B2-1. Albedo and Emissivity

1. Emissivity

While stars are good approximations of a black body, planets are not. The degree to which an object acts like a perfect black body is quantified using its emissivity.

Emissivity ($e$):

The ratio of the power radiated per unit area by a surface compared to that of a perfect black body at the exact same temperature.

$$e = \dfrac{\text{power radiated by an object}}{\text{power emitted by a black body}}$$

When applying the Stefan-Boltzmann law to an object that is not a perfect black body, the formula is modified by the emissivity factor:

$$P = e \sigma A T^4$$
  • $P$: Total power emitted by the object (W)
  • $e$: Emissivity (ranging from $0$ to $1$)
  • $\sigma$: Stefan-Boltzmann constant ($5.67 \times 10^{-8}$ W m⁻² K⁻⁴)
  • $A$: Total surface area of the object (m²)
  • $T$: Absolute temperature (K)

Exam Tip: You will be expected to remember that a perfect black body has an emissivity of exactly $1$. This is not explicitly stated in the data booklet!

2. Albedo

Albedo defines how much incoming radiation a surface reflects back into space.

Albedo ($a$):

The ratio of the total scattered (reflected) power to the total incident power of radiation arriving at a surface.

$$a = \dfrac{\text{total scattered power}}{\text{total incident power}}$$
  • Earth's average albedo is generally taken to be $0.3$, meaning $30\%$ of the Sun's rays that reach the ground are reflected back into the atmosphere.
  • An albedo of $1$ represents a surface that reflects $100\%$ of incident radiation.
  • Albedo has no units because it is a simple ratio of powers.

Factors affecting Earth's Albedo:

  • Cloud formations and seasons: Thicker cloud cover drastically increases reflection.
  • Terrain: Fresh snow has a very high albedo ($0.85$), ocean ice is around $0.60$, whereas dark asphalt is very low ($0.04$).

Example 1: Snow Reflection Ratio

Problem: The average albedo of fresh snow is $0.85$. Calculate the ratio of the energy absorbed by fresh snow to the energy reflected by it.

Solution:

  • Step 1: Define albedo. Since $a = 0.85$, the proportion of energy reflected is $0.85$.
  • Step 2: Find the proportion absorbed. If $85\%$ is reflected, then $100\% - 85\% = 15\%$ must be absorbed. The absorbed proportion is $0.15$.
  • Step 3: Calculate the ratio.
    $$\text{Ratio} = \dfrac{\text{Energy absorbed}}{\text{Energy reflected}} = \dfrac{0.15}{0.85} = \mathbf{0.18}$$