CHARACTERIZATION OF THE RED GIANT HR 2582 USING THE CHARA ARRAY

The red giant star HR 2582 (HD 50890, HIP 33243) was part of a study by Hekker et al. (2009), who used the CoRoT satellite (Auvergne et al. 2009) to observe G and K giant stars with solar-like oscillations. The distribution of these pulsating stars' seismic parameters indicted they belonged to the "red clump" of low-mass, post-flash core-He-burning, evolved stars (Miglio et al. 2009). When the effective temperature and luminosity of these stars are not well characterized, it is difficult to determine their masses, ages, and radii (Kallinger et al. 2010).

HR 2582 was of particular interest because its mass could be inferred using other data besides asteroseismology. Baudin et al. (2012, hereafter B12) used spectroscopic observations to determine the following parameters: rotational velocity vsin i (10 ± 2 km s⁻¹), effective temperature T_eff (4665 ± 200 K), surface gravity log g (1.4 ± 0.3 cm s^-2), metallicity [Fe/H] (−0.18 ± 0.14), luminosity L (log L = 2.70 ± 0.15 L_☉), and finally a radius R (34 ± 8 R_☉) using the Stefan–Boltzmann law.

HR 2582 was observed for 55 days using CoRoT during its first science run. B12 found evidence for solar-like oscillations at low frequencies (between 10 and 20 μHz) with a spacing of 1.7 ± 0.1 μHz between consecutive radial orders and noted that only radial modes are clearly visible in the data. They discovered an excess of power in the power density spectrum at ν_max = 15 ± 1 μHz and determined the star's mass using

$\begin{equation} \frac{\nu _{\rm max}}{\nu _{\rm max \odot }} \approx \frac{\frac{M}{M_\odot }}{(R/R_\odot)^2 \sqrt{\frac{T_{\rm eff}}{T_{\rm eff \odot }}}}. \end{equation} \tag{ 1 }$

Their value was 5.2 ± 2.9  M_☉, which indicates HR 2582 is more massive than the stars in the red clump group described by Hekker et al. and Miglio et al. and implies rapid evolution.

These results provide insights on the internal workings during the final evolutionary stages when the star is burning hydrogen in a shell, is burning its central helium, or is in the last stage of He-shell burning. While the star can be placed on the Hertzsprung–Russell (H-R) diagram with relative precision, it is not sufficient to distinguish between the three evolutionary stages. Still, the results provide strong constraints on stellar interior models and are a good test case for those models (B12).

The advantage interferometry brings to HR 2582 is the ability to directly measure the angular diameter of the star instead of inferring its parameters using indirect methods. Then R is determined using our angular diameter plus the distance to the star known from its parallax, and T_eff is calculated. We combine our results with those from stellar oscillation frequencies to more completely understand the star and determine its mass. Section 2 details our observing procedure; Section 3 discusses the visibility measurements and how stellar parameters were calculated, including angular diameter, radius, luminosity, and temperature; Section 4 explores the physical implications of the new measurements; and Section 5 summarizes our findings.

We observed HR 2582 using the Center for High Angular Resolution Astronomy (CHARA) Array on 2012 December 12. The CHARA Array is a six-element optical-infrared interferometer located on Mount Wilson, California. We used the Classic beam combiner in the K'-band (2.13 μm) with the 279 m S1–W1 baseline.⁵ For a full description of the instrument, and the observing procedure and data reduction process used here, see ten Brummelaar et al. (2005) and McAlister et al. (2005).

When observing using an interferometer, selecting appropriate calibrator stars is extremely important because they are the standard against which we measure the scientific target. We used two calibrators, HD 46487 and HD 49434, which are both unresolved, single stars that acted as point sources. Because the stars' angular diameters are so small, uncertainties in their apparent sizes did not affect the target's diameter calculation as much as if they had a comparable angular size. 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.

We created spectral energy distribution (SED) fits to each calibrator star to check for possible unseen close companions. We used published UBVRIJHK photometric values combined with Kurucz model atmospheres⁶ based on T_eff and log g from the literature to estimate their angular diameters. 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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3.1. Angular Diameter Measurement

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. The diameter fit to V was 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). A more realistic model of a star's disk involves limb-darkening (LD), and the relationship incorporating the linear LD coefficient μ_λ (Hanbury Brown et al. 1974) is

$\begin{eqnarray} V &=& \left({1-\mu _\lambda \over 2} + {\mu _\lambda \over 3} \right)^{-1} \nonumber\\ && \times \left[(1-\mu _\lambda) {J_1(x_{\rm LD}) \over x_{\rm LD}} + \mu _\lambda {\left(\frac{\pi }{2} \right)^{1/2} \frac{J_{3/2}(x_{\rm LD})}{x_{\rm LD}^{3/2}}} \right] \end{eqnarray} \tag{ 2 }$

where x_LD = πBθ_LDλ⁻¹. Table 2 lists the date of observation, the projected baseline B, the calibrated visibilities (V), and errors in V (σV).

Table 2. HR 2582 Calibrated Visibilities

MJD	B	V	σV
	(m)
56273.285	172.02	0.902	0.054
56273.290	174.25	0.868	0.052
56273.306	183.32	0.796	0.054
56273.343	208.48	0.812	0.021
56273.404	248.94	0.736	0.036
56273.423	259.43	0.602	0.015
56273.437	265.92	0.664	0.028

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The LD coefficient μ_K of 0.31 was obtained from Claret & Bloemen (2011) after adopting a T_eff of 4750 from Wright et al. (2003) and a log g of 2.14 cm s^-2 from Cox (2000) for a K0 III, the spectral type listed in Wright et al. The resulting θ_UD is 0.978 ± 0.020 mas and θ_LD is 1.005 ± 0.020 mas, a 2% error. Figure 2 shows the θ_LD fit for HR 2582. LD is a second-order effect in the visibility curve that appears only after the first null in the visibility curve, i.e., when the visibility drops to zero. Because we are not beyond that null, we do not expect to see LD effects in our data and do not need to incorporate it into our model fit.

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For the θ_LD fit, the errors were derived using 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. 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. The resulting χ² is 19.8 and the reduced χ² ($\chi _{\rm red}^2 =\chi ^2$/DoF) is 3.3.⁷ When the $\chi _{\rm red}^2$ is forced to be 1, χ² is 6.0 and the errors nearly double from 0.020 to 0.036 mas. However, Andrae (2010) describes why forcing $\chi _{\rm red}^2$ is not recommended: when the $\chi _{\rm red}^2$ is forced to be 1, it implies the model is completely correct, which is most often not the case. Even if the model is perfect, the DoF must be large in order to allow us to force the $\chi _{\rm red}^2$ to equal one with impunity. In this situation, our DoF is 6 so we use the errors associated with χ², not $\chi _{\rm red}^2$.

3.2. Stellar Radius, Luminosity, and Effective Temperature

HR 2582 has a parallax of 2.99 ± 0.44 mas (van Leeuwen 2007), which translates to a distance of 334.5 ± 49.2 pc. When combined with our newly measured θ_LD, this gives us the physical radius of the star: 35.76 ± 5.31 R_☉, a error of 15%. This is comparable to the radius determined by B12 of 34 ± 8 R_☉ and provides better precision over their error of 24%.

In order to determine the L and T_eff of HR 2582, we constructed its SED using photometric values published in Cousins (1962), McClure & Forrester (1981), Haggkvist & Oja (1987), Beichman et al. (1988), Golay (1972), Kornilov et al. (1991), Mermilliod et al. (1997), and Cutri et al. (2003). The assigned uncertainties for the Two Micron All Sky Survey infrared measurements are as reported, and an error of 0.05 mag was assigned to the optical measurements.

HR 2582's bolometric flux (F_BOL) was determined by finding the best fit stellar spectral template from the flux-calibrated stellar spectral atlas of Pickles (1998) using the χ² minimization technique. This best SED fit allows for extinction, using the wavelength-dependent reddening relations of Cardelli et al. (1989). The best fit was found using a K1 III template with an assigned temperature of 4656 ± 120 K, an extinction of A_V = 0.091 ± 0.042 mag, and a F_BOL of 1.48 ± 0.05 × 10⁻⁷ erg s⁻¹ cm⁻². Figure 3 shows the best fit and the results are listed in Table 3.

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Table 3. HR 2582 Stellar Parameters

Parameter	Value	Reference
From the literature
V magnitude	6.04 ± 0.01	Mermilliod (1991)
K magnitude	3.65 ± 0.28	Cutri et al. (2003)
π (mas)	2.99 ± 0.44	van Leeuwen (2007)
Distance (pc)	334.5 ± 49.2	Calculated from π
μK	0.33	Claret & Bloemen (2011)
The results of our SED fit
AV	0.09 ± 0.04
FBOL (10−7 erg s−1 cm−2)	1.48 ± 0.05
Teff, estimated (K)	4656 ± 120
θLD, estimated (mas)	0.972 ± 0.053
The results of this work
θUD (mas)	0.978 ± 0.020
θLD (mas)	1.006 ± 0.020
Rlinear (R☉)	35.76 ± 5.31
Teff (K)	4577 ± 60
L (L☉)	517.8 ± 17.5
Mass (M☉)	5.6 ± 1.7
Age (Myr)	165$^{+20}_{-15}$

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We then combined F_BOL with HR 2582's distance to estimate its luminosity where L = 4πd²F_BOL, which produced a value of 517.8  ± 17.5 L_☉. The uncertainty in L is largely due to the uncertainty in the distance. The F_BOL was also combined with the star's θ_LD to determine its effective temperature by inverting the relation

$\begin{equation} F_{\rm BOL} = {1 \over 4} \theta _{\rm LD}^2 \sigma T_{\rm eff}^4, \end{equation} \tag{ 3 }$

where σ is the Stefan–Boltzmann constant. This produces an effective temperature of 4579 ± 60 K, a 1% error. Because μ_K is chosen based on a given T_eff, we checked to see if μ_K would change based on our new T_eff and iterated. μ_K increased by 0.02 to 0.33, θ_LD increased by only 0.001 mas to 1.006 ± 0.020 mas, and T_eff decreased by 2 K to 4577 K, which are well within the errors. We adopted these θ_LD and T_eff as our final values (see Table 3). The very slight change in θ_LD did not affect the radius calculation. We also note the log g used here (2.14 cm s^-2) differs from that determined by B12 (1.4 cm s^-2). We used B12's log g to select μ_K and the resulting change in μ_K is +0.01 to 0.32 and we see above how little effect that has on the resulting θ_LD.

In Section 3.1, we compared the merits of χ² versus $\chi ^2_{\rm red}$ and leaned in favor of using χ² errors. Those are the results listed in Table 3. However, if we do assume our model is perfect, force $\chi ^2_{\rm red}$ to equal one, and use the resulting error of 0.036 mas in θ_LD, σ_TEFF increases from 60 to 91 K, an error of 2%, and σ_R remains the same at 5.31 R_☉, an error of 15%.

As a check to our measurement, we estimated HR 2582's θ_LD using two additional methods: (1) we used the SED fit as described in Section 3.2; and (2) we used the relationship between the star's dereddened (V − K) color (calculated with the extinction curve described in Cardelli et al. 1989), T_eff, and θ_LD from Blackwell & Lynas-Gray (1994). Our measured θ_LD is 1.006 ± 0.020 mas, the SED fit predicts 0.972 ± 0.053 mas, and the color–temperature–diameter relationship produces 0.926 ± 0.369 mas.

The main sources of errors for the three methods are uncertainties in visibilities for our interferometric measurement, uncertainties in the comparison between the observed and model fluxes for a given set of T_eff and log g values 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. The three θ_LD agree within the errors, and our interferometric measurements provide an error approximately 3 and 18 times smaller than the other methods, respectively.

We used our new values of T_eff and R in Equation (1) to calculate HR 2582's mass with the result of 5.6 ± 1.7 M_☉. This is slightly more massive than B12's mass of 5.2 ± 2.9 M_☉ but is well within the errors. We also used T_eff and L to estimate the age of HR 2582 using the Yonsei–Yale isochrones (Y²; Yi et al. 2001). We adopted [Fe/H] = –0.18 to be consistent with B12. The resulting age is 165$^{+20}_{-15}$ Myr (see Figure 4), which is higher than the 105.5 Myr age quoted in B12. They do not discuss how they determined the age except to note that it is one of the model outputs, and do not give an error for that parameter.

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This resulting age does not definitively answer the question of what evolutionary state HR 2582 is currently occupying. If the star is burning hydrogen in a shell on the first ascending branch, it is ∼157 Myr old. If it is burning helium in its core on the descending or second ascending branches, it is ∼163 Myr or ∼180 Myr old, respectively (B12). We lack the precision to determine exactly what is occurring in the interior of this star but it remains an excellent test case for stellar models, particularly with our more precise radius and temperature measurements.

We directly measured the limb-darkened angular diameter of HR 2582 with the CHARA Array interferometer and used our result of 1.006 ± 0.020 mas along with the parallax measurement and photometry from the literature to calculate its physical radius (35.76 ± 5.31 R_☉), luminosity (517.8 ± 17.5 L_☉), and effective temperature (4577 ± 60 K). We combined our R and T_eff values with stellar oscillation results from B12 to determine the mass, which was 5.6 ± 1.7 M_☉ and the same R and T_eff with Y² isochrones to estimate the star's age at 165$^{+20}_{-15}$ Myr.

The CHARA Array is funded by the National Science Foundation through NSF grant AST-0908253 and AST-1211129 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.