  In the last 70 years, geophysics has established that the Earth's outer core
is an FeNi alloy containing a few percent of light elements, whose nature and
amount remain controversial today. Besides the classical combinations of
silicon and oxygen, hydrogen has been advocated as the only light element that
could account alone for both the density and velocity profiles of the outer
core. Here we show how this question can be addressed from an independant
viewpoint, by exploiting the tomographic information provided by atmospheric
neutrinos, weakly-interacting particles produced in the atmosphere and
constantly traversing the Earth. We evaluate the potential of the upcoming
generation of atmospheric neutrino detectors for such a measurement, showing
that they could efficiently detect the presence of 1 wt% of hydrogen in an FeNi
core in 50 years of concomitant data taking. We then identify the main
requirements for a next-generation detector to perform this measurement in a
few years timescale, with the further capability to efficiently discriminate
between FeNiH and FeNiSi(x)O(y) models in less than 15 years.
