B.2 Greenhouse effect

Four regions of the Earth's surface are listed.

Which has the highest albedo?

Which of these is not one of the main greenhouse gases?

Which statement about the enhanced greenhouse effect is correct?

Which row correctly pairs one of the main greenhouse gases with a source of that gas that is due to human activity?

Solar radiation of intensity 900 W m⁻² is incident on the top of a forest canopy. Of this, 110 W m⁻² is scattered back upwards, 700 W m⁻² is absorbed by the leaves and the rest is transmitted to the forest floor.

What is the albedo of the canopy?

A satellite measures the albedo of the same region of open ocean at local noon on two days. The albedo on the second day is much greater than on the first day.

What is the most likely explanation?

The surface of a panel has an emissivity of 0.40.

Which statement about the surface is correct?

The absorption of infrared radiation by greenhouse-gas molecules can be explained by a resonance model and by a model based on molecular energy levels.

Which row gives the condition for strong absorption in each model?

The mean intensity of solar radiation incident on the top of Earth's atmosphere, averaged over the whole of Earth's surface, is S/4, where S is the solar constant. What is the reason for the factor of 4?

A space probe travels outwards until it is three times as far from the Sun as Earth is. At Earth's distance, the intensity of solar radiation is S. What intensity does the probe receive on a surface facing the Sun?

Which statement best describes the solar constant?

Two rocky planets orbit the same star at the same orbital radius. They have the same albedo and the same emissivity. Planet X has twice the radius of planet Y. What is the ratio \(\frac{\text{equilibrium temperature of X}}{\text{equilibrium temperature of Y}}\)?

In a simple energy-balance model of the Earth, the mean solar intensity received per square metre of the Earth's surface is taken as S/4, where S is the solar constant. Why does the factor 4 appear?

The mean temperature of a planet stays constant over many years. Which statement must be true?

A manufacturer states that the glass door of a kitchen oven has an emissivity of 0.92. What does this value mean?

A gas sample taken from a hospital operating theatre contains nitrous oxide (N₂O) from anaesthetic equipment, water vapour, carbon dioxide and nitrogen. Which of these components is not a greenhouse gas?

Greenhouse gases in the lower atmosphere absorb most strongly which type of radiation?

A fresh snowfield at a ski resort receives solar radiation of intensity 800 W m⁻². It reflects radiation of intensity 680 W m⁻². What is the albedo of the snow?

Which change would increase the albedo of the Earth?

Why do greenhouse gases such as CO₂ and H₂O absorb strongly in the infrared but let most visible light pass through?

Which of the following is not a way in which energy leaves the Earth's surface?

Solar radiation of intensity S is incident on a spherical planet of radius R.

Which row gives the power intercepted by the planet and the mean intensity when this power is averaged over the planet's whole surface?

Plates X and Y have equal surface areas. X has an emissivity of 0.40 and is at a temperature of 600 K. Y has an emissivity of 0.80 and is at 300 K.

What is the ratio (power radiated by X) / (power radiated by Y)?

Molecules of carbon dioxide in the atmosphere absorb some of the infrared radiation emitted by the Earth's surface.

Which statement about this absorption is correct?

A flat plate in space faces the Sun. Its front face absorbs solar radiation at a rate I per unit area, and the plate radiates only from this face. The emissivity of the front face is e.

What is the equilibrium temperature of the plate?

A flat region is covered by forest of albedo 0.15. After a snowfall, 40 % of the area of the region is covered by snow of albedo 0.80 and the rest is still forest. The solar intensity incident on the region is the same before and after the snowfall.

What is (solar power absorbed by the region after the snowfall) / (solar power absorbed before the snowfall)?

In the resonance model of the greenhouse effect, a molecule of a greenhouse gas is modelled as atoms joined by springs.

Which statement explains why such a molecule absorbs strongly some of the infrared radiation emitted by the Earth's surface?

In a simple energy-balance model the mean surface temperature T of the Earth is given by (1 − α)S/4 = eσT⁴, where α is the albedo, S is the solar constant and e is the effective emissivity of the Earth's surface and atmosphere.

Which row gives the effect on T of an increase in the area covered by ice and of an increase in the concentration of greenhouse gases?

The Sun is a sphere of radius R that radiates as a black body at surface temperature T. The mean distance between the Sun and the Earth is d.

What is the solar constant?

A planet has no atmosphere and radiates as a black body. Its albedo α could take any value from 0 to 1.

Which graph shows how the equilibrium temperature T of the planet would vary with α?

Instruments on a flux tower in a desert measure the mean energy flows per unit area at the ground surface during one hour around midday. The values are shown in W m⁻². The surface layer has negligible heat capacity, so the energy flows into and out of it balance.

What is the intensity X of the infrared radiation from the atmosphere that is absorbed by the surface?

Which statement about water vapour in the Earth's atmosphere is correct?

Infrared radiation of intensity I emitted by the ground passes into a thin layer of air. The layer absorbs 40 % of this radiation and transmits the rest. In a simple model, the layer re-emits all of the energy it absorbs, half upwards and half downwards. Ignore all other radiation.

What is the intensity of the infrared radiation that leaves the top of the layer?

A planet orbits the Sun at half of Earth's orbital radius and has an albedo of 0.50. The solar constant at Earth is S. What is the mean solar intensity absorbed per unit area of the planet's whole surface?

A climate model treats Earth as a black body in radiative equilibrium. If the Sun's luminosity increased by 10%, by what factor would Earth's equilibrium temperature change?

Earth has a radius of 6.37 × 10⁶ m and the solar constant is 1.36 × 10³ W m⁻². What is the total solar power intercepted by Earth?

Mars orbits the Sun at 1.5 times Earth's orbital radius. The solar constant at Earth is 1360 W m⁻². What is the mean solar intensity incident on the top of the Martian atmosphere, averaged over the whole planet?

Earth's distance from the Sun varies from 1.47 × 10¹¹ m to 1.52 × 10¹¹ m during the year. By approximately what percentage is the solar intensity above the atmosphere greater at the closest point than at the farthest point?

A space agency models a planet that moves from orbital radius d to orbital radius 2d around its star. Its albedo and emissivity do not change. What is the ratio \(\frac{\text{new equilibrium temperature}}{\text{original equilibrium temperature}}\)?

Some climate scientists propose that, in the distant past, ice may have covered the whole of the Earth's surface. Suppose such an event raised the planet's albedo from 0.30 to 0.65, with emissivity unchanged. What is the ratio \(\frac{\text{new equilibrium temperature}}{\text{original equilibrium temperature}}\)?

A student models the Earth using \(T = \left(\frac{(1-\alpha)S}{4e\sigma}\right)^{1/4}\), where e is the effective emissivity of the Earth and its atmosphere. In this model, how is an increase in the atmospheric concentration of carbon dioxide best represented?

An exoplanet orbits a star whose luminosity is 16 times that of the Sun. Its orbital radius is 4 times the Earth–Sun distance. The exoplanet has the same albedo and emissivity as the Earth. How does the equilibrium temperature of the exoplanet compare with that of the Earth?

Carbon dioxide in the atmosphere absorbs strongly the infrared radiation emitted by the Earth's surface. It is almost transparent to visible sunlight. What is the reason for this difference?

Two panels on a satellite are at the same temperature. Panel A has emissivity 0.8 and surface area 2 m². Panel B has emissivity 0.2 and surface area 4 m². What is the ratio of the power radiated by A to the power radiated by B?

The absorber plate of a rooftop solar water heater is designed to reach as high a temperature as possible. The plate operates at about 350 K. Which combination of surface properties is best?

Which statement correctly describes how greenhouse gases raise the mean surface temperature of the Earth?

Methane strongly absorbs infrared radiation of wavelength 7.7 μm. Which is the best explanation for this absorption?

A thermal camera is set to emissivity 1.00. It is pointed at a metal handrail whose true emissivity is 0.60. Reflected radiation from the surroundings is negligible. The camera calculates temperature from the measured power per unit area. The true kelvin temperature of the handrail is T. What temperature does the camera display? (0.60^1/4 ≈ 0.88)

An exoplanet orbits a star. The intensity of the star's radiation at the planet's orbit is 2400 W m⁻². The albedo of the planet is 0.25. What is the mean power absorbed per unit area of the planet's surface?

Two identical flat roofs receive the same solar intensity. One is coated white (albedo 0.80). The other is black (albedo 0.10). What is the ratio \(\frac{\text{power absorbed by black roof}}{\text{power absorbed by white roof}}\)?

A simple model of a planet makes three assumptions:
• the atmosphere is transparent to incoming sunlight;
• the atmosphere absorbs all the infrared radiation emitted by the surface;
• the atmosphere re-emits this energy equally upwards and downwards.
In equilibrium, what is the ratio of the power radiated per unit area by the surface to the solar power absorbed per unit area by the surface?

An icy moon has an albedo of 0.60. Dust from a nearby ring settles on the surface and lowers the albedo to 0.20. The emissivity and the incident intensity stay the same. What is the ratio \(\frac{\text{new equilibrium temperature}}{\text{original equilibrium temperature}}\)?

Suppose the concentration of carbon dioxide in the atmosphere suddenly increases. Before any temperatures have had time to change, which statement is correct?

Planets P and Q have no atmosphere and the same albedo, and each radiates as a black body. Q is twice as far from the Sun as P.

S is the solar constant at the position of a planet and T is the planet's equilibrium temperature. Which row gives S_Q/S_P and T_Q/T_P?

In a simple model of a planet, the atmosphere is a single layer that is transparent to solar radiation but absorbs all of the infrared radiation emitted by the surface. The atmosphere emits infrared radiation of equal intensity upwards and downwards.

The surface absorbs a mean solar intensity I and radiates as a black body at temperature T_s. The surface and the atmosphere are in equilibrium.

What is σT_s⁴?

Planets X and Y have no atmosphere and orbit the same star at the same distance. They have the same radius and the same albedo, and both radiate as black bodies.

X rotates rapidly, so its whole surface is at one uniform temperature T_X. Y always keeps the same hemisphere facing the star. In a simple model the sunlit hemisphere of Y is at a uniform temperature T_Y and its dark hemisphere radiates negligible power.

What is T_Y/T_X?

A gas-giant planet of radius R and albedo α is at a distance from the Sun where the intensity of the solar radiation is I. Energy from its slow contraction is also released inside the planet at a total rate P. The planet radiates as a black body at a uniform temperature T and is in equilibrium.

What is σT⁴?

The concentration of carbon dioxide in the atmosphere increases and then stays constant at its new value. The solar constant and the albedo of the Earth do not change. Eventually the Earth reaches a new equilibrium.

Which row compares the new equilibrium with the original equilibrium?

Two black bodies are in space at the same distance from the Sun. Body P is a thin flat plate held with one face perpendicular to the Sun's rays; it radiates from both faces. Body Q is a rapidly rotating sphere whose temperature is uniform. What is the ratio of equilibrium temperatures, T_P/T_Q?

An airless, rapidly rotating planet behaves as a black body and has equilibrium temperature T at a distance d from its star. An identical planet orbits the same star at a distance 4d. What is the equilibrium temperature of the second planet?

Two black-body objects are at the same distance from the Sun.
• Object P is a thin flat plate that always faces the Sun. Its back is perfectly insulated, so it radiates only from its sunlit face.
• Object Q is a small, rapidly rotating sphere with a uniform surface temperature.
What is the ratio \(\frac{\text{equilibrium temperature of P}}{\text{equilibrium temperature of Q}}\)?

A planet is modelled with albedo α, effective emissivity e and orbital radius d around a star of luminosity L. Each change below is made on its own. Which change produces the largest increase in equilibrium temperature?

A CubeSat in deep shadow dissipates a constant electrical power from its electronics. It loses energy only by radiation and receives no external radiation. Over time its coating degrades, and its emissivity falls from 0.80 to 0.05. By what factor does its equilibrium kelvin temperature change?

A small asteroid in sunlight has a surface that behaves as a grey body, so its absorptivity equals its emissivity at all wavelengths. Space weathering darkens the surface, and the emissivity rises from 0.30 to 0.90. The asteroid's distance from the Sun does not change. What happens to its equilibrium temperature?

A rocky planet with no atmosphere has an equilibrium surface temperature of 250 K. Suppose an atmosphere is added that is transparent to sunlight and absorbs all infrared radiation from the surface. It re-emits this energy equally upwards and downwards. The albedo is unchanged. What is the new equilibrium surface temperature?

An albedometer consists of two radiation sensors, one facing up and one facing down, mounted 1.5 m above a flat area of dry sand. The upward sensor measures the incident solar intensity I_in and the downward sensor measures the intensity I_r scattered from the sand. Readings were taken at intervals on a day with passing clouds. Each intensity has an uncertainty of ±10 W m⁻².

A student uses a satellite dataset for a region of open ocean. For each of eight months the dataset gives the mean cloud fraction f (the fraction of the area of the region covered by cloud) and the mean albedo α of the region. Each value of α has an absolute uncertainty of ±0.02; the uncertainty in f is negligible.

The student models the region as a mixture of cloud-free ocean of albedo α_o and cloud of albedo α_c, so that α = α_o + (α_c − α_o)f.

A student uses a dataset of global five-year mean values to test whether the burning of fossil fuels is the main cause of the rise in the concentration of carbon dioxide in the atmosphere. For each period the dataset gives the mean mass E of CO₂ released each year by burning fossil fuels, in gigatonnes (1 Gt = 10¹² kg), and the mean rise ΔC each year in the CO₂ concentration of the atmosphere, in parts per million (ppm). Each value of ΔC has an uncertainty of ±0.15 ppm per year; the uncertainty in E is negligible.

An increase of 1 ppm in the concentration corresponds to 7.8 Gt of CO₂ added to the atmosphere.

A student determines the emissivity e of a black paint used on spacecraft radiators. A thin square aluminium plate of side 50.0 mm, painted on both faces, is hung by two fine leads in an evacuated chamber. The walls of the chamber are kept at T_w = 290 K. A heater on the plate is adjusted until the plate reaches a steady temperature T, measured with a thermocouple to ±2 K. The electrical power P supplied to the heater is then recorded, with an uncertainty of ±0.20 W.

In the steady state P = eσA(T⁴ − T_w⁴), where A is the total area of the two faces. The graph shows P against (T⁴ − T_w⁴), with error bars, for five of the six readings.

A student determines the emissivity e of black insulating tape. A strip of the tape is stuck to an aluminium block that is warmed on a hotplate. A thermocouple inside the block measures its temperature T with an uncertainty of ±1.0 K. An infrared thermometer, set to an emissivity of 1.00, is aimed at the tape and gives a reading T_IR with an uncertainty of ±0.5 K. The room is at T_r = 293 K; treat this value as exact.

The infrared thermometer receives the radiation emitted by the tape and the radiation from the room reflected by the tape, so that T_IR⁴ − T_r⁴ = e(T⁴ − T_r⁴).

The graph shows (T_IR⁴ − T_r⁴) against (T⁴ − T_r⁴), with error bars, for five of the six readings.

Engineers test white and grey coatings for a small satellite. Each coating is applied to a thin flat coupon whose back face is insulated. The coupons are placed in a vacuum chamber whose walls are cooled by liquid nitrogen, so that the radiation they receive from the walls is negligible. A solar simulator lamp, set to give an intensity of 1361 W m⁻², shines perpendicularly onto the front face of each coupon in turn.

Every coating has an emissivity e = 0.85 for infrared radiation. The albedo α of each coating for the simulator’s radiation was measured separately with an uncertainty of ±0.01. The steady temperature T of each coupon is measured with an uncertainty of ±1 K. The graph shows T⁴ against (1 − α), with error bars, for six of the seven coupons.

A student uses a published planetary database to investigate the greenhouse effect. The table gives, for three planets, the mean distance d from the Sun in astronomical units (AU), the albedo α and the measured mean surface temperature T_s. It also gives the equilibrium temperature T_eq that each planet would have if its surface radiated as a black body (emissivity 1) and there were no atmosphere.

The solar constant at 1 AU is S = 1361 W m⁻². The intensity of solar radiation at a distance d is S/d², with d in AU.

A student uses a database of dark, airless asteroids to test a simple energy-balance model. For each asteroid the database gives its mean distance d from the Sun, in astronomical units (AU), and its mean surface temperature T, found from the infrared radiation it emits. The uncertainty in each value of T is ±3 %.

In the model, each asteroid has albedo α and emissivity e, is in radiative equilibrium, and absorbs a mean intensity equal to (1 − α)/4 times the solar intensity incident on it. The solar intensity at a distance d is S₀/d², where S₀ = 1361 W m⁻² is the solar constant at 1 AU. The student plots lg T against lg d. The graph shows the data for six of the seven asteroids.

A student at a mountain observatory estimates the solar constant S. On a cloudless morning, when the Earth is at its mean distance from the Sun, she measures the intensity I of the direct solar radiation with a radiometer pointed at the Sun. Each value of I has an uncertainty of ±10 W m⁻². For each reading she records the zenith angle z, the angle between the direction of the Sun and the vertical.

The radiation crosses a thickness of atmosphere proportional to the air mass m = 1/cos z. A model of the absorption and scattering in the atmosphere gives I = S e^−km, where k is a constant. The graph shows ln(I / W m⁻²) against m, with error bars, for seven of the eight readings.

An array of solar panels of total area 12 m² is mounted on the roof of a house. At noon on a clear day, solar radiation of intensity 900 W m⁻² is incident on the array. The array reflects 1.3 kW of the incident power and transfers 20 % of the incident power to electrical energy.

A classroom air-quality monitor measures the concentration of carbon dioxide. Infrared radiation from a small lamp passes along a tube containing air from the room and falls on a detector. A filter in front of the detector transmits only radiation of wavelength 4.26 μm, which carbon dioxide molecules absorb. The higher the concentration of carbon dioxide, the smaller the power reaching the detector.

The diagram shows four of the energy levels of a carbon dioxide molecule.

The Earth moves around the Sun in a slightly elliptical orbit. Its distance from the Sun is smallest, 1.471 × 10¹¹ m, in early January and greatest, 1.521 × 10¹¹ m, in early July. The solar constant S is the intensity of the Sun’s radiation at the Earth’s mean distance from the Sun.

Luminosity of the Sun = 3.83 × 10²⁶ W.

A horticultural glasshouse is a building with walls and roof made of glass. The glass transmits most electromagnetic radiation with wavelengths between 0.3 μm and 2.5 μm but absorbs almost all radiation with wavelengths greater than 5 μm.

The surface temperature of the Sun is 5800 K. The soil and plants inside the glasshouse are at about 300 K.

A family compares the carbon dioxide emissions of two cars.

The petrol car uses 6.0 litres of petrol for every 100 km travelled. Petrol has a density of 740 kg m⁻³ and an energy density of 46 MJ kg⁻¹. Burning 1.0 kg of petrol releases 3.1 kg of carbon dioxide.

The electric car takes 0.18 kW h of electrical energy from its charger for every kilometre travelled. The electricity is generated in a gas-fired power station of efficiency 0.50, and 92 % of the electrical energy generated reaches the charger. Burning natural gas releases 0.056 kg of carbon dioxide for every 1.0 MJ of energy released. 1 kW h = 3.6 MJ.

A solar-powered spacecraft orbits Jupiter, at a mean distance of 5.2 AU from the Sun. Its solar arrays have a total area of 60 m² and an efficiency of 0.28, and they are kept perpendicular to the Sun's rays. The solar constant at Earth (1.0 AU) is 1360 W m⁻².

The Sun has a luminosity of 3.83 × 10²⁶ W. Earth's mean distance from the Sun is 1.50 × 10¹¹ m and Earth's albedo is 0.30.

A spherical space probe of radius 0.80 m is sent to study the inner Solar System. It orbits the Sun at 0.72 AU. The probe spins slowly, so its surface temperature can be taken as uniform.

Data:
• Solar intensity at 1.00 AU = 1.36×10³ W m⁻²
• Albedo of probe surface = 0.60
• Emissivity of probe surface = 0.85
• σ = 5.67×10⁻⁸ W m⁻² K⁻⁴

Instruments on a weather satellite measure the Earth's radiation budget. Averaged over the whole surface of the Earth and over a year, they record:
• Solar constant: 1361 W m⁻²
• Mean solar intensity reflected back to space: 102 W m⁻²
• Mean surface temperature of the Earth: 288 K

σ = 5.67×10⁻⁸ W m⁻² K⁻⁴

During heavy braking in a motorsport race, a carbon brake disc reaches a temperature of 1100 K. The disc has emissivity 0.75 and an exposed radiating surface area of 0.090 m². The surroundings are at about 300 K.

Agriculture contributes significantly to the atmospheric concentration of several greenhouse gases. Water vapour, however, makes the largest contribution to the natural greenhouse effect.