Brachistochrone problem on astronomical scales?

I was revising vector calculus recently, and couldn't help but incorrectly have a train of thought about its interrelation with the calculus of variations, and the relevance of both in Mechanics. After thinking about this for a bit of time, and resolving some of my previous confusion I have ended up with a simple question. I have learned that a quarter of a cycloid is the curve that satisfies and is a solution to the Brachistochrone problem for when we assume gravity to be constant.

However, I was wondering, what kind of curve is a solution to the same problem when we don't assume g to be constant, and take the familiar newtonian expression for it? Also does this bear some reference to the study of orbits?

I would hazard a guess, that half the shape of a comet's orbit would be something close to the solution for this problem, and therefore that the curve would be a hyperbola. I'm not at all sure about this though, and am already doubting the validity of my previous sentence.

So, any ideas?