# Blue Skies…

I always like an excuse to look for blue skies, and to have others look too, even in the worst of times. I had a great reason to do it today.

Early this morning before sunrise I settled down to write my ten o’clock lecture for my Electricity and Magnetism class. On Tuesday I had ended with a computation that is the essence of the reason the sky is blue, which is a nice enough thing to talk about, but today I wanted to go more in depth on the whole thing, and show that you can in a few steps show that the blueness has a particular pattern to it. I wrote out the final equations in a few steps and looked at them for a moment or two and realized that with the sun rising at that very moment, it was the perfect situation to have! So I went outside to enjoy the beautiful Autumn day and the beauty there is in seeing an equation writ large in the sky – and it really was all there.

It is particularly at times like this that one remembers why it is that it is hard not to just love Physics! (I hope you’ll forgive my unashamed love of what I do.)

Here is the sky I saw, looking toward sunrise, and directly in the opposite direction:

Oh, you’re wondering what I am talking about..? Why is the sky blue? What pattern in the sky? Thanks for asking! There are two things I’m looking at. First, light from the sun coming to earth and passing through our atmosphere, scattering off the air molecules gets more strongly scattered (jogged from its original path) the higher frequency it is. (Red is lower frequency, blue is higher.) The effect – “Rayleigh scattering” – goes like the fourth power of the frequency and so it is pretty striking an effect. So blue light is scattered more and so the sky is preferentially blue since that is coming from more directions (due to the scattering) than red. Really?, you say. Yes, Rayleigh. (Actually, a number of other scientists figured out some of the story, not just Rayleigh…) Second, the amount of scattering per basic unit of patch of sky (solid angle we call it) is greatest along the line of light from the sun – looking toward or away – and least at 90 degrees to the incident light’s direction. Upshot? Reddish-yellow to light blue near the sunrise or in the opposite piece of sky to it. Look perpendicular to that direction however and you see the blue is strongest (deepest) there. You can see the pattern at other times of day too, but when the sun is low it is particularly dramatic to see if you have a big enough piece of sky. (You might ask why blue and why not indigo or violet which are actually of even higher frequengy than blue… this now gets into discussing what colours our eyes are more sensitive to… a longer story.)

Here is the key equation, in case you are curious to see how it looks (look in any textbook on E&M for the details):

$$\frac{d\sigma}{d\Omega}=\frac{C}{\lambda^4} \left(1+\cos^2\theta\right).$$

…And yes, I did get the students of the graduate class out there before the lecture noting down their observations of the sky before we began the lecture. I’m funny like that…

In short, just look up anytime and you’ll see: the sky gets bluer as you move away from the sun but then it gets lighter again if you go far enough. (See the pictures I took for you this morning, above.)

And yes, indeed the same scattering business is responsible for nice red sunsets and sunrises (when the sun is low there is more atmosphere involved in the scattering), and why dust or smoke (say from local pollutants, or for extra drama, from a large brush fire, or a big volcano eruption somewhere) makes them even more dramatic…

-cvj

P.S. As a bonus, here are the lyrics of one of my favourites, Irving Berlin’s Blue Skies (go out and buy Cassandra Wilson’s album of the same name for an utterly brilliant musical experience – as a bonus, her version of the song is wonderful):

I was blue, just as blue as I could be
Ev’ry day was a cloudy day for me
Then good luck came a-knocking at my door
Skies were gray but they’re not gray anymore

Blue skies
Smiling at me
Nothing but blue skies
Do I see

Bluebirds
Singing a song
Nothing but bluebirds
All day long

Never saw the sun shining so bright
Never saw things going so right
Noticing the days hurrying by
When you’re in love, my how they fly

Blue days
All of them gone
Nothing but blue skies
From now on

[2]
I should care if the wind blows east or west
I should fret if the worst looks like the best
I should mind if they say it can’t be true
I should smile, that’s exactly what I do