What is the intensity distribution of visible light over the solar disk?
What is the distribution of the visible spectrum over the solar disk
Hopefully I've answered all the different questions.
The core of the sun is extremely hot (27 million degrees Fahrenheit) compared to the surface (10,000 degrees Fahrenheit) and the atmosphere (36,000 to 2 million degrees Fahrenheit). Source: JPL NASA - "The Sun's Layers and Temperatures". To our eye the sun generally appears to have a single color and brightness due to dynamic range and color mixing.
The AM0 spectrum and intensity looks like this:
Each element emits a different multitude of wavelengths. The distribution across the surface, and out into the atmosphere, varies.
For example here is a set of images taken at different wavelengths:
That answers your question in this comment:
"Ah thank you, that is exactly what I was looking for. But how do images like these get produced? Are they adjusting the brightness to be equal througout the image before releasing it? – Jhonny007".
Images like those are produced by placing filters over the imaging sensor.
Limb darkening is caused by the different temperatures of the different layers. The center of the disk appears bumpy due to granules. Using a greater dynamic range it looks like this:
This image is a composite of 25 separate images spanning the period of April 16, 2012, to April 15, 2013. It uses the SDO AIA wavelength of 171 angstroms and reveals the zones on the sun where active regions are most common during this part of the solar cycle.
A closeup of a sunspot (a colder region, thus darker) looks like this:
The website Schurs Solar Imaging, and webpage: "Image Gallery of Solar Phenomenon - White Light Features" has a great set of photos, and one clearly showing limb darkening.
Adjusting the contrast of an image from the University of Virginia's webpage: "ASTR 1230 (Majewski) Lecture Notes" creates an image that shows limb darkening.