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Please note, I do not necessarily believe in its existence, just after a scientific (hence, non-Wikipedia) reasoning to doubt the planet's existence, other than the "we would have seen it" argument. I am, nor are the articles below, referring to the alleged 'Nemesis' planet, as in the question What are the current observational constraints on the existence of Nemesis?

According to the article Astronomers Doubt Giant Planet 'Tyche' Exists in Our Solar System (Wolchover, 2011), the planet Tyche is a hypothesised massive planet

by John Matese and Daniel Whitmire of the University of Lousiana-Lafayette, is not new: They have been making a case for Tyche since 1999, suggesting that the giant planet's presence in a far-flung region of the solar system called the Oort cloud would explain the unusual orbital paths of some comets that originate there.

Further information is found in the scientists' research paper Persistent Evidence of a Jovian Mass Solar Companion in the Oort Cloud (Matese and Whitmire, 2010), describe the hypothesised planet to be

of mass 1 − 4 M(Jupiter) orbiting in the innermost region of the outer Oort cloud. Our most restrictive prediction is that the orientation angles of the orbit normal in galactic coordinates are centered on , the galactic longitude of the ascending node = 319 and i, the galactic inclination = 103 (or the opposite direction) with an uncertainty in the normal direction subtending 2% of the sky.

They posit that the elongated orbit of Sedna is a result of the presence of this planet.

So, the question, what reason is there to doubt the existence of Tyche?

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Disproving the existence of or proving the non-existence of an object is practically impossible, so I'm going to rephrase the question a little bit. It might also get you better answers because they won't just be aimed at disproving Tyche but questioning the methods that suggest Tyche in the first place. – called2voyage Oct 18 '13 at 14:41
As an example of the practical impossiblity I suggested, someone could say that Tyche existed at one point but got pulled out of orbit (in the case that Tyche were not where it should be observed). – called2voyage Oct 18 '13 at 14:42
Now, to progress toward answering your question, the livescience article that you linked in the body of your question refers to several astronomers questioning the methods employed by Matese and Whitmire in the first place. It would be interesting to see someone actually debunk their statistics from their paper, if possible. – called2voyage Oct 18 '13 at 14:53
@called2voyage yes, that is pretty much what I am after. Good edit by the way. – user8 Oct 18 '13 at 19:59

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Generally it would be that peers of Matese and Whitmire looking at the statistics and viewing that the results are not conclusive enough to indicate the existence of Tyche. There can also be alternative reasons to explain the results mentioned in the paper other than a Jupiter sized planet in the Oort Cloud. As was mentioned in the comments, it is almost impossible to prove the non-existence of something.

At this time, the results from WISE are the best bet for locating Tyche if it exists. As it will not be visible in the visible light spectrum but more likely in the infrared and the WISE mission scanned the entire sky which means that it won't have been missed due to looking in the wrong place.

There isn't any hard science to show that Tyche doesn't exist. It is skepticism that this is the explanation for observations to date. These are the same reasons that Nemesis isn't thought to exist, the data isn't strong enough to support its existence.

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