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In this video, you can see the Falcon 9 first stage flying through the second stage's plume post-separation. I would assume that this would alter the first stage's trajectory -- potentially causing a tumble. Why not?

I suspect that either the plume is rarified enough to not cause this or the first stage has an active system (first stage nitrogen thrusters?) that is activated at that time to prevent it.

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You're correct on both counts. At the time of stage separation, the rocket is approximately 80km in altitude (although it varies based on the specific flight path) where the atmospheric pressure is around 1 Pascal, essentially a vacuum - this results in a very rarified plume which expands quickly. Combine that with the empty weight of the first stage being in excess of ~18 metric tonnes due to the fuel remaining required for the three relanding burns, and you can quickly see that any effects due to the second stage plume would be aerodynamically minimal on an object carrying that much inertia.

Additionally, following stage separation in the video you link, you can actually see the pulses of the cold Nitrogen gas thrusters actively controlling the attitude of the vehicle. Furthermore, this is a technique SpaceX has used as far back as CASSIOPE (Falcon 9 v1.1) which you can see in this video, as viewed from the second stage. Not only that, but it's been tried on Falcon 1 too!

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