5.4 Sources of systematic error internal to the spacecraft

There exist several on-board mechanisms that can contribute to the acceleration of the Pioneer spacecraft. For spinning spacecraft, like Pioneer 10 and 11, the contribution of these forces to the spacecraft’s acceleration will be primarily in the direction of the spin axis. The reason for this is that for any force that is constant in a co-rotating coordinate system, the force component that is perpendicular to the spin axis will average to zero over the course of a full revolution. Consequently, for an arbitrary force, its contribution to lateral accelerations will be limited to its time-varying component (see Section 4.4).

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.

5.4.1 Propulsive mass expulsion

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, [27Jump To The Next Citation Point] conservatively estimates that undetected gas leaks introduce an uncertainty no greater than

−10 2 σgl = ±0.56 × 10 m ∕s . (5.13 )

5.4.2 Heat from the RTGs

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 [27Jump To The Next Citation Point392Jump To The Next Citation Point].

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.

[27Jump To The Next Citation Point] 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 4π 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, [27Jump To The Next Citation Point] 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 ≈ 0.55 × 10−10 m ∕s2. The authors also add an uncertainty of the same size, to obtain a contribution from heat reflection of

−10 2 ahr = (− 0.55 ± 0.55) × 10 m ∕s . (5.14 )

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, [27Jump To The Next Citation Point] 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 995∕1005 = 0.99, or a differential emission between the half cylinders of 10 W. Therefore, the fore/aft asymmetry toward the normal would be ∫ π 10 W × 1π 0 dϕ sinϕ ≈ 6.37 W. A more sophisticated model of the fin structure resulted in the slightly smaller estimate of 6.12 W, which the authors of [27Jump To The Next Citation Point] took as the uncertainty from the differential emissivity of the RTGs, to obtain an acceleration uncertainty of

σde = 0.85 × 10−10 m ∕s2. (5.15 )

5.4.3 Nonisotropic radiative cooling of the spacecraft

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 [326Jump To The Next Citation Point].

To address the original proposal [245] and several later modifications [326327] and [2528325] developed a bound on the constancy of aP. This bound came from first noting the 11.5 year 1-day batch-sequential result, sensitive to time variation: aP = (7.77 ± 0.16) × 10− 10 m ∕s2. It is conservative to take three times this error to be our systematic uncertainty for radiative cooling of the craft,

2 σrc = ±0.48 × 10 −10 m ∕s . (5.16 )

5.4.4 Radio beam reaction force

The emitted radio-power from the spacecraft’s HGA produces a recoil force, which is responsible for an acceleration bias, brp, 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 β to denote the efficiency of the antenna, we can compute an acceleration bias as

-1- brp = mc βPrp, (5.17 )
where Prp is the transmitter’s power. The nominal transmitted power of the spacecraft is 8 W. Given the m = 241 kg 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
arp = brp ± σrp = − (1.10 ± 0.10 ) × 10− 10 m ∕s2, (5.18 )
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 [27Jump To The Next Citation Point].

5.4.5 Expelled helium produced within the RTGs

Another possible on-board systematic error is from the expulsion of the He being created in the RTGs from the α-decay of 238Pu. 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

aHe = (0.15 ± 0.16) × 10− 10 m ∕s2. (5.19 )

5.4.6 Variation between determinations from the two spacecraft

Section 5.2 presented two experimental results for the Pioneer anomaly from the two spacecraft: 2 7.84 × 10− 10 m ∕s (Pioneer 10) and 2 8.55 × 10− 10 m ∕s (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.

[27Jump To The Next Citation Point] 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: [(11.5 )(7.84) + (3.75)(8.55 )]∕(15.25) = 8.01 in units of 10− 10 m ∕s2. This result implies a bias of b2 craft = 0.17 × 10− 10 m ∕s2 with respect to the Pioneer 10 experimental result a P(exp) (see Equation (5.1View Equation)). We can take this number to be a measure of the uncertainty from the separate spacecraft measurements, so the overall quantitative measure is

−10 2 a2 craft = b2 craft ± σ2 craft = (0.17 ± 0.17) × 10 m ∕s . (5.20 )

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