Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2012 / S2 / 1

For two bodies of masses $M_1,M_2$, the relative Kepler orbit has semimajor axis $a$ and orbital period $P$ satisfying Kepler's third lawHere $a$ is the semimajor axis of the separation vector, not either star's individual barycentric orbit. In conventional stellar units the same relation is approximately $(P/\mathrm{yr}{)}^2=(a/\mathrm{AU}{)}^3/[(M_1+M_2)/M_{\odot }]$, with year, astronomical unit and solar mass as the units.

Equal radial velocity amplitudes imply equal stellar masses. Under the circular, edge-on eclipsing binary interpretation, each star moves at $K=100\mathrm{km}{\mathrm{s}}^{-1}$, and the relative speed is $2K$. ThusUsing $P=5.45$ days givesMore generally, spectroscopic binary amplitudes determine $a\sin i=P(K_1+K_2)\sqrt{1-e^2}/(2\pi )$ and $(M_1+M_2){\sin }^{3}i=P(K_1+K_2{)}^3(1-e^2{)}^{3/2}/(2\pi G)$. Thus these numerical masses and separation use $e=0$, $\sin i=1$; eclipses alone imply high inclination rather than its exact value.

TODO: the source light curve, its numerical contact phases and the meanings of the four plotted time labels are absent, so unique numerical stellar radii cannot be recovered. The central eclipse contact-radius relations explain what the missing measurements would determine.

If $t_1,\dots ,t_4$ denote the usual first through fourth contacts of a central eclipse, define $T_{14}=t_4-t_1$ and $T_{23}=t_3-t_2$, and let $R_L\ge R_S$ be the two radii. For a circular edge-on orbit,Taking half their sum and difference gives the radii. For short eclipses the familiar approximation is $R_L\simeq v_{\mathrm{rel}}(T_{14}+T_{23})/4$ and $R_S\simeq v_{\mathrm{rel}}(T_{14}-T_{23})/4$, with $v_{\mathrm{rel}}=2K$. At first and fourth contact the discs touch externally; at second and third the smaller disc is just internally tangent to the larger. Noncentral eclipses require the projected impact parameter as well.