Could a telescope see Oort objects?

Yes, a number of Oort Cloud Comets, and a small handful of 'other' Oort Cloud Objects have already been identified.

http://en.wikipedia.org/wiki/Oort_cloud#Oort_cloud...

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By "any," I am assuming you mean at least one of them.

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If you limit it to objects whose orbits are perpetually at or beyond 2000 AU, then depending on size and albedo, you are getting an object at around apparent magnitude of about 75-100, or beyond. You can easily calculate the lens diameter of a scope to collect enough light to see objects this faint. If you have unlimited money, glass, and maintenance, a telescope with enough light collecting ability could be made to do this. That being said, it is completely impractical. Attaching a smaller scope to a lang term probe would be easier to construct to get a result, it would just take longer to get the result.

If you equip a camera to a scope, you could capture objects more faint with a longer exposure time. This is how Sedna was discovered, by observing photographic plates.Your details seem to deny the use of a camera attached to the the scope, and using only the human eyeball and the telescope.

I'm not sure what you mean exactly with "seeing" those objects.

If you mean "being able to resolve details" it'll depend on the actual device built and the amount of money you're willing to spend to resolve those details. See http://en.wikipedia.org/wiki/Angular_resolution

Suppose you want to resolve a feature with size of the order of 100 meters at a distance of 1000 astronomical units. That's an angle of the order of the billionth of an arcsecond (2.5E-9 arcseconds). To resolve that for a wavelength of about 500 nanometers (about midway in the visible range of the EM spectrum), you'd need an aperture of about 1000 kilometers. You'd only achieve that with interferometry -- and that takes some SERIOUSLY sophisticated synchronization of the experiment's electronics! This is assuming that the individual telescopes used for interferometry have enough aperture to capture dim light from afar...

On the other hand, if you just want to say "oh there's something over there!" instead of actually resolving details on the object, you can already do that for the "brightest" objects. The problem here is taking photos on several days to actually spot something MOVING. If you want to estimate the distance by parallax, you'll have to patiently photograph that region of the sky for a few weeks or months at a time to be sure to track the same object between two points to measure the parallax. This is already done in some measure.

I, too, think the answer is "no". In the first place, you said "any" Oort object. Only the very large, such as the here-to-fore unproven "Tyche" (the one alledged to be 4 times the mass of Jupiter) might be picked out from the billions of tiny ice and rock debris. Good luck finding that one, using infrared technology, assuming that it even exists!

I did not count the nearer Kuiper Belt objects in my consideration.

I disagree

I think the is YES.

Given that you said "regardless of engineering or cost"

Cheers from DOWNUNDER