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How can you link telescopes to make a bigger "effective telescope"? (Intermediate)

I've read that you can translate a telescope array to the distance the telescopes are apart. For instance, say you use 10 telescopes in tandem, coast to coast in the US, you essentially get a "United States" sized telescope.

How does this work? What would adding even more telescopes do? By that I mean, if I increased the amount of telescopes to 20, but they were all still in the US, would that increase resolution or some other statistic? What does increasing the distances effective increase (say now they're spread over all of Earth...)?

Thank you for your time. (Ithaca represent!)

This is a very good question, and relevant to the recent release of the first-ever picture of gas near a black hole's event horizon! What you are describing is a technique called "interferometry". You've described it very well--telescopes that are far apart can be linked by this technique to create a "virtual telescope" with a resolution equal to a single telescope with a size equal to the distance between the linked telescopes. First, I'll answer what would happen if you (1) spread the telescopes farther apart, and (2) added more telescopes, then I'll explain why.

If you add more-distant telescopes to the array, you can see smaller details in the resulting image. That's why they used telescopes from all parts of the world to take the picture of the black hole--the black hole is very small and far away and appears extremely tiny, so we need telescopes that are very far from each other.

If you add more telescopes, but don't increase the farthest distance between any pair, you'll acomplish two things. The first is that you will have to stare at the source you're observing for less time, because more telescopes means more light-collecting area, so you collect light faster. The objects we observe with these techniques are often quite dim, meaning the exposure time for the image has to be quite high to collect enough light. The second is that the quality of the image goes up, but in an unintuitive way. The size of the smallest details that you can make out doesn't improve, but the amount of detail you can see goes up. I think I can explain best with a demo.

This is what an image of a single dot would look like if you took an image with just 2 linked telescopes. It doesn't look like a dot at all! This striped pattern is the result of the same physics that cause a pattern of dots when you shine a laser through a double-slit. These images are courtesy of Andrea Isella.

With three linked telescopes, you start seeing all kinds of dots, even though we're only taking a picture of one dot.

With four linked telescopes, there are less of these artificial dots.

With eight linked telescopes you can start to see that the dot in the center (the real dot) is brighter than all the artificial dots.

There are other ways to get rid of the artificial dots too. When you take long exsposures, the Earth will rotate while you are observing your target, which helps a lot. The image below includes the rotation of the Earth over like 6 hours.

 

The physics behind this are kind of complicated. The short version is that these arrays of telescopes measure the delay of the arrival of light between pairs of telescopes. If there is no delay, i.e. the light arrives at both telescopes at the same time, that means the line to the source has to be exactly perpindicular to the line between the two telescopes. If the delay time equals the distance between the two telescopes divided by the speed of light, that means that the source must lie on the line that connects the two telescopes. Every angle to the source corresponds to a delay time, so if you can measure this delay time, you can measure the angle to the source. If you use telescopes that are farther away from each other, the delay times are larger and easier to measure, resulting in more precise angles to the source. Combining more than two telescopes requires the use of fancy math, namely Fourier Transforms and Correlation Functions to produce images like the ones I showed above or the first images of the black hole.