Create a special white circle

I am a linux user. I installed MyPaint and GIMP a few days ago and cannot figure out the following. I would like to draw a white circle on a black background. In full-screen mode the white circle should cover most of the screen. The circle should be homogenous with the white-level uniform across the circle. This is part one and likely easy for you to do. Now the tricky part 2: the circle should now have a gradient with a centre-region with white-level (full brightness) as in part 1 and then decreasing linearly towards the limb (the edge of the circle). Basically, I would like to simulate the so-called “limb-darkening” effect (Limb darkening - Wikipedia) as seen for our Sun: bright at the centre and darker towards the limb. The limb-darkening effect can be simulated using different brightness-profile laws for the brightness change (linear, quadratic, square-root, log. etc). If we can start with the linear part and then add complexity that would be cool. [login to view URL] The final project should deliver instructions for me on how to generate pictures that simulates the limb-darkening effect for various laws.