From Stanford University News and Symmetry Magazine (The latter has a small but important correction—See the 1^st comment and the response)

It’s a mystery that presented itself unexpectedly: The radioactive decay of some elements sitting quietly in laboratories on Earth seemed to be influenced by activities inside the sun, 93 million miles away.

Is this possible?

Researchers from Stanford and Purdue universities believe it is. But their explanation of how it happens opens the door to yet another mystery.

Also discussed by Jeff Duntemann.

The article mentioned the decays Silicon-32 and Radium-226, as reported in Evidence for Correlations Between Nuclear Decay Rates and Earth-Sun Distance back in 2008. Similarly, Perturbation of Nuclear Decay Rates During the Solar Flare of 13 December 2006 indicated a solar dependence on the decay of Manganese-54. The suggestion that radioactive decay rates might in some way depend on the sun is quite extraordinary, and has prompted the reanalysis of a lot of data. Evidence against correlations between nuclear decay rates and Earth-Sun distance looked at the decay of 6 other radioisotopes without seeing any such dependence. There is no obvious dependence on atomic weight or other systematic difference between the elements.

There are different types of radioactive decay. Radium-226 decays by emitting an α particle (a Helium nucleus: 2 protons and 2 neutrons) while Silicon-32 is a case of β decay (emission of an electron). Manganese-54 decays by electron capture, which is essentially time-reversed β decay. α-decay is a manifestation of the of the strong nuclear force, while β-decay is a weak interaction. If the solar effect is real, then affects two differenct fundamental forces of nature.

John G. Cramer, in Radioactive Decay and the Earth-Sun Distance suggested that

…the Earth’s orbit has a very small eccentricity, so the annual variations in R [the Earth-Sun distance] are small. A better way of testing whether radioactive decay rates depend directly on 1/R² would be to monitor a radioactive decay process within a space vehicle in a long elliptic orbit with a large eccentricity, so that R has a very large variation. As it happens, NASA has a number of space probes that match this description, because many space probes, particularly those that venture into the outer reaches of the Solar System, are powered by radioisotope-driven thermoelectric power sources containing a strong radioactive decay source that produces enough energy as heat to power the vehicle. The power levels of such thermoelectric generators are carefully monitored because they constitute the principal power source of the vehicle.

This has been done. According to Peter Cooper, in Searching for modifications to the exponential radioactive decay law with the Cassini spacecraft

Data from the power output of the radioisotope thermoelectric generators aboard the Cassini spacecraft are used to test the conjecture that small deviations observed in terrestrial measurements of the exponential radioactive decay law are correlated with the Earth-Sun distance. No significant deviations from exponential decay are observed over a range of 0.7 – 1.6 A.U. A 90% Cl upper limit of 0.84 x 10^-4 is set on a term in the decay rate of Pu-238 proportional to 1/R² and 0.99 x 10^-4 for a term proportional to 1/R.

Less technically:

Deep-space probes usually generate power from the heat emitted by a chunk of radioactive material-plutonium-238 for the Cassini spacecraft. Cassini journeyed as close to the sun as Venus and then far back to Saturn, spanning a much wider range of distances from the sun than Earth does during its yearly orbit. If the sun had an effect on plutonium decay, the fluctuations would have been much more substantial than those seen in Earth-bound experiments. As a result, Cooper reasoned, Cassini should have measured substantial changes in its generator’s output. It didn’t.

The Stanford/Symmetry article included something new. Peter Sturrock, Professor Emeritus of Applied Physics at Stanford, suggested is that some of the variation in the Radium-226 and Silicon-32 decay rates is related to solar rotation. From Evidence for Solar Influences on Nuclear Decay Rates

Recent reports of periodic fluctuations in nuclear decay data of certain isotopes have led to the suggestion that nuclear decay rates are being influenced by the Sun, perhaps via neutrinos. Here we present evidence for the existence of an additional periodicity that appears to be related to the Rieger periodicity well known in solar physics.

Links to the research reports can be found at Variability of Nuclear Decay rates. Search for “Research papers: PERIODIC VARIATIONS: SCALE OF DAYS OR YEARS” and “Research papers: NON-PERIODIC VARIATIONS:” Subheading “Of Cosmic Origin”. Thanks to arXiv.org information about current research in physics is easily accessible.

Peter Sturrock was my first course advisor when I was a graduate student in his department at Stanford, 1972-1975. At one point there I wanted to take a course in mathematical statistics. I was a little hesitant about this, since the subject is somewhat off the main direction of graduate study in physics. To my surprise, Professor Sturrock strongly encouraged me to do so. Whatever the conclusion about the relationship between the sun and radioactive decay may be, there will be a lot of statistical analysis along the way.