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Will the New Horizons space probe be able to discover whether Pluto is an extra solar body captured by our sun or not?

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No.

First off, a Pluto-like object from outside the Solar System should look pretty much like a Pluto-like object from inside the Solar System - small and cold. When it comes down to it, one relatively tiny rock from space looks like just about any other tiny rock from space.

What could indicate whether or not Pluto came from outside the Solar System would be the presence of one of a few things:

  • A composition not consistent with that of other Kuiper Belt objects
  • A highly eccentric orbit
  • An age unlike that of other objects in the Solar System

What is known of Pluto's composition (https://en.wikipedia.org/wiki/Pluto#Structure), rock and ice, is not enough to determine whether it is from the Kuiper belt or beyond. While other objects in the Kuiper belt would have that composition, so would an object from a belt around another star or from interstellar space.

We know it doesn't have a highly eccentric orbit because of observations, that its orbit is in resonance with that of Neptune. Still, this only apply to the current era, it might have had a more eccentric orbit in the past that has become less eccentric because of the influence of the outer planets until it entered its current resonance with Neptune.

The last option is more interesting. We know the age of the Solar System through radiometric dating. Unfortunately, New Horizons will not be able to collect samples or rock from Pluto.

However, if Pluto did form outside the Solar System, chances are good that it would have the other two things - which it doesn't.

New Horizons will be able to (among other things):

  • Test the chemical composition of Pluto and Charon
  • Test the atmospheres of Pluto and Charon
  • Figure out the surface temperatures of Pluto and Charon

None of these will help figure out if Pluto was formed outside the Solar System. For details of the ability of New Horizons, see this NASA page.

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Unlikely but possible.

See the ability of New Horizons as given by NASA. Make a list of clues that could indicate that Charon-Pluto comes from outside the solar system. Then, would any of these clues be noticed?

Examples. If Pluto, Charon came close enough to one or several stars in the past, were exposed to a much stronger amount of light than it receives today for long enough, there might be signs of that on the surface and atmosphere today. Due to evaporation of methane ice and so on. Pluto and Charon are tidally locked to one another, disturbances caused by some other objects to their orbits around the other in the past could have caused detectable fissures on their surfaces. Craters, the sizes and numbers of them, a Kuiper belt object would have more impacts from smaller Kuiper belt objects, the craters would be smaller due to the low gravity of Pluto and the low orbital speed of Kuiper belt objects around the sun. There would be less heat generated by such impacts. The number of craters, there would be difference between Pluto forming 2.3 billions years ago and 11 billions years ago. Comparing Pluto's characteristics with other planetoids of the solar system, so far there is only Triton to go from. Comparing Pluto to theoretical models of Pluto/Triton sized objects from outside of the solar system, if there even are any.

An extrasolar object having the right speed and angle to enter into a somewhat stable orbit around the Sun or Neptune might seem unlikely but remember, that would have been when it worked while in other occasions, the objects would have been too fast or ejected from the solar system due to the influence of the outer planets. Given the solar system formed circa 4.5-4.6 billions years ago, rogue planets and planetoids should have passed through it since then.

Remember, these are just examples that single individual could think of in a short notice, there could be others. The full answer would need specialist/professional work. For that work to happen within NASA for example, the possibility of Pluto being an extrasolar object must have been seriously taken into account. Otherwise, any evidence(s) of that being the case go unnoticed.

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