Shape Of The Low Sun

Caribbean sunset
Click to enlarge
© The Boon Companion, 2016

Most people know why the sun looks orange-yellow when it’s rising or setting. Air preferentially scatters shorter (bluer) wavelengths of light—so the more air there is between your eye and the sun, the more short wavelengths are scattered out of the line of sight, leaving yellow/orange/red as the predominant colours reaching your eye. There’s about 38 times more air between your eye and the sun when it’s at the horizon, compared to the zenith, so it’s not surprising it looks progressively yellower the lower it is in the sky.

But what makes it change shape? The setting sun in the photograph above is only 88% as high as it is wide, and ratios down to 80% are commonly seen.

The first thing to know about the sun in that photograph is that it isn’t really there. In fact, whenever you look at the sun on the horizon, it isn’t really there. The sun is actually below the horizon, and what you’re seeing is essentially a mirage, generated by light rays curving towards your eye from beyond the geometric horizon.

Light travels slightly more slowly in denser air. The atmosphere is denser near the ground than it is at altitude. So light passing through the atmosphere is refracted—it follows a slightly curved path, concave to the denser layers of air. That means it curves around the convexity of the Earth, bringing objects into view that would be hidden below the horizon in the absence of air.Refraction at sunset, schematic

This trick of the light means that, in order to calculate the distance to the visible horizon, you need to pretend that the Earth is a little bigger than it actually is. Calculation and observation suggest that multiplying the Earth’s actual radius by 7/6 give the apparent radius produced by refraction. It also means that when objects in the sky are just beyond the curve of the geometric horizon, they are still visible, lifted above the apparent horizon by atmospheric refraction. This applies to the moon (and the stars) as well as the sun, and the tables of rising and setting times you find in newspapers and online contain several minutes allowance for the effects of refraction.

So, going back to the setting sun in the photo, the (approximate) real position of the sun is marked with the white circle below:Sunset refraction illustrated

Atmospheric refraction is lifting the lower rim of the sun into view above the horizon by an angular distance that is typically around 7/10ths of a degree. But light from the upper rim of the sun takes a trajectory with a slightly different slope through the atmosphere, and its image is lifted a little less—in my sketch, by about a twentieth of a degree less. But that slight angular difference amounts to a tenth of the apparent angular diameter of the sun, and creates the apparent  vertical flattening of its disc.

If you watch a sunset or moonset from the International Space Station, you’ll see even more dramatic flattening, because the light rays from the top and bottom edges of the disc have dipped into the atmosphere, producing the amount of flattening we’re used to seeing on the Earth’s surface, and then they’ve come back out again, experiencing a second episode of flattening. Here’s the setting moon photographed from orbit by astronaut Don Pettit:

Moonset from ISS
Image credit NASA, April 16, 2003
(From Don Pettit’s frame ISS-006-E-45773)

That’s a lot of flattening. There’s actually more than just a double dose of standard atmospheric refraction required to account for that very asymmetrical appearance.

If we go back to the original photo at the head of this post, the horizon is about 10 kilometres away. That means that light rays from the top and bottom edges of the solar disc are crossing the horizon only about 100 metres apart, vertically—although their trajectories are slightly different, they’re sampling very similar parts of the atmosphere. But in the ISS photo, the horizon (out of frame just below that flattened lunar disc) is about 2000 kilometres away. Light rays from the top and bottom edges of the lunar disc are crossing the horizon ten or twenty kilometres apart, vertically. So they’re sampling completely different parts of the atmosphere, with very different density gradients, and it’s no wonder the resulting image of the moon is very strongly distorted.

You don’t need to go into space to get that sort of dramatic distortion, however. If the lower few hundred metres of atmosphere contains layers with non-uniform density gradients (alternating bands of hotter and cooler air), then the trajectories of light rays coming from the sun will be deflected in a non-uniform way, and the neat ellipse of the sunset photo above can turn into something like this:

Sunspot mirage
Image by Brocken Inaglory
Used under Creative Commons Attribution-Share Alike 3.0 Unported licence

Notice that a single sunspot appears three times in the centre of that solar disc. Light from the sunspot is finding three different curved routes through the atmosphere that all end at the observer’s eye, coming from very slightly different directions.

There’s a critical density gradient (on Earth, corresponding to a temperature inversion of 0.11ºC/m) at which the refracted curvature of a horizontal light ray exactly matches the curvature of the Earth. In an atmosphere with this density structure, light could travel endlessly around the planet. For a while, back in the 1970s, it was thought that the planet Venus might have that sort of atmosphere. In 1975 the science fiction author John Varley wrote a short story, “In The Bowl”, set on Venus, and did a good job of describing what that might look like. (To make sense of the following, you also need to know that Venus rotates in the opposite direction to Earth, and very slowly):

I don’t like standing at the bottom of a bowl a thousand kilometers wide. That’s what you see. No matter how high you climb or how far you go, you’re still standing in the bottom of that bowl. […]
Then there’s the sun. When I was there it was nighttime, which means that the sun was a squashed ellipse hanging just above the horizon in the east, where it had set weeks and weeks ago. Don’t ask me to explain it. All I know is that the sun never sets on Venus. Never, no matter where you are. It just gets flatter and flatter and wider and wider until it oozes around to the north or south, depending on where you are, becoming a flat, bright line of light until it begins pulling itself back together in the west, where it’s going to rise in a few weeks.

But let’s go back to Earth again, and that critical value of refraction at 0.11ºC/m—this implies that, if the temperature gradient is even steeper, rising light rays can be so strongly curved by refraction that they’ll come back down again. This happens readily enough in polar regions, where cold air in contact with ice is overlain by warmer air aloft—at some critical altitude, the temperature can jump by several degrees Celsius in the space of just a few metres, creating an abrupt fall in air density.

So a polar temperature inversion can create a sort of reflective roof, allowing light to reach the observer’s eye from objects well beyond the geometric horizon. In fact, if the temperature inversion is widespread, and the terrain is flat, light rays can “bounce” several times around the curvature of the Earth on their way to the observer, bringing in distorted images from hundreds of kilometres away. If the sun moves into alignment with this “light pipe” then it can become visible despite being four degrees or more below the geometric horizon.

Novaya Zemlya Effect, schematic
Novaya Zemlya Effect, schematic (Click to enlarge)
The yellow light ray moves alternately closer to and farther from the surface of the Earth, “bouncing” off the top of a temperature inversion
(in reality, the effect plays out within a layer about 100m deep, spanning just a few degrees of Earth’s curvature)

This is called the Novaya Zemlya Effect, because it was first recorded while members of Willem Barentsz’s 1596/7 expedition were overwintering on the island of Novaya Zemlya in the Russian Arctic. Towards the end of the long polar night, on 24 January 1597, Gerrit de Veer and two other men saw a distorted image of the sun appear on the horizon two weeks before the polar night was due to end, at a time when the sun was still, geometrically, about five degrees below the horizon. This is still one of the most extreme examples of the effect ever observed. Calculations by van der Werf et al. (1.1 MB pdf) in 2003 suggest that it could have involved no less than five successive bounces from an inversion layer at about 80 metres altitude, over a distance of 400 km.

We have a drawing of what the sun looks like under these circumstances, courtesy of Fritjof Nansen, who saw an example of the Novaya Zemlya Effect during his Fram expedition to the Arctic:

Nansen's sketch of Novaya Zemlya Effect, 16 Feb 1894
Nansen’s sketch of the Novaya Zemlya Effect, 16 Feb 1894 (Click to enlarge)
From Farthest North, Vol. 1, George Newnes Ltd., 1898

Those multiple bounces along the “light pipe” completely scramble the image of the solar disc, until what remains is a stack of bright horizontal bands, all of roughly equal width.

You can get a nice impression of what it must have looked like from this beautiful video of a miraged sun, filmed for four minutes after the expected sunset time, in California:

Note:  For a lot of mathematical detail and nice informative graphs concerning sunset refraction in general and Novaya Zemlya Effect in particular, you can’t do much better than:  Van der Werf, Können & Lehn. Novaya Zemlya effect and sunsets.  Applied Optics 2003. 42: 367-78. (2MB pdf)

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