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Re-entry vehicles tend to be either mono-stable (eg soyuz) or bi-stable (apollo) which refers to the orientations in which they're aerodynamically stable. In at least one case, this has proved very useful on the soyuz where the capsule corrected orientation when guidance/control systems failed.

In how many modes are the SpaceX Dragon V1 and V2 stable?

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  • $\begingroup$ It is aerodynamically stable (At least V2), but I can't find any more details than that. $\endgroup$ – PearsonArtPhoto Apr 13 '16 at 1:24
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From this answer to this question, we can see that there are two stability modes used. The capsule by itself is mono-stable, with the heat shield pointing towards the ground. The second mode is with the trunk included, and wings extended, which is stable with the nose in the direction of movement. The second method is used in rocket abort systems, keeping the spacecraft moving with the nosecone in the direction of motion.

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