Following [24, 27, 390], we first consider forces that affect the spacecraft motion, such as those due to i) solar-radiation pressure, and ii) solar wind pressure. We then discuss the effects on the propagation of the radio signal that are from iii) the solar corona and its mismodeling, iv) electro-magnetic Lorentz forces, v) the influence of the Kuiper belt, vi) the phase stability of the reference atomic clocks, and vii) the mechanical and phase stability of the DSN antennae, together with influence of the station locations and troposphere and ionosphere contributions. Although some of the mechanisms detailed below are near the limit for contributing to the final error budget, it was found that none of them could explain the behavior of the detected signal. Moreover, some were three orders of magnitude or more too small.

[27] estimated the systematic error from solar radiation pressure on the Pioneer 10 spacecraft over the interval from 40 to 70.5 AU, and for Pioneer 11 from 22.4 to 31.7 AU. Using Equation (4.8) they estimated that when the spacecraft reached 10 AU, the solar radiation acceleration was decreasing to at 70 AU. Because this contribution falls off with the inverse square of the spacecraft’s heliocentric distance, it can bias the Doppler determination of a constant acceleration. By taking the average of the inverse square acceleration curve over the Pioneer distance, [27] estimated the error in the acceleration of the spacecraft due to solar radiation pressure. This error, in units of , is for Pioneer 10 over the interval from 40 to 70.5 AU, and six times this amount for Pioneer 11 over the interval from 22.4 to 31.7 AU. In addition the uncertainty in the spacecraft’s mass for the studied data interval also introduced a bias of in the acceleration value. These estimates resulted in the error estimates

The acceleration caused by solar wind particles intercepted by the spacecraft can be estimated, as discussed in Section 4.3.2. Due to variations with a magnitude of up to 100%, the exact acceleration is unpredictable, but its magnitude is small, therefore its contribution to the Pioneer acceleration is completely negligible. Based on these arguments, the authors of [27] concluded that the total uncertainty in due to solar wind can be limited as

Given Equation (4.30) derived in Section 4.5.1 and the values of the parameters , all in meters, [27] estimated the acceleration error due to the effect of the solar corona on the propagation of radio waves between the Earth and the spacecraft.

The correction to the Doppler frequency shift is obtained from Equation (4.30) by simple time differentiation. (The impact parameter depends on time as and may be expressed in terms of the relative velocity of the spacecraft with respect to the Earth, ).

The effect of the solar corona is expected to be small on the Doppler frequency shift, which is our main observable. This is due to the fact that most of the data used for the Pioneer analysis were taken with large Sun-Earth-spacecraft angles. Further, the solar corona effect on the Doppler observable has a periodic signature, corresponding to the Earth’s orbital motion, resulting in variations in the Sun-Earth-spacecraft angle. The time-averaged effect of the corona on the propagation of the Pioneers’ radio-signals is of order

The authors of [392] considered the possibility that the Pioneer spacecraft can hold a charge and be deflected in its trajectory by Lorentz forces. They noted that this was a concern during planetary flybys due to the strength of Jupiter’s and Saturn’s magnetic fields (see Figure 2.1). The magnetic field strength in the outer solar system, Gauss, is five orders of magnitude smaller than the magnetic field strengths measured by the spacecraft at their nearest approaches to Jupiter: 0.185 Gauss for Pioneer 10 and 1.135 Gauss for Pioneer 11. Data from the Pioneer 10 plasma analyzer can be interpreted as placing an upper bound of on the positive charge during its Jupiter encounter [269].

These bounds allow us to estimate the upper limit of the contribution of the electromotive force on the motion of the Pioneer spacecraft in the outer solar system. This was accomplished in [27] using the standard formula for the Lorentz-force, , and found that the greatest force would be on Pioneer 11 during its closest approach to Jupiter, . However, once the spacecraft reached the interplanetary medium, this force would decrease to

which is negligible.

[27] specifically studied three distributions of matter in the Kupier belt, including a uniform distribution
and resonance distributions that were hypothesized in [188]. The authors assumed a total mass of one Earth
mass, which is significantly larger than standard estimates. Even so, the resulting accelerations are only on
the order of 10^{–11} m/s^{2}, two orders of magnitude less than the observed effect. The calculated
accelerations vary with time, increasing as Pioneer 10 approaches the Kuiper belt, even with a
uniform density model. For these reasons, [27] excluded the dust belt as a source for the Pioneer
effect.

More recent infrared observations established an upper limit of 0.3 Earth masses of Kuiper Belt dust in the trans-Neptunian region [40, 363, 371]. Therefore, for the contribution of Kuiper belt gravity, the authors of [27] placed a limit of

Reliable detection of a precision Doppler observable requires a very stable frequency reference at the observing stations. High precision Pioneer 10 and 11 Doppler measurements were made using 2-way and 3-way Doppler observations. In this mode, there was no on-board frequency reference at the spacecraft; the received frequency was converted to a downlink frequency using a fixed frequency ratio (240/221), and this signal was returned to the Earth. As the round-trip light time was many hours, the stability of the frequency reference over such timescales is essential. Further, in the case of 3-way Doppler measurements when the transmitting and receiving stations were not the same, it was essential to have stable frequency references that were synchronized between ground stations of the DSN.

The stability of a clock or frequency reference is usually measured by its Allan deviation. The Allan deviation , or its square, the Allan-variance, are defined as the variance of the frequency departure (where is the frequency and is its variance during the measurement period) over a measurement period :

where angle brackets indicate averaging.The S-band communication systems of the DSN that were used for communicating with the Pioneer spacecraft had Allan deviations that are of order or less for integration times [392]. Using the Pioneer S-band transmission frequency as , we obtain

over a Doppler integration time of s. Applying this figure to the case of a steady frequency drift, the corresponding acceleration error over the course of a year was estimated [27] as

The measurement of the frequency of a radio signal is affected by the stability of physical antenna structures. The large antennas of the DSN complexes are not perfectly stable. Short term effects include thermal expansion, wind loading, tides and ocean loading. Long term effects are introduced by continental drift, gravity loads and the aging of structures.

All these effects are well understood and routinely accounted for as part of DSN operations. DSN personnel regularly assess the performance of the DSN complex to ensure that operational limits are maintained [352, 353].

The authors of [27] found that none of these effects can produce a constant drift comparable to the observed Pioneer Doppler acceleration. Their analysis, which included errors due to imperfect knowledge of DSN station locations, to troposphere and ionosphere models at different stations, and to Faraday rotation effects of the atmosphere, shows a negligible contribution to the observed acceleration:

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