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How does refraction take place in the atmosphere? Why do stars twinkle but not the planets?

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The refraction take place in the atmosphere because of the gradually changing refractive index of the different layers of atmosphere. The layers at the top is optically rare, while the layers at the bottom is optically denser.
When light travels through different layers of the atmosphere, refraction takes place. Since light passes from rarer to denser mediums, it tends to bend towards the normal.
So, the light of stars undergoes refraction continuously on entering the earth's atmosphere, before it reaches the earth. The atmosphere bends starlight towards the normal, so the apparent position of the star is slightly different from its actual position. So the apparent position of the star changes or fluctuates.
So, the position of a star as seen by us is not its true position. 
Thus, star appears slightly higher than its actual position because of atmospheric refraction.

As discussed above, the light of stars undergoes refraction continuously on entering the earth's atmosphere, before it reaches the earth.
Now we know that the apparent position of the star changes or fluctuates, so the path of rays of light coming from the star goes on changing slightly. That is why the amount of starlight entering the eye flickers. The star sometimes appears brighter and at other times, it appears fainter. This makes the star twinkle for us.
Hence the twinkling of a star is due to atmospheric refraction of light of stars.

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