7.1 Independent verifications

By now several studies of the Pioneer 10 and 11 radiometric Doppler data have demonstrated that the anomaly is unambiguously present in the trajectory solutions for both spacecraft. These studies were performed with six independent (and different!) navigational computer programs (see [24Jump To The Next Citation Point27Jump To The Next Citation Point179Jump To The Next Citation Point194274377Jump To The Next Citation Point]), namely:

These recent independent analyses of the Pioneer 10 and 11 radiometric Doppler data confirmed the existence of the Pioneer anomaly at the level reported by the JPL’s 2002 study and they also provided new knowledge of the effect. Below we review these analyses in some detail.

7.1.1 Independent verification by Markwardt

Shortly after publication of the 2002 JPL result, Markwardt [194] published an independent analysis that was unique in the sense that it utilized a separately obtained data set. Rather than using data in the form of JPL-supplied Orbit Determination Files, Markwardt obtained Pioneer 10 tracking data from the National Space Science Data Center (NSSDC) archive. This data was in the Archival Tracking Data File (ATDF) format, which Markwardt processed using tools developed for the purposes of this specific study [192193].

The Pioneer 10 data used in Markwardt’s investigation spanned the years 1987 through 1994, and his result, aP10 = (7.70 ± 0.02) × 10−10 m ∕s2 (see Figure 7.1View Image), is consistent with the JPL result. Markwardt was also the first to investigate explicitly the possible presence of a jerk (i.e., the rate of change of acceleration31, defined as j ≡ ˙a = da∕dt) term, and found that a term 2 |jP 10| < 0.18 × 10−10 m∕s ∕year is consistent with the data. Based on the studied Pioneer 10 data set, Markwardt found that the anomaly is nearly constant with time, with a characteristic variation time scale of over 70 yr, which is still too short to rule out on-board thermal radiation effects.

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Figure 7.1: Results of Markwardt’s analysis [194] show Doppler residuals as a function of time of the best fit model. The top panel shows the residuals after setting a = 0 P, and demonstrates the linear increase with time. The top panel shows all of the data, including segments that were filtered out because of interference due to the solar corona (designated by a horizontal bar with “C”) or due to general noise (designated “N”). The bottom panel shows the filtered residuals, including the best fit value of the anomalous acceleration. The equivalent spacecraft velocity is also shown.

7.1.2 Analysis by Olsen using HELIOSAT

Olsen [274] focused on the constancy of the anomalous acceleration using the HELIOSAT orbit determination program that was independently developed by him at the University of Oslo, Norway. Analysis confirmed the acceleration at the levels reported by [27Jump To The Next Citation Point] for the same segments of Pioneer 10 and 11 data that were used by JPL (see Table 7.1). The study found that systematic variations in the anomalous acceleration are consistent with solar coronal mass ejections and that the Doppler data alone cannot distinguish between constant acceleration and slowly decreasing acceleration. Specifically, the study concluded that heat dissipation cannot be excluded as a source of the anomaly.



Table 7.1: The Pioneer anomalous acceleration in units of 10–10 m/s2. This table compares the results from JPL’s ODP and the Aerospace Corporation’s CHASMP codes from [27Jump To The Next Citation Point] (see Table 5.1) to results obtained using the HELIOSAT program developed by Olsen [274].
Software Pioneer 10 (I) Pioneer 10 (II) Pioneer 10 (III) Pioneer 11
ODP/Sigma 8.00 ± 0.01 8.66 ± 0.01 7.84 ± 0.01 8.44 ± 0.04
CHASMP 8.22 ± 0.02 8.89 ± 0.01 7.92 ± 0.01 8.69 ± 0.03
HELIOSAT 7.85 ± 0.02 8.78 ± 0.01 7.75 ± 0.01 8.10 ± 0.01

7.1.3 Independent analysis by Toth

Toth [377Jump To The Next Citation Point] also studied the anomalous acceleration using independently developed orbit determination software, and confirmed that the introduction of a constant acceleration term significantly improves the post-fit residuals (Figure 7.2View Image). Toth determined the anomalous accelerations of Pioneers 10 and 11 as aP10 = (9.03 ± 0.86) × 10− 10 m ∕s2 and aP 11 = (8.21 ± 1.07) × 10−10 m∕s2 correspondingly, where the error terms were taken from [27Jump To The Next Citation Point] (excluding terms related to thermal modeling, which is the subject of on-going effort). Studying the temporal behavior of the anomalous acceleration, he was able to find a best fit for the acceleration and jerk terms of both spacecraft: aP10 = (10.96 ± 0.89) × 10−10 m∕s2 and jP10 = (− 0.21 ± 0.04) × 10−10 m ∕s2∕year (Pioneer 10) and a = (9.40 ± 1.12) × 10−10 m ∕s2 P11 and j = (− 0.34 ± 0.12 ) × 10 −10 m ∕s2∕year P11 (Pioneer 11). Toth’s study demonstrated that a moderate jerk term is consistent with the Doppler data and, therefore, an anomalous acceleration that is a slowly changing function of time cannot be excluded at present.

Toth’s orbit determination software also has the capability to utilize telemetry data. In particular, the code can be used to estimate the thermal recoil force as a function of the heat generated on-board, or conversely, to fit thermal recoil force coefficients to radiometric Doppler measurements, as discussed in Section 7.4.4.

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Figure 7.2: Results of Toth’s analysis [377Jump To The Next Citation Point]: a) best-fit residuals for Pioneer 10; b) best-fit residuals for Pioneer 10 with no anomalous acceleration term; c–d) same as a–b, for Pioneer 11.

7.1.4 Analysis by Levy et al. using ODYSSEY

Levy et al. [179Jump To The Next Citation Point] also performed an analysis of the Pioneer data using the independently developed orbit determination program ODYSSEY. The team confirmed the presence of an acceleration signal consistent with that found in other studies: for Pioneer 10, they obtained an anomalous acceleration of aP = (8.40 ± 0.01) × 10− 10 m ∕s2 (see Figure 7.3View Image). Their study shows the presence in the residual of periodic terms with periods consistent with half a sidereal day, one sidereal day, and half a year, and they investigate the possibility that these variations may be due to perturbations of unknown origin that modify the propagation of the signal.

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Figure 7.3: Best-fit Pioneer 10 residuals using the ODYSSEY orbit determination program [179Jump To The Next Citation Point]. Left: residuals after a best-fit constant acceleration of − 10 2 aP = (8.40 ± 0.01) × 10 m ∕s. Right: reconstruction of the anomalous acceleration contribution.

In view of all these studies, the existence of the Pioneer anomaly in the Pioneer 10 and 11 radiometric Doppler data is established beyond doubt. Furthermore, the analyses [179Jump To The Next Citation Point194274377Jump To The Next Citation Point] brought new knowledge about the effect, especially insofar as the temporal behavior of the anomaly is concerned. As a result, the anomalous acceleration can no longer be characterized as having a constant magnitude. Instead, the effect clearly shows temporal decrease – perhaps consistent with the decay of the radioactive fuel on board – the conjecture that needs further investigation. This recently-gained knowledge serves as a guide for new study of the effect (discussed in Section 7.3); it also points out the unresolved questions that we summarize below.


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