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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.
Calculation of limb darkeningEdit
In the figure shown here, as long as the observer at point P is outside the stellar atmosphere, the intensity seen in the direction θ will be a function only of the angle of incidence ψ. This is most conveniently approximated as a polynomial in cos ψ:
where I(ψ) is the intensity seen at P along a line of sight forming angle ψ with respect to the stellar radius, and I(0) is the central intensity. In order that the ratio be unity for ψ = 0, we must have
For example, for a Lambertian radiator (no limb darkening) we will have all ak = 0 except a0 = 1. As another example, for the sun at 550 nm, the limb darkening is well expressed by N = 2 and
(See Cox, 2000). The equation for limb darkening is sometimes more conveniently written as
which now has N independent coefficients rather than N + 1 coefficients that must sum to unity.
The ak constants can be related to the Ak constants. For N = 2,
For the sun at 550 nm, we then have
This model gives an intensity at the edge of the sun's disk of only 30% of the intensity at the centre of the disk.
We can convert these formulas to functions of θ by using the substitution
where Ω is the angle from the observer to the limb of the star. For small θ we have
We see that the derivative of cos ψ is infinite at the edge.
The above approximation can be used to derive an analytic expression for the ratio of the mean intensity to the central intensity. The mean intensity Im is the integral of the intensity over the disk of the star divided by the solid angle subtended by the disk:
where dω = sin θ dθ dφ is a solid angle element, and the integrals are over the disk: 0 ≤ φ ≤ 2π and 0 ≤ θ ≤ Ω. We may rewrite this as
Although this equation can be solved analytically, it is rather cumbersome. However, for an observer at infinite distance from the star, can be replaced by , so we have
For the sun at 550 nm, this says that the average intensity is 80.5% of the intensity at the centre.
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