Atmospheric Radiation

Interaction of solar and terrestrial radiation with atmospheric gases, clouds, and aerosols.

Conceptual context

The atmosphere is not a perfectly transparent window. Solar and terrestrial photons encounter molecules, aerosols, and droplets that can absorb or redirect them. Therefore, $I_0$ entering an atmospheric layer generally differs from $I$ leaving it. The key quantity is $\tau$, a dimensionless measure that accumulates absorption and scattering along $L$. This framework links familiar observations, such as haze or a reddened sunset, with remote sensing, climate physics, solar-energy forecasting, and radiative balance.

🟢 Essential level

$\tau$ measures how difficult it is for radiation to cross the atmosphere. It is not a distance; it summarizes the accumulated effect of absorption and scattering.

When $\tau$ is small, the atmosphere behaves almost like a clear window, so $I$ remains close to $I_0$. As it increases, every additional portion of atmosphere removes a fraction of the radiation that still remains, so the decrease is not linear. $T_r$ directly expresses the fraction that crosses the medium. The extinction coefficient describes how strongly air, clouds, or aerosols remove radiation per unit length, while the geometric path length describes how much medium is crossed. The atmosphere also emits thermal radiation according to its $T$ and effective emissivity, so a complete reading has to carefully separate three distinct things: how much radiation arrives at the top of the layer, how much is genuinely lost along the traveled path through absorption or scattering, and how much is re-emitted afterward toward the final observer on the ground or in orbit.

🔵 Formal level

Basic formulas

Attenuation of a collimated beam is represented by:

Beer-Lambert law

This relation states that $I$ is obtained from $I_0$ through an exponential factor controlled by $\tau$. The larger this optical depth, the smaller the fraction of radiation that manages to cross the atmosphere without being absorbed or scattered.

Real student questions

Is $\tau$ the same as atmospheric thickness? No. The first is dimensionless and summarizes extinction; the second is a geometric length.

Does absorbed radiation disappear? No. Its energy may become internal energy and later influence temperature or emission.

Why can a cloud break the simple model? Because clouds introduce heterogeneity, multiple scattering, and many possible paths.

Is $\epsilon$ the same as $T_r$? No. They describe thermal emission and transmission, respectively.

Why must $T$ be in kelvin? Because the thermal law uses an absolute scale raised to the fourth power.

Final synthesis

Atmospheric transmission falls exponentially as $\tau$ increases. $T_r$ gives the surviving fraction, while $\beta_{e}$ and $L$ explain how extinction accumulates. Thermal emission, controlled by $T$ and $\epsilon$, completes the radiative picture.