There are several known forces of on-board origin that can result in unmodeled accelerations. These forces, in fact, represent the most likely sources of the anomaly, in particular because previously published magnitudes of several of the considered effects are subject to revision, in view of the recently recovered telemetry data and newly developed thermal models.

On-board mechanisms that we consider in this section include: i) thruster gas leaks, ii) nonisotropic radiative cooling of the spacecraft body, iii) heat from the RTGs, iv) the radio beam reaction force, and v) the expelled helium produced within the RTG and other gas emissions.

We also review the differences in experimental results between the two spacecraft.

The attitude control subsystems on board Pioneer 10 and 11 were used frequently to ensure that the spacecrafts’ antennas remained oriented in the direction of the Earth. This raises the possibility that the observed anomalous acceleration is due to mismodeling of these attitude control maneuvers, or inadequate modeling of the inevitable gas leaks that occur after thruster firings.

The characteristics of propulsive gas leaks are well understood and routinely modeled by trajectory estimation software. Typical gas leaks vary in magnitude after each thruster firing, and usually decrease in time, until they become negligible.

The placement of thrusters (see Section 2.2.5) makes it highly likely that any leak would also induce unaccounted-for changes in the spacecraft’s spin and attitude.

In contrast, to produce the observed acceleration, any propulsion system leaks would have had to be i) constant in time; ii) the same on both spacecraft; iii) not inducing any detectable changes in the spin rate or precession. Given these considerations, [27] conservatively estimates that undetected gas leaks introduce an uncertainty no greater than

The radioisotope thermoelectric generators of the Pioneer 10 and 11 spacecraft emitted up to 2500 W of heat at the beginning of the mission, slowly decreasing to 2000 W near the end. Even a small anisotropy ( 2%) in the thermal radiation pattern of the RTGs can account, in principle, for the observed anomalous acceleration. Therefore, the possibility that the observed acceleration is due to anisotropically emitted RTG heat has been considered [27, 392].

The cylindrical RTG packages (see Section 2.2.3) have geometries that are fore-aft symmetrical. Two mechanisms were considered that would nonetheless lead to a pattern of thermal radiation with a fore-aft asymmetry.

According to one argument, heat emitted by the RTGs would be reflected anisotropically by the spacecraft itself, notably by the rear of the HGA.

[27] used the spacecraft geometry and the resultant RTG radiation pattern to estimate the contribution of the RTG heat reflecting off the spacecraft to the Pioneer anomaly. The solid angle covered by the antenna as seen from the RTG packages was estimated at 2% of steradians. The equivalent fraction of RTG heat is 40 W. This estimate was further reduced after the shape of the RTGs (cylindrical with large radiating fins) and the resulting anisotropic radiation pattern of the RTGs was considered. Thus, [27] estimated that this mechanism could produce only 4 W of directed power.

The force from 4 W of directed power suggests a systematic bias of . The authors also add an uncertainty of the same size, to obtain a contribution from heat reflection of

Another mechanism may also have contributed to a fore-aft asymmetry in the thermal radiation pattern of the RTGs. Especially during the early part of the missions, one side of the RTGs was exposed to continuous intense solar radiation, while the other side was in permanent darkness. Furthermore this side, facing deep space, was sweeping through the dust contained within the solar system. These two processes may have led to different modes of surface degradation, resulting in changing emissivities [207].

To obtain an estimate of the uncertainty, [27] considered the case when one side (fore
or aft) of the RTGs has its emissivity changed by only 1% with respect to the other
side.^{26}
In a simple cylindrical model of the RTGs, with 2000 W power (only radial emission is assumed with no
loss out of the sides), the ratio of the power emitted by the two sides would be , or a
differential emission between the half cylinders of 10 W. Therefore, the fore/aft asymmetry toward the
normal would be . A more sophisticated model of the fin
structure resulted in the slightly smaller estimate of 6.12 W, which the authors of [27] took as the
uncertainty from the differential emissivity of the RTGs, to obtain an acceleration uncertainty of

It has also been suggested that the anomalous acceleration seen in the Pioneer 10/11 spacecraft can be, “explained, at least in part, by nonisotropic radiative cooling of the spacecraft [245].” Later this idea was modified, suggesting that “most, if not all, of the unmodeled acceleration” of Pioneer 10 and 11 is due to an essentially constant supply of heat coming from the central compartment, directed out the front of the craft through the closed louvers [326].

To address the original proposal [245] and several later modifications [326, 327] and [25, 28, 325] developed a bound on the constancy of . This bound came from first noting the 11.5 year 1-day batch-sequential result, sensitive to time variation: . It is conservative to take three times this error to be our systematic uncertainty for radiative cooling of the craft,

The emitted radio-power from the spacecraft’s HGA produces a recoil force, which is responsible for an acceleration bias, , on the spacecraft away from the Earth. If the spacecraft were equipped with ideal antennas, the total emitted power of the spacecrafts’ radio transmitters would be in the form of a collimated beam aimed in the direction of the Earth. In reality, the antenna is less than 100% efficient: some of the radio frequency energy from the transmitter may miss the antenna altogether, the radio beam may not be perfectly collimated, and it may not be aimed precisely in the direction of the Earth.

Therefore, using to denote the efficiency of the antenna, we can compute an acceleration bias as

where is the transmitter’s power. The nominal transmitted power of the spacecraft is 8 W. Given the as the mass of a spacecraft with half its fuel gone, and using the 0.4 dB antenna error as a means to estimate the uncertainty, we obtain the acceleration figure of where the negative sign indicates that this acceleration is in the direction away from the Earth (and thus from the Sun), i.e., this correction actually increases the amount of anomalous acceleration required to account for the Pioneer Doppler observations [27].

Another possible on-board systematic error is from the expulsion of the He being created in the RTGs from
the -decay of ^{238}Pu. According to the discussion presented in Section 4.4.2, Anderson et al. estimate
the bias and error in acceleration due to He-outgassing as

Section 5.2 presented two experimental results for the Pioneer anomaly from the two spacecraft: (Pioneer 10) and (Pioneer 11). The first result represents the entire 11.5 year data period for Pioneer 10; Pioneer 11’s result represents a 3.75 year data period.

The difference between the two craft could be due to differences in gas leakage. It also could be due to heat emitted from the RTGs. In particular, the two sets of RTGs have had different histories and so might have different emissivities. Pioneer 11 spent more time in the inner solar system (absorbing radiation). Pioneer 10 has swept out more dust in deep space. Further, Pioneer 11 experienced about twice as much Jupiter/Saturn radiation as Pioneer 10.

[27] estimated the value for the Pioneer anomaly based on the two independent determinations derived from the two spacecraft, Pioneer 10 and 11. They calculated the time-weighted average of the experimental results from the two craft: in units of . This result implies a bias of with respect to the Pioneer 10 experimental result (see Equation (5.1)). We can take this number to be a measure of the uncertainty from the separate spacecraft measurements, so the overall quantitative measure is

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