Does the sun wobble?
Since many people come here with this question here the answer of Cornell.edu (owner and operator of astro-physics arXiv etc):
So if you're afraid it could do any harm to you: definitively not.
But scientifically there is a more than huge difference between a stationary sun and a wobbling sun. This new knowledge changed our view to the universe completely. A whole new era started in astronomy leading to an explosion of our knowledge about the universe.
With the new knowledge of wobbling stars mankind learned that there are billions of other planetary systems out there in the universe. And perhaps a second earth too (New analysis assumes that there are at least 17 billion Earth-sized worlds in the Milky Way. See also here. Read especially the conclusions section.). These new found planets are called 'Exoplanets' and are among the hottest topics in today's science. It is the essence of scientific achievements to replace ancient crude approximations. Science knows no 'good enough approximation for most purposes'. Especially since this ignorance prevented the biggest scientific progress of this and the last century, that our galaxy is teeming with planetary systems. And this in contrast to the fact that just 25 years ago all scientists world wide agreed that our planetary system is the sole planetary system in the whole universe! Electricity has been known since the Greeks, but it was the scrutiny of men like Volta, Ampere, Ohm, Faraday and Maxwell who made electricity the easy to handle force we are using everyday in our houses, cities, trains, machinery.... If these men would have been satisfied with 'good enough approximations for most purposes' we would still live in candle lighted houses and steam-engines would lift our elevators to the 36. floor, our cities would still be gas-lighted and television and computers would be dreams of the far future..
Tau Ceti now delivered in the end of 2012 the all important proof: this star, which is only 12 light years away from us has been the target of one of the earliest searches for extraterrestrial life in the 60ths and 70ths. But astronomers couldn't find any hints of orbiting planets in the 60s through 90s. A recently completed re-examination, which especially considered the wobbling of the central star discovered five potential planets ranging from two to nearly seven times the mass of Earth, with orbits ranging from 14 to 640 Earth days long.
But the most exciting fact about this news is, that this discovery was made by purely re-analysing the old data, no new data was won. And mathematics hasn't changed a bit since the sixtieth of last century. It was the new knowledge about wobbling stars that led to the discovery of these exoplanets. Two of these planets could potentially be the closest habitable exoplanets yet discovered. So we will likely see soon many new discoveries on many more earlier examined stars.
The new knowledge that the barycentre of our solar system is not identical with the sun's centre has and had quite naturally many more scientific and practical implications and consequences (if you haven't heard about any of these important changes in the press or television you're not alone. Many physicists are not aware of them since these changes happen very secretly and very silently and unnoticed by the vast majority of the physics and even astronomy community. So the confusion is after more than twenty years still alive and still big: http://www.cv.nrao.edu/~rfisher/Ephemerides/times.html#TDB. This makes necessary even in 2010 such explanations as this one or this one from end of 2011.)
A rather recent (nov. 2013) overview on the state of research on solar dynamics see here or this paper ( arXiv link, the arXiv file seems to be very busy): "A growing body of empirical evidence suggests that solar activity on monthly to millennial time scales may be modulated by gravitational and magnetic planetary harmonic forces"
Always the latest news concerning the search for a second
earth you find on
1) try yourself to search for it in astronomy blogs or science journals
2) what concerns "relativistic corrections" see "How to read the JPL Ephemeris and Perform Barycentering C. Markwardt 28 Jun 2001":
The "geometric" correction is the primary effect, which is essentially the time for the photon to travel from earth to the SSB.
(geometric) = (robs . n)/c where
robs - is the vector from the SSB to the observatory
n - is a unit vector pointing to the astrophysical source
The value of n depends on the position of the source on the sky (ie, RA and DEC), is assumed to be known already, or is solved for in using an iterative process."
3) Since the first exoplanet was detected now 18 years ago well over 7000 papers have been published and about 3000 conference proceedings. In the last years about 1000 papers per year are being published, steadily rising.