 Solution

ID: 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/s2/1/3/solution

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.

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} \geq R_{S}

be the two radii. For a circular edge-on orbit,

R_{L} + R_{S} = a sin \frac{π T _{14}}{P}, R_{L} - R_{S} = a sin \frac{π T _{23}}{P} .

Taking half their sum and difference gives the radii. For short eclipses the familiar approximation is

R_{L} ≃ v_{rel} (T_{14} + T_{23}) /4

and

R_{S} ≃ v_{rel} (T_{14} - T_{23}) /4

, with

v_{rel} = 2 K

. 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.

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