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What is the intensity of sunlight required to form a rainbow. I read in the love of physics by Walter Lewin than

"“For you to see a rainbow, three conditions need to be met. First, the Sun needs to be behind you. Second, there must be raindrops in the sky in front of you—this could be miles or just a few hundred yards away. Third, the sunlight must be able to reach the raindrops without any obstruction, such as clouds.”

I did not understand the third point.

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  • $\begingroup$ If you're interested in reading more about the physics of rainbows and other atmospheric phenomena, an excellent book is Rainbows, Halos and Glories by Robert Greenler: books.google.co.uk/books/about/… $\endgroup$
    – tea-and-cake
    Commented May 22, 2021 at 18:19
  • $\begingroup$ Do note that Rain clouds consisting of condensed water droplets are too thick to let sunlight through easily, so they do not contribute to a rainbow and also block any rainbow light originating behind them. $\endgroup$
    – Nilay Ghosh
    Commented May 23, 2021 at 3:41

3 Answers 3

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The formation of the rainbow results from geometric optics, so the incoming photons need to have the same direction to make a bright rainbow appear. Thus you need direct sunlight. In case of clouds you get diffuse illumination (photons hitting the water droplets under different angles).

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  • $\begingroup$ what is diffused illumination $\endgroup$
    – Parth Deodhar
    Commented May 22, 2021 at 13:15
  • $\begingroup$ Can you also explain why Alexanders Dark Band in formed even if there are other water droplets to scatter light in area above the rainbow and increase its brightness. Why isn't the effect of rest of water molecules scattering considered during rainbow $\endgroup$
    – Parth Deodhar
    Commented May 22, 2021 at 13:18
  • $\begingroup$ @Parth Deodhar The diffuse light is caused by the cloud (covering the sun), the photons encounter single and multiple scattering effects inside and as a result leaving the cloud essentially in all possible directions. For your second question, photons reflected one more inside the raindrop produce a second rainbow under a different angle. A nice overview explaining this with images is here <scijinks.gov/rainbow>, but without mathematical calculations of the angles. $\endgroup$
    – Charles Tucker 3
    Commented May 22, 2021 at 14:40
  • $\begingroup$ What is the minimum intensity of light required to form a-rainbow $\endgroup$
    – Parth Deodhar
    Commented May 23, 2021 at 13:18
  • $\begingroup$ @Parth Deodhar The rainbow is just a consequence of the physical processes happening to the photons, so I cannot image that there is a minimum intensity required (of course there is also absorption in the atmosphere etc., so some number of photons are necessary, but that's not really a lower limit below the formation of the rainbow isn't possible). And your eyes require of course a minimum intensity to actually see something at all. $\endgroup$
    – Charles Tucker 3
    Commented May 23, 2021 at 14:51
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The answer from Charles is right and to see why the photons need to come from the same direction for a clear rainbow...

enter image description here

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One can always use a garden hose to make a small, but very own rainbow.

This way, it is possible to try and check all of these conditions.

  1. The Sun must be behind you. In theory, higher-order rainbows (created by multiple reflections inside the droplets) can exist in the Sun direction, but they are at least as faint as the first-order one and the Sun is much brighter, so one needs pretty unusual conditions to see them.

  2. You need water droplets to send the properly colored light in your direction, or no rainbow for you.

  3. These droplets must receive direct sunlight. If they are in a shadow, no rainbow either.

p.s. the rainbow is a profoundly distorted image of the Sun. Water droplets are simply not a good mirror.

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