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 limb 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. 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 limb 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. Optical depth combines with effective temperature gradients inside the star to produce limb darkening. The light seen is approximately the integral of all emission along the line of sight modulated by the optical depth to the viewer (i.e. 1/e times the emission at 1 optical depth, 1/e2 times the emission at 2 optical depths, etc.). Near the center of the star optical depth is effectively infinite, causing approximately constant brightness. Near the edge of the star the optical depth decreases due to lower gas density and shorter distance through the sun until the effective optical depth becomes zero at the apparent edge of the sun. The effective temperature of the photosphere also decreases for an increasing distance from the center of the star. The radiation emitted from a gas is approximately black-body radiation, which is proportional to the fourth power of the temperature. Thus, even before the optical depth in a direction becomes finite, the emitted energy comes more from cooler parts of the photosphere, resulting in less total energy reaching the viewer. The temperature in the atmosphere of a star is not always decreasing with increasing height. For certain spectral lines, the optical depth is greatest in a region of increasing temperature. In that case the phenomenon of 'limb brightening' is seen instead; 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, lies 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.