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Assuming that the existence of Planet 9 is confirmed and it really exists, how does it affect the orbits of the Kuiper belt objects?

My understanding is that the discovery came in part from the observation that the Kuiper belt objects all orbit to one side of the invariable plane. Are they always on one side or are they on opposite sides depending on whether Planet 9 is at perihelion or aphelion? What would be the difference in their orbits?

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According to wikipedia the orbital period of Planet 9 is 10,000–20,000 years. This means it would take a minimum of 5,000 years to switch between perihelion and aphelion.

So - even our best historical observation data of kuiper belt objects is just a tiny fraction of the time it takes for Planet 9 to move in it's orbit. There's not really sufficient data available to answer your question. This answer would rely on some pretty advanced modeling of the solar system over 20,000+ years.

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