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For example: if a satellite in orbit around earth had its prograde rotational speed increased from 60 rpm to 600 rpm, would its orbit be affected?

Another example would be if earth’s rotational speed were increased, so that a day only lasted 1 hour, would its orbit around the sun change?

Also, would it matter from where the rotational energy is given to it? (energy from outside vs from within the object)

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In brief "no", at least not unless the objects are so dense, massive or rapidly rotating, or you measure the orbit with such extreme precision, that general relativity becomes a significant factor. The centre of mass of the satellite (or of the Earth) follows its elliptic or circular orbit regardless of how the rest of the satellite (or planet) is rotating around the centre.

Of course some external forces can affect both the rotation of the object and its orbit -- anything that delivers a net force not through the centre of mass will do this.

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    $\begingroup$ Atmospheric density also plays a role for large objects, when density is much lower at the upper parts than on the lower parts. Image something like a large water wheel - rotating this fast enough could generate some propulsion. $\endgroup$ – asdfex Feb 6 at 14:31
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    $\begingroup$ How fast does the object have to rotate that general relativity becomes a significant factor? $\endgroup$ – undefined Feb 6 at 15:28
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    $\begingroup$ @undefined. Basically, fast enough that something is moving at a decent fraction of light speed $\endgroup$ – Steve Linton Feb 6 at 22:03
  • $\begingroup$ The precision with which we can observe things let's you say "no". But the long answer is a yes, correct? $\endgroup$ – Mazura Feb 6 at 22:24
  • $\begingroup$ ... regardless of whatever fraction of the speed of light. $\endgroup$ – Mazura Feb 6 at 22:31
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If the question is making implicit reference to the conservation of angular momentum, then, as so often the case, "it depends". The source of the change in rotation cannot strictly come from "energy inside the satellite". The satellite can make its exterior rotate faster by means of an internal gyro rotating in the opposite direction. This case would not affect its orbital angular momentum at all.

However, the satellite can use rockets to expel mass to increase its rotational velocity. In this case, well, it depends on how much momentum the rockets impart to the satellite. They could be tuned to keep the overall orbital angular momentum what it was, or they could simply fire in a way which would conserve the satellite's linear momentum and the orbital angular momentum will be affected.

External forces would act like the rockets: "it depends". An alien simply slapping an extended solar panel would increase the angular momentum around the satellite's center of mass and also would increase the linear momentum, which would change the orbit. A pair of aliens situated on opposite sides could make it spin around more without changing the linear momentum.

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  • $\begingroup$ This is the best answer so far as it educates in what realistic ways a satellite can and can not change its angular momentum, and how those mechanisms can impact the orbit. $\endgroup$ – uhoh Feb 6 at 23:45
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In most cases, yes. Spinning a satellite up will not immediately change its orbit, but tidal forces will slowly transfer energy from the satellite to its orbit until it becomes tidally locked with its primary.

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  • $\begingroup$ I say "in most cases" because I'm not sure what would happen if an already tidally locked satellite was spun around the normal of its primary. $\endgroup$ – Fax Feb 6 at 17:19
  • $\begingroup$ If it's spinning around the normal, then in a quarter revolution it won't be (because it will be spinning about the prograde/retrograde axis instead). $\endgroup$ – Kevin Feb 6 at 20:33
  • $\begingroup$ +1 I didn't want to believe this is right at first, but for a lumpy satellite and a lumpy Earth, I suppose it is! It would take a really long time for a small artificial satellite, and probably differential drag on the two sides of the rotating satellite might mask this effect. $\endgroup$ – uhoh Feb 6 at 23:48

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