THE VISUAL ORBIT OF THE 1.1 DAY SPECTROSCOPIC BINARY σ² CORONAE BOREALIS FROM INTERFEROMETRY AT THE CHARA ARRAY

In addition to our own, four other radial-velocity data sets have been published in the literature (Harper 1925; Bakos 1984; Duquennoy & Mayor 1991; SR03). Except for the more recent one, the older data are generally of lower quality and contribute little to the mass determinations, but they do extend the time coverage considerably (to nearly 86 years, or 27, 500 orbital cycles), and can be used to improve the orbital period. Because of our concerns over possible systematic differences among different data sets, particularly in the velocity semiamplitudes but also in the velocity zero points, we did not simply merge all these observations together indiscriminately, but instead we proceeded as follows. We considered all observations simultaneously in a single least-squares orbital fit, imposing a common period and epoch of maximum primary velocity in a circular orbit, but we allowed each data set to have its own velocity semiamplitudes (K_p, K_s) as well as its own systematic velocity zero-point offset relative to the reference frame defined by the CfA observations. Additionally, we included one more adjustable parameter per set to account for possible systematic differences between the primary and secondary velocities in each group. These were statistically significant only in the observations by SR03. Relative weights for each data set were determined by iterations from the rms residual of the fit, separately for the primary and secondary velocities. The resulting orbital period is P = 1.139791423 ± 0.000000080 days, and the time of maximum primary velocity nearest to the average date of the CfA observations is T = 2, 450, 127.61845 ± 0.00020 (HJD). We adopt this ephemeris for the remainder of the paper.

Our measured radial velocities enable us to derive the three remaining spectroscopic orbital elements, namely, the center-of-mass velocity (γ) and the radial velocity semi-amplitudes of the primary and secondary (K_p and K_s, respectively). To check for consistency with prior efforts, we used the velocities published in SR03 to derive a second orbital solution. The calculated radial velocities for the derived orbits are shown in Figures 1 and 2 (the solid and dashed curves for the primary and secondary, respectively) along with the measured radial velocities and residuals for the primary (filled circles) and secondary (open circles). The corresponding orbital solutions are presented in Table 3 along with the related derived quantities. For comparison purposes, we have also included the values presented in SR03, which are consistent with our orbit generated using their velocities. However, the orbit obtained using our velocities is statistically different from that obtained using SR03 velocities. While the primary's velocity semi-amplitude matches within the errors between these two solutions, the secondary's differs by over 5σ, resulting in a 4σ difference in the mass ratios.

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One possible explanation of the difference in the orbital solutions could be the velocity residuals for the orbit using SR03 data (Figure 2), which show an obvious pattern for both components. Those observations were obtained on four nights over a five-day period. To further examine these patterns, we display the residuals for each of the four nights in Figure 3, as a function of time. Clear trends are seen on each night, which are different for the primary and secondary components and have peak-to-peak excursions reaching 4 km s⁻¹ in some cases, significantly larger than the velocity errors of 0.1–1.2 km s⁻¹ (SR03). On some, but not all, nights there appears to be a periodicity of roughly 0.20–0.25 days. The nature of these trends is unclear, particularly because this periodicity is much shorter than either the orbital or the rotational periods. Instrumental effects seem unlikely, but an explanation in terms of the considerable spottedness of both stars is certainly a distinct possibility. The Doppler imaging maps produced by SR03 show that both components display a very patchy distribution of surface features covering the polar regions. Individual features coming in and out of view as the stars rotate could easily be the cause of the systematic effects observed in the radial velocities, and the effects would not necessarily have to be the same on both stars, just as observed. Slight changes in the spots from one night to the next could account for the different patterns seen in Figure 3. The relatively large amplitude of the residual variations raises the concern that they may be affecting the velocity semi-amplitudes of the orbit, depending on the phase at which they occur. We do not see such trends in the CfA data, perhaps because our observations span a much longer time (more than 7 years, and ∼2200 rotational cycles), allowing for spots to change and average out these effects. We therefore proceed on the assumption that possible systematic effects of this nature on K_p and K_s are lessened in the CfA data.

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The basic measured quantity from an interferometric observation is visibility, which evaluates the contrast in the fringe pattern obtained by combining starlight wave fronts from multiple apertures, filtered through a finite bandwidth. For a single star of angular diameter θ, the interferometric visibility V for a uniform disk model is given by

$$\begin{equation} V = {2 J_1 (\pi B \theta / \lambda) \over \pi B \theta / \lambda }, \end{equation} \tag{ 1 }$$

where J₁ is the first-order Bessel function, B is the projected baseline length as seen by the star, and λ is the observed bandpass central wavelength. The interferometric visibility for a binary, where the individual stars have visibilities V_p (primary) and V_s (secondary) per Equation (1), is given by

$$\begin{equation} V = {\sqrt{\big(\beta ^2 V_{\rm p}^2 + V_{\rm s}^2 + 2\beta V_{\rm p}V_{\rm s} \cos ((2\pi / \lambda)\bf {B\cdot s}) \big)} \over 1+\beta }, \end{equation} \tag{ 2 }$$

where β is the primary to secondary flux ratio, B is the projected baseline vector as seen by the binary, and s is the binary's angular-separation vector in the plane of the sky.

Using our measured interferometric visibilities and the above equations, we are able to augment the spectroscopic orbital solutions to derive a visual orbit for σ² CrB. Adopting the period and epoch of nodal passage from Section 2.1, we now derive the parameters that can only be determined astrometrically: angular semimajor axis (α), inclination (i), and longitude of the ascending node (Ω). We also treat the K'-band magnitude difference as a free parameter in order to test evolutionary models.

For a circular orbit, the epoch of periastron passage (T₀) is replaced by the epoch of ascending nodal passage (T_node), defined as the epoch of fastest secondary recession, in the visual orbit equations (Heintz 1978). Accordingly, we translate the T value listed in Section 2.1 by one-half of the orbital period to determine the epoch of the ascending nodal passage as T_node = 2, 450, 127.04855 ± 0.00020 (HJD) for use in our visual orbit solution. The 1σ errors of this and other adopted parameters listed in Table 4 have been propagated to our error estimates for the derived parameters.

The angular diameters of the components are too small to be resolved by our K'-band observations. We therefore estimate these based on the components' absolute magnitudes and temperatures as described below. We first estimate the Johnson V-band magnitude of σ² CrB using its Tycho-2 magnitudes of B_T = 6.262 ± 0.014 and V_T = 5.620 ± 0.009 and the relation V_J = V_T − 0.090(B_T − V_T) from the Guide to the Tycho-2 Catalog. Then, using the V-band flux ratio from Section 2 and the Lestrade et al. (1999) parallax, we obtain absolute magnitudes of M_V = 4.35 ± 0.02 for the primary and M_V = 4.74 ± 0.02 for the secondary. These magnitudes lead to linear radius estimates of 1.2 R_☉ for the primary and 1.1 R_☉ for the secondary using the tabulation of stellar physical parameters in Popper (1980) and Andersen (1991). Finally, using the Lestrade et al. (1999) parallax, we adopt component angular diameters of θ_p = 0.50 mas and θ_s = 0.45 mas, propagating a 0.05 mas uncertainty in these values for deriving the uncertainty of our orbital elements. Diameter estimates using the temperatures of the components from Section 2 are consistent with these values.

We conduct an exhaustive search of the parameter space for the unknown parameters mentioned, namely, α, i, Ω, and ΔK'. The orbital inclination is constrained by the asin i from spectroscopy, the free-parameter α, and the Lestrade et al. (1999) parallax. We impose this constraint during our exploration of the parameter space along with its associated 1σ error. We explore the unknown parameters over many iterations, by randomly selecting them between broad limits and using Equation (2) to evaluate the predicted binary visibility for the baseline and binary positions at each observational epoch. The orbital solution presented here represents the parameter set with the minimum χ² value when comparing the predicted and measured visibilities.

Figure 4 shows the measured visibilities (plus signs) with vertical error bars for each of the 26 observations, along with the computed model visibilities (diamonds), and Table 2 lists the corresponding numerical values of the observed and model visibilities along with the residuals of the fit. Table 4 summarizes the visual orbit parameters for σ² CrB from our solution and Figure 5 plots the visual orbit in the plane of the sky. As seen in Figure 5, we have a reasonably good phase coverage from our observations.

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As mentioned in Section 4.1, star spots can create systematic effects in the data obtained on this binary. These effects are especially significant for data obtained over a short time baseline, as seen for the SR03 spectroscopic solution. While our interferometric data span 73 days, allowing for some averaging of these effects, the bulk of the data used were obtained over 12 days, justifying an exploration of this effect. Specifically, the separation between the stars derived from our visibility data would represent the separation of the centers of light rather than that of mass. As discussed in Hummel et al. (1994), heavily spotted stars will incur a systematic shift in the center of light from rotational and orbital motions, perhaps inducing an additional uncertainty in the orbital elements derived. We assume a spot-induced change in the angular semimajor axis of 2% of the primary's diameter, or 0.01 mas. This is less than the uncertainty of our derived semimajor axis, and at our baselines of 270–330 m translates to a 0.005–0.011 change in the visibility. While the uncertainties of our measured visibilities are an order of magnitude larger than this, we ran a test orbital fit by adding a 0.010 uncertainty to the visibility errors as a root-sum-squared. While, as expected, the χ² of the fit improved, the values and uncertainties of the derived parameters remained unchanged, leading us to conclude that this effect, while real, is too small to affect our results.

We determine the 1, 2, and 3σ uncertainties of each visual orbit parameter using a Monte Carlo simulation approach. We compute the orbital fit for 100, 000 iterations, where for each iteration, we randomly select the adopted parameters within their respective 1σ intervals and the model parameters around their corresponding best-fit solution, generating a multi-dimensional χ² "surface." Then, we project this surface along each parameter axis, resulting in the plots shown in Figures 6–9. The figures show the χ² distribution around the best-fit orbit and enable estimation of 1, 2, and 3σ errors for each parameter based on a χ² deviation of 1, 4, and 9 units, respectively, from its minimum value. The horizontal dashed lines in the figures from bottom to top mark the minimum χ² value and those corresponding to 1, 2, and 3σ errors, and Table 4 lists the corresponding numerical 1σ errors of the model parameters.

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Our angular semimajor axis obtained from interferometry translates to 0.0279 ± 0.0003 AU or 5.99 ± 0.07 R_☉ using the Lestrade et al. (1999) parallax. Newton's generalization of Kepler's third law then yields a mass sum of 2.227 ± 0.073 M_☉ for the pair, and using the mass ratio from our spectroscopic solution of 0.9586 ± 0.0047, we get individual component masses of 1.137 ± 0.037 M_☉ and 1.090 ± 0.036 M_☉ for the primary and secondary, respectively. As noted in Section 4.1, the SR03 velocities yield a significantly different mass ratio of 0.9746 ± 0.0016, but this 4σ difference is not enough to influence the mass estimates significantly. The uncertainty in our masses is dominated by the cubed semimajor axis factor in estimating the mass sum, resulting in about a 3% uncertainty in mass sum corresponding to a 1% uncertainty in the semimajor axis. The high precision of the mass ratio from the spectroscopic solution results in final masses of 3% uncertainty as well. Component mass estimates using the SR03 velocities are 1.128 ± 0.037 and 1.099 ± 0.036, in excellent agreement with the masses using our velocities. These masses along with other physical parameters derived are listed in Table 5.

Assuming synchronous and co-aligned rotation of spherical components, reasonable given the short orbital period and evidence from SR03 of unevolved stars contained within their Roche limits, we can estimate the component radii from the measured spectroscopic vsin i. As mentioned in Section 2, our spectra yield vsin i = 26 ± 1 km s⁻¹ for both the primary and secondary. These values and uncertainties are identical to those in SR03. Using the inclination from our visual orbit, and adopting the orbital period from spectroscopy as the rotational period, we get identical component radii of 1.244 ± 0.050 R_☉ for the primary and secondary. This translates to an angular diameter of 0.509 ± 0.020 mas for each component using the Lestrade et al. (1999) parallax, in excellent agreement with our adopted diameter for the primary and a 1σ variance for the secondary, given our associated 0.05 mas errors for these values. These radii estimates, along with the effective temperatures from Section 2 and the relation L ∝  R² T⁴_eff, lead to a luminosity ratio of 0.89 ± 0.16. Alternatively, using bolometric corrections from Flower (1996) of BC_p = −0.038 ± 0.017 and BC_s = −0.064 ± 0.020 corresponding to the components' effective temperatures, the V-band flux ratio of 0.70 ± 0.02 from spectroscopy translates to a total luminosity ratio of 0.68 ± 0.20, a 1σ variance from the estimate above. Conversely, our estimates of effective temperature and luminosity ratio require a radius ratio of 0.88 ± 0.14, again at a 1σ variance from the 1.00 ± 0.06 estimate from the identical vsin i values of the components.

We allowed the K'-band magnitude difference to be a free parameter for our visual orbit fit, obtaining ΔK' = 0.19 ± 0.19, consistent with the 0.18 estimate from the mass–luminosity relations of Henry & McCarthy (1993).¹¹ The uncertainty in ΔK' is large because visibility measurements of nearly equal mass, and hence nearly equal brightness, pairs are relatively insensitive to the magnitude difference of the components (Hummel et al. 1998; Boden et al. 1999). Using Equation (2), we have verified that a 10% change in ΔK' for σ² CrB results in only 0.1% change in visibility. This, along with the poor-quality K magnitude listed in 2MASS (for σ² CrB, K = 4.052 ± 0.036, but flagged as a very poor fit), thwart any attempts to use these magnitudes for checking stellar evolution models. However, we can revert to V-band photometry to explore this topic.

In Section 4.2, we derived the absolute V-band magnitudes of the components of σ² CrB as M_V = 4.35 ± 0.02 for the primary and M_V = 4.74 ± 0.02 for the secondary. For σ¹ CrB, we similarly use the Tycho-2 magnitudes and the Lestrade et al. (1999) parallax to obtain M_V = 4.64 ± 0.01. SR03 had a smaller magnitude difference for the components of σ² CrB, and the corresponding results using their spectroscopy are also included in Table 5 along with the values from their paper. Figure 10 plots these three stars on a Hertzsprung–Russell (H–R) diagram using our magnitude and temperature estimates, along with isochrones for 0.5, 1.5, 3.0, and 5.0 Gyr ages (left to right) from the Yonsei–Yale isochrones (dotted, Yi et al. 2001) and the Victoria–Regina stellar evolution models (dashed, VandenBerg et al. 2006) for solar metallicity (see Footnote 8).

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Wright et al. (2004) estimate an age of 1.8 Gyr for σ¹ CrB based on chromospheric activity, and Valenti & Fischer (2005) estimate an age of 5.0 Gyr from spectroscopy with limits of 2.9–7.8 Gyr based on 1σ changes to log L. SR03 identify a much lower age, of a few times 10⁷ years, by matching pre-main-sequence evolutionary tracks and point to their higher Li abundance as supporting evidence. While abundance determinations in double-lined spectroscopic binaries are particularly difficult and more prone to errors, the high-Li abundance of 2.60 ± 0.03 (SR03) for the slow-rotating single-lined companion σ¹ CrB does argue for a young system. Each point along the isochrones plotted in Figure 10 corresponds to a particular mass, allowing us to use our mass estimates for the components of σ² CrB to further constrain the system's age. Our mass, luminosity, and temperature estimates indicate an age for this system of 0.5–1.5 Gyr, with a range of 0.1–3 Gyr permissible within 1σ errors.

Our mass estimates for the components of σ² CrB allow us to constrain the mass of the wider visual companion σ¹ CrB as well. Scardia (1979) presented an improved visual orbit for the AB pair based on 886 observations spanning almost 200 years of observation, yielding P = 889 years, a = 59, i = 318, e = 0.76, and Ω = 169. However, he did not publish uncertainties for these parameters, and given the long period, his less than one-third phase coverage leads to only a preliminary orbital solution, albeit one that convincingly shows orbital motion of the pair. He further uses parallaxes available to him to derive a mass sum for the AB system of 3.2 M_☉. We used all current WDS observations, adding almost 200 observations since Scardia (1979), to update this orbit and obtain uncertainties for the parameters. Our visual orbit is presented in Figure 11, along with the Scardia orbit for comparison, and Table 6 lists the derived orbital elements. Adopting the Lestrade et al. (1999) parallax of the A component, we estimate a mass sum of 3.2 ± 0.9 M_☉, resulting in a B-component mass estimate of 1.0 M_☉, consistent with its spectral type of G1 V (Gray et al. 2003). Valenti & Fischer (2005) estimate a mass of 0.77 ± 0.21 M_☉ based on high-resolution spectroscopy, but we believe that they systematically underestimate their uncertainty by overlooking the log e factor in converting from uncertainty in log L to uncertainty in L. Using the log e factor, we followed their methods for obtaining a mass estimate of 0.77 ± 0.44 M_☉. The mass error is dominated by the uncertainty of the Gliese & Jahreiß (1991) parallax used by Valenti & Fischer (2005). Adopting the higher precision Lestrade et al. (1999) parallax of the primary, we follow their method, and using the log e factor, get a mass estimate of 0.78 ± 0.11 M_☉. This mass is too low for the spectral type (as well as our own estimate of the effective temperature; see Section 2) and the expectation from the visual orbit. A possible contamination of the secondary's spectral type from the 7'' distant primary is unlikely, as determined by Richard Gray at our request from new spectroscopic observations (R. Gray 2008, private communication).

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The inclination and longitude of the ascending node for this visual orbit are similar to those of the inner (σ² CrB) orbit, suggesting coplanarity. For the outer visual orbit, we can use our radial velocity estimate for σ¹ CrB, our derived systemic velocity for σ² CrB, and the speckle observations to unambiguously determine the longitude of the ascending node as Ω = 280 ±  05. Using the equation for the relative inclination of the two orbits (ϕ) from Fekel (1981), we get ϕ = 47 or 603, given the 180°ambiguity in Ω for the inner orbit, confirming coplanarity as a possibility.