B.2 Greenhouse Effect

Solution outline

1. $\alpha_A = \dfrac{150}{500} = 0.30$
2. $\alpha_B = \dfrac{350}{500} = 0.70$
3. $\text{Surface B has higher albedo and therefore reflects more energy, keeping cooler on average.}$

Solution outline

1. $\text{The Earth intercepts sunlight over a disk of area } \pi R^{2} \text{ but spreads the energy over its full surface area } 4\pi R^{2}.$
2. $I_{\text{in}} = \dfrac{\text{total power intercepted}}{4\pi R^{2}} = \dfrac{S \pi R^{2}}{4\pi R^{2}} = \dfrac{S}{4}.$

Solution outline

1. $I_{\text{in}} = \dfrac{S}{4}(1-\alpha) = \dfrac{1.36\times10^{3}}{4}\times0.70 = 238\ \mathrm{W\,m^{-2}}$
2. $\text{At equilibrium, absorbed power per unit area equals emitted power per unit area:}$
3. $I_{\text{in}} = \sigma T^{4}$
4. $T = \left(\dfrac{238}{\sigma}\right)^{1/4} \approx 255\ \mathrm{K}\ (about\ -18\ ^{\circ}\mathrm{C})$
5. $\text{This illustrates that the natural greenhouse effect is needed to raise the actual mean surface temperature.}$

Solution outline

1. $P_{\text{emit}} = e\sigma A T^{4}$
2. $\text{At the same } T,\ P_{\text{emit, concrete}} > P_{\text{emit, snow}} \text{ because } e_{\text{concrete}} > e_{\text{snow}}.$
3. $\text{However, temperature also depends on how much solar radiation is absorbed; snow has high albedo and absorbs less, so it can still stay cooler overall.}$

Solution outline

1. $\text{Earth’s surface emits infrared radiation corresponding to its temperature.}$
2. $\text{Greenhouse gases have vibrational and rotational energy levels that resonate with certain IR wavelengths, so they absorb this outgoing radiation.}$
3. $\text{Excited molecules then re‑emit radiation in random directions, sending some energy back towards the surface and lower atmosphere.}$
4. $\text{This repeated absorption and re‑emission traps energy and raises the mean temperature of the climate system.}$

Solution outline

1. $\text{CO}_{2}: \text{Burning fossil fuels in power stations, transport and industry adds large amounts of carbon dioxide to the atmosphere.}$
2. $\text{CH}_{4}: \text{Rice agriculture and livestock farming release methane through anaerobic decay and enteric fermentation.}$
3. $\text{Enhanced greenhouse effect leads to higher mean surface temperatures, which contribute to sea-level rise, more frequent heatwaves and shifts in weather patterns.}$