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A spacecraft orbits a planet. It extends a mass about a horizontal direction on a long tether. At this point there is not much gravitational gradient between the craft and the ejected mass so there is not much tension in the tether. Later the gravitational gradient will tidally align this tethered two body system vertically now there is a tension in the tether due to the gradient.

To retract the ejected mass the spacecraft start pulling back the tether. Since there is a tension on the tether it will need to spend energy to do so. The question where this energy will go? Would it boost the orbit?

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  • $\begingroup$ Pretty interesting question, but could you please make a simple sketch of this situation? I think people might misunderstand what you mean by 'extends a mass about a horizontal direction on a long tether' and 'the gravitational gradient will tidally align the tether'. $\endgroup$ – João Vítor G. Lima Feb 16 at 15:39
  • $\begingroup$ @JoãoVítorG.Lima Ok I will draw something when I arrive home. $\endgroup$ – Calmarius Feb 16 at 16:24
  • $\begingroup$ It can't be used to boost the orbit because angular momentum would not be conserved: the initial & final states (with the mass wound in) must have the same angular momentum, but orbits of different radii don't. $\endgroup$ – tfb Feb 16 at 16:42
  • $\begingroup$ @tfb The Moon is slowly getting farther away from Earth because the tides slowing the Earth down, so the Earth's angular momentum becomes the Moon's orbital one. I wouldn't rule out the same possibility with the spacecraft I devise. When the tethered two body system stabilizes it has one rotation per orbit, it will have angular momentum from this rotation especially when the tether is long. And this angular momentum may be converted to the orbital one when the tether is retracted. $\endgroup$ – Calmarius Feb 16 at 19:31
  • $\begingroup$ @Calmarius: the only place the thing can exchange AM with is the Earth's spin, and that effect is going to be absurdly tiny. $\endgroup$ – tfb Feb 16 at 19:37
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In terms of energy the situation I describe is very much like letting a bucket down on a rope. The gravity gradient will pull on the rope and pull the mass away and can be connected to a generator to generate some energy. And this energy is used up when we retract the rope.

In terms of angular momentum the orbiting spacecraft has lots of angular momumentum (high radius, high speed). When the tethered mass is released on a long rope the tidal forces will align it so it will rotate once per orbit. This angular momentum must come from the orbital angular momentum, so the orbit will decay a bit as this rotational momentum is not too large compared to the orbital one.

When the cable is retracted this rotational momentum wants to be conserved, and the rotational speed increases as the cable is retracted. So this tethered system can be used to launch objects with a precise $\Delta v$. This is the operating principle of the rotating sky hooks.

If the retraction happens slowly it's possible to keep it in sync the stabilizing forces. In this case the orbital angular momentum is recovered.

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