STELLAR DIAMETERS AND TEMPERATURES. I. MAIN-SEQUENCE A, F, AND G STARS

The CHARA Array is an optical/infrared interferometric array located at Mount Wilson Observatory in the San Gabriel mountains of southern California (a detailed description of the instrument can be found in ten Brummelaar et al. 2005). Briefly, the CHARA Array consists of six, 1 m aperture telescopes in a Y-shaped configuration spread across the mountaintop of the Observatory. With the six telescopes, there are 15 available baseline combinations, ranging from 34 to 331 m, at a variety of position angle orientations ψ. The CHARA Array currently is the longest baseline operational optical/infrared interferometer in the world.

There are several beam combiners available for the CHARA Array, and for this project, the observations were made using the CHARA Classic beam combiner in two-telescope mode. CHARA Classic is a pupil-plane beam combiner, which is used in the K' band, empirically measured in Bowsher et al. (2010) to have a central wavelength of $\lambda _{K^{\prime }} = 2.141 \pm 0.003 \,\mu$m. Fringes are detected and recorded on the Near Infrared Observer (NIRO) camera, which is based upon an HgCdTe PICNIC Array readout at high speed. Nearly all (98.5%) of the observing for this work was performed remotely from Georgia State University's Cleon Arrington Remote Operations Center (AROC) in Atlanta, GA during the 2007, 2008, and 2009 observing seasons.

The main goal of this survey is to determine the angular diameters of a large number of stars to high precision. We limited the stars observed by selecting only those for which the predicted precision of the measured angular diameter will be better than 4%. The precision of a measurement of the stellar angular diameter depends on how far down the visibility curve one is able to sample. The expression of the uniform disk visibility function of a single star (Equation (1)) is dependent on B the projected baseline, θ the angular diameter of the star, and λ the wavelength of observation:

$$\begin{equation} V = \frac{2 J_1(x) }{x}, \end{equation} \tag{ 1 }$$

where

$$\begin{equation} x = \pi \emph {B} \theta \lambda ^{-1} \end{equation} \tag{ 2 }$$

and J₁ is the first-order Bessel function. By knowing the λ and B utilized in a given observation, we can estimate the optimum resolution range resulting from the precision with which we can measure the object visibility. To ensure that we reach the precision goal of σθ < 4% for our observations, we find the approximate limiting resolution is θ ∼ 0.65 mas for K'. In reality, this number will vary (for example, see Baines et al. 2008), but its use to establish a first-order cutoff for a preliminary sample selection is appropriate.

Each star we chose to observe was then hand selected from a Hipparcos catalog query. This process was initiated from assumptions of the nominal linear size of a main-sequence star from Cox (2000) based on B − V colors, in order to determine a maximum distance to be sampled for each spectral type before reaching the θ = 0.65 mas resolution limit. Additionally, the luminosity class of the star was restricted by apparent V magnitudes to only admit roughly main-sequence stars (Cox 2000). The declination limit to this survey was restricted to targets above −10° declination. All stars were easily within the magnitude limits for observing with the CHARA Array, with magnitudes ranging from 2.5 < V < 6.4 and 1.6 < K < 4.4.⁸ Table 1 lists the names, coordinates, spectral types, magnitudes, metallicity [Fe/H], and Hipparcos distance of the stars observed.

Table 1. Target Sample of A, F, and G Dwarfs

HD	HR	HIP	Other	R.A.	Decl.	Spectral	Spectral	V	K	(B − V)	[Fe/H]d	π ± σe
			Namea	(hh mm ss.xx)	(dd mm ss)	Typeb	Typec	(mag)	(mag)	(mag)		(mas)
4614	219	3821	24 η Cas A	00 49 06.29	57 48 54.67	F9V	G0V	3.46	2.05	0.587	−0.30	168.01 ± 0.48
5015	244	4151	GJ 41	00 53 04.20	61 07 26.29	F8V	F8V	4.80	3.54	0.540	0.00	53.35 ± 0.33
6582	321	5336	34 μ Cas A	01 08 16.39	54 55 13.22	G5Vb	G5Vp	5.17	3.36	0.704	−0.83	132.40 ± 0.60
10780	511	8362	GJ 75	01 47 44.84	63 51 09.00	G9V	K0V	5.63	3.84	0.804	0.05	99.34 ± 0.53
16895	799	12777	13 θ Per A	02 44 11.99	49 13 42.41	F7V	F7V	4.10	2.78	0.514	−0.12	89.88 ± 0.23
19373	937	14632	ι Per	03 09 04.02	49 36 47.80	G0IV-V	G0V	4.05	2.70	0.595	0.09	94.87 ± 0.23
20630	996	15457	κ Cet	03 19 21.70	03 22 12.71	G5V	G5Vvar	4.84	3.34	0.681	0.00	109.39 ± 0.27
22484	1101	16852	10 Tau	03 36 52.38	00 24 05.98	F9IV-V	F9V	4.29	2.93	0.575	−0.09	71.60 ± 0.54
30652	1543	22449	1 π3 Ori	04 49 50.41	06 57 40.59	F6IV-V	F6V	3.19	2.07	0.484	−0.03	123.94 ± 0.17
34411	1729	24813	15 λ Aur	05 19 08.47	40 05 56.59	G1V	G0V	4.69	3.24	0.630	0.05	79.18 ± 0.28
39587	2047	27913	54 χ1 Ori	05 54 22.98	20 16 34.23	G0IV-V	G0V	4.39	2.97	0.594	−0.16	115.42 ± 0.27
48737	2484	32362	31 ξ Gem	06 45 17.37	12 53 44.13	F5IV-V	F5IV	3.35	2.30	0.443	0.01	55.55 ± 0.19
56537	2763	35350	54 λ Gem	07 18 05.58	16 32 25.38	⋅⋅⋅	A3V	3.58	3.27	0.106	⋅⋅⋅	32.36 ± 0.22
58946	2852	36366	62 ρ Gem	07 29 06.72	31 47 04.38	⋅⋅⋅	F0V	4.16	3.32	0.320	−0.31	55.41 ± 0.25
81937	3757	46733	23 h UMa	09 31 31.71	63 03 42.70	⋅⋅⋅	F0IV	3.65	2.82	0.360	0.06	41.99 ± 0.16
82328	3775	46853	25 θ UMa	09 32 51.43	51 40 38.28	F5.5IV-V	F6IV	3.17	2.02	0.475	−0.12	74.18 ± 0.13
82885	3815	47080	11 LMi	09 35 39.50	35 48 36.48	G8+V	G8IV-V	5.40	3.70	0.770	0.06	87.96 ± 0.32
86728	3951	49081	20 LMi	10 01 00.66	31 55 25.22	G4V	G3V	5.37	3.82	0.676	0.20	66.47 ± 0.32
90839	4112	51459	36 UMa	10 30 37.58	55 58 49.93	F8V	F8V	4.82	3.54	0.541	−0.16	78.26 ± 0.29
95418	4295	53910	48 β UMa	11 01 50.48	56 22 56.74	A1IV	A1V	2.34	2.35	0.033	0.06	40.89 ± 0.16
97603	4357	54872	68 δ Leo	11 14 06.50	20 31 25.38	A5IV(n)	A4V	2.56	2.26	0.128	0.00	55.82 ± 0.25
101501	4496	56997	61 UMa	11 41 03.02	34 12 05.89	G8V	G8V	5.31	3.53	0.723	−0.12	104.03 ± 0.26
102870	4540	57757	5 β Vir	11 50 41.72	01 45 52.98	F8.5IV-V	F8V	3.59	2.33	0.518	0.11	91.50 ± 0.22
103095	4550	57939	CF UMa	11 52 58.77	37 43 07.24	K1V	G8Vp	6.42	4.38	0.754	−1.36	109.98 ± 0.41
109358	4785	61317	8 β CVn	12 33 44.55	41 21 26.93	G0V	G0V	4.24	2.72	0.588	−0.30	118.49 ± 0.20
114710	4983	64394	43 β Com	13 11 52.39	27 52 41.46	G0V	G0V	4.23	2.90	0.572	−0.06	109.53 ± 0.17
118098	5107	66249	79 ζ Vir	13 34 41.59	−00 35 44.95	A2Van	A3V	3.38	3.11	0.114	−0.02	44.01 ± 0.19
126660	5404	70497	23 θ Boo	14 25 11.80	51 51 02.68	F7V	F7V	4.04	2.78	0.497	−0.14	68.83 ± 0.14
128167	5447	71284	28 σ Boo	14 34 40.82	29 44 42.47	F4VkF2mF1	F3V	4.47	3.52	0.364	−0.36	63.16 ± 0.26
131156	5544	72659	37 ξ Boo	14 51 23.38	19 06 01.66	G7V	G8V	4.54	2.96	0.720	−0.33	149.03 ± 0.48
141795	5892	77622	37 Ser	15 50 48.97	04 28 39.83	kA2hA5mA7V	A2m	3.71	3.43	0.147	⋅⋅⋅	46.28 ± 0.19
142860	5933	78072	41 γ Ser	15 56 27.18	15 39 41.82	F6V	F6V	3.85	2.66	0.478	−0.19	88.85 ± 0.18
146233	6060	79672	18 Sco	16 15 37.27	−08 22 09.99	G2V	G1V	5.49	3.55	0.652	−0.02	71.93 ± 0.37
162003	6636	86614	31 ψ Dra	17 41 56.36	72 08 55.84	F5IV-V	F5IV-V	4.57	3.43	0.434	−0.17	43.79 ± 0.45
164259	6710	88175	57 ζ Ser	18 00 29.01	−03 41 24.97	F2V	F3V	4.62	3.66	0.390	−0.14	42.44 ± 0.33
173667	7061	92043	110 Her	18 45 39.73	20 32 46.71	F5.5IV-V	F6V	4.19	2.89	0.483	−0.15	52.06 ± 0.24
177724	7235	93747	17 ζ Aql	19 05 24.61	13 51 48.52	A0IV-Vnn	A0Vn	2.99	2.92	0.014	−0.68	39.27 ± 0.17
182572	7373	95447	31 b Aql	19 24 58.20	11 56 39.90	⋅⋅⋅	G8IV	5.17	3.53	0.761	0.33	65.89 ± 0.26
185144	7462	96100	61 σ Dra	19 32 21.59	69 39 40.23	G9V	K0V	4.67	2.78	0.786	−0.24	173.77 ± 0.18
185395	7469	96441	13 θ Cyg	19 36 26.54	50 13 15.97	F3+V	F4V	4.49	3.52	0.395	−0.04	54.55 ± 0.15
210418	8450	109427	26 θ Peg	22 10 11.99	06 11 52.31	⋅⋅⋅	A2V	3.52	3.22	0.086	−0.38	35.34 ± 0.85
213558	8585	111169	7 α Lac	22 31 17.50	50 16 56.97	⋅⋅⋅	A1V	3.76	3.75	0.031	⋅⋅⋅	31.80 ± 0.12
215648	8665	112447	46 ξ Peg	22 46 41.58	12 10 22.40	F6V	F7V	4.20	2.87	0.502	−0.24	61.37 ± 0.20
222368	8969	116771	17 ι Psc	23 39 57.04	05 37 34.65	F7V	F7V	4.13	2.89	0.507	−0.08	72.91 ± 0.15

Notes. ^aBayer-Flamsteed or GJ (Kostjuk 2004). ^bGray et al. (2001, 2003). ^cSIMBAD (Wenger et al. 2000). ^dHolmberg et al. (2007), when available. For stars without metallicity estimates from Holmberg et al. (2007), the [M/H] values from Gray et al. (2003, 2006; HD 82885, HD 97603, HD 118098, HD 131156, HD 177724, HD 210418), and Takeda et al. (2005; HD 182572) are used. Stars with no metallicity measurements are HD 56537, HD 141795, and HD 213558. ^evan Leeuwen (2007).

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We follow the standard routine for interferometric observing where a calibrator star with known size is observed before and after every observation of a science object,⁹ which enables us to remove the instrumental response of the system as well as effects from local seeing conditions. A description of the ideal calibrator and the propagation of errors to the true visibility measurements of the science star can be found in van Belle & van Belle (2005). The calibrator stars used in this work were initially selected from the getCal web interface¹⁰ and were restricted by angular distance in the sky to the object, magnitude, and its estimated angular size. In most cases, the calibrator star was less than 6° from the object, and never lying further than ∼10°. The closeness of the object to the calibrator is crucial for quickly acquiring brackets, where under typical observing conditions, there is approximately 4–5 minutes between observations. The magnitudes of the calibrator stars were chosen from observability limits with our telescopes and instrument. Finally, the estimated angular size of the calibrator was the attribute that is most weighted in the calibrator searching process.

With CHARA's long baselines, unwanted biases may be introduced if the calibrator is resolved. If possible, calibrators were chosen to have a angular diameter less than 0.45 mas, following the example described in van Belle & van Belle (2005) for CHARA's maximum baseline of 331 m. In order to estimate the calibrator's angular size, θ_SED, a Kurucz model spectral energy distribution (SED) was fit to Johnson UBV (Mermilliod 1997), Strömgren uvby (Hauck & Mermilliod 1998), and Two Micron All Sky Survey (2MASS) JHK (Cutri et al. 2003) flux-calibrated photometry.¹¹ The calibrators used in this work are listed in Table 2. Table 2 also shows the calibrator Johnson, Strömgren, and 2MASS magnitudes, and θ_SED fit for each star. The last column lists the science object(s) observed with that calibrator.

Two-thirds of the calibrators used meet the θ_SED < 0.45 mas criteria.¹² In many cases, the science stars were observed with more than one calibrator, on more than one occasion, in order to reduce the likelihood of biases being introduced (for example, see HD 30652; Table 3). In some instances, we were not so fortunate to observe stars with multiple calibrators having the ideal angular size of θ_SED < 0.45 mas. For instance, the stars HD 5015, HD 6582, and HD 10780 were observed with only one calibrator, HD 6210 (θ_SED = 0.519 mas). However, we note that the observations of the star HD 4614 were also obtained with this calibrator as well as two additional calibrators, one of which meets the ideal criterion for being completely unresolved (HD 9407, θ_SED = 0.430 mas). We find that all the calibrated data sets for HD 4614 agree flawlessly in the final diameter fits. Thus, using the slightly resolved calibrator star HD 6210 is not a cause of concern. Only four stars, HD 97603, HD 185144, HD 126660, and HD 114710 have observations with only one calibrator with 0.45 mas < θ_SED < 0.57 mas, but consistency in the calibration process seen with stars observed with more than one calibrator (such as HD 4614) provides assurance that this issue is not a problematic one. Additionally, the maximum CHARA baseline of 331 m was never reached for a significant amount of these observations, thus pushing the theorized ideal maximum angular size for the calibrator to larger sizes.

The observing log is shown in Table 3 and lists the identifications of the 44 stars observed in this work (Column 1), UT date (Column 2), CHARA baseline (Column 3), number of brackets (Column 4), and the calibrator(s) used on that date (Column 5). The specifics of the CHARA baseline configurations are displayed in Table 4. We include the observations of HD 6582, HD 10780, and HD 185144 here for completeness, because they are an original part of this survey, but their results have been previously presented in Boyajian et al. (2008).

3.1.1. A Stars

3.1.2. Multiplicity

Angular diameters of a few dozen main-sequence stars measured with the Palomar Testbed Interferometer (PTI) were presented by van Belle & von Braun (2009). Their work provides measurements of 14 stars in common with the CHARA stars measured here and is the only alternate source of direct angular diameter measurements of our program stars. The longest baseline obtainable with PTI is 110 m, a factor of three shorter than those of the CHARA Array, and accurate measurements were quite difficult with this instrument due to the small angular sizes of these stars.

Table 6 lists the 14 stars in common with van Belle & von Braun (2009), the limb-darkened angular diameters and errors, and how many σ the two values differ from each other. For these stars, the errors on the PTI angular diameters are anywhere from 2 to 12 times (with an average of 6.5 times) the errors on the CHARA angular diameters. However, this comparison can still point to any systematic offsets in the results from each instrument. Comparing the angular diameters from this work and van Belle & von Braun (2009), we find that the weighted mean ratio of CHARA to PTI diameters is θ_CHARA/θ_PTI = 1.052 ± 0.062. Van Belle & von Braun (2009) make this same comparison of their diameters compared to diameters from Baines et al. (2008), who used the CHARA Array to measure the diameters of exoplanet host stars, and find that the ratio of the four stars they have in common is θ_CHARA/θ_PTI = 1.06 ± 0.06, very similar to the results found here, indicating again that there is a slight preference for smaller PTI diameters, and larger CHARA diameters, although the displacement is at the <1σ level.

The preference to larger diameters compared to other interferometric measurements was also seen in Boyajian et al. (2009), where the diameters of the four Hyades giants were measured with the CHARA Array. In that work, two of the stars, Tau and δ¹ Tau, were measured previously with other interferometers (Mark III, NPOI, and PTI), all of which lead to smaller diameters than those measured with CHARA. However, Boyajian et al. (2009) find that models for the Hyades age and metallicity match flawlessly with the CHARA observations, and the smaller angular diameters from other works in turn lead to temperatures that are too hot for these stars.

A main distinction that could lead to offsets in measured diameters is the estimated size of the calibrator stars. For instance, van Belle & von Braun (2009) discuss the calibrator selection in their work compared to Baines et al. (2008). Van Belle & von Braun (2009) set a limit to a sufficiently unresolved calibrator at CHARA to be <0.5 mas in diameter, a criterion which all but a few calibrators in this work meet. However, to investigate the possibility that the estimated size of the calibrators in this work is offset to the calibrators used in van Belle & von Braun (2009), we compare the estimated sizes of the calibrators in the Palomar Testbed Interferometer Calibrator Catalog (PTICC; van Belle et al. 2008) to the ones derived here. Twenty-nine of the 63 calibrators used in this work are included in the PTICC. Overall, the ratio of the estimated diameter of the calibrator in this work to the PTICC is 0.97 ± 0.06, a less than 1σ difference.

Twelve of the 14 stars in common with van Belle & von Braun (2009) were observed with calibrators whose diameters are also included in the PTICC. For each of these 12 calibrators, the estimated angular diameter θ_SED is presented in Table 7, along with the ratio of the CHARA to PTI SED diameters. The object employing the calibrator is also listed in Table 7 along with the ratio of the CHARA to PTI measured limb-darkened diameters. Here, there is no pattern in the calibrator SED diameter ratio and the object diameter ratio. In fact, the effects of a slight offset in the calibrator's estimated diameter listed above (ratio θ_CHARA/θ_PTI = 0.97 ± 0.06) would actually contribute counterproductively to the slight offset in the diameter measurements (ratio θ_CHARA/θ_PTI = 1.05 ± 0.06). For instance, in the case of our data, the size of the calibrator θ_SED is typically smaller, thus the true visibility of the calibrator would be greater (i.e., it would be more unresolved). If the true visibility of the calibrator is greater, it would in turn make the true visibility of the object larger in the calibration process. Thus, the object would appear more unresolved (having larger calibrated visibilities) if we were using a SED diameter of the same calibrator but with a larger value. Because we do not see the case of smaller CHARA diameters, this indicates that the calibrators are not the cause of any offset, if present, in each data set.

In Figure 2, we show a plot of the measured limb-darkened angular diameters presented here compared to the computed SED diameters for the stars in this survey. The 14 stars in common with the van Belle & von Braun (2009) work are highlighted in color, red indicating a CHARA measurement and blue for a PTI measurement. It is apparent here that the majority of the PTI diameters are deceptively low compared to the SED estimates.

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Figure 3 shows the fractional difference of the measured limb-darkened versus SED angular diameter as a function of (B − V) color index. For stars with a (B − V) <0.4, the measured θ_LD is always larger than the estimated θ_SED. There is a lot of scatter for stars redder than (B − V) ∼0.4, but no systematics are seen. However, in this region the two most metal-poor stars, HD 6582 and HD 103095, are the most extreme outliers, where θ_LD > θ_SED in both cases. Additionally, HD 182572 (B − V = 0.761; [Fe/H] = 0.33) is the most metal-rich star observed in this work and shows the largest disagreement here where θ_LD < θ_SED. Perhaps there is a lack of sufficient data to show if there is in fact a trend in either the color index or metallicity of a star and the estimated θ_SED.

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We present new measures of the bolometric flux F_BOL for each star in this work that follows the method used previously in support of interferometric observations (van Belle et al. 2008; van Belle & von Braun 2009). The fit involves a collection of flux-calibrated photometry for each source available in the literature (see Table 8) that is subsequently fit with a template spectra from the Pickles (1998) library. Reddening is for the most part absent for the sample;¹⁶ however, we fit it here to the photometric data using reddening corrections based upon the empirical reddening determination described by Cardelli et al. (1989), which differs little from van de Hulst's theoretical reddening curve number 15 (Johnson 1968; Dyck et al. 1996).

Uncertainties in F_BOL are for our data, on average 1.9%, and tend to be dominated by uncertainty in the reddening fit and poor photometry in the 0.6–1.2 μm range. Each science object is listed in Table 9 along with the fitting parameters: Pickles (1998) template spectral type, number of photometry points, and reduced χ² of the fit, as well as the resulting integrated F_BOL and A_V. In Figure 4, we show an example SED fit.

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These F_BOL measurements are then simply used in combination with the Hipparcos distance d to solve for the absolute luminosity L:

$$\begin{equation} L = 4 \pi d^2 F_{\rm BOL}. \end{equation} \tag{ 3 }$$

Additionally, we can express the effective temperature of a star as defined through the Stephan–Boltzmann law:

$$\begin{equation} F = \sigma T_{\rm EFF}^{4}, \end{equation} \tag{ 4 }$$

where F is the total emergent flux of the star and σ is the Stefan–Boltzmann constant. Transforming this equation to observables at Earth, we arrive at the expression

$$\begin{equation} F_{\rm BOL} = \frac{1}{4} \theta ^{2} \sigma T_{\rm EFF}^{4}. \end{equation} \tag{ 5 }$$

The effective temperature T_EFF is found by solving Equation (5) in terms of F_BOL and θ_LD, where in Equation (6), F_BOL is in units of 10⁻⁸ erg s⁻¹ cm⁻² and θ_LD is in units of mas. This yields the relation

$$\begin{equation} T_{\rm EFF} = 2341 (F_{\rm BOL}/\theta ^2)^{0.25}. \end{equation} \tag{ 6 }$$

Finally, using Hipparcos parallaxes from van Leeuwen (2007), we transform the measured angular diameters of these stars into linear radii R. Each of these derived parameters are presented in Table 10. Graphical representations of these data sets are presented in Figures 5 and 6.

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With the results in Table 10, we may begin to define a foundation of empirically based color–temperature relations. These relationships are extremely useful in extending our knowledge to a larger number of stars, at distances too far to accurately resolve their sizes. For giants and supergiants, temperature scales accurate to the 2.5% level are obtainable, and are currently limited by the distances to these objects, not the sensitivities of our interferometric observations (van Belle et al. 1999, 2009). Recently, and more similarly comparable to this work, van Belle & von Braun (2009) present relations for main-sequence stars, ranging from (V − K) ∼ 0.0 to 4.0 (spectral types of ∼A-M). These scales are slightly better than the aforementioned evolved classes of stars and are accurate to the ∼2% level. The authors note here that the limitations to these relations are in the angular diameter measurements themselves.

5.1.1. (B − V)–T_EFF

Here, we derive color–temperature relations based on the precise T_EFF measurements presented in Table 10 in the form of a sixth-order polynomial. The solution for the (B − V) color–temperature relation is expressed as

$$\begin{eqnarray} \log T_{\rm EFF}& =&3.9680 \pm 0.0025 - 0.2633 \pm 0.0876 (B-V)\nonumber \\ && -\, 3.2195 \pm 1.3407 (B-V)^2 + 15.3548\nonumber \\ && \pm 6.8551 (B-V)^3 - 27.2901\pm 15.7373 (B-V)^4\nonumber \\ && +\, 19.9193 \pm 16.8465 (B-V)^5 - 4.5127\nonumber \\ && \pm\, 6.8539 (B-V)^6\quad {\rm for} \; 0.05 < (B-V)< 0.80\nonumber \\ && {\rm and} \; {\rm [Fe/H]} > -0.75. \end{eqnarray} \tag{ 7 }$$

This solution, in the form of a sixth-order polynomial, defines the shape of the data inflection point at (B − V) ∼0.3 better than a lower order polynomial function, as well as a power-law function (see Figure 7). We are also cautious that the stars with low metallicity will affect the (B − V)–T_EFF transformation too severely to be useful for stars of solar-type abundances, and we refrain from using these in this analysis.¹⁷ A preliminary fit to the data yields a median deviation in temperature of 68 K, and a median deviation in temperature of 55 K is found for an identical solution if we omit three obvious outliers lying more than 5σ away, HD 210418, HD 182572, and HD 162003 (offset of −6.7σ, −5.7σ, and +6.5σ, respectively). The statistical summary for the solution to the polynomial can be found in Table 11.¹⁸

Table 11. Solutions to T_EFF Relations

	(B − V)	(V − K)	Spectral Type
Equation in text	(8)	(9)	(10)
n ...	39	44	41
Range ...	0.05–0.80a	0.0–2.0	A0–K0a
Reduced χ2 ...	5.97	6.04	12.1
Median dTEFF ...	55	64	90

Notes. Refer to Section 5.1 for details. ^aMetallicity [Fe/H] >−0.75.

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In Figure 7, we show our data and the solution for the fit. We also show the solution from several other sources, Code et al. (1976), Gray (1992), and Lejeune et al. (1998). All the solutions shown here are approximately identical in the range of (B − V) >0.45.

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Similar to our work, the results from Code et al. (1976) are derived solely on empirical measurements. In that milestone paper, Code et al. (1976) measure the diameters of 32 stars using the NSII, all being objects hotter than the Sun and most having evolved luminosity classes. We show in Figure 7 the nine data points from Code et al. (1976) that have a (B − V) >0 (∼A-type and later) with luminosity class V or IV (eight A-type objects and one F-type object), as well as the fit from Code et al. (1976). Comparing this fit to ours, we note that it is a few hundred Kelvin hotter than our own for 0.05 < (B − V) <0.3, converging at the bluest range of (B − V) ∼0, as well as the reddest range (B − V) ∼0.4. The offset is likely strongly connected to the fit's dependence on the sparse amount of data in this intermediate range.

The function presented in Gray (1992) also uses a sixth-order polynomial to fit (B − V) to T_EFF (their Equation (15.14)). This is a fit to a compilation of data, including the NSII data from Code et al. (1976), but mostly data obtained via the infrared flux method (IRFM; see Blackwell & Shallis 1977), yielding a semi-empirical approach in obtaining the angular diameter, and consequentially the effective temperature, of a star. There is an impressive correlation with our fit throughout the range of (B − V) colors. A deviation only begins to appear at the bluest colors ((B − V) ∼0). Our sample stops here, but it can be readily seen in Gray (1992) that there is another inflection point at this color index, producing a steeper curve at more negative color indices. Because of this, we caution against the use of the relation presented here for colors bluer than (B − V) <0.05.

As an update, the work presented in Casagrande et al. (2010) presents an excellent analysis and improvement to the IRFM temperature scales that have been proposed over the years. In Figure 7, we also show the Casagrande et al. (2010) solution for solar metallicity for comparison (valid in the ranges of (B − V) > 0.3). This relation is distinguishable from ours only when 0.3 < (B − V) < 0.4, where at this point, the Casagrande et al. (2010) relation begins to predict ∼100 K higher temperatures. However, we suspect that the likely explanation for this difference is that their solution is in the form of a third-order polynomial, which has been shown not to model the inflection point in this region properly.

For an additional comparison on the (B − V) to T_EFF relations, we also show the entirely model-based solution from Lejeune et al. (1998; dashed line). This solution is intermediate to ours (as well as the one in Gray 1992) and the one from Code et al. (1976) for the mid-A- to mid-F-type stars.

The median error in the temperature measurement for these stars (45 K) is lower than the median deviation to the fit, suggesting the potential for improvement. Iterating on the fit and removing outliers that lie further than 3σ gives an identical solution, but reduces the number of points used from 39 to 33. However, it does not improve the error of the fit by much. The scatter in the points with a (B − V) >0.6 are clearly the cause of the ill-correlated relation, as illustrated in Figure 7.

5.1.2. (V − K)–T_EFF

We also present a fit of the temperature versus (V − K) color index (Equation (9) and Figure 8), the benefit here being that (V − K) colors are less sensitive to the stellar abundances than the (B − V) colors. A solution including all stars gives a median deviation in temperature of 67 K, with only one extreme outlier, HD 162003 (+5.8 σ).¹⁹ Omitting this star, we arrive at our final solution, practically identical from the first, with a median temperature deviation of 64 K. The statistical overview for this solution can be found in Table 11, and Figure 8 illustrates the fit. We note that the solution is identical when clipping stars that lie more than 3.5σ from the fit (instead of 5-σ) and has an improved median deviation in temperature of 56 K. As expected, the temperatures for the two previously mentioned metal-poor stars, HD 6582 and HD 103095, agree exceptionally well with the temperature–(V − K) relation:

$$\begin{eqnarray} \log T_{\rm EFF} &= &3.9685 \pm 0.0034 + 0.0830 \pm 0.0570 (V-K)\nonumber \\ &-&1.8948 + 0.2874 (V-K)^2+4.0799 \nonumber \\ &\pm& 0.5438 (V-K)^3 - 3.7353 \pm 0.4739 (V-K)^4\nonumber \\ &+&1.5651 \pm 0.1936 (V-K)^5 - 0.2472 \nonumber\\ &\pm& 0.0301 (V-K)^6\quad {\rm for} \; 0.0 < (V-K) < 2.0.\quad\quad \end{eqnarray} \tag{ 8 }$$

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5.1.3. Spectral-type–T_EFF

We next derive a useful (albeit less accurate) relation between spectral type and temperature. We do this by converting the spectral types for each star into a numerical value, following the scheme: A0, A1, A2 ... F0, F1, F2 ... G0, G1, G2, ..., K0 → 0, 1, 2, ..., 10, 11, 12, ..., 20, 21, 22, ..., 30. Again, we omit the two metal-poor stars previously mentioned and fit a fourth-order polynomial to arrive at the relation

$$\begin{eqnarray} T_{\rm EFF}&=&9393.59 \pm 60.45 - 490.25 \pm 28.79 {\rm ST} + 36.44\nonumber \\ && \pm 3.62 {\rm ST}^2 - 1.44 \pm 0.17 {\rm ST}^3 + 0.0208 \pm 0.0027 {\rm ST}^4\nonumber\\ &&{\rm for} \; {\rm A0} < {\rm ST} < {\rm K0} \; {\rm and} \; {\rm [Fe/H]} > -0.75, \end{eqnarray} \tag{ 9 }$$

where in this equation, the variable ST refers to the numerical value for the spectral-type index. The fit for Equation (10) has a median absolute deviation of 90 K and is plotted in Figure 9. In the range from F6 to G5, we also show the fit from the data in Table 7 from van Belle & von Braun (2009; Figure 10 is a close-up view of this range) and our results are consistent with each other.

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In the literature, three surveys exist of nearby stars that include objects that overlap with the majority of stars in this sample: Allende Prieto & Lambert (1999), Holmberg et al. (2007), and Takeda (2007). The number of stars in common with each survey respectively are 37, 34, and 25. We compare our results to these works in this section.

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Allende Prieto & Lambert (1999, hereafter APL99) derive fundamental parameters for the stars in their survey by fitting model evolutionary tracks from Bertelli et al. (1994) to observed (B − V) photometry and absolute V-band magnitude M_V. The Geneva–Copenhagen survey of the solar neighborhood II, done by Holmberg et al. (2007, hereafter GCS07), utilizes the Padova models (Girardi et al. 2000; Salasnich et al. 2000) to derive the stellar parameters based on Strömgren uvby calibrations to T_EFF and M_V. Last, Takeda (2007, hereafter Tak07) use the Yonsei–Yale (Y²) stellar isochrones (Yi et al. 2001; Kim et al. 2002; Yi et al. 2003; Demarque et al. 2004) to fit their spectroscopically determined T_EFF along with the photometrically derived L (from the absolute magnitude and bolometric correction; see Takeda et al. 2005 for details). GCS07 demonstrate that these model isochrones (among others) show minimal differences when compared to each other, also seen in Boyajian et al. (2008). However, we choose to compare all three sources since the target overlap is not identical or complete with respect to our own, as well as the fact that different data sets and photometric indices were used for each group.

In Figure 11, we compare our temperatures to the temperatures of the stars in common in each reference. The most apparent discrepancies appear when comparing our results to those in APL99. For most cases seen here, APL99 overestimate the effective temperature of the star through the entire range of effective temperatures by about 5%, up to 15%. GCS07 and Tak07 are less drastic in comparison, but there is still a tendency of the models to overestimate the temperatures by a couple percent. Figures 12 and 13 plot the fractional offset of the stellar temperature from each method versus (B − V) color index and metallicity. For the hottest/bluest of stars (T_EFF > 6500, B − V < 0.4, a range mostly covered by APL99 only), a positive offset is seen for all but one measurement. Likewise, temperatures of stars with the lowest metallicity are also overestimated in each reference compared to the temperatures presented here, but no trend can be identified with the sparse quantity of data available in this range.

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The stars with the largest offsets in the effective temperatures of GCS07 are HD 103095 (5.5%), HD 146233 (5.8%) and HD 162003 (6.4%). Interestingly enough, these stars also have high discrepancies between their SED diameter and the limb-darkened diameter measured with CHARA. However, stars such as HD 10780 and HD 109358 also have high deviation in the SED diameter versus the limb-darkened diameter measured by CHARA, but their agreement with the temperature from GCS07 is at the ∼1% level. It is interesting to note that the star HD 146233 (18 Sco), which was first identified by Porto de Mello & da Silva (1997) to be a solar twin, is one of these stars with a large offset in effective temperature.

The agreement in temperature with Tak07 is better than both the APL99 and GCS07 surveys, but there is still a preference to higher temperatures than what we measure. The largest outliers in temperature offsets compared to Tak07 are HD 103095 (6.6%) and HD 128167 (5.2%). Comparing these outliers to the GCS07 outliers, there are no two stars in each that show large deviations from the model versus CHARA temperature, with the exception of the very metal poor star, HD 103095. Again, it does not appear that a star's metallicity or color index is related to the deviation in temperatures of each source.

We determine an estimate of the stellar mass and age by fitting our temperatures and luminosities presented in Section 4.2 to the Y² stellar isochrones (Yi et al. 2001; Kim et al. 2002; Yi et al. 2003; Demarque et al. 2004). We are able to do this for most stars observed for this work, but unfortunately, useful results are not obtainable when fitting isochrones to stars with L < 0.75 L_☉, because sensitivity to age in this region is minimal.²⁰ To run the model isochrones, input estimates are required for the abundance of iron [Fe/H] (Table 1) and α elements [α/Fe].²¹ For each star, isochrones were generated in increments of 0.1 Gyr, in the range of 0.1–15 Gyr. We fit the model isochrones in the theoretical temperature–luminosity plane, where the solutions from the model are purely from the theory of stellar structure. This eliminates any dependence of the color table used in transforming the model isochrone temperature to observed photometric colors (Lejeune et al. 1998). Once a best-fit isochrone is established, an age along with the associated mass for this best-fit isochrone is recorded for each star. We show an example of this routine in Figure 14, and the results are listed in Table 12.

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We calculate the errors on the best-fit isochrone mass and age using the 1σ temperature and luminosity maximum offset for the isochrone solution. It is worth noting that the metallicity input for the model isochrones has an impact on the derived age (and in turn also on the derived mass). Lower metallicity isochrones shift down and to the left on these diagrams, so for a star with a true metallicity less than the input value, a higher isochrone age would be found. The opposite is true for stars with higher values of metallicity, where a younger age would result. For our isochrone fits, we adopt fixed values of metallicity measured from a uniform source (in this case Holmberg et al. 2007) that are used in the model input for computations. Thus, relative ages may be correct while absolute ages are highly uncertain.²²

With the linear radii known for all stars in the CHARA sample, we are able to determine the mass of a star using log g estimates published in APL99 and Tak07 using the relation

$$\begin{equation} g_{\ast } = G M_{\ast } / R_{\ast }^{2}, \end{equation} \tag{ 10 }$$

where G is the gravitational constant, M_* is the mass of the star, R_* is the radius of the star, and g_* is the surface gravity of the star. Hereafter, we will refer to the mass derived in this manner as M_g–R.

Masses derived from isochrone fitting (M_Iso) and those obtainable when the surface gravity and radius are known through Equation (10) (M_g–R) are compared in Figure 15 for the stars with gravity measurements published by APL99 and Tak07. There is significant scatter; however, we find that for stars greater than ∼1.3 M_☉, the masses are overpredicted when using gravities from APL99. It is possible that the offset links back to the temperature offsets discussed in the previous section. For instance, if the model temperature used to fit the spectral lines to determine log g values for the stars is offset, this will in turn lead to spurious values of log g. Presumably, these overestimated temperatures will lead to a slightly more massive star because hotter stars on the main sequence are more massive than their cooler counterparts. Sure enough, this trend can be seen in Figure 16 when comparing the deviation in temperatures and masses for these stars. Ages derived are also affected in the sense that the stars will appear younger if their temperature or log g is artificially offset to higher values. In fact, a slight trend may be seen in Figure 17 to support this when comparing our ages to the ages in GCS07 and Tak07 (APL99 did not publish ages), where our ages are typically greater.

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The discussion in Holmberg et al. (2007) also gives several examples of how an offset in effective temperature will, in turn, offset the metallicity and argues that these effects double up when determining the ages of the stars, thereby producing false age–metallicity relations.

For further comparison, we introduce the data set presented in Andersen (1991). Andersen (1991) provides a compilation of data on all non-interacting eclipsing binaries (EB) known at the time—a total of 90 stars, most of which are on the main sequence. Section 4 in Andersen (1991) argues that the motivation for compiling the EB data is to aid in the prediction of single-star properties where masses and radii are unobtainable by direct measurements for a large number of stars. We use these data for eclipsing binaries to compare with our results for single stars in this section.

In Figure 18, we show the relation between (B − V) color index and stellar mass. The mass–luminosity relation is shown in Figure 19. EB data from Andersen (1991) are plotted as well as the masses for stars in this project derived from the Y² isochrones, and masses derived from the combination of the CHARA radii and log g estimates from each source (APL99 or Tak07).

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The Y² masses we found are in excellent agreement with the sample of EB from Andersen (1991) with respect to (B − V) color index and luminosity (Figures 18 and 19). However, comparing the M_g–R data points to the EB sample, we see that these masses are biased to larger masses, forcing them to appear underluminous compared to the EB sample as well as the Y² data points. The effect is again most apparent in the range where the APL99 gravities are used, where the systematics appear for stars with masses greater than ∼1.3 M_☉.