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How was the 10th planet's orbit established when it was observed for only a short time?

DICK SWARTWOUT, PENN LAIRD, VIRGINIA
The first calculation of a newly discovered transneptunian object's (TNO) orbit is essentially a guess. It's sometimes possible to obtain a reliable result (from observations on a single night) for an object zooming close to Earth. And astronomers can make a comparable computation for an asteroid in the main belt, between Mars and Jupiter, as it moves a few Moon-diameters relative to the background stars over the course of several nights.

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