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In the case of visual binaries, with the inclination and the relative radial velocities of the two components known, we can evidently obtain the masses, the dimensions of the orbit, and the parallax, but not the diameters and densities of the two stars.
In the other case, the one under consideration, where the spectroscopic binary revolves in a plane so nearly in the line of sight that the stars mutually eclipse one another twice in the period, then, if an accurate light curve is available, not only the inclination but the relative dimensions of the two bodies in terms of the major axis of the system can be determined. When, in addition, both spectra can be measured, complete information can be obtained about the actual diameters, masses, densities, and distance apart of the two bodies. This is of importance in view of the small number of stars for which we have such data.
In the three orbits to be discussed here, both spectra have been observed in U Ophiuchi and RS Vulpeculae in which consequently complete dimensions are available. With TW Draconis the second spectrum is invisible and the dimensions in this case must depend upon an assumption as to the ratio of the masses of the two bodies.
The eclipsing variable U Ophiuchi (a, 17h 11.4m; 8, +1° 19′ 1900, Vis. Mag. 5.7, Spectral Type B9) was placed under observation for radial velocity on March 19, 1919, and observations were continued until June 4. In all 18 plates were obtained, of which 14 were used in determining the spectroscopic elements. The remaining four plates were so near the minimum or zero velocity in the orbit that the doubled lines could not be separated, with the result that their velocities are unreliable and were not used.
The lines of the spectrum are rather wide and are diffuse and faint, lacking in contrast, this latter probably due to the superposition of the continuous spectrum of the one star on the absorption lines of the other. One plate, No. 1983, obtained at principal minimum shows the spectrum with single lines. This spectrum is principally but not wholly, as it is only a partial eclipse, of the fainter star, and hence gives a truer idea of the character and type. From this spectrum the type should be classed as B5 but is rather
abnormal in the breadth of the hydrogen lines. In all nine doubled lines have been measured: Hy, H8; He 4472, 4388, 4144, 4026; C 4267; Mg 4481; Ca, K, 3934. Owing to the lack of contrast and the diffuseness of the lines, the measures are difficult and necessarily not of very high accuracy. In many cases some of the lines of the weaker spectrum could only be recognized and bisected by their showing a faint lightening of the spectrum on each side of the wire. Nevertheless, I believe the measures are reliable, as there is fair interagreement among the values for the different lines, and the general dimensions of the system are substantially correct.
TABLE I-OBSERVATIONS OF U OPHIUCHI.
In the above table the first three columns give the plate number, the date and Julian date of the observations. The fourth column gives the phase from primary minimum of the photometric orbit computed from the initial phase J.D. 2,418,026.703 with the period 1.6773476 days. The fifth and sixth columns give the observed velocities and the eighth and ninth the residuals from the final orbit of the brighter and fainter components respectively, while the seventh column contains the number of doubled lines measured on each plate, with the exception of plate 1983 on which six single lines were measured.
As the photometric orbit does not show any ellipticity and as the observations seem to follow sine curves as closely as can be expected, the orbit was assumed to be circular and consequently only K, the half amplitude, and y, the velocity of the system, remain to be determined. From smooth curves drawn through the observations K was assumed 182.0 km., K, 210.0 km. and y, -12.0 km.
The velocity at any phase is given by Vy+K sin @ and to apply least squares corrections we have dVdy+dK sin 0.
The least squares correction gives the final elements
while the probable error of a single plate for each spectrum was the same and equal to ±8.3 km. Considering the character of the spectrum and the difficulty of measurement, these results are satisfactory, the uncertainty of K and consequently of the dimensions of the system being only slightly over one per cent.
Applying these values by the well known formulae we have
In Shapley's photometric orbit* three solutions are obtained, two assuming uniformly illuminated discs and a third postulating darkening towards the limb. Shapley's second and third solutions are applied here, one uniform, one darkened. These give inclinations of 85° 42' and 83° 58',respectively, and make the radius of each star 0.252 of the semi-axis of the relative orbit in each solution. The dimensions, masses and densities of the system at once follow.
The volumes of the stars are computed assuming the elliptical
forms given by the photometric orbit and
*Contributions from Princeton Observatory, No. 3, p. 84.
If we assume the surface intensity of a B5 star to be -2.5 magnitudes*, the sun's absolute magnitude to be 4.86, the absolute magnitude of the brighter component of U Ophiuchi whose apparent magnitude is 6.35, is -0.19 and hence the parallax is 0.0049.
A graph showing the sine curves for the two components with the velocities represented by open circles is shown in Fig. 1.
The eclipsing variable RS Vulpeculae (a, 19h 13.4m, 8, +22 16', 1900, Type A, Vis. Mag. 7.35) was placed under observation April 26, 1919, and the last plate used in the principal orbit was obtained on July 20, 1919. A fine-grained Seed 23 plate was obtained on July 30 and used for the second spectrum. Fourteen plates were obtained in this interval, all of which have been measured. and used. The spectrum is of Type B8 instead of A and the magnitude as judged by comparison with U Coronae and TW Draconis appears considerably brighter than given by Nijland or Stewart.
*Astrophysical Journal, 40, p. 415, 1914.
This binary is especially interesting on account of the great disparity in size of the two components, the fainter star being five times the diameter of the brighter and nine-tenths as bright. It would consequently be expected, when the two stars are of nearly equal brightness, that the second spectrum would be plainly visible, but it can only be seen and measured with great difficulty and it is estimated to be of only about one-fourth the intensity of the brighter spectrum. Nevertheless, it was measured on six plates and although the residuals are in some cases rather large, the probable error of the measures of the second spectrum on a single plate being ±8.1 km., the lines and plates are in fair interagreement. Con
sequently, there can be no doubt of the reality of the second spectrum even though the mass of the fainter body is only 0.31 that of the brighter, a greater difference in masses when both spectra appear than has previously been found. Why the second spectrum. should be relatively so faint, when the two stars are of nearly equal brightness, is not apparent. It may be that, although the continuous parts of the spectra are of nearly equal intensity, the absorption lines of the second body are fainter than those of the primary, rendering them difficult to see when the continuous spectrum of the primary is superposed. Or again, a more likely explanation is that the lines are widened so much by the rotation of the large-diameter fainter star as to be made relatively very faint. One plate was