There are few direct methods for determining the radii of stars. One of the most
important uses eclipses of binary star systems. By carefully examining the light
curve, it is possible to determine how long it takes one star to pass in front
of the other. This can be used to infer the sizes of the stars.
Geometry of a Binary Eclipse
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At point 1 the eclipse starts and the light begins to decrease. At point 2 the eclipse becomes total and the light output is constant until point 3 where the smaller star begins to emerge and the light output begins to grow until finally at point 4 the eclipse is complete.
Diameters from Timing
From point 1 to point 2 the smaller star moves a distance on its orbit equal to its diameter D. But if the magnitude of the velocity v for the smaller star can be determined from Doppler methods and we carefully measure the time t from point 1 to point 2, the diameter follows from multiplying the velocity by this time: D = v t.By similar reasoning, in the time from when the eclipse begins until it ends, the smaller star covers a distance equal to the diameter of the larger star. Thus, by carefully measuring this interval and multiplying by the velocity we can determine the diameter of the larger star.
Diameters and Partial Eclipses
In the example shown here we assume for simplicity of illustration that the eclipses are complete and the orbits are circular. Even if the eclipses are only partial and the orbits are not circular, an analysis of the detailed shape of the light curve may still allow diameters to be measured.
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