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What approximate size range, considering the different construction and composition, must meteoroids be to become meteorites on Earth, Mars, and the Moon?

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There isn't any, because we know of homogenous materials that have molecular and atomic bonds stronger than maximum dynamic pressure per their surface area that they would have to endure as they enter the Earth's atmosphere, are subject to gravitational forces and chemical interaction with the atmosphere. I'm specifically saying this for Earth, since it has the densest atmosphere and is the most massive object of the three in your question, and Martian atmosphere doesn't have any chemicals not already present in Earth's atmosphere.

For example, we know that in celestials of sufficient mass and carbon content, solid diamonds form in regions of required internal pressure and temperature. For example inside Jupiter, regions of its interior sustain tremendous pressure that is sufficient to form diamond lattice crystal of carbon atoms. In fact, Jupiter is even too massive and these diamonds are later subject to even stronger pressure they can no longer withstand, and they melt into a tar-like material. Anyway, chances that any of these diamonds would be free-floating in space are rather remote, but not impossible, if e.g. there was a cataclysmic collision between two such celestials tearing them apart, sending some of its innards at escape velocities on trajectories towards other celestials.

Science fiction? Perhaps, but there are other homogenous materials of sufficient strength too that would be perfectly capable of entering Earth's atmosphere and not lose a single atom through ablation as they are subject to tremendous surface friction with the atmosphere, being torn apart by tidal forces, evaporate or interact chemically. Crystal structures are particularly strong and inert as the atoms form perfect of near perfect bonds between each other, but some polymer molecular bonds can be nearly as strong and don't change state to evaporate easily. Of course, most meteoroids aren't homogenous, and they might have structural weaknesses that would cause them to disintegrate, but the point is that there isn't any minimum size all the way down to a single atom. Not with Earth's atmosphere, and even less so on Mars and its tenuous atmosphere, and no atmospheric restrictions on the Moon whatsoever, since its exosphere isn't dense enough to even act like gasses do (molecules colliding with each other, and forming surface pressure).

One thing about meteoroids that is often misunderstood is that they tend to "burn" in the atmosphere. This is a simplification, and there isn't any burn in conventional sense, and bolides and meteoroids don't form flames. What happens is that they enter the atmosphere at such immense speeds that the friction with it causes its surface to heat up, ablate or gas out and all of these high energy particles ionize the atmosphere surrounding it into a gaseous plasma. The level of such interaction would depend on the total surface area of all its parts, velocity and the rate of ablation (macro scale) and evaporation (micro scale). Chemical burn is actually extremely remote. Meteoroids of strong enough surface materials and internal structure wouldn't lose any of their mass during such interaction with the atmosphere at all, even though the bolide / meteoroid would still appear as if it's "burning", as it would still create an ionized trail along its path. How they would fragment on impact is another matter, but that depends on the hardness of the surface they impact with, and their velocity would be greatly reduced on their way through the atmosphere (where present).

So this boils down to structural integrity of the impacting meteoroid, not so much its size. Sure, as the size increases, the surface area subject to friction, tidal forces and chemical interaction with the atmosphere will increase, and so will internal stretching and pulling on bonds holding it together, but we're already going in the opposite direction now, describing their maximum size before being torn apart (still not preventing impact, mind you), not their minimum one.

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