Saturday, 8 October 2016

Physics of Rainbows; Moonbows; Fogbows - All You Need to Know - an Outreach Feature for the Community

Blog Contents and Who am I?

"The rainbow is one of the most beautiful phenomenon in nature. It has inspired art and mythology in all people and (to describe) it has been a pleasure and challenge to mathematical physicists..."  ...Vande Hulst

Rainbow is a bow-shaped display in the sky of the colours of the spectrum, caused by the refraction, reflection and dispersion of the Sun's rays through rain and mist.  ...Colin's English Dictionary 

The first satisfactory explanation of the formation of rainbows was given by Rene Descartes in 1637 based on the laws of reflection and refraction as enunciated by Snell in 1621.  That white light consists of rainbow colours was not known at the time. Isaac Newton subsequently explained the colours of the rainbow. 
Newton's explanation of the colours of the rainbow horrified poet John Keats.  Keats complained that by reducing the rainbow to its prismatic colours, Newton had robbed these marvels of nature of their magic. 

The magical rainbow  -  bright, elusive and heavenly -  is overpowering in its presence and has featured in mythology in all ancient societies.  Without the scientific tools, historic man described the rainbow with awe and inspiration. Rainbows are often portrayed as bridges between people and supernatural beings.  The mythologies developed around the rainbow are absolutely fascinating - I use slides 1 and 2  to describe some of them. You can look at 1, 2, 3 for many more.
The reason I decided to write a blog about rainbows is because, over the years many people have asked me how rainbows happen.  It is a subject people are curious about. The difficulties in explaining how rainbows are formed to somebody with no science background are obvious - I felt that it better to provide a detailed analysis with the basics written down too.  Such an analysis is not available in published literature at one place.  Because the blog is prepared as an outreach piece, I have repeated myself on few occasions  - just to make sure that the pace is not too quick.  Hope this works.  First I list the contents of the blog which are along the lines that people have asked me questions on this topic.


CONTENTS


Rainbows in Mythology
The Primary Rainbow (1st Order Rainbows)
At What Angle in the Sky is a Rainbow Seen?
Background Physics (may be skipped w/o loss of continuity)
Rainbow Formation for a Single Colour
Rainbow Formation with White Light
Why is Red at the Top of the Primary Rainbow?
Why is Sky Brighter Under the Primary Rainbow Arc?
Why is there a Dark Region Between the Rainbows?
The Secondary Rainbow (2nd Order Rainbows)
Rainbows and Water Drop Size
Higher Order Rainbow
Final Word


Click on the slide to view its full page image, press Escape to return to Text
Slide 1:
Slide 2:
The Primary Rainbow
In most cases, we see one rainbow - we call it the primary or 1st order rainbow and it is formed by one internal reflection inside of the raindrops. Standing on solid ground the rainbow we see is part of a circle.  From a high mountain or from a flying plane, sometimes it is possible to see the full circle of the rainbow.  
At times, you can see a second rainbow (secondary rainbow) that is at a higher elevation and is formed by two internal reflections within the raindrops. The two slides show such rainbows in their splendid beauty.
Slide 3:

Slide 4:

At what angle in the sky is a Rainbow seen?   

First we look at rainbow formation for light of one colour - monochromatic light. Myriads of raindrops are formed in rain clouds, and if in another part of the sky the Sun is shining, then parallel rays of sunlight will fall on the raindrops.  If you stand with your back to the Sun then the situation will be as in the diagram - a rainbow is observed when the raindrop deflects the incident rays from the Sun by 138 degrees. 
Slide 5:
Slide 6:

Slides for Background Physics

The science behind the formation, colours and structure  of rainbows requires some background in the way light propagates in different media - air and water in our case. Many of you would be familiar with this science - however, for completeness of description I have included some slides to explain these.  You can miss the next four slides (#7 to 10) - if you are familiar with Snell's laws of reflection and refraction of light.
Slide 7:
 Slide 8:
 Slide 9:
 Slide 10:

Rainbow Formation with a Single Colour  

Light rays of different colours behave differently at the boundary of two media.  First we analyze what happens to a single colour (monochromatic light) and then generalize how different colours disperse to form a rainbow when white light passes through raindrops.
Slide 11 follows the light ray as it enters a raindrop at point A, reflects at the back of the drop at B and then emerges at C in a direction that is different from its original direction.  The deflection angle d depends on the angle of incidence i and the refractive index n; dependence of d on i for water is plotted in slide 12. 
Slide 11:
 Slide 12:

The deviation of the direction of the incident ray passes through a minimum  for an incidence angle of 60 degrees and then d is 138 degrees.  From the figure in slide 12 we also notice that rays for i from about 50 to 70 degrees are bunched around 138 degrees and will emerge from the raindrop along 180-138 = 42 degrees relative to the original rays of light  - this is shown in slide 14.
Slide 13:  
 Slide 14:
Slide 14 shows that a lot more intensity is concentrated around the minimally deviated  ray (number 7 in slide 14) and if you look towards the raindrops along this direction (180 - 138 = 42 degrees) relative to the initial direction of sunlight (antisolar line) then you will notice that the sky is more intense than it is in other directions. As discussed before, the intensity is along a circular arc and is the rainbow you would see.  
Rainbow Formation with White Light

White light consists of a range of colours (see slide 9) - these are separated in the process of refraction because each colour has a different refractive index and bends differently from other colours (see slides 9 and 10).  Each colour ray behaves as monochromatic light and produces its own rainbow.  Slide 15 shows the case for red and violet rays while slide 16 shows the full spectrum. 
Slide 15:


Slide 16:


The colourful light produced by raindrops is received by the eye to perceive the arc of the rainbow.  We notice from slide 16 that in the spectrum red colour rays are deviated the least and should appear at the bottom of the rainbow.  But slide 3 shows that red arc is at the top of the primary rainbow. How?
Why is Red Colour at the Top of the Primary Rainbow?

When we observe a rainbow, our eye is at a fixed position in space and rays of light, whatever colour, must enter our eye for us to perceive them. Slides 17 and 18 show how light rays from different raindrops reach us.

Slide 17:
Slide 18:


Each drop sends a unique colour ray to the eye.  Because violet is deviated more than the red, raindrops generating violet colour are lower in sky than those producing red (see slides 17 and18).  The linear distance of the drops from the eye does not matter, only the angle determines what colour will reach us. That is why we see the colouful arcs of the rainbow with red arc positioned at the top.
Interestingly, raindrops are falling towards the Earth and continuously being replaced by others.  The rainbow we see is generated by new set of raindrops continuously replacing the previous set - it is really unique to the observer and ever-changing.
Why is the Sky Brighter under the Rainbow?
Why is there a Dark Region Between the Rainbows?
Slides 7, 17 and 18 provide a ready explanation why the sky is brighter under the rainbow.
Slide 19:

This is because primary rainbow is formed by one internal reflection of light in raindrops. Slide 7 demonstrated that the rainbow ray is the least deflected ray and raindrops generating these lie at about 42 degrees to the antisolar light. All other rays lie above the rainbow ray and to reach the eye, raindrops must be at a lower elevation than the rainbow arc. This internally reflected light makes the sky under the rainbow arc look brighter.  Slide 7 also shows that the angles that the various rays are travelling cover a wide angular range - this means that light rays of different colours overlap each other and the colour definition is washed out making the sky appear white.
Also little or no light is sent to the eye by raindrops above the primary rainbow arc and the region above the rainbow appears darker (Alexander's dark region). 
The Secondary Rainbow
Primary rainbows are formed by light internally reflecting once inside raindrops.  If the light reflects two times inside a raindrop then the dispersion in constituent colours still happens and the angle of deviation is different.  The rainbow is seen at an elevation of 51 degrees from the antisolar line.  Slide 20 is from hyper-physics website and demonstrates the formation of the secondary rainbow. 
Slide 20:
In secondary rainbows the order of colours is reversed with violet colour at the top of the rainbow.  The secondary rainbow is fainter - only about 10% as intense as the primary -  because it is more spread out and also at each reflection and refraction point, some light intensity is lost.


Rainbows and Water Drop Size
A bow's appearance depends on the size of the raindrops. Raindrops ~ 1 mm diameter produce bright narrow rainbows.  Smaller drops produce duller, broader rainbows.      Raindrops are never exactly identical in size.  Drops greater than 5 mm diameter are not common - as collisions between drops break them up. Surface of bigger drops also oscillates and degrades the sharpness of any rainbows produced by them.
The rainbow description had assumed that drops were spherical; which is certainly correct for drops up to about 0.3 mm diameter.  Larger drops tend to have flattened shapes due to air drag as they fall under earth's gravity. Flattened drops affect the distribution of the brightness in the rainbow arc.
For very small drops, 0.05 mm diameter, different colours start overlapping and the appearance of the rainbow takes a whitish hue.  This happens in fogs and clouds where the rainbows are practically white in colour.  Slide 21 shows a couple of examples of fogbows.

Slide 21:


Higher Order Rainbows


A detailed description of higher order rainbows is given in the August 2016 Review by Professor Haussmann.  We know that the primary (1st order) and secondary (2nd order) rainbows are produced by one and two internal reflections inside raindrops.  There is no reason that three or more internal reflections should not happen and produce the corresponding order rainbows. 
For red light, the calculated minimum deviation of incident rays (the rainbow ray) is as follows (angles in degrees)

Rainbow        Minimum       Rainbow          rainbow
  order            Deviation       Elevation         Angular 
                                                                        Size  
    1                  137.8               42.2                 0.9
    2                  129.3               50.7                 1.7
    3                    41.9             138.1                 2.4
    4                    43.4             136.6                 3.1
    5                  127.9               52.1                 3.7
    6                  148.1               31.9                 4.4
Rainbow Elevation is the direction with respect to the antisolar line that the rainbow should be visible.  We can notice the difficulty in viewing 3rd and 4th order rainbows. They are located in in the backward direction - direction facing the sun, they are also wider and less intense than the 1st and 2nd order rainbows.  5th order rainbow lies just below the 2nd order and is in the Alexander's band (the dark region between the 1st and 2nd order).  This makes the observation of higher order (3 and above) rainbows extremely difficult.
Slide 22:


Recently, however, 3rd and 4th order rainbows have been photographed. I show the only example that I can find in published literature:
Slide 23:

I refer you to Professor Haussmann's review for a detailed discussion of the observation of even higher order rainbows.


Moonbows

If the Moon is bright and similar conditions to normal rainbow formation are present - looking at rain clouds with the Moon shining behind you, then a bow may be observed.  In rare cases, the refracted light in raindrops may be intense enough to make the moonbow visible.
Because moonlight is much weaker than sunlight, a moonbow is not observed very often and the separation of colours is not clearly defined - the moonbow appears whitish. 
On 16th October 2016, we were lucky to have the super-moon light the skies with just the right conditions for observing a bow.  The Moon was bright and even the colours of the spectrum may be inferred in the moonbow.  I have used the following pictures of moonbows from the BBC website - they are truly remarkable photographs.


Final Word: Rainbows have been objects of awe and fascination since historic times.  Their vast size and majestic appearance has given rise to many myths and legends.  
How rainbows are produced may be explained at various levels of sophistication - I have provided the simplest description using ray optics.  Wave optics is required to explain the appearance of some dark bands at the edge of a rainbow  - I have considered this to be outside the scope of the present blog. A detailed mathematical description of rainbows may be found in a review (~140 pages) by Adams.
Other good source for rainbows is the Hyperphysics web site.

I am grateful to Professor Haussmann for some very useful communications about this wonderful subject.

Hope you have enjoyed reading this blog about rainbows.  I would be happy to discuss/explain any questions about this topic.