phenomenological formula [@Anderson2008] and some modified models of gravity [@Acedo2015; @Acedo2017three].

Orbital model {#sec:2}
=============

In this section we discuss the development of an orbital model optimized for the region around the perijove. Our problem is summarized in the set of Newtonian equations of motion: $$\begin{aligned}
\label{eqmotionr}
\displaystyle\frac{d {\bf r}}{d t}&=&{\bf v} \; ,\\
\noalign{\smallskip}
\label{eqmotionv}
\displaystyle\frac{d {\bf v}}{d t}&=&-\mu_J \displaystyle\frac{{\bf r}}{r^3}+\bm{\mathcal{F}}_{\mbox{tidal}}+\bm{\mathcal{F}}_{\mbox{zonal}} \; ,\end{aligned}$$ where ${\bf r}$ and ${\bf v}$ are the position and velocity vectors of the spacecraft, respectively, $\mu_J$ is Jupiter’s mass constant, ${\bm{\mathcal{F}}}_{\mbox{tidal}}$ is the perturbing tidal force exerted by the Sun and Jupiter’s satellites, and $\bm{\mathcal{F}}_{\mbox{zonal}}$ are the corrections arising from the known zonal harmonics of the planet. Concerning the mass constants of Jupiter, the Sun and the Galilean satellites we have the following values [@DE431; @Satellites]: $$\begin{array}{rcl}
\mu_J&=&126712764.800000 \; \mbox{km$^3/$s$^2$}\; , \\
\noalign{\smallskip}
\mu_{\mbox{Sun}}&=&132712440041.939400\; \mbox{km$^3/$s$^2$}\; , \\
\noalign{\smallskip}
\mu_{\mbox{Io}}&=&5959.916\; \mbox{km$^3/$s$^2$}\; , \\
\noalign{\smallskip}
\mu_{\mbox{Europa}}&=&3202.739\; \mbox{km$^3/$s$^2$}\; , \\
\noalign{\smallskip}
\mu_{\mbox{Callisto}}&=&7179.289\; \mbox{km$^3/$s$^2$}\; , \\
\noalign{\smallskip}
\mu_{\mbox{Ganymede}}&=&9887.834\; \mbox{km$^3/$s$^2$}\; . 
\end{array}$$ The tidal force on the reference frame of Jupiter exerted by the Sun or any of Jupiter’s satellites is given by: $$\label{Ftid}
\bm{\mathcal{F}}_{\mbox{tidal}}=\mu \left(-\displaystyle\frac{{\bf R}}{R^3}+\displaystyle\frac{{\bf R}-{\bf r}}{\left(r^2+R^2-2 {\bf r} 
\cdot {\bf R} \right)^{3/2}}\right)\; .$$ Here ${\bf R}$ is 