Flyby anomaly
The flyby anomaly is a discrepancy between current scientific models and the actual increase in speed (i.e. increase in "kinetic energy") observed during a planetary flyby (usually of Earth) by a spacecraft. In multiple cases, spacecraft have been observed to gain greater speed than scientists had predicted, but thus far no convincing explanation has been found. This anomaly has been observed as shifts in the S-band and X-band Doppler and ranging telemetry. The largest discrepancy noticed during a flyby has been 13 mm/s. Observations. Gravitational assists are valuable techniques for Solar System exploration. Because the  success of such flyby maneuvers depends on the exact geometry of the trajectory, the position and velocity of a spacecraft during its encounter with a planet is continually tracked with great precision by the Deep Space Network (DSN). The flyby anomaly was first noticed during a careful inspection of DSN Doppler data shortly after the Earth flyby of the "Galileo" spacecraft on 8 December 1990. While the Doppler residuals (observed minus computed data) were expected to remain flat, the analysis revealed an unexpected 66 mHz shift, which corresponds to a velocity increase of 3.92 mm/s at perigee. Investigations of  this effect at the Jet Propulsion Laboratory (JPL), the Goddard Space Flight Center (GSFC) and the University of Texas have not yielded a satisfactory explanation. No such anomaly was detected after the second Earth flyby of "Galileo" in December 1992, where the measured velocity decrease matched that expected from atmospheric drag at the lower altitude of 303 km. However, the drag estimates had large error bars, and so an anomalous acceleration could not be ruled out. On 23 January 1998 the Near Earth Asteroid Rendezvous (NEAR) spacecraft experienced an anomalous velocity increase of 13.46 mm/s after its Earth encounter . "Cassini–Huygens" gained around 0.11 mm/s in August 1999, and "Rosetta" gained 1.82 mm/s after its Earth flyby in March 2005. An analysis of the "MESSENGER" spacecraft (studying Mercury) did not reveal any significant unexpected velocity increase. This may be because "MESSENGER" both approached and departed Earth symmetrically about the equator (see data and proposed equation below). This suggests that the anomaly may be related to Earth's rotation. In November 2009, ESA's "Rosetta" spacecraft was tracked closely during flyby in order to precisely measure its velocity, in an effort to gather further data about the anomaly , but no significant anomaly was found. The 2013 flyby of Juno on the way to Jupiter yielded no anomalous acceleration. In 2018, a careful analysis of the trajectory of the presumed interstellar asteroid ʻOumuamua revealed a small excess velocity as it receded from the Sun. Initial speculation suggested that the anomaly was due to outgassing, though none had been detected. Summary of some Earth-flyby spacecraft is provided in table below. Anderson's empirical relation. An empirical equation for the anomalous flyby velocity change was proposed in 2008 by J. D. Anderson et al.: where "ω" is the angular frequency  of the Earth, "R" is the Earth radius, and "φ" and "φ" are the inbound and outbound equatorial angles of the spacecraft. This formula was derived later by Jean Paul Mbelek from special relativity, leading to one of the possible explanations of the effect. This does not, however, consider the SSN residuals – see "Possible explanations" below. Possible explanations. There have been a number of proposed explanations of the flyby anomaly, including: Future research. Some missions designed to study gravity, such as MICROSCOPE and STEP, will make extremely accurate gravity measurements and may shed some light on the anomaly. Flyby anomaly