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A planet is an object that:

  a) orbits a star, and is not a star nor a brown dwarf itself;
     i) a star is an object that is massive enough and hot enough to conduct protium fusion in its core
    ii) a brown dwarf is an object that is massive enough to conduct deuterium fusion, but insufficiently massive (and hence insufficiently hot) for protium fusion  
  b) has attained hydrostatic equilibrium, that is, it is spheroidal;
  c) it has cleared its local neighbourhood of 'rival planets' since its birth;
Ergo, Jupiter is a planet. It is a certain type of planet—a gas giant—but a planet nonetheless.

There is some controversy about when giant planets stop being giant planets and start being brown dwarfs—typically around 30 Jupiter masses—but this is an active area of research, to understand the fusion ignition in these objects. As I mentioned in another comment, classifications are useful. Of course they are arbitrary and they are biased—they were decided by humans. But the entire point of science is to push the edges of our understanding and reduce these biases and make our models more general and less arbitrary.

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Their claim is that it fails (a) because the Sun and Jupiter orbit a common barycenter, and hence Jupiter does not orbit the Sun (or any star).
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Everything orbits a barycentre. The position of that barycentre within or without the primary is not really relevant within several orders of approximation—and when it does become relevant, then that system becomes interesting in itself. Chances are the planet in question is so massive relative to its primary that we need to ask a more fundamental question: are we even looking at a star-planet relationship, or is it closer to a binary star system? Then we can investigate how that system formed and developed, and generalise and improve our model of stellar/planetary system development.
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There are plenty of binary star systems where this orders of magnitude standard is not met .

There is no academic literature or accepted consensus barycenter has to be orders of magnitude away - the accepted consensus is that a body orbiting another means internal barycenter, if they have external barycenter then they orbit each other period.

The fact that Charon and Pluto do not have internal barycenter is why Pluto-Charon (also Eris-Dysonmia) are considered dwarf planet systems ; and that was used partly in support (while not used in definition) of arguing for dwarf status of Pluto .

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Yes — their point is that as-written, it orbits a barycenter external to the star: hence doesn’t orbit the star, but they’re in a mutual orbit.

Thus the suggestion for a new dwarf category — precisely because we do label times the barycenter is outside the star differently, as you yourself note.

You were so busy trying to “correct” me, you failed to address the topic.

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'In orbit around' is frequently used as short-hand to refer to a secondary with mass << primary, to a first order approximation. In the Solar System, even for satellite transfers, the effect of Jupiter on the Sun can and is ignored.

The topic is pointless hair-splitting and nostalgia. It's like saying we can't ever use F = ma to describe free-body diagrams; we need a full relativistic per-particle QFT simulation to accurately and precisely model every single interation. That is impractical, at least for now. Likewise, we come up with classifications that are useful for the present time. If 'dwarf planet' is a catch all for all the small roundish bodies in our planetary system, then so be it. I suspect the IAU will make things more fine grained in the relatively near future. We even classify the asteroids as more than just rocky lumps.

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We have QFT precisely because mechanics improperly categorized events.

We can similarly say that “cold dwarf binary systems behave similarly to planetary systems, due to the proximity of the barycenter to the star”.

And that’s more informative than calling Jupiter a planet.

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I agree with this picture as the barycenter argument seals it for me. The only leap left is for me to believe Ol'Jove as a failed star (it still emits heat in excess of what it derives from Sol) and we find ourselves cozily herded about by a binary star system.
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