1. INTRODUCTION

About thirty years ago the author began a pro-gram of observations of the radial velocities of late-type stars in the Supplement to the Yale Bright Star Catalog (^{Hoffleit et al. 1983}), for which no radial velocity was listed in that publication. That pro-gram makes use of the coudé spectrograph of the Dominion Astrophysical Observatory’s 1.22-m tele-scope, at a dispersion of 0.24 nm mm⁻¹. The three binaries whose orbits are presented here are among the first to be discovered in this program, in part due to their relatively short periods and consequent rapid changes in radial velocity.

2. BACKGROUND INFORMATION

3. OBSERVATIONS

Initially the radial velocities were determined at the telescope with the radial-velocity spectrometer (RVS) (^{Fletcher et al. 1982}). Numerous observations of IAU standard stars have been used to adjust the zero-point of the RVS system to the scale defined by ^{Scarfe (2010)}. Since 2004, when the spectrometer was decommissioned, observations have been made through the same spectrograph optics, but using a CCD as detector, and reduced with a ‘pipeline’ pro-gram developed by D. Bohlender. Radial velocities were determined by averaging the results from cross-correlation with standard stars. The choice of standard stars for each binary is based on the precision of the cross-correlation, which in turn depends on the match in spectral type between the binary and the standards. The standards for each binary discussed here are listed in Table 1, where the velocities are on the scale and zero-point of Scarfe (2010), although the velocities differ very slightly from those in that paper, owing to the inclusion of slightly more data since its publication. The uncertainties are those of the mean value, and the binaries for which each standard was used are indicated.

For HD 121212, 95 observations were obtained with the RVS between JD 2447216 and 2452880, and 68 with the CCD between JD 2453117 and JD 2457497. The total span of 10281 days is just over 87 orbital cycles, so that the period is deter-mined quite precisely, with a formal uncertainty of less than three minutes. The RVS velocities are pre-sented in Table 2, and those obtained with the CCD in Table 3.

For HD 148434, 42 observations were obtained with the RVS between JDs 2447303 and 2452886, followed by 76 with the CCD between JDs 2453117 and 2457941. The span of 10638 days is just over 50 orbital cycles, and as a result the period of this binary is also determined fairly precisely, with a formal uncertainty of a little more than an hour. The RVS data are presented in Table 4 and those from the CCD in Table 5.

For HD 148912, 68 observations were obtained with the RVS between JDs 2447263 and 2452886, followed by 82 with the CCD between JDs 2453117 and 2457941. The span of 10678 days is over 22 orbital cycles, and as a result of this and the relatively high eccentricity, the period of this binary is also determined fairly precisely, with a formal uncertainty of a little less than an hour. The RVS data are presented in Table 6 and those from the CCD in Table 7.

4. ORBITS

For all three binaries, solutions were obtained from the RVS and CCD data separately, and weights based upon the standard deviations from those solutions were assigned for a solution from the combined data. The weight for the CCD data was al-ways set to 1.0. In all cases the weight for the RVS data was lower than for the CCD data, being 0.25, 0.18 and 0.10 respectively for HD 121212, HD 148434 and HD 148912. The orbital elements are presented in Table 8 for all three binaries, and their velocity curves are given in Figures 1, 2 and 3.

Fig. 1 The observed radial velocities of HD 121212, with the curve derived from the adopted elements drawn through them. The DAO CCD data are shown as filled circles and the spectrometer data as open circles. Rejected RVS velocities are shown as plus signs.  

Fig. 2 The observed radial velocities of HD 148434, with the curve derived from the adopted elements drawn through them. The symbols have the same meaning as in Figure 1, except that there are no rejected data.  

Fig. 3 The observed radial velocities of HR 148912, with the curve derived from the adopted elements drawn through them. The symbols have the same meaning as for Figure 1, except that a rejected CCD velocity is shown as a cross. 

5. DISCUSSION

All three binaries discussed in this paper show no trace of faint companions in our data, and it must be concluded that the companions must be at least 2.5 magnitudes fainter than the primaries, especially if their spectral types are similar, since then they are likely to cause some slight distortion of the cross correlation function. Moreover, if the secondaries are main sequence stars a difference of only 2.5 magnitudes would indicate that the secondaries are of type about A0 for HD 121212 and HD 148912, and about F5 for HD 148434. The former would have mass near 3 M_⊙, and the latter, 1.3 M_⊙. Either should be detectable in the system’s colour indices, and in-deed in the spectra of HD 121212 and 148912. Thus we conclude that the secondaries are either main-sequence stars that are fainter than the primaries by more than 2,5 magnitudes, or slightly evolved sub-giants, in which case their masses are only a little less than those of the primaries, given the evolutionary timescales of stars once they leave the main sequence. One final possible alternative is that the secondaries are white dwarfs, which would again be very much fainter than the primaries, offering little hope of detection.

Despite these considerations, it might be possible to detect the companions interferometrically. The best chance of doing so, as suggested in the preceding paragraph, is in the case that they are evolved objects only slightly less massive than the primaries. We estimate below values of the angular size of the major semiaxes of the systems’ orbits. As an example, we make the plausible assumption that the

primary star’s mass is twice that of the sun for all three systems. We make use of the following two equations. The first is the equation of the mass function, in which M₁ is the primary’s mass, M ₂ is that of the secondary, and q = M ₂ /M ₁ .

The second equation is for the projected angular size α of the major semiaxis, in terms of the parallax

π, the mass ratio q, the inclination i, and the product a₁ sin i, determined from the orbit.

As indicated above we assume that despite the luminosity differences between the stars, they are simi-lar in mass and surface temperature, as was found for the components of HR 6046 by ^{Scarfe et al. (2007)}, where the companion was detected with difficulty, despite being almost equal in mass to the primary. Here, let us assume that q = 0.9 for all three binaries. Then we solve equation 1 for the inclination of each orbit, and insert its cosecant into equation 2 along with the values of a₁ sin i from the orbit solutions, the parallaxes and the assumed values of q to find the projected major semiaxes.

An alternative, more direct, approach is to estimate the angular sizes of the systems’ major semi-axes, using Kepler’s Third Law and their parallaxes, by means of equation (3), in which a is the angular major semiaxis, π is the parallax, P is the period in years and M is the total mass of the system in solar units.

A note of caution - choice of too small a value for M , coupled with too low a value of the mass ratio q, may lead to values of sin i that exceed unity when the mass function is large, as it is for HD 121212. For example our choice of 3.8 solar masses for the total mass of that system does not permit the mass ratio to be as small as 0.6. One can avoid this pitfall by checking with equation 1.

Since our choice of primary mass and system mass ratio are the same for both the method using the mass function and that using Kepler’s Third Law, it is not surprising that both methods yield the same results for the angular major semiaxes, 1.7 mas for HD 121212, 9.1 mas for HD 148434, and 5.4 mas for HD 148912. The angular separations at any orbital phase should be of this order of magnitude for our representative choice of masses, and any plausible masses and mass ratios are unlikely to yield very different results, as indicated by the weak dependence of equation 3 on the total mass. High inclinations like that of HD 121212 will make the separation significantly smaller at times, and large eccentricities like that of HD 148912 will give occasionally larger separations. The major axes estimated here would make resolution of the pairs challenging for Gaia and very difficult, if not impossible, for earlier instruments. Nor do any of the inclinations estimated here offer much hope of detecting eclipses, despite the likely angular sizes of the giant primaries. Determination of the actual masses thus remains a challenge.

The author would like to acknowledge the National Research Council of Canada for the provision of observing time at the Dominion Astrophysical Observatory for a period now exceeding half of the century of that observatory’s existence, and especially to thank members of the DAO staff, in particular D. Bohlender, D. Monin and the late L. Saddlemyer for their greatly appreciated support and assistance. He would also like to thank a kindly referee for some helpful advice.