Introduction
Now that we have a warm and fuzzy for solar radiation coming from the sun, lets look at what happens when sunlight reaches our atmosphere. If Earth didn’t have an atmosphere, the sun would always appear white (but nobody would be alive to see it). However, everybody who can see knows that sunlight looks different based on whether it’s sunrise, sunset, or somewhere in between.
Topics of this Sky Dive include
- Atmospheric Window
- Absorption
- Scattering
- Reflection
- Emission
- Kirchoff’s Law
- Beer’s Law
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Updated: 2019-08-26
Radiative Transfer
Particles in the atmosphere – such as smoke, dust, or oxygen – interact with sunlight (i.e. photon’s or radiation). Absorption, reflection, and trasmissivity must always equal one because energy cannot be created nor destroyed, only transferred. The main equation describes what happens to a beam of light passing through the atmosphere is:
dI_\lambda = -I_\lambda k_\lambda \rho r ds \\~\\where
- dI_\lambda =Change of Irradiance (light)
- \lambda = Wavelength
- I = Irradiance
- k_\lambda = Absorption or Scattering Efficiency
- \rho = Density of air
- r = Mass of absorbing/scattering gas per unit mass of air
- ds = Change in path length
The extinction coefficient K_\lambda = k_\lambda \rho r . Extinction coefficients are additive for gasses as well as absorption and scattering.
This Sky Dive will cover the concepts mentioned above, starting with the atmospheric window.
Earth’s Atmospheric Window
A single window by itself isn’t the easiest way to understand Earth’s atmospheric window. Imagine a row of windows where each wavelength band has its own window. For example, one window represents all x-ray wavelengths, one for visible light, one for radio waves, and so on. Some windows are fully closed and don’t let in radiation. Some are wide open, and the others are somewhere between fully open and closed.
In the image above, opacity (blockage) of 100% means that no radiation of the respective wavelength reaches Earth’s surface. Nearly all of the dangerous radiation at wavelengths shorter than visible light (to the left in the figure) are blocked by our atmosphere. Think of x-rays and gamma rays as having their windows closed. Ultraviolet wavelengths (slightly shorter wavelengths than visible light) are nearly all blocked which means the window is slightly cracked open. Almost all visible light reaches Earth’s surface, so we can say the window is mostly open. I think you are starting to get the idea. Wavelengths slightly longer than visible light are in various stages of open, and the window is wide open for radio waves.
The next logical question to ask is, why? Why is the x-ray window completely closed, but the radio wave window is completely open?
Absorption of Sunlight
The reason for the varying amount of opacity (blockage) is that molecules in our atmosphere absorb radiation. Are you ready for a very brief discussion about quantum mechanics? Too bad, here it comes.
There are four main methods a molecule can change energy states. They can rotate, vibrate, move (i.e. translate, or have kinetic motion) and electrons can jump to a higher energy, or orbital, level (Wallace and Hobbs 2006). A photon can even break the chemical bond of a molecule. If a photon doesn’t meet the change in energy required to excite a molecule (change its energy state), then the photon passes by and continues its journey to Earth’s surface. How do we know what wavelengths of radiation (I use radiation and photons interchangeably) change a molecules energy level? Absorption spectra.
Each molecule has what we call absorption bands (or absorption spectra). These bands (spectra) are wavelengths of radiation that are absorbed by a molecule. Each spike in the absorption band shows where radiation provides enough energy to allow a molecule to jump from one energy level to another one. Below you will see absorption spectra of the main absorbing molecules in Earth’s atmosphere (molecule absorption spectra are normalized for comparison purposes).
As you can see in the figure above, water vapor absorbs photons in wide swaths of the electromagnetic spectrum. The scale is cut off, but water vapor continues to absorb well into radio wavelengths. Oxygen and ozone absorb ultraviolet radiation, and carbon dioxide primarily absorbs radiation in the infrared wavelengths.
Now that we have covered absorption, it’s time to talk about scattering.
Scattering
The atmosphere is full of aerosols (e.g. dust, smoke, cloud droplets, etc.) which can scatter photons (sunlight). Scattering efficiency depends on particle size where aerosols/particles can be thought of as tiny spheres. Three main regimes describe scattering based on aerosol/particle size and wavelength of radiation (Wallace and Hobbs 2006):
- Rayleigh Scattering
- Mei Scattering
- Geometric Optics
When aerosols are really small, they scatter according to the Rayleigh scattering regime and their scattering efficiency is equivalent to \lambda^{-4} where \lambda = wavelength. Scattering is divided evenly between forward and backward directions from the aerosol. When scattering efficiency is above 1, radiation is primarily scattered forward from the aerosol. Mei scattering is for intermediate sized aerosols/molecules with a scattering efficiency oscillating around 2. Geometric Optics is for large aerosols/molecules which have slightly oscillating scattering efficiencies a little above 2 (Wallace and Hobbs 2006).
Color of the Sky
Well, that was a lot of information. Is it useful? We can use this information to explain why the sky is blue during midday, and sometimes red/orange near sunrise and sunset. Blue light has a wavelength of roughly 0.45 micrometers and red light is about 0.63 micrometers. Using the equation above (scattering efficiency = \lambda^{-4}) we can see that the scattering efficiency of blue light (24.39) is about 4 times that of red light (6.35). This explains why the sky is blue. Blue light is scattered more efficiently which allows us to see it in the sky.
In the evening, light has to travel through a longer path to reach Earth’s surface. If there are sufficiency aerosols (such as dust or smoke), the blue end of the visible spectrum can be completely scattered which only leaves shorter wavelengths, such as red, for our eyes to see.
Have you ever wondered about why clouds look white? Cloud droplets are a similar size as the wavelength of visible light. Since droplets are a similar size, they scatter all visible wavelengths. We learned previously that when all of the visible wavelengths are available we get the color white. But why can clouds look very dark and almost black? This occurs when tall/deep clouds scatter nearly all of the incoming sunlight.
Reflection
Cloud droplets are the main mechanism of reflection in the atmosphere. Clouds are made up of enormous amounts of tiny cloud droplets (~ 20 micrometers, about the width of a single cotton fiber) that are much smaller than raindrops (~ 2000 micrometers, thickness of a Nickle). A droplet does not simply reflect all radiation that it intercepts. Instead, a cloud droplet results in diffuse reflection. This means light is reflected at multiple angles.
Emission
We have already discussed how molecules can absorb radiation, but they also emit radiation at the same wavelength (i.e. absorb and emit radiation at 500 nm). This concept is one of Kirchoff’s Law’s which states that emissivity must equal absorptivity at a specific wavelength.
The reason carbon dioxide is discussed heavily in climate change is because it absorbs radiation in thermal infrared wavelengths (we know this from carbon dioxides absorption spectra). This also means that carbon dioxide radiates infrared energy equally in all directions. Some of the thermal infrared radiation emitted by Earth’s surface that would have escaped our atmosphere is instead redirected back to earth. Water vapor absorbs over a wider range of the thermal infrared wavelengths. A warmer atmosphere can hold more water vapor thanks to the Clausius Clapeyron equation. An increase in carbon dioxide concentration allows for an increase of total water vapor (due to increased air temperature) which can further increase air temperature. This process keeps feeding/reinforcing itself. The sciency term is a positive feedback.
Without the natural greenhouse effect, Earth’s mean near-surface air temperature would be uninhabitable for humans (~0 degrees Fahrenheit).
Transmissivity
But wait! What about transmissivity!? Remember our original equation?
dI_\lambda = -I_\lambda k_\lambda \rho r ds \\~\\Now you do. Transmissivity is simply the amount of radiation/photons that pass by a molecule/aerosol. To figure this out we can use Beer’s Law.
MATH ALERT!
Even though it sounds like one equation, Beer’s Law is three main equations. We start with in initial equation and convert ds (slanted path) to dz (straight path).
ds = sec \;\theta \; dz \\~\\Where:
- sec = secant (1/cosine) = Hypotenuse/Adjacent
- \theta = theta = zenith (sun) angle
- dz = Change in z, or height
The equation then becomes:
dI_\lambda = -I_\lambda \; k_\lambda \; \rho r \; sec \;\theta \: dz \\~\\Now we rearrange
\frac {1}{I_\lambda}dI_\lambda = -k_\lambda \; \rho r \; sec \;\theta \: dz \\~\\And integrate both sides
lnI_{\lambda \infty} - lnI_\lambda = sec \; \theta \int_z^\infty k_\lambda \rho r \: dz \\~\\Then take the antilog of both sides
I_\lambda = I_{\lambda \infty} e^{-sec \; \theta \int_z^\infty k_\lambda \rho r \: dz} \\~\\Substituting
\tau_\lambda = \int_z^\infty k_\lambda \rho r \: dz\\~\\yields
I_\lambda = I_{\lambda \infty} e^{- \tau_{\lambda} \; sec \; \theta}\\~\\Now we have something! Transmission is defined as:
T_\lambda = e^{- \tau_{\lambda} \; sec \; \theta} \\~\\\tau_\lambda is optical thickness for a particular wavelength. Optical thickness can be thought of as the total reduction of a radiation beam as it passes through a gas (Wallace and Hobbs 2006). These equations can be used to calculate how much a beam of radiation (at a specific wavelength) is reduced by scattering and absorption as it travels through the atmosphere.
Next Step
Now that sunlight has made its way to Earth’s surface, our next Sky Dive will look at what happens at the surface.
