5.5 Computational systematics

The third group of effects was composed of contributions from computational errors (see Table 5.2). The effects in this group dealt with i) the numerical stability of least-squares estimations, ii) accuracy of consistency/model tests, iii) mismodeling of maneuvers, and that of iv) the solar corona model used to describe the propagation of radio waves. It has also been demonstrated that the influence of v) annual/diurnal terms seen in the data on the accuracy of the estimates was small.

5.5.1 Numerical stability of least-squares estimation

The authors of [27Jump To The Next Citation Point] looked at the numerical stability of the least squares estimation algorithms and the derived solutions.

Common precision orbit determination algorithms use double precision arithmetic. The representation uses a 53-bit mantissa, equivalent to more than 15 decimal digits of precision [151]. Is this accuracy sufficient for precision orbit determination within the solar system? At solar system barycentric distances between 1 and 10 billion kilometers (1012 – 1013 m), 15 decimal digits of accuracy translates into a positional error of 1 cm or less. Therefore, we can conclude that double precision arithmetic is adequate in principle for modeling the orbits of Pioneer 10 and 11 in the outer solar system in a solar system barycentric reference frame. However, one must still be concerned about cumulative errors and the stability of the employed numerical algorithms.

The leading source for computational errors in finite precision arithmetic is the addition of quantities of different magnitudes, causing a loss of least significant digits in the smaller quantity. In extreme cases, this can lead to serious instabilities in numerical algorithms. Software codes that perform matrix operations are especially vulnerable to such stability issues, as are algorithms that use finite differences for solving systems of differential equations numerically.

While it is difficult to prove that a particular solution is not a result of a numerical instability, it is extremely unlikely that two independently-developed programs could produce compatible results that are nevertheless incorrect, as a result of computational error. Therefore, verifying a result using independently-developed software codes is a reliable way to exclude numerical instabilities as a possible error source, and also to put a limit on any numerical errors.

In view of the above, given the excellent agreement in various implementations of the modeling software, the authors of [27Jump To The Next Citation Point] concluded that differences in analyst choices (parameterization of clocks, data editing, modeling options, etc.) give rise to coordinate discrepancies only at the level of 0.3 cm. This number corresponds to an uncertainty in estimating the anomalous acceleration on the order of 8 × 10–14 m/s2, which was found to be negligible for the investigation.

Analysis identified, however, a slightly larger error to contend with. After processing, Doppler residuals at JPL were rounded to 15 and later to 14 significant figures. When the Block 5 receivers came online in 1995, Doppler output was further rounded to 13 significant digits. According to [27Jump To The Next Citation Point], this roundoff results in the estimate for the numerical uncertainty of

2 σnum = ±0.02 × 10 −10 m ∕s . (5.21 )

5.5.2 Model consistency

The accuracy of navigational codes that are used to model the motion of spacecraft is limited by the accuracy of the mathematical models employed by the programs to model the solar system. The two programs used in the investigation – JPL’s ODP/Sigma modeling software and The Aerospace Corporation’s POEAS/CHASMP software package – used different parameter estimation procedures, employed different realizations of the Earth’s orientation parameters, used different planetary ephemerides, and different data editing strategies. While it is possible that some of the differences were partially masked by maneuver estimations, internal consistency checks indicated that the two solutions were consistent at the level of one part in 1015, implying an acceleration error ≤ 10−4aP [27Jump To The Next Citation Point].

The consistency of the models was verified by comparing separately the Pioneer 11 results and the Pioneer 10 results for the three intervals studied in [27Jump To The Next Citation Point]. The models differed, respectively, by (0.25, 0.21, 0.23, 0.02) m/s2. Assuming that these errors are uncorrelated, [27Jump To The Next Citation Point] computed the combined effect on anomalous acceleration aP as

−10 2 σconsist∕model = ±0.13 × 10 m ∕s. (5.22 )

5.5.3 Error due to mismodeling of maneuvers

The velocity change that results from a propulsion maneuver cannot be modeled exactly. Mechanical uncertainties, fuel properties and impurities, valve performance, and other factors all contribute uncertainties. The authors of [27Jump To The Next Citation Point] found that for a typical maneuver, the standard error in the residuals is σ0 ∼ 0.095 mm ∕s. Given 28 maneuvers during the Pioneer 10 study period of 11.5 years, a mismodeling of this magnitude would contribute an error to the acceleration solution with a magnitude of δaman = σ0∕τ = 0.07 × 10−10 m ∕s2. Assuming a normal distribution around zero with a standard deviation of δa man for each single maneuver, a total of N = 28 maneuvers yields a total error of

σman = δa√man- = 0.01 × 10−10 m ∕s2, (5.23 ) N
due to maneuver mismodeling.

5.5.4 Annual/diurnal mismodeling uncertainty

In addition to the constant anomalous acceleration term, an annual sinusoid has been reported [27Jump To The Next Citation Point390Jump To The Next Citation Point]. The peaks of the sinusoid occur when the spacecraft is nearest to the Sun in the celestial sphere, as seen from the Earth, at times when the Doppler noise due to the solar plasma is at a maximum. A parametric fit to this oscillatory term [27Jump To The Next Citation Point392Jump To The Next Citation Point] modeled this sinusoid with amplitude vat = (0.1053 ± 0.0107) mm ∕s, angular velocity ω = (0.0177 ± 0.0001 ) rad∕day at, and bias b = (0.0720 ± 0.0082) mm ∕s at, resulting in post-fit residuals of σT = 0.1 mm ∕s, averaged over the data interval T.

The obtained amplitude and angular velocity can be combined to form an acceleration amplitude: aat = vatωat = (0.215 ± 0.022) × 10− 10 m ∕s2. The likely cause of this apparent acceleration is a mismodeling of the orbital inclination of the spacecraft to the ecliptic plane [27Jump To The Next Citation Point392Jump To The Next Citation Point].

[27Jump To The Next Citation Point] estimated the annual contribution to the error budget for aP. Combining σT and the magnitude of the annual sinusoidal term for the entire Pioneer 10 data span, they calculated

2 σat = 0.32 × 10−10 m∕s . (5.24 )
This number is assumed to be the systematic error from the annual term.

[27Jump To The Next Citation Point] also indicated the presence of a significant diurnal term, with a period that is approximately equal to the sidereal rotation period of the Earth, 23h56m04s.0989. The magnitude of the diurnal term is comparable to that of the annual term, but the corresponding angular velocity is much larger, resulting in large apparent accelerations relative to a P. These large accelerations, however, average out over long observational intervals, to less than − 10 2 0.03 × 10 m ∕s over a year. The origin of the annual and diurnal terms is likely the same modeling problem [27Jump To The Next Citation Point].

These small periodic modeling errors are effectively masked by maneuvers and plasma noise. However, as they are uncorrelated with the observed anomalous acceleration (characterized by an essentially linear drift, not annual/diurnal sinusoidal signatures), they do not represent a source of systematic error.

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