Somewhere Over the Rainbow

Rex Saffer the AstroDoc
10 min readSep 14, 2024

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Saturday, September 7 was a mostly cloudy day across southeastern Pennsylvania, accompanied by scattered afternoon showers, some of them fairly heavy. I was driving home from my bridge club late in the afternoon and popped into my local grocery for a few items from my basic food pyramid: cookies, Snickers and M&M’s, milk (whole of course, never reduced–fat), ice cream, salty carbs, you know, all the essentials.

As I came out into the parking lot with my purchases, I was astonished to see one of the rarest sights in the daytime sky. It was a spectacular double rainbow, spanning almost an entire semi–circle from horizon to horizon. Awestruck, I admired it for a few minutes before whipping out my smartphone and taking a half dozen pics. I was not alone; there must have been eight or ten other people doing the same. Figure 1 is the best of my images, which even so conveys only a meager taste of the splendor and magnificence of its in situ glory:

Figure 1. A Dazzling Double Rainbow

We’ll delve into the scientific details of this breathtaking work of art by Mother Nature in a little while, but to get started, let us describe some of its observable features:
1) The outer or upper part of the primary bow is red, and the inner or lower part is blue (or violet).
2) The sky and clouds inside the primary bow are significantly brighter than outside it.
3) Both bows are nearly complete semi–circles.
4) The secondary bow is much fainter than the primary bow, and the color gradient is reversed with respect to the primary bow (violet on top, red at bottom).

Next, let’s consider some basic properties of light, how it reflects and refracts, and the separation of white light into the colors of the visible spectrum when it passes through a transparent medium. We will then have a foundation to discuss the physics of rainbows in general and our double rainbow in particular.

Isaac Newton and the History of Geometric Optics

Isaac Newton is most famously known for his Theory of Uniform Gravitation, wherein every mass in the universe exerts a gravitational force on every other mass. The force is proportional to the product of the masses and inversely proportional to the square of the distance between them:

Newton’s Law of Universal Gravitation

where G is a constant, m₁ and m₂ are the masses, and r is the separation between them. In plain language, if one of the masses is doubled, the force is doubled. If the separation between the masses is doubled, the force is reduced by a factor of four.

Gravitation is the foundation of Newton’s first major published work, Philosophiae Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy, 1687). But that is not the topic of this essay, and we need not pursue any of its details further. That’s a good thing, because the Principia is completely unreadable by most of us and largely so by the rest of us.

Newton’s second major work, Opticks (yes, with a k), was published in 1704, just after his election to the Presidency of the Royal Society. Newton also held the Lucasian Chair of Mathematics at the University of Cambridge. This is the same professorship held by Steven Hawking until his death in 2018. The Royal Society is the oldest continuously existing scientific academy in the world and, arguably, remains to this day as the most prestigious of all scientific fellowships.

In contrast with the Principia, Opticks is one of the most readable of all the great classics of physical science. In it, Newton presents a comprehensive survey of early 18th century knowledge of light and describes his experiments with spectroscopy, colors, lenses, reflection, refraction, and more, in language lay readers can still follow. (Disclaimer: I lifted this bit from a sirisaacnewton.info website.)

The Nature of Light

Newton thought of light as a stream of particles, but in the late 17th century Dutch physicist Christiaan Huygens proposed that light was made up of so–called transverse waves that vibrate perpendicularly to the direction of motion. Widespread experimental confirmation soon followed, but it was almost two hundred years later before it received powerful support in James Maxwell’s unified theory of electromagnetic radiation, where the electric and magnetic forces were shown to be manifestations of a single underlying force of nature.

This view persisted until the early 20th century, when quantum mechanical theory and experimental observations not only restored an understanding of light as a particle but showed that light could be considered to be both a particle and a wave! Which one is observed depends on the type of experiment being conducted; look for a particle and you’ll find a particle, look for a wave and you’ll find a wave. Again, we need not go into further details of this counterintuitive paradox.

The Basis of Geometric Optics

The reflection of light from flat or modestly curved shiny surfaces is familiar to all of us — just look in a mirror:

Figure 2. The Hubble Space Telescope’s Primary Mirror

The refraction of light may not be nearly as well known to some of us, until we consider the passage of a ray of white light through a glass prism:

Figure 3. The Reflection and Refraction of Light

This is nothing new — all of the qualitative physics was known by Newton and published in Opticks. The direction of propagation of the white light ray is changed, but the ray is also spread out into the colors of the visible spectrum. This is due to a physical property called dispersion, where the amount or angle of refraction depends on color. We are now in a position to discuss how a rainbow is produced in the first place, and why it appears the way it does in the sky.

Raindrops Keep Fallin’ On My Head

1) Shortly after a local rain shower, small raindrops are still suspended in the air and slowly falling. When a ray of sunlight hits a drop at just the right angle, the ray is first refracted at the upper interface from air into water and disperses into its component colors. Blue light is refracted more strongly than red light, so red is initially at the top and violet at the bottom.

Figure 4. Refraction With a Single Internal Reflection

The refracted rays next internally reflect at the rear surface of the drop, then they refract a second time at the lower interface and emerge from water into air. The internal reflection reverses the ordering of the colors so that when the rays emerge, violet is now at the top and red at bottom, making angles with respect to the horizontal of 40° (violet) and 42° (red).

But hold on there! If violet is on top and red at bottom as the rays emerge from the drop, why is the upper part of the primary bow red and the lower part violet??

Consider a vertical sheet of raindrops and the differential refraction of red and violet rays as seen by an observer looking at the top and bottom of the sheet:

Figure 5. Color Variation with Line of Sight

When the observer looks at the top of the sheet, the line of sight preferentially is along red rays emerging from the drops at the steeper 42° angle, while violet rays at the shallower 40° angle pass above the line of sight and are not observed. Lower down, the effect is reversed, with the light of sight along violet rays and red rays passing below. So as observed, in the primary bow red is on top, violet at bottom! The Figure also shows that for a rainbow to be visible at all, the sun must be behind us.

2) What about the brightness difference between the sky below the primary bow and the sky above it?

Figure 6. Rays Refracted Below the Primary Bow

Consider any ray entering the drop other than the one that produces the primary bow. The green color is arbitrary and represents any of the dispersed rays. Most of these rays will lie below the primary bow ray and will be refracted and reflected such that the emerging ray is at a shallower angle than any of the dispersed rays forming the bow. As Figure 5 above shows, we must look lower down to observe shallower rays, so most of these will appear below the bow. There are many more of these than any that might enter the drop above the bow ray, resulting in increased brightness below the bow.

3) Why does our rainbow appear as a nearly complete semi–circle? If we were high enough above the ground, a rainbow would actually appear as a complete circle:

Figure 7. A 360° Bow Seen From High Above the Ground

Recall that the Sun is behind us, so that the center of the circle, called the Anti–Solar point, lies on the extension of a straight line from the Sun passing through our heads. The lower we are with respect to the ground, or equivalently, the higher the Sun is in the sky behind us, the lower the Anti–Solar point is with respect to the horizon in front of us.

So, when we observe rainbows from the ground, a full semi–circle can only occur when the Sun is just above the horizon, near either sunrise or sunset. At any other time of day, the Sun is higher in the sky, the Anti–Solar point is lower (below the horizon), and we can observe only part of a full semi–circle.

Figure 8. Double Rainbow Over Monterey Bay

4) Finally, what about the secondary bow in a double rainbow? It is produced by a similar mechanism to the primary bow, but there are two internal reflections instead of just one. For this to happen, the incident solar ray must enter a raindrop below its midpoint, as shown in Figure 9. The image is a graphic, not a photograph:

Figure 9. Refraction With a Double Internal Reflection

The second reflection reinverts the color gradient of the dispersed rays so that red rays now emerge on top and violet rays at bottom. Referring to Figure 5, this is why the secondary bow colors are seen by an observer to be violet (steeper) at the outer edge of the bow and red (shallower) at the inner edge. Further, since only a small fraction of the rays incident on the drop are low enough to produce a double internal reflection and refract into the observer’s line of sight at a suitable angle, the secondary bow is much fainter than the primary bow, by about a factor of ten.

Closing Thoughts

The heavens declare the glory of God;
and the firmament sheweth his handywork.
Day unto day uttereth speech,
and night unto night sheweth knowledge.
There is no speech nor language,
where their voice is not heard.
Their line is gone out through all the earth,
and their words to the end of the world.
In them hath he set a tabernacle for the sun,
Which is as a bridegroom coming out of his chamber,
and rejoiceth as a strong man to run a race.
His going forth is from the end of the heaven,
and his circuit unto the ends of it:
and there is nothing hid from the heat thereof.
Psalm 19:1–6 (King James Version)

I am a scientist specializing in astrophysics, and perpetually grateful for that. It gives me great joy to be able to understand something of the glory of the heavens and the firmament, as the Psalmist so eloquently proclaims, and to share that with others. But if my profession has taught me nothing else, it is that none of us know much at all about anything. I used to tell my students,

“In high school, I learned a little about a lot. In college, my knowledge became significantly narrower and deeper. In graduate school, even more so. As a Postdoctoral Fellow, I found myself knowing more and more about less and less. And now that I am a Professor, I know everything about nothing.”

But even if I knew nothing at all about the physics of rainbows, I could still be rendered temporarily immobile, mute, awe–full, and mesmerized at the sight of one in all its splendor. Our contemporary use of the word “awesome” has become a pale shadow stripped of its intended power. The primary definition of awe is something like, “An overwhelming feeling of reverence, admiration, fear, etc., produced by that which is grand, sublime, extremely powerful, or the like.” It is synonymous with adoration, wonder, veneration, worship, and so on.

The natural world is full of such wonders, endlessly so, and all we have to do to experience them is to raise our eyes from the handheld screens that enslave us and take an attentive look around. The more I do so, the more I become convinced that there is a pervasive order to the universe, an underlying unifying principle, so that we are inescapably woven into its fabric and with each other. Some identify this order and principle with God. Through evolution, the universe has made us into conscious creatures that somehow are able to interpret the mathematical language of the tapestry of existence and to unlock its grand design, at least to some extent.

On the other hand, there is much we do not know, much more even than what we do know, and which we may never be able to know completely. This suits me just fine. A world without mystery would not interest me at all.

Miraculously, we have become the eyes and ears of creation, the means by which the universe comes to know and understand itself. Quite literally, we are the world, and we should act that way, toward the world itself and all its creatures, especially our fellow humans. Nothing can divide us from us but ourselves.

All the best,
From Broomall, PA
On Friday, September 13
Rex

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Rex Saffer the AstroDoc

Retired Physics Professor, Motorcyclist, Bridge Player, Voracious Reader, Philosopher, Essayist, Science/Culture Utility Infielder