A PHOTOMETRIC ANALYSIS OF SEVENTEEN BINARY STARS USING SPECKLE IMAGING

Since the RIT filters differ from Bessel filters, the Fernie (1983) transformation equations used in Horch et al. (2001) can be improved. In addition, B − V colors and apparent V magnitudes for the binaries are in the standard Johnson filter set (Johnson 1965), which also have slight differences with the Bessel standard filters. Similarly, for WIYN data, a system of transformation equations was needed to convert between the narrow instrumental filters and the Johnson system. The approach here has been a four-step process, which is described in further detail below, and is summarized as follows. (1) The Pickles spectral library (Pickles 1998) was used to develop the transformations to convert the system V magnitudes and B − V colors to instrumental values for the desired speckle filters. (2) The V and B − V values for each system were then converted to instrumental values using these results. (3) These were then used in combination with the speckle magnitude differences to obtain component magnitudes and colors in the speckle filters. (4) Finally, these values were converted back to the Johnson system. The final results in the Johnson system could then be used for isochrone fitting.

The Pickles spectral library (Pickles 1998), containing 131 sample spectra, was used to simulate magnitude measurements, along with the filter and atmospheric transmission curves and the quantum efficiency curve for the CCD. For a filter x, the stellar flux in that filter, f_x, is therefore expressed as

$$\begin{equation} f_{x}\ =\ \int ^{\infty }_{0}SAF_{x}Q\ d\lambda, \end{equation} \tag{ 1 }$$

where S is the star's flux, A is the atmospheric transmission curve, F_x is the filter transmission curve, and Q is the quantum efficiency of the CCD. In addition, there are losses in the optical system, but these are assumed small enough to ignore in this study. From such a flux, a magnitude is then determined.

The filter transmission data for the Johnson UBV filters are available at the GCPD Web site. Magnitude values for stars in the Pickles library were calculated in the Johnson filter set and then compared with the values in Schmidt–Kaler (1982). Since these standard literature values assume a quantum efficiency curve that is independent of wavelength, the quantum efficiency curve was omitted from the calculations of colors in the Johnson filter set. This led to a near one-to-one relationship, as seen in Figure 2. For magnitude calculations in both the RIT and WIYN filter sets it was of course necessary to use the quantum efficiency curve, as our observations were taken using the Kodak CCD.

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Upon calculating magnitudes for all the sample spectra in each of the essential filters, color–color plots were created. To achieve more accurate relations between colors, and to reduce error, magnitude values were also separated based on luminosity class prior to fitting. A least-squares fit was then performed leading to a polynomial function, in most cases of order three or less, for each luminosity class. This means that, based on the position of the system on the H–R diagram, a preliminary assignment of the luminosity class of the components of each binary was made; in the few cases where the final determination was in conflict with the initial choice, the calculations were redone for that system with the final luminosity class determinations of the first calculation. This then always yielded consistent results in these cases. Also, the Pickles library consists of spectra of single stars, not binaries. By constructing binary spectra of known input magnitude and color difference between the components, we were able to verify that the color–color relations derived based on single stars do not produce errors of more than a few hundredths of a magnitude when used on synthetic binary systems. Since this is in most cases lower than the uncertainties in our input magnitude differences, this source of error can be ignored at this point. However, in the future, when magnitude differences measures are more numerous and the average result in each filter has greater precision, it may be necessary to devise an iterative scheme for ensuring that magnitude and color calibrations do not dominate the errors in the final results.

System colors in the instrumental filter set were used, along with the speckle magnitude differences, to determine magnitudes for the individual stars in all instrumental filters. This is done using the magnitude difference formula

$$\begin{equation} m_{b}-m_{a}\ =\ 2.5\log {\frac{f_{a}}{f_{b}}}\ =\ \Delta m, \end{equation} \tag{ 2 }$$

where m_a and m_b are the magnitudes of the primary and secondary, respectively, and f_a and f_b are the fluxes for the primary and secondary, respectively. We can then write the same expression for the binary system as a whole and for a zero-magnitude star. Individual magnitudes for the primary and secondary can then be found, through simple calculation. Once all the individual magnitudes have been calculated, colors for the primary and secondary can be trivially derived. Finally, with instrumental parameters in hand for each component of the binary, the V magnitude and B − V color are obtained with the appropriate inverse transformations, obtained again from the Pickles spectral library.

Isochrone models were created using Y² isochrones and the interpolator program from Yale University (Yi et al. 2001, 2003; Kim et al. 2002; Demarque et al. 2004). Solar metallicity isochrones were generally used since, when known, the iron abundance is usually no more than ∼0.1 dex away from solar, and the [Fe/H] determinations are probably uncertain on nearly that same level in most cases. In the cases of three stars, we did elect to interpolate the Y² isochrones to the metallicity matching the measured iron abundance. These are BU 1163 and COU 2674, which are significantly metal poor, and STF 2912 which is significantly metal rich. Once the metallicity to be used was decided, we created 1800 isochrones for ages of 0.1 to 1.0 Gyr in steps of 0.001 Gyr, and for 1.0 to 9.99 Gyr in steps of 0.01 Gyr. The smaller increments in the younger isochrones were used to provide more precise results for younger systems.

To determine the best fitting, along with upper and lower limit isochrones, we compared our 1800 isochrone models against the primary and secondary positions for a given system. For the purposes of this study, we will refer to the 1σ errors associated with the primary and secondary positions as producing an error "rectangle" in the H–R diagram. The isochrones were first linearly upsampled, by a factor of 100, to fill in any spaces between points, as the models can have large gaps between points in areas where the isochrones are relatively straight, i.e., on the main sequence. Next, the isochrones were checked for points falling within the error rectangles. For isochrones which had points lying within the error rectangles, the shortest overall distance of the primary and secondary to the isochrones were determined. The best-fitting isochrone is simply the one which had the shortest overall distance. The age range for a system was simply determined by finding the youngest and oldest isochrones which passed through both error rectangles.

The best-fitting isochrone was then used to estimate the mass range for both the primary and secondary, by determining the portion of the isochrone which falls within the corresponding 1σ error. From this we estimate the average mass for both primary and secondary, which are only based on the best-fitting isochrone, and combined them to obtain a total mass for the system. We can then compare this against the total dynamical mass from orbital information. The mass fraction,

$$\begin{equation} B\ =\ \frac{M_{B}}{M_{A}+M_{B}}, \end{equation} \tag{ 3 }$$

is also calculated, again using only the best-fitting isochrone, from the average mass of the primary, M_A, and secondary, M_B.

Table 3 contains the derived absolute V magnitudes and B − V color magnitudes, in the Johnson filter set, for the components of each system observed at CTIO. Specifically, the columns here give (1) the Discoverer Designation and (2) the Hipparcos Catalog number; (3) the best-fit isochrone age; (4) the age range determined as discussed above; (5) the component identifier for subsequent columns in the table, either A or B; (6) the derived B − V color and (7) absolute V magnitude determination; (8) the luminosity class determination; and (9) the mass derived for the component.

Results were derived for all the systems using speckle ΔV and ΔR magnitudes. For the systems STF 186 (HIP 8998), JC 8AB (HIP 14913), and AC 4 (HIP 32677), a second result was determined using speckle ΔB and ΔV magnitudes. In these cases, observations were not combined to produce a single result in order to examine any differences which arise from observations in the B and R filters, which will be discussed below. While the B − V color values of the components of these systems are generally different by the order of a tenth of a magnitude, the uncertainties are also typically of the same order. This can be traced back to the uncertainty in individual speckle magnitude differences at the small aperture, as well as the number of observations, ⩽8 in every case. Nonetheless, it can be seen in the final columns of Table 3 that the mass values still have very good precision. This is due to the fact that the color of a component gives leverage mainly on the age of the system, whereas the absolute magnitudes derived affect the mass more than the age in the isochrone fitting. The uncertainties in absolute magnitude depend also on distance, but in general translate into modest uncertainties in the masses of the components.

Systems and components were then separately plotted on the H–R diagrams seen in Figure 3. The solid line is the isochrone which best fits the primary and secondary locations, while the dashed lines represents the isochrones that are the upper and lower age limits. Lines marking the zero age main sequence, subgiant sequence, and giant sequence are also drawn.

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For those systems having two sets of results, the H–R diagrams appear in Figure 4. Differences between using ΔB and ΔR become obvious when comparing these plots. From our results, it is generally the case that the secondary has a redder position in the H–R diagram when the analysis uses the B and V filters as opposed to the V and R filters. Assuming that the components of a system formed at the same time, the results of the R observations would appear to be more realistic. This is particularly noticeable in STF 186 (HIP 8998) and JC 8AB (HIP 14913), which have system locations near the main sequence, and where the components would be expected to have positions more or less on the main sequence, especially for the secondary. One possible explanation for this is that the isoplanatic patch is expected to be smaller as the wavelength of observation decreases, thus affecting the B filter the most. As discussed in Horch et al. (2001), any decorrelation between primary and secondary speckle patterns would lead to an overestimate of the derived magnitude difference. If this effect is most noticeable in the B filter, a redder color could be derived for the secondary star. Since there are only three systems which had sufficient observations in both B and R filters, we cannot draw any definitive conclusions about this; however, the results we have indicate observations taken in the R filter produce position locations which are more consistent with evolutionary calculations.

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For the objects observed at CTIO, typically data in only two filters were available. This meant that usually only one color–color transformation was needed in order to complete the analysis described above. In contrast, the WIYN objects have been observed in at least three, and up to seven, filters. This presents additional work in completing the analysis, though essentially the same method described above was simply used multiple times. For every filter pair that could reasonably be used to generate a color, color–color transformations were created from the Pickles spectral library, and estimates of the component M_V and B − V values were obtained. Once all possible estimates were obtained, average values were formed.

In Table 4, we show the results for the component V magnitudes and B − V colors as well as the results from the isochrone fitting for objects observed at the WIYN Telescope. The column headings are exactly the same as in Table 3. In this case, it is possible to see the advantage of the larger aperture, particularly on the derived B − V values for the components. Values for the component masses are likewise relatively precise, with estimated uncertainties under 0.1 solar masses for the primaries in all cases. The H–R diagrams for these systems are shown in Figure 5.

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Several systems in the data set deserve special mention.

• 1.  
  BU 395 (HIP 2941). Due to the calculation of Pourbaix (2000), we have a very good orbit, listed as Grade 1 in the Sixth Orbit Catalog¹⁴ (Hartkopf et al. 2001) and a mass uncertainty of only ∼0.1 M_☉. Of the systems observed at CTIO, this is the smallest dynamical mass uncertainty. Our total mass value matches that derived from the Pourbaix orbit almost exactly, although there is a discrepancy in the mass fraction. This will have to be monitored and reevaluated in the coming years as more data become available.
• 2.  
  STT 98 (HIP 23879). The dynamical mass sum of this object is 5.45 M_☉ (with an uncertainty of well over 1 M_☉, considering that the orbital parameters were published without uncertainties), and the B − V color for the system is 0.334. Given that the components appear to be near the main sequence (with the primary slightly evolved) and that the system has a modest magnitude difference of approximately 1 mag, one would expect spectral types of perhaps F0 and F5. Using data in Schmidt–Kaler, this would yield masses of approximately 1.6 and 1.4 solar masses, that is, a mass sum of approximately 3 solar masses. The orbit is given a grade of 2 in the Sixth Orbit Catalog, but the period is 197 years, so that modern astrometric techniques do not cover the majority of the orbit. We suggest that the mass sum deduced from this orbit may be an overestimate. In our previous work (Horch et al. 2001), we derived preliminary colors and temperatures of the components; the work here is consistent with those earlier results.
• 3.  
  BU 151AB (HIP 101769). As mentioned above, this system consists of two evolved stars, allowing for a relatively precise age determination from both CTIO and WIYN data. The orbit of Alzner (1998) is of excellent quality, easily grade 1 in the Sixth Orbit Catalog, and was produced from high-quality speckle data spanning over a full orbit. This represents one of the most interesting tests of our method, since the uncertainty in the dynamical mass sum is only about three tenths of a solar mass, and the components are evolved. The agreement between the dynamical value and those of both the CTIO and WIYN results is nearly perfect.
• 4.  
  STT 535AB (HIP 104858). This main-sequence system has a very precise orbit determination, published with uncertainties. The uncertainty in the mass sum is only a tenth of a solar mass, making this object the best pure test of our photometrically derived mass values for WIYN data. Our value is well within 1σ of the dynamical value. Our mass fraction also agrees, though we still have an uncertainty of over 10% in this number at present.
• 5.  
  BU178 (HIP 113184). This system has the largest discrepancy between the dynamical mass sum and that derived from the photometric analysis presented here. We find the primary to be a giant, which is consistent with the red system B − V color of ∼0.9, with the secondary still on or near the main sequence. The orbit has a period of 97 years, and the dynamical mass sum is 8.67 ± 2.22 M_☉, with the uncertainty being an underestimate due to the lack of error estimates for the orbital elements. The orbit leading to this mass sum is given a grade of 4 in the Sixth Orbit Catalog, but interestingly does include one early interferometric observation of Maggini in 1923 quoted in the 4th Interferometric Catalog. We note that an earlier orbit of Baize (1981), when combined with the Hipparcos parallax, gives a mass sum of nearly 10 M_☉, so that at least the accumulation of speckle data since 1981 has decreased the mass somewhat. Nonetheless, we suggest that the dynamical mass value should not be given significant weight at this time.
• 6.  
  SEE 492 (HIP 116436). In this case there is a nearly 2σ discrepancy between the mass sum obtained here and that deduced from the dynamical information. Again, the orbit is listed as grade 4 in the Sixth Orbit Catalog, with a period of 78.5 years. We also note that the most recent observations appear to deviate from the orbit prediction, suggesting that an orbit revision may be needed in the coming years.