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To deploy 10 satellites, does the rocket do a series of burns, or do the satellites have enough propulsion to deploy themselves to separate orbits?

The satellites do it themselves. The deployments from second stage are staggered with pauses of 100 seconds between each to allow them to safely drift apart from the initial separation jolt. After that, the sats do their own thing.

The deployment from the dispenser just puts the satellites in the same orbit with a little bit of offset between them, enough for comfort. The satellites then raise their orbits just a hair using their own propulsion and then use phasing to adjust their positioning so that they are equidistant along the orbit. That part takes a long time (hours) but it doesn't require a lot of propulsive capabilities so the satellites do it themselves.

The required propulsion is minimal. They're all going into the same orbit, just spaced around it. A tiny speed difference is enough to do that, since it adds up over time.

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