 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, ${\lambda }_{\max }T\simeq 2.90\times 10^{-3}\mathrm{m}\mathrm{K}$, 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 $T_{\mathrm{eff}}=[4F_{\mathrm{bol}}/({\sigma }_{\mathrm{SB}}{\Theta }^2){]}^{1/4}$. 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.