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The way Newton solved the Planet system is even more basic than what we use here. The quasi particles are used (planets one at a time) the Coulomb potential of the nucleus is used (the Gravitational potential of the sun) But the next two terms are generally neglected. I have never in my physics education used the influence of other planets on the orbit of the Earth. I’m a bit unsure how to split V_H and V_xc for the planet system…
V_H would be the average position of the planet so maybe a ring of mass M_planet2?
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