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It is known that all the mass in the Solar system moves around the Barycenter.

For the two focal points in Kepler's law; is the first focal point $F_1$ a Heliocenter? Or is it actually a Barycenter?

If the $F_1$ is a Heliocenter, then, can the same exact ellipse and eccentricity of orbit be maintained by adjusting the second focal point $F_2$ and taking Barycenter as $F_1$ instead of Heliocenter?

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Planetary orbits are normally described as heliocentric, but it is possible to describe them from a barycentric point of view. JPL Horizons (https://ssd.jpl.nasa.gov/horizons.cgi) provides for both possibilities.

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It depends!

If we plotted the orbits of the innermost planets in our solar system, they would be closer to ellipses with the Sun at one focus.

If we plotted the orbit of Jupiter, the biggest "gravitational bully" in the solar system (it messes with everything!) it would be closer to an ellipse with the Sun-Jupiter barycenter at one focus.

It gets a little complicated out there because the next three planets (Saturn, Uranus and Neptune) also push the Sun around a lot. We might think that lightweight Neptune wouldn't do much, but it's larger distance makes up somewhat for its smaller mass, since the center of mass is weighted by the distance*mass product.

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