The Revised TESS Input Catalog and Candidate Target List

The most basic quantity required for every TIC object, aside from its position, is its apparent magnitude in the TESS bandpass, which we represent as T. As we did in TICv7, we derived a relation based on the PHOENIX model atmospheres (Husser et al. 2016). We adopted the most up-to-date TESS passband available (R. Vanderspek 2019, private communication) and the Gaia passbands for G, ${G}_{\mathrm{BP}}$, and ${G}_{\mathrm{RP}}$ from Gaia Collaboration et al. (2018). Note that we did not apply the small corrections to the Gaia bandpasses from Maíz Apellániz & Weiler (2018) as these were not available prior to our construction of the base TIC.

The relation that we adopt,

$$\begin{eqnarray}\begin{array}{rcl}T & = & G-0.00522555{\left({G}_{\mathrm{BP}}-{G}_{\mathrm{RP}}\right)}^{3}\\ & & +\ 0.0891337{\left({G}_{\mathrm{BP}}-{G}_{\mathrm{RP}}\right)}^{2}\\ & & -\ 0.633923({G}_{\mathrm{BP}}-{G}_{\mathrm{RP}})+0.0324473,\end{array}\end{eqnarray} \tag{ 1 }$$

is valid for dwarfs, subgiants, and giants of any metallicity, and the formal scatter is 0.006 mag. The fit and residuals of the relation are shown in Figure 3. Strictly speaking, this relation is valid for −0.2 < ${G}_{\mathrm{BP}}$ − ${G}_{\mathrm{RP}}$ < 3.5, but we extrapolate it to −1.0 < ${G}_{\mathrm{BP}}$ − ${G}_{\mathrm{RP}}$ < 6.0 because by NASA requirement every star in the TIC must have a T magnitude. Even though the relation degrades considerably for M dwarfs (${G}_{\mathrm{BP}}$ − ${G}_{\mathrm{RP}}$ > 2), we consider it to be the best available estimator from the G band. Most importantly, the refined T magnitudes provided in the specially curated Cool Dwarf list override the magnitudes computed by the relation above.

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Of course, the use of stellar atmosphere models introduces some systematic error, so the true errors in the predicted T are likely to be larger than 0.006 mag. To estimate this, we used the same atmosphere models to derive a relation between V magnitude and the magnitudes in the Gaia passbands, and we compared our relation with the empirical relation reported by Evans et al. (2018). Their empirical relation is based on real V magnitudes for stars from various catalogs and from the measured G magnitudes for the same stars from Gaia DR2. Figure 4 compares the two relations. The comparison here is based on the set of stars from TICv7 that had spectroscopic ${T}_{\mathrm{eff}}$, so they are not the same set of stars used by Evans et al. (2018); however, this should not affect our conclusions significantly. A slight difference is evident between our model fit and the Evans et al. (2018) empirical fit, which is to be expected. However, the largest difference between the two relations is only ∼0.1 mag, providing confidence in our adopted relation and suggesting that the true uncertainties in our derived T are likely to be at most ∼0.1 mag in most cases.

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The TESS magnitude relation above is strictly valid for unreddened stars. Because the measured magnitudes and colors from Gaia are typically affected by extinction, we first deredden the colors and correct the G magnitudes for extinction, then apply the relation to obtain a dereddened T magnitude, and finally add extinction back into T to obtain an apparent magnitude. The extinction coefficients required in each band are provided below in Section 2.3.3.

Finally, we have developed simple relations for stars that either have colors beyond the formal validity limits of the above relation or that do not have fluxes reported for all three Gaia passbands. For stars that are bluer or redder than the limits of the above relation, we simply extrapolate the same polynomial, but to be conservative we increase the formal errors by 0.1 mag (added in quadrature). For stars with no valid ${G}_{\mathrm{BP}}$ − ${G}_{\mathrm{RP}}$ colors, but which have a valid G magnitude, we use the following simple offset:

$$\begin{eqnarray}&&T=G-0.430.\end{eqnarray} \tag{ 2 }$$

This offset is the value that specifically corresponds to a star like the Sun, with a color of ${G}_{\mathrm{BP}}$ − ${G}_{\mathrm{RP}}$ = 0.82 based on the PHOENIX stellar atmosphere models. As a very conservative error, we assign 0.6 mag, which should be valid for all but the reddest M dwarfs, which are in any case dealt with via the specially curated Cool Dwarf list.

For completeness and for maximum usability, we compute the apparent V magnitude for TIC stars that do not possess a measured V from TICv7 but which possess Gaia photometry, as follows, using the relations provided by the Gaia team (Evans et al. 2018):

$$\begin{eqnarray}\begin{array}{rcl}V & = & G+0.01760+0.006860({G}_{\mathrm{BP}}-{G}_{\mathrm{RP}})\\ & & +\ 0.1732{\left({G}_{\mathrm{BP}}-{G}_{\mathrm{RP}}\right)}^{2},\end{array}\end{eqnarray} \tag{ 3 }$$

which has a reported scatter of about 0.046 mag.

Because we estimate stellar ${T}_{\mathrm{eff}}$ principally from an empirical color relation involving the Gaia ${G}_{\mathrm{BP}}$ − ${G}_{\mathrm{RP}}$ color (see Section 2.3.4), which is susceptible to reddening effects, it is necessary to first apply a dereddening correction, as we now describe.

When creating TICv7, we were limited by the available dust maps to estimates of the reddening along the full line of sight through the Galaxy in any particular direction, and we also were unable to estimate reddening within about 15 degrees of the Galactic plane. Here, we adopt the newly released three-dimensional, empirical, nearly all-sky dust maps from Pan-STARRS (Green et al. 2018), which provides an ability to estimate the reddening on a star-by-star basis according to the star's position in three dimensions (coordinates on the plane of the sky together with the distance). For the region of the sky not covered by Pan-STARRS (decl. below −30°), we continue to use the Schlegel et al. (1998) map, now with an adjustment to the total line-of-sight extinction for distance (from Gaia), assuming a standard exponential model for the disk with a scale height of 125 pc (see, e.g., Bonifacio et al. 2000). In both cases, we apply a recalibration coefficient of 0.884 to the E(B − V) values, as prescribed by Schlafly & Finkbeiner (2011).

As an initial sanity check on the Pan-STARRS reddening estimates, we compared the E(B − V) reddening values reported by the Pan-STARRS map against that estimated from the dust maps that we used in TICv7 (Schlegel et al. 1998). As shown in Figure 5, the agreement is in general quite good, with the E(B − V) agreeing to within ∼0.05 mag for Galactic latitudes $| b| \gt 15$ deg.

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We require a relation to convert the E(B − V) values that are provided by the reddening maps into E(${G}_{\mathrm{BP}}$ − ${G}_{\mathrm{RP}}$) and A_G for dereddening the Gaia ${G}_{\mathrm{BP}}$ − ${G}_{\mathrm{RP}}$ and G observed colors. As the most common type of star in the TIC is similar to the Sun, we used a synthetic solar-like spectrum as the source (${T}_{\mathrm{eff}}$ = 5800 K, log g = 4.5, [Fe/H] = 0.0, the closest PHOENIX model spectrum to the Sun) to compute the effective wavelengths for the various passbands following Equation (18) of Casagrande & VandenBerg (2014). We then used these mean wavelengths with the Cardelli et al. (1989) extinction law, which yields E(${G}_{\mathrm{BP}}$ − ${G}_{\mathrm{RP}}$) = 1.31E(B − V) and A_G = 2.72E(B − V). We also used this procedure to determine the relation of E(B − V) to A_T, the extinction in the TESS bandpass, for use in calculating the T magnitudes (see Section 2.3.1), which gives A(T) = 2.06E(B − V).

Applying the estimated reddening to the determination of ${T}_{\mathrm{eff}}$ via the photometric colors has the effect generally of making apparently cool stars hotter, and that hotter ${T}_{\mathrm{eff}}$ in turn has the effect of implying a larger radius and mass from our empirical relations described in Section 2.3.5. Therefore it is important to assess the quality of the dereddened ${T}_{\mathrm{eff}}$. Figure 6 (top) shows a comparison of the ${T}_{\mathrm{eff}}$ obtained from dereddened colors versus ${T}_{\mathrm{eff}}$ measured spectroscopically for ∼2 million stars from TICv7 with the relevant quantities available. As with all of the spectroscopic ${T}_{\mathrm{eff}}$ values that we adopt in the TIC, we limited the sample to stars whose spectroscopic ${T}_{\mathrm{eff}}$ values have reported uncertainties less than 300 K to ensure that the spectroscopic ${T}_{\mathrm{eff}}$ does not dominate the comparison of errors. The comparison overall is quite good, with a mean difference of 20 K and a 95th percentile range of −410 to +220 K. The slight skew toward negative ${T}_{\mathrm{eff}}$ differences (i.e., dereddened photometric ${T}_{\mathrm{eff}}$ being slightly cooler than the spectroscopic ${T}_{\mathrm{eff}}$) suggests that in some cases the reddening is underestimated (not enough dereddening correction applied), but this is at the margins of the overall distribution. Interestingly, as shown in Figure 6 (bottom), both the Pan-STARRS and Schlegel et al. (1998) dust maps appear to slightly overestimate the extinction toward the southern Galactic pole, resulting in our inferring a marginally elevated ${T}_{\mathrm{eff}}$ at b ≲ −70°. Note that both G and T are broad photometric bands, which can complicate extinction corrections. In particular, the ratio of total to selective extinction, R_G ≡ A_G/E(B − V), is a function of ${T}_{\mathrm{eff}}$ as well as the overall extinction A_V. The variation with A_V is small (dR_G/dA_V ≈ −0.03) and can be ignored, but the variation with ${T}_{\mathrm{eff}}$ is slightly larger. This could also be a reason for the systematic trends seen in Figure 6 (top); see Figure A of Sanders & Das (2018) for further details and for one way to account for this.

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Also shown in Figure 7 are the ${T}_{\mathrm{eff}}$ differences as a function of Galactic coordinates, where the effect of larger errors within ∼10 degrees of the plane is clear. Note that the Schlegel et al. (1998) dust maps have been shown to overestimate extinction for E(B − V) > 0.15 by a factor of about 1.4 (Arce & Goodman 1999; Cambrésy et al. 2005); a correction is given in Equation (24) of Sharma et al. (2014). In any event, we reduce the CTL priority by a factor of 0.1 for stars within 10 degrees of the Galactic plane (see Section 3.3).

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Finally, based on the comparisons above between the Pan-STARRS and Schlegel et al. (1998) dust maps, we find that only 1% of the ∼4 million stars compared have applied E(B − V) values that disagree by more than 2.5 mag. This means that a star that in reality has a very low reddening could appear with E(B − V) > 2.5 if the dust map at that location is such an extreme outlier. Thus, we adopt 2.5 as the maximum permissible E(B − V) from the Schlegel et al. (1998) dust map (effectively a cap on A_V of ≈7.8) in cases where a reliable Pan-STARRS value is not available. In cases where there is not a reliable Pan-STARRS value and our adopted value from Schlegel et al. (1998) has been capped, we report the values, but do not apply any reddening in calculating the T magnitude, and we also do not attempt to provide any derived stellar parameters.

We derived a new empirical relation between T_eff and the Gaia ${G}_{\mathrm{BP}}$ − ${G}_{\mathrm{RP}}$ colors, based on a set of 19,962 stars having spectroscopically determined T_eff and being within 100 pc so as to avoid reddening. We have ignored possible binarity, which is generally unknown for these reference stars and for stars in the larger TIC. After removing obvious outliers, we fit a spline function by eye. Figure 8 shows the fit, with the spline nodes marked with circles. Table 2 lists the nodes as (${G}_{\mathrm{BP}}$ − ${G}_{\mathrm{RP}}$, ${T}_{\mathrm{eff}}$) pairs.

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Table 2.  Spline Nodes for Relation in Figure 8

${G}_{\mathrm{BP}}$–${G}_{\mathrm{RP}}$	${T}_{\mathrm{eff}}$
−0.21	15,000
−0.17	13,500
−0.12	12,000
−0.085	11,000
−0.02	10,000
0.1	8849.803711
0.3	7709.192383
0.5	6875.640137
0.7	6172.216309
0.9	5532.801758
1.1	5017.910156
1.3	4618.64209
1.5	4327.293457
1.7	4048.811523
1.9	3935.294434
2.1	3780.993652
2.3	3652.275635
3.00	3200
3.50	3000

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The residuals of the fit (see Figure 6, top panel) show a near-zero mean offset, with an rms scatter of 122 K. This seems quite reasonable: given the ∼100 K uncertainties typical of the spectroscopic ${T}_{\mathrm{eff}}$, this would imply a true scatter in the photometric relation of ∼70 K. The range of validity of the relation is seen in the figure and is ${G}_{\mathrm{BP}}$ − ${G}_{\mathrm{RP}}$ = [−0.2, +3.5], although the predictions are likely to be less reliable for hot stars with ${G}_{\mathrm{BP}}$ − ${G}_{\mathrm{RP}}$ < 0 because of the very steep slope of the relation, and also for M stars with ${G}_{\mathrm{BP}}$ − ${G}_{\mathrm{RP}}$ > 2. As discussed below, the CTL relies mainly on the specially curated Cool Dwarf list for cool dwarf stars and draws ${T}_{\mathrm{eff}}$ estimates for those stars from that list.

We note that metallicity has been ignored in deriving the above relation, largely because it is unknown for any given field star. The relation may therefore return somewhat biased values for stars with compositions very different for solar. For example, referring back to the top panel of Figure 6, we see the average shift between the photometric and spectroscopic temperatures is about +160 K if we restrict the comparison to stars more metal-poor than [Fe/H] = −1, and −80 K for stars more metal-rich than [Fe/H] = +0.35.

This relation provides a continuous color–${T}_{\mathrm{eff}}$ relation from 3000 to 15,000 K. For stars with ${G}_{\mathrm{BP}}$ − ${G}_{\mathrm{RP}}$ outside of this range of validity, the TIC reports ${T}_{\mathrm{eff}}$ = Null, unless a spectroscopic ${T}_{\mathrm{eff}}$ is available or if a ${T}_{\mathrm{eff}}$ is available from the CTL associated with TICv7. The final ${T}_{\mathrm{eff}}$ errors reported in the TIC from the above polynomial relation include the 122 K scatter added in quadrature to the uncertainties from the photometric errors.

We compute the stellar radii using the Gaia parallaxes, which we now have for every star in the CTL, according to the standard expression from the Stefan–Boltzmann relation:

$$\begin{eqnarray}\begin{array}{rcl}\mathrm{log}(R/{R}_{\odot }) & = & \displaystyle \frac{1}{5}\left[4.74-5+5\mathrm{log}D-G\right.\\ & & -\ \left.10\mathrm{log}({T}_{\mathrm{eff}}/5772)-{\mathrm{BC}}_{G}\right]\end{array}\end{eqnarray} \tag{ 4 }$$

where D is the distance based on the Gaia parallax from Bailer-Jones et al. (2018), G is the observed Gaia magnitude corrected for extinction (G_obs − A_G), T_eff is the temperature from either spectroscopy or from dereddened colors, and BC_G is the bolometric correction in the Gaia passband as a function of ${T}_{\mathrm{eff}}$ (corrected for reddening if from colors). The above relation assumes that the object is a single star, and it will in general return biased values for the radius if it is a binary because the Gaia photometry will be affected by the companion. To be conservative, for ${T}_{\mathrm{eff}}$ from spectroscopy, we add 100 K in quadrature to the ${T}_{\mathrm{eff}}$ uncertainty if the catalog-reported ${T}_{\mathrm{eff}}$ uncertainty is less than 100 K when computing the resulting mass and radius uncertainties.

In order to develop a relation for BC_G as a function of ${T}_{\mathrm{eff}}$ for the widest possible ${T}_{\mathrm{eff}}$ range, we have adopted the following prescription (see Figure 9):

• 1.  
  For the range 3300–8000 K, we adopt the polynomial formulae for BC_G reported by the Gaia team (Andrae et al. 2018, see their Equation (7) with coefficients in their Table 4), which are based on MARCS stellar atmosphere models within 0.5 dex of solar metallicity. Those relations come in two parts: 3300–4000 K³⁴ and 4000–8000 K. We have added a minor correction to the cooler segment—a shift of +0.0036 mag in the a₀ coefficient—in order to achieve continuity between the two segments. Andrae et al. (2018) also provide additional polynomials to describe the error in BC_G as a function of ${T}_{\mathrm{eff}}$, based essentially on the scatter as a function of log g; we adopt these errors as well.
• 2.  
  Above 8000 K, and up to the 12,000 K limit available in the PHOENIX library of stellar atmosphere models, we fit a cubic polynomial to the bolometric corrections from the models, restricted to metallicities within 0.5 dex of solar, as above. We chose to consider only log g values above 3.0, as our bolometric corrections here are intended for deriving radii of stars in the CTL only, from which we intentionally exclude giants. We shifted this polynomial by +0.01036 mag to match up exactly with the one from Gaia at 8000 K. For this hotter segment, we have adopted a constant error in BC_G of 0.04 mag based on the scatter as a function of log g, which also provides continuity with the Gaia uncertainties.

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The complete set of relations described above is therefore as follows:

For the range 3300–4000 K, and where X ≡ ${T}_{\mathrm{eff}}$ − 5772 K,

$$\begin{eqnarray}\begin{array}{rcl}{\mathrm{BC}}_{G} & = & 1.7454+1.977\times {10}^{-3}X+3.737\times {10}^{-7}{X}^{2}\\ & & -\ 8.966\times {10}^{-11}{X}^{3}-4.183\times {10}^{-14}{X}^{4}\,\mathrm{mag}.\end{array}\end{eqnarray} \tag{ 5 }$$

The uncertainty in BC_G is given by

$$\begin{eqnarray}\begin{array}{rcl}{\sigma }_{{\mathrm{BC}}_{G}} & = & -2.487-1.876\times {10}^{-3}X+2.128\times {10}^{-7}{X}^{2}\\ & & +\ 3.807\times {10}^{-10}{X}^{3}+6.570\times {10}^{-14}{X}^{4}\,\mathrm{mag}.\end{array}\end{eqnarray} \tag{ 6 }$$

For the range 4000–8000 K,

$$\begin{eqnarray}\begin{array}{rcl}{\mathrm{BC}}_{G} & = & 0.0600+6.731\times {10}^{-5}X-6.647\times {10}^{-8}{X}^{2}\\ & & +\ 2.859\times {10}^{-11}{X}^{3}-7.197\times {10}^{-15}{X}^{4}\,\mathrm{mag}\end{array}\end{eqnarray} \tag{ 7 }$$

with an uncertainty given by

$$\begin{eqnarray}\begin{array}{rcl}{\sigma }_{{\mathrm{BC}}_{G}} & = & 2.634\times {10}^{-2}+2.438\times {10}^{-5}X-1.129\times {10}^{-9}{X}^{2}\\ & & -\ 6.722\times {10}^{-12}{X}^{3}+1.635\times {10}^{-15}{X}^{4}\,\mathrm{mag}.\end{array}\end{eqnarray} \tag{ 8 }$$

For the range 8000–12,000 K,

$$\begin{eqnarray}\begin{array}{rcl}{\mathrm{BC}}_{G} & = & -3.70485+1.32935Y-0.144609{Y}^{2}\\ & & +\ 0.00457793{Y}^{3}\,\mathrm{mag}\end{array}\end{eqnarray} \tag{ 9 }$$

where Y ≡ ${T}_{\mathrm{eff}}$/1000, with a constant uncertainty in BC_G of 0.04 mag. Note that the independent variable is different here, for numerical reasons.

This last polynomial should not be extrapolated beyond 12,000 K, so we are not able to compute BC_G (and therefore radius using the parallax) for ${T}_{\mathrm{eff}}$ > 12,000 K. For stars cooler than 3300 K, the Gaia polynomial relation could in principle be extrapolated by a small amount, although in practice we adopt the stellar parameters for M dwarfs from the specially curated Cool Dwarf list.

We can infer stellar mass from ${T}_{\mathrm{eff}}$ for stars that are on the main sequence or not too far evolved from it. Therefore, we only apply our ${T}_{\mathrm{eff}}$–mass relation if the stellar radius places the star below the red giant branch and above the white dwarf sequence, as defined in Section 3.1 (see Figure 11). Note that we implicitly are reporting a mass for stars that are subgiants; these should be regarded with caution. However, the luminosities that are reported for these subgiants are expected to be reliable, as the luminosities depend only on radius and ${T}_{\mathrm{eff}}$ (see Section 2.4).

We have revised slightly the spline relations that we developed for stellar mass as a function of ${T}_{\mathrm{eff}}$ by Stassun et al. (2018), with the result that the formal errors are now somewhat smaller. Table 3 gives the spline nodes for the mean relation (unchanged from TICv7; Stassun et al. 2018) and the new nodal points for the lower and upper error bars, as functions of ${T}_{\mathrm{eff}}$. Approximate spectral types are also provided for convenience.

Table 3.  Updated Spline Relation for TICv8

Approx. Spectral Type	${T}_{\mathrm{eff}}$	Mean Mass	Lower Limit	Upper Limit
⋯	55,000	91.052	81.0	100.5
O5	42,000	40.0	36.0	44.0
B0	30,000	15.0	13.5	17.0
⋯	22,000	7.5	6.7	8.5
B5	15,200	4.4	3.95	4.95
B8	11,400	3.0	2.65	3.4
A0	9790	2.5	2.2	2.85
A5	8180	2.0	1.75	2.35
F0	7300	1.65	1.45	2.00
F5	6650	1.4	1.23	1.70
G0	5940	1.085	0.965	1.22
G5	5560	0.98	0.87	1.11
K0	5150	0.87	0.78	0.98
K5	4410	0.69	0.615	0.77
M0	3840	0.59	0.51	0.662
M2	3520	0.47	0.395	0.535
M5	3170	0.26	0.21	0.30
⋯	2800	0.117	0.091	0.14
⋯	2500	0.056	0.042	0.07

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As an independent check on our derived stellar radii, we have compared our radii determined as above with those determined from fitting to stellar spectral energy distribution (SED) models by Deacon et al. (2019) and asteroseismically by Huber et al. (2017) for stars in common in the Kepler field. For Huber et al. (2017), we find very good agreement, as shown in Figure 10, with a mean difference of 0.64% and rms scatter of 7.03%, though with a slight skew toward larger radii in the TIC.

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Comparing to Deacon et al. (2019) finds very good agreement as well. For ∼843,000 stars in common, the mean radius from CTLv8 is 3.3% larger, which is about a 0.6σ offset. We observed a small number of large outliers due mainly to differences in the adopted distances; while TICv8 uses the Bayesian distance estimator from Bailer-Jones et al. (2018), Deacon et al. (2019) use the Gaia parallax directly, which can differ significantly for very small parallaxes.

We follow the same procedures as in the original CTL (Stassun et al. 2018), with the same assumed parameters. Briefly, contaminants are searched for within 10 TESS pixels of the target, and the contaminating flux is calculated within a radius that depends on the target's TESS magnitude and uses a PSF that is based on prelaunch PSF measurements of the field center (note that the PSF model does not attempt to account for bleed trails from very bright stars). The flux contamination reported is simply the ratio of the total contaminant flux to the target star flux (Figure 13). See Section 3.2.3 of Stassun et al. (2018) for more details.

We have implemented a Monte Carlo based approach to improve the final uncertainty estimates for the stellar parameters reported in the CTL, in which we perturb each observed quantity 1000 times and carry the perturbed values through the calculations to obtain a distribution for each derived quantity. We then report the 16th and 84th percentiles of those distributions as the corresponding lower and upper error bars. We have done this for two main reasons. First, we wish to be able to report asymmetric errors to better reflect the nature of the parameter posteriors. Second, a simple summation in quadrature of the underlying parameter errors overestimates the final uncertainty. This overestimation of the final uncertainty is particularly severe for the stellar radius because the distance error enters four times (D, G, T_eff, BC_G), the errors from the reddening maps enter three times (G, T_eff, BC_G), the error from the ${T}_{\mathrm{eff}}$ calibration enters twice (T_eff, BC_G), and photometric errors in G_BP − G_RP enter twice (T_eff, BC_G).

In what follows, we represent Monte Carlo perturbed quantities with primed (or double-primed) symbols and nominal values with unprimed symbols. In addition, ${ \mathcal N }$ represents a normal Gaussian deviate (mean = 0, σ = 1) that is used to perturb the nominal quantities. Once a quantity is perturbed, we use the same perturbed value throughout the procedure in order to preserve parameter correlations. For perturbing quantities with asymmetric error bars, we assume each side is reasonably well represented by a Gaussian distribution, and we use the lower error bar if the Gaussian deviate is negative or the upper error bar otherwise.

The procedure follows the steps below, in sequence:

• 1.  
  Perturb the stellar distance: $D^{\prime} =D+{ \mathcal N }\times \{{\sigma }_{D,\mathrm{low}},{\sigma }_{D,\mathrm{high}}\}$.
• 2.  
  Perturb the reddening, which involves two contributions: one from the distance, and another from the intrinsic dust map errors. Query the dust map(s) with D' to obtain a perturbed reddening E(B − V)''. For Pan-STARRS, find the 16th and 84th percentiles of the reddening distribution at the nominal distance, and subtract from reddening at nominal distance to obtain the intrinsic dust map errors, {σ_red,low, σ_red,high}. For Schlegel et al. (1998), adopt σ_red,low = σ_red,high = 0.01 mag, derived from typical Pan-STARRS errors for stars outside the Galactic plane.³⁵ Then compute $E(B-V)^{\prime} \,=E(B-V)^{\prime\prime}$ + ${ \mathcal N }\times \{{\sigma }_{\mathrm{red},\mathrm{low}},{\sigma }_{\mathrm{red},\mathrm{high}}\}$, and calculate reddening and extinction in the Gaia passbands with E(${G}_{\mathrm{BP}}$ − ${G}_{\mathrm{RP}}$)' = 1.31E(B − V)' and A(G)' = 2.72E(B − V)'.
• 3.  
  Perturb the Gaia color and magnitude using the photometric errors: $({G}_{\mathrm{BP}}-{G}_{\mathrm{RP}})^{\prime} =({G}_{\mathrm{BP}}-{G}_{\mathrm{RP}})$ + ${ \mathcal N }\,\times {\sigma }_{{G}_{\mathrm{BP}}}$ + ${ \mathcal N }\times {\sigma }_{{G}_{\mathrm{RP}}}$, and $G^{\prime} =G+{ \mathcal N }\times {\sigma }_{G}$.
• 4.  
  Deredden the perturbed Gaia color and magnitude: $({G}_{\mathrm{BP}}-{G}_{\mathrm{RP}}{)}_{\mathrm{dered}}^{{\prime} }$ = (${G}_{\mathrm{BP}}$ − ${G}_{\mathrm{RP}}$)' − E(${G}_{\mathrm{BP}}$ − ${G}_{\mathrm{RP}}$)' and ${G}_{\mathrm{dered}}^{{\prime} }$ = G' − A(G)'.
• 5.  
  Compute the perturbed and dereddened TESS magnitude ${T}_{\mathrm{dered}}^{{\prime} }$ with the expression in Section 2.3.1 using ${G}_{\mathrm{dered}}^{{\prime} }$ and $({G}_{\mathrm{BP}}-{G}_{\mathrm{RP}}{)}_{\mathrm{dered}}^{{\prime} }$. Calculate the perturbed extinction in the T band as A(T)' = 2.06E(B − V)'. Then apply the extinction and compute the final perturbed apparent TESS magnitude (i.e., affected by extinction) as $T^{\prime} ={T}_{\mathrm{dered}}^{{\prime} }\,+A(T)^{\prime}$ + ${ \mathcal N }\times {\sigma }_{T}$, where σ_T is the scatter of the T calibration.
• 6.  
  Compute the perturbed ${T}_{\mathrm{eff}}$: ${T}_{\mathrm{eff}}^{{\prime} }={T}_{\mathrm{eff},\mathrm{dered}}^{{\prime} }$ + ${ \mathcal N }\times {\sigma }_{{T}_{\mathrm{eff}}}$, where ${T}_{\mathrm{eff},\mathrm{dered}}^{{\prime} }$ is the perturbed temperature derived from the perturbed dereddened color, and ${\sigma }_{{T}_{\mathrm{eff}}}$ = 122 K (Section 2.3.4).
• 7.  
  Compute the perturbed radius (R') from the perturbed bolometric correction (BC'), ${T}_{\mathrm{eff}}^{{\prime} }$, D', and ${G}_{\mathrm{dered}}^{{\prime} }$, where $\mathrm{BC}^{\prime} =\mathrm{BC}({T}_{\mathrm{eff}}^{{\prime} })$ + ${ \mathcal N }\times {\sigma }_{\mathrm{BC}}$, and σ_BC is a function of ${T}_{\mathrm{eff}}^{{\prime} }$.
• 8.  
  Compute the perturbed mass as $M^{\prime} =M({T}_{\mathrm{eff}}^{{\prime} })+{ \mathcal N }$ × $\{{\sigma }_{M,\mathrm{low}},{\sigma }_{M,\mathrm{high}}\}$, where σ_M is a function of ${T}_{\mathrm{eff}}^{{\prime} }$. Then compute the perturbed log g, luminosity, and mean density as log g' = 4.4383 + log M' − 2 log R', L' = (R')² (${T}_{\mathrm{eff}}^{{\prime} }/5772$)⁴, and ρ' = M'/(R')³.