Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2016 / S2 / 1

The first law gives elliptical relative orbits with the center of force at a focus; the second gives equal swept areas in equal times, expressing angular-momentum conservation. For circular binary motion, let $r_1+r_2=a$ and $M_1{r}_1=M_2{r}_2$. The force equation $M_1(2\pi /P{)}^2{r}_1=G{M}_1{M}_2/a^2$ givesThe corresponding relation for ellipses uses the relative semimajor axis rather than instantaneous separation.

Eclipses justify an approximately edge-on orbit, and equal velocity amplitudes imply equal masses. The relative speed is 200 km s${}^{-1}$, so $a=(v_1+v_2)P/(2\pi )\simeq 1.50\times 10^{10}$ m, or $\boxed{a\simeq 0.100\mathrm{AU}}$. Substitution in the binary period law gives $M_1+M_2\simeq 4.52M_{\odot }$, hence $\boxed{M_1=M_2\simeq 2.26M_{\odot }}$. Inclination corrections scale the dynamical masses as ${\sin }^{-3}i$ if exact edge-on viewing is relaxed.

Assume a central eclipse and nearly uniform projected speed during it. Let $r$ be the smaller radius and $R$ the larger. First-to-second contact requires travel $2r$, whereas the totality interval requires $2(R-r)$. With relative speed 200 km s${}^{-1}$,The light curve has ingress, flat minimum and egress lasting 1.7, 0.8 and 1.7 hours. At the four contacts the projected center separations are $R+r,R-r,R-r,R+r$. A noncentral chord would require an impact parameter and changes the inferred radii; the data alone do not fix that geometry exactly. For comparable masses in an ordinary coeval system, a component larger than its main sequence companion suggests expansion away from the main sequence, conventionally toward a subgiant/early giant track. That is a conditional evolutionary interpretation: neither effective temperature nor a bolometric flux is supplied, so a unique position on the Hertzsprung-Russell diagram cannot be inferred from these radii alone. Binary interactions and the assumed central chord also matter; these numbers do not establish an ordinary single-star evolutionary model.