How deep can a 'deep field' image be? Hubble's famous deep field image was created by pointing the telescope at the same spot for 10 days continously. This aggregates photons and creates a coherent image - If I understand what's going on correctly.
If the focus was for 100 days or 1000 days would we have had a clearer imager with more detail? Or is there a theoretical limit beyond which we do not get any more information?
 A: The earliest light we can see is the cosmic microwave background. That is light that was emitted when the universe cooled enough to allow hot gas to form instead of ionized plasma. The gas was transparent. The plasma was not. Both emitted light, but light that traveled through gas is still around. That happened about 300,000 years after the big bang. The Big Bang was 13.7 billion years ago. So the deepest possible deep field is 13.7 billion light years deep.
At the time it was the glow of white-hot gas. But the universe is expanding, and has stretched the wavelength by a factor of 1100. We see it easily with a microwave antenna.
The JWST sees shorter wavelengths, so it cannot see that far back. But the earliest object it has seen so far is 13.1 billion light years away. https://www.nasa.gov/image-feature/goddard/2022/nasa-s-webb-delivers-deepest-infrared-image-of-universe-yet
One problem with distant objects is that they are very faint. Telescopes have two ways to overcome that. One is with a longer exposure to gather more light. The other is to build a larger diameter telescope to gather more light. That will make details bright enough to see. Clarity - the ability to distinguish two nearby details - requires more.
The JWST deep field image comes from a very small patch of sky. You could cover it with a grain of sand held at arm's length. JWST and Hubble are very good at resolving light coming from 2 stars separated by a very small angle.
A distant star appears so small that it is indistinguishable from a point. Light from that point spreads out in a sphere. If the star is light years away, the sphere has a radius of light years by the time it reaches us. Like the surface of the earth. that sphere is indistinguishable from a plane. Light from a distant star is a plane wave.
A telescope takes an incoming plane wave and focuses it to a point on its sensor. Two plane waves arriving from slightly different angles must be focused on two different points. The slightest blurring will make the light focus to small circles that overlap. You wouldn't be able to distinguish one star from the other. There are a number of sources of blurring.
For telescopes on Earth, the atmosphere is the biggest problem. It isn't perfectly uniform. Temperature variations make it bend light slightly. Winds make temperature variations move around. Star twinkle when you look at them. This is a prime reason to put telescopes in space.
Another problem can be vibration. The telescope must not change its angle during its long exposure. On Earth, a lot of effort has gone into slowly moving big telescopes as the Earth rotates to keep it pointing to the same stars. In space, this isn't a problem. But extreme care has gone into eliminating vibrations from moving parts and designing a system to keep the telescope aligned.
Imperfect optics is the next problem. Lenses are made of glass. The index of refraction is part of what determines the shape of the lens. But the index of refraction changes at different wavelengths. You can make a compound lens out of different types of glass where one lens element corrects the problems of the other. But you can't make a lens that works perfectly at all wavelengths. This is why telescopes use mirrors instead of lenses.
The perfect shape for a telescope mirror is very close to a sphere. A spherical mirror is much easier to make with the extreme accuracy required for a good focus. If you are going to put a giant telescope in space, you take the trouble to make the perfect not-quite-spherical shape. So that isn't a limitation.
The limitation that you can't get around comes from the fact that light is a wave. If you pass a wave through a pinhole, it diffracts and spreads out. The aperture of a telescope is a giant pinhole. The amount of spreading out is tiny. But if all other problems are solved, it is the limiting factor in image quality. In this case, the image is said to be diffraction limited. The smallest angle that can be resolved by a diffraction limited mirror is given by
$$\theta = 1.27 \frac{\lambda}{D}$$
where $\lambda$ is the wavelength of light and $D$ is the diameter of the mirror.
The diameter of the Hubble mirror is $2.4$ meters - a very big pinhole indeed. Hubble is sensitive to UV, visible, and near infrared wavelengths, $0.1$ to $2.5$ microns. The JWST mirror is $6.5$ meters across. It is sensitive to the mid - far infrared, $0.8$ to $28$ microns. So the clarity for both depends on the wavelength. See https://webb.nasa.gov/content/about/comparisonWebbVsHubble.html
A: You don't get a "clearer image", in the sense of angular resolution, because that is set by the quality of the telescope optics, the size of the mirror and the size of the pixels on the detector. In fact, taking a longer exposure might degrade the angular resolution is there is motion in the telescope or if you are co-adding lots of images and there are small errors in the alignment of each image.
What you might get is an image with a higher signal-to-noise ratio, which allows you to pick out more detail in the faint smudges of distant galaxies. This is the real purpose of the Hubble Deep Fields.
However, there is a limit to how far you can play this game. As well as increasing your signal, you are also increasing the noise in the background image. That is because most of the "noise" in a deep image is actually the countless even fainter galaxies that lie both in front of and behind the galaxies you are interested in or faint glows associated with (e.g.) zodiacal or geocoronal light. This means there are diminishing returns in observing for longer and longer if you have a fixed angular resolution for your telescope.
The solution to this is either you need a telescope with higher angular resolution or you need to observe at wavelengths where the very distant objects you are interested in are more luminous and might offer a better contrast with the background or you place your telescope where there might be fewer local sources of contaminating background. This is the idea behind JWST, which observes in the near and mid-infrared but has an angular resolution that matches or even slightly exceeds that of  HST and is placed far away from the Earth's atmosphere.
