The Energy Transfer Process in Planetary Flybys
  We illustrate the energy transfer during planetary flybys as a function of time using a number of flight mission examples. The energy transfer process is rather more complicated than a monotonic increase (or decrease) of energy with time. It exhibits temporary maxima and minima with time which then partially moderate before the asymptotic condition is obtained. The energy transfer to angular momentum is exhibited by an approximate Jacobi constant for the system. We demonstrate this with flybys that have shown unexplained behaviors: i) the possible onset of the "Pioneer anomaly" with the gravity assist of Pioneer 11 by Saturn to hyperbolic orbit (as well as the Pioneer 10 hyperbolic gravity assist by Jupiter) and ii) the Earth flyby anomalies of small increases in energy {\it in the geocentric system} (Galileo-I, NEAR, and Rosetta, in additioon discussing the Cassini and Messenger flybys). Perhaps some small, as yet unrecognized effect in the energy-transfer process can shed light on these anomalies. 
John D. Anderson, James K. Campbell, and Michael Martin Nieto