WILLIAM & DEBORAH HILLYARD
There are a number of anomalies in gravitational theory that, as yet, have no complete explanation.  It is possible that the observations themselves are in error, or  
that our understanding of gravity is incomplete.  Certainly, when I see one anomaly predicting gravity falling off more slowly than theory predicts, and another  
predicting gravity increasing in strength, I reserve judgement!  The anomalies include:
The Pioneer Anomaly
The two Pioneer spacecraft leaving the solar system, and slowing down due to the Sun's gravity.  The rate at which they are slowing is more than would be expected.  One suggested explanation is, of course, errors in the measurements or observations.  Other suggestions include that they are radiating heat in a non-uniform way or that there is leakage of helium gas from the spacecraft, though it would be strange that this affected both spacecraft identically.  Some observers suggest that there is an error in current theory, but it is interesting that the outer planets do not seem affected in the same way, so the jury is still out on this one.  The New Horizons spacecraft, heading to Pluto, the Kuiper Belt and beyond, may provide further information. 
The Flyby Anomaly
During Earth flybys of spacecraft, there appears to be an unexplained velocity increase.  It is possible, though by no means certain, that this is linked to the Pioneer Anomaly.  It was first observed in 1990 on the Galileo mission where a small change of of about 4 mm/second occurred.  In 1998, a change of more than 13 mm/second occurred in the Near Earth Asteroid Rendezvous (NEAR) spacecraft during an Earth flyby.  Since then, observed velocity changes in spacecraft flybys have been much smaller.  Explanations include the affect of dark matter or a possible large scale Casimir effect. 
The Anomalous Increase in Planetary Orbits
As the Sun radiates energy, it slowly looses mass.  This means that the orbits of the planets slowly widen.  Observations of this expansion, about 7 meters (nearly 23 feet) per century for the Earth, are more than expected based on the rate of solar radiation.  Attempts to explain this anomaly to date have been very exotic, including abandoning the whole Big Bang cosmology!  We await developments with interest. 
Hydrogen Clouds & Dark Flow
At very large distance scales, hydrogen appears far more "clumped" than expected based on the time available since the Big Bang.  In addition, the Dark Flow of galaxy clusters towards a huge, unobserved mass is difficult to explain as it is unlikely that such a large mass could have formed in the time since the Big Bang.  Taken together, some observers suggest that gravity falls off more slowly than the inverse square law over these very large distances. 
Highly Energetic Photons
When photons travel for great distances through galaxy clusters, they should gain energy going into the cluster and then lose it on the way out.  The accelerating expansion of the universe prevents return of all the energy.  However, photons from the cosmic microwave background appear to gain twice as much energy as expected, which suggests that gravity falls off faster than the inverse-square law on large distance scales.

Anomalies

Physics

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Gravity

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