Specifically, what is in an ice core that can be used to study carbon dioxide levels in air from so long ago?

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Literally Air including carbon dioxide is what is in Ice cores. The deeper the core sample the longer ago it was at the surface as water (solid or liquid). when the water was gaseous as clouds before it fell to earth all the atmospheric gasses (Air) dissolved into the water, when it froze those gasses became trapped when it was then covered by layer upon layer of subsequent ice/snow it compacted even further trapping those gasses. When we measure the composition of that air and compare it with current levels we see that CO2 levels are higher than they have been in thousands if not a couple hundred of thousands of years.

Explanation:

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