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Some interesting excerpts from Is the Sun still shining?, F .E. Close, Nature Vol. 284, 10 April 1980 (pgs. 507-508)

THE conjunction of cosmology and particle physics as the result of recent developments in high energy physics formed the subject of a Wolfson Invitation Lecture given at the University of Oxford by J. Ellis of CERN.

… Argon-37 undergoes K-capture decay with a 35 day period. Almost all these K capture events are detected and so the amount of argon is inferred and in turn the incident neutrino flux responsible for creating it in the first place can be calculated. This quantity is expressed in solar neutrino units (SNU). But although theoretical understanding of the Sun's interior requires that about 6 ± 2 SNU be observed, experiment has detected only a fraction of this. (A meeting in Rome during early February gave 2.2 ± 0.3 as the most up-to-date result).
One possibility is that our theoretical understanding of the Sun is badly wrong somehow. The next source of blame is in the inferences drawn from the experimental rate of detection.

Another possibility is that the Sun has stopped shining. If the thermonuclear processes in the heart of the Sun have run down, there will be a significant drop in the neutrinos produced and a low flux will be detected on Earth. This is because the Sun is almost transparent to neutrinos and so they stream out from the core unhindered. In contrast, the Sun is quite opaque to photons and those that irradiate the Earth have come from the Sun's outer reaches. It would be some hundreds of thousands of circumstances in the Sun's core reached its surface. Thus we could be somewhere in this period, waiting for the surface photons to be turned off - the low neutrino flux is our advance warning of a real energy crisis.
Fortunately there is another possible cause for the low flux which is of particular interest in light of recent discoveries in particle physics.
Before 1976 two varieties of neutrino were known … Since 1976 a third variety of neutrino has been discovered associated with the heavy lepton tau(T) …

… the eccentricity of the Earth's orbit gives a natural variation to the distance that neutrinos have travelled en route from the Sun. Neutrino masses of the order of 10-5 eV could account for the solar neutrino anomaly, and provide an annual periodic variation in intensity of electron neutrinos … would indirectly support attempts to unify the natural forces and would indeed resolve the solar neutrino anomaly, avoiding the catastrophic alternative.

Who knew: solar neutrino fluctuation / variability ?

I believe current thinking is that neutrinos oscillate between the three flavours (electron-neutrino, muon-neutrino, tau-neutrino), which requires them to have a small mass and not travel at the speed of light.
Indeed, it's a great story - the mystery stood for rather a while but has been resolved in broad terms. This is a good read:
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