Anderson etal [gr-qc/0104064]
have announced an anomalous acceleration of
the Pioneer spacecraft toward the Sun. This acceleration is not seen in
the planets and would be easily detectable. The acceleration of Neptune
toward the Sun is 6*10^{-4} cm/sec^{2}
while the anomalous acceleration is 8*10^{-8} cm/sec^{2}.
The increased acceleration would lead to an increased orbital velocity
with *dv/v = 0.5*da/a = 0.7*10 ^{-4}*,
which would shorten the orbital period by
the same fraction, and lead to a mistake in the distance of Neptune of

Aladar Stolmar claims that the anomalous acceleration of the Pioneer 10 and 11 is due to a tired light explanation of the Hubble law. If this were true the redshift produced would be

v = cz = HDand the time derivative of this effect would be

a = dv/dt = cdz/dt = d[HD]/dt = HvThe Pioneer 10 is moving outward at

Note that the gravitational redshift effect on the clocks has no effect on this measurement because the data are from two-way Doppler ranging. In any case the Pioneers do not carry clocks that are good enough to measure this effect.

The two-way ranging doubles the size of the frequency shift produced by any cosmological effect, but it also doubles the frequency shift produced by an acceleration. Thus the ratio of anomalous acceleration to H is just as computed above.

The anomalous acceleration implies a thrust on the spacecraft that corresponds to the force produced by a desk lamp: a "photon rocket" with a power of just 60 watts. The radioactive thermoelectric generators [RTGs] on the Pioneers produce 2000 watts of thermal power. Most of this is radiated as waste heat. If this were radiated slightly anisotropicly, with 1030 W going outward and 970 W going inward, the whole anomalous acceleration would be explained. The radiator fins have been exposed to space for many years, with one side always in sunlight and the other side always in shadow. I think that a difference in the emissivity of the radiator fins of this magnitude is quite possible. No tests were done on the spacecraft to measure this anisotropy before launch, and there is no way to retrieve them for a calibration measurement.

The Cassini spacecraft shows a LARGER non-gravitational acceleration than the Pioneer's. But since it was designed to radiate RTG waste heat primarily out the back, this is not considered to be anomalous.

Spacecraft Mass Acceleration P(rtg) mac/P Cassini 5200 kg -3E-9 m/s^2 10000 W 50% Pioneer 258 kg -8E-10 m/s^2 2000 W 3%F = ma is the force, and Fc = mac is the power needed to give this force in a photon rocket. Cassini is radiating its waste heat with a 50% asymmetry factor, while Pioneer only needs to be 3% asymmetric to explain its "anomalous" acceleration.

Therefore this extraordinary claim is not supported by extraordinary evidence and should be treated with caution.

Update 19 Apr 2008:
Turyshev *et al.* (2008) have done a thermal model of the
Pioneer 10 & 11 spacecraft and get a 23 W asymmetric flux, which
explains part of the anomaly. They presented this work at the April 2008
American Physical Society meeting.

Update 15 Jul 2011:
Turyshev
*et al.* (2011, PRL accepted)
have recovered and analyzed old tracking
files on Pioneer 10 and 11, more than doubling the time span of
the available data. They find that the
anomalous acceleration is decaying with time in a manner consistent
with the anisotropic waste heat emission model.

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