  The recently approved Juno mission will orbit Jupiter for one year in a highly eccentric (r_min=1.06R_Jup, r_max=39R_Jup) polar orbit (i=90 deg) to accurately map, among other things, the jovian magnetic and gravitational fields. Such an orbital configuration yields an ideal situation, in principle, to attempt a measurement of the general relativistic Lense-Thirring effect through the Juno's node Omega which would be displaced by about 570 m over the mission's duration. Conversely, by assuming the validity of general relativity, the proposed test can be viewed as a direct, dynamical measurement of the Jupiter's angular momentum S which would give important information concerning the internal structure and formation of the giant planet. The long-period orbital perturbations due to the zonal harmonic coefficients J_L, L=2,3,4,6 of the multipolar expansion of the jovian gravitational potential accounting for its departures from spherical symmetry are a major source of systematic bias. While the Lense-Thirring node rate is independent of the inclination i, the node zonal perturbations vanish for i=90. In reality, the orbit injection errors will induce departures \delta i from the ideal polar geometry, so that the zonal perturbations will come into play at an unacceptably high level, in spite of the expected improvements in the low-degree zonals by Juno. A linear combination of Omega, the periJove omega and the mean anomaly M cancels out the impact of J_2 and J_6. A two orders of magnitude improvement in the uncanceled J_3 and J_4 would be needed to reduce their bias on the relativistic signal to the percent level; it does not seem unrealistic because the expected level of improvement in such zonals is three orders of magnitude. 
Juno, the angular momentum of Jupiter and the Lense-Thirring effect