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At the current pace of technological advancement, when would humans be able to build an interplanetary telescope array?

This telescope doesn't have to be that huge; building one spanning Earth, Moon and Mars would be enough to call it a day.

Bonus question: how would such a project fare against the Webber space telescope in terms of usefulness?

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A good follow-up question would be whether it could be useful; telescopes in space seem to have greater purpose. If we can put it on the moon/Mars, wouldn't we be better off just putting it in space? I'm genuinely curious. – Mikey Feb 14 at 23:47
    
Does this have to be an optical telescope? – HDE 226868 Feb 14 at 23:58
    
Why would you want to put them on planets? Think about how much better the Hubble Space Telescope is, in many respects, than much-larger Earthbound telescopes that use technology that wasn't even invented when Hubble was built. – jamesqf Feb 15 at 4:32

We can do it now

We could in theory do that now.

The problem isn't one of capability, it is cost and time.

Solar Focus Mission

Solar Focus Point:
Solar Focus Point

Even better, we could theoretically fly a mission to the Sun's focal point (about 550 AU) and use the Sun as a giant telescope lens. It would give you a 200 meter resolution at Alpha Centauri.

Even using a nuclear powered craft (such as nuclear pulse propulsion) expect the mission to take decades. For conventional/ less practical propulsion (chemical, solar sail, etc.), expect it take centuries.

TAU

The Thousand Astronomical Unit (TAU) mission would be similar in scope and cost to the Solar Focus mission. It would be a great trial run for an interstellar mission and you would fly through the Solar Focus.

The ship would not be recovered.

As with most space related projects, the hurdle is not technological, it is the tremendous cost of these ambitious programs.

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+1, I came here to tell that we can do it now, only no one is willing to pay. So, well, you beat me to it. – Mołot Feb 14 at 22:55
    
What do those missions have to do with arrays? – JDługosz Feb 15 at 16:06
    
The first accomplishes the same goal - providing vastly improved telescopy. The second is just an example of the type of capability that humanity could probably accomplish. – Jim2B Feb 15 at 18:01

Let us consider how incredibly useful such a telescope is

The maximum angular resolution of a telescope is:

$$\theta =1.22 \frac{\lambda}{D}$$

where $\lambda$ is the wavelength of the light, and $D$ is the diameter of the telescope.

For our test, we place a banana (fruit, 20 cm long) on Pluto (not fruit, on average 39.5 AU away)

The wavelength of a bananas colour, #EFFA00, is approximately $580nm$, and the diameter of a Moon telescope is $3476km$

Angular size of a banana: $3.386×10^{-14}$ radians

Angular resolution of telescope: $2.036×10^{-13}$ radians

Not even close.

Let us upgrade to an Earth sized telescope, and a banana with a larger blue shift.

Angular resolution of an Earth sized telescope looking at a blue banana: $4.495×10^{-14}$ radians

Close, but no cigar

But, we can do better. You said the Moon, Earth and Mars? Turns out we can combine those telescopes with astronomical interferometry. Surprisingly, it is as simple as just combining the diameters of the individual telescopes.

Resolution of wasted terrestrial planets telescope: $2.495×10^{-14}$ radians.

Finally, no pixelated blue banana can hide on Pluto.

blue banana.

Given a telescope building economy comparable in size with the current market price of bananas (0.58$/pound), times the total world production of bananas (139 million metric tons), we have a budget of 178 billion dollars to build telescopes for.

The cost per area of a telescope array is a bit tricky, but the one hectare telescope seems to end up at approximately 80 million dollar.

That results in a total cost of covering a hemisphere of Earth with telescopes of $2×10^{18}$ dollars. (we just need one hemisphere, the other one can be used for say, growing bananas).

Divided by our budget, that is 11.2 million years of building time. From a geological point of view, that is not so bad.

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How do you keep them aimed at the same thing on 3 frames of reference? – King-Ink Feb 15 at 0:43
    
In that case why don't we take a picture every night with the same telescope run the the math and get a 93 million mile (149.59787 km) diameter telescope? With a lot better light gathering (365 x the telescope in question) and use the telescopes we already have. – King-Ink Feb 15 at 1:03
    
@King-Ink I had to actually look that particular aspect up, and it turns out the interferometry measurements must be done simultaneously. A smaller telescope with multiple observation can detect a higher number of photons, but is still limited in angular resolution. – Hohmannfan Feb 15 at 1:06
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+1 for Pluto not being a fruit. – FraserOfSmeg Feb 15 at 7:09
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Well, a banana is technically speaking a dwarf fruit, and Pluto a berry. – Hohmannfan Feb 15 at 8:14

The only issue with spreading telescopes across the countryside (or solar system) is while interferometers can have the angular resolution of a telescope the size of the baseline, the light gathering is still limited to the size of the mirrors on the actual telescopes.

In principle, you could raid Toys'R'Us for a bunch of telescopes suitable for introducing children to astronomy and create an interferometer the size of the United States by placing one in each state and combining the inputs. You could do even better by going to another store, making a deal with SpaceX and placing the extra telescopes on the Moon and Mars (as in the opening question).

You now have a fantastically powerful interferometer, but the light gathering of even several hundred children's telescopes would barely match a single mirror on a mid sized observatory.

So if you are going to all the expense to create a planetary or even interplanetary interferometer, then the individual telescopes need to be as large as possible, given the resources you can apply to the project.

Of course, one good thing about creating an interferometer is you can build it in stages. Even starting with two large telescopes in the Western Hemisphere already creates a continent wide "mirror", and you can add incrementally add telescopes around the world, and in LEO, HEO, the Moon and so on as budgets allow, gradually improving the power of the device until (as Hohmanfan says) even bananas on Pluto become clearly visible.

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