A REVISED EFFECTIVE TEMPERATURE SCALE FOR THE KEPLER INPUT CATALOG

We took griz photometry from the KIC (Brown et al. 2011); photometric uncertainties were taken as 0.01 mag in gri and 0.03 mag in z. Errors were taken from the quadrature sum of uncertainties in the individual filters. JHK_s photometry was taken from the All Sky Data Release of the 2MASS Point Source Catalog (Skrutskie et al. 2006)⁷ and checked against complementary information in the KIC itself.

For our sample we chose long-cadence targets in the KIC; our initial source had 161, 994 candidates. We selected stars with griz photometry detected in all of the bandpasses. This sample was nearly complete in the 2MASS catalog. We excluded a small number of sources with 2MASS photometry quality flags not equal to AAA (N = 3602) and stars with colors outside the range of validity of either the IRFM or SDSS scales (N = 11, 830), leaving us with a main sample of 146, 562 stars. We then further restricted our sample by excluding stars with log g estimates below 3.5 dex in the KIC (N = 19, 663) for a dwarf comparison sample of 126, 899. We illustrate the distribution of stars in the sample in 100 K bins in Figure 1, both in the initial catalog (top panel) and the revised one in this paper (bottom panel). We did not use the giants in our comparison of the dwarf-based temperature scale (Section 3), but we do employ theoretical log g corrections to the photometric temperatures for the purposes of the main catalog (see Sections 3.2 and 4.2).

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We adopted three primary color indices (g  −  r, g  −  i, and g  −  z) as our temperature indicators for the SDSS filter system. A preliminary comparison of colors yielded surprising internal differences and trends as a function of mean T_eff in the relative temperatures inferred from these color indices (see below). Because the A09 color–color trends were calibrated using SDSS photometry of M67, this reflects a zero-point difference between the KIC and SDSS photometry in the color–color plane. It is not likely that this difference is caused by extinction or stellar population differences because all three colors have similar sensitivities to extinction and metallicity. Initially, we suspected problems with the SDSS calibration (see An et al. 2008 for a discussion of zero-point uncertainties). However, the differences seen were outside of the error bounds for the SDSS photometry. For a fraction of the targets (about 2%) the temperatures inferred from different color sources (SDSS versus IRFM from 2MASS colors) are also discordant by more than three standard deviations, in some cases by thousands of degrees in T_eff. We examine both phenomena below.

About 10% of the stars in the Kepler field are covered in the most recent data release (DR8) of the SDSS imaging survey (Aihara et al. 2011). There is an overlap in the two photometric sets at 14 ≲ r ≲ 18. We compare photometry for stars in common in Figure 2. With a 1'' search radius, we found that the median differences (in the sense of the SDSS minus KIC), after rejecting stars with differences greater than 0.2 mag on both sides, are Δg = −0.040, Δr = −0.028, Δi = −0.045, and Δz = −0.042.

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Inspection of Figure 2 shows that these differences are also functions of color. Solid lines are a linear fit to the data after an iterative 3σ rejection. The linear transformation equations are as follows:

$$\begin{eqnarray} &&g_{\rm SDSS} = g_{\rm KIC} +0.0921 (g\,{-}\, r)_{\rm KIC} -0.0985, \end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray} &&r_{\rm SDSS} = r_{\rm KIC} +0.0548 (r\,{-}\, i)_{\rm KIC} -0.0383, \end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray} &&i_{\rm SDSS} = i_{\rm KIC} +0.0696 (r\,{-}\, i)_{\rm KIC} -0.0583, \end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray} &&z_{\rm SDSS} = z_{\rm KIC} +0.1587 (i\,{-}\, z)_{\rm KIC} -0.0597, \end{eqnarray} \tag{ 4 }$$

where the subscripts indicate either SDSS or KIC photometry.

It is possible that the SDSS photometry in the Kepler field has some zero-point shifts with respect to the main SDSS survey database; the SDSS photometry pipeline can fail to work properly if the source density is too high. To check this, we compared the KIC and the DAOPHOT crowded-field photometry (An et al. 2008) in the NGC 6791 field. Although the sample size is smaller, a comparison of DAOPHOT and KIC photometry in the NGC 6791 field yields systematic offsets in the same sense as the field mean in all bandpasses. Given that the cluster fiducial sequence from DAOPHOT photometry matches that from an independent study (Clem et al. 2008) relatively well (An et al. 2008), it is unlikely that the offsets seen in Figure 2 are due to zero-point issues in the SDSS photometry.

We also checked the standard star photometry in Brown et al. (2011), which is originally from the SDSS DR1 photometry for 284 stars outside of the Kepler field. We compared with the SDSS DR8 photometry, but did not find the aforementioned trends outside of those expected from random photometric errors: the mean differences (SDSS minus KIC values) were −0.009, −0.004, +0.004, and +0.012 mag in griz, respectively, with an error in the mean of the order of 0.001 mag. Therefore, revisions of the standard magnitude system (SDSS DR1 versus DR8) do not appear to be the explanation either. We also investigated the possibility of a zero-point difference between the faint and bright stars in the KIC, which had different exposures. However, we found that magnitude offsets with respect to SDSS are similar for both samples, and that the internal dispersions of the KIC temperature estimates are essentially the same. Regardless of the origin, the differences between the SDSS and KIC photometry are present in the overlap sample, and we therefore adjusted the mean photometry to be on the most recent SDSS scale.

Inspection of Figure 2 also reveals another problem in the KIC photometry: A sub-population of stars are much brighter in the KIC than in the SDSS even after photometric zero-point shifts have been accounted for. We do believe that unresolved background stars explain the occasional cases where different colors predict very different temperatures. In Figure 2 there are many data points that have KIC magnitudes brighter than the SDSS ones. We attribute these stars to blended sources in the KIC. The mean FWHM of SDSS images is 14, while that of KIC photometry is 24.

To check on this possibility, we cross-checked 13, 284 stars in common between DR8 of the SDSS and our KIC sample. Three hundred twelve stars had a resolved SDSS source within 24, while 20 have two or more such blended sources; 2.5% of the stars would therefore have resolved blends between the resolution of the two surveys. If we assume that the space density of blends is constant, we can use the density of blends to estimate the fraction present even in the higher resolution SDSS sample. When this effect is accounted for, we would expect 3.8% of the KIC sources to have a blended star within the resolution limit of the KIC. The average such star was 2.85 mag fainter than the KIC target, sufficient to cause a significant anomaly in the inferred color–temperature relationships. A comparable fraction of the catalog is likely to have similar issues. A significant contribution from background stars would in general combine light from stars with different temperatures. As a result, one would expect different color–temperature relations to predict discordant values. We therefore assess the internal consistency of the photometric temperatures as a quality control check in our revised catalog to identify possible blends (Section 2.4).

To identify blended sources in the KIC, we further performed a test using the separation index (Stetson et al. 2003), which is defined as the logarithmic ratio of the surface brightness of a star to the summed brightness from all neighboring stars (see also An et al. 2008). However, we found that applying the separation index to the KIC does not necessarily provide unique information for assessing the effects of the source blending.

We adopted stellar isochrones in A09 for the estimation of photometric temperatures. Interior models were computed using YREC, and theoretical color–T_eff relations were derived from the MARCS stellar atmospheres model: see A09 and An et al. (2009a) for details. These model colors were then calibrated using observed M67 sequences as in our earlier work in the Johnson-Cousins system (Pinsonneault et al. 2003, 2004; An et al. 2007a, 2007b). The empirical color corrections in ugriz were defined using M67 at its solar metallicity, and a linear ramp in [Fe/H] was adopted so that the color corrections become zero at or below [Fe/H] <−0.8. A detailed test on the empirical color corrections will be presented elsewhere (D. An et al. 2012, in preparation).

As a base case of this work, we adopted the mean metallicity recorded in the KIC of [Fe/H] =−0.2. This metallicity is comparable to, or slightly below, that in the solar neighborhood. For example, the Geneva-Copenhagen Survey (Nordstrom et al. 2004) has a mean [Fe/H] of −0.14 dex with a dispersion of 0.19 dex; a recent revision by Casagrande et al. (2011) raises the mean [Fe/H] to −0.07 dex, which is a fair reflection of the systematic uncertainties. The bulk of the KIC dwarfs are about 100 pc above the galactic plane, and thus would be expected to have somewhat lower metallicity. In the following analysis, we assumed [Fe/H] =−0.2 when using griz–T_eff or IRFM color–T_eff relationships, unless otherwise stated.

Table 1 shows our base model isochrone at [Fe/H] =−0.2 and the age of 1 Gyr. All colors are color calibrated as described above. Note that the isochrone calibration is defined for the main sequence only; the relevant corrections for the lower gravities of evolved stars are described separately in Section 3.2. The SDSS photometry did not cover the main-sequence turnoff region of M67 because of the brightness limit in the SDSS imaging survey at r ∼ 14 mag. As a result, the M67-based griz color calibration is strictly valid at 4000 K ⩽ T_eff ⩽ 6000 K (see Figure 17 in A09).

The choice of 1 Gyr age in our base model isochrone has a negligible effect on the color–T_eff relations. The difference between 1 Gyr and 12 Gyr isochrones is only less than 5 K near main-sequence turnoff. However, younger age of the models enables the determination of photometric T_eff over a wider range of colors at the hot-T_eff end.

From Table 1 we derived polynomial color–T_eff relations of our base model for convenience of use. The following relationship was used over the temperature range 4080 K ⩽T_eff(YREC) < 7000 K:

$$\begin{equation} 5040/T_{\rm eff} = a_0+a_1 x+a_2 x^2+a_3 x^3+a_4 x^4+a_5 x^5, \end{equation} \tag{ 5 }$$

where x represents g  −  r, g  −  i, or g  −  z, and a₀–a₅ are coefficients for each color index as listed in Table 2. Difference in T_eff inferred from these polynomial equations compared to those found in Table 1 from interpolation in the full tables is at or below the 6 K level.

In Table 3 we provide the metallicity sensitivity of the color–T_eff relations in the model isochrones at several [Fe/H]. To generate this table, we compared 1 Gyr old isochrones at individual [Fe/H] with our fiducial model (Table 1) at [Fe/H] =−0.2 for each color index, and estimated the T_eff difference at a given color (individual models minus the fiducial isochrone). We include the sensitivity to metallicity predicted by atmospheres models, but do not include an additional empirical correction below [Fe/H] =−0.8 because the cluster data did not require one. The T_eff at a fixed color generally becomes cooler at a lower [Fe/H]. We use the metallicity corrections in the comparisons with spectroscopic T_eff where we have reliable [Fe/H] measurements (see Section 3.4), but do not apply corrections to the KIC sample (see Sections 3.1 and 4.2).

The stellar parameters for the KIC were generated using a Bayesian method (see Brown et al. 2011 for a discussion). We adopt a less ambitious approach focused on KIC stars identified as dwarfs. The three key assumptions in our work are that we define T_eff at a reference [Fe/H] and the model log g (Table 1), and that we adopt the map-based E(B  −  V) in the KIC as a prior. Within this framework we can then derive independent temperature estimates from the griz photometry and infer the random T_eff errors. Uncertainties in the extinction, the impact on the colors of unresolved binaries, and population (metallicity and log g) differences can then be treated as error sources. In the latter case, we can compute correction terms to be used if there is an independent method of measurement. This approach is not the same as the one that we have employed in earlier studies, so a brief justification is in order.

The traditional approach to photometric parameter estimation is to take advantage of the fact that different filter combinations respond to changes in metallicity and extinction. If one has the proper template metallicity and extinction, for example, the answers from the various colors will agree within photometric errors; if not, the pattern of differences can be used to solve for them (see An et al. 2007a, 2007b).

The particular problem for the KIC is that the available color combinations in griz are rather insensitive to both over the narrow metallicity range and the modest mean extinctions (0 ≲ E(B  −  V) ≲ 0.2) in the field (see An et al. 2009a for a discussion of griz-based estimates). In other words, all the color combinations in griz produce similar metallicity sensitivities of color–T_eff relations. Therefore, even though the absolute change of photometric T_eff can be significant by the error in the adopted metallicity, it is difficult to infer photometric metallicities based on the available filter combinations in griz alone.

The temperature estimates in Lejeune et al. (1997, 1998), which were used as the basis color–T_eff relations in our prior color calibration in the Johnson-Cousins system (An et al. 2007a, 2007b), are insensitive to log g near the main sequence, and the IRFM scale in C10 does not include an explicit log g dependence for the temperatures. As a result, we believe that the most fruitful approach is to define a benchmark temperature estimate. If additional color information or spectroscopic [Fe/H] data become available, the relevant corrections can be applied, and we present methods below to do so (Section 3).

The KIC gravities for cool stars are precise enough to separate dwarfs (KIC log g > 3.5) from giants (KIC log g ⩽ 3.5) and to be used as a basis for corrections to the temperatures for giant stars (Sections 3.2 and 4.2). The KIC metallicities are more problematic, and we do not use them for temperature corrections. Instead the metallicity sensitivity is included as an error source in our effective temperature estimates.

We adopted the map-based KIC catalog extinction estimates (A_V) and the Cardelli et al. (1989) standard extinction curves with A_V = 3.1 E(B  −  V). Extinction coefficients in griz were derived in A09: A_g = 1.196 A_V, A_r = 0.874 A_V, A_i = 0.672 A_V, and A_z = 0.488 A_V. We further took A_J = 0.282 A_V, A_H = 0.180 A_V, $A_{K_s}=0.117\, A_V$, and $A_{V_T}=1.050\, A_V$, where V_T represents the Tycho V passband (An et al. 2007a).

For a given extinction-corrected set of griz magnitudes, we searched the best-fitting stellar template in the model isochrone for each star in the KIC. The mean T_eff was obtained by simultaneously fitting the models in griz, assuming 0.01 mag error in gri and 0.03 mag error in z. We also estimated T_eff using the same model isochrone, but based on data from each of our fundamental color indices (g  −  r, g  −  i, and g  −  z), which is simply a photometric T_eff estimation from a single color–T_eff relation. Its purpose is to readily identify and quantify the internal consistency of our primary temperature determination from the multi-color–T_eff space.

In Figure 3 we plot the internal dispersion and the mean trends of T_eff from a given color index with respect to the average error-weighted temperature from griz for all of the dwarfs in our sample. The top panel shows the case of the original KIC data, and the bottom panel shows the one for the corrected KIC photometry. The magnitude corrections described in Section 2.2 were motivated by concordance between SDSS and the KIC. Nevertheless, the results when using the recalibrated KIC photometry as temperature indicators were extremely encouraging.

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Although the internal agreement is not complete, the remaining differences in the bottom panel of Figure 3 are comparable to the zero-point uncertainties discussed in An et al. (2008). We view this as strong supporting evidence for the physical reality of the magnitude corrections illustrated in Figure 2. We therefore recommend that the zero points of the KIC photometry be modified according to Equations (1)–(4). In the remainder of the paper, we use magnitudes and colors adjusted using these equations.

We have three native temperature scales to compare: the one in the KIC, our isochrone-based scale from griz (hereafter SDSS or griz-based scale unless otherwise stated), and one from the (J  −  K_s)-based IRFM. Below we compare the mean differences between them and compare the dispersions to those expected from random error sources alone. We find an offset between the KIC and the other two scales. The IRFM and SDSS scales are closer, but some systematic differences between them are also identified. In this section, we examine various effects that could be responsible for these differences, and finish with an overall evaluation of the error budget.

We computed IRFM and SDSS T_eff estimates assuming [Fe/H] =−0.2. In terms of the temperature zero point, adopting this metallicity led to mean shifts of +20 K in J  −  K_s, and −40 K in the griz-based T_eff estimate, relative to those which would have been obtained with solar abundance. In other words, changes in the adopted mean metallicity would cause zero-point shifts of ∼60 K in the overall T_eff scale comparison. On the other hand, a scatter around the mean metallicity in the Kepler field is another source of error that would make the observed T_eff comparison broader. We discuss this in Section 3.6 along with other sources of uncertainties.

In the comparisons below we repeatedly clipped the samples, rejecting stars with temperature estimates more than three standard deviations from the mean, until we achieved convergence. This typically involved excluding about 1% of the sample. Such stars represent cases where the extinction corrections break down or where the relative colors differ drastically from those expected for single unblended stars.

Random errors were taken from the photometric errors alone and yield a minimum error in temperature. For the SDSS colors we also computed the internal dispersion in the three temperature estimates from g  −  r, g  −  i, and g  −  z, and used the larger of either this dispersion or the one induced by photometric errors as a random uncertainty. Median random errors for the SDSS and IRFM temperatures were 40 K and 171 K, respectively. These estimates are consistent with expectations from the observed dispersions of the colors (see Figure 17). We then compared stars at fixed KIC temperature and computed the average T_eff difference between those inferred from the IRFM, those inferred from griz, and the scale in the KIC itself. For a limited subset of stars, we also had Tycho photometry and computed temperatures from V_T  −  K_s. This sample is small, so we used it as a secondary temperature diagnostic.

In Figures 4–5 we illustrate the differences between KIC and the IRFM and SDSS, respectively. For the IRFM scale in Figure 4, we compare T_eff from J  −  K_s. In Figure 5 we compare the mean SDSS temperatures inferred from griz to that in the KIC. In both cases we see a significant zero-point shift, indicating a discrepancy between the fundamental effective temperature scale and that adopted by the KIC.

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The T_eff from the IRFM and the SDSS from individual color indices (g  −  r, g  −  i, and g  −  z) are compared in Figure 6. The IRFM scale for the Tycho V_T and 2MASS K_s is used in the bottom right panel. The central result (that the KIC scale is too cool) is robust and can also be seen in comparisons with high-resolution spectroscopic temperature estimates (see Section 3.4 below). In Section 4.2 we provide quantitative tabular information on the statistical properties of the sample.

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The two fundamental scales (IRFM and SDSS) are close, but not identical, for cooler stars; they deviate from one another and the KIC above 6000 K (on the SDSS scale). As discussed in Section 3.6 below, the total internal dispersion in the griz temperature estimates is also consistently larger for cool stars than that expected from random photometric uncertainties alone, and there are modest but real offsets between the two fundamental scales even for cool stars. We therefore need to understand the origin of these differences and to quantify the random and systematic uncertainties in our temperature estimates.

Open clusters provide a good controlled environment for testing the concordance of the SDSS and IRFM scales. The SDSS scale was developed to be consistent with Johnson-Cousins-based temperature calibrations in open clusters, so a comparison of the An et al. and IRFM Johnson-Cousins systems in clusters will permit us to verify their underlying agreement. As we show below, the two scales are close for cool stars when B  −  V, V  −  I_C, or V  −  K_s indices are employed in the temperature determinations, but exhibit modest but real systematics for the hotter stars. The IRFM relation in J  −  K_s, on the other hand, is found to have a systematic difference from those of these optical-2MASS indices. For the reasons discussed in the following section, we therefore adopt a correction to our SDSS temperatures for hot stars, making the two photometric systems consistent.

We can also check our methodology against spectroscopic temperature estimates and need to consider uncertainties from extinction, binary companions, and metallicity. We therefore begin by defining an extension of our method to giants, which can be checked against spectroscopy. We then look at open cluster tests, spectroscopic tests, binary effects, and the overall error budget.

Our YREC T_eff estimates are based on calibrated isochrones (Table 1), which do not include evolved stars. About 14% of the KIC sample are giants and subgiants with log g ⩽ 3.5 as estimated in the KIC, so a reliable method for assigning effective temperatures to such stars is highly desirable. Fortunately, this is feasible because the color–temperature relations for the bulk of the long-cadence targets are not strong functions of surface gravity. For the purposes of the catalog we therefore supplement the fundamental dwarf scale with theoretical corrections for the effect of surface gravity on the colors.

Theoretical model atmospheres can be used to quantify the log g dependence of the color–temperature relations by comparing the spectral energy distributions of dwarfs and giants. Figure 7 shows color–temperature relations along a 1 Gyr solar-abundance isochrone for g  −  r, g  −  i, g  −  z, and J  −  K_s for illustrative purposes. The model isochrone was taken from the web interface of the Padova isochrone database (Girardi et al. 2002; Marigo et al. 2008).⁸ As seen in Figure 7 the model color–T_eff relations are moderately dependent on log g, and illustrate that our photometric T_eff needs to be adjusted for giants.

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We corrected for the difference in log g by taking theoretical log g sensitivities in griz colors from the ATLAS9 model atmosphere (Castelli & Kurucz 2004). The choice of these models seems internally inconsistent with our basis model isochrone with MARCS-based colors. Nevertheless, we adopted the ATLAS9 log g–color relations, primarily because our cluster-based empirical calibration of the color–T_eff relations has not been performed for subgiant and giant branches due to significant uncertainties in the underlying stellar interior models at these evolved stages. Therefore, it is just a matter of choice to adopt the ATLAS9 color tables instead of that of MARCS. Since we generated MARCS color tables in An et al. (2009b) with a specific set of model parameters for dwarfs (log g ⩾ 3.5), we simply opted to take the ATLAS9 colors, and estimate a relative sensitivity of theoretical log g–color relations.

We convolved synthetic spectra with the SDSS griz filter response curves⁹ and integrated flux with weights given by photon counts (Girardi et al. 2002). Magnitudes were then put onto the AB magnitude system using a flat 3631 Jy spectrum (Oke & Gunn 1983). We created a table with synthetic colors from log g = 0.0 to 5.0 dex with a 0.5 dex increment, and from 4000 K to 6000 K with a 250 K increment at [M/H] = −1.0, −0.5, +0.0, and +0.2. Because YREC T_eff values were estimated at the fiducial metallicity, [Fe/H] =−0.2, we interpolated the color table to obtain synthetic colors at this metallicity. Note that Castelli & Kurucz (2004) adopted the solar mixture of Grevesse & Sauval (1998), as in our YREC isochrone models (A09), so we assumed that [M/H] in Castelli & Kurucz (2004) is the same as the [Fe/H] value.

Figure 8 shows the correction factors in T_eff as computed from synthetic colors as a function of colors in g  −  r, g  −  i, and g  −  z. We used our base isochrone to compute ΔT_eff at Δlog g = 0.5, 1.0, 1.5, 2.0, 2.5, and 3.0 dex, where Δlog g represents the difference between YREC log g and the log g in the KIC. The sense of ΔT_eff is that giants with lower log g than the base model generally tend to have lower T_eff than main-sequence dwarfs in the color range considered in this work.

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In Figure 8 we used a linear ramp over 4800 K < T_eff < 5800 K (0.42 < g  −  r < 0.82), so that the theoretical ΔT_eff becomes zero at T_eff > 5800 K. Otherwise the amplitude of theoretical T_eff variations on the blue side (g  −  r ≲ 0.6) would be similar to that of the red colors. Although this is not strictly true if the Δlog g is large for blue stars, those stars are rare because stars on the giant branch (with the largest Δlog g) have g  −  r ≳ 0.5 at near solar metallicity. The correction factors are tabulated in Table 4. If one wishes to adopt a different log g scale than in our base isochrone, tabulated ΔT_eff factors can be used to correct for the log g difference. More importantly, Table 4 can be used to infer T_eff for giants, since our base isochrone (Table 1) covers stellar parameters for main-sequence dwarfs only.

The biggest ΔT_eff in Figure 8 is ∼100 K. However, the effects of the log g corrections are moderate in the KIC. If we take the mean ΔT_eff correction in g  −  r, g  −  i, and g  −  z, the mean difference in T_eff between KIC and YREC decreases from 190 K to 166 K for stars with log g ⩽ 3.5. The log g corrections are insensitive to metallicity. The ΔT_eff in Figure 8 was computed at [Fe/H] =−0.2, but these corrections are within 10 K away from those computed at [Fe/H] =−0.5 (∼1σ lower bound for the KIC sample) when Δlog g = 1.

The statistical properties of the SDSS giant temperatures are compared with spectroscopic data in Section 3.4 and with the KIC in Section 4.

The IRFM technique provides global color–metallicity–T_eff correlations using field samples, while clusters give snapshots at fixed composition, which define color–T_eff trends more precisely. Deviations from color to color yield the internal systematic within the system, as the color–temperature relationships defined in An et al. (2007b) are empirical descriptions of actual cluster data. The A09 SDSS system, by construction, agrees with the An et al. (2007b) Johnson-Cousins system; but we can check the concordance between the two scales within the open cluster system.

We have two basic results from this comparison. First, (J  −  K_s)-based temperatures from the IRFM are different from other IRFM thermometers. J  −  K_s is also the only IRFM diagnostic available for the bulk of the KIC sample. When accounting for the offset in J  −  K_s relative to other IRFM indicators, the underlying IRFM system and the SDSS system are in excellent agreement for stars below 6000 K. Second, there is a systematic offset between the IRFM and SDSS scales above 6000 K. We therefore correct the high end temperature estimates for the SDSS to put them on the IRFM scale, which yields an internally consistent set of photometric temperature estimates.

Figures 9–11 show how the IRFM T_eff determinations are internally consistent in the Johnson-Cousins–2MASS system in B  −  V, V  −  I_C, V  −  K_s, and J  −  K_s using stars in four well-studied clusters: The Hyades (red circles in Figure 9), Praesepe (blue triangles in Figure 9), the Pleiades (Figure 10), and M67 (Figure 11). All of the stars shown in these figures are likely single-star members of each cluster after excluding known (unresolved) binaries. In Figure 11, we show results based on the two independent sets of M67 photometry from Montgomery et al. (1993, blue triangles) and Sandquist (2004, red circles). The compilation and individual sources of the cluster photometry can be found in An et al. (2007b).

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To construct Figures 9–11 we corrected observed magnitudes for extinction using E(V  −  I_C)/E(B  −  V) = 1.26, E(V  −  K_s)/E(B  −  V) = 2.82, and E(J  −  K_s)/E(B  −  V) = 0.53 (An et al. 2007a). Foreground reddening values of E(B  −  V) = 0.000 ± 0.002, 0.006 ± 0.002, 0.032 ± 0.003, and 0.041 ± 0.004 mag were used for the Hyades, Praesepe, the Pleiades, and M67, respectively (An et al. 2007b). The IRFM T_eff equations in C10 include metallicity terms, and we adopted [Fe/H] =+0.13 ± 0.01, +0.14 ± 0.02, +0.04 ± 0.02, and +0.00 ± 0.01 dex for the Hyades, Praesepe, the Pleiades, and M67, respectively, based on high-resolution spectroscopic abundance analysis (see references in An et al. 2007b). Only the (B  −  V)-based estimates are significantly impacted by metallicity corrections, and the relative abundance differences in these well-studied open clusters are unlikely to be substantial enough to affect our results.

The ±1σ error bars in Figures 9–11 are those propagated from the photometric errors only. Mean differences in the IRFM T_eff and the errors in the mean are provided in Table 5. Global differences are shown for stars at 4000 K < T_eff ⩽ 7400 K, and those cooler and hotter than 6000 K are shown in the table. The σ_sys represents a total systematic error in this comparison from the reddening and metallicity errors (summed in quadrature); however, systematic errors are less important than random errors because of the precise E(B  −  V) and [Fe/H] estimates of these well-studied clusters.

The low-mass stars in the Pleiades are known to have anomalously blue colors related to stellar activity in these heavily spotted, rapidly rotating, young stars (Stauffer et al. 2003). The temperature anomaly for B  −  V at T_eff ≲ 5000 K in Figure 10, which is ∼200 K larger than that for more massive stars, reflects this known effect and therefore is not a proper test of internal consistency in old field stars (such as those in the KIC). The M67 data may also be inappropriate for the test of the IRFM internal consistency, but with a different reason. Two independent photometry sets lead to a different conclusion: Montgomery et al. (1993) photometry shows internally less consistent IRFM T_eff for M67 stars than Sandquist (2004). A similar argument was made in An et al. (2007b), based on the differential metallicity sensitivities of stellar isochrones in different color indices (see Figure 11 in the above paper); see also VandenBerg et al. (2010) for an independent confirmation of the systematic zero-point issue with the Montgomery et al. (1993) photometry.

Our cluster tests based on the Hyades and Praesepe demonstrate the internal consistency of the C10 color–T_eff relations in B  −  V, V  −  I_C, and V  −  K_s. The mean differences in T_eff among these color indices are typically few tens of degrees for both hot and cool stars (Table 5). Some of these mean differences could be systematic in nature, but they are generally consistent with the scatter in the C10 IRFM calibrations. However, the (J  −  K_s)–T_eff relation tends to produce hotter T_eff than those from other color indices for these cluster stars (see the bottom panel in Figure 9). The mean differences between T_eff(V  −  I_C) and T_eff(J  −  K_s) are 95 K and 44 K for the Hyades and Praesepe, respectively. There is also a hint of the downturn in the comparison for the hot stars in these clusters, where (J  −  K_s)–T_eff produces cooler temperatures than (V  −  I_C)–T_eff relation. The ∼100 K offset between the hot and the cool stars roughly defines the size of the systematic error in the IRFM technique of C10 in J  −  K_s.

The Pleiades stars show a weaker systematic T_eff trend for the cool and the hot stars than the Hyades and Praesepe. In spite of this good agreement, we caution that this could be a lucky coincidence because the Pleiades low-mass stars probably have slight near-IR excesses in K_s (Stauffer et al. 2003). The main-sequence turnoff of M67 is relatively cool, so the difference is only suggestive.

Figure 12 shows comparisons between the IRFM and YREC T_eff estimates. The left panel shows the comparisons for the Hyades and Praesepe stars, while the right panel shows those for the Pleiades and M67 stars. The IRFM T_eff on top panels was computed based on the (V  −  I_C)–T_eff relation in C10, just as those used for a principal T_eff estimator in the above comparisons (Figures 9–11). The YREC T_eff was estimated using An et al. (2007b) isochrones, which have the same underlying set of interior models as those used in the current analysis. The model T_eff was computed at a constant M_V of individual stars, assuming (m  −  M)₀ = 3.33 ± 0.01, 6.33 ± 0.04, 5.63 ± 0.02, and 9.61 ± 0.03 mag for the distance moduli of the Hyades (550 Myr), Praesepe (550 Myr), the Pleiades (100 Myr), and M67 (3.5 Gyr), respectively (see references in An et al. 2007b).

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Table 5 lists weighted mean differences between YREC and IRFM T_eff. The mean difference between the (V  −  I_C)-based IRFM and the luminosity-based YREC T_eff for cool stars (T_eff < 6000 K) is less than 20 K, but the differences rise above 6000 K to the 50 K level. The difference between the (J  −  K_s)-based IRFM and M_V-based YREC T_eff shows different offsets for the cool and hot stars; this trend is consistent with the above comparison between (J  −  K_s)-based IRFM and other IRFM determination.

The bottom panel in Figure 12 shows comparisons between the YREC T_eff and the average IRFM T_eff from B  −  V, V  −  I_C, and V  −  K_s. Our results using J  −  K_s as a thermometer are consistent with our earlier finding in Section 3.1 that C10 (J  −  K_s)-based T_eff values are systematically cooler than those from the griz-based YREC models for hot stars (above about 6000 K). The (J  −  K_s)-based T_eff differ both from other IRFM diagnostics and the values inferred from SDSS colors for cooler stars, while the mean values inferred from the IRFM are close to SDSS for the cooler stars.

We therefore conclude that the cool star temperature scales are consistent, while there is evidence for a systematic departure at the hot end. A similar pattern emerges when we compare with spectroscopy, as discussed in the next section. Caution is therefore required in assigning errors for stars with formal temperature estimates above 6000 K.

Systematic T_eff differences are shown in Figure 13. The red line represents the difference with the (J  −  K_s)-based IRFM T_eff for the open cluster sample (Hyades and Praesepe), while the orange line shows that with respect to the mean IRFM values from B  −  V, V  −  I_C, and V  −  K_s. Error bars indicate ±1σ error in the mean difference. The difference between the average IRFM scale and the SDSS scale in the clusters is less than 25 K on average from 4000 to 6000 K, which we take as a conservative systematic temperature uncertainty in that domain. The differences are moderately larger for the IRFM J  −  K_s temperature alone, but that diagnostic is also different from other IRFM thermometers for cool stars.

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The differences in the hot cluster stars reflect actual differences in the calibrations, not issues peculiar to the photometry, extinction, or blending. We therefore attribute the comparable differences seen in the KIC stars (gray band) as caused by calibration issues in J  −  K_s rather than as a reflection of systematics between the IRFM and SDSS systems. Furthermore, the SDSS calibration was based on M67 data, where the hotter turnoff stars (T_eff > 6000 K) were saturated. As a result, we believe that an adjustment closer to the IRFM scale is better justified.

A simple correction term, of the form

$$\begin{eqnarray} T_{\rm eff,SDSS} < 6000\, {\rm K} &:& T_{\rm eff,corr} = T_{\rm eff}{\rm (SDSS)}, \end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray} && 6000\, {\rm K} \le T_{\rm eff,SDSS} < 7000\, {\rm K}: T_{\rm eff,corr} \nonumber\\ &&\quad = 0.8\ (T_{\rm eff}{\rm (SDSS)} - 6000\, {\rm K}) + 6000\, {\rm K}, \end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray} T_{\rm eff,SDSS} \ge 7000\, {\rm K} &:& T_{\rm eff,corr} = T_{\rm eff}{\rm (SDSS)} - 200\, {\rm K} \end{eqnarray} \tag{ 8 }$$

brings the two scales into close agreement across their mutual range of validity. This empirical correction is indicated by the black dashed line in Figure 13. Below we find offsets similar in magnitude and opposite in sign between the IRFM and spectroscopic temperatures for hotter stars. Although this does not necessarily indicate problems with the fundamental scales, it does imply that systematic temperature scale differences are important for these stars.

Spectroscopy provides a powerful external check on the precision of photometric temperature estimates. Spectroscopic temperatures are independent of extinction, and can be less sensitive to unresolved binary companions and crowding. In this section we therefore compare the photometric and spectroscopic temperature estimates for two well-studied samples in the Kepler fields. Bruntt et al. (2012, hereafter B12) reported results for 93 stars with asteroseismic data, including 83 stars in our sample. Molenda-Żakowicz et al. (2011, hereafter MZ11) reported results for 78 stars, including 45 targets in common with our sample. The MZ11 data for cool stars are mostly subgiants and giants, while the bulk of the dwarf sample is hotter than 6000 K. The B12 sample is similarly distributed, with the transition from the cool evolved to the hot unevolved sample occurring at 5500 K.

All comparisons below are for the corrected photometric scale, adjusted for concordance with the IRFM at the hot end. We compare spectroscopic methods both with the fixed-metallicity ([Fe/H] =−0.2) temperatures in the catalog and the refined temperature estimates made possible with the addition of metallicity information and theoretical metallicity corrections. We excluded outliers in the following statistical comparisons using a 3σ outlier rejection.

As demonstrated below, we find that the two spectroscopic samples have different zero points with respect to both the SDSS and KIC samples, indicating the importance of systematic errors in such comparisons. The photometric scale for the cool dwarfs and giants are in good agreement with the B12 scale, while both are offset relative to MZ11. The situation is different for hot dwarfs. The IRFM scale was cooler than the uncorrected SDSS scale. The spectroscopic samples are cooler than both. We interpret this as evidence of additional systematic uncertainties for the F stars and discuss possible causes.

The stellar parameters for the MZ11 sample were derived using the Molenda-Żakowicz et al. (2007) template approach. The spectra were compared with a library of reference stars. The surface gravity, effective temperature, and metallicity were derived from a weighted average of the five closest spectral matches in the catalog. B12 used asteroseismic surface gravities and derived effective temperatures from traditional Boltzmann–Saha consistency arguments.

We compare the spectroscopic and photometric temperature estimates in Figure 14. The top, middle, and bottom panels compare spectroscopic temperatures to those of the KIC, IRFM (J  −  K_s), and SDSS, respectively. Left panels show comparisons for dwarfs (KIC log g > 3.5), while the right panels show those for giants (KIC log g ⩽ 3.5). Filled circles are the B12 data, while open circles are the MZ11 data. In total, 83 out of 93 sample stars in B12 were used in this comparison; the remaining 10 stars do not have griz photometry in all passbands, so were not included in our KIC subsample. For the same reason, we initially included 45 spectroscopic targets from MZ11, but later excluded eight more stars with g  −  r < 0.1 or g  −  r > 1.0. Triangles in the bottom two panels represent stars flagged as having internally inconsistent effective temperature estimates (Section 4.2.4). Error bars show the expected random errors, with a 70 K error adopted in the temperature for the individual B12 sample stars.

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In the above comparisons, we corrected the IRFM temperature estimates for the spectroscopic metallicity measurement of each sample, although the T_eff corrections in C10 were negligible (ΔT_eff ≈ 18 K) in J  −  K_s. We also used individual stellar isochrones at each spectroscopic metallicity to estimate SDSS T_eff from griz, assuming a constant age of 1 Gyr at all metallicity bins. However, the net effect of these corrections was small (ΔT_eff ≈ 25 K), because griz–T_eff relations are relatively insensitive to metallicity and the mean metallicities of the spectroscopic samples are close to our fiducial value (〈[Fe/H]〉 = −0.07  ±  0.02 and −0.11  ±  0.03 for the B12 and MZ11 samples, respectively). The SDSS T_eff values for giants were corrected for the log g difference from the dwarf temperature scale as described in Section 3.2.

Both spectroscopic samples for dwarfs are systematically hotter than the KIC (top left panel in Figure 14). The weighted average difference between the B12 sample and the KIC, in the sense of the KIC minus spectroscopic values, is −170 K with a dispersion of 116 K, after a 3σ outlier rejection. The MZ11 sample is closer to the KIC, with a −82 K mean difference and a dispersion of 172 K. This difference of 88 K is a reflection of the systematic errors in the spectroscopic temperature scales. In the above comparisons, we did not include stars with inconsistent SDSS temperature measurements (triangles in Figure 14).

The weighted average difference between the B12 sample and the SDSS (in the sense SDSS–Spec) for dwarfs is 85 K with a 95 K dispersion, after excluding those flagged as having discrepant T_eff(YREC) values. If the metallicity corrections to the SDSS values were not taken into account (i.e., based on models at [Fe/H] =−0.2), the mean difference becomes 73 K, but the dispersion increases to 111 K.

However, there is a strong temperature dependence in the offset. Below 6000 K the mean difference is 50 K with a dispersion of 47 K. For the hotter stars the mean difference is 101 K with a dispersion of 118 K. The blue line in Figure 13 shows a moving averaged difference between the B12 spectroscopic values and SDSS T_eff without the hot-end T_eff corrections (Equations (6)–(8)).

Although the size of the dwarf sample in MZ11 is small, it is found that the effective temperatures are systematically cooler than the SDSS values, with a weighted mean offset of 152 K (SDSS–Spec) and a dispersion of 175 K. The difference is temperature dependent, being 53 K for the stars below 6000 K and 178 K above it. These differences are 3 K and 77 K larger, respectively, than the results from the B12 sample. The temperature differences between photometry and spectroscopy are therefore smaller than the differences between the spectroscopic measurements and the KIC, while there is a real difference at the hot end even when systematic differences between the two spectroscopic samples are accounted for.

The B12 sample includes only two giants (KIC log g ⩽ 3.5), but their spectroscopic temperatures are consistent with both IRFM and SDSS temperatures (see the middle and bottom right panels in Figure 14). On the other hand, the MZ11 sample shows a large offset from IRFM (ΔT_eff = 245 K) and SDSS (ΔT_eff = 206 K), while the KIC and the MZ11 values agree with each other (ΔT_eff = 9 K).

The cool MZ11 stars are mostly subgiants and giants, while the B12 cool sample includes a large dwarf population between 5000 K and 6000 K. The difference between the two cool end results—good agreement with B12 for cool dwarfs, but not with MZ11—is real. This could reflect systematic differences between the dwarf and corrected giant results for the SDSS or the templates adopted by MZ11 for the evolved and unevolved stars. The scatter between the MZ11 results and the photometric ones is substantially larger than that between B12 and photometric temperature estimates. It would be worth investigating the zero point of the templates used in the former method, as well as the random errors, in light of the results reported here.

In the section above we have focused on differences between the scales; it is fair to ask how both might compare to the true temperatures. The photometric scale is at heart simply an empirical relationship between color and the definition of the effective temperature itself (L = 4πR²σT⁴_eff), and therefore the scale itself should be sound where the photometric relations are well defined. However, the photometric methods can fail if there is more than one contributor to the photometry, or if the reddening is incorrectly measured. Spectroscopic temperatures measure physical conditions in the atmosphere, and are only indirectly tied to the fundamental flux per unit area, which defines the effective temperature. There are also systematic uncertainties between different methods for inferring effective temperatures, for example, fitting the wings of strong lines, or the use of Boltzmann–Saha solutions based on ionization and excitation balance. Finally, both photometric and spectroscopic estimates are only as good as their assumptions; stars with large surface temperature differences will be poorly modeled by both methods.

Our primary conclusion is therefore that the various dwarf temperature methods, spectroscopic and photometric, are in good agreement for the cooler stars. Systematic effects are at or below the 50 K level. The hotter stars in the sample have real systematic differences between spectroscopic and photometric temperatures, and similar discrepancies are also present between the photometric methods themselves. This is further evidence that work is needed to tie down more precisely the temperature scale above 6000 K, and that larger systematic errors should be assigned in this domain until such an analysis is performed. We have less data for the giants, but there does appear to be a real difference between the photometric results and the temperatures inferred for the MZ11 sample.

Unresolved binaries in the sample could bias a color-based T_eff estimate. Unless the mass ratio of the primary and secondary components in the binary system is close to either unity (twins) or zero (negligible contributions from the secondary), composite colors of the system are redder than those from the primaries alone, leading toward systematically lower T_eff. It is difficult to directly flag potential binaries given the filters available to us, and as a result we do not include star-by-star corrections in the table. However, such a systematic bias will be important when evaluating the bulk properties of the KIC sample. In this section, we therefore estimate the size of the bias due to unresolved binaries in the KIC, and provide statistical corrections for the effect of unresolved binary companions on average effective temperature estimates.

Binary contamination effects on the color–T_eff relations were derived by performing artificial star tests. We used a 1 Gyr old Padova models at solar abundance (Girardi et al. 2004). These models include stellar masses down to 0.15 M_☉, allowing us to include low-mass systems outside the formal range of the SDSS color calibration. The absolute color–T_eff relations in these models are not exactly the same as in our base calibration, and the adoption of a solar-metallicity isochrone is not strictly self-consistent with our application of the base model at [Fe/H] =−0.2. However, our main purpose is to evaluate the relative temperature errors induced by companions, and the effects of these offsets are presumably small.

We assumed a 50% binary fraction with 10, 000 single stars and 10, 000 binary systems. Primary masses were randomly drawn from a Salpeter mass function, while we explored three different choices for the relative masses of the secondaries: Salpeter, flat, and one drawn from the open cluster M35 (Barrado y Navascués et al. 2001). A flat mass function is expected for short-period binaries, which will be a minority of the sample; this is thus a limiting case. In the artificial star simulations, we derived empirical color–color sequences in g  −  i, g  −  z, and J  −  K_s with g  −  r as the principal color index. We simulated photometric errors by injecting dispersions of 0.01 mag in gri, 0.03 mag in z, 0.024 mag in J, and 0.028 mag in K_s. These 2MASS errors are median values of the actual photometric errors in the KIC sample.

The result of these binary simulations is presented in Figure 15, which shows the mean deviations in g  −  i, g  −  z, and J  −  K_s from those with primaries alone. For Figure 15 we fitted a Gaussian for each g  −  r bin to estimate the mean color offset and the uncertainty as shown by circles and error bars. The three curves indicate results from three different relative mass functions for secondaries.

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The sizes of these color shifts are shown in Table 6. The systematic color shift due to unresolved binaries is less strongly dependent on the choice of secondary mass functions. Typical sizes of these color shifts are ∼0.003 mag, 0.008 mag, and 0.010 mag in g  −  i, g  −  z, and J  −  K_s, respectively. To correct for the unresolved binaries in the KIC, the above color shifts should be subtracted before estimating T_eff. The last four columns in Table 6 list the average T_eff difference between a population with a 50% unresolved binary fraction and that of primaries alone. The sense is that unresolved binary stars have lower temperatures than expected from primaries alone. Different SDSS color indices have similar binary sensitivities, and temperatures based on these filters are less affected by unrecognized companions than those derived using J  −  K_s. These color shifts are small for any given star, but significant when applied to the entire catalog. We therefore recommend including them when using large samples of photometric effective temperature estimates, and include this effect in our global error budget below.

We can assess our overall errors by comparing the real to the observed dispersions in the color–color plane. Photometric errors, unresolved binaries, and metallicity all induce scatter; so would extinction uncertainties. Significant mismatches between the two reflect unrecognized or overestimated error sources.

Figure 16 shows the observed color–color diagrams in the KIC, after the extinction corrections and the zero-point adjustment as described in Section 2.2. From Figure 16, we estimated the standard deviation of the color dispersion from a fiducial line (fit using a fifth-order polynomial) in g  −  i, g  −  z, and J  −  K_s at each g  −  r bin. These observed dispersions with good T_eff estimates are shown as solid black curves with closed circles in Figure 17. Here the criteria for the good T_eff are that the standard deviation of individual T_eff from three color indices (g  −  r, g  −  i, and g  −  z) is less than 130 K or that the difference between SDSS and IRFM measurements is no larger than three times the random errors of these measurements (see also Section 4.2.4). There is a strong overlap between the two criteria. Since the formal random SDSS errors are of order 40 K, and the systematics between the colors are typically at that level as well, differences of 130 K represent clear evidence of a breakdown in the color–temperature relationships, likely from unresolved blends. Excluding extreme outliers is essential because they would otherwise dominate the dispersion measure, and we are interested in testing the properties of the majority of the sample.

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Other lines in Figure 17 represent contributions from random photometric errors, unresolved binaries or photometric blends, metallicity, and dust extinction as described below. Red lines with open circles are the quadrature sum of all of these error sources.

We assumed 0.01 mag errors in gri, 0.03 mag errors in z, 0.024 mag in J, and 0.028 mag in K_s to estimate color dispersions from photometric errors alone (red curve in Figure 17). To perform this simulation in grizJK_s, we combined our base model (Table 1) with our earlier set of isochrones in the 2MASS system (An et al. 2007b)¹⁰ at the same metallicity ([Fe/H] =−0.2) and age (1 Gyr) as those for the base isochrone. As with the binary simulations described in the previous section, we employed a 1 Gyr old, solar-metallicity Padova model (Girardi et al. 2004) to generate color–color sequences with a 50% binary fraction based on the M35 mass function for secondaries. Again, running this isochrone in the simulation is not strictly consistent with the usage of our base model, but the relative effects induced by unresolved companions would be rather insensitive to the small metallicity difference. The dispersion induced by unresolved binaries is shown in a red dashed curve in Figure 17.

The KIC sample has a mean [Fe/H] =−0.2 with a standard deviation of 0.28 dex. If the KIC [Fe/H] values are accurate enough for these stars, this metallicity spread would induce a significant spread in T_eff. The color dispersion due to metal abundances was estimated by taking the color difference between our base model ([Fe/H] =−0.2) and the models at [Fe/H] =+0.1 and −0.5 as an effective ±1σ uncertainty. The metallicity error contribution is shown in blue solid curves. The KIC sample has a wide range of reddening values (0 ≲ E(B  −  V) ≲ 0.2). We took 0.02 mag error as an approximate ±1σ error in E(B  −  V), roughly equivalent to a 15% fractional uncertainty for a typical star. Stars on the simulated color–color sequence were randomly displaced from their original positions assuming this E(B  −  V) dispersion. The resulting color dispersion is shown with the blue dashed curves in Figure 17.

In Figure 17 there is a color-dependent trend in the error budget, where observed color dispersion increases for cooler stars in g  −  i and g  −  z. On the other hand, the simulated dispersions (open circles connected with solid red curves) are essentially flat. Our results are consistent with expectations in J  −  K; if anything, the random errors appear to be overstated. This is probably caused by correlated errors in J and K_s, which were treated as uncorrelated in the temperature error estimates.

Based on this exercise, we conclude that our error model is reasonable for the hot stars in the sample, especially when the stars most impacted by blends are removed. There is excess color scatter for red stars, which correspond to effective temperatures below ∼5000 K in our sample. About 16% of the sample are found in this temperature domain. This could reflect contamination of the dwarf sample by giants, which have different color–color relationships; or a breakdown in the photometric error model for red stars. It would be useful to revisit this question when we have a solid estimate of the giant contamination fraction for the cool dwarfs in the sample.

We present results for the long-cadence sample with the overall properties of the catalog and systematic error estimates in this section. We have not provided corrected values for the entire KIC, because the additional quality control is outside the scope of our effort. However, our method could be applied in general to the KIC, employing the following steps.

• 1.  
  Correct the KIC griz photometry onto the SDSS DR8 system using Equations (1)–(4).
• 2.  
  Apply the KIC extinctions and the extinction coefficients in Section 2 to obtain dereddened colors.
• 3.  
  Use our griz–T_eff polynomials (Table 2) or the original isochrone (Table 1) to obtain temperature estimates. If complementary IRFM estimates are desired, use the C10 polynomials (for V_TJK_s).
• 4.  
  Adjust hot-end temperatures above 6000 K using Equations (5)–(7). The polynomials in Table 2 are for the original SDSS temperature calibration (Table 1) without the hot-end adjustment described in Section 3.3.
• 5.  
  In Table 7 we adopted a metallicity [Fe/H] =−0.2 and a dispersion of 0.3 dex for error purposes. We also adopted a fractional error of 15% in the extinction.
• 6.  
  The SDSS temperatures are inferred from the weighted average of the independent color estimates using the photometric errors discussed in Section 2, and the random uncertainties are the maximum of the formal random errors and the dispersion in those inferred from different griz colors.
• 7.  
  If the metallicity is known independent of the KIC, the SDSS temperatures can be corrected using the values in Table 3 and if desired the IRFM temperatures can be corrected for metallicity by adopting star-by-star metallicities in the C10 formulae.
• 8.  
  Apply gravity corrections in Table 4 for giants with log g(KIC) ⩽ 3.5.
• 9.  
  Outside the temperature range of the SDSS calibration, zero-point shifts of 223 K at the hot end and 150 K at the cool end should be applied to the KIC T_eff to avoid artificial discontinuities in the temperature scale at the edges of validity of the method.
• 10.  
  In our revised T_eff table, we did not apply statistical corrections for binaries, but the current Table 6 could be employed to do so, and this should be included in population studies.
• 11.  
  We expect about 4% of the sample to have photometry impacted by blends. Such stars could be identified as those having an excess dispersion from individual SDSS colors on the order of 130 K or more, and/or as those showing more than a 3σ deviation from the mean difference between the IRFM and SDSS temperatures.

Our main result, the revised T_eff for 161, 977 stars in the long-cadence KIC, is presented in Table 7. All of our revised T_eff estimates in the catalog are based on the recalibrated griz magnitudes in the KIC (Section 2.2). In addition to the griz-based SDSS T_eff, Table 7 contains (J  −  K_s)-based IRFM T_eff using the original C10 relation, and KIC values along with log g and [Fe/H] in the KIC. The null values in the SDSS T_eff column are those outside of the color range in the model (4043 K <T_eff < 7509 K). Similarly, the C10 IRFM T_eff are defined at 0.07 ⩽ (J  −  K_s)₀ ⩽ 0.80.

Statistical properties of our final temperature estimates are listed in Table 8 for dwarfs and for giants, separately. The relative KIC, IRFM, and SDSS temperatures for dwarfs and giants in the final catalog are compared in Figure 18. These comparisons include the adjustment to the hot end published SDSS scale described in Section 3.3. We did not correct the IRFM temperature estimates for gravity effects in the giants. The discrepancy between the two scales for the cool giants is consistent with being caused by this effect, as can be seen from the gravity sensitivity of (J  −  K_s) in Figure 7.

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Below we describe each column of Table 7 and provide a summary on how to correct T_eff for different log g, binarity (blending), and metallicity.

4.2.1. Error Estimates in T_eff

4.2.2. Corrections for Different log g

4.2.3. Corrections for Binaries

4.2.4. Quality Control Flag

In addition to our main catalog in Table 7, we also present in Table 9 the IRFM T_eff in Tycho V_T and 2MASS JHK_s colors for 7912 stars. These stars are a subset of the long-cadence KIC sample, which are bright enough to have V_T magnitudes, and can be used as an independent check on our T_eff scale (see the lower left panel in Figure 6). The IRFM T_eff values are presented using V_T  −  J, V_T  −  H, V_T  −  K_s, and J  −  K_s, with both random (σ_ran) and total (σ_tot) errors. As in Table 7, random errors are propagated from photometric uncertainties, and total errors are a quadrature sum of random and systematic errors (15% error in reddening and 0.3 dex error in [Fe/H]).

Our main result is a shift to higher effective temperatures than those included in the existing KIC. We have employed multiple diagnostic tools, including two distinct photometric scales and some high-resolution spectroscopy. In the case of cool (below 6000 K) dwarfs, the various methods for assigning effective temperature have an encouraging degree of consistency. The Johnson-Cousins measurements of An et al. (2007a) are in good agreement with the independent IRFM temperatures from C10 in star clusters. In Table 5, for example, the V  −  I_c results agree within 15 K for all clusters if we adopt the Sandquist (2004) data set for M67. The SDSS-based A09 system is constructed to be on the same absolute scale as the An et al. (2007a) system, so a similar level of agreement is expected between the IRFM and the temperatures that we derive from the SDSS filters. A comparison of the IRFM and SDSS temperatures in the KIC confirms this pattern, with agreement to better than 100 K for the cool stars. Even this level of disagreement overestimates the underlying accord in the systems, because the IRFM (J  −  K_s) diagnostic that was available to us in the KIC has systematic offsets relative to other IRFM thermometers even in the open clusters. When we correct for these offsets, the agreement for cool stars between the SDSS-based method of A09 and the IRFM (J  −  K_s) temperatures is very good, with average differences below 25 K and maximum differences below the 50 K level. Our cool dwarf temperatures are also within 50 K on average when compared with the spectroscopic results from B12. The spectroscopic sample of MZ11 is cooler at the 88 K level, which we take as a measure of systematic uncertainties in the spectroscopic scale (see Bruntt et al. 2010 for a further comparison of the spectroscopic and fundamental temperature scales).

For hotter dwarfs the revised temperature estimates are higher than in the KIC, but the magnitude of the offset is not consistent between the two photometric scales and the spectroscopic data. Motivated by this offset, we adjusted the SDSS-based system of A09 to be cooler on average by 100 K between 6000 K and 7000 K on the IRFM system. The consistency between photometric and spectroscopic scales degrades for stars in this range. This could reflect defects in the fundamental temperature scale for hotter stars; the existing fundamental data for the IRFM include relatively few solar-abundance dwarfs above 6000 K. There could also be errors in photometric or spectroscopic temperature estimates from the onset of rapid rotation above 6300 K, or color anomalies from chemically peculiar hot stars. On the spectroscopic side, it would be valuable to compare the atmospheric temperatures inferred from Boltzmann and Saha constraints to fundamental ones; as discussed in C10, there can be significant systematic offsets between these scales for some systems. This issue deserves future scrutiny and additional fundamental data would be very helpful.

In the case of evolved stars we also found a hotter temperature scale than in the KIC. We had to employ theoretical estimates of gravity sensitivity, however, to temperature diagnostics derived for dwarfs. An extension of the fundamental work to giants has been performed for other colors in the past, and it would be beneficial to test the theoretical predictions against actual radius data.

There are some significant drawbacks of the existing catalog, and care is required in its proper application. Binary companions will modify the colors and temperatures of stars; we have provided tables for statistical corrections, but have not included this in the tabulated effective temperatures. Blending can also impact colors, and there is clear evidence of some blended objects in our comparison of the KIC to SDSS DR8 data with superior resolution. The major error source for the temperature estimates is the uncertainty in the extinction. We have adopted a global percentage value based on typical errors in extinction maps, but there could be larger local variations. The color combinations available to us have limited diagnostic power for star-by-star extinction and binary corrections. For population studies, the stars in the long-cadence KIC sample were selected for a planet transit survey, and do not represent an unbiased set of the underlying population.

The KIC abundance estimates have significant errors, largely because the filters with the greatest metallicity sensitivity were not available. As a result, we have adopted metallicity insensitive temperature diagnostics, but the temperatures should be corrected for individual metallicities if available. These effects are at the 100 K dex⁻¹ level, and will therefore be smaller than the extinction uncertainties for most stars in the sample. The log g values for hot stars are not well constrained in the KIC, but we have adopted KIC gravities for cool stars. Our results would be affected at a modest level by changes in the derived gravities, and the appropriate corrections should be made if precise values are available.

There are two open areas for further discussion as well: the appropriate temperature scale for the hot dwarfs and errors in the photometry. In the former case, we recommend adjustments above 6000 K to the SDSS scale. For the entire domain we also note inconsistencies between the J  −  K_s calibration and the other color–temperature relationships in the IRFM. Even after putting the fundamental photometric temperature scale on a common system, however, there is a difference between it and the spectroscopic scale for stars above 6000 K. Until it is resolved we recommend inclusion of systematic temperature errors in this domain. The impact of the log g determinations on the extinction estimates for the hot stars should be investigated as well. The gravity diagnostics for the hot stars are not well measured, and asteroseismic gravities confirm this expected lack of precision. The KIC catalog included this as an ingredient in the distance estimates, but it is difficult to reconstruct the weights and importance of this uncertainty after the fact. Star-by-star extinctions would be useful for this purpose.

The origin of the differences between the SDSS (DR8) and KIC photometry should also be tracked down, and there may be spatially dependent or magnitude-dependent terms. We also noted some cases with severe internal inconsistency in the photometric temperature diagnostics and flagged those which we identified. We believe that unresolved blends are a promising candidate, but further work on this front is warranted. In a small fraction of cases these photometric issues can cause severe errors in the temperatures. Effective temperatures for stars where different colors return very different estimates should be treated with caution.

Despite these reservations, we believe that the addition of temperatures more closely tied to the fundamental scale will significantly improve the reliability of inferences about the underlying stellar populations.

A shift to higher effective temperatures will have consequences for both planetary and stellar science. On the main sequence, hotter stars will be on average more massive and larger. This would imply larger planet radii on average for such objects. The radii of evolved stars require more information (especially from surface gravity effects), and the consequences of the temperature scale shift for them are more difficult to predict from first principles. Stars of known asteroseismic radius will be on average more luminous, which could partially explain discrepancies in the mass–radius relationship for evolved stars (Chaplin et al. 2011). Asteroseismic parameters defined with scaling relationships will also be impacted. A more precise absolute effective temperature scale will also permit more stringent constraints on asteroseismic properties from detailed modeling of the frequency spectrum (see Metcalfe et al. 2010).

However, the full potential will be realized as complementary information becomes available on the Kepler sample. Blue data (such as Johnson UB or SDSS u) could be employed to infer more reliable photometric metallicities; Johnson-Cousins UBV(RI)_C data would enable more reliable extinction estimates, binary discrimination, and broader application of the IRFM directly to stars in the sample. Photometric systems naturally designed for F-type stars, such as Strömgren, would be useful for addressing the temperature and surface gravity scales in that regime.

A more robust set of input data would provide an important control sample for the measured planet population; it will be challenging to obtain spectroscopic temperatures of both the planet candidates and the background stellar population. A better calibration of the fundamental temperature scale is possible once asteroseismic radii are combined with parallaxes in the Kepler field, either via Kepler data or through the Gaia mission. The time domain data from the satellite are exquisite; a proper application of complementary tools from stellar astrophysics is now essential to fully realize their considerable scientific promise.