How rapid do the attractive properties of GEO and lagrange points deteriorate with the distance from optimum? How "large" are such locations?
At what distance would station keeping require, say, twice the delta-v per year?
Is there any reason for spacecrafts to try to stay put in a lagrangian point, or can they as well orbit them (like Gaia does) and if so, how large could such orbits be and does this mean that the L-points are practically non-scarce locations?
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$\begingroup$ This seems like two separate questions, since the orbital dynamics of GEO and Lagrange points are substantially different. On the later point, every mission to a Lagrange point I've read about consistently adopts an orbit around it; I'd like to see a concise explanation of the whys & wherefores. $\endgroup$– Jerard PuckettCommented Oct 6, 2014 at 13:08
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First, it is nothing special with the geostationary orbit regarding gravity, this is just the location where an object seems to hoover in the same spot in the sky. That is dependent on the rotation of the Earth, and has thus no forces attracting it. A normal station keeping budget for a satellite in GEO is typically 50 m/s of delta-v a year. If I interpret your question as how much delta-v it is going to cost us to still hover, even though we are a little bit off, the delta-v cost is 0.5 m/s per year per meter. This is not exactly scalable, but accurate enough within a few hundred kilometres.
As for L-points, they come in two types. Stable and unstable. In the stable points, like L4 and L5, an object that is a little bit off is still hold in place and therefore "orbiting". In the unstable points, an error is growing until the object escapes. Imagine this as the difference of balancing marbles on a sphere.
Because any error grows over time, the unstable points do not have a size, they are just points. But the delta-v cost does not have to be high, and satellites can indeed orbit them for some time, like the Genesis mission did. This is necessary if you want to accommodate multiple satellites in the same L-point.
As for the size of the L4 and L5 points, this question has been asked before, see Maximum Amplitude of a Lissajous Orbiting Object in a L4 or L5 Position at astronomy. A particular amplitude is not stated though, but from the common illustration of L-points, it seems like an object can go almost back to the L3 point in an orbit.
Note that for example Venus is at sometimes closer to the SEL4 and SEL5 points than the Earth, thereby causing significant perturbations, necessary to compensate.
answered Dec 21, 2015 at 10:52
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$\begingroup$ Are disturbances from Venus (or Jupiter) really relevant to SEL4, L5 instability? Earth's orbit should be as unstable in that case, as any object anywhere along Earth's orbit. I asked a related question here. But the collinear Lagrange points L1, L2, L3 are of more interest to this question. How fast does stability decrease with distance from them? Linearly, exponentially, chaotically? $\endgroup$ Commented Dec 21, 2015 at 12:31
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$\begingroup$ @LocalFluff Of course those perturbations is from a relative point of view, you can place an object in the Earth's orbit and not in L4 and L5, and it would stay put for decades. As for increase in instability, it does in general increase quadratically, but at small distances it can be approximated as linear. If you want exact three-body solutions, the picture is more complicated $\endgroup$ Commented Dec 21, 2015 at 12:53
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$\begingroup$ @Hohmannfan How do you quantify instability in this case? Do you define some kind of orbit-crossing timescale? $\endgroup$ Commented Feb 19, 2016 at 17:39
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$\begingroup$ @AtmosphericPrisonEscape Instability is here quantified as the acceleration away from the point. 'linear' or 'polynomial' is here referring to the acceleration as a function of distance to the point. $\endgroup$ Commented Feb 19, 2016 at 17:57