2022 August 29

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

So far we have been using temperatures without any discussion about
what temperature is or what it measures. Though the technical details of temperature are tricky, in
everyday life and for our purposes we can consider
**temperature** as a measure of how much thermal energy is
in an object. There are two important properties of temperature: adding
heat to an object causes its temperature to increase^{1}One
major exception is phase changes like ice melting or liquid water
evaporating; another of course is black holes, as mentioned
earlier., and when two objects are allowed to freely
exchange energy then heat will flow from the hotter object to the colder
object. As we discussed in part 2, if these two properties were not true
then life would not be possible.

We say an object is in **thermal equilibrium** with its
surroundings if its temperature is not changing over time. This is the
same thing as saying there is no net flow of heat in or out of the
object. What happens if an object was at thermal equilibrium but is
transiently heated up? The object is hotter than before and will
therefore lose heat faster than before; for example, through blackbody
radiation, which is greater the hotter an object is. Since the object
will now have a net flow of heat outwards, it will cool down over time.
Similarly, if an object was at thermal equilibrium and is transiently
cooled down, it will experience a net flow of heat inwards and heat up
over time.

Therefore we expect that in general an object will have a unique
**equilibrium temperature**, that is, the one temperature
at which it would be at thermal equilibrium with its surroundings. Above
this temperature, the object will cool down, and below this temperature,
the object will heat up, and at this temperature the object will be in
thermal equilibrium. The equilibrium temperature depends on the object’s
surroundings and the interactions between the object and its
surroundings. If these change, then the equilibrium temperature can also
change. For example, consider a room heated by a heater on a cold day;
the room reaches some steady temperature at which the heat it gains from
the heater is balanced by the heat lost to the outside. Opening a window
decreases the equilibrium temperature, so the room will cool off until
it approaches this new equilibrium temperature.

By analyzing the equilibria of a system we often find it much easier
to understand how the system changes over time. The most direct approach
to studying the changes in a system, which I would call the “dynamic”
method, is to determine the current state of the system and how that
state is changing. For example, if we know the current temperature of an
object, and we know how the temperature is changing, then we can predict
the future temperature of the object. Alternatively we can analyze the
equilibria of a system, which I call the “static” method. The static
method loses information about transient fluctuations or detailed
behavior of the system, but it often is better at giving an overall
understanding of the behavior of the system. The static method is also
usually easier and more robust to model error^{2}**Model error** refers to details
of the real-world system which are omitted in the mathematical model.
Regardless of how detailed and precise the model is, for a model to be
useful there will always be
some further detail that is missing..

In the specific example of studying the temperature of the Earth, the dynamic method requires knowing the current temperature of the Earth, exactly how much energy the Earth is receiving from the Sun at each point in time, and exactly how much energy the Earth is losing to space at each point of time. Knowing these three things we can calculate the temperature of the Earth at any point in the future. Of course, this is totally infeasible because, for example, the amount of sunlight reflected back into space depends on how many clouds there are and what shape they have, which rapidly changes within hours. If we tried to make predictions in this way they would become wildly inaccurate almost immediately because any small error compounds upon itself (Which is not to suggest that such predictions are impossible, just that this naive approach is not viable.).

Alternatively, using the static method we first measure how much energy the Earth is typically receiving from the Sun, and then calculate the temperature at which the Earth would emit as much energy as it is receiving. This calculation tells us the equilibrium temperature of the Earth, that is, the temperature at which it radiates the same amount of energy as it receives, so that the Earth’s temperature does not change. While this method does not explain any oscillations or fluctuations around the predicted equilibrium temperature, it does capture the most important features that are relevant to the Earth’s temperature.

We now have the pieces to understand what temperature the Earth would
have in the absence of an atmosphere. The Earth gains energy through
light received from the Sun; some of this light is reflected back to
space but most of it is absorbed by the Earth’s surface. The proportion
of light reflected back to space is called the **albedo**
of Earth, and it equals approximately 30%^{3}Of
course, if the Earth had no atmosphere it would have a significantly
different albedo, among many other major differences.. The
Earth loses energy by radiating it to space due to the blackbody effect,
that any warm object emits light; the amount of energy lost this way
depends on the Earth’s temperature. The equilibrium temperature of the
Earth is the temperature at which the energy gains from the Sun are
equal to the energy losses due to the blackbody effect.

The rate at which energy is received from the Sun is equal to

where is the radius of the Earth, is the albedo of the Earth, and is the **insolation** of the Earth
(the amount of energy in sunlight received by the Earth, per area and
per time). The reason for the factor of is that, from the perspective of the Sun,
the Earth appears to be a disc of radius , so that is the total *effective* area
illuminated by the Sun^{4}While
the surface area of the Earth is and half of that is exposed to sunlight
at any time, the amount of sunlight a location receives depends on the
angle the Sun is above the horizon, and the average illuminated location
receives half as much light as it would receive under direct, full
sunlight.. Since the albedo is the proportion of light reflected back to
space, is the proportion absorbed.

From the Stefan-Boltzmann law, the rate at which energy is radiated from the Earth due to the blackbody effect is

where again is the radius of the Earth, is the Stefan-Boltzmann constant, and is the temperature of the Earth. Here is the surface area of a sphere with radius .

At the equilibrium temperature the energy received and emitted is equal, so

If we know the values of and we can solve for :

Observe that the radius of the Earth has no effect; if the Earth were larger, it would absorb more sunlight and also emit more blackbody radiation.

If we use sensible values^{5}We
take albedo , insolation , and Stefan-Boltzmann constant . As
per a previous note, we use an emissivity . With a more realistic , we get K. for and , we compute the equilibrium temperature

or -18 C or -1 F.

The temperature of an astronomical body computed in this way, by
ignoring all atmospheric effects, is called the **effective
temperature**; specifically, we find the effective temperature of
an object by measuring how much light it emits and using the
Stefan-Boltzmann law to determine the temperature needed to emit that
much light. In fact, when we previously spoke about the temperature of
the surface of the Sun, we really meant the effective temperature of the
Sun^{6}Like the gas planets, the Sun does not have a
solid surface, but instead gradually becomes denser and more opaque
closer to the center. The “surface” is defined somewhat arbitrarily in
terms of a certain level of opacity. The exact temperature at this depth
would be difficult to measure, but is likely very close to the effective
temperature.. A distant astronomer attempting to measure
the temperature of the Earth would be measuring its effective
temperature.

The effective temperature of the Earth differs from the true temperature of the surface of the Earth in two important ways:

- The effective temperature is a single number, while the true temperature varies with location.
- The surface temperature is affected by the atmosphere, in particular the greenhouse effect.

The effective temperature can be thought of as a suitable
average^{7}Rather than the typical arithmetic mean, the
fourth root of the arithmetic mean of the fourth powers is the suitable
average. This is always warmer than the arithmetic mean, although not
significantly so for the Earth. of the temperature across
all locations. These spatial variations are important to weather and the
climate but do not directly pertain to the greenhouse effect, so we will
not discuss them here.

One major exception is phase changes like ice melting or liquid water evaporating; another of course is black holes, as mentioned earlier.↩︎

**Model error**refers to details of the real-world system which are omitted in the mathematical model. Regardless of how detailed and precise the model is, for a model to be useful there will always be some further detail that is missing.↩︎Of course, if the Earth had no atmosphere it would have a significantly different albedo, among many other major differences.↩︎

While the surface area of the Earth is and half of that is exposed to sunlight at any time, the amount of sunlight a location receives depends on the angle the Sun is above the horizon, and the average illuminated location receives half as much light as it would receive under direct, full sunlight.↩︎

We take albedo , insolation , and Stefan-Boltzmann constant . As per a previous note, we use an emissivity . With a more realistic , we get K.↩︎

Like the gas planets, the Sun does not have a solid surface, but instead gradually becomes denser and more opaque closer to the center. The “surface” is defined somewhat arbitrarily in terms of a certain level of opacity. The exact temperature at this depth would be difficult to measure, but is likely very close to the effective temperature.↩︎

Rather than the typical arithmetic mean, the fourth root of the arithmetic mean of the fourth powers is the suitable average. This is always warmer than the arithmetic mean, although not significantly so for the Earth.↩︎

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