OBSERVATIONS OF BINARY STARS WITH THE DIFFERENTIAL SPECKLE SURVEY INSTRUMENT. III. MEASURES BELOW THE DIFFRACTION LIMIT OF THE WIYN TELESCOPE^*

To form the list of observations under consideration for the current work, we reviewed the archive of WIYN speckle data from the RYTSI period to the present and identified possible sub-diffraction-limited observations. We then used the same method of reduction and analysis as in our previous papers (most recently described in Paper II), i.e., the observations selected conform to the same data quality cuts as normal observations above the diffraction limit. This is a Fourier-based method, where a fringe pattern is fitted to the object's spatial frequency power spectrum deconvolved by that of a point source. The region over which the fit is made is approximately an annulus in the Fourier plane. The inner region (representing the lowest spatial frequencies) was not fit due to the fact that it is dominated by the seeing disk, and small differences in seeing (at high signal-to-noise ratio) can greatly affect the final reduced-χ² of the fit. On the other hand, the highest spatial frequencies (near and beyond the diffraction limit) are dominated by noise and can likewise affect the final fit in an adverse way. The outer boundary of the fit annulus is therefore set as a contour of constant signal-to-noise ratio.

In previous work, we applied a data quality cut such that the effective outer radius of the fit annulus times the separation was required to be above a certain value. This ensured that the observation was at or above the diffraction limit in high-quality observations, and for lower quality observations, it ensured that the observation displayed at least three fringes (a central and both first-order fringes) within the fit annulus, which we determined was needed to make certain that lower signal-to-noise observations had high-quality astrometry. Obviously, in the current work this particular data cut was relaxed, as even high-quality observations would exhibit only a central fringe before the diffraction limit was reached in the Fourier plane. Because of this, it is not unreasonable to expect that some loss of astrometric precision may occur in sub-diffraction-limited observations.

Based on this reduction scheme, 222 observations were identified for consideration for this work. For RYTSI data, three filters are represented: 550 nm, 698 nm, and 754 nm. For DSSI, the data were taken in three filters of slightly different wavelengths: 562 nm, 692 nm, and 880 nm. The basic properties of this sample are illustrated in Figures 1 and 2. The first of these shows the magnitude difference obtained as a function of both (1) separation and (2) system magnitude. Figure 1(a) illustrates that the sensitivity to large magnitude difference systems decreases with decreasing separation, as can be expected since the fringe depth becomes shallower with increasing magnitude difference, and therefore the sub-diffraction-limited separations become harder to identify. Also of note is the fact that the envelope of this plot sits at approximately 3 mag when the measures are near the diffraction limit and matches extremely well with what we have found for systems just above the diffraction limit in previous papers (see, e.g., Figure 1(a) of Paper II). Figure 1(b) shows that system magnitudes of as faint as V = 10 can be identified, though again the sensitivity to magnitude difference falls off at fainter magnitudes. This can be understood in terms of signal-to-noise ratio and compared directly with Figure 1(b) of Paper II. The current figure has a very similar appearance though it appears shifted to the left (or in other words, toward brighter magnitudes) by approximately two magnitudes relative to work above the diffraction limit. We conclude that speckle observations below the diffraction limit are less sensitive both in terms of limiting magnitude and magnitude difference than those above the diffraction limit.

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In Figure 2, we explore the astrometric repeatability of the sample by pairing observations wherever possible, either by using the simultaneous observations in the case of DSSI or sequential observations in pre-DSSI observations. (In the latter case, the second observation was only required to be during the same observing run as the first observation, not directly sequential in time.) Figure 2(a) shows the behavior of the position angle differences between each pair, while Figure 2(b) shows the separation differences. Both are plotted as a function of the average separation obtained. The mean value for the position angle difference is $\overline{\Delta \theta } = -6\mbox{$.\!\!^\circ $}7 \pm 2\mbox{$.\!\!^\circ $}8$, while the subset of observation pairs taken in different filters, this is reduced to $\overline{\Delta \theta } = -2\mbox{$.\!\!^\circ $}3 \pm 1\mbox{$.\!\!^\circ $}9$. For the subset of observation pairs taken in different filters and simultaneously, the result is $\overline{\Delta \theta } = -1\mbox{$.\!\!^\circ $}6 \pm 1\mbox{$.\!\!^\circ $}6$. In separation, the average differences for the same three samples are $\overline{\Delta \rho } = 0.1 \pm 1.3$ mas, $\overline{\Delta \rho } = -0.7 \pm 0.6$ mas, and $\overline{\Delta \rho } = -0.6 \pm 0.6$ mas, respectively. Turning now to the standard deviations for these three samples, we obtain σ_Δθ = 225 ± 20, σ_Δθ = 131 ± 13, and σ_Δθ = 93 ± 11 in position angle and σ_Δρ = 10.4 ± 0.9 mas, σ_Δρ = 4.3 ± 0.4 mas, and σ_Δρ = 3.5 ± 0.4 mas. In general, these values appear to indicate that better repeatability is achieved when the observations are obtained simultaneously. There is also basic consistency between the position angle and separation values, as the average separation of the sample is approximately 30 mas and, at that separation, a linear measurement difference of 3.5 mas represents an angle difference of approximately $\arctan (3.5/30) \sim 7^\circ$, compared with the measured value of ∼9°. The fact that the measured value is slightly larger than the linear prediction is easily explained by the smallest separation systems, where the predicted angle would be much larger than that of the average separation.

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Taking the 3.5 mas figure as the best-case scenario, we note that this figure should be approximately equal to $\sqrt{2}$ times the true standard deviation of the sample, since in the subtraction, the sample standard deviation is added in quadrature with itself (assuming Gaussian errors). Furthermore, if the astrometry from the two observations is averaged, then this would decrease the sample standard deviation by another factor of $\sqrt{2}$. Therefore, the best case of the precision value for paired, averaged astrometry is 3.5/2 = 1.8 mas. This is somewhat higher than what we have recently found for observations above the diffraction limit (1.1 mas in Paper II), but given the more challenging nature of sub-diffraction-limited work, still good enough to be quite useful even at very small separations.

Of the 222 observations initially identified as of interest for this project, 90 are of objects with orbits in the Sixth Catalog of Visual Binary Star Orbits (Hartkopf et al. 2001b). If we consider only objects with ephemeris separation below the diffraction limit at the time of observation and having published uncertainties in the orbital elements, 66 observations remain. This provides an excellent sample with which to study the measurement accuracy and precision in the sub-diffraction-limited case. We can first study the observed minus ephemeris residuals from the orbits for these measures, treating each measure singly, that is, not pairing any observations, even if two were taken at the same time. This is shown in Figure 3. In calculating the ephemeridal uncertainties δθ and δρ in each case from the published uncertainties in the orbital elements, we find a large range of values. This highlights the fact that the orbits themselves have a range in quality, but if we consider the highest quality orbits as those with δθ ⩽ 120 and δρ ⩽ 5 mas, then we obtain a mean residual of $\overline{\Delta \theta } = -8.4 \pm 5\mbox{$.\!\!^\circ $}1$ with standard deviation of σ_Δθ = 34.0 ± 36. For separation, the results are $\overline{\Delta \rho } = +4.5 \pm 1.4$ mas with standard deviation of σ_Δρ = 10.2 ± 1.0 mas. The largest residuals in both cases occur at the smallest ephemeris separations, below ∼20 mas. If only observations above this value are considered, then the mean and standard deviations are $\overline{\Delta \theta } = -7.0 \pm 3\mbox{$.\!\!^\circ $}4$ and σ_Δθ = 24.8 ± 24 for position angle, and $\overline{\Delta \rho } = +0.4 \pm 1.2$ mas and σ_Δρ = 6.6 ± 0.9 mas in separation. The standard deviation values contain both the uncertainty in the ephemeris and the error, both random and systematic, from our measures. Nonetheless, these results indicate that it is unwise to report measures below 20 mas because there is at minimum a systematic overestimate of the separation in these cases. Above this ephemeris separation, on the other hand, there is no evidence for a significant offset in either coordinate.

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Next, we can pair observations and examine the residuals in this case. We consider three types of pairs: (1) pairs where both observations are taken during the same telescope pointing and are in different filters (sequentially if taken before DSSI was completed in 2008 and simultaneously if taken with DSSI), (2) pre-DSSI pairs that are not taken on the same pointing but are from the same run and are in different filters, and (3) observations taken in the same pointing but in the same filter. These residuals are shown in Figure 4, with observation Type 1 drawn as filled circles, observation Type 2 as open circles, and observation Type 3 shown as crosses. The horizontal axis used in these plots is the difference in secondary position between the two observations in arcseconds divided by the mean separation, which represents a dimensionless consistency parameter characterizing the observation pair. The plots demonstrate that requiring consistency between the two colors (if the observation pair is taken in two filters) does help to distinguish between observations affected by systematic error (most likely residual dispersion) and those that are more trustworthy. On the other hand, the one-filter pairs can have a large residual but a small abscissa, indicating that the color information is indeed necessary to make this determination. Note that there will be some duplication in these plots due to cases that can be considered in either Type 2 or Type 3, depending on how the data files for a given run are paired.

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Figure 4 suggests that the following simple approach can be used in the analysis of our sub-diffraction-limited observations:

• 1.  
  Wherever possible, an observation should be paired with another taken in a different filter. That is, observation Type 1 defined above is most desirable, followed by observation Type 2. For such observation pairs, calculating the difference in secondary position divided by the average separation and applying the data cut at 0.7 will ensure high data quality without significant systematic error.
• 2.  
  For observations that cannot be paired with another observation in a different filter, one must apply a data cut in observed separation at 20 mas, and not report measures below this value, as these are susceptible to systematic error. Observed separation is essentially a proxy for ephemeris separation in this context, since many observations will not have an orbital prediction. As a consequence, this data cut may not eliminate systematic error completely since the effect is to increase the observed separation (possibly above the limit of 20 mas); however, it is the only observable available for this purpose.

We study the final astrometric accuracy and precision in the same way as described above for the full set of observations, that is, by comparing to the ephemeris position of those objects with orbits in the Sixth Orbit Catalog. We confine our attention to only those orbits which have published uncertainties for the orbital elements, shown in Table 3. The astrometric properties of the observations in the two final tables are detailed in Table 4 and in Figure 5. In the former, we show the number of measures, average residual (observed minus ephemeris), and standard deviation in both separation and position angle for five subgroups of data: (1) all unpaired observations (i.e., those appearing in Table 1), (2) observations that are paired but which were taken in different telescope pointings, (3) those paired but taken during the same telescope pointing, (4) all paired observations (i.e., those appearing in Table 2), and (5) the paired observations of the objects with the highest quality orbits (with ephemeris uncertainty of less than 5 mas in separation or less than 12° in position angle, respectively. The average residuals of these subsamples show a scatter around 0 of up to ∼2σ in the worst case; nonetheless, the sample sizes are not large here and the unpaired observations as well as the sample of all paired observations do not appear to have values that differ significantly from zero. The standard deviations are larger for the unpaired sample than for the all-paired sample; this is at least partly due to the fact that we have averaged the astrometry in the case of the paired observations. However, error from the ephemerides is also included here.

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Table 4. Measurement Precision Results

Data Group	Observed	Number	Average	Standard	Avg. Eph.	Subtracting
	Parameter	of Meas.	Residual	Deviation	Uncertainty	in Quad.
Unpaired observations (Table 1)	ρ	14	3.2 ± 2.3 mas	8.7 ± 1.6 mas	5.5 ± 1.4 mas	6.7 ± 2.4 mas
Paired, diff. pointing	ρ	11	2.2 ± 1.4 mas	4.5 ± 1.0 mas	3.5 ± 1.4 mas	2.8 ± 2.4 mas
Paired, same pointing	ρ	6	−2.0 ± 1.9 mas	4.6 ± 1.3 mas	3.8 ± 1.6 mas	2.6 ± 3.3 mas
All paired (Table 2)	ρ	17	0.7 ± 1.2 mas	4.9 ± 0.8 mas	3.6 ± 1.0 mas	3.3 ± 1.6 mas
All paired with δρeph < 5 mas	ρ	14	0.8 ± 1.2 mas	4.4 ± 0.8 mas	1.8 ± 0.3 mas	4.0 ± 0.9 mas
Unpaired observations (Table 1)	θ	14	−68 ± 55	206 ± 39	155 ± 40	136 ± 75
Paired, diff. pointing	θ	11	56 ± 28	92 ± 20	76 ± 26	52 ± 52
Paired, same pointing	θ	6	−45 ± 77	189 ± 55	88 ± 36	167 ± 65
All paired (Table 2)	θ	17	20 ± 33	138 ± 24	80 ± 20	112 ± 33
All paired with δθeph < 120	θ	13	53 ± 27	97 ± 19	39 ± 09	89 ± 21

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To obtain an estimate of the true measurement uncertainty, we compute the average ephemeris uncertainty and subtract this in quadrature from the standard deviation, in essence assuming that the measurement errors here and those of the orbital elements are uncorrelated. (Since all of the orbits used here have uncertainties in orbital parameters listed in the Sixth Catalog, we can use these to compute uncertainties in the observables ρ and θ for a desired observation date.) The final values for the measurement uncertainty in separation are 6.7 mas for the observations in Table 1 and 3.3 mas for those in Table 2. The other values in the same (rightmost) column of the table indicate that there is no significant advantage in precision when pairing observations taken on the same telescope pointing (either sequentially for pre-DSSI observations or simultaneously with DSSI) and those taken on different pointings but during the same observing run. This provides the justification for combining all such pairings into Table 2. For the position angle, we find values of 136 for the unpaired observations and 112 for the paired observations. These may be converted into an estimate of the linear measurement uncertainty orthogonal to separation by computing the arctangent and multiplying by the average separation; in doing so, we find that the unpaired observations have value 7.7 mas and the all-paired sample has value 5.8 mas. Finally, since these values are measured orthogonal to the separation and therefore represent independent values, we can average these with those mentioned above to obtain a final linear measurement precision. For unpaired observations (Table 1), the result is 7.2 mas and for the all-paired sample (Table 2) it is 4.6 mas. Recalling that the measures in Table 2 are the average of those obtained in two filters, we would expect the difference in precision to be a factor of $\sqrt{2}$ between the two samples; indeed, $7.2 / \sqrt{2}$ = 5.1 mas, very similar to 4.6 mas. However, it is important to emphasize that the paired observations also represent a sample that includes separations at and below 0.25 of the diffraction limit, while the unpaired sample is limited to somewhat larger separations.

To give a feel for the data used in this study, we show three of the orbits used in the study in Figures 6 and 7. In Figure 6(a), we plot existing and new data for YR 6Aa,Ab together with our own recent orbit determination (Paper II). The new data presented here fall very close to the predicted orbital path, although it should be stated that all data to date has been reported by our group, and a greater diversity of observers would be desirable in order to make certain that no systematic trends exist. Figure 6(b) shows the orbital data of BLA 1Aa,Ab, where the orbit is that of mason (1997). In this case, there is more scatter in the orbital points most likely owing to the contributions of several observers, but again, despite the small scale of the orbit by speckle standards, the data quality of the points presented here is reasonably good.

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In Figure 7, we show the multiple system HIP 85209 = HD 157948. This is a hierarchical quadruple system, where the widest component (COU 1142) has a separation of approximately 2 arcsec and is not shown. (This component has shown little motion over the last 20 years according to data in 4th Interferometric Catalog.) However, the innermost pair, a spectroscopic binary whose orbit was determined by Latham et al. (1992) and updated by Goldberg et al. (2002), was resolved and measured several times by Horch et al. (2006b) using the Fine Guidance Sensors (FGSs) on the Hubble Space Telescope. The FGS observations also revealed the presence of an intermediate-separation component (HSL 1Aa,Ac) that has been easily monitored with speckle observations at WIYN over the past few years. This component has a number of measures in Tables 1 and 2 and is the most frequently measured component that is well above the diffraction limit, due to our interest in the spectroscopic pair here. The plot of the orbital data shows that HSL 1Aa,Ac has undergone significant orbital motion during this period of time, with rapidly decreasing separation. The data in hand end with the 2010 sub-diffraction-limited measure appearing in Table 2. This measure and the one for the spectroscopic pair of the same observation date were obtained with a triple-star fit to the power spectrum resulting in the two sub-diffraction-limited separations (with the fourth component just off of the chip). It is of course possible to fit the data of HSL 1Aa,Ac to an orbit, and we have done so, obtaining a period of approximately 18 years. However, we feel that it is premature to report the other orbital elements at this stage since the 2010 observation has a quadrant ambiguity that affects the period substantially. In any case, the best approach for this system would be to incorporate all of the data available for the system in a simultaneous orbit fit for both HSL 1Aa,Ac and the inner pair. We hope that the data presented here will encourage other observers to work on this system over the next few years.

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Our standard method for estimating the accuracy and precision of our differential photometry in previous papers has been to compare with the space-based magnitude differences appearing in the Hipparcos Catalogue. We have generally considered only speckle observations taken in a filter with properties similar to the H_p filter. However, for objects presented here, there are few that have values listed in the Catalogue, owing to their generally very small separations. Those that were measured by Hipparcos have large uncertainties in ΔH_p, typically 0.2 mag or more, much worse than typical for Hipparcos data. Nonetheless, with the sample for which the comparison can be made (12 objects from the paired sample, and 4 from the unpaired), we find observed minus Hipparcos residuals that differ from zero by less than 1σ in both cases, and standard deviations in the 0.4–0.5 magnitude range. However, the mean error of the ΔH_p values in both cases is also in the same range. Therefore, we conclude that the measurement error in Δm for sub-diffraction-limited measures is certainly much lower than 0.4 mag, and that there is no evidence at this time that it is significantly larger than what we have previously reported for WIYN speckle data above the diffraction limit, roughly 0.1 mag per observation.

In Table 5, we show new orbital elements for two systems for which the observations presented here, together with other relatively recent observations in the 4th Interferometric Catalog, permit modest orbit revisions. To calculate the orbital elements, we have used our own orbit fitting routine, described in MacKnight & Horch (2004). We do not anticipate that these orbits are dramatically better in quality than those published earlier; nonetheless, since they are small-separation systems, the data used span a more complete range in position angle and provide an up-to-date dynamical picture prior to discussing the evolutionary status of the components of these systems.

The first of these binaries is FIN 350(= HIP 64838 = HR 5014), where the previous orbit (which is Grade 2) was computed by Hartkopf et al. (1996). Since that time, several observations have appeared in the literature, including our measures presented here. Our orbit increases both the semi-major axis and the period slightly while decreasing the uncertainties of both substantially. The total mass, when computed with the revised Hipparcos parallax (van Leeuwen 2007), therefore changes from 3.3 ± 3.0 M_☉ to 3.4 ± 0.3 M_☉. Given that this is an F0V system with at most a small magnitude difference, a total mass of approximately 3.0–3.2 M_☉ is expected from the photometry, in excellent agreement with the current orbit. To make the conversion from spectral type to stellar mass, we have used a standard table from the literature (Schmidt-Kaler 1982).

For MCA 40(= HIP 71729 = HD 129132 = HR 5472), the orbit currently listed in the Sixth Catalog is also Grade 2, that of Baize (1989), which we improve upon here at least by estimating uncertainties for the elements. From these we can deduce a total mass of 6.7 ± 1.4 M_☉. However, the spectral type in SIMBAD⁷ is listed as G0V difficult to reconcile with this result. The absolute magnitude derived from an apparent magnitude of 6.23 and revised Hipparcos parallax of 8.60 ± 0.61 mas is +0.83, much too bright for a G-type dwarf pair. (An extinction estimate, though less than 0.1 mag, was included using the NASA/IPAC reddening and extinction map available on the IPAC Web site.⁸) We suggest therefore that at least the primary is evolved and, given the fact that the magnitude differences observed to date are not terribly large (though with considerable scatter), it may be that both components have left the main sequence. If so, this system could provide quite a sensitive test of stellar evolution theory with more high-quality differential photometry. Graphical representations of our orbits for both FIN 350 and MCA 40 are shown in Figure 8.

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With the astrometric data in hand from Tables 1 and 2 and in the literature, it is possible to calculate first orbits for three objects, with the caveat that more data will clearly be needed to make the elements definitive. These are shown in Figure 9. However, these orbits, together with photometric and spectroscopic information, permit a useful discussion of the status of these systems at present. The orbital elements we derive are shown in Table 6, and the astrometric data and residuals are shown in Table 7. Here again we have used the fitting routine of MacKnight & Horch (2004).

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The first of these systems is HDS 1962(= HIP 68380 = HD 122106). Although the latest version of the Geneva–Copenhagen Catalogue (Holmberg et al. 2009) gives the iron abundance of this system as slightly metal-rich, [Fe/H] = +0.13, it does not give a mass ratio. There is a non-detection at 1988.163 by McAlister et al. (1993) for which our orbital elements predict a separation of 52.5 mas. This may be an indication that the data to date produce a period that is slightly too large. If we compute the ephemeris position with P = 10.2 years (1σ lower that the value presented), then the separation is 26 mas, well below the stated limit of the observation of 38 mas. Nonetheless, combining the period and semi-major axis obtained here with the revised Hipparcos parallax of van Leeuwen (2007), the mass sum is 1.6 ± 0.5 M_☉. On the other hand, this system has spectral type of F8V in the SIMBAD database, but the absolute magnitude that we calculate from the apparent magnitude, parallax, and extinction (again from the NASA/IPAC online map) is +1.8, too bright by over 1.5 mag to be explained by a pair on the main sequence with that spectral type. The speckle and Hipparcos magnitude differences available in the 4th Interferometric Catalog suggest a value near 1 in V, so perhaps an F7IV–F8V pair comes closer to matching the photometry here. If so, this suggests a mass sum of perhaps 2.5 M_☉, somewhat higher than that obtained from the orbit, but within 2σ. If the true value of the period is lower than that of our orbit as the McAlister non-detection suggests, this would of course increase the mass sum, making it more consistent with the 2.5 M_☉ value.

The second orbit we present is that of HDS 2554(= HIP 88852 = HD 166409). The Hipparcos data point is in the third quadrant, and subsequent observations have been in the first and second quadrants, thus the position angles available now cover nearly a full orbit since the discovery observation in 1991. This object is slightly below the solar abundance ([Fe/H] = −0.10 according to Holmberg et al. 2009), and once again no mass fraction appears in the Geneva–Copenhagen Catalogue. The system has spectral type F5 in SIMBAD, and the differential photometry that exists at present supports a modest magnitude difference, approximately 0.5 mag. The implied absolute magnitude using the revised Hipparcos parallax is +1.6, which is approximately a magnitude too bright for a main-sequence pair and would seem to suggest that the primary may be slightly evolved. If it is composed of an F(4–5)IV primary and an F(7–8)V secondary, then this implies a total mass in the range of perhaps 2.6–3.0 M_☉, whereas the orbital elements in combination with the same parallax value give 3.1 ± 0.5 M_☉.

Finally, we have the case of HDS 2932(= HIP 101181 = HD 195397), a system with spectral type F8. Of the three systems discussed here, this is the most metal-poor, with [Fe/H] = −0.17, and the mass fraction in the Geneva– Copenhagen Catalogue is m₂/m₁ = 0.578 ± 0.037. The magnitude difference appears to be approximately 1, given four measures in the 4th Interferometric Catalog; however, the Hipparcos measure has a large uncertainty and there is significant variation in the three remaining measures. The absolute magnitude derived from the apparent magnitude and revised Hipparcos result is relatively consistent with a main-sequence or near-main-sequence system, so allowing for a sizeable range in secondary spectral type due to the uncertainty in the magnitude difference, perhaps we have an F(6–8)V primary with a G(1–6)V. This implies masses of ∼1.26 ± 0.10 M_☉ and 0.96 ± 0.06M_☉, so that is a mass ratio of 0.76 ± 0.07. While the mass ratio is larger than that in the Geneva–Copenhagen Catalogue, the total mass agrees quite well with that obtained from our orbital parameters in Table 6 and the parallax, namely 2.0 ± 0.5 M_☉. One aspect of the analysis here is difficult to explain: the non-detection by mason et al. in 1998, even though the same group did successfully resolve the system about a year before. We explored orbits which place the secondary below the diffraction limit at their observation date, but this reduces the period significantly, and in view of the photometry and the distance information available, unrealistically. Several of our own measures of this system taken over the past few years were judged to be too poor in quality to report, so more work will be needed to fully understand the nature of this difficulty.