Small telescopes are very useful for searches for extremely faint, diffuse galaxies because it turns out that these searches are surface brightness limited rather than being magnitude limited like most surveys. Surface brightness is a little counterintuitive because the minimum surface brightness a survey can find is actually independent of telescope size. Using a 8-m telescope is no better for finding a diffuse galaxy than a telephoto lens. The only thing that can improve your signal to noise ratio is observing time and observing time is much cheaper on a small telescope.
The basic reason for this is that as you increase the size of your telescope, not only does the collecting area increase, but the area of the image increases as well. So if you're looking at a diffuse source like a galaxy, the increased number of photons gets spread out over a larger area and the total number of photons per pixel remains the same. This isn't a problem for point sources (like stars) because even as the image size increases, all the photons from a point source still fall in the same pixel so the source appears brighter to the detector.
As peter303 also mentioned, refracting telescopes don't suffer from the same artifacts that reflecting telescopes do, which is why these surveys use small telephoto lenses rather than small reflectors.
So would you get similar surface brightness performance out of a larger telescope by decreasing the size of the detector, by adding a couple lenses to focus the light onto a smaller area? It's been a while since I took astronomy courses and I try to keep up with the basic physics.
Yes and no. In principle it would be possible to do something like that, but you could only do it for a part of the image and you would lose the focus. You'd basically be trying to transform an extended source into a point source. In practice it's easier just to use a smaller telescope for a longer period of time.
Incidentally, this is related to a puzzle: can you burn a paper with moonlight if you have a sufficiently big magnifying glass? It turns out that you cannot (at least as long as you're using a normal magnifying glass that brings light to a focus), because the highest temperature you can produce is the temperature at the surface of the moon. In the limit of having an infinitely big magnifying glass, the view from the sheet of paper would be the same as the view from the moon, and so the temperatures would be the same as well [1].
[1]: Making various assumptions about radiative thermal equilibrium, etc.
I had not seen that xkcd article before! Thanks! It's the most lucid explanation of the problem that I've seen.
The only other interesting surface-brightness-related fact I can think of at the moment is that surface brightness fluctuations can be used to measure the distance to galaxies. This is one of the cases where it helps to have a really big telescope.
The basic idea is that although galaxies are extended sources, they're really just a collection of point sources --- they're just a bunch of stars grouped together in an area of the sky. Now suppose you have two similar galaxies (at least they have similar stellar densities) and one of them is close by and the other is far away. The nearby galaxy will have relatively few stars per pixel and the distant galaxy will have many more stars per pixel. Since the stars are randomly distributed, the number of stars in any given pixel will be given by a Poisson distribution. A consequence of this is that in the nearby galaxy there will be a lot of variation in the flux from one pixel to the next, whereas in the distant galaxy the image will be much more smooth. So even though the average flux per pixel from both galaxies is the same, you can still tell which one is close and which one is distant based on the surface brightness fluctuations.
But without actually imaging stars, how do you know that two galaxies have similar stellar densities? The OP was about low-star/high-darkmatter galaxies. I guess you would have to hope that the entire thing was rotating, and at an angle, that doppler effects could determine the size/mass.
That's where you have to make some assumptions given the galaxy type. As with a lot of distance techniques in astronomy, there is a lot of uncertainty. This is one of the reasons that surface brightness fluctuations are not a very popular method.
This has been popularized by the XKCD article, but it's simply incorrect.
What the physics in the article prove is that you cannot generate a higher temperature than the color temperature of the black body whose light you are refracting.
The moon is not a black body emitter, it is a diffuse reflector. I will grant that you cannot start a fire by reflecting the blackbody emissions of the moon during a complete lunar eclipse.
However, I experimentally proved to myself that I can start a fire with sunlight reflected off an ordinary mirror. The mirror was about room temperature, but the magnifying glass produced a much higher temperature. I never got around to taking a giant parabolic reflector out on a full moon night to measure the temperature change, but I remain convinced that it would work. Seeing this repeated here makes me think I should.
A diffuse reflector like the moon behaves differently than a specular reflector like an ordinary mirror. A specular reflector will preserve the image but a diffuse reflector will destroy it.
To get a little technical, if the Sun is reflected by either a diffuse or specular reflector, the integrated surface brightness over 4 pi steradians surrounding the reflector will be equal to the integrated surface brightness around the same point if the reflector had not been present.
The difference is that in the case of the diffuse reflector, the specific intensity becomes isotropic, but in the case of a specular reflector, the specific intensity is unidirectional. Now, from your perspective somewhere off to the side looking at the reflector, you only intercept a small fraction of the magnitude of the specific intensity in the case of a diffuse reflector. But you intercept the entire magnitude of the specific intensity if the alignment is right, and zero otherwise. So the surface brightness you're able to observe is unchanged in the case of a specular reflector (if the alignment is right), but is reduced by the solid angle subtended by the reflector as seen by the source divided by 4 pi.
So it is not surprising that if you take a magnifying glass to an image of the Sun in a mirror you can start a fire. But this doesn't mean that you'd be able to use a diffuse reflector like the moon to do the same thing. After all, a white baseball is also a diffuse reflector and is reflecting the Sun's light on a sunny day. But I doubt you will be able to put a magnifying glass to it and light a piece of paper on fire. But you should get a big Fresnel lens and go out the next time there's a full moon and prove it to yourself! (The big Fresnel lens will be fun for a lot of other experiments, too.)
Would it be possible to software process images taken by large numbers of amateurs of the same point in the sky to get even better results even cheaper?
Yes, in principle. In practice, there would be so much variation in the sensitivities of the detectors that it would be a nightmare to do the image reduction. In normal observing you have to do "flat fielding" every night (or at least every few nights) where you take an image of the sky at twilight so that you can calibrate the different sensitivities of the different pixels in the detector. Then all the pixel scales will be different (i.e., a single pixel on one telescope will correspond to a slightly different area than on another telescope), so you'll have to figure out how to scale all the images to overlay them all. It's just easier to do it all yourself.
An additional explanation I heard is that the more common reflector telescope possesses diffraction artifacts inherently caused by the mounts of secondary mirrors. These artifacts interfere with seeing the faint magnitude of galaxy halos. Refractor telescopes don't have these artifacts.
I've worked across a wide range of quantitative science fields and one thing I've noticed is that while the bulk of the funding goes to massive facilities, often a bit of clever engineering and $100K can make a huge difference. This is true in computing; one of the things that's interesting about computer science is that it can turn a complicated physics equation that is O(n4) or worse into an O(nlogn) approximation that is "good enough" to justify not using brute force.
The basic reason for this is that as you increase the size of your telescope, not only does the collecting area increase, but the area of the image increases as well. So if you're looking at a diffuse source like a galaxy, the increased number of photons gets spread out over a larger area and the total number of photons per pixel remains the same. This isn't a problem for point sources (like stars) because even as the image size increases, all the photons from a point source still fall in the same pixel so the source appears brighter to the detector.
As peter303 also mentioned, refracting telescopes don't suffer from the same artifacts that reflecting telescopes do, which is why these surveys use small telephoto lenses rather than small reflectors.