the position vector of the third body and ${\bf r}$ is the position vector of the spacecraft with respect to the mass center of Jupiter. The contribution to the gravitational potential of the quadrupole, octupole and higher order terms is given by: $$\label{Upot}
U(r,\theta)=-\displaystyle\frac{\mu_J}{r} \, \displaystyle\sum_{n=2}^N \, J_n \, \left( \displaystyle\frac{R_J}{r} \right)^n P_{n}(\cos \theta) \; ,$$ where $J_n$ are the zonal harmonics coefficients [@Vallado], $P_n(x)$ are the Legendre polynomials and $\theta$ is the colatitude of the spacecraft (the angle formed by the spacecraft’s position vector and the axis of the planet). The reference radius is $R_J=71492$ km and the known zonal harmonics [@Transfer] are given by: $$\label{Jcoeff}
\begin{array}{rcl}
J_2&=&0.01469645 \; , \\
\noalign{\smallskip}
J_4&=&-0.00058722\; , \\
\noalign{\smallskip}
J_6&=&0.00003508\; ,
\end{array}$$ so, we take $N=6$ in Eq. (\[Upot\]) and we consider also only the coefficients of even order. In spherical coordinates, the components of the perturbing force corresponding to the potential in Eq. (\[Upot\]) are: $$\begin{aligned}
\label{Fgeor}
{\mathcal F}_r &=&-\displaystyle\frac{\mu_J}{r^2} \, \displaystyle\sum_{n=2}^N \,
J_n (n+1)\left( \displaystyle\frac{R_J}{r} \right)^n \, P_n(\cos \theta) \; ,\\
\noalign{\smallskip}
\label{Fgeot}
{\mathcal F}_\theta &=&-\displaystyle\frac{\mu_J}{r^2}\, \displaystyle\sum_{n=2}^N  \, J_n \left(\displaystyle\frac{R_J}{r} \right)^n\, P^{'}_n(\cos \theta) \, \sin\theta\, \; .\end{aligned}$$ So the perturbing force arising from the zonal harmonics terms is $\bm{\mathcal{F}}_{\mbox{geo}}=
\mathcal{F}_r 