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Can the location of an unknown planet that is absent from a planetary system be calculated from the orbits of the remaining planets?

Space Exploration Asked on November 10, 2021

In the book ‘Seeker’ by Richard McDevitt, two characters search for a lost planet. They know

  1. the star system currently has a number of celestial bodies in orbit around it and their orbits are accurately known

    A) two gas giants at 10 AU and 14 AU

    B) an earth like planet at 1 AU with a highly eccentric orbit

      Perihelion 40 million kilometers
      Aphelion   400 million kilometers
    

    C) a derelict starship(#1) in an extremely eccentric solar orbit
    Aphelion 7.2 AU

    D) another derelict starship(#2)

    E) a space dock also in an eccentric orbit

    F) a moon in an eccentric solar orbit with a period of 735 years

  2. There is unlimited computational capacity and the orbit of the moon and space dock
    can accurately be extrapolated back in time 9000 years ago.
    The calculated proximity of the moon, the earth like planet and
    the space dock 9000 years ago, puts them all very close together

  3. The earth-like planet was presumed to be in orbit around the star 9000 years ago
    In an ‘earth-like’ orbit with little eccentricity

  4. The moon is presumed to have had a ‘moon-like’ orbit around the earth-like planet 9000 years ago

  5. The space dock and starships are presumed to have been in orbit around the earth like planet at an altitude similar to our ISS.

  6. a rogue celestial body is presumed to have entered the system 9000 years ago,
    passed close to the planet and moon and disrupting their orbits. The mass of
    That celestial body is not known, nor is its location or origin

We know the orbits of the all the known celestial bodies today.

We can predict with high accuracy the locations of the presently known celestial bodies 9000 years ago.

We assume the orbits of the earth-like planet and moon around the star were very similar to our solar system with minimally eccentric orbits.

Can the current location of the rogue celestial body be estimated?

I am asking if there is a unique event that results in a specific perturbation of a set of orbits of celestial bodies if you know their masses and orbits today and also at the time the event happened?

My sense is that you also need to know the mass of the rogue planet, but maybe not since you know the masses and orbits of all the remaining knowing celestial bodies(a planet, a moon, a space dock, and two derelict spaceships) I don’t know what additional information might be needed to find the rogue planet.

EDIT: I thought about the problem above and tried to distill it to a simpler 2D example. Consider a billiard ball that is moving and then its trajectory suddenly changes due to a collision with an invisible object. Is there only a single unique impact that could have caused the change in trajectory? My sense is that the momentum of the impacting object might be unique but that would still allow for many combinations of mass and velocity. However gravitational perturbations are not strictly an impact so momentum transfer might not be the correct way to solve the problem.

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