Empirical Bolometric Fluxes and Angular Diameters of 1.6 Million Tycho-2 Stars and Radii of 350,000 Stars with Gaia DR1 Parallaxes

In principle, if were were only interested in inferring the angular diameters of our sample of stars (and from these diameters inferring radii using the Gaia parallaxes), we could simply adopt empirically calibrated color-angular diameter relations (e.g., Boyajian et al. 2014). However, we chose to instead derive ${T}_{\mathrm{eff}}$ and ${F}_{\mathrm{bol}}$ individually from separate empirical color–${T}_{\mathrm{eff}}$ and color–${F}_{\mathrm{bol}}$ relations, for two reasons. First, we can compare our inferred estimates of ${T}_{\mathrm{eff}}$ with spectroscopic measurements, thus validating our inferred values and allowing us to use the spectroscopic ${T}_{\mathrm{eff}}$ as constraints. Second, we do not know the extinction to the stars in our sample a priori. A significant extinction would bias the broadband photometry we use to infer the angular diameters, thereby leading to a bias in the angular diameters and radii. We must therefore estimate the extinction as well.⁹

In order to infer ${T}_{\mathrm{eff}}$, we apply the Casagrande et al. (2010) color–${T}_{\mathrm{eff}}$ and color–${F}_{\mathrm{bol}}$ relations from the infrared flux method to obtain effective temperatures for the Tycho-2 stars. As noted above, to do so however, it is first necessary to de-redden the photometric measurements. As we do not know the extinction A_V a priori, we estimate the extinction and the effective temperature as follows.

• 1.  
  We step through ${N}_{i}=\,{A}_{V,\max }/{\rm{\Delta }}{A}_{V}$ extinction values ${A}_{V,i}$ in increments of ${\rm{\Delta }}{A}_{V}=0.01$ over the range ${A}_{V}\in \{0,\,{A}_{V,\max }\}$, where $\,{A}_{V,\max }\equiv {E}_{B-V}{R}_{V}$ is the maximum line-of-sight total extinction, estimated from is the maximum line-of-sight color excess (selective extinction) ${E}_{B-V}$ determined from the Schlegel et al. (1998) dust maps, and adopting R_V = 3.1 for the ratio of total to selective extinction. Hence, for ${A}_{V}\geqslant 0.1$, we sample over 10 extinctions; exploring fewer values would generate a coarser sampling of high-extinction regions.We note that, despite the caution indicated by Schlegel et al. (1998), we adopt their maximum color excess even for stars within 10° of the Galactic plane. We will provide qualitative tests of the validity of our inferred extinctions for stars in this region of the sky in a later section.
• 2.  
  At each value of ${A}_{V,i}$, we de-redden the photometric magnitudes using the (Cardelli et al. 1989) extinction law. We then calculate ${T}_{\mathrm{eff},j}$ from each of $j=0,1,\,\ldots \,{N}_{j}$ applicable Casagrande et al. (2010) empirical relations for which the de-reddened color and the metallicity are within the applicable ranges. The uncertainty ${\sigma }_{{T}_{\mathrm{eff},j}}$ on the ${T}_{\mathrm{eff},j}$ derived from each relation is calculated as the square-root of the quadrature sum of the standard deviation about the empirical relation listed in Table 4 of Casagrande et al. (2010) and the (linearly) propagated uncertainty due to the (assumed to be independent) errors on [Fe/H] and the photometric measurements. If no empirical relations apply, e.g., because all de-reddened colors lie outside of the suitable ranges for all relations, then we skip to the next ${A}_{V,i}$.
• 3.  
  We calculate the weighted mean ${T}_{\mathrm{eff},\mathrm{mean},i}$ for each ${A}_{V,i}$ from all N_j applied relations, where we weight each ${T}_{\mathrm{eff},j}$ by the square of the uncertainty in each relation ${\sigma }_{{T}_{\mathrm{eff},j}}$. We then reject ${T}_{\mathrm{eff},j}$ if ${T}_{\mathrm{eff},j}-{T}_{\mathrm{eff},\mathrm{mean},i}\gt 3{\sigma }_{{T}_{\mathrm{eff},j}}$. We then re-calculate the weighted mean ${T}_{\mathrm{eff},\mathrm{mean},i}$ and iterate until no outliers remain or until only one relation remains.
• 4.  
  We calculate a "merit function" ${\chi }_{\mathrm{ICT}}^{2}\equiv {\sum }_{j=0}^{{N}_{j}}{({T}_{\mathrm{eff},j}-{T}_{\mathrm{eff},\mathrm{mean},i})}^{2}/{\sigma }_{{T}_{\mathrm{eff},j}}^{2}$, which essentially quantifies how well all of the N_j inferred values of ${T}_{\mathrm{eff},j}$ are consistent with a constant value of ${T}_{\mathrm{eff}}$, given their respective uncertainties ${\sigma }_{{T}_{\mathrm{eff},j}}$ and the assumed value of ${A}_{V,i}$.
• 5.  
  We add a penalty term ${\chi }_{\mathrm{spec}}^{2}\equiv {({T}_{\mathrm{eff},\mathrm{mean},i}-{T}_{\mathrm{eff},\mathrm{spec}})}^{2}$ $/{\sigma }_{{T}_{\mathrm{eff},\mathrm{spec}}}^{2}$ if a spectroscopic ${T}_{\mathrm{eff}}$ exists. If ${\chi }^{2}=0$ because there is no spectroscopic effective temperature for this star and only one calibration relation applies, then we skip to the next ${A}_{V,i}$. Because each color relation is applicable over a certain range of de-reddened color values, different relations can be applicable at different extinctions. For example, the fainter stars whose Tycho magnitudes are unreliable and/or highly uncertain, the three previous steps serve to exclude the ${T}_{\mathrm{eff}}$ determined by the ${B}_{T}-{V}_{T}$ color when calculating the error-weighted ${T}_{\mathrm{eff}}$ if it is a significant outlier, or suppress the statistical contribution of the ${B}_{T}-{V}_{T}$ color to the final estimate of ${T}_{\mathrm{eff}}$.
• 6.  
  We select the value ${A}_{V,i}$ and the error-weighted mean ${T}_{\mathrm{eff},\mathrm{mean},i}$ corresponding to the minimum ${\chi }^{2}\equiv {\chi }_{\mathrm{ICT}}^{2}\,+{\chi }_{\mathrm{spec}}^{2}$. If no merit function was calculated for any value of ${A}_{V,i}$, then we have insufficient information to inform our choice of effective temperature; therefore, we drop this star from the sample and move onto the next star.
• 7.  
  We re-scale the uncertainty on ${T}_{\mathrm{eff},\mathrm{mean},i}$ such that ${\chi }_{\nu }^{2}=1$ for the minimum ${\chi }^{2}$, where the ${\chi }_{\nu }^{2}$ is reduced ${\chi }^{2}$ with ν degrees of freedom. Explicitly, the scale factor is

  $$\begin{eqnarray}&&\sqrt{\displaystyle \frac{1}{{N}_{j}+1}\left[{({T}_{\mathrm{eff},\mathrm{mean},i}-{T}_{\mathrm{eff},\mathrm{spec}})}^{2}+\displaystyle \sum _{j=0}^{{N}_{j}}{({T}_{\mathrm{eff},j}-{T}_{\mathrm{eff},\mathrm{mean},i})}^{2}\right]}\end{eqnarray} \tag{ 4 }$$

  if there is a spectroscopic ${T}_{\mathrm{eff}}$ and

  $$\begin{eqnarray}&&\sqrt{{\sigma }_{{T}_{\mathrm{eff},\mathrm{mean}}}\equiv \displaystyle \frac{1}{{N}_{j}}\displaystyle \sum _{j=0}^{{N}_{j}}[{({T}_{\mathrm{eff},j}-{T}_{\mathrm{eff},\mathrm{mean},i})}^{2}]}\end{eqnarray} \tag{ 5 }$$

  if not. We re-calculate ${T}_{\mathrm{eff},\mathrm{mean},i}$ using these rescaled uncertainties.
• 8.  
  We then estimate the uncertainty on the extinction by taking the range of extinctions corresponding to ${\rm{\Delta }}{\chi }^{2}=1$, using this scaled uncertainty. If no extinctions lie within this range—which is the case for 5% of the 1.6 million stars in the angular diameters sample—then the extinction range corresponding ${\rm{\Delta }}{\chi }^{2}=1$ is smaller than our step-size, so we adopt our step-size, ${\rm{\Delta }}{A}_{V}=0.01$, as a conservative uncertainty. For the remaining 95% of stars, the mean uncertainty in A_V is 0.046, the median uncertainty is 0.03, and the standard deviation is 0.064.If no color–${T}_{\mathrm{eff}}$ relations apply for any extinction, then we drop the star from the sample. Additionally, for the stars without spectroscopic temperatures, if there exists an extinction for which only one color–${T}_{\mathrm{eff}}$ relation applies, then the merit function ${\chi }^{2}=0$ by definition for that extinction. In these cases, we infer that we do not have enough information to determine both extinction and ${T}_{\mathrm{eff}}$, and so we also drop such stars from our sample.

We estimate the unextincted bolometric flux ${F}_{\mathrm{bol}}$ as follows.

• 1.  
  We de-redden the magnitudes with the extinctions obtained in Section 3.1.1 and apply all ${N}_{{F}_{\mathrm{bol}}}$ applicable Casagrande et al. (2010) bolometric flux relations.
• 2.  
  For each relation, we calculate the mean error-weighted bolometric flux, ${F}_{\mathrm{bol}}$, and the weighted uncertainty on the flux, ${\sigma }_{{F}_{\mathrm{bol}}}$, where the weights are the quadrature sums of the scatter about each relation as cited by Casagrande et al. (2010) and the (linearly) propagated uncertainties due to the uncertainties on the magnitudes, extinction, and [Fe/H], assuming all uncertainties are independent. As with the effective temperature procedure, if no relations were applicable, e.g., because the de-reddened color lies outside the ranges of all relations, we drop the star from our sample.
• 3.  
  We calculate ${\chi }_{{F}_{\mathrm{bol}}}^{2}\equiv {\sum }_{j=0}^{{N}_{{F}_{\mathrm{bol}}}}{({F}_{\mathrm{bol},{\rm{j}}}-{F}_{\mathrm{bol},\mathrm{mean}})}^{2}/{\sigma }_{{F}_{\mathrm{bol},{\rm{j}}}}^{2}$ and scale the uncertainties by a constant factor such that ${\chi }_{\nu ,{F}_{\mathrm{bol}}}^{2}=1$, where $\nu ={N}_{{F}_{\mathrm{bol}}}$. We then re-calculate ${F}_{\mathrm{bol},\mathrm{mean}}$ and ${\sigma }_{{F}_{\mathrm{bol},\mathrm{mean}}}$ using these rescaled uncertainties.

As one way of validating the results of our iterative procedure outlined in Section 3, we check that our method prefers reasonable extinction values. First, we examine the distribution of extinctions across Galactic latitude b; most of the highly extincted stars should lie in the Galactic disk—roughly $| b| \lt 10^\circ$—where there is more dust along a typical line of sight. Figure 1 shows these distributions for the four subsamples; indeed, the most highly extincted stars are those in the disk.

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For the subsample of our stars for which we calculate radii (see Section 4.2), we compare our iterative-color derived V-band extinctions to the extinctions from the Green et al. (2015) three-dimensional Milky Way dust map. For each star, we calculate its distance modulus from the Gaia DR1 parallax distance and our V-band extinctions. We then find the Pan-STARRS best-fit reddening at this distance modulus by interpolating linearly between distance moduli along the line of sight. Finally, we multiply the reddening by 3.1 to get the V-band extinction. Figure 2 contains histograms showing the difference between ICT and Pan-STARRS V-band extinctions. Our extinctions for both the stars with and without spectroscopic parameters agree well with the Pan-STARRS extinctions, though we note that the extinction difference distribution for the non-spectroscopic stars has a larger tail toward higher ICT extinctions.

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Figure 3 shows the iterative-color technique-derived effective temperatures versus the spectroscopic values for the resulting sample of 64345 LAMOST stars, 214,707 RAVE stars, and 14360 APOGEE stars.

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This excludes stars with reduced ${\chi }_{\nu }^{2}\gt 100$, which effectively removes stars for which the iterative-color temperatures and spectroscopic temperatures differ by ${ \mathcal O }({10}^{3}\,{\rm{K}})$. Notably, these include many giants, for which the iterative-color relations were not calibrated. The iterative-color effective temperatures are positively correlated with the spectroscopic temperatures; stars with discrepant temperatures tend to have higher ${\chi }^{2}$ values, while the iterative-color temperatures for stars with lower ${\chi }^{2}$ values tend to agree with the spectroscopic values. Moreover, our technique appears to systematically underestimate the effective temperatures for stars with spectroscopic temperatures above 7000 K.

Figure 4 shows our effective temperature distributions inferred using the iterative-color technique for the stars in our sample that also have spectroscopically measured effective temperatures. In the RAVE sample, Kordopatis et al. (2013) use a grid of synthetic stellar spectra to derive the stellar parameters; hence, the peaks in the histogram correspond to the ${T}_{\mathrm{eff}}$ grid points and are separated by the grid resolution of 250 K. As shown in Figure 3, the iterative-color technique infers an excess of stars with ${T}_{\mathrm{eff}}\gt 7000$ K relative to the spectroscopically determined temperature distributions for the stars with LAMOST, RAVE, and APOGEE spectra. These stars typically have very few photometric measurements with which to infer ${T}_{\mathrm{eff}}$, ${F}_{\mathrm{bol}}$, and A_V, and our inferences about their properties are thus highly suspect. Moreover, only three of the 423 stars used to calibrate Casagrande et al. (2010) relations have ${T}_{\mathrm{eff}}\gt 7000$ K. Therefore, we urge the reader to use extreme caution when applying our results for stars hotter than about 7000 K.

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As discussed below, the uncertainty on the Gaia parallax distances begin to dominate the radius error budget beyond ∼100 pc (see also Stassun et al. 2016). The uncertainty in radius due to the error on the projected end-of-mission Gaia parallax should be greatly improved; for example, at 100 pc, the parallax signal is 10 milliarcseconds, so a projected 10 microarcsecond uncertainty translates to a 0.1% uncertainty on the radius.

As mentioned in Section 2.1, for the stars without measured metallicities, we adopt a metallicity and uncertainty equal to the median [Fe/H] and dispersion from the Geneva–Copenhagen Survey of late-type, solar neighborhood stars (Casagrande et al. 2011), as described in Section 3. To determine what effect our choice of metallicity has on the recovered effective temperatures and bolometric fluxes, we repeat our method for the subset of stars with spectroscopic parameters, this time using the Geneva–Copenhagen median and $1\sigma$ metallicity instead of the LAMOST metallicities. As Figure 5 illustrates, our decision to use the median and dispersion of the Geneva–Copenhagen survey as a proxy for the metallicity of stars without directly measured metallicities has a negligible effect on the temperature and bolometric flux we infer for these stars from our iterative-color procedure.

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Nominally, the color–${T}_{\mathrm{eff}}$ and color–${F}_{\mathrm{bol}}$ relations of Casagrande et al. (2010) are parameterizations of stellar spectral energy distributions (SEDs). As another check on our effective temperatures, bolometric fluxes, and angular diameters, we generate SEDs for 293 stars in common between our sample and the Casagrande et al. (2010) sample. We use Kurucz model atmospheres (Kurucz 2013) to fit SEDs to Tycho-2 ${B}_{T}{V}_{T}$ (Høg et al. 2000), UBV (Mermilliod 2006), Strömgren by Paunzen (2015), 2MASS JHK_S (Cutri et al. 2003; Skrutskie et al. 2006), GALEX NUV (Bianchi et al. 2011) and WISE (Cutri et al. 2014) photometry when available. As before, we adopt the listed measurement uncertainties unless these uncertainties are smaller than 0.01 mag (or 0.1 mag for the GALEX NUV and WISE4 bands); in these cases, we adopt 0.01 mag (0.1 mag) as the measurement uncertainties. In addition, to compensate for an artifact in the Kurucz atmospheres at 10 μm, we artificially inflate the WISE3 uncertainty to 0.3 mag unless the reported uncertainty is already larger than 0.3 mag; see Stassun & Torres (2016a) for a more detailed discussion. Additionally, we iteratively clip $5\sigma$ outlier measurements.

We sample the effective temperature at the Casagrande et al. (2010) listed value as well as $\pm 1\sigma$. We fix $\mathrm{log}{g}_{\star }$ and [Fe/H] to the Casagrande et al. (2010) values, rounded to the nearest values for which a Kurucz model atmosphere exists. We sample 10 extinctions from A_V = 0 to the maximum line-of-sight extinction from the Schlegel et al. (1998) dust maps. We show our SEDs in the Appendix. Figures 6–10 compare the iterative-color extinctions, effective temperatures, bolometric fluxes, and angular diameters, respectively, to those from the SEDs. We find generally good agreement between the two methods. We note that, for the majority of stars, the SED and iterative-color extinctions agree quite well—as seen in the bottom two panels of Figure 7—though we overestimate the extinctions relative to the SED values for a few systems in the tail of the aforementioned histogram. Correspondingly, we overestimate the effective temperatures and bolometric fluxes for these stars and underestimate the angular diameters relative to the model SEDs at the level of a few percent. We see no trend with unscaled reduced ${\chi }^{2}$. The model SED fits to the photometry are shown in Figure 20 and can be seen to fit the data well overall.

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