# Explanation of the Pioneer 10 and 11 Acceleration Anomaly

In the last 10-15 years or so, it has emerged that the tracking data for the Pioneer 10 and 11 space probes indicate an unexpected slowdown of their velocity on their way out of the solar system, i.e. an anomalous acceleration towards the sun which can apparently not be explained by known forces (see the paper by Anderson et al.). The figure below (which is taken from this paper) gives a plot of the observed unexplained part of the acceleration against distance from the sun. It shows that the acceleration is apparently about constant over the whole range where the error bars are small enough so that the data can be considered as significant (i.e. between about 20 and 50 AU). Several theories to explain this anomaly have been suggested, but none of these appear to give a conclusive explanation (at least, all explanations in terms of conventional physics are practically ruled out in the study of Anderson et al.). However, one obvious cause has apparently not been considered, and this is a mis-modelling of the earth's rotation:

the earth's rotation corresponds to an acceleration at the equator of about 3 cm/sec2. With the acceleration vector always directed towards the earth's center from the observing station, its projection (or the projection of any error in the assumption for the rotational acceleration) on the direction towards the spacecraft will thus always have the same sign, i.e. the effect will not simply be sinusoidal but result in a net acceleration that does not average out with time. As the space craft is not always near the zenith but on average presumably in all parts of the sky (and furthermore the observing station not at the equator), the average projected acceleration will be somewhat less than the maximum value, let's say 1.5 cm/sec2. Now although the earth's rotation is obviously being taken into account in the model, the point is that both the earth's radius and angular velocity are only known with relative accuracies of 1.5.10-8 and 1.4.10-8 respectively (see http://hpiers.obspm.fr/eop-pc/models/constants.html). Combined, this results in a relative error for the rotational acceleration of about 3.10-8, and if one assumes that this is in the sense of an overestimation, this would thus correspond to an apparent acceleration of the spacecraft with 4.5.10-8 cm/sec2 towards earth. If one multiplies this by two (because the analysis uses two-way Doppler data and the error will occur both for the up- and down-link), this yields an acceleration anomaly of 9.10-8 cm/sec2 towards earth, in good agreement with the measurements (it is generally claimed that the acceleration is towards the sun, but this can actually not be distinguished due to the beam width of the Pioneer antenna).

There is also another aspect which may be of interest here, namely a discrepancy between the 'light distance' and the actual geometrical distance to the spacecraft due to an incorrect interpretation of the invariance of the speed of light. Interestingly, this leads to an apparent outward acceleration which is exactly identical to the contribution of the radiation pressure in Anderson et al.'s model (see the page Speed of Light and Anomalous Acceleration). Although it does on its own not resolve the known anomaly, it might call for a re-modelling of the radiation pressure force in general.

Note: recent publications by Turyshev et al., Francisco et al. and others suggest that the Pioneer anomaly can be explained in terms of the radiation pressure (recoil) due to the thermal emission of thr spacecraft alone. This shouldn't invalidate the issue addressed on this page though, especially as the radiation pressure theory can be questioned on fundemental theoretical grounds (as explained here). The uncertainty of the figure for the earth's rotation rate would still leave any observed acceleration data for space probes uncertain by a corresponding amount (at least if time averaged data are used like in the original 2002 paper by Anderson et al.).

Thomas Smid (M.Sc. Physics, Ph.D. Astronomy)
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See also my sister site https://www.plasmaphysics.org.uk