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Why can I look directly at the sun during a sunset (with a lot less pain and unconfort) but not at noon?

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At Noon, sunlight is coming from nearly directly above you, and goes through ~2 miles (4km) of reasonably dense atmosphere to reach you (Atmospheric pressure at 2 miles is about one half of sea level pressure). At sunset (and sunrise), since the sunlight is traveling mostly parallel to the surface, it travels through dense atmosphere for considerably longer. When traveling through dense atmosphere it's subjected to more scattering, which decreases the observed intensity of the light.

Or, considered a different way, when looking at a sunset/sunrise, you don't see the sun as a single well defined point. It's hazy, with the bits of sky near to the sun also giving off a fair amount of light. That additional light is some of the light that was scattered from an otherwise direct beam. If it were all re concentrated to a single point, it would be brighter/more painful to look at (though not as bright as the noon sunlight- for shaky statistical and light absorption reasons).

Creative Commons image of a Sunset... by... Mario Antonio?

Using an Earth Curvature Calculator, sunlight at the horizon travels through ~120 miles of 'dense' atmosphere (ie has an altitude <2 miles). That will fall off rather quickly the higher above the horizon the sun is.

(dense atmosphere in this refers partly to just thicker air; but also to particulates/dust in the air, haze, mist, fog, etc)

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    $\begingroup$ What is the relevance of Atmospheric pressure at 2 miles is about one half of sea level pressure in this answer, and what do you mean by has an altitude of <2 miles? $\endgroup$ – gerrit Mar 11 '16 at 11:31
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    $\begingroup$ I think it's because the atmosphere never really ends, it just gets less and less dense. So he's using the 2 miles as a marker where there's about half the atmospheric density to say we can ignore anything from there up. $\endgroup$ – zundi Mar 11 '16 at 13:19
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    $\begingroup$ @CoderTao is there an equation for this? For example, at what angle above the horizon would the sun be about 50% as bright as at noon? $\endgroup$ – zundi Mar 11 '16 at 13:21
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    $\begingroup$ @TheVoid -- Stop posting nonsense, and in this case, dangerous nonsense. It is not safe to look at the Sun for any extended period of time, even when the Sun is right on the horizon. It is not safe to look at the Sun, period, with the naked eye once the Sun is even a few degrees above the horizon. $\endgroup$ – David Hammen Mar 11 '16 at 15:04
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    $\begingroup$ @sandy Sadly I don't have an equation for it, though David Hammen's link to the wiki article on Air Mass would be good reading, and does seem to have a few equations for magnitude vs angle above horizon. And yes, 2 miles was chosen as a rough 'going straight up, we've passed through half the air particles we're going to see'. $\endgroup$ – CoderTao Mar 13 '16 at 9:59
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An excellent explaination found here: https://www.youtube.com/watch?v=ZD1roP5wISU

Goes into detail about Rayleigh dispersion.

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