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1916Aj 29. .173V the Astbonomical Journal .173V 29. THE ASTBONOMICAL JOURNAL 1916AJ FOUNDED BY B. A. GOULD. NO. 694 VOL. XXIX ALBANY, N.Y., 1916, JULY 15 NO. 22 MASS-RATIOS IN VISUAL BINARY STARS, By G. VAN BIESBROECK. In recent years several successful attempts have been a constant term depending on the initial conditions. made to find the masses of binary stars. Combining From measures of one or other of the components at visual, spectroscopic and even photometric measures different epochs we deduce linear equations of condi- or the fairly well known parallaxes of some of them tion containing the three unknowms : ¡jl, A and K, and the linear dimensions have been determined in a num- we see that there will be no indeterminateness in the ber of binaries and consequently we have good approx- problem as long as there is no proportionality between imations to the total mass of the system. The ques- r and t. Three sufficiently distant epochs will fulfill all tion that we are considering here is another one: requirements in such a case. namely, the partition of that total mass among the The question is further simplified in the majority of two components opening thus a way to find masses the stars to be considered because g is sufficiently well of individual stars. The solution of this problem is known from a long series of meridian observations. If found by measuring the fraction of the total relative so two unknowns only are to be considered: A and K. motion that is to be attributed to each or at least to one Two epochs, let us call them 1 and 2, will furnish all of the components. It is a very simple one theoreti- the required data. From the equations in C for the cally. Let us call A and B the masses of the two com- two epochs we find: ponents of a binary, A referring to the brighter one or the one that is taken as the origin for the relative A = M (t2 ~ él) ~ (C2 ~ Cl) motion. Consider their absolute positions (eventually r2 — n corrected for parallax,) on the celestial sphere. We measure these positions from a fixed origin and in a The best determination of A will be obtained when the certain fixed direction, be it in right ascension, in denominator r2 — n is as large as possible. That con- declination, or in any other chosen direction. We will dition fixes the most advantageous direction for the suppose measurement. The relative displacement r2 — r1} will A A B = 1 be a maximum when the measures are made in a direc- tion nearly parallel to the relative displacement be- the masses meaning then the fractions of the mass of tween the two epochs. If for instance at the time con- the system that belongs to each component. We will sidered the companion is moving principally in declina- call a and b the absolute coordinates of the two com- tion relative to the principal star the measures in ponents — if necessary corrected for parallax — and r declination will give the best possible result. Measures the relative coordinate,—B relative to A,—at the same in right ascension would be of no value, as in that case time t. Then we shall have: the denominator in the relation giving A will be nearly equal to zero. a = ¡dt ~ (1 — A) r A K or b = ¡¿t A Ar A K We see that the mass A is given as a simple ratio of directly measured quantities, and does not contain the with the condition r — C — a. elements of the orbital motion of the binary considered when the two components can be measured so as to give In these relations ¡i represents the proper-motion of the the values of r. Even when the period is entirely un- center of gravity of the system, measured also along known the formula can give good results provided the the same direction as the one chosen for a and C; K is relative motion, that enters in the denominator, is (173) © American Astronomical Society Provided by the NASA Astrophysics Data System .173V 29. 174 THE ASTRONOMICAL JOURNAL N°- 694 large enough during the time covered by the two epochs. star is a case where the period is still very doubtful The influence of an error in the assumed proper-motion and where the relative motion was found from the will be small if the interval of time t — h is relatively measures of the companion. For the three other 1916AJ 2 short, but as the differences C2 — Ci, and r2 — n, systems -generally only one component could be meas- become small at the same time the highest accuracy ured — at least at the time when the plates were taken is required in the measurement of these quantities. — so that the elliptic elements of the motion had to For that purpose measures of photographs obtained be taken from the best sources. For the last star it with long-focus instruments, such as are used in stellar was found necessary to recompute better elements of parallax work, give an enormous advantage over the orbital motion for establishing the coefficients in meridian or visual micrometer work.' Aitken briefly the equations of condition. called attention to that point when he published his “ Observing List for the Determination of the Relative t? Cassiopeiae £ 60 (3m.6 — 7m.9) Masses in Visual Binary Systems/’ [Lick Obs. Bull. 0h 43m +57° 17' (1900) No. 208, Vol. VII, p. 7]. It is the aim of this note to present some results obtained from plates available at As a first application of the preceeding remarks we this observatory, most of them not taken for this par- consider this well known system, that has been ob- ticular purpose. The striking advantage of photogra- served many years. The period is long and still un- phy in this case is shown especially by the small num- certain by at least a century. The relative motion in ber of years required for obtaining data that are far recent years was larger in right ascension than in superior to long series of meridian or micrometric ob- declination. But as we found on hand among the servations where personal and systematic errors conceal unfinished series of parallax-plates taken for this star, most of the small displacements. In photographic two good plates separated by an eleven year interval work the determinations can be made in a purely the relative motion was thought to be large enough in differential way, as is done in parallax work, and the both coordinates. The two plates, which show meas- chances of vitiating the results by systematic or per- urable images of both components, were therefore sonal errors are reduced to a minimum. measured in both right ascension and declination, so In looking over the plate material available at this as to yield two independent determinations. observatory, a number of stars were found where de- terminations of relative masses seemed possible. By a The following plates have been used: more careful consideration of the observational data some of these stars had to be discarded for various Number Date Gr. M. T. Parallax Factors reasons, the most frequent one being the shortness of In a In $ time covered by the plates; in other cases the two h m components were sufficiently bright to produce an Schles. 473 1904 Sept. 26 17 44 + 0.161 -0.839 image, but their distance was too small (such as OA’ Tr 2194 1915 Aug. 18 20 44 +0.678 -0.479 235, £ Scorpii, ß 80) making measures on the individual components difficult. In some instances (61 Cygni, As the proper-motion of this star is large in both co- 2J 2398, . ) the time covered by the plates was large ordinates, it is important to orient the plate carefully: enough so far as the relative motion is concerned, but this was done by means oí AG stars reduced to the as in such cases the proper-motion of the center of mean of the two dates 1910.2. Five comparison stars gravity cannot be assumed, longer intervals of time have been used and the measures reduced by the de- proved to be necessary. Finally four binary stars pendence method. The solutions or measured dis- were retained as is illustrated hereafter. The first placements of the two components were found: Measured Displacement Corrected for Parallax Principal Star Companion Principal Star Companion In right ascension + 12".81 + 10".51 + 12" ,72 + 10".42 In declination - 5 .99 - 4 .40 - 6 ,05 - 5 .69 For the parallax we adopted Peter’s value 0".18 de- proper-motion of the system we have adopted the duced from heliometer measures. From photographic values given in Boss’ P. G. C. The corresponding determinations Russell found 0".19. Finally for the displacements in the 10.897 years’ interval between © American Astronomical Society • Provided by the NASA Astrophysics Data System .173V 29. N°- 694 THE ASTRONOMICAL JOURNAL 175 the two plates are given at the same time: s 1916AJ + 0 . 1392 + 12".28 Proper motions (1910.2) r/ Displacements — 0 .523 - 5".69 From these data we find: + 12.28 - 10.42 12.72 - 10.42 0.81 from right ascensions Mean 0.79 - 5.69 + 4.46 0.77 from declinations - 6.05 + 4.46 As only one plate has been used for each epoch the whence he adopts 0.57, but the result is considered as agreement must be considered as satisfactory. A very uncertain “owing to the independence of the ex- larger number of plates taken so as to avoid or to de- pressions for orbital and proper-motion respectively." termine the parallax would give a higher weight to the In the present determination the proper-motion was result.
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