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3

1) and 2) are easy to show, the bonus is very hard and I will not attempt it. A $L$iberation point can be seen as a balance between three accelerations in a rotating frame of reference. Gravity from $M_1$ Gravity from $M_2$ Centrifugal acceleration. For $L_2$, the first two are $-\frac{(1 - \delta)M_1}{(R + r_2)^2}$ and $-\frac{M_2}{r_"^2}$ respectively. ...

5

L2 isn't a super stable point, but only quasi-stable. Things can't stay at that point for a long period of time without some work to stay there. Estimates say that number is around 5-16 m/s, depending on the object exactly. Sufficeth to say, most objects don't thrust to stay there naturally, so there isn't a lot of dust actually trapped at any particular L2 (...

2

In fact it could be arbitrarily low. But wait before you rejoice. The Moon has an elliptic trajectory. This elliptic motion perturbs any orbit of a satellite in LEO. If you first assume your satellite is in the same plane as the Moon and describe the system as a time dependent Hamiltonian system with two degrees of freedom, a process called Arnold diffusion ...

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