Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2015 / S2 / 2

The effective temperature is defined by the Stefan–Boltzmann law applied to a star's total luminosity:Here $R$ is the adopted photospheric radius and ${\sigma }_{\mathrm{SB}}$ is the Stefan-Boltzmann constant. This defines the temperature of a black body radiating the same total flux per unit surface area; it does not assert that every wavelength of the stellar spectrum follows a perfect Planck spectrum.

The parallax gives $d=1/0.00763\simeq 131\mathrm{pc}$. Ignoring extinction, the distance modulus givesThe bolometric magnitude relation then yields $L/L_{\odot }=10^{0.4(4.72-M_{\mathrm{bol}})}\simeq 2.53\times 10^4$. With $T_{\odot }\simeq 5772\mathrm{K}$,A star this cool and luminous belongs to the red-supergiant region of the Hertzsprung-Russell diagram. Extinction and the adopted effective temperature are important systematic inputs; this calculation uses the supplied temperature to infer the radius.

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.