Why is it okay to watch a sunset but not an eclipse?
During a sunset, the Sun is lower in the sky than during most of the day - much lower. Therefore, light from the Sun travels through about 120 miles of dense atmosphere, compared to the roughly 2 miles it travels through from straight up. Here's a rough sketch (not to scale) to demonstrate this. It is clear that $B>A$:
Light scatters in the atmosphere; in fact, one type of scattering is why the sky is blue. The longer travel distance means that there is much more scattering of ultraviolent light, which in turn means that the light you see is less intense.
Okay, you say. But doesn't the eclipse still block a lot of light? Well, unless there's 100% coverage - totality - there's still plenty of light coming from the uncovered part of the Sun, and that matters. The uncovered part is as bright as it normally is, and looking at that part is just as dangerous with or without the eclipse.
There's one more thing to consider, which is that people watching a sunset don't look at the Sun; they look at the clouds and sky around the Sun. If you look directly at the Sun, your eyes will be damaged, no matter what's happening, an eclipse or a sunset.
Thank you for your answer. If the Earth is surrounded by atmosphere, why is there 120 miles of atmosphere when the sun is closer to the horizon, as opposed to ~2 miles at its apex? If this needs to be a separate question, let me know, and I'll pose it as one.
@RockPaperLizard I've added a diagram to make that clearer; hopefully, the geometry helps a bit.
I don't understand what "2 miles" you are referring to. The atmosphere is at least 30 times thicker than that. And I cannot see in your diagram how B could be 60 times longer than A.
@MartinArgerami The words "Not to scale" are important, and while definitions usually put the atmosphere at about 300 miles high vertically, the majority of it is actually within 10 miles. The density is relevant.
There is also a second reason for this. The density of the atmosphere decreases as you go up in the atmosphere. At sunset, the sun's rays hit the atmosphere at an angle and refract through the atmosphere. The refraction is proportional to the incident angle, so it happens more at the horizon than during the day when the sun is high in the sky. The refraction also causes dispersion of light. This is part of the reason why the sun becomes red at sunset. The redness is also caused by the increasing scattering of blue light as the light passes through more atmosphere. The other colors, including ultraviolet radiation are systematically removed from the sun as it drops below the horizon. Because the light is refracted, the sun's image appears above the horizon even though the sun has actually already gone below the horizon. At this point, the damaging ultraviolet light can no longer reach your eyes.