Limb darkening

Limb darkening is an optical effect seen in stars (including the Sun), where the center part of the disk appears brighter than the edge or of the image. Its understanding offered early solar astronomers an opportunity to construct models with such gradients. This encouraged the development of the theory of radiative transfer.

Crucial to understanding limb darkening is the idea of optical depth. A distance equal to one optical depth is the thickness of the absorbing gas from which a fraction of 1/e photons can escape. This is what defines the visible edge of a star, since it is at a few optical depths that the star becomes opaque. The radiation reaching us is closely approximated by the sum of all the emission along the entire line of sight, up to that point where the optical depth is unity. In particular, if the intensity of radiation in the star varies linearly with optical depth, then the radiation reaching us will be of the intensity at an optical depth of unity. When we look near the edge of a star, we cannot "see" to the same depth as when we look at the center because the line of sight must travel at an oblique angle through the stellar gas when looking near the limb. In other words, the solar radius at which we see the optical depth as being unity increases as we move our line of sight towards the limb.

The second effect is the fact that the effective temperature of the stellar atmosphere is (usually) decreasing for an increasing distance from the center of the star. The radiation emitted from a gas is a strong function of temperature. For a black body, for example, the spectrally integrated intensity is proportional to the fourth power of the temperature (Stefan–Boltzmann law). Since when we look at a star, at first approximation, the radiation comes from the point at which the optical depth is unity, and that point is deeper in when looking at the center, the temperature will be higher, and the intensity will be greater, than when we look at the limb.

In fact, the temperature in the atmosphere of a star does not always decrease with increasing height, and for certain spectral lines, the optical depth is unity in a region of increasing temperature. In this case we see the phenomenon of "limb brightening"; for the Sun the existence of a temperature minimum region means that limb brightening should start to dominate at far-infrared or radio wavelengths. Outside the lower atmosphere, and well above the temperature-minimum region, we find the million-kelvin solar corona. For most wavelengths this region is optically thin, i.e. has small optical depth, and must therefore be limb-brightened if spherically symmetric.

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