On the convergence of the three body disturbing function

[Maher]

For the three-body-disturbing function expanded in multipolar orders with respect to the ratio of the semi major axes, the function converges for small ratios, how to check the convergence for a certain set of parameters? I'm using the work of the Laskar in his paper "Explicit expansion of the three-body disturbing function for arbitrary eccentricities and inclinations".

 

[lippyka]

The convergence of the three-body disturbing function expanded in multipolar orders can be determined by calculating the residuals of the sum of the terms in the expansion. The residuals should decrease as the order of the multipole increases, until they reach a small enough value that indicates that the series is converging. In the paper by Laskar, the convergence is tested by calculating the maximum absolute residual for each order and checking if it decreases below a certain threshold with increasing order. If this threshold is not reached for a certain set of parameters, then the series is not converging.