= Solution
A first distance-independent method fits the continuum shape or a calibrated <colour index> to an atmosphere model. A black-body approximation uses <Wien's displacement law>, $\lambda_{\max}T\simeq2.90\times10^{-3}\,\mathrm{m\,K}$, or ratios of the <Planck spectrum> at two wavelengths. It requires a known extinction correction and sufficient treatment of molecular absorption and line blanketing; a cool supergiant need not have a black-body colour temperature equal to its <effective temperature>.
A second method fits temperature-sensitive <spectral lines> or molecular bands. <Boltzmann excitation> and the <Saha equation> explain why excitation and ionization ratios change with temperature; detailed atmosphere calculations also include surface gravity, composition and departures from <local thermodynamic equilibrium>. TiO molecular bands are particularly useful for cool stars. Line ratios remove the overall distance normalization, but their calibration remains model dependent.
The direct flux-radius method can be made distance independent as well: an independently measured angular diameter $\Theta$ gives $T_{\rm eff}=[4F_{\rm bol}/(\sigma_{\rm SB}\Theta^2)]^{1/4}$. For a large nearby supergiant with a resolved photosphere, good bolometric flux and a limb-darkened <interferometric angular diameter>, this is often the most direct determination. Its advantage depends on those measurement uncertainties and a consistent radius definition. The radius inferred in the preceding part cannot itself provide an independent temperature measurement, because the supplied temperature was already used to calculate it.
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