Except for those pesky photos posted of the sun & the moon in the sky at the same time during a lunar eclipse.
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The photos to which you refer do not show the sun and moon both visible in the sky during the eclipse.
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The moon is visible, with a copper hue ("blood moon") but the sun is not above the horizon. Only the glow of pre-dawn is seen.
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However, by abusing the exposure of the camera the picture is overexposed near where the sun will soon rise and you can't see what is there due to the overexposure. But the narrator says "there is the sun!" even though the sun is not there yet.
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The same deception is used in videos which claim that the sun gets smaller as it sets in the west by overexposing the footage when the sun is high in the sky making it look like the sun is very large, but all it is is an overexposed sun that seems to be about 5 times the diameter it would be if the proper filters were used. The photographer did not use filters because he was attempting to skew his image to make the sun appear 5 times larger than it actually should look.
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Flat-earthers are prone to frequent deception, for whatever reason. Probably because they can't provide anything truly convincing.
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If it is an emulsion film camera you probably won't cause any damage, unless the internal light meter gets too much sunlight.
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But if you're using a digital camera, you could damage it by overexposing it to the harmful effect of focused direct sunlight.
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Not to mention the damage you can do to your own eye if you're stupid enough to be looking through the viewfinder while filming the sun, even if you can't quite see the sun before sunrise.
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