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Consider a solar system with 1 sun and two planets revolving around the sun in a 2D Euclidean space. While time continues, the sun moves forward, while the two planets revolve around the sun (move up and down). However,

  • we know nothing about the sun,
  • but the mass equivalents of our planets ($m_1$ and $m_2$),
  • their distances to the sun ($d_1$ and $d_2$) as well as
  • the forces acting on it (or 1st derivative of the distances, $d'_1$ and $d'_2$) are known.

For some reasons (i.e. external forces acting on it) the sun changes its trajectory (moving up or down).

Is it possible to derive the position of the sun ($y$-coordinate; $d_{sun}$) as well as its 1st derivative ($d'_{sun}$) from the masses, distances and distance changes (distance derivatives) of the two planets? How? Which formulas can be used to calculate/estimate it? Would you be so kind and give a numerical example on how to calculate it?

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I am assuming you know the positions of the two planets at any given time.

If the sun and the planets are confined to a two dimensional plane then this is just a geometry problem. There are (at most) two points in the plane that are at a distance $d_1$ from planet 1 and a distance $d_2$ from planet 2 (think of the intersection of two circles). Therefore the sun must lie at one of these two points. Find these two points for several different times and the trajectory of the sun should become clear.

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  • $\begingroup$ Dear @gadalf61, yes, mayb just geometry?! Nevertheless, this question then is at any given time, how to derive the sun's position. I guess it's position $d_{sun}$ (relative to $y = 0$ is just the mean of the positions of both planets, weighted by its weights. But what about the sun's $d'_{sun}$. I doubt it can be that easily calculated - but maybe I'm just too much thinking in a physical world?! Another, more geometrical way I could think of this problem is of moving averages. The two planets are two moving averages with different calculation periods, revolving around a line (the sun)?! $\endgroup$
    – Anti
    Commented Dec 30, 2022 at 9:50

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