5.6 Error budget and the final 2002 result

Table 5.2: Error budget: a summary of biases and uncertainties as known in 2002 [27Jump To The Next Citation Point]. Values that are the subject of on-going study are marked by an asterisk.
Item Description of error budget constituents Bias Uncertainty
    10–10 m/s2 10–10 m/s2
1 Systematics generated external to the spacecraft:  
a) Solar radiation pressure and mass +0.03 ± 0.01
b) Solar wind   ± < 10–5
c) Solar corona   ± 0.02
d) Electro-magnetic Lorentz forces   ± < 10–4
e) Influence of the Kuiper belt’s gravity   ± 0.03
f) Influence of the Earth’s orientation ± 0.001
g) Mechanical and phase stability of DSN antennae   ± < 0.001
h) Phase stability and clocks   ± < 0.001
i) DSN station location   ± < 10–5
j) Troposphere and ionosphere   ± < 0.001
2 On-board generated systematics:  
a) Radio beam reaction force +1.10* ± 0.11
b) RTG heat reflected off the craft –0.55* ± 0.55
c) Differential emissivity of the RTGs   ± 0.85
d) Nonisotropic radiative cooling of the spacecraft   0.00* ± 0.48
e) Expelled Helium produced within the RTGs +0.15 ± 0.16
f) Gas leakage   ± 0.56
g) Variation between spacecraft determinations +0.17 ± 0.17
3 Computational systematics:  
a) Numerical stability of least-squares estimation ± 0.02
b) Accuracy of consistency/model tests   ± 0.13
c) Mismodeling of maneuvers   ± 0.01
d) Mismodeling of the solar corona   ± 0.02
e) Annual/diurnal terms   ± 0.32
  Estimate of total bias/error +0.90 ± 1.33

The results of the 2002 study [27Jump To The Next Citation Point] are summarized in Table 5.2. Sources that contribute to the overall bias and error budget are grouped depending on their origin: external to the spacecraft, generated on-board, or computational in nature. Sources of error are treated as uncorrelated; the combined error is the root sum square of the individual error values.

The contribution of effects in the first group in Table 5.2, that is, effects external to the spacecraft to the overall error budget is negligible: −10 2 σexternal ∼ 0.04 × 10 m ∕s. The second group (on-board effects) yields the largest error contribution: 2 σon-board ∼ 1.29 × 10−10 m ∕s. Lastly, computational systematics amount to σcomp ∼ 0.35 × 10− 10 m ∕s2.

Similarly, the largest contribution to bias comes from on-board effects: b ∼ 0.87 m ∕s2 on-board, a value that is dominated by the radio beam reaction force. External effects contribute a bias of 2 bexternal ∼ 0.03 m ∕s, while computational systematics contribute no bias.

Note that several items in Table 5.2 are marked with an asterisk, indicating that these items are the subject of an on-going new investigation of the Pioneer anomaly (discussed in Section 7).

The bias (third column) and error (fourth column) in Table 5.2 give the final acceleration result in the form

aP = aP(exp) + bP ± σP , (5.25 )
aP (exp) = (7.84 ± 0.01) × 10−10 m ∕s2 (5.26 )
is the reported formal solution for the Pioneer anomaly that was obtained with the data set available prior to 2002 [27Jump To The Next Citation Point]. Specifically, after accounting for the systematics listed in Table 5.2 and using Equations (5.25View Equation) and (5.26View Equation), the authors of [27Jump To The Next Citation Point] presented the final result of their study as
aP = (8.74 ± 1.33) × 10−10 m ∕s2. (5.27 )
This 6-σ effect is clearly significant and, as of 2009, still remains unexplained.

The 2002 analysis demonstrated that after accounting for the gravitational and other large forces included in standard orbit determination programs [24Jump To The Next Citation Point27Jump To The Next Citation Point392Jump To The Next Citation Point], the anomaly in the Doppler frequency blue shift drift is uniformly changing with a rate of f˙P = (5.99 ± 0.01) × 10− 9 Hz ∕s [391Jump To The Next Citation Point] (see Figure 5.2View Image). Let us denote the frequency of the signal observed by a DSN antenna as f obs, and the predicted frequency of that signal after modeling conventional forces and other signal propagation effects as fmodel. Then, for a one-way signal, the observed anomalous effect to first order in v ∕c is given by fobs − fmodel = − f˙Pt. This translates to

[fobs(t) − fmodel(t)] = − f0aP-t, (5.28 ) DSN c
where f0 is the DSN reference frequency [27Jump To The Next Citation Point391Jump To The Next Citation Point393Jump To The Next Citation Point] (for a discussion of the DSN sign conventions, see Endnote 38 of [27Jump To The Next Citation Point]).

Since the publication of the 2002 study [27Jump To The Next Citation Point], many proposals have been put forth offering theoretical explanations of the anomaly. These are reviewed in the next section (Section 6). On the other hand, our knowledge of the anomaly also improved. The existence and magnitude of the anomalous acceleration has been confirmed by several independent researchers. Others have attempted to model the thermal behavior of the spacecraft, arguing that the magnitude of thermal recoil forces might have been underestimated by the 2002 study. The recovery of essentially all Pioneer 10 and 11 telemetry, as well as large quantities of archived project documentation, raised hope that it might be possible to construct a sufficiently accurate thermal model of the spacecraft using modeling software, and properly estimate the magnitude of the thermal recoil force. This remains one of several open, unresolved questions that, hopefully, will be answered in the near future as a result of on-going study, as detailed in Section 7.

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