What is the reason, mathematical and/or physical, that the further away something is the smaller it looks? We know stars are humungous, but they look like tiny dots in the sky.
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3$\begingroup$ Ross Millikan's answer is great. Geometrically, check out this picture and think about what happens as "D" gets large, but "S" remains fixed. "V" would need to get smaller, which corresponds to the object covering less surface area in our field of vision. en.wikipedia.org/wiki/Visual_angle#mediaviewer/… $\endgroup$– Kaj HansenAug 11, 2014 at 4:27
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2$\begingroup$ Perhaps this is a helpful related question to think about: why is it that moving your hand closer to the front of a light-source makes the resulting shadow bigger? $\endgroup$– Ben GrossmannAug 11, 2014 at 15:44
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1 Answer
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If an object is twice as far away, the angle it subtends (in the small angle approximation) is half as large. The solid angle it covers is then one quarter as large. This reflects the fact that the surface angle of a sphere of radius $R$ is $4\pi R^2$ so as $R$ doubles the portion of the sphere that the object covers is reduced by a factor $4$
answered Aug 11, 2014 at 4:12
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2$\begingroup$ Thank you, this is exactly an answer I was looking for. I'm not sure why I'm getting all these down votes! $\endgroup$ Aug 11, 2014 at 4:13
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1$\begingroup$ Small coda on this (excellent) answer: The small angle approximation presumes space is (approximately) flat. In a positively-curved space it's generally untrue that apparent angular size decreases with distance. (!) $\endgroup$ Aug 11, 2014 at 16:23