1. INTRODUCTION

The three binary stars discussed in this paper came to the author’s attention in different ways. 6 Per was interesting because of its high, and at the time poorly determined, eccentricity. This, coupled with an argument of periastron close to 270^◦, leads to a rapid increase in radial velocity, most of which is accomplished in only one percent of the orbital period, which is close to 4.3 years. HR 8078 was in a list of stars claimed to be possibly variable in velocity, whose author had been unfortunate enough to miss the short interval, about a year in a period a little over seven years, in which most of the excursion of radial velocity occurs, and this led to the uncertainty. HD 212989 was one of a handful of stars with no published radial velocity in a list used by the author many years ago. Since then an orbit has been published by others, but their regrettable distribution of observations led them to deduce the wrong period. The correct one is a little over five years.

Observations of all three systems have been obtained using the coudé spectrograph of the Dominion Astrophysical Observatory’s 1.22-m telescope, at a dispersion of 0.24 nm mm⁻¹. 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 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’ program developed by D. Bohlender. Radial velocities were determined by averaging the results from cross-correlation with the standard stars listed in Table 1, where the velocities are on the scale and zero-point of ^{Scarfe (2010)}, and the binaries for which each was used are indicated.

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 always set to 1.0. In all cases the weight for the RVS data was lower than for the CCD data, being 0.35, 0.20 and 0.25 respectively for 6 Persei, HR 8078 and HD 212989. The orbital elements are presented in Table 10 for all three binaries.

2. 6 PERSEI

The bright star 6 Persei (HR 645, HD 13530, HIP 10366, α = 2^h 13^m 36^s, δ = 51^◦ 3^′ 57^′′ (J2000)) lies very close to the southern edge of its constellation, near the boundary with Andromeda. Its UBV magnitude and colours have been determined several times, with results clustering around the values V = 5.31, B−V = 0.93, U−B = 0.62, consistent with its spectral type of G8 III, given in the Bright Star Catalog.

The Hipparcos parallax of 6 Per (^{van Leeuwen 2007}) is π = 15.4 ± 0.5 mas, which yields M_v = 1.25 ± 0.07, as might be expected from its spectral classification. Hipparcos also measured proper motions of 347 mas yr⁻¹ in right ascension and -171 mas yr⁻¹ in declination. Together with the parallax these yield a velocity of 17.1 km s⁻¹ perpendicular to the line of sight, almost as large as the systemic radial velocity.

The first known spectroscopic orbit of 6 Per was published by ^{Christie (1936)}, based on observations from DAO and Lick Observatory. It gave only rough values of the elements and was graded ‘e’ (poor) by ^{Batten et al. (1989)}. But it was not until 1999, when the present observations were well under way, that a much better orbit was published by ^{de Medeiros and Udry (1999)}. By then, however, the author’s observations with the radial velocity spectrometer yielded an orbit of quality comparable to theirs, so observations were pursued in the hope of making further improvements.

For 6 Persei, 155 observations were obtained with the RVS between J.D. 2444257 and 2453040, and 68 with the CCD from J.D. 2453213 to 2457295. The total span is a little more than eight orbital cycles. The RVS and CCD velocities are presented in Tables 2 and 3 respectively.

For 6 Persei, the elements in Table 10 differ only slightly from those obtained by ^{de Medeiros and Udry (1999)}, and only the period is determined here with much greater precision. The most notable difference is that our values of K, and hence of a₁ sin i and f(M), are significantly smaller than theirs. The velocity curve derived from the new elemsts is presented in Figure 1.

Fig. 1 The observed radial velocities of 6 Persei, 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. A rejected CCD velocity is shown as a plus sign. 

3. HR 8078

The bright star HR 8078 (HD 200817, HIP 103956, α = 21^h 3^m 48^s, δ = 53^◦ 17^′ 9^′′ (2000)) lies in the northern part of Cygnus, about two degrees south of the boundary with Cepheus. Its spectral type is given as K0 III, and its BV magnitude and colour are V = 5.92, B−V = 1.01, according to the Tycho-2 Catalogue. Its Hipparcos parallax (^{van Leeuwen 2007}) is π = 8.05 ± 0.43 mas, which yields M_v = 0.46 ± 0.12, consistent with its spectral classification. Hipparcos also measured proper motions of 59 mas yr⁻¹ in right ascension and 11 mas yr⁻¹ in declination. Together with the parallax these yield a velocity of 34.4 km s⁻¹ perpendicular to the line of sight, comparable to the systemic radial velocity.

HR 8078 came to the author’s attention in a list of stars with possibly variable velocity published by ^{Beavers (1985)}. HR 8078 has only a small variation for nearly six years of its seven-year period, and the author was lucky that the variation in the seventh year occurred very soon after he began observing, while Beavers was unfortunate in missing such a year, and was therefore unable to show variability with certainty. After following the variation for several years the author presented a preliminary orbit to the Canadian Astronomical Society. In the meantime, however, Prof. R.F. Griffin had also begun to obtain observations through the rapidly variable portion of the cycle. But on seeing the author’s abstract in print (^{Scarfe 2001}) he very kindly offered to contribute his data to the determination of a definitive orbit.

For HR 8078, 164 observations were obtained with the RVS between J.D. 2444548 and 2452971, and 43 with the CCD from J.D. 2453144 to 2457295. The total span for HR 8078 is about 4.75 orbital cycles. The RVS and CCD velocities are presented in Tables 4 and 5 respectively. In addition, the 19 observations covering one of the minima near periastron, that were obtained by Griffin and kindly contributed by him to the orbital solution, are given in Table 6. It should be noted that he provided his data to a precision of only 0.1 km s⁻¹, but their very small residuals sugggest that they appear to merit a second decimal digit. They were included in the solution with unit weight, after a small adjustment of their zero poit.

The elements are presented in Table 10 and the velocity curve derived from them is drawn through the data in Figure 2.

Fig. 2 The observed radial velocities of HR 8078, with the curve derived from the adopted elements drawn through them. The DAO data are shown as in Figure 1. Griffin’s data are shown as filled triangles, and a rejected RVS velocity is shown as a cross. 

4. HD 212989

HD 212989 (HIP 110900, α = 22^h 28^m 7^s, δ = 12^◦ 14^′ 55^′′ (2000)) is to be found in Pegasus, almost on a direct line between ǫ and α Peg, and slightly closer to the latter. Its UBV magnitude and colours are (^{Argue, 1966}) V = 7.06, B-V = 0.90, U-B = 0.56, consistent with its MK type, K0 V. It should be noted that HD 212989 is the primary of a visual binary, known as ADS 15972 or WDS J22281+1215, with a period of 554 years. The secondary is 2.28 magnitudes fainter than the primary, but no doubt contributes a little to the above magnitude and colours, since the current separation is close to one arc second.

The Hipparcos parallax of HD 212989 is π = 15.87 ± 1.18 mas, which yields an absolute visual magnitude of M_v = 3.06 ± 0.16, more luminous than would be expected from its spectral type, by more than can be accounted for by the light of the secondary, suggesting that the primary star is somewhat evolved from the main sequence. Hipparcos also measured proper motions of 196 mas yr⁻¹ in right ascension and -9 mas yr⁻¹ in declination, which in turn yield a velocity of 57.3 km s⁻¹ perpendicular to the line of sight.

HD 212989 first came to the author’s attention when he was still a graduate student, when it was one of a small handful of stars on his observing list that had no known radial velocity. He did not try to remedy that lack until after the radial velocity spectrometer became available in the early 1980s. The primary’s binary nature was soon discovered and the system has been followed ever since. However in the meantime it also came to the attention of other observers, including ^{Massarotti et al. (2008)}, who included HD 212989 in a long list of binaries for which they determined orbits. Unfortunately in the case of HD 212989 the distribution of their data led them to determine an incorrect period. The correct value is derived in this paper from the author’s observations and shown to be consistent with the data of Massarotti et al. More recently Halbwachs et al. (2014) detected a very faint spectroscopic secondary component and obtained a rather uncertain mass ratio of 0.72 for the system, although they appear not to have redetermined its orbit. They also find that the star’s absolute magnitude is 3.0, and suggest that it be reclassified as K0 IV. A more recent value of the parallax obtained by GAIA, 11.14 mas, would yield an even brighter absolute magnitude.

For HD 212989, 169 observations were obtained with the RVS between J.D. 2444823 and 2452971, and 51 with the CCD from J.D. 2453180 to 2457295. The total span in this case is over 6.5 orbital cycles. The RVS and CCD data are presented in Tables 7 and 8 respectively. Because a separate solution was also made with the CCD data and the CfA data of ^{Massarotti et al. (2008)}, those CfA data are also reproduced in Table 9. In that solution the CfA data were given weight 0.5, but no adjustment of their zero point was necessary.

In all tables the residuals are those from the final solution, except for the CfA data, where they are from the CfA-CCD solution mentioned above and included in Table 10. The velocity curves from the solution from DAO data only, and from the CfA-CCD solution, are drawn through the corresponding data in Figures 3 and 4 respectively. As can be seen the CfA-CCD solution is very similar to that from DAO data only, attesting to the very satisfactory way in which the CfA data conform to the new solution. In fact it is clear that the origin of the erroneous solution presented by ^{Massarotti et al. (2008)} is simply that they were unfortunate in placing their final observation on the wrong branch of the velocity curve.

Fig. 3 The observed DAO radial velocities of HR 212989, with the curve derived from the adopted elements drawn through them. The symbols have the same meaning as for Figure 1, except that there are no rejected data. 

Fig. 4 The observed DAO CCD radial velocities of HR 212989, and those from CfA that were used by ^{Massarotti et al. (2008)} with the curve derived from the elements obtained from all those data drawn through them. The DAO data are shown as filled circles, and the CfA data as open circles. 

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. Despite this, it might be possible to detect the companions interferometrically. We estimate below values of the likely angular separation of the stars at apastron, on two different assumptions. For both, we make the plausible assumption that the primary’s mass is twice that of the Sun in the cases of 6 Per and HR 8078, and 1.2 solar masses for HD 212989, for want of evidence to the contrary. 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, M₂ is that of the secondary, and q = M₂/M₁.

The second equation is for the projected separation ρ at apastron, in terms of the parallax π, the mass ratio q, and the eccentricity e, the argument of periastron ω, the inclination i, and a₁ sin i.

Our first method begins by assuming that despite the luminosity differences between the stars, they are similar in mass, as was found for the components of HR 6046 by ^{Scarfe et al. (2007)}, where the companion was detected with difficulty. Here, let us assume that q = 0.9 for all three binaries. In that case equation (1) gives inclinations of 43^◦.6, 31^◦.2, and 26^◦.9 for 6 Persei, HR 8078 and HD 212989 respectively. These values, the relevant elements, and the Hipparcos parallaxes, entered into equation (2), yield angular separations at apastron of 87, 20 and 81 mas for the three binaries.

The second method is follow the procedure used by ^{Scarfe and Griffin (2012)} for HD 23052, namely to assume for all three binaries the average value of sin³ i, which for a random distribution is 0.589 (^{Scarfe 1970}). This turns equation (1) into a cubic equation in q, which yields mass ratios of 0.68, 0.47 and 0.405 for the three stars in the above order, and then equation (2) yields angular separations at apastron of 63, 12 and 44 mas for each. These results are slightly smaller than those for q = 0.9, but similar in order of magnitude.

We can perhaps do a little better in the case of 6 Persei. ^{Pourbaix and Boffin (2003)} made use of the Hipparcos intermediate astrometric data to find an inclination of 104^◦, and assuming a primary mass twice that of the Sun, as we have done above, found q = 0.56. Using these results in equation (2) yields an apastron separation of 27 mas. This smaller result is primarily due to the high inclination, so despite the very eccentric orbit of 6 Per, the angular separation may be larger at phases other than apastron.

For HD 212989 we can choose to adopt the preliminary mass ratio of 0.72 found by Halbwachs et al. (2014), which yields an inclination of 32^◦.7 and a separation at apastron of 73 mas.

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

As it happens all the previously described methods yield cube roots of the total mass close to 1.5 M⊙13 for both 6 Per and HR 8078, and 1.25 M⊙13 for HD 212989. Those values lead to angular major semiaxes of 61 mas for 6 Per, 46 mas for HR 8078 and 60 mas for HD 212989.

All of these approaches yield angular separations that are amenable to modern interferometry, depending, of course, on the correctness of our underlying assumption that the secondary is a mainsequence star and not a more compact object. That assumption appears to be correct in the case of HD 212989, from the work of ^{Halbwachs et al.}, but is unproven for the other two systems. Interferometric observations would yield much valuable information on all three. It is to be noted that the above discussion is intended only to give the order of magnitude of the likely separations, since phases other than apastron may give larger values, and better parallaxes from GAIA or other sources may alter the details of the results, but not the conclusion that interferometric observation is possible. Ultimately, reliable detection of the secondaries will permit confident determination of masses and other fundamental properties.

The author is grateful to R.F. Griffin for generously contributing his observations of HR 8078 to this work. He would also like to acknowledge the greatly appreciated support and assistance of members of the DAO staff, in particular D. Bohlender, D. Monin and the late L. Saddlemyer. Thanks are also due to a referee for some helpful comments and suggestions.