In A Mathematician's Miscellany, Littlewood offers this item:
To determine the orbit of a planet or comet 3 observations, each of two (angular) co-ordinates and the time t, suffice. It is actually the case that to any set of observations (point the telescope anyhow at any 3 times) an orbit corresponds. Imagine a speck on the telescope's object glass ; this satisfies the observations, and it also describes an orbit (that of the earth). Now (some sordid details being taken for granted) the equations for the elements of the orbit are reducible to the solution of an equation of the 8th degree. This has accordingly one real root. But since the degree is even it must have a second real root.
This to all intents rigorous argument is a test of taste. Incidentally the joke is in the mathematics, not merely about it.
Can you please explain the joke?
I'm not perfectly sure, but here is how I see this: If you observe any object at three times, then you have the required data. You are searching for the orbital elements of an object which was in the line of sight of the telescope in each of these occasions.
But apart from the object you are observing, there is one other object which is also always in the line of sight of the telescope, namely the telescope itself. I assume that the discourse about the speck might be a hint in this direction. So one other real solution would always be the orbit of earth itself.
However, this argument would break down if the question refers to not any observable object but indeed the speck. In that case, I can see no way to describe another real solution besides the orbit of earth. Not sure if the question was meant this way.
I'm also wondering, if my interpretation is right and the orbit of earth will always be a solution, shouldn't it be possible to factor that known solution out of the equation to obtain one of degree $7$ only? I haven't seen the equation he's referring to (and I haven't got the time to research that just now), but theoretically that should be possible.