Past exam of the short options paper of the physics Prelims of the University of Oxford / 2022 / S02 / 1

Assume spherical stars, central eclipses, uniform disc surface brightness, and negligible third light. Let $s$ denote the smaller star and $\ell$ the larger one, with $r=R_s/R_{\ell }\le 1$ and ${\ell }_s=L_s/L_{\ell }$. Normalize the out-of-eclipse flux to one, and denote the deep and shallow fractional losses by $D$ and $d$. The graph gives approximately $F_{\mathrm{out}}\simeq 0.98$, $F_{\mathrm{deep}}\simeq 0.57$, $F_{\mathrm{shallow}}\simeq 0.90$, so $D\simeq 0.42$, $d\simeq 0.08$; the last digits are graphical estimates.

At total occultation of the smaller star, the lost flux is $L_s/(L_s+L_{\ell })$. At its central transit across the larger star, the lost flux is $r^2{L}_{\ell }/(L_s+L_{\ell })$. There are two identifications:

These are ratios of luminosities in the observed band. Treating them as bolometric luminosities requires similar band corrections or an explicitly bolometric light curve. Limb darkening and finite exposure times limit the precision of these elementary estimates.

The Stefan–Boltzmann law gives $L=4\pi R^2\sigma T_{\mathrm{eff}}^4$ for bolometric luminosity. HenceWith the central values above, this is approximately $1.51$ for the first geometry and $0.67$ for the second. If only one-band luminosities are measured, bolometric corrections are needed before using this formula for effective temperatures.

Two direct methods are fitting the continuum stellar spectrum or several photometric colors to a stellar atmosphere model, and measuring temperature-sensitive absorption lines or line ratios in a stellar spectrum. The former includes the approximate black-body radiation estimate ${\lambda }_{\max }T=b$ from Wien's displacement law; the latter uses ionization and excitation populations. Observations during total eclipse or spectral disentangling help separate the two stellar spectra.

For comparable composition and accessible fuel fractions, the main-sequence lifetime is proportional to available nuclear fuel divided by luminosity, $t_{\mathrm{MS}}\propto M/L$. Estimate the masses from a calibrated mass-luminosity relation $L\propto M^{\alpha }$ in the appropriate mass range. ThenFor illustration $\alpha =3.5$ gives about $1.27$ and $5.6$ for the two graphical luminosity ratios. The common main sequence approximation is additional physics; eclipse depths alone do not determine masses or ages. A main sequence consistency check would also favor the second geometry because ordinary main-sequence stars have radius and effective temperature increasing with mass.