Why is the Earth's shadow blue?

Question

After every sunset, once the sun is gone, I always enjoy seen the belt of venus rising in the other side, followed by the Earth's shadow.

This is a picture I took (you will find better ones online). The Pinkish-purple band on top is the Belt of Venus, and the blue band on the bottom is the Earth's shadow.

This figure from Sky & Telescope explains it well:

While contemplating these beauties, I've had hard time understanding why we see the colors we do. To my eyes, the belt of Venus looks purple, which didn't makes sense to me, as the very short wavelength of purple light should have been scattered long before arriving back there. Then all made sense when I realized that if you mix blue and red light you get purple. So the color comes from the blue mixed with the red light backscattered by the high atmosphere that is still illuminated by the Sun.

However, I can't still figure out why the Earth's shadow looks blue, and I haven't been able to find an answer (here some links about the Earth's shadow but without an explanation of the color: 1, 2, 3).

We know the sky is blue and the sunset red to due the preferential scattering of blue light by the air. But: How can the air in the Earth's shadow scatter blue light if it doesn't get any direct illumination, and any indirect illumination getting back there should be VERY depleted in blue (i.e. mostly red light)?

I thought the fact that the belt of Venus scatters red instead of blue as the rest of the atmosphere, is precisely because back there there is no more blue light left, so the weaker scattering of red light becomes noticeable (please let me know if I'm wrong there).

Or is the air actually emitting blue light in some sort of fluorescence/glowing?

PS: As this is a purely atmospheric phenomena I though it would be better suited here than in the Astronomy SE.

Answer 1

To my eyes, the belt of Venus looks purple, which didn't makes sense to me, as the very short wavelength of purple light should have been scattered long before arriving back there. Then all made sense when I realized that if you mix blue and red light you get purple.

You got that part right. Purple is not violet. Violet is a spectral color at the high frequency end of the visible portion of the spectrum, while purple is a non-spectral color on the line of purples that spans from nearly ultraviolet to nearly infrared. We see color as a wheel rather than as a line. One possible explanation is that the long wavelength ("red") cones have a secondary response in the violet end of the spectrum. Spectral violet light excites both the short wavelength and long wavelength cones, but so does a non-spectral mix of red and blue light.

And that's what you're seeing in the Belt of Venus -- a muted mix of red and blue, or a purple (or perhaps magenta, or a rosy pink).

So the color comes from the blue mixed with the red light backscattered by the high atmosphere that is still illuminated by the Sun. However, I can't still figure out why the Earth's shadow looks blue.

The backscattered red light doesn't come from the high atmosphere. It comes from the troposphere. The high atmosphere (the stratosphere and above) contains very few particulates. The upper atmosphere is the only part of the atmosphere that remains sunlit above the eastern horizon after sunset. The sunlight hitting the upper atmosphere is Rayleigh scattered, so you see that very faint blue scattered light. You don't see the backscattered reddish light from the particulate-containing lower atmosphere at the eastern horizon because that part of the atmosphere is fully in the Earth's shadow above the eastern horizon.

Answer 2

How can the air in the Earth's shadow scatter blue light if it doesn't get any direct illumination, and any indirect illumination getting back there should be VERY depleted in blue (i.e. mostly red light)?

You're completely right. If you simulate light scattering only, you'll get exactly this result: a sandy-colored sky, with a bit redder belt of Venus, and a gray shadow of the Earth:

But our stratosphere contains a small amount of ozone, which absorbs in the orange region of the spectrum in what is known as Chappuis band:

At daytime the solar light propagates at a small angle to the atmosphere, crossing a low amount of ozone, so the color of the sky is not affected (see also the question at Chemistry.SE: What exact color does ozone gas have?). But at twilight the sunlight goes through a large layer of atmosphere horizontally, thus the Chappuis absorption is very prominent.

The same simulation as above, but with ozone included (and with a bit higher exposure), will look like this:

Why is the shadow more blue (less orange) than the other parts of the sky? It's because in the shadow, there's no light from the single scattering: a ray from the Sun must be scattered at least twice to get to the earthly observer. This makes the Rayleigh scattering multiplier $\propto\lambda^{-4}$ apply twice, thus increasing the blue component of the inscattered light relatively to the red component. OTOH, the other atmosphere parts being not in the shadow, means that they do have contribution from the light scattered once, which has only a first power of this multiplier.