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# 12.15: The Size of Stars

A parsec is the distance from theÂ Sunto anÂ astronomical objectÂ which has aparallaxÂ angle of oneÂ arcsecond. (1 AU and 1 pc are not to scale (1 pc = 206265 AU)). Click here for original source URL.

The size of Betelgeuse compared to our solar system. Click here for original source URL.

Size is a fundamental property of a star. It is extremely difficult to measure the size of a star by direct observation. We can use the small angle equation to see why. Imagine a star like the Sun at a distance of a parsec. The angle subtended by a solar-type star at that distance is 206,265 (d / D), where the diameter of the Sun, d, is 1.4 × 106 kilometers, and D = 3 × 1013 kilometers. We see that the star is only 0.01 seconds of arc across. That’s the distance to the nearest stars so that’s the largest a star will appear! Most stars will appear much smaller. Since the Earth’s atmosphere blurs the images of all stars by about 0.5 to 1 second of arc, we have no hope of resolving such a distant star from the ground. A useful analogy to this problem is the resolution limit of a computer screen – a point that is smaller than one pixel in size would be smeared to one pixel. Even from space, a large telescope would be needed to make an image of most stars with high enough resolution to see surface features. The Hubble Space Telescope was barely able to resolve the relatively close super giant star Betelgeuse, in the constellation Orion, in ultraviolet light.

Beyond measuring a star's size from its angular extent on the sky, it is also possible to derive a star's size if it’s in an eclipsing binary system. From stellar spectroscopy, we can measure the velocity of the stars. By combining Kepler's laws, velocity data and the length of time between eclipses, it becomes possible to determine the size of the stellar orbits. Once the distance between the stars is known, the amount of time it takes one start to eclipse the other yields the ratio of the stellar radii.

Graphic representation of Wein's Law. Click here for original source URL.

Hertzsprung-Russell diagram showing color and size of stars. Click here for original source URL.

There is an indirect method for calculating a star’s diameter. Once we know a star’s temperature and luminosity, we can apply the Stefan-Boltzmann law to measure its diameter. All objects emit thermal radiation with a peak wavelength that relates to the temperature. Hotter objects emit shorter wavelength radiation (Wien’s law). The Stefan-Boltzmann law says that, in addition, hotter objects emit more photons per second. That is, as an object heats up, both the frequency and the amount of thermal radiation increase. This important relationship allows us to determine a star’s diameter. The equation used relates the luminosity of a star to its temperature and radius, L = 4 π R2 σ T4, where σ is the Stefan-Boltzmann constant, which has a value 5.67 × 10-8 when luminosity is measured in Watt, size in meters, and temperature in Kelvin. Such measures indicate a vast range of stellar diameters, from less than the size of the Earth to huge stars bigger than the diameter of Jupiter’s orbit!