scarcity of the data and the absence of dedicated missions to study this phenomenological issue. On the other hand, this does not prevent us from defining a clear-cut research objective in experimental gravity and space research: Are highly elliptical and hyperbolic orbits with periapsis close to the main body well described by our current theories of gravity and spacecraft navigation models ?. Starting with Anderson et al. [@Anderson2008] there are many researchers who think that we face a problem in this case and that further research is necessary to obtain as accurate predictions as our current technology allows.

The objective of this paper is to develop an orbital model specially suited for the perijove time-frame. This model should take into account, at least, the tidal effects of Jupiter’s Galilean satellites and the known zonal harmonics of the planet. By comparing with the telemetry data we disclose a small, but significant, anomalous acceleration whose components in spherical coordinates are of the order of magnitude of a few mm$/$s$^2$ and decay below the measurement error bars after a period of $30$ minutes before or after the perijove. As we will see this is compatible with the expected order of magnitude from Anderson’s 