A flyby anomaly for Juno? Not from standard physics

An empirical formula recently appeared in the literature to explain the observed anomalies of about $\Delta\dot\rho\approx 1-10$ mm s$^{-1}$ in the geocentric range-rates $\dot\rho$ of the Galileo, NEAR and Rosetta spacecraft at some of their past perigee passages along unbound, hyperbolic trajectories. It predicts an anomaly of the order of $6$ mm s$^{-1}$ for the recent flyby of Juno, occurred on 9 October 2013. Data analyses to confirm or disproof it are currently ongoing. We numerically calculate the impact on the geocentric Juno's range rate of some classical and general relativistic dynamical effects which are either unmodelled or mismodelled to a certain level in the software used to process the data. They are: a) The first even zonal harmonic coefficient $J_2$ of the multipolar expansion of the terrestrial gravitational potential causing orbital perturbations both at the $\left.{\rm a}^{'}\right)$ Newtonian ($J_2$) and at the $\left. {\rm a}^{''}\right)$ first post-Newtonian level ($J_2 c^{-2}$) b) The post-Newtonian gravitoelectric (GE) Schwarschild-like component of the Earth's gravitational field c) The post-Newtonian gravitomagnetic (GM) Lense-Thirring effect. The magnitudes of their mismodeled and nominal range-rate signatures are: $\left. {\rm a}^{'}\right)$ $\Delta\dot\rho_{\sigma_{J_2}} \approx 1$ $\mu$m s$^{-1}$ $\left. {\rm a}^{''}\right)$ $\Delta\dot\rho_{J_2 c^{-2}} \approx 0.015$ $\mu$m s$^{-1}$ b) $\Delta\dot\rho_{\rm GE} \approx 25$ $\mu$m s$^{-1}$ c) $\Delta\dot\rho_{\rm GM} \approx 0.05$ $\mu$m s$^{-1}$. If a flyby anomaly as large as a few mm s$^{-1}$ will be finally found also for Juno, it will not be due to any of these standard gravitational effects. (Abridged)