B2-2. The Solar Constant

1. Defining the Solar Constant

Because life on Earth is entirely dependent on the Sun's energy, we must quantify exactly how much of its energy reaches the very top of our atmosphere.

The Solar Constant ($S$):

The intensity of the Sun's radiation arriving perpendicularly to the Earth's atmosphere when the Earth is at its mean distance from the Sun.

The average value of the solar constant is $1.36 \times 10^3$ W m⁻².

Why the Solar Constant Varies:

  • The Earth is in an elliptical orbit. It receives slightly more intensity when it is physically closer to the Sun and slightly less when it is further away.
  • The Sun's own energy output fluctuates by about $0.1\%$ during its $11$-year sunspot cycle.

2. Incoming Radiative Power

The solar constant ($S$) measures the intensity hitting a flat, perpendicular cross-section of the Earth. However, the Earth is a spinning sphere, meaning this energy must be distributed over its entire surface area.

  • A planet intercepts solar radiation over a flat cross-sectional area of $\pi r^2$.
  • However, the planet rotates and redistributes this heat across its total spherical surface area of $4 \pi r^2$.
  • Therefore, the mean radiative intensity spread across the entire planet is exactly one-quarter of the solar constant:
$$\text{Average Intensity} = S \times \dfrac{\pi r^2}{4\pi r^2} = \dfrac{S}{4}$$

Example 1: Calculating the Solar Constant

Problem: The Sun emits $4 \times 10^{26}$ J of energy in one second. The mean distance from the Earth to the Sun is $1.5 \times 10^{11}$ m. Using this data, calculate the solar constant.

Solution:

  • Step 1: List knowns. Power output $P = 4 \times 10^{26}$ W. Distance $r = 1.5 \times 10^{11}$ m.
  • Step 2: Model the geometry. As light leaves the Sun, it spreads uniformly outward through a massive spherical shell whose radius is the distance to Earth. The surface area of this massive sphere is $4 \pi r^2$.
  • Step 3: Apply the intensity equation ($I = P / A$).
    $$S = \dfrac{P}{4\pi r^2}$$
  • Step 4: Calculate.
    $$S = \dfrac{4 \times 10^{26}}{4\pi(1.5 \times 10^{11})^2} = \mathbf{1415 \text{ W m⁻²}}$$ (or $1.4$ kW m⁻²).