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Since Jupiter is very massive, it is the only planet (in our solar system) that has a center of mass with the Sun that lies outside the volume of the Sun. (Source)

If Jupiter was a star, they would form a « binary star ».

If the Sun was a planet, they would form a « double planet ».

Since the Sun is a star and Jupiter is a planet, does this have a particular name?

Does Jupiter have a special status or a particular effect in our solar system because of its heavy mass?

Since Jupiter-Sun's center of mass lies outside the volume of the Sun, that means that the Sun moves around that center of mass. Does this have an effect on Mercury, Venus, Earth and Mars orbits?

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Systems composed of other objects besides stars are currently ill-defined. For example, Pluto-Charon should technically be a binary dwarf planet system, but it is yet to be recognized as such by the IAU. –  called2voyage Dec 11 '13 at 16:53
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I have problem understanding what the question is. Status as defined by whom and for what purpose? –  Noordung Dec 11 '13 at 18:09
    
Since Jupiter-Sun's center of mass lies outside the volume of the Sun, that means that the Sun moves around that center of mass All objects orbit at the barycenter between the two objects, not just Jupiter. –  asawyer Dec 11 '13 at 18:43
    
@TildalWave called2voyage cleverly renamed my question. I hope it's clearer. –  Thibault Dec 11 '13 at 21:03
    
@asawyer I know, but in this case, the center of mass is outside the Sun because Jupiter is very massive. Which make it a special case. See en.wikipedia.org/wiki/Double_planet#Center-of-mass_definition –  Thibault Dec 11 '13 at 21:06

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I'm not sure I understand your question entirely, but i'll do my best to offer a decent answer. It's true that the composition of Jupiter is very similar to that of the Sun (very similar approx. $H$ and $He$ abundance and pretty similar in density). The problem is that Jupiter is not nearly massive enough to have the internal pressure and temperature to undergo nuclear fusion. Jupiter doesn't have any particular special status aside from being the King of Planets in our solar system.

As for the last part of your question, all objects orbit around a center of mass. Though because the Sun is much more massive, the center of mass lies very close to the center of the star (except in the case of Jupiter, where the CoM lies outside of the sun and is approx the length of its radius). This is why all planets in our solar system orbit around the sun. This will indeed cause a slight perturbation of orbital alignment, but I don't believe it's significant.

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I know, but in this case, the center of mass is outside the Sun because Jupiter is very massive. Which make it a special case. See en.wikipedia.org/wiki/Double_planet#Center-of-mass_definition Thank you for your answer. –  Thibault Dec 11 '13 at 21:07
    
You're right Jupiter is indeed the only exception (i'll go ahead and edit that in the answer just to be clear). Though the Sun is not a planet, so i'm not certain the double planet classification holds. Jupiter is also not a star from the reasons stated above so it's not a binary star system. I don't believe a special status name exists for this type of phenonmenon. –  Lame-Ov2.0 Dec 11 '13 at 21:17

The Sun burns hidrogen, so it is a star.

Jupiter does not, so it is not a star. Not even a brown dwarf.

Since a non-star body orbits a star, it is a planet, a dwarf planet or an asteroid/comet. Since Jupiter has cleared his orbit, it is a planet.

Why do I say that Jupiter orbits a star? This is where I go to the center of your question.

The orbit of the Sun around the galactic center is (more or less) a circle. If you draw Jupiter's orbit around the galactic center it has some retrogradations and is not always convex.

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No, it is not since Jupiter is not a star. We call the system of a star with its planets surrounding the center of mass a simply "solar System".

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