Cherubini and Mashhoon  discuss the possibility of nongravitational acceleration of the Sun, orthogonal to the ecliptic plane, but they found that it is necessary for the Sun to emit all electromagnetic radiation in the opposite direction.
The possibility of an unknown interaction of the Pioneer radio signals with the solar wind was considered in . In addition, there were ideas to invoke a model for superstrong interaction of photons or massive bodies with the graviton background .
Wilson and Blome  derived the equations of motion for an accelerated, rotating observer in a Gödel universe, and calculated the contribution of the universal cosmic rotation or vorticity. However, they found that this term cannot account for the observed Pioneer acceleration.
Trencevski  considered a time-dependent gravitational potential that could explain the Pioneer anomaly without causing planetary perihelion precessions that are in conflict with observation. Mansouri, Nasseri, and Khorrami  argued in favor of an effective time variation of the gravitational constant; a similar analysis was performed by Sidharth .
Kowalski-Glikman and Smolin  proposed triply special relativity: a generalization of Einstein’s theory of special relativity with three invariant scales, the speed of light , a mass , and a length , as a means to address several observational puzzles, including the Pioneer anomaly.
Wiltshire  considered the effects of decoupling bound systems from the global expansion of the universe. While he found that this cannot be responsible for the Pioneer effect, he also proposed an intriguing test case: a cosmic microwave background imager flown on a trajectory similar to that of Pioneer 10 and 11, with sufficient sensitivity to determine if an anomaly, if observed, is due to a clock effect related to gravitational energy.
Capozziello and Lambiase  argued that neutrino oscillations in Brans–Dicke theory could produce a phase shift with observable effects on astronomical or cosmological time scales.
Attempting to make sense of the Pioneer anomaly, one can easily discover several numerical coincidences that could mislead the discussion. We review some common misconceptions in the hope that this helps researchers navigate more easily through a formidable volume of literature.
The numerical value of the anomalous acceleration gave rise to much speculation. First, it was noticed that where is the speed of light and is Hubble’s constant at the present epoch. However, an anomalous acceleration that is somehow connected to the cosmic expansion rate would necessarily point away from, not toward, the Sun. Furthermore, an acceleration with the magnitude would only be observed for distant objects at a cosmic redshift of . Additionally, the numerical coincidence is only approximate, not exact; good agreement with the observed value of requires , well above the presently accepted value of .
Then there were suggestions that the Pioneer anomaly may be related to galactic gravity. This claim is not true. The Milky Way, as most spiral galaxies, if approximated as a point mass using a nominal value of as the distance from the Earth to the galactic bulge, and a nominal mass of , yields an acceleration of , a number that is almost identical to . Yet this coincidence is misleading. Galactic gravity acts on all bodies in the solar system, including the Sun, the Earth, and flying spacecraft. The observed value of is an acceleration relative to the Sun, so if it is caused by galactic gravity, it would have to be a result of a difference between the galactic gravitational force on the Sun vs. the spacecraft, i.e., a tidal force. Even at d = 100 AU, this tidal acceleration amounts to only , which is about seven orders of magnitude less than .
Another numerical coincidence exploited in some proposals that have been communicated to the authors concerns the value or some variation thereof, where is the heliocentric velocity of the Pioneer spacecraft. Evaluated numerically using the heliocentric velocity of Pioneer 10, we obtain . While this value is very close numerically to , it has the dimensions of a velocity, not acceleration, and the numerical coincidence is valid only so long as seconds are used as the unit of time.
Mäkelä  notes that , where is the proton mass and is the proton’s Compton wavelength, is very close in value to the Pioneer acceleration. The factor of may arise as the result of an unspecified quantum field theoretical calculation.
As described in Section 5, early data indicates that there was an “onset” of the anomalous acceleration of Pioneer 11 at around the time of its encounter with Saturn, which also marks this spacecraft’s transition from an elliptic orbit to an hyperbolic escape trajectory. Some authors [254, 258] attempted to draw far reaching conclusions based solely on this fact. However, this observation cannot be used to draw any firm conclusions, as analysis of the early data is highly suspect: it was not developed using consistent methods and instead is an amalgamation of results derived by different analysts who used to work on Pioneer navigation at a particular time. So at best, it is only an indication of the temporal dependence of the anomaly, and quite likely, the “onset” is simply an artifact that will vanish when data is re-analyzed using a consistent strategy.
Lastly, the flyby anomaly [22, 34] must be mentioned, as it has been related to the Pioneer anomaly by some authors [22, 23]. Several spacecraft, including Galileo, NEAR and Rosetta, demonstrated a small, unexplained change in kinetic energy during a gravity-assist Earth flyby. The origin of this flyby anomaly remains unknown, although the possibility of a systematic origin cannot be excluded .
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