The falling temperature of the photosphere with height is responsible for the effect known as limb darkening. If we look close to the limb of the Sun, to reach a set depth in the photosphere we will look through more material, or for the same amount of material we would see less far into the photosphere at the limb than we would at the sun centre. As the light we observe comes from a constant optical depth, this means that the limb radiation is characteristically cooler than that from the sun centre and hence will be less intense (and slightly redder) than that from the sun centre.
An approximate expression for the intensity of light coming from the surface of the star is
q = angle between the normal to the stellar surface
and the line of sight to the observer.
Io = the intensity of light in the center of the stellar disk (q=0)
w = is the wavelength dependent limb darkening coefficient (<1, ~0.3)
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