Dr John D. Anderson, of Jet Propulsion, tried to use the Doppler shifts from the Pioneers to calculate Jupiter’s mass as they were accelerated by the planet’s gravity. However, the radio signals from both spacecraft kept shifting in an unexpected way. After years of analysis, Anderson calculated that if the anomaly is caused by motion, then both craft experienced a sun-ward acceleration of 8.74 ± 1.33 × 10^-10 m/s². http://arxiv.org/pdf/gr-qc/0104064v5.pdf

The Pioneers used a transponder that multiplies the received radio frequency by the ratio 240/221 to generate the reply frequency. The up frequency came from NASA’s hydrogen maser clocks on earth (2.113 GHz) so the returning signal from the spacecraft was ~2.295 GHz (modified by the Doppler shifts from the relative motion between the Earth and the spacecraft). A Doppler counter back on Earth adds up the difference between the up-link frequency and the returned signals. JPL navigators calculate that the range changes by one wavelength for each Doppler count, after correcting for Earth’s motion. The Pioneer anomaly correlates with distance and also shows sinusoidal yearly and daily (diurnal) variations. http://arxiv.org/pdf/0906.0399v1.pdf  This suggests that the anomaly points toward the Earth, not the Sun. If it were a result of unexpected  gravity, it should point towards the Sun.

Scientists have proposed numerous explanations for the Pioneer anomaly. Perhaps gravity increases slightly with distance. However, we do not observe the planets slowly falling towards the Sun. Perhaps the spacecraft are colliding with thin gases. However, the measured density of the interplanetary medium is far too low.

Slava Turyshev and Viktor Toth tried to explain the Pioneer Anomaly using heat radiating from the radioisotope thermoelectric generators (RTGs). They thought that infrared from the RTGs reflects off the back of the dish antenna and this might be slowing down the craft. (Imagine sitting in the bow of a rowboat throwing bean bags against the transom. Only the ones that do not hit the boat can accelerate it. This is because the reaction of throwing the bean bags is cancelled by the impact on the transom). The RTGs on the Pioneers are on trusses out past the dish antenna and the fins are edge on the back of the dish.

Two other spin stabilized spacecraft, Ulysses and Galileo, had a similar Doppler anomaly, although they have a different weight and their RTGs are not positioned the same. Since neither went beyond Jupiter’s orbit, the anomaly on these spacecraft are not as pronounced as on the Pioneers. The Voyagers, Cassini and newer spacecraft use inertial wheels to maintain their orientation, which introduces unaccounted accelerations.  New Horizons is spin stabilized during its cruise phase and may someday provide useful information to supplement the Pioneer data.

Anderson proposed another possible explanation, that clocks everywhere accelerate. If NASA’s hydrogen maser clocks are speeding up, the up-link radio signal would bed slower than the down-link signal and would correlate with distance. Anderson’s estimate for the acceleration of all clocks is (2.92±0.44)×10^-18 s/s². Edwin Hubble found that the farther away a galaxy, the more its light is shifted relative to the frequencies of local atoms. According to Hubble, the distance to a galaxy is measurable by its Doppler shift. The Hubble ratio, H_0, is supposed to represent the rate of the universe’s expansion. As telescopes improved, astronomers discovered that early galaxies would need to move faster than light to fit Hubble’s explanation. Astronomers now imagine that galaxies are standing still relative to their local vacuum as they move away due to the expansion of the intergalactic vacuum. Astronomers speculate that the light form billions of ancient galaxies is shifted by the expanding vacuum, something never observed in any experiment. The Hubble ratio multiplied by the speed of light is 6.7 × 10^-10 m / sec². This is just slightly lower than estimates for the Pioneer Anomaly.

The landing anomaly may support the acceleration of clocks. Landings on Mars always are downrange of the target. Pathfinder landed 27 km, Spirit 13.4 km, Opportunity 14.9 km, Phoenix 21 km and Curiosity 2km – all downrange. (NASA used extraordinary efforts for Curiosity, yet it still landed downrange). If clocks are speeding up as radar signals make the round trip to a planet, the orbital distances would be slightly shorter than estimates and the spacecraft would likely land long.

The flyby anomaly may represent another manifestation of the Pioneer Anomaly. NASA occasionally uses Doppler to navigate spacecraft back to Earth for a gravity assist. When the craft slingshots around Earth, they sometimes achieve unexpected speeds (Galileo, Cassini and Near) in the range of several mm per second. Anderson suggests that the flyby anomaly may be related to the rotation of the Earth. Spacecraft that slingshot along the ecliptic, rather than over the poles, experience different flyby effects. http://arxiv.org/pdf/astro-ph/0608087v2.pdf 

Atomic clocks are really two clocks connected by a feedback circuit. The first clock radiates cesium 133 with microwaves. The second clock counts 9,192,631,770 microwave pulses from the relaxing cesium, which by scientific definition is a second. The feedback circuit adjusts the frequency of the input clock for a maximum signal strength at the sensors of the output counter. Hydrogen maser atomic clocks have a similar feedback circuit, although they count only 1,420,405,751.786 cycles as a second. If atoms are changing relationally, all atomic clocks would keep on tracking with the changing atoms. Seconds are the primary scientific unit from which meters, velocities, accelerations, energies, and the laws of physics depend. Tens of thousands of empirical definitions, measuring techniques, mathematical formulas and “constants” were contrived on the notion that clocks measure linear time.

When trying to understand the universe, one should first consider the first principle of physics, which you can read of on this website: The First Principle of Physics.

Hundreds of billions of galaxies shine from all directions, to the creation age. Not a single one of them shines with the frequencies of modern atoms.  The earliest galaxies, bright enough to obtain a spectrum with current telescopes, shine at about 7% of the frequencies of modern atoms. We also observe chains of emerging star globs around early spiral galaxies. This is Hubble Ultra Deep galaxy 4491, that shines at about 48% of the frequencies of modern atoms. Notice the stream of blue stars globs emerging from the redder core. Throughout visible cosmic history, the star streams accelerate outward along with the accelerating atomic clocks.