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I assume performing spectroscopy on meteors would give us information about the composition of the outer layers of meteoroids, which we can't study on finds because it has already burned up. This would increase our knowledge on the small bodies of the solar system.

There are programs, like CAMS or FRIPON, that watch the entire sky to spot meteors. These are just regular cameras.

We can perform spectroscopy for many points in the sky simultaneously, like with MUSE, so it would be possible to monitor most of the sky continuously to get spectroscopic data of meteors.

Has this been done? If not, why not?

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You can do this kind of thing reasonably easily, even as amateur astronomer. See e.g. this link. There also have been programmes by the Americal Meteor society and there are a lot of published papers to that end, e.g. here

Two reasons it has not been done more widlely:

It's a difficult and very costly endevour to make high-resolution spectroscopy at very short time intervals - at the time you have for the meteor to show. There is no reasonable technique to justify a setup which points a large telescope within sub-second time to an arbitrary location as identified by a full-dome camera scanning the whole sky - to which you will have to attach the spectrograph.

At the same time there is little reason to believe that the outer material of meteorites which ablates in the upper atmosphere is different to what we find as meteorite on Earth. The surface is altered, and the processes are well-understood. Thermal conductivity and inertia are very small so that below this very shallow crust the material is unaltered. That said, missions like the Apollo missions as well as StarDust and similar did in-situ analysis of interplanetary material which has not been thermally altered in Earth's atmosphere. General statistics on type of meteorites, including their chemical composition and origin can also be done with the finds from Earth's surface.

So in summary: the cost to gain ratio for such endevour might be much smaller than you seem to anticipate.

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