The expansion in multipoles J _ { \ell } , \ell = 2 , \ldots 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 J _ { 2 } 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 ( J _ { 2 } = 1.47 \times 10 ^ { -2 } ) .
The semi-major axis a and the perijove \omega of Juno are expected to be shifted by \Delta a \lesssim 700 - 900 m and \Delta \omega \lesssim 50 - 60 milliarcseconds ( mas ) , respectively , over 1 - 2 year .
A numerical analysis shows also that the expected J _ { 2 } c ^ { -2 } range-rate signal for Juno should be as large as \approx 280 microns per second ( \mu m s ^ { -1 } ) 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 \approx 1 - 5 \mu m s ^ { -1 } 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 J _ { 2 } c ^ { -2 } effects can not be responsible of the flyby anomaly in the gravitational field of the Earth .
A dedicated spacecraft in a 6678 km \times 57103 km polar orbit would experience a geocentric J _ { 2 } c ^ { -2 } range-rate shift of \approx 0.4 mm s ^ { -1 } .