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I'm an amateur trying to take a shot at recreational stargazing. I've read about how looking a some eclipses without the appropriate instruments (i.e, with naked eyes) is bad for the eyes. To what extent is this true? Is staring at the phenomena bad or is even a quick glance dangerous?

Are there any more phenomena like eclipses I have to watch out for to protect myself?

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The only thing you need to protect your eyes from is the Sun. A lunar eclipse, no matter what phase it is in, is not dangerous to look at. A partial or an annular solar eclipse are dangerous because the Sun is still visible. A total solar eclipse is perfectly safe to look at—however, as the phenomenon starts and ends with a partial eclipse (as the Moon slowly blocks the view, then slowly moves away), those parts of the (otherwise) total solar eclipse are dangerous to look at directly.

DO NOT LOOK DIRECTLY at the Sun with the naked eye, even though some people have done it for VERY SHORT times (seconds, at most) without permanent damage. Again, DON’T DO THIS! The problem is the eye doesn’t have pain receptors, so you wouldn’t know when you had been looking at the Sun for long enough to be hurting your eyes. This can range from a small “blind spot” in your field of view, that goes away after a few days or weeks, to complete, permanent blindness.

There are safe ways to view the Sun through an optical instrument (which, because it collects more light than the eye, would be even MORE dangerous to look through without protection!), and some of them are really affordable financially. The best, however, is a solar filter that goes in front of the objective, NOT at the eyepiece (as the temperature there is so high, it could melt and make you permanently blind INSTANTLY)! Such filters come in many different qualities and many different prices, but are generally rather affordable as well. They are made by applying a special metallic coating to an optical-grade piece of glass or even plastic (Mylar).

DO NOT LOOK AT THE SUN THROUGH AN EXPOSED PHOTOGRAPHIC EMULSION (yes, those still exist!) and DO NOT LOOK AT THE SUN THROUGH A POLARIZED FILTER (unless you’re also using a safe filter as described above).

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    $\begingroup$ The reason you need a proper solar filter (or a very dark welder's mask) is that many of the homemade filters only block visible light, and infrared and ultraviolet light are both quite capable of cooking your eyes. $\endgroup$ – Mark Dec 26 '20 at 2:45
  • $\begingroup$ Indeed, and photographic emulsions and polarized filters DON’T block UV and IR light. Also, projection is completely safe for the eyes, but slight prolonged misalignments of the optical tube on the Sun can project its image on internal surfaces, heating them to the point of inflammation. I have a slightly damaged eyepiece because of that, and I have heard of telescopes that have burnt down completely because they were left in the sun without surveillance. $\endgroup$ – Pierre Paquette Dec 26 '20 at 16:12
  • $\begingroup$ Excellent answer. Besides the sun, looking at the moon through a telescope is not dangerous but it is bright enough that it can be unconfortable on a large diameter telescope (maybe 9” or higher?); sometimes a neutral density filter (“moon filter”) is advisable to reduce the brightness to comfortable levels. $\endgroup$ – Euro Micelli Dec 26 '20 at 18:02
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Basic rule: do not ever have a light-gathering instrument between your eye and the Sun, and think very carefully if pointing it somewhere where it might pick up specular reflections of the Sun (e.g. casually abusing it to look at some distant paragliders, where you can suddenly get the reflection of the Sun on a window).

Do not trust filters or aperture restrictors in any point of the optical path. "Everybody knows" that eyepiece filters are dangerous because they can shatter, but irrespective of the type of filter there's always the risk that you (or a bystander) might think the filter is correctly in place but it's not (yet) been fitted.

I'd not go so far as the vicar of a small Devon town where I once lived who prayed for clouds lest any eclipse-watcher injure his eyes, but the only safe safe way of looking at a bright source is by projection onto a piece of card.

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