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Given that Planet Nine is hypothesized to be 10-20 times the mass of the Earth orbiting the sun some 200-2500 AU out, what kind of planetary-mass object would it be?

Even given that it has apparently "influenced" several Trans-Neptunian objects, since its orbit is so large and so slow (approximately 25,000 years) it would take an awful long time to clear out its own orbit (especially given that we do not know how long it has been in that orbit), so it seems like it would not qualify as an actual planet under the rules that re-classified Pluto.

Since it orbits our sun, it wouldn't be an exoplanet nor a rogue planet. And it sure seems too big to be called a dwarf planet.

A captured planet seems closest in other attributes, but it probably was not either a capture nor a re-capture, but more likely a partial ejection.

So what classification would it fall into?

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It is a planet - either a gas giant core, a mini-Neptune, or a super-Earth.

First off, Mike Brown has stated outright

“It is a planet—there’s virtually no doubt,” he said. “What we now call planets are objects that can gravitationally dominate their neighborhood. Pluto is a slave to the gravitational influence of Neptune. By area, Planet Nine dominates more of the solar system than any other known planet—it’s only because of this that we can infer its existence. And because of this we’re pretty sure it’s not a small object: it’s at least ten times more massive than Earth and five thousand times more massive than Pluto. In many ways, you could argue that this is more of a planet than anything else in the solar system.”

I wrote an answer to How was the hypothetical ninth planet kicked so far out of the Solar System? that discussed the possibility that the ninth planet is the hypothetical 5th gas giant proposed by Nesvorný (2011). The details can be found in that answer, but basically, Brown & Batygin's reference to previous papers by Morbidelli et al. suggest that they think that this is what remains of an ice giant, like Uranus and Neptune, that formed near Saturn and Jupiter but was ejected in a series of gravitational interactions that led it to the Kuiper Belt and beyond.

That said, Batygin's estimates place the ejection of this planet way before previous models of a 5th gas giant predict, so either we accept models with a 6th gas giant (also an ice giant) and say that both of these were ejected independently of one another - not impossible - or we go for some other option. The bottom line is that this is a planet, if we go with the above assumptions.

Two other guesses have been made based on the planet's mass, estimated by Brown & Batygin at about 10 Earth masses. One is that it is a very massive super-Earth, hanging around the proposed upper mass limit for those objects. This implies that the planet is rocky, but does not imply anything else. The other guess is that it is a mini-Neptune, with a predominantly gaseous composition but at about the same mass. The super-Earth hypothesis arose in part because of claims a long time ago about the ejection of a 5th terrestrial planet.

I've used the term "planet" throughout this entire answer, and that's because this object may be one - and most likely was in the past. Let's look at the IAU's three major criteria to be a planet:

  • It orbits the Sun. This is definitely true, if it exists. Also note that if this is a captured rogue planet, as some have proposed, then we can still call it a "planet", because it now orbits the Sun.
  • It is in hydrostatic equilibrium. This is most likely true, and is true if any of the three models I suggested above are correct.
  • It clears its neighborhood. This is a bit iffy. It clearly influences objects around it, but we don't know much about those influences besides observations of the six Trans-Neptunian Objects (TNOs) Batygin & Brown studied. That said, an object with a mass of 10 Earth masses will most likely clear the neighborhood around its orbit if it has a stable orbit, which should be the case here. See the quote at the start of this answer for more information.
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  • $\begingroup$ The part you call "Iffy" is the critical part of my question: What if it has not yet cleared it's orbit? What is classified as then? $\endgroup$ – RBarryYoung Jan 23 '16 at 20:48
  • $\begingroup$ @RBarryYoung That's something I deliberately skirted around, mostly because it seems highly unlikely that a 10-Earth-mass object would not have clearer its neighborhood. I remember reading something about Mike Brown asserting that it would have; I'll see if I can dig that up. Yep, I found it. Answer edited. $\endgroup$ – HDE 226868 Jan 23 '16 at 20:51
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It would either be a super-Earth or an ice giant like Uranus and Neptune. If it is a planet. In September 2019 astrophysicists suggested that it actually may be a primordial black hole. Such black holes are extremely tiny and if it is one, it would have a diameter of about 2 inches (5 cm). It would reportedly also have an accretion disc likely. And it could have own planets of course.

And please forget the 2006 definition of a planet which became selected by 4% of the IAU; according to it, there would be no planet at all.

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