e. rapidly varying stars—the problem admitted of a definite and reliable solution. Lest my own observations, obtained by the method of sequences, might contain within them some systematic error leading to a false result, I asked Professor Pickering to permit me to make use of the very fine series of observations of the rapidly varying star U Pegasi made by Professor O. C. Wendell with the Harvard Meridian Photometer. With his usual kind- ness Professor Pickering at once sent me the whole of Professor Wendell’s observations of U Pegasi. I think it will be regarded as a sufficiently complete exposition of the problem we are considering if at this stage of its development we simply deal with two suitable stars, one in the northern hemisphere and one in the southern hemisphere. The northern variable star TJ Peg asi ¡ both because of its rapid period and because of the fulness and accuracy with which its light-curve has been determined by Professor Wendell, presents a very favourable object for the investigation we have in purpose. Then among southern stars over 10,000 observations have been secured of RR Centauri, the mean light-curve resulting from this mass of detail being sufficiently refined to warrant an inquiry regarding the absolute dimensions of the system. The mode of observing both stars is so different in principle and in art that the possibility of a common systematic error, personal or instrumental, influencing the observations is remote. © Royal Astronomical Society • Provided by the NASA Astrophysics Data System Jan. 1906. Dimensions of an Algol Variable Star, 125 II. Statement of the Problem. In the two following figures are given the light-curves of 1906MNRAS..66..123R V Pegasi and RR Centauri respectively. Before dealing directly with the relation which exists between the velocity of light and certain portions of the unsymmetric variation exhibited in these light-curves it will be necessary to distinguish and define, as far as we can, the more important circumstances which operate in producing the regular ebb and oh I 2 3 4 5 6 7 8 9 h oh i 2 3 4 5 6 7 8 9 10 ii 12 i i sh flow of brightness represented in the light-curves under consideration, as well as in all light-curves of the same type. (1) It seems reasonable to regard the major portion of the variation of both U Pegasi and RR Centauri as due to the eclipse of one star by its fellow. If we consider the component bodies to move in a closed orbit, to be of regular and unchanging figure, then expressions may be readily deduced to exhibit the relation that exists between the symmetrical portion of an Algol star’s variation and the chief elements of the system. L 2 © Royal Astronomical Society • Provided by the NASA Astrophysics Data System 126 Dr. Boberts, On Determining the Absolute lxvi. 3, As the purpose of this paper is to deal with a wider extension of the problem than the simple determination of the elements of orbital movement, I may here just refer to the fact 1906MNRAS..66..123R that a fairly full exposition of the relation connecting movement and variation is given in Monthly Notices^ vol. Ixiii. pp. 531-534. Those curious regarding the preliminary portion of the present investigation are referred to this article. (2) The formulae given in Monthly Notices, vol. Ixiii. p. 534, presume that the eccentricity of the system is zero. In order to correct any error arising from this assumption it is necessary to introduce into the equations defining the variation terms depending on the eccentricity. Let P = Period of variable in minutes. £ = Eccentricity of orbit. X = Longitude of periastron reckoned from line of sight* 6 = True angular distance of eclipsing star from line of sight at time T. Am = Change in magnitude per minute at time T. Then it is evident that ?-^-esin(0 + X) ... ... ... (1) 7T expresses the difference (in minutes) between the true anomaly and the mean anomaly at any time T if the eccentricity be small. Consequently • sin (0 + X) (2) 7T is the correction in magnitude to be applied to the mean places, computed on the basis of a circular orbit, to reduce them to true places. Putting £ sin \ = p £ cos X == q expression (2) becomes ^ eosöjp+i—-— sm djq (3) In the case therefore of the eccentricity and the position of the apsidal line being unknown, the above expression (3) will afford a means of determining their values. (3) An examination of the light-curves of U Pegasi and IŒ Uentauri, given above, indicates that eclipse alone will not account for all the phenomena of variation. © Royal Astronomical Society • Provided by the NASA Astrophysics Data System Jan. 1906. Dimensions of an Algol Variable Star. - 127 For example, we have distinct evidence in the case of both U Pegasi &nà RE Centauri of two unequal maxima. This, as well as certain other irregularities, can be explained 1906MNRAS..66..123R readily on the assumption that the eccentric movement of the component stars produces tidal perturbations, and consequent change of figure. Physical movements of this nature must of necessity produce a measurable variation in the intrinsic brightness of the stars, as well as a definite and rhythmic alteration in their form. If we regard these changes to be completed in the period of the star then a simple expression of the type g sin (0 + Z) may be taken as covering all the outstanding periodic variation due to physical causes.

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