How to fill radial array elements with gradient like this:
No idea how to make it happen automatically, rather than element by element.
Start with the flat pieces of the star. Make certain they are a compound shape by selecting them all and choosing
Object > Compound Path > Make from the menu. (It doesn’t matter what color they are.)
Now draw a circle which crosses roughly at the center of the shapes.
Change the stroke to be a gradient and very thick. Thick enough so all your shapes fall within the stroke’s boundaries. And move it behind the shapes. You may need to rotate the circle itself so the gradient will start/stop between two of the shapes.
Select the shapes and the stroke and choose
Object > Clipping Mask > Make.
(Note gradient strokes requires Illustrator CS6 or better.)