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Considering a space shuttle re-entering Earth, what's the minimum velocity that would cause the vehicle to start burning? Is there an equation to calculate that?
I presume it varies according to the materials that compose the shuttle and of the atmosphere.
Spacecraft usually do not burn on re-entry, in the usual sense of burning. Burning implies a combustion reaction, which does not happen on a spacecraft designed for it, and even spacecraft not designed for it usually won't burn, because they're not made of flammable materials; they just break apart under the thermal and physical stresses.
However, upon re-entry from orbit, there is a lot of re-entry heating. This is caused by adiabatic compression, a process in which air heats up as it is compressed. The air heats up a lot from this compression, which is what causes the bright glow of re-entry. This will happen in the vicinity of any object which is traveling at hypersonic speed (speeds greater than around Mach 5), as, at that speed, the air literally cannot get out of the way in time.
This video does a good job of explaining re-entry heating and measures used to protect spacecraft from it, if you want to learn more.
