It is important to understand that limb darkening depends on
the color - the wavelength - you are looking at. So if you look at the Sun in the blue or in the red color, the limb darkening is different. This is what I would like to illustrate here. The x-axis shows the position on the disk, with zero being in the center and one being at the edge. The y-axis shows the brightness of the star divided by the value in the center.
The diagram visualizes how
limb darkening looks like for the continuum, which basically
means averaged over a large wavelength interval which in this case is in the
visual wavelength regime. The colors indicate the central wavelength of the used interval, each interval is 10 nm wide. So the 380 nm line (blue) shows how limb darkening looks like on average over the wavelengths from 375 to 385 nm.
The bottom line is: For all colors a star is brighter at the center than at the edge. But for longer wavelengths (red colors) the limb darkening is weaker than for short wavelengths (blue); the difference in brightness between center and limb is smaller for longer than for shorter wavelengths.
The bottom line is: For all colors a star is brighter at the center than at the edge. But for longer wavelengths (red colors) the limb darkening is weaker than for short wavelengths (blue); the difference in brightness between center and limb is smaller for longer than for shorter wavelengths.
For comparison I also draw one line in the ultraviolet (100 nm) and one in the infrared (2500 nm). In these cases, which are imperceptible by the eye, limb darkening really looks very different. In the IR the disk is only 20 % darker at the edges than in the center, whereas for UV the limb of the star is virtually dark.
Especially interesting is the fact that the radius of the star is not equal for all colors. Where the distance from the disk center equals one the stellar disk is supposed to end and the brightness should be zero. However, this is not true. In the UV the brightness is still non-zero well beyond the 'defined' edge of the star. Zooming into that region we would see that this is true for all colors - in each color the star has a slightly different 'size'. Although this might seem to be weird when thinking about it the first time, it actually is not very surprising. What we call the edge or surface of the Sun is not a real boundary, the Sun does not suddenly stop within a meter or even a kilometer. What we see as surface of the Sun is the region in the atmosphere where the visible light is coming from - the photosphere - which is several hundreds of kilometer thick. Above that layer comes the chromosphere, and above that the corona. So the Sun, and other stars too, does not have a sharp edge. This, maybe, makes it easier to understand why in the diagram the Sun does not have the same radius for all colors.
