B.2 · IB Physics
Planet energy balance
Change the solar intensity, albedo and emissivity to see how incoming, reflected and emitted power per square metre balance, and watch the surface temperature settle to its new equilibrium as greenhouse absorption increases.
Absorbed = (1 − α) I / 4
Emitted = ε σ T⁴
C dT/dt = (1 − α) I / 4 − ε σ T⁴
Equilibrium: T = [ (1 − α) I / (4 ε σ) ]^(1/4)
σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴
C = 4.00 × 10⁸ J m⁻² K⁻¹ (ocean layer)
Emitted = ε σ T⁴
C dT/dt = (1 − α) I / 4 − ε σ T⁴
Equilibrium: T = [ (1 − α) I / (4 ε σ) ]^(1/4)
σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴
C = 4.00 × 10⁸ J m⁻² K⁻¹ (ocean layer)
Try this
- Start from Earth today and press “Add greenhouse gas” three times. How much does the equilibrium temperature rise, and roughly how many years does the surface take to get most of the way there?
- Halve the solar intensity from 1361 to about 680 W m⁻². Predict the new equilibrium temperature using the factor 2^(−1/4) ≈ 0.841, then check your prediction.
- Raise the albedo to 0.35 and find the emissivity that brings the equilibrium temperature back to 288 K.
Exam tip: the mean incoming power per square metre is I/4 because the planet intercepts sunlight on a disc of area πR² but radiates from a sphere of area 4πR², so include the 4 and work in kelvin before taking the fourth root.