The Pioneer 10 spacecraft launched from Cape Canaveral, Florida on March 2, 1972. Its planned mission was to be the first man-made craft to traverse the asteroid belt and image Jupiter up close. Pioneer 10’s power plant design was required to last for a minimum of two years in space, but the craft was still traveling and transmitting on January 23, 2003—30 years, 10 months, and 22 days later—when the signal became too weak to pick up on Earth. It was 80 AU (12 billion kilometers) away, at the edge of the solar system.

After Pioneer 10 and its younger sibling, Pioneer 11, passed the 20 AU mark, radio-metric Doppler data showed that there was a deviation in the acceleration of the spacecraft from the numbers predicted by scientists and astronautical engineers. This was first noticed in 1980, but did not receive significant attention until the mid 1990s. It was found that if a tiny, constant, sunward acceleration of (8.74±1.33)x10^{-10} m/s^{2} was applied to the predicted orbital model, the discrepancy between the model and the data would go away. However, the source of this acceleration has remained an open question that has had many possible explanations, from the mundane to what was termed “new physics.”

Recent analysis suggests that some form of “new physics”—such as the breakdown of the inverse square law of gravity as described by Newton—will have to wait, as the more pedestrian answer of thermal recoil solves the problem. Thermal recoil force is the (ever so slight) force that results from thermal photons being emitted from a surface—it’s Newton’s “for every action there is an equal and opposite reaction” in a straightforward form. If heat emissions are distributed unevenly on the spacecraft, then more thermal photons emanating from a given area will impart more of a force in the opposite direction—enough to account for the acceleration anomaly.

To figure out if this is possible here, a team of researchers from Caltech’s Jet Propulsion Laboratory and the Applied Science Laboratory built a highly detailed finite-element (FE) thermal model of the Pioneer 10 spacecraft. Many modern FE packages can import CAD drawings of the surface or solid you wish to model; however, a spacecraft designed in the 1960s predates the use of CAD (it was done by hand with old school isometric drawings and instrumentation diagrams). The final thermal FE model consisted of about 3,300 surface elements, 3,700 nodes, and 8,700 linear conductors plus the thermal and radiative properties for each—putting all that together shows off the fun an intern or first year graduate student can look forward to.

The Pioneer 10 spacecraft was powered by four radioisotope thermoelectric generators (RTG), each one holding about 100 grams of ^{238}Pu. As they generated heat, it would not be distributed evenly around the spacecraft, and the FE analysis was used to determine the actual distribution of temperature. To determine the magnitude of the recoil force, the spacecraft model was placed at the center of a large black sphere with a radius of 40 times the diameter of the high gain antenna on the craft.

The amount of radiative emission absorbed by the sphere’s surface then corresponded to the amount of momentum carried in that direction. The authors were able to model the RTG thermal power output to within one percent of the known value, and temperatures that were always within ±2K.

Combining this data with the equation that relates the thermal power emitted in a given direction with the resulting acceleration, the team came up with a two parameter model that describes the acceleration due to thermal recoil force. The two terms in the equation relate the acceleration derived from the RTGs, and the acceleration that results from the thermal energy of the electronic equipment on board the spacecraft. The two free variables are the net efficiencies of each process, i.e. how much energy is really turned into acceleration. They come up with η_{RTG} = 0.0104 and η_{Elec} = 0.406, with an RMS error of only 0.78 W, a value well below the inherent error in the telemetry.

To complement the novel thermal model presented in the paper, the researchers also incorporated their thermal recoil force model into a Doppler analysis of the actual craft’s travels through the solar system. This allowed them to carry out a completely independent sanity check of their work to see if the values they obtained for the efficiency parameters from the thermal model would agree with another method. Here, they find an η_{RTG} = 0.0144 and η_{Elec} = 0.480—very close to the numbers obtained by the thermal analysis, but producing about a 20 percent higher force. To determine if this was a statistically significant difference, the authors examined the η_{RTG} and η_{Elec} parameter space, and plotted out the region within it that included a 1σ deviation. They found that at a 1σ level, these two estimates of the parameters overlapped each other, indicating that they were statistically similar.

The net result: accounting for this heat emission seems to bring the predicted trajectory much more in line with the craft’s actual trajectory.

The authors finish their paper by listing areas of research that could further improve these numbers; better estimates of certain thermal properties of various coatings on the craft, out gassing of various components over the years, and even a repeat of the analysis with the Pioneer 11 spacecraft. They conclude with the following simple statement: “the anomalous acceleration of these spacecraft is consistent with known physics.” Just another nail in the coffin for those hoping alternate theories of gravity like MOND may yet hold true