8. Appendix

  

Figure: An illustration of the discontinuity extraction procedure. Top left: the graphic of the derivative of the integral average of $1/D$ with respect to $\omega$, as a function of $\omega ,e$; there are two lines of discontinuity, corresponding to crossings at the ascending node with the Earth and at the descending one with Venus. Top right: the derivative of the integral average of $1/d$ with respect to $\omega$; note the discontinuity at the ascending node with the Earth. Bottom left: the difference between the two previous functions shows that the discontinuity due to the Earth has been removed; note that the difference function is continuous, but not differentiable along the ascending node crossing line. The discontinuity at the descending node with Venus is still present. Bottom right: the level curves of the integral average of $1/D$.

This appendix contains the analytical formulas representing the discontinuity of the derivatives of the averaged perturbing function $\overline {R}$.

If we choose, for instance, the direction from the $\{d^+_{nod}>0\}$to the $\{d^+_{nod}<0\}$ area, the difference in the twofold definition of the derivatives of the averaged perturbing function are

 

where

$$det(\underline{\underline {\bf A}}) = {a^2 a'^2\over (1-e^2)}\cdot(\sin^2I (1 +2 e\cos\omega + e^2) + e^2\cos^2I \sin^2\omega)$$

(see [Gronchi and Milani 1998] for the notation).

In case of crossing at the descending node, we have to modify the previous formulas as follows:

$$det(\underline{\underline {\bf A}}) = {a^2 a'^2\over (1-e^2)}\cdot(\sin^2I (1 -2 e\cos\omega + e^2) + e^2\cos^2I \sin^2\omega)$$