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For the purpose of this post, full spectrum means a non-trivial number of frequencies within a non-trivial band. So human eyes aren't full spectrum within the visual range (each cone is wide band, they overlap and there are only three) but ten or twenty relatively narrow non-overlapping channels covering the full width would. I'm defining it up-front so that it's clear what I'm discussing.

I know that you can link radio telescopes over thousands of miles with a collecting area of a square kilometre. It's called SKA and it's currently being built.

Likewise, I know that you can build optical interferometers, but currently, none are capable of resolving a visual image.

Max Tegmark built a huge interferometer (either microwave or infrared, I'm unsure), the Omniscope, for looking at the cosmic background radiation.

But here you run into the first problem. The average distance to the asteroid belt from the sun is 3.2 AU, so we can treat our disk of radio telescopes as having a diameter of 6.4 AU and a circumference of 32.2 AU. Even if you processed the data on Earth, half that disk isn't visible, so you've got to transmit the data over unreliable, non-deterministic, low-bandwidth, high-latency links for 34.2 AU (distance to a common transmitter since there's only one deep space network plus distance to Earth). The non-determinism is the potential killer as you have no means of determining how to overlay the data.

The second problem is that even optical interferometry is limited. For full-spectrum, you've got to get it through UV and into X-Ray, and telescopes have to look over much narrower bands. I don't know if such telescopes are possible.

Given that a greater range of telescopes complicates data delivery (you've got more complicated paths to get the data from A to B because telescopes want to transmit their own data, bandwidth is constrained because you're using radio telescopes and interferometry still has to patch the data together), it's reasonable to theorize that you have a minimum number of relay stations elsewhere in the belt for the number of telescopes.

But you've now added the number of places that can collide with other objects, that can fail due to hard radiation in space, and that move unpredictably (N-body problem) relative to the telescopes they're relaying.

So we can say that there should be an upper limit, a bound beyond which either the telescopes can't be linked as an interferometer due to communications problems, where there's just no added value (an interferometer of half the size and twice the time base will see more), or where the probability of failure from any cause exceeds the value of the data obtained in the mean time between failures. The exact cause of the limit is irrelevant, although if there is published science on this, it would be good to see.

We can also say that there is an upper frequency beyond which interferometry is impossible with any known science. The reason doesn't matter, just the bound, although, again, the science would be good to see if published.

Because the asteroids move relative to each other, the change in the relative position of each obviously impacts the timebase (unless you create yet another mechanism for tracking position, with the unreliability that creates). The tools used in synthetic aperture receivers might be useful since you can in principle treat the motion as simply receiving on different spots on your fixed virtual dish.

If there is a function tying maximum size to maximum frequency, that would be wonderful, as then you can plot the full range of possibilities.

Otherwise, how large of a telescope over how large of a range of frequencies over how many bands could you have? Would you need to create an original ringworld (disconnected platforms in a ring) to build this, or can you utilize the asteroid belt with minimal impact?

(To clarify, this last bit is the question of interest.)

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This question asks for hard science. All answers to this question should be backed up by equations, empirical evidence, scientific papers, other citations, etc. Answers that do not satisfy this requirement might be removed. See the tag description for more information.

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    $\begingroup$ @Mołot Well; interferometer principles, theory and design looks to me like something there would exist real-world citable scientific works on. I'll absolutely grant that scientific papers on asteroid-belt-scale interferometers are unlikely to exist, but that doesn't necessarily imply that relevant citable material doesn't exist. This looks to me like it might be answerable to a hard-science standard, and if not, I'd personally rather see the question being relaxed from hard-science to maybe science-based, than start out science-based and only later be changed to hard-science. $\endgroup$ – a CVn Dec 6 '17 at 19:36
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    $\begingroup$ I'm not an expert but I'm actually reading up on this so I do have some limited understanding of the topic. Here are some thoughts: 1. Currently the largest optical array has 6 telescopes here on Earth. Having an array of hundreds of thousands of telescopes out in the asteroid belt is a couple of levels of magnitude more difficult. 2. The rotation of the belt will actually give you different baselines and having many different baselines results in better quality image. 3. Since your telescopes are in space you won't get any atmospheric interference. $\endgroup$ – ventsyv Dec 6 '17 at 21:22
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    $\begingroup$ You may want to read about en.wikipedia.org/wiki/FOCAL_(spacecraft) That's a project that would use Sun gravitational lens for building giant telescope and that is possible(borderline) with existing technology. $\endgroup$ – Vashu Dec 6 '17 at 23:12
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    $\begingroup$ I don't see why the need of using asteroids. They could place the telescopes over a regular, empty orbit, and it will be easier and cheaper - landing on an asteroid is tricky. $\endgroup$ – Rekesoft Dec 7 '17 at 11:34
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    $\begingroup$ kingledion - A few months is fine, look how long NASA has waited! :) Ok, leaving it hard science. $\endgroup$ – Imipak Dec 7 '17 at 22:12
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Now, forgive me if I am missing some things, but what you are looking for in this post does not seem to be that challenging. I'll propose a solution with near-future technology.

Summary

  • Use two large-ish asteroids in the belt in locations that make them unlikely to be hit by space debris.
  • You can't use a space station due to vibration problems, you need to build into an asteroid with some serious mass.
  • Do not use the interferometer for objects in the plane of the solar system, instead use it for objects closer to the axis of rotation of the solar system.
  • Use multiple telescopes/apertures to get the spectrum you desire
  • Coordinate image taking with a constellation of navigation satellites
  • Aggregate and post-process information later

Sources

Method

Site selection

The asteroid belt is relatively sparse. Estimates of the number of asteroids over 1 km range from 1 to 2 million. Gladman, 2009, finds the power law scaling of asteroids with size at in this range to be -2.5; so the number of asteroids is $N \propto r^{-2.5}$; this would put our 100 m asteroid estimate at 300-600 million.

The inner part of the asteroid belt is distributed between approximately 2.2 and 3.3 AU from the sun, at an inclination up to 20 degrees. This corresponds with a torus with major radius 2.75 AU and minor radius 0.55 AU. This gives a volume of about 16 cubic AU, or $5.5\times10^{25}$ km$^3$.

For an assumed 500 million asteroids of 100m or more, this gives a density of $9.1\times10^-18$ km$^{-3}$; or, assuming a random distribution, an average distance between objects of 500,000 km; more than the distance from the Earth to the moon.

For objects over 100 m diameter, the density is as low as the density of moon-sized objects near the Earth. Since the Earth is in no great danger of being hit by the moon, our interferometer is not in particular danger of being hit or otherwise affected by another asteroid. For objects smaller than 100 m diameter, these are approaching the size of objects that we do move in space. If we are able to bring a large telescope installation to the asteroid belt, we should be able to deflect an asteroid of this size.

Vibration management

A space station will not have the optical resolution required due to vibration. I share some vibration information from ISS in this post. The sort of vibrational stability needed to resolve a milliarcsecond with a 100m baseline receiver is aobut 0.5 $\mu$m; the ISS vibrates with an amplitude of about 4 mm.

How can we get a stable enough platform? Well, the Earth is obviously stable enough for giant interferometers like LIGO. The asteroids we need to pick will be intermediate in stability, since they are between the size of the Earth and the ISS. I could not find reasonable information or calculations to perform regarding the stability of a platform built on or into an asteroid, but we will assume that an asteroid must be selected with the proper characteristics of a stable platform. I would imagine we would chose an asteroid of 1km or greater diameter, if possible. The larger, the more stable.

Directing the interferometer

If you have two points on opposite sides of the asteroid belt, then it makes sense that you will not be able to resolve the objects which lie in the plane of the solar system. There will be too much interference from other asteroids, or the sun or what have you.

The solution is simply to restrict your observations to one or the other hemisphere. For example, you can build your telescopes on one side of the asteroids so that nearly the entire celestial northern hemisphere (roughly the same northern hemisphere we would see from Earth) is visible to both telescopes at all times. Since most of the mass of the solar system is in the plane (ok, most is in the sun, but the rest is in the plane), there should be little in your way. There are many asteroids in the main belt with high orbital inclinations, so you would have to account for this in the site selection phase, and perhaps make some efforts to move a few of them out of the way.

Now, a key to stabilization will be to rotate the asteroid. This will take some time and a lot of fuel, but by slowly rotating both asteroids at exactly the same speed, you will both improve stability of your optical platform, and provide a constant motion for each telescope relative to the other one. Again, this serves to restrict your field of view somewhat to one or other hemisphere.

Lastly, if you have enough money, you could mount separate telescopes on both sides of the asteroids, so that you can look at the northern and southern hemispheres at the same time with separate instruments. There are two ends of the axis of rotation, so you can be looking at both sides at once.

Multiple telescopes

If you want non-trivial frequencies with a non-trivial band, why not use a non-trivial number of telescopes? Since we're setting up shop on an asteroid of at least 1 km radius, and preferably more, there should be room for a variety of instruments.

The Hubble telescope has multiple instruments but only one mirror. Without going into specifics of what frequencies you are interested in, I think it is plausible to have two sets of instruments, one in the visual and near-infrared range, and another in the UV and/or X-ray range, each with their own optical mirror to focus on a variety of specialized instruments.

Position and timekeeping

The solution to your issues with combining the pictures from so far apart is to use high precision stationkeeping and timekeeping devices. For this purpose, a fleet of satellites similar to Earth's GPS system will do.

For example, a satellites could be set up in two orbits, one inside and one outside the Asteroid Belt. You will need enough satellites that at least two in each orbit are visible to each telescope observatory at all times. I believe that you will only need three in each orbit, but possibly four.

Using these satellites like GPS, if you are getting four signals at the same time, you can calculate your position accurately in fourspace (x, y, z, t). The principles of operation are the same as GPS satellites. These satellites already use atomic clocks and relativity adjustments for accuracy, so they will provide the location, direction, and timekeeping metadata to accompany each picture taken by your telescopes.

Post processing

With accurate enough 4-d orientation in spacetime, it becomes a relatively trivial matter to combine the pictures at a later time. The pictures and their metadata can all be beamed back to Earth for post-processing (the way that our deep space probes like New Horizons do now).

Conclusions

The only part of this which is not within our current technological capabilities is the heavy lift of dragging 100 m optical, IR, or X-ray mirrors 3 AU away to a suitable asteroid.

The 'image' combination technology is not much different from what LIGO is using for its disparately spaced detectors (Washington state and Louisiana); the only difference our orientation satellites need from GPS satellites is more power to push their signals over AU distances. And the telescopes in whatever bands you are interested in don't have to be any more powerful than the best we have on Earth (although, they do need to work in a vacuum, I suppose).

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For the purpose of this post, full spectrum means a non-trivial number of frequencies within a non-trivial band. So human eyes aren't full spectrum within the visual range (each cone is wide band, they overlap and there are only three) but ten or twenty relatively narrow non-overlapping channels covering the full width would. I'm defining it up-front so that it's clear what I'm discussing.

Not a big deal. Multi spectral charge-coupled device (CCD) cameras are out there.

Likewise, I know that you can build optical interferometers, but currently, none are capable of resolving a visual image.

I am not an expert on the subject, but as far as I can tell here, optical interferometers that resolve visual images either exist or are in prototyping.

But here you run into the first problem. The average distance to the asteroid belt from the sun is 3.2 AU, so we can treat our disk of radio telescopes as having a diameter of 6.4 AU and a circumference of 32.2 AU. Even if you processed the data on Earth, half that disk isn't visible, so you've got to transmit the data over unreliable, non-deterministic, low-bandwidth, high-latency links for 34.2 AU (distance to a common transmitter since there's only one deep space network plus distance to Earth). The non-determinism is the potential killer as you have no means of determining how to overlay the data.

From what I've read, the solution used in other large setups is to take time stamped snapshots of the data. The trick would be keeping the clocks in sync, which is not an unreasonable problem. The imagery wouldn't be live. Would that be a problem?

Given that a greater range of telescopes complicates data delivery (you've got more complicated paths to get the data from A to B because telescopes want to transmit their own data, bandwidth is constrained because you're using radio telescopes and interferometry still has to patch the data together), it's reasonable to theorize that you have a minimum number of relay stations elsewhere in the belt for the number of telescopes.

You could use a mesh network. The most radial nodes passing their information inward. It would require less power and be more fault tolerant.

But you've now added the number of places that can collide with other objects, that can fail due to hard radiation in space, and that move unpredictably (N-body problem) relative to the telescopes they're relaying.

The nodes could determine their own positions by Kalman filtering inputs of :

  • on-board accelerometer reading, plus last known velocity and position
  • latency to nearest mesh network node (or multiple nodes) (this technique is used by your cell phone to determine location as a supplement to GPS)
  • range and bearing (determined by luminance) to the sun
  • latency of timestamped pulses from Earth or other emitters on your system (GPS), which can also be used to measure clock drift

So we can say that there should be an upper limit, a bound beyond which either the telescopes can't be linked as an interferometer due to communications problems, where there's just no added value (an interferometer of half the size and twice the time base will see more), or where the probability of failure from any cause exceeds the value of the data obtained in the mean time between failures. The exact cause of the limit is irrelevant, although if there is published science on this, it would be good to see.

We can also say that there is an upper frequency beyond which interferometry is impossible with any known science. The reason doesn't matter, just the bound, although, again, the science would be good to see if published.

I'm not certain that either of these is true.

If there is a function tying maximum size to maximum frequency, that would be wonderful, as then you can plot the full range of possibilities.

I haven't seen one, but I'll take a stab.

  • Assuming your communication channel (with some factor of safety) can reliably communicate 'n' bits per second.
  • Assuming your CCD has a fixed number of 'p' cells/pixels which will be exposed to 'f' filters at different frequencies, and that the total size in bits of any frame is 'P' = f p
  • Assuming that each observer node can take 'i' time-stamped observations per second (in synch with the other nodes), the total transmission size (in bits per second) of each observer node 'I' = P i = f p i
  • Given 'N' is the maximum number of nodes
  • Assuming processing time is not a consideration, and that the communication pipeline is the primary limitation

For the thing to work, it must be true that n >= I N. You can set n = I N, and solve for the variables you like.

Otherwise, how large of a telescope over how large of a range of frequencies over how many bands could you have? Would you need to create an original ringworld (disconnected platforms in a ring) to build this, or can you utilize the asteroid belt with minimal impact?

The asteroid belt seems like an unsafe place (to me) to place observation stations. I would think you could just place your nodes in orbits in what we tend to think of as empty space. Your choice, of course.

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