Two stars shaped like rugby balls

Adapted from IOAA 2009 (Tehran), theory problem 15. We have restated it in our own words and kept the original data. The solution was last checked on 21 October 2026.

A close eclipsing binary has two giant stars of the same size, pulled by each other's gravity into prolate spheroids with $a=2b$, where $a$ and $b$ are the semi-major and semi-minor axes, and their long axes always lie along the line between them. A prolate spheroid is the shape made by turning an ellipse about its long axis, like a rugby ball. The orbit is seen edge on. Ignoring any differences in temperature over the surfaces and darkening towards the edges, find the amplitude of the variation of the light, in magnitudes.

AnswerThe light varies by 1.5 magnitudes from brightest to faintest. The official solution writes $\mathrm{\Delta }m=-1.5$, but an amplitude is quoted as a positive number of magnitudes.