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I've looked at the design of the James Webb Space Telescope and I got curious about something, some years ago, it seems that the international space station was hit by micro-meteorites. I'm wondering if the same couldn't happen in the James Webb Space Telescope.

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If you covered the mirrors then they wouldn't be mirrors anymore. –  Jeremy Jun 22 at 9:23
    
@Jeremy Yes, but in the case of the Hubble, the mirror seems to be more protected somehow. –  Igäria Mnagarka Jun 22 at 9:24
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No. Not too unprotected, as you put it. There are several misconceptions that I find common about the JWST, that need to be addressed:

JWST primary mirror elements are not made of glass and do not shatter on impact

It's primary hexagon mirror elements are made out of Beryllium powder pressed into blocks, that were later cut in half to create two mirror blanks and had most of their back side cut away leaving reinforcing rib structures and thin and light front mirror surfaces. Those front surfaces were later precisely shaped to their specification at JWST operational temperature (-400°F, -240°C) and polished to a mirror finish. Beryllium was specifically chosen due to its lightness, strength and ability to maintain shape at mentioned cryogenic temperatures. Read more about this process on JWST - The Primary Mirror (includes images and videos).

Unlikely impacts with micrometeorites (see below) would result in bullet holes, not fractures and shattered mirror surfaces, and those punctures are something that can be established during mirror calibration mode and corrected for with image post-processing. No large mirror is free from slight defects throughout their lifetime, and this is something astronomers are well used to dealing with. The important thing is that the primary mirrors still maintain their shape, together with secondary mirror maintain focus, and most of their joint surface doesn't distort end image too much with any such imperfections. Smaller problem areas can be corrected for in software, or otherwise adjusted to in hardware.

JSWT target halo orbit around Sun-Earth Lagrange point 2 is not littered with debris

Sun-Earth Lagrange point 2, or SEL2 for short, the point around which JWST will be stationkeeping in a halo orbital regime (its own orbital plane around SEL2 is perpendicular to the Sun-Earth plane), is one of the least gravitationally attractive points in the near-Earth space, roughly 1.5 million km more distant to the Sun than the Earth, and with the same orbital period as Earth itself. Think of this point either as the top of a hill, or perhaps better yet (since it's not really a "hill" per se, that would require negative gravitational attraction, or anti-gravitational point), as a nearly flat surface with increasingly steep slope towards its parent massive bodies, in our case the Sun and the Earth. In another analogy, where is the safest place to stand if you're at risk of being swept by an avalanche? At the flat top of otherwise a steep hill, of course.

This is a lot different to orbiting around the Earth in LEO (Low Earth Orbit), where the International Space Station (ISS) is;

JWST is not defenseless against collisions with debris and micrometeorites

Still, even with all said, collision with objects in transit through the SEL2 region are not excluded, so there are a few defense mechanisms that JWST will have available to it and its mirrors;

  • The most obvious one is that it will deploy a large, multi-layered sunshield that will be kept pointing towards the Sun, as the name suggests. The aft of the craft will not merely be these deployable sail-like sunshields though, but will also host more robust structures, such as observatory's communications subsystem, navigation subsystem and engines, and so on. All of them should protect the mirrors from roughly half of the impact vectors, while they point in the other direction.

  • Halo orbit is, as mentioned, inclined roughly 90° to the Sun-Earth plane, which means that the JWST would only pass the plane at which most transient heliocentric dust would orbit at that orbital altitude two times during each of its large 800,000 km (500,000 mi) radius orbit, and only for relatively short duration. Any of these heliocentric interplanetary dust transiting SEL2 would also have small relatively velocity to the observatory. SEL2's radial velocity is roughly 30.08 km/s ($v_o \approx {2 \pi a \over T}$), while orbital speed at Earth + 1.5 million km heliocentric altitude is roughly 29.64 km/s ($v_o \approx \sqrt{\frac{GM}{r}}$). So we're in theory talking of impact velocity in the range of 440-445 m/s, or about 1.5 times the speed of sound at sea-level Earth. In terms of possible relative velocities of man-made debris in LEO, this is roughly 99.9% smaller kinetic potential ($E_\text{k} =\tfrac{1}{2} mv^2$) per same debris mass than in LEO with retrograde debris hitting prograde satellites at up to 15.4 km/s (e.g. ISS orbits at roughly 7.7 km/s, or 4.8 mi/s).

  • Larger Near Earth Objects (NEO) will of course continue to be tracked by Earth-based and in-orbit observatories, and needless to say JWST can detect even the darkest surface NEO on its own that are hard to detect in visual spectrum (albeit that won't be its mission, most new NEO detections are still by chance), since it'll be doing observations in the infrared range, and can detect these objects' radiated and reflected heat signature. In case a collision with a larger object would be predicted, and this object's trajectory well established, JWST could complete collision avoidance maneuvers with its own thrusters, if needs be.

And there are other risk management options available to JWST, including preparing conjunction analysis for potential collisions with other satellites stationkeeping in SEL2 or intersecting it (won't be many, but it'll still be done, no question about it), planning its stationkeeping and attitude maneuvers to avoid hazards and protect its most sensitive parts against debris, interplanetary dust, micrometeorites, solar and cosmic events, and other detected threats, when, where, and as needed. But in a general sense, SEL2 is a relatively safe place to be compared to near-Earth orbits, as far as orbital debris and micrometeorite impacts go, as per your question.

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