determine every parameter, at least 6 frequency terms need to be measured in the TTV signal. This gives six equations for phases and six for amplitudes, i.e., 12 equations and 12 unknowns, as it should be.

Well, what seems simple in theory is not easy in practice, mainly because the equations are ugly and difficult to deal with. Rather than struggling with the analytic solution, I realized that it would more practical to do things numerically, and my goals shifted. Still, while my original plan somewhat predictably failed, I learned many things from this exercise. It occurred to me, for example, that the analytic method can be used to greatly speed up the whole process of inversion.

To appreciate that, let’s take a modern viewpoint on this issue and consider a purely numerical method. Say that an efficient N-body integrator is instructed how to compute the transit times for any given planetary system. The code is interfaced with some smart algorithm that knows how to maximize the likelihood of the fit. All that is OK, but the basic difficulty is that the algorithm must search in parameter space of 12 dimensions, which is a lot of dimensions. The program may 