systematic error added in quadrature to each synthetic magnitude to account for systematic errors in the conversion to observed magnitudes.

For each trial combination of these parameters we use the empirical colour – effective temperature relations by @2013ApJ...771...40B to predict the apparent magnitudes for each star in each of the observed bands. We used the same transformation between the Johnson and 2MASS photometric systems as Boyajian et al.. We used the Cousins I$_{\rm C}$ band as an approximation to the DENIS Gunn i$^{\prime}$ band and the 2MASS K$_{\rm s}$ band as an approximation to the DENIS K band [see Fig. 4 @2005ARA+A..43..293B]. We used interpolation in Table 3 of @2000PASP..112..961B to transform the Johnson B, V magnitudes to Tycho-2 B$_{\rm T}$ and V$_{\rm T}$ magnitudes. We assume that the extinction in the V band is $3.1\times {\rm E}({\rm B}-{\rm
V})$. Extinction in the SDSS and 2MASS bands is calculated using A$_{\rm r} =
2.770\times {\rm E}({\rm B}-{\rm V})$ from and extinction coefficients relative to the r$^{\prime}$ band from @2014MNRAS.440.3430D.

We use the transformation from Sloan g$^{\prime}$, r$^{\prime}$ and i$^{\prime}$ magnitudes by @2011AJ....142..112B to estimate Kepler Kp magnitudes for each star in the system. This enables us to include the flux 