The biggest challenge to the mechanisms that use to explain the anomaly either the unseen matter distribution in the outer solar system or modifications of gravitational theory (see in Section 6.2) is the existence of a large volume of high-precision solar system data. Any naive modification of gravitational theory, for instance, that is designed to account for the anomalous acceleration of the Pioneer spacecraft would also likely induce changes in the predicted orbits of the outer planets.
A gravitational source in the solar system as a possible origin for the anomaly has been considered by . According to the equivalence principle, such a gravitational source would also affect the orbits of the planets. In the case of the inner planets, which have orbits determined with great accuracy, they show no evidence of the expected anomalous motion if the source of the anomaly were located in the inner solar system. In fact, the anomalous acceleration is too large to have gone undetected in planetary orbits, particularly for Earth and Mars. Indeed, the authors of  observed that if a planet experiences a small, anomalous, radial acceleration, , its orbital radius is perturbed byl is the orbital angular momentum per unit mass and is the Newtonian acceleration at distance .
For Earth and Mars, is about –21 km and –76 km. Take the orbit of Mars, for example, for which range data provided by the Mars Global Surveyor and Mars Odyssey missions have yielded measurements of the Mars system center-of-mass relative to the Earth to an accuracy of one meter . However, the anomaly has been detected beyond 20 AU (i.e., beyond Uranus, 19 AU), and the orbits of the outer planets have been determined only by optical methods, resulting in much less accurate planet ephemerides.
The idea of looking at the impact of a Pioneer-like acceleration on the orbital dynamics of the solar system bodies was originally put forth in [24, 27]. These papers, however, considered the motion of the Earth and Mars finding no evidence of any effect induced by an extra acceleration like the Pioneer anomaly. In particular, a perturbation in produces a perturbation to the orbital angular velocity of.
More interestingly, the authors of  have investigated the effect of an ever-present, uniform Pioneer-like force on the long-period comets. The authors of  proposed to use comets and asteroids to assess the gravitational field in the outer regions of the solar system and thereby investigate the Pioneer anomaly.
Furthermore,  concluded that available planetary ranging data limit any unmodeled radial acceleration acting on Earth and Mars to less than . Consequently, if the anomalous radial acceleration acting on spinning spacecraft is gravitational in origin, it is not universal. That is, it must affect bodies in the 1000 kg range more than bodies of planetary size by a factor of 100 or more. As the authors of  said: “This would be a strange violation of the Principle of Equivalence”.
Attempts to detect observable evidence of unexpected gravitational effects acting on the orbits of the outer planets have not yielded any positive results yet. Hence, the authors of  used parametric constraints to the orbits of Uranus and Neptune and found that the reduced solar mass to account for the Pioneer anomaly would not be compatible with the measurements. A similar result was obtained in  based on the Gauss equations to estimate the effect of a gravitational perturbation in terms of the time rate of change on the osculating orbital elements. These authors argue that the perturbation would produce long-period, secular rates on the perihelion and the mean anomaly, and short-period effects on the semimajor axis, the eccentricity, the perihelion and the mean anomaly large enough to be detected.  also considers the effect on the path of the outer planets of a disturbance on a spherically-symmetric spacetime metric, and rules out any model of the anomaly that implies that the Pioneer spacecraft move geodesically in a perturbed spacetime metric. A recent test of the orbits of 24 Trans-Neptunian Objects using bootstrap analysis also failed to find evidence of the anomaly in these objects .
Nevertheless, the absence of support for a perturbation of the planetary orbits in the outer solar system is weak and inconclusive (primarily due to the lack of precision range measurements). For example, the authors of  conclude that such anomalous gravitational disturbances would not be detected in the orbits of the outer planets. Therefore, efforts to find a gravitational explanation continue, as in the case of a recent paper  that proposes an azimuthally symmetric solution to Poisson’s equation for empty space to explain qualitatively the Pioneer anomaly. This solution results in a gravitational potential dependent on the distance and the polar angle, and it also has implications for the planetary orbits, albeit not yet tested with ephemeris data yet.
Many of these considerations are based on simplified models. As such, they must be contrasted with efforts that incorporate a radial acceleration into computations of planetary ephemerides.
Using the planetary ephemerides, Standish [358, 357] investigated modifications to the laws of gravitation that can explain the anomaly and still be compatible with the known motion of the planets from Saturn and outward. Out of five different acceleration models, four were shown to be not viable.
Using radiometric tracking data from the Cassini spacecraft, Folkner  established an upper bound on Saturn’s radial acceleration as less than 10–14 m/s2. This is around just one tenth the anomalous acceleration of Pioneer 10 and 11.
Using the latest INPOP08 planetary ephemerides, Fienga et al  determined that a radial acceleration greater than 1/4 times the observed Pioneer anomaly at distances beyond 20 AU is not consistent with planetary orbits.
These investigations represent serious challenges to any attempt to explain the Pioneer anomaly using unseen mass distributions in the solar system or modifications of the theory of gravity. Such attempts, in addition to accounting for the anomalous acceleration of the two spacecraft, must also be able to explain why an acceleration of the same magnitude is not readily seen in the orbit of Saturn and in the known ephemerides of the outer planets.
One reason that the Pioneer anomaly generated so much interest is that two spacecraft on very different trajectories yielded similar anomalous acceleration values. However, the two spacecraft are nearly identical in design, and although their trajectories are different, they both follow hyperbolic escape trajectories. This raises the obvious question: is it possible to confirm the presence of an anomalous acceleration using i) different spacecraft, or ii) spacecraft on different trajectories?
The Pioneer 10 and 11 spacecraft are sensitive to such a small, anomalous acceleration term because they are very far from the Sun, and therefore, the effects of solar radiation pressure are much smaller than the anomalous acceleration. For spacecraft located in the inner parts of the solar system (e.g., within the orbit of Jupiter), solar radiation pressure is significant; for Pioneer 10 and 11 at 5 AU, solar radiation results in an acceleration that exceeds the anomalous acceleration by nearly a full order of magnitude. This is a clear indication that the only candidates for investigating accelerations as small as the Pioneer anomaly are spacecraft in deep space, preferably at or beyond the orbit of Jupiter.
The most obvious choice that comes to mind are the twin Voyagers. Voyager 1 and 2, like Pioneer 10 and 11 before them, are traveling on hyperbolic escape trajectories after encounters with the gas giants in the outer solar system. Unfortunately, unlike Pioneer 10 and 11, Voyager 1 and 2 are not spin stabilized. The three-axis stabilization employed by these spacecraft requires the use of small attitude stabilization thrusters several times a day. The noise produced by unmodeled small accelerations that arise as a result of uncertainties in the magnitude and exact duration of thruster firings and possible leakage afterwards reduce the sensitivity of these spacecraft to small accelerations such that an anomalous acceleration of order 10–9 m/s2 is completely undetectable.
Next on the list was Galileo. This spacecraft, designed to orbit Jupiter, featured an innovative design comprising a spinning and a nonspinning section. However, due to mechanical problems between the two sections, the spacecraft was often configured to use the all-spinning mode. The malfunction of Galileo’s high gain antenna opening mechanism put this antenna out of commission, and therefore, Galileo communicated with the Earth using an omnidirectional low-gain antenna at much lower than planned data rates, at a very low signal-to-noise ratio. Despite these difficulties, navigational data was obtained from this spacecraft with sufficient accuracy for precision orbit estimation. A combined analysis of Doppler and ranging data by The Aerospace Corporation from a 113-day period beginning shortly before Galileo’s second Earth encounter (Galileo flew a complicated series of flyby orbits before it achieved a transfer orbit that took it to Jupiter) does show the presence of a possible acceleration term of the correct magnitude .
Ulysses, a joint project between NASA and the European Space Agency (ESA), flew on a highly elliptical heliocentric orbit tilted about 80° from the plane of the elliptic. Although Ulysses flew relatively close to the Sun and therefore, solar radiation had a significant effect on its trajectory, its varying distance from the Sun, combined with variations in the Earth-spacecraft-Sun angle make it possible in principle to separate the effects of solar radiation from any anomalous acceleration. This is how an estimate of (12 ± 3) × 10–10 m/s2 was obtained using JPL’s ODP software. Although this value remains highly correlated with solar radiation pressure , it is compatible with the value of measured for Pioneer 10 and 11.
Yet another deep-space craft, Cassini, is three-axis stabilized like Voyager 1 and 2. Nevertheless, the sophisticated tracking capabilities of this spacecraft could offer a potential contribution to the research of the anomalous acceleration .
The recently (2006) launched New Horizons mission to Pluto is potentially a better candidate for researching the anomalous acceleration. Like Pioneer 10 and 11, New Horizons is spin stabilized, using only infrequent thruster maneuvers for attitude stabilization. Unfortunately, current mission plans call for New Horizons to spend much of the time between Jupiter and Pluto in “hibernation” mode, with only infrequent communications with the Earth. Therefore, during the flight to Pluto, little or no precision Doppler data may be collected [254, 380, 391].
As a result, attempts to verify the anomaly using other spacecraft have proven disappointing [27, 28, 32, 260], because the Voyager, Galileo, Ulysses, and Cassini spacecraft navigation data all have their own individual difficulties for use as an independent test of the anomaly. In addition, many of the deep space missions that are currently being planned will not provide the needed navigational accuracy and trajectory stability of under 10–10 m/s2.
The acceleration regime in which the anomaly was observed diminishes the value of using modern disturbance compensation systems for a test. For example, the systems that are currently being developed for the LISA (Laser Interferometric Space Antenna) and LISA Pathfinder missions are designed to operate in the presence of a very low frequency acceleration noise (at the mHz level), while the Pioneer anomalous acceleration is a strong constant bias in the Doppler frequency data. In addition, currently available accelerometers are a few orders of magnitude less sensitive than is needed for a test [32, 260, 263, 391, 394]. To enable a clean test of the anomaly there is also a requirement to have an escape hyperbolic trajectory. This makes a number of other currently proposed missions less able to directly test the anomalous acceleration.
A number of alternative ground-based verifications of the anomaly have also been considered; for example, using VLBI astrometric observations. However, the trajectories of Pioneers, with small proper motions in the sky, make it presently impossible to use VLBI in accurately isolating an anomalous sunward acceleration of the size of .
This work is licensed under a Creative Commons License.