Earth's Black-Body Temperature

Hello! Welcome to your first lesson in the "Understand climate change impacts" course.

Given your extensive background in economics, statistics, and systems design, you'll find that many concepts in climate science rely on similar principles of modeling complex systems, dealing with uncertainty, and understanding equilibrium states.

This first module, "Foundational Climate Physics," lays the groundwork for the entire course. We'll start by building simple, physics-based models of the Earth's climate. These models, while simplified, are powerful tools for understanding the fundamental mechanisms that govern our planet's temperature.

Today's lesson focuses on our first learning outcome: Apply the Stefan-Boltzmann law to calculate the effective temperature of the Earth, treating it as a black-body and ignoring the atmosphere.

We will build the simplest possible climate model by treating the Earth as a single point in space, governed by a fundamental principle: energy balance.

1. Thermal Radiation and the Black Body Concept

Any object with a temperature above absolute zero ($0\text{K}$) emits energy in the form of electromagnetic radiation. This is called thermal radiation. To quantify this, physicists use an idealized model called a black body. A black body is a theoretical object that absorbs 100% of the radiation that hits it and, crucially, is a perfect emitter of thermal radiation.

The amount of energy a black body emits is determined solely by its temperature. This makes it a very useful starting point for modeling planetary temperatures.

To get a solid grasp of these concepts, please watch the beginning of the following video.

2. The Stefan-Boltzmann Law

As the video explained, the Stefan-Boltzmann law gives us a precise mathematical relationship between temperature and radiated energy for a black body. The law states that the total energy radiated per unit surface area, per unit time (i.e., the power flux), is proportional to the fourth power of the object's absolute temperature.

The formula for the emissive power ($E$), or energy flux, is:

$E=\sigma T^4$

Where:

• $E$ is the energy radiated per square meter (${\text{W/m}}^2$).
• $\sigma$ (sigma) is the Stefan-Boltzmann constant, approximately $5.67\times 10^{-8}{\text{W m}}^{-2}{\text{K}}^{-4}$.
• $T$ is the absolute temperature in Kelvin ($\text{K}$).

The total power ($P$) radiated by an object with surface area $A$ is:

$P=E\cdot A=\sigma A{T}^4$

The $T^4$ relationship is extremely important. It means that even a small increase in temperature leads to a much larger increase in radiated energy. This non-linear relationship is a cornerstone of climate physics.

3. A Simple Climate Model: Earth's Energy Balance

We can now build our first model. The core assumption is that the Earth is in thermal equilibrium. This means the energy it absorbs from the Sun is equal to the energy it radiates back into space.

Energy In = Energy Out

This is a steady-state assumption, analogous to an economic system where inflows equal outflows to maintain a stable stock.

Energy In: Solar Radiation Absorbed by Earth

1. Solar Constant ($S_0$): The Sun's energy arrives at the top of Earth's atmosphere with an intensity of about $S_0=1368{\text{W/m}}^2$. This is the solar constant.

2. Absorption Area vs. Emission Area: The Earth intercepts solar radiation over its cross-sectional area, which is a disk of area $\pi R^2$. However, it radiates energy from its entire surface, which is a sphere of area $4\pi R^2$. To find the average incoming solar radiation over the entire surface, we must divide the total intercepted energy by the total surface area. This introduces a factor of $\frac{\pi R^2}{4\pi R^2}=\frac{1}{4}$. So, the average incoming solar flux is $S_0/4$.

3. Albedo ($\alpha$): Not all incoming sunlight is absorbed. A fraction, called the albedo, is reflected back to space by clouds, ice, and other bright surfaces. Earth's average albedo is approximately $\alpha =0.3$. The fraction of energy absorbed is therefore $(1-\alpha )$.

Combining these factors, the average power absorbed by the Earth per unit area is:

$\text{Energy In}=\frac{S_0}{4}(1-\alpha )$

Energy Out: Thermal Radiation Emitted by Earth

Here, we apply the Stefan-Boltzmann law. We treat the Earth as a black body with a single effective temperature, $T_e$. The power it radiates per unit area is:

$\text{Energy Out}=\sigma T_e^4$

The Equilibrium Equation

By setting Energy In equal to Energy Out, we get our model's governing equation:

$\frac{S_0}{4}(1-\alpha )=\sigma T_e^4$

4. Calculating Earth's Effective Temperature

Now, let's use this equation to calculate $T_e$. The following readings walk through this exact calculation, providing the necessary values and steps.

Let's perform the calculation ourselves.

Given:

• $S_0=1368{\text{W/m}}^2$
• $\alpha =0.3$
• $\sigma =5.67\times 10^{-8}{\text{W m}}^{-2}{\text{K}}^{-4}$

Step 1: Calculate absorbed energy.
$\text{Energy In}=\frac{1368{\text{W/m}}^2}{4}(1-0.3)=342{\text{W/m}}^2\times 0.7=239.4{\text{W/m}}^2$

Step 2: Set absorbed energy equal to emitted energy.
$239.4{\text{W/m}}^2=\sigma T_e^4$

Step 3: Solve for $T_e$.
$T_e^4=\frac{239.4}{5.67\times 10^{-8}}\approx 4.22\times 10^9{\text{K}}^4$
$T_e=(4.22\times 10^9{)}^{1/4}\approx 255\text{K}$

To convert this to Celsius, we subtract 273.15:
$T_e=255-273.15=-18.15^{\circ }\text{C}$

This value, 255 K or -18°C, is the effective radiating temperature of the Earth. It's the temperature our planet would have if it were a simple black body with no atmosphere.

5. Practice and Interpretation

The actual observed global average surface temperature is about 288 K or 15°C.

Our simple model predicts a temperature that is about 33°C colder than reality. This large discrepancy is not a failure of the model; rather, it's a measurement of the impact of something we deliberately ignored: the atmosphere.

To test your understanding, consider this problem:

Mars has a solar constant of approximately $590{\text{W/m}}^2$ and an albedo of about $0.25$. Calculate its effective temperature $T_e$. How does it compare to Earth's?

You can follow the same three steps we used for Earth. This exercise will help solidify the relationship between solar input, albedo, and temperature. The resource from the University of Washington (95124) contains a worked example for Mars in its final section if you wish to check your answer.

Conclusion

In this lesson, we've established the foundational principles for modeling planetary climate.

Key Takeaways:

• The principle of energy balance (Energy In = Energy Out) is the starting point for all climate models.
• The Stefan-Boltzmann law ($E=\sigma T^4$) is the physical law that governs the emission of thermal energy.
• A simple model treating Earth as a black body without an atmosphere yields an effective temperature of 255 K (-18°C).
• This calculated temperature is significantly colder than the actual surface temperature, which immediately highlights the crucial warming role of the atmosphere.

This result sets the stage perfectly for our next lesson, where we will address the learning outcome: "Quantify the magnitude of the greenhouse effect by comparing the Earth's actual average surface temperature to its effective black-body temperature." We will explore why our simple model is incomplete and how the atmosphere accounts for the 33°C of warming that makes life on Earth possible.