Specifically, the following open questions are important for understanding the physical nature of the Pioneer anomaly and are a subject of on-going investigation:
In the following subsections, we review these unresolved questions in detail, including information that has become available since 2002 as a result of the on-going efforts of several teams.
The direction of the anomalous acceleration vector has been the subject of many discussions. If this direction was precisely known, it would allow one to establish the possible cause(s) of the anomaly; at present, all causes must be considered.
The canonical value of was developed using the hypothesis that the anomalous acceleration of the two spacecraft is Sun-pointing, and it finds that the hypothesis is consistent with the data. However, this does not exclude the possibility that equally good solutions can be obtained by postulating an hypothetical acceleration vector in some other direction.
Thus, we must consider at least four possible directions for the anomaly (see Figure 7.4), all indicating a different physical mechanism. Because these directions differ by at most a few degrees during the period of time from which Doppler data was studied, they cannot be distinguished easily. However, each of these four directions implies very different physics.
Specifically, if the acceleration was: i) in the direction towards the Sun, this would indicate a force, likely gravitational, originating from the Sun, likely signifying a need for gravity modification; ii) in the direction towards the Earth, this would indicate a time signal anomaly originating in the DSN hardware or introduced by the space flight-control methods; iii) in the direction of the velocity vector, this would indicate an inertial force or a drag force providing support for a media-dependent origin; or, finally, iv) in the spin-axis direction, this would indicate an on-board systematic, which is the most plausible explanation for the effect. As seen on the left of Figure 7.4, the corresponding directional signatures of these four directions are distinct and could be easily extracted from the data [260, 391, 397].
The navigation of the Pioneer spacecraft relied on an S-band radio-Doppler observable (no ranging capabilities!), which is not very accurate for the purposes of a 3-dimensional orbit reconstruction. To maintain communications with the Earth using the narrow beam of their HGA, an Earth-pointing attitude was necessary. During the flight through the inner solar system, this required frequent attitude correction maneuvers to re-orient the spin axis. At this time, the Sun-craft-Earth angle was relatively large, as was the angle between these and the direction of motion (Figure 7.4). Even so, the Pioneer data from distances up to 27 AU indicate an approximately sunward directional anomaly for both craft. However, at distances farther than 40 AU, both the Sun and the Earth were within the 3° of the antenna radiation pattern (set by 10 dbm range in the antenna gain), thus limiting accuracy in directional reconstruction.
Given the importance of the direction of the anomalous acceleration, it is perhaps surprising that this direction has not yet been established. This is due to the fact that the data that has been investigated to date is from the late cruise phase of the Pioneer 10 and 11 missions, when Pioneer 10 was over 40, and Pioneer 11 was over 22 AU from the Sun. From this distance, the Earth-spacecraft-Sun angle is less than 3°. The angle between the spacecraft spin axis and the Earth-spacecraft line is also small, as the spacecraft was always oriented towards the Earth in order to maintain continuous radio communication. Lastly, at these distances from the Sun the hyperbolic orbits of the spacecraft are nearly asymptotic, and the direction of motion corresponds closely with the Sun-spacecraft direction.
The discussion above indicates that the true direction of the Pioneer anomalous acceleration still remains poorly determined. Based on the Pioneer data analyzed to date, a more correct statement about the direction would be that the anomaly is directed towards the inner part of the solar system – the region that includes both the Sun and the Earth. This question will be re-examined with the extended data set that is now available for new investigation (see Section 3.3).
The magnitude of the anomalous acceleration has been confirmed by several studies [179, 194, 274, 377], all of which show results that are consistent with JPL’s “canonical” value of . However, the constancy of the Pioneer anomalous acceleration remains a subject of serious debate.
For example, if the anomaly was of thermal origin, its magnitude would decay with time, consistent with the decreasing amount of heat generated on-board. Initially  it was assumed that the decay of a thermal recoil force would be consistent with the half-life of the 238Pu fuel (87.74 years), but later it was realized [164, 245, 327] that other effects, such as regulation of the electric power on board, can mask at least some of this decay.
Meanwhile, Markwardt (; see also Section 7.1.1) and later, Toth (, see Section 7.1.3) demonstrated that the Doppler data are, in fact, consistent with a change-of-acceleration (jerk) term for both spacecraft, and that the magnitude of this jerk term is consistent with the decay of the on-board thermal inventory.
Early Pioneer 10 and 11 data (before 1987) were never analyzed in detail, especially with regard to systematics. However, by about 1980 the Doppler navigational data had began to indicate the presence of an anomaly. At first this was considered to be only an interesting navigational curiosity. But even so, samples of data a few months long in duration were periodically examined by different analysts. By 1992 an interesting string of data-points had been obtained; they were gathered in a JPL memorandum , and are shown in Figure 5.1. (More details on this issue are in [24, 27, 392].)
For Pioneer 10, an approximately constant anomalous acceleration seems to exist in the data as close in as 27 AU from the Sun. For Pioneer 11, beginning just after Jupiter flyby, the early navigational data show a small value for the anomaly during the Jupiter-Saturn cruise phase in the interior of the solar system. But right at Saturn encounter, when the craft passed into an hyperbolic escape orbit, there was an apparent fast increase in the anomaly, whereafter it settled into the canonical value.
The data, therefore, indicate the possibility that there was an “onset” of the anomaly at around this time, as the anomalous acceleration component was much smaller prior to Saturn encounter. However, this is likely a premature conclusion. The observations shown in Figure 5.1 do not represent a systematic set of measurements obtained using a consistent, common editing strategy. The apparent onset of the anomaly that is seen in Figure 5.1 has been used to justify theoretical work that predicted a modification of gravity or other nongravitational forces that affect only objects at sufficient distance from the Sun, or objects in hyperbolic orbits. Yet it must be emphasized (as indeed, the authors of the 2002 study  emphasized in a footnote) that this early detection data cannot be viewed as a measurement; confirmation of the onset of the anomaly requires re-analysis of early Doppler data using a consistent editing strategy.
Other factors, such as the greater frequency of maneuvers, also put into question the extent to which one can rely on the apparent pattern represented by the first few data points in Figure 5.1. Also, during this time period, solar radiation pressure produced an acceleration that was several times larger than the anomaly. Scheffer [325, 327] pointed out that the onset that is seen in the data may, in fact, be an artifact of solar model calibration. If an anomalous sunward acceleration was present but not accounted for at the time the solar model coefficients were measured when the spacecraft were still near the Sun (i.e., if the anomalous acceleration was absorbed into estimates of solar model coefficients), later, as solar radiation decreased, one would observe an apparent onset as a result of this miscalibration.
As indicated in Section 5.5.4, even after a best fit solution is obtained, the resulting residuals contain clearly discernible annual and diurnal signatures. These small, approximately sinusoidal contributions do not affect the determination of the (approximately constant) anomalous acceleration, as they are uncorrelated with it. However, the origin of these sinusoidal terms remains unknown.
The authors of [27, 390] expressed the belief that these terms are due to orbital mismodeling, notably mismodeling of the orbital inclination of the spacecraft to the ecliptic plane. Other possibilities also exist: for instance, the annual term may be related to mismodeling of the effects of solar plasma on the radio signal, whereas the diurnal term may be related to atmospheric effects on the signal.
Among the effects considered by the authors of , the radio beam reaction force produced the largest bias to the result (see Section 5.4.4). This bias is due to the fact that the spacecraft are continuously transmitting a highly collimated radio beam in the direction of the Earth, and as a result, experience a proportional recoil force, resulting in an acceleration of . As the force exerted by the radio beam necessarily points away from the Earth and the Sun (as indicated by the negative sign in the preceding equation), the correction of the measured data increases the amount of the observed anomalous attractive force and makes the Pioneer effect larger.
Two open questions remain concerning the actual magnitude of the radio beam recoil force. First, according to the recovered flight telemetry, the radio beam reaction force might not have been constant. Flight telemetry indicates that during much of the mission, the radio beam was more powerful than the nominal 8 W, exceeding the nominal value by 1 W and more. Near the end of Pioneer 10’s mission, however, the transmitter power may have decreased by as much as 3 W (see Figure 2.19). As this apparent decrease coincides with a drop in the main bus voltage on board (due to the depletion of the spacecraft’s 238Pu power supply), the decrease may be an artifact of a failing telemetry system (see also Section 2.4.5).
Second, Scheffer  argues that assuming a typical antenna design, as much as 10% of the power emitted by the high gain antenna (HGA) feed would have missed the parabolic dish altogether, and would have produced a reaction force in the opposite direction, at an approximately 45° angle. Although the HGA is discussed in detail in the recovered project documentation, no attempt has yet been made to establish more precise estimates on the efficiency with which the transmitter’s power is converted into a recoil force.
The spin rates of the Pioneer 10 and 11 spacecraft, nominally 4.8 revolutions per minute (rpm), were in fact changing with time. As we showed in Section 2.3.7, the two spacecraft exhibited markedly different behavior, with unique features.
Pioneer 10 was slowly spinning down, with three discernible phases in its spin history, described in detail in Section 2.3.7. The changes in Pioneer 10’s spin rate approximately coincide with unexplained readings from its propulsion tank (see Section 2.3.6).
Meanwhile, Pioneer 11 was spinning up, although a detailed examination of its spin history reveals that spin-up events coincided with attitude correction maneuvers, and between maneuvers the spacecraft was spinning down, albeit at varying rates (see Section 2.3.7).
It is possible that a constant or near constant rate of spin change is of thermal origin: if thermal radiation is emitted by the spacecraft not just anisotropically but also asymmetrically with respect to the spacecraft’s center-of-gravity, a torque acts on the spacecraft.
No convincing explanation has yet been offered for the anomalous change in the spin change rate during the history of Pioneer 10. The approximate coincidence of this change with the anomalous change in fuel tank pressure readings may be significant; on the other hand, it must be emphasized that this coincidence is only approximate, and may very well be accidental.
In contrast, the spin rate change of Pioneer 11 exhibits no dramatic changes, and appears to be a combination of three effects: A spindown that may be similar to that of Pioneer 10 and may be of thermal origin, instantaneous spinups that occur at each maneuver and may be a result of a small thruster misalignment, and a third effect that changes the spindown rate after each maneuver, and may be related to thruster leaks and outgassing.
It should be noted that these considerations are qualitative in nature, and no detailed study of the Pioneer 10 and 11 spin history has taken place to date. In particular, we do not know if the spin rate change is related to the anomalous acceleration of these spacecraft.
The effect of rejected thermal radiation was the second largest bias/uncertainty that has been the most critical systematic bias to quantify (see Section 4.4.3). If heat generated by the on-board power sources was asymmetrically reflected by the body of the craft, an acceleration along the spin axis could be produced causing the measured anomaly.
The Pioneer spacecraft were powered by SNAP-19 (Space Nuclear Ancillary Power) RTGs mounted on long extended booms (designed to protect the on-board electronics from heat and radiation impact) [24, 27, 28]. It was recognized early that, in principle, there was more than enough heat available on the craft to cause the anomaly (see Section 2.4). In addition to heat from the RTGs, additional waste heat was produced by electrical instrumentation, Radioisotope Heater Units (RHUs), and the propulsion system.
However, it was assumed that the spacecraft’s spin-stabilized attitude control, special design of the RTGs and the length of the RTG booms that resulted in a relatively small spacecraft surface available for the preferential heat rejection significantly minimized the amount of heat for the mechanism to work. These considerations led to an estimated acceleration not exceeding . This result is now being reconsidered, in view of on-going work on Pioneer thermal modeling, which suggests that the acceleration due to asymmetrically reflected heat from the RTGs may have been several times this value.
As the spacecraft is in an approximate thermal steady state, heat generated on board must be removed from the spacecraft . In deep space, the only mechanism of heat removal is thermal radiation: the spacecraft can be said to be radiatively coupled to the cosmic background, which can be modeled by surrounding the spacecraft with a large, hollow spherical black body at the temperature of 2.7 K.
The spacecraft emits heat in the form of thermal photons, which also carry momentum , in accordance with the well known law of , where is the photon’s frequency, is Planck’s constant, and is the velocity of light. This results in a recoil force in the direction opposite to that of the path of the photon. For a body that emits radiation in a spherically symmetric pattern, the net recoil force is zero. However, if the pattern of radiation is not symmetrical, the resulting anisotropy in the radiation pattern yields a net recoil force.
The magnitude of this recoil force is a subject of many factors, including the location and thermal power of heat sources, the geometry, physical configuration, and thermal properties of the spacecraft’s materials, and the radiometric properties of its external (radiating) surfaces.
The total thermal inventory on board the Pioneer spacecraft exceeded 2 kW throughout most of their mission durations. The spacecraft were in an approximate steady state: the amount of heat generated on-board was equal to the amount of heat radiated by the spacecraft.
The mass of the Pioneer spacecraft was approximately 250 kg. An acceleration of is equivalent to a force of acting on a 250 kg object. This is the amount of recoil force produced by a 65 W collimated beam of photons. In comparison with the available thermal inventory of 2500 W, a fore-aft anisotropy of less than 3% can account for the anomalous acceleration in its entirety. Given the complex shape of the Pioneer spacecraft, it is certainly conceivable that an anisotropy of this magnitude is present in the spacecrafts’ thermal radiation pattern.
The possibility that heat from the RTGs can be responsible for the Pioneer anomaly was first proposed by Katz [26, 164]. Murphy [25, 245] pointed out the potential significance of anisotropic rejection of electrically generated heat. Scheffer [325, 327] attempted to account for all possible sources of the thermal recoil force and estimated that the total recoil force is more than sufficient to produce an anomalous acceleration of the observed magnitude. Unfortunately, none of these studies benefited from detailed information about the spacecrafts’ design or from thermal and electrical telemetry. These data became available for study in 2005.
Key questions concerning the thermal recoil force that have been raised during the study of the Pioneer anomaly include [164, 245, 327]:
A presently (2009) on-going effort is to build a comprehensive thermal model of the Pioneer spacecraft from design documentation. The model is to be validated by telemetry, and evaluated at different heliocentric distances at different times during the lifetime of the spacecraft. If successful, this effort will yield a high-accuracy estimate of the thermal recoil force, which can then be incorporated into future trajectory models.
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