Greenhouse effect

The radiation **power received per unit area** (perpendicular to the rays). Unit: **W m⁻²**.

$I = \dfrac{P}{A}$ — power ÷ area. **Given** in the data booklet.

**W m⁻²** (watts per square metre).

$I = \dfrac{P}{4\pi d^{2}}$ — the power spread over a sphere of radius d (so I ∝ 1/d²).

The **intensity of the Sun's radiation arriving at Earth's distance** (just above the atmosphere): **S = 1.36 × 10³ W m⁻²**.

**1.36 × 10³ W m⁻²** — given in the data booklet.

A fixed power spreads over an ever-larger **sphere** (A = 4πd²); same power ÷ bigger area = smaller intensity.

It drops to a **quarter** (× 1/4), because I ∝ 1/d² (inverse-square law).

Multiply by the whole sphere area: **P = I × 4πd²**.

Incident **intensity × panel area × efficiency** (efficiency as a decimal).

**Per m²** — it is an intensity (W m⁻²) at Earth's distance, not the Sun's total power (W).

A perfect **absorber and emitter** of radiation — it absorbs every wavelength that hits it and, when hot, radiates over all wavelengths. Stars are a good model.

The total power (luminosity) radiated by a black body is $L = \sigma A T^4$ — surface area × temperature to the fourth power × the Stefan-Boltzmann constant. **Given** in the data booklet.

The **Stefan-Boltzmann constant**, σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴ (given).

As **T⁴** — doubling the kelvin temperature multiplies the power by 2⁴ = **16**.

The peak wavelength and absolute temperature multiply to a constant: $\lambda_{max} T = 2.9 \times 10^{-3}$ m K. **Given** in the data booklet.

They are **inversely** related — a **hotter** body has a **shorter** peak wavelength (bluer light).

**Kelvin (K)** — never °C. Convert with K = °C + 273.

It gets **taller** (more total power, Stefan-Boltzmann) and its **peak shifts to a shorter wavelength** (Wien).

$\lambda_{max} = \dfrac{2.9 \times 10^{-3}}{5800} = 5.0 \times 10^{-7}$ m (500 nm).

Write $L = \sigma A T^4$ for each and **divide** one by the other — σ cancels, leaving a ratio of areas and T⁴.

Rising T shifts the spectrum's peak to shorter wavelengths (Wien) and adds power across all wavelengths (Stefan-Boltzmann), so the visible colour shifts red → orange → white.

The sphere's surface area, $A = 4\pi r^2$, so $L \propto r^2 T^4$.

The **fraction of incident sunlight that a surface reflects** (scatters back). A number between 0 and 1, with no unit.

$\text{albedo} = \dfrac{\text{total scattered power}}{\text{total incident power}}$ — the reflected fraction. **Given** in the data booklet.

**0.70** — the absorbed fraction is 1 − albedo.

About **0.30** — roughly 30% of sunlight is reflected back to space.

**High:** fresh snow/ice (~0.8), thick cloud (~0.7). **Low:** dark ocean (~0.06), forest/asphalt (~0.1–0.2).

Sunlight lands on the **disc** Earth shows the Sun (πr²) but is shared over the whole **sphere** (4πr²): πr² ÷ 4πr² = 1/4.

About **240 W m⁻²**: (1 − 0.30) × (S ÷ 4) = 0.70 × 340 ≈ 240 W m⁻².

At a steady temperature the power **absorbed** from the Sun equals the power **radiated** away. Energy in = energy out.

The **surface** (ice/cloud high, ocean/forest low), **cloud cover**, and the Sun's angle — so **latitude** and **time of day**.

Treating albedo as the **absorbed** fraction. Albedo is the **reflected** fraction; absorbed = 1 − albedo.

A real surface radiates **emissivity ×** the black-body value. Use it when the surface is not a perfect black body (emissivity < 1).

Greenhouse gases let **sunlight in** but absorb the **infrared** the warm surface radiates out, sending some **back down** — so the surface stays **warmer**.

**Carbon dioxide (CO₂)**, **methane (CH₄)**, **water vapour (H₂O)** and **nitrous oxide (N₂O)**.

**Outgoing infrared** from the warm surface. Incoming sunlight (mostly visible) passes straight through.

Greenhouse gases **absorb** the **infrared** the surface emits, then **re-emit** it in all directions, so some returns **back down** to the surface, keeping it warmer.

CO₂/CH₄ bonds **resonate** (vibrate) at infrared frequencies, so they absorb infrared; the simple N₂/O₂ bonds do not.

The infrared radiation's frequency **matches** the natural vibration frequency of the gas molecule's bonds, so the bond absorbs the energy.

**Natural** = warming from gases always present (Earth ~33 °C warmer, needed for life). **Enhanced** = **extra** warming from human-added gases.

**Burning fossil fuels** (coal, oil, gas), which releases extra **CO₂**.

About **33 °C** warmer than it would be with no atmosphere — without it, Earth would be far too cold for life.

Saying the gases block **incoming sunlight**. They don't — sunlight passes in; the gases trap the **outgoing infrared**.

**Farming** (cattle), **rice fields**, **landfill** and **gas leaks** — a strong infrared absorber per molecule.