This from a friend / college math professor Dr. Michael Coleman at U of Albany:

"Now we hit the bad news - since we've got both inverse trig and some square roots floating around, the easiest way to envision the solution - just substitute in until it's a single variable problem, and solve - is bad news. VERY bad news. We need to solve:

d = sqrt((y-sqrt(r^2-a^2))^2 - ((((r^2-a^2)-y*sqrt(r^2-a^2))/a)^2)) + arccos(-a/r)*r

which is a nasty enough symbolic manipulation problem that the solvers I have access to on my home PC are throwing a **"computation time exceeded" error**, and I'm suspicious there's at least one branch in there. So I'm going to see if I can be more clever and present the problem in a way that doesn't cause that. Realistically, though, this says a numerical approximation is your best bet."

So, while there (likely) IS a direct solution, what Michael Bowen suggests is a more practical approach mathematically.

Knowing all this, I will stick with about 6 brute force successive approximations.