B.2.1 Conservation of energy in Earth’s atmosphere

Energy Balance: Emissivity and Albedo

The Earth’s energy balance is a delicate equilibrium between the energy it receives from the Sun and the energy it radiates back into space.

Two key concepts—emissivity and albedo—play critical roles in this balance.

Emissivity: How Efficiently a Surface Radiates Energy

1. A black body is a perfect emitter with an emissivity of 1, meaning it radiates the maximum possible energy at a given temperature.
2. Real surfaces have emissivity values between 0 and 1.
3. The formula for emissivity is:

$$e=\frac{\text{Power radiated per unit area}}{\sigma T^4}$$

Where:

• $e$ is the emissivity.
• $\sigma$ is the Stefan-Boltzmann constant ($5.67\times {10}^{-8}{\text{W m}}^{-2}{\text{K}}^{-4}$).
• $T$ is the absolute temperature of the surface in Kelvin.

How Emissivity Affects Radiation

• High Emissivity: Surfaces with emissivity close to 1, like black bodies, radiate energy efficiently.
• Low Emissivity: Surfaces with emissivity near 0, like polished metals, radiate much less energy.

Emissivity in the Real World

Different surfaces have varying emissivity values:

• Oceans: High emissivity (~0.8), radiating energy efficiently.
• Ice: Low emissivity (~0.1), radiating less energy.

Albedo: How Much Energy is Reflected

1. It is the fraction of incoming radiation that is reflected back into space.
2. The formula for albedo is:

$$\alpha =\frac{\text{Total scattered power}}{\text{Total incident power}}$$

Albedo values range from 0 (no reflection) to 1 (all radiation is reflected).

How Albedo Affects Energy Balance

• High Albedo: Surfaces like snow and ice reflect most of the incoming radiation, contributing to cooler temperatures.
• Low Albedo: Surfaces like oceans and forests absorb more radiation, contributing to warmer temperatures.

Factors Influencing Albedo

• Surface Type: Snow and ice have high albedo, while forests and oceans have low albedo.
• Cloud Cover: Clouds increase albedo by reflecting sunlight.
• Angle of Incidence: Radiation striking a surface at a shallow angle is more likely to be reflected.

Relationship Between Emissivity and Albedo

For a surface that only absorbs or reflects radiation (without transmitting it), the sum of emissivity and albedo is 1:

$$e+\alpha =1$$

This relationship highlights the balance between absorption, emission, and reflection of energy.

Solar Constant ($S$): The Sun’s Energy Reaching Earth

Its value is approximately 1,400 W m⁻².

Calculating the Solar Constant

1. The Sun emits a total power $P=3.9\times {10}^{26}\text{W}$.
2. To find the intensity of solar radiation at Earth, imagine a sphere with radius $d$, the average distance from the Sun to Earth ($1.5\times {10}^{11}\text{m}$).
3. The intensity $I$ at this distance is:

$$I=\frac{P}{4\pi d^2}$$

Substituting the values gives:

$$I=\frac{3.9\times {10}^{26}}{4\pi (1.5\times {10}^{11}{)}^2}\approx 1,400{\text{W m}}^{-2}$$

This is the solar constant.

Average Intensity on Earth’s Surface

1. Since Earth is a rotating sphere, the solar energy is distributed over its entire surface.
2. To find the average intensity received by Earth, divide the power passing through the cross-sectional area ($\pi R^2$) by the total surface area ($4\pi R^2$):

$$I_{\text{avg}}=\frac{S}{4}$$

This accounts for day-night cycles and varying angles of sunlight.

Energy Balance and Equilibrium Temperature

Energy Balance Equation

• For Earth to maintain a stable average temperature, the energy it absorbs must equal the energy it radiates back into space.
• The absorbed energy per unit area is:

$$I_{\text{in}}=(1-\alpha )\frac{S}{4}$$

The radiated energy per unit area, assuming Earth behaves as a black body, is:

$$I_{\text{out}}=\sigma T^4$$

At equilibrium, these two intensities are equal:

$$(1-\alpha )\frac{S}{4}=\sigma T^4$$

Calculating Earth’s Equilibrium Temperature

Solving for $T$ gives:

$$T={\left(\frac{(1-\alpha )\frac{S}{4}}{\sigma }\right)}^{1/4}$$

Reflection and Broader Implications

Understanding emissivity, albedo, and the solar constant is crucial for studying Earth’s climate and energy balance.