Greenhouse Effect

A black-body is an idealised object that absorbs every wavelength of electromagnetic radiation that falls on it and, at any given absolute temperature, emits thermal radiation at the maximum possible rate. The total power it radiates (its luminosity $L$) follows the Stefan-Boltzmann law:

$$L = \sigma A T^4$$

Doubling the temperature therefore multiplies the radiated power by $16$. The spectrum of a black-body depends only on its temperature, and the wavelength at which emission peaks is given by Wien's displacement law:

$$\lambda_{\text{max}} T = 2.90 \times 10^{-3}\ \text{m K}$$

Hotter bodies peak at shorter wavelengths, which is why a heated iron bar glows red and then white as it gets hotter.

Real surfaces are imperfect emitters. Emissivity $e$ compares the power per unit area a surface actually radiates with what a black-body at the same temperature would radiate, so $P = e \sigma A T^4$ with $0 \le e \le 1$.

A matt black surface like charcoal has $e$ close to $1$, while polished aluminium can be below $0.1$. The Sun and stars are modelled well as black-bodies, whereas Earth's surface radiates with $e$ slightly below $1$.

You should be able to:

• sketch black-body spectra at different temperatures (higher curve, peak shifted left for hotter bodies)
• use both laws in calculations
• estimate a star's surface temperature from its peak wavelength

Albedo $\alpha$ is the fraction of incident electromagnetic power that a surface scatters:

$$\alpha = \frac{P_{\text{scattered}}}{P_{\text{incident}}}$$

It is dimensionless and varies hugely with surface type. Fresh snow scatters most of the sunlight hitting it ($\alpha \approx 0.8$), while the open ocean absorbs almost everything ($\alpha \approx 0.06$). Averaged over clouds, ice, land, and sea, Earth's mean albedo is about $0.30$, meaning roughly $30\%$ of incoming solar radiation is scattered straight back to space without ever heating the planet.

Earth's energy balance treats the planet as a system in thermal equilibrium: the average solar intensity absorbed must equal the average intensity radiated away as infrared. The absorbed side is $\frac{S(1-\alpha)}{4}$ and the emitted side is $e \sigma T^4$, so setting them equal lets you predict a planet's equilibrium temperature. For Earth this simple model gives about $255\ \text{K}$, roughly $33\ \text{K}$ colder than the observed mean surface temperature of about $288\ \text{K}$; the difference is the natural greenhouse effect.

You must be able to:

• construct this balance equation
• solve it for temperature
• explain qualitatively how a change in albedo (for example, melting sea ice exposing dark water) shifts the equilibrium and can create a feedback loop

The solar constant $S$ is the mean solar power arriving per unit area on a surface held perpendicular (normal) to the Sun's rays at Earth's average orbital distance, measured above the atmosphere. Its value is about $1360\ \text{W m}^{-2}$.

It follows from the inverse-square spreading of the Sun's output: a star of luminosity $L$ spreads its power over a sphere of area $4\pi d^2$ at distance $d$, so:

$$S = \frac{L}{4\pi d^2}$$

A planet closer to its star therefore receives a larger 'solar constant', and Mars, farther out than Earth, receives a smaller one.

The most important subtlety is the factor of $4$. A planet intercepts sunlight over a flat disc of area $\pi R^2$ facing the star, but it has a total spherical surface area of $4\pi R^2$ over which that energy is effectively shared (the planet rotates and radiates from its whole surface).

The average incident intensity over the whole globe is therefore $\frac{S}{4}$, and after scattering the average absorbed intensity is $\frac{S(1-\alpha)}{4}$.

You should be able to:

• calculate $S$ from a star's luminosity and orbital distance
• explain the origin of the $\frac{S}{4}$ factor using the disc and sphere areas
• use $S$ in planetary energy-balance problems

The atmosphere is mostly transparent to the short-wavelength visible and near-ultraviolet radiation arriving from the Sun, which passes through and warms the surface. The surface, at around $288\ \text{K}$, re-radiates this energy as long-wavelength infrared (peak near $10\ \mu\text{m}$ by Wien's law).

Greenhouse gases such as $CO_2$, $H_2O$, $CH_4$, and $N_2O$ absorb strongly in this infrared region, trapping energy that would otherwise escape to space.

The mechanism is molecular. These gases are multi-atomic molecules with polar bonds that can undergo vibrational motion. When such a molecule absorbs infrared radiation, the incoming photon excites a transition between molecular vibrational energy levels; this quantum transition is only possible if the vibration causes a temporary change in the molecule's electric dipole moment.

Because the natural frequencies of these vibrations fall within the infrared part of the electromagnetic spectrum, the gases show strong resonance absorption there: the photon energy matches the energy difference between vibrational states. Once excited, the molecules quickly decay back to lower energy states, re-emitting infrared photons in random directions, so a significant fraction of the outgoing thermal energy is redirected back towards Earth's surface.

You should be able to:

• name the main greenhouse gases
• explain the resonance mechanism
• stress the contrast between transmitted short-wavelength solar radiation and absorbed long-wavelength terrestrial radiation

The natural greenhouse effect keeps Earth's mean surface temperature about $33\ \text{K}$ warmer than the bare energy-balance prediction, and life depends on it. The enhanced greenhouse effect is the *additional* warming caused by human activity raising the concentrations of greenhouse gases above their natural levels: burning fossil fuels and deforestation increase $CO_2$, agriculture and livestock add $CH_4$ and $N_2O$.

Physically, a higher greenhouse-gas concentration means a larger fraction of the surface's outgoing infrared radiation is absorbed and re-emitted back downwards. The planet then radiates less to space than it absorbs from the Sun, so it is temporarily out of thermal equilibrium and its internal energy rises.

The surface temperature increases until the outgoing radiation again balances the absorbed solar power, but this new equilibrium sits at a higher temperature. Warming also triggers feedback effects. Melting ice lowers Earth's albedo so more sunlight is absorbed, and warmer air holds more water vapour, itself a greenhouse gas; both are positive feedbacks that amplify the initial change.

You should be able to:

• distinguish clearly between the natural and enhanced greenhouse effects
• identify the human sources of each major gas
• explain in energy-balance terms why increased absorption of infrared radiation shifts the equilibrium surface temperature upwards