How big can we make a telescope?

Question

I've mentioned in earlier posts that it will be much cheaper to build immense space-based observatory instruments than to even come close to launching an interstellar expedition.

There are several figures of merit including total light gathering area and separation of light gathering points.

Imagine nanotechnology that "grows" a module from material in the asteroid belt, and then dispatches it out beyond the dust of the inner solar system. The technology can record the visible and infrared light waveforms in sufficient resolution to combine them from different modules and synthesize an image from a mirror the size of the separation. (This kind of recording delayed synthesis has long been a thing for radio frequency observation.)

How small of details on exoplanets could be seen? Is there a diminishing return when making the distributed modules ever bigger, or can resolution go up indefinitely?

Does the light gathering capacity matter as well? What is the right order of magnitude to match the magnification? Off hand, I expect the target to be lit as bright as daylight, just very tiny; does the total light gathered change with the apparent target size?

Today, a star-shield is needed to prevent a planet from being washed out from the nearby star. Would a narrow enough field of view make that simply unnecessary, or is there some optical effects related to absolute separation of the targets?

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summary

1. How small of details on exoplanets could be seen? math: separation between modules, individual module size, and resolving power; resolving power to exoplanet distance and ground feature size.
2. Is there a diminishing return when making the distributed modules ever bigger, or can resolution go up indefinitely?
3. What is the relationship between light gathering capability, brightness of the image, and size of the imaged object?
4. Use with star shield?

Answer 1

There's a couple of different questions in there, and I'm going to take a stab at some of them.

First, you seem to be generally familiar with aperture synthesis, as you imply it in one of your questions. That basically means if you have a bunch of comparatively small data sources (telescopes) you can merge them into one image as if you had on telescope the diameter of the distance between any two of them.

This gives you a much better angular resolution than you might expect. Unfortunately, you seem to be correct about the diminishing returns aspect. Take a look at this, which graphs total diameter of a telescope against its angular resolution for given wavelengths of light. It's on a log scale, and only goes up to 10,000 km. On an astronomic scale, that's roughly the size of the earth.

This introduces quite a problem. Humans already do something like this by taking advantage of the Earth's annual trip around the sun. By taking measurements six months apart (from anywhere on Earth), you get an effective dish size of around 300 million kilometers. And as it turns out, with a dish that large the closest star is still only 0.772 arc-seconds across. Fortunately, that seems to be within a reasonable range (that is, appears on) at least one axis of the chart I linked earlier.

My conclusion is either I'm really misunderstanding something here¹, or you should be able to get a reasonable picture (in some wavelength of light) by creating a satellite cloud between Earth and Mars. If you have to build it bigger, that should actually help.

As a side note, when I started researching this question, I expected to need a telescope the size of the solar system or larger, implying the limit to the size of the telescope is actually how long you can keep the satellite powered (shout out to the Voyagers!) before they stop giving you data back. I was pleasantly disappointed on that.

¹Some math would really help here for verification.

Answer 2

It warms the cockles of my heart to find someone else who realised space-based observatories are much, much cheaper than interstellar expeditions. They would gather the same or possibly more information than our first generation of interstellar probes. Perhaps, hypertechnological telescopes are one solution to the Fermi paradox. Aliens aren't visiting us or are wandering up and down the galaxy because their telescopes do all that work for them.

Aperture synthesis is definitely one way to go. Both optical and radio telescopes can ramp up the size to truly heroic sizes. However, there is an alternative astronomical observatory that will yield a cornucopia of information about so much of the galaxy. Basically it exploits the gravitational lens formed by the Sun and this is located outside of the solar system. The proposed FOCAL mission doesn't need new technology. Although it is way beyond the space technology because the extremely long mission time. The Sun's gravitational lens is roughly 550 AU away. That's a cool 82,500,000,00 distant. More information can be found at the following links.

http://www.centauri-dreams.org/?p=785

https://en.wikipedia.org/wiki/FOCAL_(spacecraft)

http://www.newyorker.com/tech/elements/the-seventy-billion-mile-telescope

The FOCAL mission can involve both optical and radio telescopes. Radio telescopes are capable of gathering information which is orders of magnitude more information than in any photograph. It's possible to imagine a future ring of FOCAL observatories strung around the solar system studying the galaxy in fine detail. They would still be cheaper than sending just one relativistic starship.