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Context. The paths followed by the known extreme trans-Neptunian objects(ETNOs) effectively avoid direct gravitational perturbations from the fourgiant planets, yet their orbital eccentricities are in the range between0.69-0.97. Solar system dynamics studies show that such high values of theeccentricity can be produced via close encounters or secular perturbations. Inboth cases, the presence of yet-to-be-discovered trans-Plutonian planets isrequired.  Aims. If the high eccentricities of the known ETNOs are the result ofrelatively recent close encounters with putative planets, the mutual nodaldistances of sizeable groups of ETNOs with their assumed perturber may still besmall enough to be identifiable geometrically. In order to confirm or rejectthis possibility, we used Monte Carlo random search techniques.  Methods. Two arbitrary orbits may lead to close encounters when their mutualnodal distance is sufficiently small. We generated billions of random planetaryorbits with parameters within the relevant ranges and computed the mutual nodaldistances with a set of randomly generated orbits with parameters consistentwith those of the known ETNOs and their uncertainties. We monitored whichplanetary orbits had the maximum number of potential close encounters withsynthetic ETNOs and we studied the resulting distributions.  Results. We provide narrow ranges for the orbital parameters of putativeplanets that may have experienced orbit-changing encounters with known ETNOs.  Conclusions. Our calculations suggest that more than one perturber isrequired if scattering is the main source of orbital modification for the knownETNOs. Perturbers might not be located farther than 600 AU and they have tofollow moderately eccentric and inclined orbits to be capable of experiencingclose encounters with multiple known ETNOs.

Memories of past close encounters in extreme trans-Neptunian space: Finding unseen planets using pure random searches