Back in 1998, researchers at the Super-Kamiokande neutrino detector released this image:
of the sun using their neutrino detector using approximately 500 days of data. This picture covered 90x90 degrees of the sky compared to the sun's disk angular size of about a half degree.
Is it possible that as more neutrino detectors become available, we could collect data from detectors around the world over a longer period of time to get a statistical estimate on the size of the core of the sun? Perhaps this would involve releasing data on incidental events of suspected solar neutrinos which then could be released to the public domain and pooled for statistical analysis?