Anomalous Acceleration of Pioneer 10 and 11

James F. Carter1


The path of the Pioneer 10 spacecraft is not reproduced completely accurately by Newtonian mechanics[1]; there is an excess acceleration toward the Sun. Members of galaxies orbit consistently faster than expected from the estimated baryonic matter inside their paths[2]. Milgrom has modelled these deviations empirically under the name MOND[3].

Hubble discovered that spectra of distant galaxies are red-shifted proportional to the independently estimated distance to the galaxies. A plausible model of this effect is that the universe is expanding uniformly. Freedman et al.[4] recently estimated the expansion rate (Hubble constant) H as 728 km per second per megaparsec, or 2.330.2610-18 sec-1.

Newton modelled orbital motions in the Solar System through a gravitational force between objects of

where F= the force of gravity, mi= the objects' masses, r= the distance between them, and G=6.67310-11 m3/kg s2. For computing the trajectory of spacecraft through the Solar System, experience shows that this model is extremely accurate. The Pioneer 10 spacecraft, at a distance from the Sun of about 67 astronomical units (AU) or 1.01013 m, experiences a measured acceleration toward the Sun of 1.3210-6 m/s2, which is 8.741.3310-10 m/s2 less than the Newtonian model[1, 5].

Galaxies are gravitationally bound, and their outer members revolve considerably faster than might be expected from the detectable matter in the galaxies. Milgrom[3] has also pointed out that a similar adjustment to gravity, oriented to the center of the individual galaxy, can explain their speed.

Because the universe is expanding, the spacecraft has an unmodelled velocity of 2.3310-5 m/s. It takes light and gravitation 3.34104 sec (0.4 day) to travel from the Sun to the spacecraft. In that time the spacecraft moves away by 0.78 m. In general, a distance of r is inflated by dr=r2H/c, where c=speed of light. To be precise about reference frames, we measure (half) the round-trip lightspeed time to the spacecraft, and report a history of its postion 0.4 days in the past, which will be r+dr where r refers to the distance when the telemetry started out (and neglecting or subtracting off Newtonian modelled motion)[6]. However, gravity propagates under the same rules as light, and the spacecraft will experience an acceleration as if its distance from the Sun were r. In other words, gravity is the gradient of a potential computed thus:
φ(y ,t)=
ρ(x,t- | y-x | /c)

where y is the position of the spacecraft, ρ(x,t)=the density of matter as a function of location and time, and integration is potentially over the whole universe. The density of matter is recognized as it was before the lightspeed delay, when it was closer to the spacecraft than the measured r+dr. As a result, the Newtonian modelled acceleration will be too small by 2dr/r, which for Pioneer 10 comes out to 2.0510-19 m/s2. This correction is far too small to explain the anomalous acceleration.

Anderson et al.[1] point out that the anomalous acceleration could be produced if the tracking receiver's master oscillator slowed down relative to the transmitter's oscillator per the Hubble constant. Specifically, the tracking system reports an acceleration of a=hf(c/f) where h is the fractional frequency drift rate in inverse seconds, f is the telemetry frequency in Hz (2.21109 or 2.40109Hz for the up and down links), and c is the speed of light in m/s. Solving for h and using the downlink frequency, we find h=2.920.4410-18 s-1 which is plausibly close to the currently measured value of Hubble's constant. This result is independent of the distance to the spacecraft.

However, the uplink and downlink oscillators are the same hydrogen maser, stable to 10-15f when integrated over 1000 seconds. If the master oscillators at different tracking stations drifted at the Hubble constant rate, it would have been noticed within hours. On board the spacecraft, the uplink frequency is multiplied by a rational number 240/221 to give the downlink, but if the downlink oscillator unlocked and ran free, drifting at a rate h, the effect could be explained, but a much larger and randomly changing drift would be expected, which would be painfully obvious. Thus it is not credible that actual frequency shifts in the equipment could have occurred.

Within experimental error the anomalous acceleration is constant between radial coordinates of 15 to 45 AU (later extended to 67 AU), yet is not significantly different from 0 in the range of 5 to 10 AU. A variation of 20% was looked for (in connection with a possible source in the Radioactive Power Generators, whose output decreases with time) but was not found. Pioneer 10 and 11 show effects of the same size directed toward the Sun (or Earth), though they travel in nearly opposite directions. Thus, it is not credible that the acceleration, or appearance of acceleration, is caused by an effect happening as the telemetry propagates through space, which would be proportional to distance.

As for unmodelled masses in the Solar System, a constant acceleration could be produced by a spherically symmetric cloud of matter whose density varies as 1/r. Inside 67 AU its mass would have to be 1.31027 kg or about 200 Earth masses. This is not credible.

Although the tie-in with Hubble's constant attracts attention, a credible connection cannot be discovered. Anderson et al.[1] believe that the most likely cause of the acceleration is either unmodelled gas escape, or the momentum of thermal photons from the Radioactive Power Generators, differing from the model used.


John D. Anderson, Philip A. Laing, Eunice L. Lau, Anthony S. Liu, Michael Martin Nieto, Slava G. Turyshev, ``Study of the anomalous acceleration of Pioneer 10 and 11'', Phys.Rev. D65 (2002) 082004.
need reference
Milgrom, M., ``A modification of the Newtonian dynamics - Implications for galaxies'', ApJ, 270, 371 (1983).
W. L. Freedman, B. F. Madore, B. K. Gibson, L. Ferrarese, D. D. Kelson, S. Sakai, J. R. Mould, R. C. Kennicutt, Jr., H. C. Ford, J. A. Graham, J. P. Huchra, S. M. G. Hughes, G. D. Illingworth, L. M. Macri, P. B. Stetson, ``Final Results from the Hubble Space Telescope Key Project to Measure the Hubble Constant", Astrophys.J. 553 (2001) 47-72.
The acceleration is toward the center of mass of the Solar System; transverse acceleration is 0210-10 m/s2. Within experimental error it is the same from a radial position of 15 to 67 AU, but from 5 to 10 AU it is different, not significantly different from 0. The locations of the planets, Luna and larger asteroids are modelled in detail. Minor gravitational effects are accounted for generically. Numerous effects are accounted for in the position of the telemetry antenna (such as Earth's rotational irregularities, tides in the solid Earth, and continental drift), nongravitational forces on the spacecraft (such as Solar radiation pressure, radiation pressure from the telemetry transmission, and thruster leakage), and conditions on the propagation path (such as interplanetary plasma and atmospheric refraction). The quoted error of 1.3310-10 m/s2 is the combined uncertainty in all these effects.
As for arc length, the outbound path is r+dr/2 whereas the inbound path is r+3dr/2, and while the resulting phase skew could potentially be measured, the position coordinate of the spacecraft does not go by the arc length needed to reach it.
Very useful in preparing this document was this link page belonging to Stacy McGaugh at University of Maryland.

Department of Mathematics, University of California, 405 Hilgard Avenue, Los Angeles, CA 90095-1555, USA. E-mail:

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