I often get questions about the Pioneer anomaly and our on-going research. All too often, the questions boil down to this: what *percentage* of the anomaly can <insert theory here> account for?

This is a very bad way to think about the anomaly. It completely misses the fact that the Pioneer anomaly is not an observed sunward acceleration of the Pioneer spacecraft. The DATA is not a measured acceleration; what is measured is the frequency of the spacecraft’s radio signal.

I already capitalized the word DATA in the previous paragraph; let me also capitalize the words MODEL and RESIDUAL, as these are the right terms to use when thinking about the Pioneer anomaly.

As I said above, the DATA is the Doppler measurement of the spacecraft’s radio frequency.

The MODEL is a model of all forces acting on the spacecraft including gravity, on-board forces, solar pressure, etc; all effects acting on the spacecraft’s radio signal, including the Shapiro delay, solar plasma, the Earth’s atmosphere; and all effects governing the motion of the ground stations participating in the communication.

The RESIDUAL is the error, the difference between the MODEL’s prediction of the Doppler measurement vs. the actual measurement. This RESIDUAL basically appears as noise, but with characteristic signatures (a diurnal and an annual sinusoid along with discontinuous jumps at the time of maneuvers) that suggest mismodeling.

The goal is to make this RESIDUAL “vanish”; by that, we mean that only random noise remains, any diurnal, annual, or maneuver-related signatures are reduced to the level of background noise.

The RESIDUAL can be made to vanish (or at least, can be greatly reduced) by incorporating new contributions into the MODEL. These contributions may or may not be rooted in physics; indeed, orbit determination codes typically have the ability to add “unmodeled” effects (basically, mathematical formulae, such as a term that is a quadratic or exponential function of time) to the MODEL, without regard to the physical origin (if any) of these effects.

Anderson et al. found that if they add an unmodeled constant sunward acceleration to the MODEL, they can make the RESIDUAL vanish. This is the result that has been published as the Pioneer anomaly.

If one has a physical theory that predicts a constant sunward acceleration, it is meaningful to talk in terms of percentages. For instance, one may have a physical theory that predicts a constant sunward acceleration with magnitude *cH* where *c* is the speed of light and *H* is Hubble’s constant at the present epoch; it then makes sense to say that, “using the widely accepted value of *H* ~= 71 km/s/Mpc, the theory explains 79% of the Pioneer anomaly,” since we’re comparing two numbers that represent the same physical quantity, a constant sunward acceleration.

However, note (very important!) that the fact that a constant sunward acceleration fits the data does not exclude alternatives with forces that are not constant or sunward pointing; the DATA admits many different MODELs.

Now let’s talk about the thermal recoil force. It is NOT constant and it is NOT sunward pointing. As we recompute this force, incorporating the best thermal model that we can compute into the MODEL and re-evaluate it, we obtain a new RESIDUAL. There are, then, the following possibilities:

- Suppose that the new RESIDUAL is as free of a mismodeling signature as the constant acceleration model and that its magnitude cannot be reduced by adding any unmodeled effects (i.e., we reached the level of our basic measurement noise.) Does it then make sense to speak of percentages? OK, so the thermal recoil force is 30%, 70%, 130%, you name it, of the constant sunward acceleration. But the thermal recoil force is neither constant nor sunward, and by incorporating it into the MODEL, we got a different trajectory than the constant sunward acceleration cas. Yet the RESIDUAL vanishes, so the MODEL fits the DATA just as well.
- Suppose that the new RESIDUAL is half the original RESIDUAL at least insofar as the apparent mismodeling is concerned. What does this mean? Does this mean that the thermal recoil force and the resulting acceleration is half that of the constant sunward value? Most certainly not. Say it’s 65%. Now did we explain 50% of the anomaly (by reducing the RESIDUAL to 50%) or did we explain 65% of the anomaly (by producing a thermal recoil acceleration that’s 65% of the published constant sunward value?)

Instead of playing with percentages, it makes a lot more sense to do this: after applying our best present understanding insofar as thermal recoil forces are concerned, we re-evaluate the MODEL. We compute the RESIDUAL. We check if this residual contains any signatures of mismodeling. If it doesn’t, we have no anomaly. If it does, we characterize this mismodeling by applying various unmodeled effects (e.g., a constant sunward force, exponential decay, etc.) to check if any of these can characterize the RESIDUAL. We then report on the existence of a (revised) anomaly with the formula for the unmodeled effect as a means to consisely characterize the RESIDUAL. If this revised anomaly is still well described by a constant sunward term, we may use a percentage figure to describe it… otherwise, it’s probably not helpful to do so.