The greenhouse effect part 2: Physics of light and temperature

2022 August 22
the greenhouse effect
  1. Part 1: What is the greenhouse effect? An accessible, scientific introduction
  2. Appendix A: What is the atmosphere?
  3. Appendix B: Ozone
  4. Part 2: Physics of light and temperature
  5. Part 3: Temperature of the Earth without an atmosphere
  6. Part 4: A model of the greenhouse effect
  7. Part 5: Differences between model and reality


Light, also called radiation or electromagnetic radiation, is a familiar aspect of our daily lives. Light can be emitted, absorbed, reflected, transmitted, or refracted by objects, and travels in a straight line otherwise. When light strikes our eyes, it is absorbed and gives us information about the object that emitted or reflected that light, which we call “seeing” the object. Light is a type of energy.

A beam of light is made of many individual photons, which are indivisible parcels of light. While the brightness of a beam of light depends on how many photons are in it, not all photons are the same. The properties of a photon can be described with a single number, its wavelength; two photons with the same wavelength are indistinguishable1Instead of wavelength, photons are sometimes described by their frequency or energy. These are related by E = h \nu = h c / \lambda, where E is the energy of the photon, \nu is the frequency, \lambda is the wavelength, h = 6.626 \cdot 10^{-34} \text{J}\cdot\text{s} is Planck’s constant and c = 3 \cdot 10^8 \text{m}/\text{s} is the speed of light. For consistency we will only use wavelength..

The wavelength of a photon affects what objects it can interact with. For example, radio waves, which have a wavelength of 1 meter or more, can pass through walls but will interact with the antenna of a radio receiver; and the microwaves in a microwave oven, which have a wavelength of 12.2 centimeters, are easily absorbed by the water in food to heat it up but cannot pass through the small holes in the metallic screen covering the door2The interior of the microwave oven is surrounded on all sides by metal, forming a Faraday cage from which microwaves cannot escape..

In particular, certain wavelengths of light are able to interact with the light-sensitive cells (the “rod” and “cone” cells) in our eyes to produce sight; light of these wavelengths is called visible light, while light of other wavelengths cannot be seen3Except that very strong x-rays shown directly in the eye can appear faintly blue; this was discovered in 1895 before the dangers of x-rays were known.. The various wavelengths of visible light appear to our eye as different colors. Just as the chemicals in our food are perceived by us as having different tastes, the different wavelengths of visible light are perceived as different colors; and just as some chemicals are tasteless, some wavelengths of light cannot be seen at all.

While wavelength and color are closely related, the wavelength of a photon is a single number (a length) that describes its physical properties, but the color of a beam of light is a far more complicated property that depends on how light interacts with the human eye and how the brain interprets this interaction, and therefore is also slightly different from person to person. The longest wavelength a photon can have and be detectable to the human eye is about 0.7 microns4A micron is one millionth of a meter. “Micron” is short for “micrometer”, which is also written 1 \upmum. A micron is about a hundred times smaller than the thickness of a sheet of paper, and is a bit smaller than the typical bacterium. We will usually use microns to describe wavelengths of light.; light of this wavelength appears dull red. Longer wavelengths of light are infrared, microwave, and radiowaves, which are invisible to the eye. The shortest visible wavelength is around 0.39 microns, which appears deep blue or violet; shorter wavelengths are ultraviolet, x-rays, and gamma rays. Light with wavelengths between 0.39 and 0.7 microns are visible.

Rainbows are formed by separating the photons in a beam of light according to their wavelength; so if all the photons in a beam of visible light have the same wavelength, then its color will be one of the colors of the rainbow. Other colors, such as pink, magenta, or brown, can only be formed by a beam of light containing a mixture of photons of different wavelengths. Two different mixtures can appear to be the same color; for example, light at 0.58 microns (which appears yellow) cannot be made by a standard computer monitor, but a monitor can make a mixture of red and green that appears that same yellow color to the human eye. Some colors, called impossible or imaginary colors, cannot be produced by any beam of light, but can be seen for example using afterimages.

The mix of wavelengths that are in a beam of light is called the spectrum of that beam. We can graph a spectrum by showing how much energy the beam has at different wavelengths; where the spectrum is low, the beam has very few photons around that wavelength, and where the spectrum is high, the beam has many photons around that wavelength. Two simple examples of spectra are shown below.

To illustrate the usefulness of spectra for understanding light, the following figure shows real-world observations of the spectra of light from four commercially available light bulbs. The shaded region indicates visible wavelengths of light. We can see from the spectra that the incandescent and halogen bulbs emit many more photons of long wavelengths; they will appear “warmer” (that is, with tones of yellow or red). In comparison, the fluorescent and LED bulbs will appear “cooler” (more white or blue). We can also see that the incandescent and halogen produce a great deal of infrared light that is not visible to the eye. While more than 99% of the light produced by the LED is visible light, only 15% of the light produced by the incandescent bulb and 10% of the light produced by the halogen bulb is visible. This contributes to the much greater efficiency of fluorescent and LED lighting.

Spectra of real-world observations of four commercially available light bulbs. The shaded region indicates the portion of the spectrum that is visible to the eye; the other light is wasted for purposes of illumination. The spectra have been scaled so that all four contain the same amount of energy in the visible region; each of the four bulbs would have the same brightness to the eye. The LED and fluorescent lights emit almost all of their light as visible radiation, which contributes to their excellent efficiency. By contrast, the incandescent and halogen bulbs waste a great deal of energy emitting infrared light.

The simplest possible bulb, the incandescent light, works by heating an object (the filament) until it is so hot that it glows visibly due to the blackbody effect, which is discussed below. The close match of the observed spectrum of the incandescent light with the theoretical spectrum of a blackbody at a temperature of 2950 K (2667 C, 4850 F), also shown in the figure, suggests that the filament of the incandescent bulb used in the experiment had a temperature near 2950 K. The only commonly available materials with a melting point above that are carbon and tungsten; while the first light bulb filaments used carbon, since around 1906 tungsten filaments have been used almost exclusively.

The poor match of the observed spectrum of the LED and fluorescent spectra with a theoretical blackbody spectrum shows that those two bulbs use a different physical process to emit light.

The relationship between temperature and light: blackbody radiation

It has long been understood that hot objects feel hot from a distance, although a scientific explanation of this phenomenon has only come recently. This process is different from conduction, which is the movement of heat from a hot object to an object it is touching. For example, a fire in a fireplace feels warm from a distance even before the air in the room has begun to warm up, so the warmth is not due to conduction via the air. Furthermore, a person near the fireplace feels warmer on the side facing the fire even though the air on both sides is the same temperature. This also can be observed with a hot stovetop, which feels warm from the side even though rising hot air should only be felt directly above the stovetop.

This relationship is because every object emits light according to its temperature, which is called the blackbody effect; light emitted in this way is called blackbody radiation5The first person to study blackbodies was Gustav Kirchhoff in 1860, who was unable to determine the formula for blackbody radiation but called it “a problem of the highest importance”; this proved true when the discovery of the formula led to the discovery of the photon by Max Planck and Albert Einstein around 1905, for which Einstein received the Nobel Prize.. Since light is a form of energy, emitting light causes an object to cool down6With a very few exceptions; for example, black holes warm up when they lose energy., and absorbing light causes an object to warm up.

It is a necessary fact of life that the hotter an object is the more blackbody radiation it emits; so if two objects of different temperature are placed near each other then the hotter object emits more light than it absorbs, cooling down, and the cooler object absorbs more light than it emits, warming up. If this were not true and instead hotter objects emitted less blackbody radiation, then hotter objects would get hotter and hotter, while cooler objects would get cooler and cooler, until everything in the universe would be either extremely hot or extremely cold7Since black holes become colder when they absorb energy, and they start colder than their surroundings, they just get even colder over time until they approach absolute zero; for example the black hole in the center of the Milky Way, called Sagittarius A^*, is approximately 1.7 \cdot 10^{-14} K. The rest of the universe is currently about 2.7 K, so Sgr A^* is gaining energy from its surroundings and continuing to cool down. Eventually the rest of the universe will cool down until it is even colder than Sgr A^*, so Sgr A^* will start losing energy over time and warm up until it ultimately explodes..

An exact formula for the amount of light emitted by an object by the blackbody effect was found in 1879, called the Stefan-Boltzmann law. The amount of light emitted is

A \sigma T^4,

where A is the area of the object, T is its temperature relative to absolute zero (for example, using Kelvin), and \sigma is a constant8Real-world objects actually emit slightly less light than indicated by this formula; the proper formula is A \epsilon \sigma T^4 where \epsilon is the emissivity of the object, a number between 0 and 1 that depends on the material the object is made out of and the wavelength of light that we are interested in. Most objects in daily life have an emissivity around 0.9 to 1 in infrared wavelengths. The surface of the Earth has an average emissivity of about 0.96 in infrared wavelengths. For simplicity we take \epsilon = 1 from now on.. As an example, doubling the temperature of an object multiplies its emissions by 2^4 = 16. Using this law we can calculate the temperature of an object by measuring its blackbody radiation; this is how infrared thermometers work. In fact, Josef Stefan used this law to give the first accurate estimate for the temperature of the surface of the Sun, 5700 K, by comparing sunlight to light emitted from a hot object with a known temperature. The true temperature is 5772 K.

As another example, we can use this law to estimate how much light is given off by an electric stovetop. A hot stovetop (or any object) begins to visibly glow a very dull red when it reaches around 800 K (527 C, 980 F), called the Draper point. This compares to room temperature of about 300 K (27 C, 80 F). Using the Stefan-Boltzmann law, we see that the hot stovetop emits (800 / 300)^4 \approx 50 times more light than it would at room temperature. From this, one may be able to crudely estimate that you need to be within \sqrt{50} / 2 \approx 3 stovetop-lengths of the stovetop to significantly feel the light from a red-hot stovetop compared to surroundings at room temperature.

As well as knowing the total amount of blackbody radiation emitted at a specific temperature, we also know what wavelengths blackbody radiation has (that is, the radiation’s spectrum). An exact formula for how much each wavelength occurs in blackbody radiation is called Planck’s law9Discovered in 1900, and directly leading to Einstein’s prediction of the photon.. Planck’s law tells us that blackbody radiation is mostly near a specific wavelength, and that the hotter an object is the shorter that wavelength is. A room temperature object will mostly emit light near 14 microns, which is infrared light, and an object at the Draper point of 800 K will mostly emit light nearer 5 microns, which is still infrared. The Sun, however, mostly emits light of wavelengths near 0.7 microns, so it is easily visible to the eye.

Above we see a comparison of the emissions from a blackbody (such as a stovetop, which is a good approximation to a blackbody) at a temperature of 800 K against the same object with a temperature of 300 K; the object emits 50 times more light when hot. Almost all of the light emitted is infrared; the hot object emits 10 million times more infrared light than visible light, which is marked with the shaded region of the diagram. While we can’t see the infrared light at all, there is so much of it that we can feel it as heat when nearby. However, we have no trouble seeing the visible emissions of the stovetop because our eyes are tremendously sensitive at detecting visible light – so much so that laboratory tests have found that people can sometimes detect a single photon in ideal conditions.

An idealized comparison of the sunlight that reaches the Earth and of the light that the Earth emits to space is given above. While the amount of energy that enters the Earth equals the amount of energy that leaves the Earth, causing the Earth to be in balance, the wavelength of the incoming and outgoing light is different.

In the context of climate science and the greenhouse effect, the term longwave radiation is used to describe blackbody radiation emitted by the Earth (including the Earth’s surface, oceans, the atmosphere, clouds, etc.) and the term shortwave radiation is used to describe blackbody radiation emitted by the Sun. Since objects on the Earth tend to be much colder than the surface of the Sun, longwave radiation has much longer wavelength than shortwave radiation. Specifically, since objects on the Earth are about 20 times colder than the surface of the Sun, they emit radiation that has a wavelength about 20 times longer.

Because of the large gap between the wavelengths of longwave and shortwave radiation, certain chemicals will interact strongly with one type of radiation but not with the other. The most common gases in our atmosphere, oxygen and nitrogen, are transparent to both longwave and shortwave radiation. However greenhouse gases such as water vapor, carbon dioxide and methane are opaque to longwave radiation but transparent to shortwave radiation. These gases interfere with longwave radiation emitted by the Earth so that it does not cool as effectively, but do not interfere with shortwave radiation from the Sun, so they cause a net warming effect called the greenhouse effect. The details of this process will be explored in the following parts. Conversely, sulfur dioxide in the stratosphere can form sulfate particles that, like clouds, are partially reflective of sunlight and thus cool the Earth by decreasing how much shortwave radiation it receives.

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