1.3 Solar Radiation
The Sun provides virtually all the energy that drives Earth's climate system. Understanding how solar radiation is absorbed, scattered, and reflected is fundamental to atmospheric science.
Solar Constant
S₀ = 1361 W/m²
Total solar irradiance at Earth's mean orbital distance
Average Insolation
$$\bar{S} = \frac{S_0}{4} = 340 \text{ W/m}^2$$
Factor of 4 from spherical geometry
Solar Luminosity
$$L_\odot = 3.83 \times 10^{26} \text{ W}$$
Total power output of the Sun
Blackbody Radiation Laws
Planck Function
The spectral radiance of a blackbody at temperature T as a function of wavelength:
$$B(\lambda, T) = \frac{2hc^2}{\lambda^5} \; \frac{1}{e^{\,hc/(\lambda k_B T)} - 1}$$
where $h = 6.626 \times 10^{-34}$ J s is Planck's constant, $c$ is the speed of light, and $k_B$ is Boltzmann's constant
Wien Displacement Law
The wavelength of peak emission is inversely proportional to temperature:
$$\lambda_{\max} \, T = 2898 \; \mu\text{m} \cdot \text{K}$$
For the Sun ($T \approx 5778$ K): $\lambda_{\max} \approx 0.50 \; \mu$m (visible green). For Earth ($T \approx 255$ K): $\lambda_{\max} \approx 11.4 \; \mu$m (thermal infrared).
Absorption & Scattering
Rayleigh Scattering
Scattering by molecules (λ ≪ particle size). Blue light scattered more than red, giving the sky its blue color.
The Rayleigh scattering cross-section is:
$$\sigma_R = \frac{8\pi^3 (n^2-1)^2}{3 N^2 \lambda^4}$$
where $n$ is the refractive index and $N$ is the number density. The key feature is the strong $\lambda^{-4}$ wavelength dependence.
Mie Scattering
Scattering by aerosols (λ ~ particle size). More forward scattering, less wavelength dependent.
Absorption
O₃ absorbs UV (200-320 nm). H₂O and CO₂ absorb infrared. O₂ absorbs far UV (<200 nm).
Solar Geometry & Albedo
Solar Zenith Angle
The angle between the Sun and the local vertical depends on latitude, solar declination, and hour angle:
$$\cos\theta_z = \sin\phi \, \sin\delta + \cos\phi \, \cos\delta \, \cos h$$
where $\phi$ is latitude, $\delta$ is solar declination ($\pm 23.45°$), and $h$ is the hour angle
Albedo
The fraction of incident radiation reflected by a surface:
$$\alpha = \frac{F_{\text{reflected}}}{F_{\text{incident}}}$$
Typical values: ocean ~0.06, forest ~0.15, desert ~0.35, fresh snow ~0.85, Earth mean ~0.30
Earth's Energy Budget
~30%
Reflected (Albedo)
~23%
Absorbed by Atmosphere
~47%
Absorbed by Surface
Effective Temperature
$$T_e = \left[\frac{S_0(1-\alpha)}{4\sigma}\right]^{1/4} = 255 \text{ K}$$
Without greenhouse effect (actual surface ~288 K, +33 K from GHG)
Energy Balance Derivation
At equilibrium, absorbed solar radiation equals emitted thermal radiation:
$$\underbrace{\pi R^2 \, S_0 (1-\alpha)}_{\text{absorbed}} = \underbrace{4\pi R^2 \, \sigma T_e^4}_{\text{emitted}}$$
The factor $\pi R^2$ is the cross-sectional area intercepting sunlight; $4\pi R^2$ is the total emitting surface area.
Interactive Simulation: Planck Blackbody Radiation
PythonPlots the Planck function B(lambda, T) for multiple temperatures, showing Wien displacement law peaks and the visible spectrum region. Demonstrates how hotter objects emit more intensely and at shorter wavelengths.
Click Run to execute the Python code
Code will be executed with Python 3 on the server