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I'm not a science guy, but I did some research and I gathered that the rocket's upper stage spins before releasing the satellites stack in such a way that each satellite is given a slightly different speed. When the stack is released, the satellites are spaced out precluding or minimizing any possibility of collision between themselves.

If the satellites are stacked together, how does the spin give each satellite a different speed?

And, the starlink "train", which we can see for a few days after deployment, is it the result of this spin, or of the satellites firing their own thrusters into a single file , or both?

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If the satellites are stacked together, how does the spin give each satellite a different speed?

That's just basic geometry: imagine a spinning playground carousel, let's say for the sake of argument it spins at 1 revolution per second (hey, I didn't say it is a safe playground carousel!). Every point on the carousel spins at the same angular rate: it will complete 1 revolution per second.

However, the circumference on the outside of the carousel is much bigger than the circumference near the center of the carousel. Ergo, since the points at the outside of the carousel go around the bigger circumference in the same time that the points on the inside of the carousel go around the smaller circumference, the points on the outside of the carousel have a higher tangential velocity.

Let's say the carousel has a radius of 2m. Then a point at the outside moves at $2*2\pi\,\frac{m}{s}\approx12.6\,\frac{m}{s}$, whereas a point at half the radius moves at $2\pi\,\frac{m}{s}\approx6.3\,\frac{m}{s}$, In other words: the tangential velocity scales linearly with the distance from the center of the rotation.

As soon as the tensioning rod lets go of the Starlink satellite stack, there is no force which forces the satellites to spin together with the rocket anymore, so according to Newton's First Law of Motion, they will all continue in a straight line at their current velocity. And since this velocity is different for every satellite, they will slowly spread out over time, even without any force acting on them.

Note that the rocket spins end-over-end, i.e. it "somersaults", which is different from the direction we normally mean when we say a rocket "spins". This confused me in the beginning when I heard of the "spin release" deployment for the first time.

And, the starlink "train", which we can see for a few days after deployment, is it the result of this spin, or of the satellites firing their own thrusters into a single file, or both?

I guess that depends on when, exactly, you see the train and how long the checkout phase is until they start maneuvering using their Hall effect thrusters.

But even without using the thrusters, over time, they would spread out from a pretty close clump directly after deployment to the typical "train" with an ever growing length and distance between the satellites.

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