I get what you want to do. however applying a gradient to a selection like that would be linear, not exactly what you are looking for, or so I believe.
Instead make your black circle larger so that it fill about half of what that selection would be.
Now feather the circle this can be done by using the feather command in the selection menu, or better yet, use a Gaussian blur on the circle.
Or better yet, Instead of enlarging the circle, you use that selection, but change it to include the circle (this can be done by using the modifier keys and clicking on the circle's thumbnail in the layers panel) and create a mask using that selection, then fill that selected portion with white. Finally apply a Gaussian blur to that mask. (you may need to fix the edge of the mask by painting with a brush or use a selection tool to make sure the edge is pure white and not blurred)
As you can see there are several ways to accomplish the same thing, you could even try using styles. The point is try to make it as simple as possible but also try to make it non-destructible and there fore re-editable and repeatable.