Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2017 / S2 / 2

The effective temperature is defined by $\boxed{L=4\pi R_{\ast }^2\sigma T_{\mathrm{eff}}^4}$, with bolometric luminosity $L$, photospheric radius $R_{\ast }$ and Stefan-Boltzmann constant $\sigma$. It is the blackbody temperature giving the same total surface flux, not a claim that each spectral line forms at that temperature.

The parallax gives $d=20$ pc and the distance modulus gives $M_{\mathrm{bol}}=-0.641-5{\log }_{10}2=-2.146$. Therefore $L/L_{\odot }=10^{(4.72-M_{\mathrm{bol}})/2.5}\simeq 558$. Taking $T_{\odot }\simeq 5772$ K,This is a cool luminous giant, in the red-giant region of the Hertzsprung-Russell diagram. On the usual first-ascent interpretation, luminosity comes predominantly from hydrogen shell burning around an inert helium core, with gravitational adjustment also contributing. The two measured quantities alone do not uniquely determine evolutionary history: a later giant phase can overlap this region, so they do not prove the precise core-burning state.

A continuum peak gives a color temperature through Wien's law, provided the continuum is approximately Planckian and extinction is corrected. Broad-band colors or a fitted spectral energy distribution compare flux ratios with model atmospheres; they require a reddening estimate, abundance and gravity assumptions, and calibrated passbands. Spectroscopic line ratios or excitation/ionization balances provide another thermometer, requiring atomic data and a suitable atmosphere model, often local thermodynamic equilibrium.

For this cool giant, line blanketing and molecular absorption substantially distort a simple continuum peak, so Wien's law is the least reliable of these approximations. A detailed spectroscopic analysis, checked against a multi-band atmosphere fit, would generally be the strongest of the three if good data and model parameters are available. There is no unconditional accuracy ranking without measurement errors and model systematics; an accurately measured angular diameter plus bolometric flux would offer a more direct effective-temperature determination, if obtainable.