# How deep does impact heating penetrate?

Let's say I find a cold rocky body with no atmosphere, and I bombard its surface with breathable air, thus converting the gas's GPE into heat. This will warm the body somewhat (and might initiate further temperature changes e.g. by releasing trapped greenhouse gases, but let's just focus on primary warming). This is an easy exercise in $$\Delta T=Q/(mc)$$ in theory, except I don't know how much mass absorbs the heat.

I suspect the heat won't spread all the way to the core if the body is large; in other words, I conjecture a skin depth effect. Given a spherical body of radius $$R$$ much larger than a thickness $$\tau$$ such an effect involves, only a proportion $$3\tau/R$$ of the volume absorbs the heat. But how large would $$\tau$$ be? This feels like a heat equation exercise, so dimensional analysis suggests something like $$\tau=\alpha/v$$, with $$\alpha$$ the thermal diffusivity and $$v$$ the speed of sound.

However, I suspect this is too low a value. Based on some back-of-the-envelope assumptions, I concluded that adding enough atmosphere to Mars to match Earth's surface pressure would warm the planet about a tenth of a Kelvin if all its mass absorbs the heat, but more like a million Kelvin if only that thick of a skin (a few centimetres) is relevant. Part of me suspects the real answer would be somewhere in between.

For all I know, other variables are relevant. For example, if the mass of material ejected in impact crater formation isn't proportional to the impactor mass, and if it is the ejected mass which absorbs the heat, the exact consequences would depend on how the new atmosphere is split into payloads, and perhaps this could be fine-tuned to ensure the full surface is heated an average of once each, to whatever depth implies the temperature rise is whatever we want.

• The question in the title is an interesting question that is difficult to answer. The bulk of the text is effectively a ramble. Bombarding the surface with a breathable atmosphere is deflecting focus on what's important: a bolide hits a sizeable cosmic body, a crater is created, material is ejected from the crater, the site of the impact is heated by the impact, how much heat does the impact produce & how deep does the heat penetrate. ...
– Fred
Nov 4, 2021 at 9:54
• ... This question ... on the Physics Forum will give you an idea of some of the things to consider.
– Fred
Nov 4, 2021 at 9:55
• @Fred That problem assumes the heat evaporates some water (and only just makes it warm enough to evaporate) and isn't dumped into any other matter, so $\Delta T$ sets $m$ rather than the other way round. Would we expect the same viz. rock evaporation for a dry impact site? You imply the issue is more complex than that.
– J.G.
Nov 4, 2021 at 10:33