  The expansion in multipoles of the gravitational potential of a rotating body
affects the orbital motion of a test particle orbiting it with long-term
perturbations both at a classical and at a relativistic level. In this
preliminary sensitivity analysis, we show that, for the first time, the J2 c^-2
effects could be measured by the ongoing Juno mission in the gravitational
field of Jupiter during its yearlong science phase (10 November 2016-5 October
2017) thanks to its high eccentricity (e=0.947) and to the huge oblateness of
Jupiter (J2=1.47 10^-2). The semi-major axis a and the perijove \omega\ of Juno
are expected to be shifted by \Delta a =700-900 m and \Delta\omega = 50-60
milliarcseconds, respectively, over 1-2 yr. A numerical analysis shows also
that the expected J2c^-2 range-rate signal for Juno should be as large as 280
microns per second during a typical 6 h pass at its closest approach.
Independent analyses previously performed by other researchers about the
measurability of the Lense-Thirring effect showed that the radio science
apparatus of Juno should reach an accuracy in Doppler range-rate measurements
of 1-5 microns per second over such passes. The range-rate signature of the
classical even zonal perturbations is different from the 1PN one. Thus, further
investigations, based on covariance analyses of simulated Doppler data and
dedicated parameters estimation, are worth of further consideration. It turns
out that the J2 c^-2 effects cannot be responsible of the flyby anomaly in the
gravitational field of the Earth. A dedicated spacecraft in a 6678 km X 57103
km polar orbit would experience a geocentric J2 c^-2 range-rate shift of 0.4 mm
s^-1.
