Limb darkening

Look at a picture of the Sun carefully. Compare the appearance of the center of the solar disk to the edges.

It looks brighter in the center, doesn't it? A color picture shows another difference:

The limbs of the Sun are not only dimmer than the center, they are redder, too. Why?

Part of the answer lies in the gradient of temperature within the solar photosphere. The deeper you go, the hotter it gets.

The other part of the answer involves optical depth. Light rays can escape from the Sun's photosphere when they are about 1 optical depth away from empty space. That means that we see photons which come from 1 optical depth "below the surface":

When we look at the center of the solar disk, we see light rays which are coming radially outwards. They originate relatively deep in the photosphere, where the temperature is relatively high. When we look at the limbs, we see light rays which must skim through the photosphere at a shallow angle to reach the Earth. They originate in the upper reaches of the photosphere, where the temperature is somewhat lower.

Since the source spectrum is (approximately) a blackbody in both instances, the radiation coming from the hot material near the center of the disk will be more intense and have a shorter peak wavelength. If we know exactly how temperature changes with radius, and how opacity and density change with radius, we can predict the change in surface brightness as a function of angular distance away from the center of the disk. See the example on page 292 of your textbook, which derives an expression of the form

We can also measure the brightness as a function of position very easily -- for the Sun:

Be careful when comparing limb darkening. Some studies, like the one above, use angular distance from the center of disk as the independent variable (on the horizontal axis). Others use the cosine of this angle.

Now, apart from the Sun, there are very, very few stars which are big enough, and close enough, that astronomers can resolve their photospheres and measure their limb darkening. One of the very few which can be studied directly is the star Betelgeuse; but look at how few measurements we can make across its disk:

Apart from Betelgeuse and 3 or 4 others, stars exhibit such small angular diameters that we have no way to resolve their disks, and so no way to measure their limb darkening.