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I remember one time a while back looking at the night sky through night vision goggles. One interesting thing I noticed is that there appeared to be many very faint meteors in the sky practically all the time. I assume this is dust burning up in the atmosphere, or possibly an unusual effect produced by the night vision goggles themselves. I more suspect that it is dust burning faintly in the atmosphere because it was seen only in the sky. My main question is: Is space dust constantly entering the atmosphere as small meteors at a fairly high rate? If I remember right, it was a few every second, which is much more frequent than normal "brighter" meteors, even during a meteor shower.

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The answer is yes. The mass (and hence brightness) of meteorites follows some sort of power-law relationship such that there are many more small particles than big lumps of rock. This means that there are indeed many more faint meteorites than bright ones, though I am struggling to find any detailed study that goes below the magnitude range you can see with the naked eye. This paper by Kresakova (1966) seems thorough and representative. It suggests that the number of meteorites goes up by a factor of 3 for each unit increase in astronomical magnitude (i.e. decreasing brightness).

The paper referred to by Barry Carter in his comments is Cook et al. (1980), which finds that log of the cumulative number of meteors is proportional to about half the astronomical magnitude. i.e. $$\frac{d \log \phi}{dm} \simeq 0.5 $$ This means that the number of meteorites increases by a factor of $10^{0.5}$ for each unit increase in magnitude - i.e. also about 3.

It is a separate issue as to whether you see more with night vision goggles on. I guess you would because these would have light amplification properties that would enable you to see fainter objects - that's the whole point of them.

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