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2

Wikipedia actually goes into some detail regarding the components of Earth's gravitational force field. In the Wikipedia reference we do not have $J_{m,n}$ for $m>n>0$. Rather this term is broken up into a cosine component $C_n^m$ and a sine component $S_n^m$. This table gives results from one on a series of models called the JGM models, developed by ...

2

Does J22 represent the rest of Earth's quadrupole moment? With a more usual notation ldegree, morder, there are the zonal l2m0, tesseral l2m1 and sectoral l2m2 spherical harmonics, but you probably already know. What does J22 look like? What is its shape and symmetry? Here's what I get for l2m2: For comparison, here's what I get for your "acceleration ...

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. ...

2

The answer to this question can be directly extracted from the definition of an osculating orbit. the osculating orbit of an object in space at a given moment in time is the gravitational Kepler orbit (i.e. an elliptic or other conic one) that it would have around its central body if perturbations were absent. The previous question that was linked here, ...

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Here is a theoretical physicist's answer to this problem (see below for what I mean by this). This should be adequate to get a handle on the relative positions of satellites when they are distant from each other, and when the time after some known position & velocity of the satellites is not too long. The theoretical physicist's approach First of all ...

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