6.1 Unmodeled forces external to the spacecraft

The trajectory of the Pioneer spacecraft, while governed primarily by the gravity of the solar system, is nevertheless a result of a complex combination of gravitational and nongravitational forces, all of which must be taken into account for a precision orbit determination. What if some of those forces were not properly accounted for in the model, resulting in an unmodeled acceleration of the observed magnitude? Several authors considered this possibility.

6.1.1 Gravitational forces due to unknown mass distributions and the Kuiper belt

Of course, one of the most natural mechanisms to generate a putative physical force is the gravitational attraction due to a known mass distribution in the outer solar system; for instance, due to Kuiper belt objects or interplanetary dust. Anderson et al. [27Jump To The Next Citation Point] have considered such a possibility by studying various known density distributions for the Kuiper belt and concluded these density distributions are incompatible with the discovered properties of the anomaly. Even worse, these distributions cannot circumvent the constraint from the undisturbed orbits of Mars and Jupiter.

The possibility of a gravitational perturbation on the Pioneer paths has also been considered by the authors of [5694153253], who studied the possible effects produced by different Kuiper Belt mass distributions, and concluded that the Kuiper Belt cannot produce the observed acceleration.

Nieto [253] studied several models for 3-dimensional rings and wedges whose densities are either constant or vary as the inverse of the distance, as the inverse-squared distance, or according to the Boss–Peale model. It was demonstrated that physically viable models of this type can produce neither the magnitude nor the constancy of the Pioneer anomaly. In fact, the results emphasized the difficulty in achieving a constant acceleration within a finite cylindrically-symmetric distribution of matter. The difficulties are even stronger if one considers the amount of mass that would be needed to mimic the Pioneer anomaly.

The density of dust is not large enough to produce a gravitational acceleration on the order of aP [27Jump To The Next Citation Point56253] and also it varies greatly within the Kuiper belt, precluding any constant acceleration. In particular, Bertolami and Vieira [56] obtained the largest acceleration when the Kuiper belt was represented by a two-ring model. In this case, the following magnitude of a radial acceleration arad could be obtained (using spherical coordinates (r,𝜃,ϕ )):

∫ 2π 2 ----GM------- ∑ -------r −-Ri-cos-𝜃cosϕ------ arad(r,𝜃) = − 2π(R1 + R2 ) 0 Ri(r2 + R2i − 2rRi cos 𝜃cosϕ )3∕2d ϕ, (6.1 ) i=1
where R1 = 39.4 AU (3:2 resonance) and R2 = 47.8 AU (2:1 resonance) are the radii of the two rings, M is the total mass of the Kuiper belt, and G is the gravitational constant. Equation (6.1View Equation) can be evaluated numerically, yielding a nonuniform acceleration that is at least an order of magnitude smaller than the Pioneer anomaly. Other dust distributions, such as those represented by a uniform disk model, a nonuniform disk model, or a toroidal model yield even smaller values. Hence, a gravitational attraction by the Kuiper belt can, to a large extent, be ruled out.

6.1.2 Drag forces due to interplanetary dust

Several nongravitational, conventional forces have been proposed by different authors to explain the anomaly. In particular, the drag force due to interplanetary dust has been investigated by the authors of [56253]. The acceleration a drag due to drag can be modeled as

κρv2A adrag = − ------, (6.2 ) m
where ρ is the density of the interplanetary medium, v is the velocity of the spacecraft, A its effective cross section, m its mass, while κ is a dimensionless coefficient the value of which is 2 for reflection, 1 for absorption, and 0 for transmission of the dust particles through the spacecraft.

Using Equation (6.2View Equation) as an in situ measurement of the “apparent” density of the interplanetary medium, one obtains ρ ≃ 2.5 × 10 −16 kg ∕m3. This is several orders of magnitude larger than the interplanetary dust density (∼ 10−21 kg∕m3) reported by other spacecraft (see discussion in [262]).

The analysis of data from the inner parts of the solar system taken by the Pioneer 10 and 11 dust detectors strongly favors a spherical distribution of dust over a disk. Ulysses and Galileo measurements in the inner solar system find very few dust grains in the 10− 18 –10− 12 kg range [262]. The density of dust is not large enough to produce a gravitational acceleration on the order of aP [27Jump To The Next Citation Point]. The resistance caused by the interplanetary dust is too small to provide support for the anomaly [262], so is the dust-induced frequency shift of the carrier signal.

The mechanism of drag forces due to interplanetary dust as the origin of the anomaly was discussed in detail in [262]. In particular, the authors considered this idea by taking into account the known properties of dust in the solar system, which is composed of thinly scattered matter with two main contributions:

In [262] these properties were used to estimate the effect of dust on Pioneer 10 and 11 and it was found that one needs an axially-symmetric dust distribution within 20 – 70 AU with a constant, uniform, and unreasonably high density of ∼ 3 × 10 −16 kg ∕m3 ≃ 3 × 105 (ρ + ρ ) IPD ISD. Therefore, interplanetary dust cannot explain the Pioneer anomaly.

One may argue that higher densities are present within the Kuiper belt. IR observations rule out more than 0.3 Earth mass from Kuiper Belt dust in the trans-Neptunian region. Using this figure, the authors of [56] have noted that the Pioneer measurement of the interplanetary dust density is comparable to the density of various Kuiper belt models. Nonetheless, the density varies greatly within the Kuiper belt, precluding any constant acceleration.

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