Geological/Planetary differentiation is one of those topics that gets glossed over in intro Geo courses. We know that differentiation occurs over deep time (and will never end) and is a result of the heating of rock. Heating in this case can be driven by collision, accretion (planet formation), and radioactivity. So, we know that the Earth, Mars, and Moon have all differentiated. On the other hand, smaller asteroids will never differentiate. Are there any set/known size requirements on a body before it attains a high level of differentiation like the Earth?
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Not really.
Assuming even minor igneous bodies differentiate (stocks), and that differentiation is basically a process of decantation of the solid magmatic phase (a process known as fractional crystalization) and fractionation of the fluid magmatic phase, virtually any solid mixture can differentiate. If you're not convinced, here's another argument.
Let's take an analogous situation to differentiation: exsolution. They're not the same. Exsolution involves ionic migration in a solid structure. Differentiation involves ionic migration in a fluid (magma), which lacks structure. But both involve an outcome of different chemical phases. And exsolution can happen in microscopic sizes of milimetric crystals. So its fluid-ambient counterpart would, as a logical conclusion, be also able to happen on such minute conditions, and perhaps on smaller even.
And I'm not sure what you mean that differentiation will never end? On a long term geological scale - I'm talking about hundreds of millions to billions of years - that idea is perhaps correct, assuming the upper mantle and crust are constantly changing their compositions. But if you're saying that differentiation is a process that happens widespread (the entire crust, or the entire planet), rather than on specific locations (where partial melting occurs), under a small timescale (millions of years), then you're not correct, because differentiation only occurs in mixtured fluids (and I'm not sure whether it's used for any other fluid than magma).
