The Science Behind Rainbows: Explaining the Center of a Rainbow | Gooood Morning

[Trying2Learn]

TL;DR Summary

What would we see, if we could?

Gooood Morning

I was traveling with my son and he noted the rainbow out the window and asked why we see only half.

I informed him that the earth blocks the other half and if we were in an airplane, we could see the entire circle.

Set that aside.

When we look at the electromagnetic spectrum, we "see" blue on one side (and if we continue that way, ultraviolet and so on with small and smaller wavelengths). On the other side, red, and then infrared with longer and longer wavelengths.

So then, when looking at the inner band of the rainbow, as blue, my son asked me: "If we were creatures that could 'perceive' the other wavelengths, what would we see at the center of the circular rainbow?"

I suppose this is tantamount to asking if there is a limit on how small a wavelength can possibly be discernible by an "imaginary" creature. However, in reality, I was left dumbstruck by the question of a 15 year old boy and have no answer.

May I ask your thoughts?

 

[Baluncore]

At the centre of the rainbow is the shadow of the observer's head.

The line from the Sun, through the observer's head, defines the axis of the rainbow. Raindrops on that line are in the shade of the head, so do not reflect sunlight.

 

[Trying2Learn]

At the centre of the rainbow is the shadow of the observer's head.

The line from the Sun, through the observer's head, defines the axis of the rainbow. Raindrops on that line are in the shade of the head, so do not reflect sunlight.

Yes, that I get, but if we turned the spectrum into a circle, what happens to the blue then ultraviolet and so on...

It seems to me there is no limit on the longer wavelengths (beyond infrared and longer and longer for a greater "radius", but there appears to be a limit on the shorter wavelengths: the center of the circle (unless I am totally confused). In other words, remove the "viewer" from the picture and just focus on the shorter wavelengths: what happens at the center?

 

[Baluncore]

The refractive index of water is wavelength dependent, which results in the angular separation of the colours. The rays of light enter the drop, then are reflected from the inside back wall of the drop. Different reflection path combinations give two or three rainbows.

The available spectrum of colours is limited, first by the transmission of light through the atmosphere, then by attenuation of light in the water drop, then by your ability to detect the reflected light with your eye.

If you image the rainbow with a monochrome CMOS camera, while changing filters to remove the IR and UV, you will see wavelengths not visible to human eyes.

 

[Trying2Learn]

Yes, I understand the refractive issue and the ability to detect the light, but that is not the issue. The issue is: is there a limit to how small the wavelength could be of the spectrum (which, couched in my son's langage is: "what if a creature COULD detect it, what would they see?"). My son looked it up and said there might be a lower limit due to Plancks constant.

 

[Baluncore]

There is not a 1:1 mapping of wavelength to angle. The spectrum is folded several times onto the radius of the rainbow.

The wavelength limit is determined by the transmission of light through the air, and through the water drop.

 

[Trying2Learn]

There is not a 1:1 mapping of wavelength to angle. The spectrum is folded several times onto the radius of the rainbow.

The wavelength limit is determined by the transmission of light through the air, and through the water drop.

Ah! I see. Thank you. Now I understand... I forgot about the double and triple rainbows folding onto each other.

 

[kuruman]

t seems to me there is no limit on the longer wavelengths

Not so fast. One has to consider the interaction of E&M radiation with water which does not necessarily result in specular reflection. Microwave ovens typically run at a frequency of 2450 MHz which corresponds to a wavelength of about 12 cm. I am sure you know what happens to water droplets inside a microwave oven.

On the short wavelength side of the spectrum, you know that x-rays mostly go through the water in the human body with more absorption in bones and teeth than flesh which makes x-ray pictures possible. How likely is it to have internal reflection of x-rays and specular back-scattering as in a visible spectrum rainbow? Never mind the frequency-dependent photoelectric effect, Compton scattering and pair production.

Your son should be commended for asking a perceptive question which, alas, does not have a simple answer.

 

[Ibix]

Ah! I see. Thank you. Now I understand... I forgot about the double and triple rainbows folding onto each other.

I'm not sure you're understanding the comment. There are double and triple rainbows, but those are due to multiple reflections in a single drop, and this isn't quite what we're talking about.

The point is that a rainbow happens because the refractive index of water for blue light is higher than that of red light, so it is bent more and the red and blue light separate. It's reasonable to guess that there's a "near infrared" stripe of the rainbow just outside the red stripe, but whether that guess is correct depends on whether the refractive index of water in near infrared is lower than in red. That turns out to be true, but once you get to 2.85$\mathrm{\mu m}$ the refractive index stops falling and starts to rise again - so wavelengths below and above 2.85$\mathrm{\mu m}$ will be bent the same amount. So you wouldn't see stripes outside that "colour" and you'd actually see multiple stripes overlapping if you could see both above and below that wavelength.

You can also find that some wavelengths are absorbed by water so you get no reflection at that wavelength.

The wiki article that lists refractive indices for water shows that the refractive index increases and decreases several times over the listed wavelength range. That means that if you could see all of that wavelength range you would see a lot of overlapping colours.

 

[renormalize]

Yes, I understand the refractive issue and the ability to detect the light, but that is not the issue. The issue is: is there a limit to how small the wavelength could be of the spectrum (which, couched in my son's langage is: "what if a creature COULD detect it, what would they see?"). My son looked it up and said there might be a lower limit due to Plancks constant.

Keep in mind that there is negligible illumination from the sun below the ultraviolet:

(Source: https://theory.labster.com/light/)
That means that any creature with eyes sensitive to UV would indeed see the rainbow with an additional ring of UV inside the blue ring that we humans can perceive. But inside of the UV ring, the creature would only see what's behind the rainbow (the landscape, sky, etc.).

Also, the formation of rainbows is explained by classical electromagnetic scattering, which doesn't involve Planck's constant.

 

[DaveC426913]

 

[A.T.]

Here are some infos and even code to simulate this effect (glory):
https://www.itp.uni-hannover.de/fileadmin/itp/emeritus/zawischa/static_html/miecolor.html

More:
https://johnedevans.wordpress.com/2019/01/27/the-physics-of-glories/
https://www.hko.gov.hk/en/education/earth-science/optical-phenomena/00348-the-mysterious-glory.html
https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1002/2014GL059650

 

[Hornbein]

Suppose all this realistic physics didn't apply, there were no limit on the angle the light can be bent, and this increased as the wavelength increased. There would then be some wavelength that focused on the middle. For wavelengths even shorter one would get a circle again, and such circles would increase in diameter as the wavelength decreased. In short, there is nothing special about the center as far as wavelengths are concerned.

 

[Drakkith]

This is actually a far, far move complicated phenomenon than most people understand. I just spend the last hour or two looking into it and it's a fairly complex issue without an easy, intuitive answer. A few of the key issues:

1. Dispersion. Even without dispersion we would get a very bright 'sunbow' of white light. It would look similar to an optical caustic in the sky, being very, very bright in the middle and then sort of fading out on both sides of the bright bow. Dispersion takes this bright bow and splits it into colored bows of much lower intensity.

2. Curvature of the droplet. The curvature of the droplet gives rise to an optical 'turning point' of sorts. Rays entering the droplet near the center, near to normal with the water's surface, end up exiting at a shallow angle. As we gradually move 'up' the droplet (further away from the optical axis) the angle of incidence upon entering and leaving both increase. Up until a point. There is a point where moving further along the droplet's surface causes the light's exiting angle of incidence to decrease. This means that there is a small region where the light stops being spread out by the droplet and is instead 'focused' somewhat. This is the reason you see a rainbow at all instead of just seeing a white mist. It allows for someone viewing the droplet from a certain angle to see a bright band of color. Without this effect there would be no preferential angle you could view a raindrop from. Every angle would look the same.

3. Angle of reflection off of the back surface. As the frequency increases, the first angle of refraction increases, which changes the position where the ray hits the back surface of the droplet which in turn changes the angle that the back surface makes with the ray and alters its final angle and path through the droplet before being refracted out. If the refractive index is high enough (possibly corresponding to a very high frequency ray) the ray ends up being reflected back very close to where it came in. However the refractive index required for this is very high and probably not possible to reach. A very rough calculation tells me you'd need n > 3 at least.

So it's a little difficult to simply say, "You'd see extreme UV at the center" or something. You might. Or you might see multiple overlapping bands.

 

[A.T.]

Suppose all this realistic physics didn't apply, there were no limit on the angle the light can be bent,

The light isn't just bent, but also internally reflected, once or twice. So you would have at least 2 different wavelengths in the center, based on the assumption that you can extend the two visible rainbows to the center.

 

[Orthoceras]

"If we were creatures that could 'perceive' the other wavelengths, what would we see at the center of the circular rainbow?"

Another way to look at that question is these two graphs. The first one shows the radius of the secondary rainbow is zero (which corresponds to the center of the rainbow) for an index of refraction of about 1.17; the radius of the primary rainbow does not tend to zero for any n. The second graph shows the refractive index of water is 1.17 for λ = 2.8, 11, and 13 µm.

 

[hutchphd]

Suppose all this realistic physics didn't apply, there were no limit on the angle the light can be bent, and this increased as the wavelength increased. There would then be some wavelength that focused on the middle. For wavelengths even shorter one would get a circle again, and such circles would increase in diameter as the wavelength decreased. In short, there is nothing special about the center as far as wavelengths are concerned.

This is not correct. I fear you misunderstand the geometry.

1. Dispersion. Even without dispersion we would get a very bright 'sunbow' of white light. It would look similar to an optical caustic in the sky, being very, very bright in the middle and then sort of fading out on both sides of the bright bow. Dispersion takes this bright bow and splits it into colored bows of much lower intensity.

This is exactly so.
It is not difficult to show for the primary bow $$sin {\theta} =\sqrt {\frac {4-n^2} 3}$$ where $n=n(\lambda)$ and theta the angle from central axis. Except for regions of anomalous dispersion n will be monotonic. Should n be near two the angle subtended by the bow gets small. For water this should give 42 deg

 

[Baluncore]

At the very centre of the rainbow is the shadow of the observer. Surrounding that is a bright white area that is due to retro-reflection by spherical droplets, the cat's eye effect.
https://en.wikipedia.org/wiki/Retroreflector#Cat's_eye
Retro-reflective paint includes small clear glass spheres. We do not notice the rainbow in white retro-reflective paint, partly because cars have two headlights, and we are looking ahead. The rainbow tinting is often visible if you look to the side.

 

[Drakkith]

@hutchphd Interesting. From your link:

Materials with an index of refraction of 2.00 or more do not produce primary rainbows. Diamond, for all its lovely optical effects, fails the primary rainbow test. Its index of refraction is 2.42. But, if rendered into microspheres, it could produce secondary and higher-order rainbows.

 

[sophiecentaur]

It seems to me there is no limit on the longer wavelengths (beyond infrared and longer and longer for a greater "radius", but there appears to be a limit on the shorter wavelengths: the center of the circle (unless I am totally confused). In other words, remove the "viewer" from the picture and just focus on the shorter wavelengths: what happens at the center?

It's not really to do with perception. There is nothing relevant about the centre of your circle to the wavelength, either. I suggest you are looking some 'pattern' between the centre and the wavelength and there is no wavelength that would give a measurable centre. The limit to the blue direction of the rainbow is set by how the water affects the spectral components of the solar radiation. Yes, a bit of UV sensitivity would give the rainbow a smaller diameter but not much. The UV would be attenuated and the refractive index of the water gets higher for short wavelengths so the equations go astray. X rays would go straight through the water surfaces. Also, the size of the droplet, for longer wavelengths would mean that simple ray optics won't work and diffraction effects would be relevant for high order bows.