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At the edge of the solar system, where the Sun is barely brighter than other stars, icy boulders, trans-Neptunian objects, circle around. Their orbits resemble hoops scattered by someone. But look closer: twelve of these hoops are not just floating in the void. They align in nearly the same plane, as if an invisible dancer had spun them into a single roundelay.

This dance is no accident. The authors built a mathematical model based on 12 objects — six from earlier observations and six more selected by similar but broader criteria. The model showed that the common plane of these bodies' orbits, with high precision, sets the inclination and longitude of the ascending node for an invisible partner: 19.5° and 107.7°. The argument of perihelion came out to 307.5°. It's as if, by watching autumn leaves swirl, you could deduce where the invisible groundskeeper with a broom is standing.

And there's more. Four, possibly five, of the dozen objects are in resonance with the planet: they return to the same point relative to it after an integer number of orbits. Such a coincidence is not just a pattern but a hard clue about the mass. It turns out to be 7.1 times the mass of Earth. An orbital eccentricity of 0.39 draws a highly elongated ellipse: the planet dives toward the Sun to 311 astronomical units and then flies far into the darkness.

So where to look for the invisible one? The calculations point to two patches of sky: the lower part of Taurus and the upper region of Libra. The planet's wide orbital arc sweeps across half the orbits of those icy bodies near their perihelia, like a hoop inside which little rings run around.

Thus, twelve icy wanderers helped sketch a portrait of the one that governs their motion. All that remains is to point telescopes in the right direction and catch the reflected light of a distant giant.