The Curvature of the Twilight Wedge
Updated: Feb 14, 2021
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In January 2010 I shot a 260-degree panorama at dawn from the summit of 14,265-foot Quandary Peak. Quandary Peak Panorama features a spectacular twilight wedge, the blue shadow of the Earth projected onto the western sky directly opposite the sun just before sunrise. Above the band of blue lies a swath of pink. The pink band occurs because light from the rising sun, which is still just barely below the horizon, takes a tremendously long path through the atmosphere, during which Rayleigh scattering causes the blue light to scatter out of the beam. The pink light that remains is finally scattered in its turn and returns to your eye. The shadow is blue because no direct light is reaching that part of the atmosphere. The only light you see coming from the direction of the shadow is the blue light scattered from the beam of direct sunlight reaching the atmosphere above.
As I examined the completed panorama, I noticed that the top of Earth's shadow was curved when compared to the straight horizon. Why? Could it be that I was actually seeing proof that the Earth was round? The shadow of the Earth, if projected onto a hypothetical flat “screen” positioned perpendicular to my line of sight, would appear circular. Could it be I was seeing just the edge of the circular shadow of the Earth rising above the straight horizon? I decided to find out.
In my initial reasoning, it seemed that the only way that the upper edge of Earth's shadow could appear curved, while the physical horizon appeared straight, would be if the light from the shadow was coming to my eyes from a much greater distance than the horizon. Any spherical object will appear to have a more pronounced curvature if viewed from a greater distance. The horizon is still curved when seen from a Kansas cornfield, but the deviation from a straight line is absolutely imperceptible. Seen from Earth orbit, however, the curve of the horizon is unmistakable. After much head-scratching and trigonometry, I concluded that the "screen" on which the shadow was falling was indeed much further away than the physical horizon. Granted, there’s no screen out there on which the shadow falls. There’s only air. Nonetheless, there has to be some average distance to the molecules which had scattered the light back to my eyes. If the “screen” was much farther than the physical horizon, that must mean that I had seen the curvature of the Earth in the form of Earth’s shadow. I was ecstatic.
I was also wrong. Or, to be more precise, I was right, but I am now convinced that my initial explanation was mostly wrong. Here's my best explanation now: the curvature of the top of the twilight wedge is indeed evidence that Earth is a sphere, but not primarily for the reasons I supposed. Imagine that you are on top of a mast on a sailing ship on a calm day in the middle of the ocean. The distance to your horizon is the same in every direction. That means that the distance through the atmosphere to outer space as you look at the horizon is also roughly the same in every direction. (If you’re looking straight up, the distance through the atmosphere is very different.) Since the distance through the atmosphere is the same looking anywhere along the horizon, the scattering properties of the atmosphere are also essentially the same. This means that the “screen” on which Earth’s shadow is projected just before sunrise is actually more like the inside surface of a cylinder, set perpendicular to your line of sight, rather than a flat plane. However, if you project a straight line onto a curved surface, such as the inside of a cylinder, the line will appear curved unless the observer is in the plane defined by that curved line. Since we observe the twilight wedge when the sun is just below the horizon, we are below that plane and therefore looking up at the upper edge of Earth’s shadow projected onto a curved surface. The primary reason that the top of Earth's shadow appears curved is the projection of a nearly straight line onto a curved surface.
Does the curvature of the horizon contribute directly and significantly to the curvature of the top of the twilight wedge? Is my original explanation at least part of the answer? Let's take a closer look.
The distance from the summit of 14,259-foot Longs Peak to the horizon as you look east toward the plains 9,000 feet below is about 125 miles. Even at that distance, the horizon’s deviation from a straight line is imperceptible to the human eye, in part because you have nothing perfectly straight to compare it to. Over a 90-degree field of view (measured horizontally) the "bulge" of the horizon above a straight line would only measure about 0.5 degrees ‒ too small to be noticeable. However, I measured the maximum height of the twilight wedge at the moment I shot the photo at only about 2.5 degrees. If you are viewing the twilight wedge by looking west at sunrise from the summit of Longs Peak, that very slightly curved eastern horizon is the geographic feature that creates the Earth's shadow. The curvature of the eastern horizon does contribute directly to the curvature of the top of the twilight wedge, but is only a small fraction of the cause.
This is the best explanation I can give right now for the curvature of the twilight wedge. I'd like to thank professional astronomer Roger Clark for pointing me in the right direction, but I also have to point out that the analysis of the curvature of the horizon in the paragraph above is mine, and only I deserve the blame if it's wrong. Are there any experts among my readers who can confirm or refute my explanation? If so, I'd love to hear from you!
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