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:

sunrise_scattering_1 sunrise_scattering_2

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):

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

…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

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19 Responses to Blue Skies…

  1. Carol says:

    Excellent. Love Cassandra Wilson’s Blue Skies!

  2. robert says:

    On high ground, with the rising sun behind you and looking out over the morning mist in the valley below – that’s when you will see the glory in all its wonder. The same underlying equations apply, though in a rather different limit; do you get to see it out in California?

  3. Rob says:

    Very nice pictures of the sky. I love it.

  4. Plato says:

    Clifford,

    Something crossed my mind as I watch your article update on my Blog listing.

    Cosmologically, we see evidence of high energy photons, in events as the Crab Nebula or M87 Jet as synchrotron radiation? Is it possible then to see this effect “from the inside?”

    Also, “the Rainbow,” quite similar here, in perspective as that you show by example?

    Please feel to correct the assumptions put here that one might find in relation to your topic?

    Oh there is definitely nothing wrong with showing feelings in this way about what we can “abstractly perceive,” can be seen in natures own way. After all it is kind of a romance with nature that can stir “such feelings.”

    Best,

  5. Clifford says:

    Plato,

    I’m sorry, I tried hard, but I’ve no clue what you’re talking about. I don’t understand the stuff in quotes, I don’t know what “from the inside?” means, I don’t know what the quotes mean, I don’t understand the connection you are making and I don’t understand if there is a question in the rest of what you wrote at all for me to answer, or ….

    -cvj

  6. Plato says:

    Sorry Clifford,

    I’m having a hard time knowing even myself what I am saying.:)

    It’s the angle of information, as your explaining the horizon, let’s say on my left and how the sun location rises to over head. Where the sky is blue?

    So it’s a “matter of perspective” where I am thinking the sun is directly overhead versus the information I might receive from those cosmological events.

    Then I switch myself to the location of the cosmological events or the sun and the angle with which this information is released. I am confusing even myself by not understanding at which location I am speaking from.

    [long repeating quote snipped out for length…cvj]

    Then for instance we place Fermi between us on earth and the cosmological event, calorimetry measure of ultra relativistic photons again points to “angle of incidence” as that information is received.

    Pop Quiz in Guardian Comment 41

    Inside, was a wrong word to use while I was not clear from which location I was speaking and how we might see photon information from the sun as it appears directly across from, this appeared to me mistakenly as looked from earth “inside the event.” I have to be clearer about this for sure and think harder about what I referring.

    Best,

  7. Supernova says:

    Also, Rayleigh scattering creates polarization, whose magnitude depends on the scattering angle. So if you have polarized sunglasses, you can observe that the blue sky is strongly polarized at about 90 degrees away from the Sun. Another fun observation to make on a clear day…

  8. Clifford says:

    Indeed… we observed and derived that too… only one student had polarized sunglasses to see the effect, but he did.

    Indeed,

    [tex]\Pi(\theta)=\frac{\sin^2\theta}{1+\cos^2\theta}[/tex]

    is written in the sky too!

    Cheers,

    -cvj

  9. kim says:

    well can you explain to us (the general public) how the equation above is written in the sky?

  10. Clifford says:

    Well… did you read the post? The big paragraph under the pictures has it all in. In the equation you can replace [tex]\lambda[/tex] on the bottom line with frequency on the top line. Then the words fit the equation more or less exactly. [tex]\theta[/tex] is the scattering angle – how much the light deviates from the direction it would have stayed on having come from the sun had it not been diverted by particles in the sky. Looking directly at the sun it is zero, directly away from the sun it is 180 degrees and half way in between is 90 degrees.

    Thanks,

    -cvj

  11. Kortney says:

    Love the scientific and colourful descriptions of the East/West views of the sky on a crisp Fall morning! Well composed and so complimented by the lyric’s of Irving Berlin. You are fortunate to have much creative energy and so little need for sleep! Enjoy the blog and all your many insights!
    Cheers!

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  13. Nige Cook says:

    “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. …

    “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.” – Clifford

    Your explanation of the first point is excellent. Your explanation of the second point by producing an equation and stating that the amount of scattering per steradian should be greater close to the radial line from sun to observer, ignores a couple of mionr points. First, the maximum blueness of the clear sky occurs vertically overhead, regardless of where the sun is. Maybe theta in your equation is the angle from the vertical?

    The reason why the blue is less deep towards the horizon is simply that the light from nearer the horizon has a larger slant distance of low-altitude air to penetrate through, and the low air contains more particles of dust. Rayleigh scattering (producing blue sky) requires small particles like air molecules, which are small in size compared to the ~0.4-0.8 micrometres wavelength of visible light. Dust particles bigger than the wavelength of the light waves will ignore Rayleigh’s law and will often scatter all wavelengths similarly, so you get more greyish light from the horizon. This insight is in Philip J. Dolan’s research compilation on atmospheric transmission of visible radiation from large explosions in the Nevada desert (pp. 3-7 and 3-8 of DNA-EM-1, 1978):

    “The sky is blue because most of the scattering at high altitudes is by air molecules … Near the surface, scattering by haze particles contributes more to the light in the air than does scattering by air molecules. The sky still appears blue, but the color is not as deep as it would be on a clearer day. Distant hills and the sky near the horizon appear to be more gray than blue, which indicates that the lower atmosphere is scattering all wavelengths of light about equally.”

    Dolan’s PDF report: http://nige.files.wordpress.com/2009/02/em1-ch-3-ada955387.pdf

  14. Nige Cook says:

    The other minor point is that <100% of sunlight is scattered along the line from sun to observer (there is some contribution of unscattered, direct solar radiation along that particular line). But exactly 100% of sunlight coming from all other directions has to be scattered (or refracted or otherwise deflected) by the atmosphere in order to reach your eye. This means that the sun itself doesn’t appear blue.

    Summary: near the horizon you get sunlight that is mainly scattered by dust particles of a few microns in diameter, so you get non-Rayleigh scattering, giving greyish white light (due to a path that is predominantly through low-altitude, dusty air). From vertically overhead, you get sunlight that is mainly scattered by tiny air molecules, hence a deep blue due to Rayleigh’s law.

  15. Nige Cook says:

    There is blue sky and sunshine here, so I’ve just had a look and it turns out that you’re right, the sky near the sun (and not just near the horizon) is lighter blue than it is at the same elevation in other directions. Maybe light coming on paths near the line from sun to observer is simply less scattered than light coming on paths further from the sun, so the blue is less deep around the sun due to the smaller amount of scattering needed to reach your eye.

  16. Clifford says:

    “…and it turns out that you’re right”

    Good Heavens!

    (Does this now mean you’ll stop writing endlessly long chains of comments of dubious accuracy? 😉 )

    Best,

    -cvj

  17. Nige Cook says:

    Sorry!

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