# Mathematics

Proving an idea correct or not is often much harder than one originally assumes. The process for doing so can often be categorised into three leaky buckets: proposing a theory, showing it is valid, and finally testing whether or not it is actually true.

Applied mathematics can be seen as a tool box for building scale, or dummy, models of proposed phenomena. It is objective. It does not often require expensive kit or laboratory space. But it is not absolute, for it can only test the assumption placed in – itself under the assumption that they have been implemented correctly. In regards this project, our application of mathematics as published in Evatt et al. (2016) was to show that under hypothetical Antarctic conditions (temperature, solar energy, ice ablation), iron-rich meteorites melt at a faster rate than iron-poor (stony) meteorites. And that they could melt faster than the rate of ice moving upwards, and do so for long enough to mean they never reach the surface. Hence, the notion of a hidden layer of meteorites encapusulated at depth within the ice.

For those familiar with undergraduate level mathematics, the precise tools we used involved a set of free-boundary partial differential equations. Here, the ‘free boundary’ relates to the position of a boundary condition, which is itself solved by another equation. The upshot being that the main equation appears to have more boundary conditions than the highest derivative term would first suggest. In modelling the englacial meteorite, we had two free boundaries: the bottom of the meteorite, which was melting the underlying ice, and above the meteorite, between the ice and the meteorite/ melted water.

To apply such techniques to a full 2-d or 3-d model would have been exceptionally difficult, (and possibly more complex and time consuming than actually going to Antarctica to find them!). As such, we reverted to solving the system in a 1d system. Reducing the dimensionality down allows for significant clarity in regards how a system works. And crucially, testing it’s validity.

With the developed model, we could then apply real parameter values to see if it works as one intended. But to avoid the case that it worked out of pure coincidence, we also applied the model to a controlled laboratory setting (itself another validation test). Here we tested the model against pre-buried englacial meteorites under very controlled settings, and found that our model matched very well against the empirical evidence of englacial melting. With that control test successfully completed, the validity of our theory was firmly established.

Our resulting equations (see Evatt et al. 2016):

And how they performed (solid black line) against our laboratory measurements of ice meteorites (open circle symbols and dashed line) (see Evatt et al. 2016 for details):

And will our lost Antarctic meteorite hypothesis actually be true? Or will something we have not yet considered cause them to not be there?

Well, this is what we’re are going to Antarctica to find out. In the mean-time we continually revisit our original model to see if we’ve missed anything. But despite this important dose of self-criticism, at least we’ve done what we can to show there is a valid hope of finding a lost layer of meteorites trapped within blue ice fields in Antarctica.