FUNDAMENTAL PARAMETERS OF THE EXOPLANET HOST K GIANT STAR ι DRACONIS FROM THE CHARA ARRAY

Frink et al. (2002) announced the discovery of a substellar companion to ι Draconis (K2 III, HD 137759) with a period of 536 days and a minimum mass for the companion of 8.9 M_Jupiter. They used a stellar mass of 1.05 M_☉ from Allende Prieto & Lambert (1999), who compared the absolute visual magnitude and (B − V) color from Hipparcos data with theoretical isochrones from Bertelli et al. (1994). However, Frink et al. acknowledge that evolutionary tracks for a range of masses are close together on the H-R diagram, so any slight change in the evolutionary model can have a large impact on the derived mass.

Zechmeister et al. (2008, hereafter Z08) observed ι Dra in search of stellar oscillations using three separate instruments over almost eight years in order to refine the orbital parameters of the planet and determine the mass of the central star. They found low-amplitude, solar-like oscillations with a frequency of 3.8 day⁻¹ in two of the data sets and derived a stellar mass of 2.2 M_☉ using the equations

$$\begin{equation} \nu _{\rm osc} = \frac{L / L_\odot }{M / M_\odot } \times 0.234 \; {\rm m\,s^{-1}} \end{equation} \tag{ 1 }$$

and

$$\begin{equation} f_{\rm max} = 62 \; {\rm {day}}^{-1} \frac{(T_{\rm eff} / {\rm 5777 \; K})^{3.5}}{\nu _{\rm osc} / \; {\rm m \; s^{-1}}} \;, \end{equation} \tag{ 2 }$$

where ν_osc is the oscillation velocity amplitude, f_max is the frequency of the strongest mode, and T_eff is the effective temperature. They used a luminosity of 64.2 ± 2.1 L_☉ from the Hipparcos catalog and T_eff = 4490 K from McWilliam (1990). Z08 then compared their 2.2 M_☉ value to those derived using T_eff, surface gravities (log g), and metallicities ([Fe/H]) from the literature and the PARAM stellar model by Girardi et al. (2000) and da Silva et al. (2006).⁴ The masses ranged from 1.05 ± 0.36 M_☉ based on values from Allende Prieto & Lambert (1999) to 1.71 ± 0.38 M_☉ based on values from Santos et al. (2004). Because Z08's mass was significantly higher than those derived using the model, they chose a mass of 1.4 M_☉ to calculate the minimum mass of the companion, which they list as 10.3 M_Jupiter.

A more accurate way to estimate the star's mass would be to investigate the frequency splitting (Δf₀) using the equation

$$\begin{equation} \Delta f_{\rm 0} = \sqrt{\frac{M / M_\odot }{(R / R_\odot)^3}} \times 11.66 \; {\rm day}^{\rm -1}, \end{equation} \tag{ 3 }$$

combined with an interferometrically measured radius, but unfortunately Z08's data set was not suitable for measuring Δf₀.

The advantage interferometry brings is the ability to directly measure the angular diameter of the star. Then, the physical radius can be determined using the distance from the parallax, and T_eff can be calculated. We combine our results with those from stellar oscillation frequencies to more completely understand the system through a description of the central star's and exoplanet's masses and the extent of the habitable zone. Section 2 details our observing procedure, Section 3 discusses how ι Dra's angular diameter and T_eff were determined, and Section 4 explores the physical implications of the new measurements.

Observations were obtained using the Center for High Angular Resolution Astronomy (CHARA) Array, a six-element Y-shaped optical–infrared interferometer located on Mount Wilson, California (ten Brummelaar et al. 2005). We used the CHARA Classic and CLIMB beam combiners in the K' band (2.13 μm) while visible wavelengths (470–800 nm) were used for tracking and tip/tilt corrections. The observing procedure and data reduction process employed here are described in McAlister et al. (2005). We observed ι Dra over four nights spanning four years with two baselines: in 2007 and 2008, we used the longest telescope pair S1–E1 with a maximum baseline of 331 m, and in 2011, we used a shorter telescope pair W1–W2 with a maximum baseline of 108 m.⁵

Choosing proper calibrator stars is vital because they act as the standard against which the scientific target is measured. We selected four calibrators (HD 128998, HD 139778, HD 141472, and HD 145454) because they are single stars with expected visibility amplitudes >80% so they were very nearly unresolved on the baseline used. This meant uncertainties in the calibrators' diameters did not affect the target's diameter calculation as much as if the calibrator stars had a significant angular size on the sky. We interleaved calibrator and target star observations so that every target was flanked by calibrator observations made as close in time as possible, which allowed us to convert instrumental target and calibrator visibilities to calibrated visibilities for the target.

To check for possible unseen close companions that would contaminate our observations, we created spectral energy distribution (SED) fits based on published U BV RI JHK photometric values obtained from the literature for each calibrator to establish diameter estimates. We combined the photometry with Kurucz model atmospheres⁶ based on T_eff and log g values to calculate angular diameters for the calibrators. The stellar models were fit to observed photometry after converting magnitudes to fluxes using Colina et al. (1996, UBVRI) and Cohen et al. (2003, JHK). The photometry, T_eff and log g values, and resulting angular diameters for the calibrators are listed in Table 1. There were no hints of excess emission associated with a low-mass stellar companion or circumstellar disk in the calibrators' SED fits (see Figure 1).

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The observed quantity of an interferometer is defined as the visibility (V), which is fit with a model of a uniformly illuminated disk (UD) that represents the observed face of the star. Diameter fits to V were based upon the UD approximation given by V = 2J₁(x)/x, where J₁ is the first-order Bessel function and x = πBθ_UDλ⁻¹, where B is the projected baseline at the star's position, θ_UD is the apparent UD angular diameter of the star, and λ is the effective wavelength of the observation (Shao & Colavita 1992). Table 2 lists the Modified Julian Date (MJD), projected baseline at the time of observation (B), projected baseline position angle (Θ), calibrated visibility (V), and error in V (σV) for ι Dra.

Figure 2 shows the UD fit to the observed visibilities and it is clear that this is not a sufficient model. A more realistic model of a star's disk involves limb darkening (LD), particularly in this case because most of the observations are on the third lobe of the visibility curve where secondary effects such as LD play a more important role than in the curve between B = 0 and the first null. Lacour et al. (2008) analyzed multiple LD prescriptions for the K1.5 III star Arcturus and found the power-law model to be a sufficient approximation. We chose to use this model because Arcturus' spectral type is very close to that of ι Dra. The model was based on Hestroffer (1997):

$$\begin{equation} I(\mu)/I({\rm 1}) = \mu ^\alpha, \end{equation} \tag{ 4 }$$

where I(μ) is the brightness of a point source at wavelength λ, I(1) is the brightness at the center, $\mu = \sqrt{1-({\rm 2}r/\theta _{\rm LD})^2}$, r is the angular distance from the star's center, and θ_LD is the limb-darkened angular diameter. In terms of the star's visibilities, the power-law prescription becomes

$$\begin{equation} V(v_r) = \sum _{k \ge 0} \frac{\Gamma (\alpha /2 + 2)}{\Gamma (\alpha /2 + k + 2) \Gamma (k + 1)} \left(\frac{-(\pi v_r \theta _{\rm LD})^2}{4} \right) ^k, \end{equation} \tag{ 5 }$$

where v_r is the radial spatial frequency, Γ is the Euler function (Γ(k+1) = k!), α = 0.258 ± 0.003 (Lacour et al. 2008), and k is the number of terms used. We tried a range of k values to check its effect on θ_LD and found that after 10 terms, θ_LD remained steady. Therefore we used k = 11.

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The resulting UD and LD angular diameters and their errors (<1%) are listed in Table 3. Additionally, the combination of the interferometric measurement of the star's angular diameter plus the Hipparcos parallax (van Leeuwen 2007) allowed us to determine the star's physical radius, which is also listed in Table 3.

For the θ_LD fit, the errors were derived via the reduced χ² minimization method (Wall & Jenkins 2003; Press et al. 1992): the diameter fit with the lowest χ² was found and the corresponding diameter was the final θ_LD for the star. The errors were calculated by finding the diameter at χ² + 1 on either side of the minimum χ² and determining the difference between the χ² diameter and χ² + 1 diameter. Figure 3 shows the LD diameter fit for ι Dra.

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Once θ_LD was determined interferometrically, the T_eff was calculated using the relation

$$\begin{equation} F_{\rm BOL} = {1 \over 4} \theta _{\rm LD}^2 \sigma T_{\rm eff}^4, \end{equation} \tag{ 6 }$$

where F_BOL is the bolometric flux and σ is the Stefan–Boltzmann constant. F_BOL was determined in the following way: ι Dra's V and K magnitudes were dereddened using the extinction curve described in Cardelli et al. (1989) and its interstellar absorption (A_V) value was from Famaey et al. (2005). The intrinsic broadband color (V − K) was calculated and the bolometric correction (BC) was determined by interpolating between the [Fe/H] = 0.0 and +0.2 tables from Alonso et al. (1999). They point out that in the range of 6000 K ⩾T_eff ⩾ 4000 K, their BC calibration is symmetrically distributed around a ±0.10 mag band when compared to other calibrations, so we assigned the BC an error of 0.10. The F_BOL was then determined by applying the BC and the T_eff was calculated. The star's luminosity (L) was also calculated using the absolute V magnitude and the BC. See Table 3 for a summary of these parameters.

We estimated the limb-darkened angular diameter for ι Dra using two additional methods as a check for our measurement. First, we created an SED fit for the star as described in Section 2, where UBV photometry was from Mermilliod (1991), RI photometry was from Monet et al. (2003), and JHK photometry was from Cutri et al. (2003). Figure 4 shows the resulting fit. Second, we used the relationship described in Blackwell & Lynas-Gray (1994) between the (V − K) color, T_eff, and θ. Our measured θ_LD is 3.596 ± 0.015 mas, the SED fit estimates 3.81 ± 0.23 mas, and the color–temperature–diameter relationship produces 3.63 ± 0.53 mas. Because ι Dra is so bright in the K band (0.7 mag) and because Two Micron All Sky Survey (2MASS) measurements saturate at magnitudes brighter than ∼3.5 even when using the shortest exposure time,⁷ we used the K magnitude from the Two-Micron Sky Survey (Neugebauer & Leighton 1969) for the color–diameter determination.

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The main sources of errors for the three methods are uncertainties in visibilities for the interferometric measurement, uncertainties in the comparison between observed fluxes and the model fluxes for a given T_eff and log g for the SED estimate, and uncertainties in the parameters of the relation and the spread of stars around that relation for the color–temperature–diameter determination. All three diameters agree within their errors but our interferometric measurements provide an error approximately 15 and 35 times smaller than the other methods, respectively.

With our newly calculated T_eff, we were able to estimate the mass of the central star using Equations (1) and (2) and obtained a mass of 1.82 ± 0.23 M_☉. We then calculated the exoplanet's minimum mass using the orbital parameters presented in Kane et al. (2010) and the equation

$$\begin{equation} f({\rm m}) = \frac{(m \sin i)^3}{(M + m)^2} = \frac{P}{2 \pi G} (K \sqrt{1-e^2})^3, \end{equation} \tag{ 7 }$$

where the period P was 510.72 ± 0.07 days, the amplitude K was 306.0 ± 3.8 m s⁻¹, and the eccentricity e was 0.713 ± 0.008. Our calculation produced a minimum mass of 12.6 ± 1.1 M_Jupiter, which converged in two iterations.

Our T_eff of 4545 ± 110 K is within the range listed in Z08, which spans 4466 ± 100 K (Allende Prieto & Lambert 1999) to 4775 ± 113 K (Santos et al. 2004). On the other hand, the stellar mass derived here is lower than that calculated in Z08 (2.2 M_☉) and slightly higher than the range presented in their paper (1.2–1.7 M_☉), though it overlaps within errors with the mass derived from the T_eff and log g in Santos et al. (2004). Hopefully, future observations of this star will determine the frequency splitting (see Equation (3)), which will allow for the direct measurement of the star's mass when combined with the interferometrically measured radius.

Using the following equations from Jones et al. (2006), we were also able to calculate the size of the system's habitable zone:

$$\begin{equation} S_{b,i}(T_{\rm eff}) = \big(4.190 \times 10^{-8} \; T_{\rm eff}^2\big) - (2.139 \times 10^{-4} \; T_{\rm eff}) + 1.296 \end{equation} \tag{ 8 }$$

and

$$\begin{equation} S_{b,o}(T_{\rm eff}) = \big(6.190 \times 10^{-9} \; T_{\rm eff}^2\big) - (1.319 \times 10^{-5} \; T_{\rm eff}) + 0.2341, \end{equation} \tag{ 9 }$$

where S_{b, i}(T_eff) and S_{b, o}(T_eff) are the critical fluxes at the inner and outer boundaries in units of the solar constant. The inner and outer physical boundaries r_{i, o} in AU were then calculated using

$$\begin{equation} r_i = \sqrt{ \frac{L/L_\odot }{S_{b,i}(T_{\rm eff})} } \; \; \; \; \; {\rm and} \; \; \; \; \; r_o = \sqrt{ \frac{L/L_\odot }{S_{b,o}(T_{\rm eff})} }. \end{equation} \tag{ 10 }$$

We obtained habitable zone boundaries of 6.8 AU and 13.5 AU. ι Dra's planet has a semimajor axis of 1.34 AU (Z08), so there is no chance that the planet orbits anywhere near the habitable zone.

The CHARA Array is funded by the National Science Foundation through NSF grant AST-0606958 and by Georgia State University through the College of Arts and Sciences, and by the W.M. Keck Foundation. S.T.R. acknowledges partial support by NASA grant NNH09AK731. This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France. This publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation.