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Does the size of the atom place a theoretical limit on a telescope's focal length (and thus, resolution)?

It seems to me that that the allowed margin of error for parabolic mirrors drops with an increasing focal length. It's already like 25 nm for advanced Earth-based telescopes - which is just 50-250 atoms.

If we try building mirrors with an ever increasing focal length (e. g. as a part of a huge space-based interferometer), will we eventually reach a point where the precision required will be less than the size of the atom and thus no longer achievable? If yes, how far away are we from this limit now?

If we had unlimited resources, would it be possible to construct a space-based telescope for imagining extrasolar planets in great detail, or would this be not even theoretically possible due to the fixed atom size?

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    $\begingroup$ My understanding is that the allowed error (1/2 wave total for all surfaces) depends only on the wavelength of the light you are trying to focus, not on the focal length. As a practical matter, it is difficult to get that precision over a very large mirror, but I don't think the required precision changes as the mirror gets larger. $\endgroup$ – antlersoft Jul 18 '18 at 19:30
  • $\begingroup$ @antlersoft I agree. I'll see if I can come up with an answer that satisfies both intuition and logic. $\endgroup$ – Florin Andrei Jul 19 '18 at 0:55
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    $\begingroup$ The mirrors used by LIGO are smooth at an atomic level. $\endgroup$ – zephyr Jul 19 '18 at 2:41
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Ok, finally I can explain it. Not sure why it took so long.

Bottom line, for visible light atoms don't matter. Let's say the blue side of the visible spectrum of light has a wavelength of 400 nm (round number for simplicity). Let's assume the distance between two atoms in glass is 0.2 nm.

The thing is, light is a wave. Anything smaller than about 1/4 of a wavelength becomes essentially invisible to that wave. It's not a sharp threshold, but as things get smaller and smaller, the wave "sees" them less and less. It just goes around the object as if it's not even there.

1/4 wave of visible light is 100 nm, which is 500 glass atoms.

I make telescope mirrors. I can still see the effects of surface errors down to about 1/20 ... 1/25 of a wave. Beyond that the image is not really distinguishable from a geometrically perfect mirror, either in real-life tests under the stars, or in the lab using the equipment. Maybe an interferometer could push that limit a bit further, but not a whole lot.

Let's say the absolute measurable limit is 1/50 wave, which is 8 nm, which is 40 atoms. That's plenty of atoms. Any defects smaller than that simply do not change the image created by the mirror - when talking about regular telescope mirrors.

The surface level difference from a single layer of atoms does not even register in visible light. It takes far more thickness than that to make a visible difference.

So your ultra-long focal length mirror would be almost flat, slowly adding extra atomic layers towards the edge. As long as it approximates the desired ideal surface (paraboloid, etc), then it will be fine.

I see one exception. If the focal ratio is indeed so big that there isn't enough curvature in the surface to require even a single extra layer of atoms at the edge. Then, indeed, the mirror would not work, because it's flat down to the atomic level. Keep in mind though, that would be a mirror with a humongous focal ratio - and indeed with a cosmic-scale focal length. There are many issues with such a system before you even begin thinking about surface errors. Probably not feasible with current technology.

But then what you would do is make a segmented mirror, increasing the apparent diameter and reducing the focal ratio. You could even have multiple flat segments (flat down to the atomic level) but far from each other to massively increase the total diameter, and slightly tilted according to the shape of the ideal surface. If you can afford a cosmic-scale focal length, then you can surely afford a cosmic scale diameter, and that's how you reduce the focal ratio. Maintaining that precise tilt angle between segments would be a tremendous feat of engineering.

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Does the size of the atom limit the focal length of telescopes?

Does the size of the atom place a theoretical limit on a telescope's focal length (and thus, resolution)?

No, the size of an atom does not limit the focal length of a telescope.

Halfway through this answer I further explain why the size of an atom isn't relevant and that instead we can use the size of an electron. Hydrogen is the smallest atom, it has one electron, one proton and no neutrons. An electron is 1/1836th the size of a proton; thus it is much smaller than an atom of $^1$H.

See Wikipedia's proton-to-electron mass ratio:

$$\mu = m_p/m_e = 1836.15267389(17).$$

The number enclosed in parentheses is the measurement uncertainty on the last two digits. The value of $\mu$ is known to about 0.1 parts per billion.

For an explanation of what actually does limit the focal length see: What limits the usable focal length of telescopes currently?, to see why an atom does not limit the resolution of a telescope skip halfway through this answer.

Arne wrote

Visual resolution of a telescope is directly proportional to the aperture of the telescope. The focal length, and hence the magnification that can be achieved, is then just following on the visual resolution.

The telescopes today are usually so well build that they are diffraction limited, which means optical resolution due to diffraction is the limiting factor. If you want to have "higher magnification" in a telescope, you always want to have a larger aperture. The longer focal length may help, but is not quite necessary.

And, as Jeremy said, the limiting resource in this is money. There are some engineering problems with building extremely large telescopes, but most of these can be solved, given enough money, time and resources.

It is diffraction that limits resolution of a telescope, not the size of an atom. That part of your question is essentially what differentiates your question from the other.

See Wikipedia's page "diffraction-limited system":

The resolution of a given instrument is proportional to the wavelength of the light being observed, and inversely proportional to the size of its objective. For telescopes with circular apertures, the size of the smallest feature in an image that is diffraction limited is the size of the Airy disk. As one decreases the size of the aperture in a lens, diffraction increases. At small apertures, such as f/22, most modern lenses are limited only by diffraction.

In astronomy, a diffraction-limited observation is one that is limited only by the optical power of the instrument used. However, most observations from Earth are seeing-limited due to atmospheric effects. Optical telescopes on the Earth work at a much lower resolution than the diffraction limit because of the distortion introduced by the passage of light through several kilometres of turbulent atmosphere. Some advanced observatories have recently started using adaptive optics technology, resulting in greater image resolution for faint targets, but it is still difficult to reach the diffraction limit using adaptive optics

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Log-log plot of aperture diameter vs angular resolution at the diffraction limit for various light wavelengths compared with various astronomical instruments. For example, the blue star shows that the Hubble Space Telescope is almost diffraction-limited in the visible spectrum at 0.1 arcsecs, whereas the red circle shows that the human eye should have a resolving power of 20 arcsecs in theory, though normally only 60 arcsecs.

Why is the size of an atom not a consideration?

See Wikipedia's page: "Calorimetric Electron Telescope":

The CALorimetric Electron Telescope (CALET) is a space telescope being mainly used to perform high precision observations of electrons and gamma rays. It tracks the trajectory of electrons, protons, nuclei, and gamma rays and measures their direction, charge and energy, which may help understand the nature of dark matter or nearby sources of high-energy particle acceleration.

For more information see: "The CALorimetric Electron Telescope (CALET) for high-energy astroparticle physics on the International Space Station" (O Adriani et al 2015 J. Phys.: Conf. Ser. 632 012023):

Figure 1

Figure 1. Drawing of the CALET payload. On the ISS the payload will be installed in such a way to have open sky on top, Earth on bottom.

Figure 2

Figure 2. Drawing of the CALET-CAL instrument, with a typical shower of secondary particles generated by a 1 TeV electron incident from top.

Page 2: "2.1. CALET-CAL instrument

The CALET calorimeter (see figure 2) is composed of a charge detector (CHD), a pre-shower imaging calorimeter (IMC) and a total absorption calorimeter (TASC). It is optimized for particle identification and energy measurement of several cosmic-ray species:

  • electrons and positrons in the 1 GeV - 20 TeV energy range (with no charge sign discrimination);

  • photons from few GeV to ∼ 10 TeV;

  • nuclei up to the Fe region, with energy from tens of GeV to ∼ 1 PeV;

  • ultra-heavy (Z > 28) nuclei with E > 600 MeV/nucleon (in this case, with no energy measurement).

The CHD, positioned on top of the IMC structure, is composed of two layers of plastic scintillator with mutually orthogonal segmentation in 14 bars (SciBars), each of dimensions 3.2 × 1.0 × 44.8 cm$^3$ and read by a photomultiplier tube (PMT, Hamamatsu Photonics R7400U-06) and front-end circuit (FEC) with charge sensitive amplifier, for a total of 28 channels. The CHD determines the charge absolute value |Z| of the incoming charged particle, through the Z$^2$ dependence of the specific ionization loss. The very low uncertainty in the Z measurement (0.1 for light nuclei up to B, 0.3 in the Fe region) allows for resolving individual chemical elements with Z from 1 to 40.

The IMC is a finely segmented sampling calorimeter, with surface area of 45 × 45 cm$^2$ and total thickness of 3 radiation lengths X$_0$; internally, 8 double layers of scintillating fibres (SciFi, 1 mm$^2$ cross-section) are interleaved with a sequence of 7 tungsten plates: 5 of thickness 0.2 X$_0$ and 2 of thickness 1.0 X$_0$. The fibres of each double layer are mutually orthogonal and arranged in belts, each read by a 64-channel multi-anode PMT (MAPMT, Hamamatsu Photonics R7600-M64) and VA front-end ASIC circuit, for a total of 7168 channels. The IMC fine granularity allows for precise determination of the incoming particle trajectory, localization of the starting point of the secondary shower possibly generated, discrimination of the primary incident particle against possible backscattering from the shower developing in the pre-shower imaging calorimeter (IMC) and underlying total absorption calorimeter (TASC).".

Thus we can see a tiny particle, if it has enough energy, from as far as it can travel.

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  • $\begingroup$ I think you misunderstood the topic of the question. It's related to the performance of the instrument as related to geometry errors in the mirror surface. $\endgroup$ – Florin Andrei Jul 19 '18 at 0:53
  • $\begingroup$ The OP refers to the fact that telescope mirrors cannon be made infinitely smooth, because they are made of atoms. At some scale the surface of the mirror becomes "pixelated". That means you can't make mirrors with infinite precision. The OP believes long focal length mirrors require more precision than regular mirrors - which is where I believe they are wrong. I didn't have time to think of a proper answer, but feel free to take a stab at it if you're so inclined. $\endgroup$ – Florin Andrei Jul 19 '18 at 17:49
  • $\begingroup$ @Florin Andrei - Yes, my concern is exactly the possible precision of mirrors. Let's make a thought experiment and imagine a mirror 10^6 meters in diameter with a focal length of 10^9 meters. If we build this mirror from 1m x 1m segments, these segments will be indistinguishable from flat mirrors, because the maximum difference from a flat surface will be about the size of an atom. And it appears to me that a mirror constructed of such flat segments will not focus a point source into a point - because each one of these flat segments will reflect the point source onto a surface about its size. $\endgroup$ – cuckoo Jul 19 '18 at 21:48
  • $\begingroup$ @cuckoo - You will need to use numbers much bigger than that to get close to making your point. I have explained that the electron telescope can detect particles way smaller than a hydrogen atom, but let's use that much bigger size. --- 1.2 * (10^(-10)) m * (10^6) = 0.00012 meters --- If the biggest possible telescope were only one meter and you talk about making one a million times bigger and not being able to make it flat, that it would be 1mX1m segments, no - it would be less than 0.00012x0.00012 meter sized segments and what were viewing would be a million times bigger anyways. $\endgroup$ – Rob Jul 19 '18 at 22:03
  • $\begingroup$ @cuckoo - Another example: The JWST is only 6.5 meters across, made from only 18 segments each 1.32 meters edge-to-edge, and divided into 3 different optical prescriptions. --- Our Solar system is 4.571 billion years old, and the Milky Way is 13.51 billion years old. The JWST can see over 13 billion light years, so we will be able to see outside of our creation. The reason it is tiny is because it needs to be placed in orbit due to the frequencies being observed. It's frequency and not size that matters. $\endgroup$ – Rob Jul 19 '18 at 22:51

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