IB Physics SLTopic B.2 — Climate & the Greenhouse EffectPaper 1 & 2Energy Balance · One-Layer Model~8 min read
Energy Balance Calculations
Everything from this topic — albedo, emissivity, the solar constant, greenhouse gases — comes together here. If you can balance incoming and outgoing radiation for a planet or an atmosphere layer, you can calculate an equilibrium temperature from scratch.
📘 What you need to know
A body is in energy balance when the intensity it absorbs equals the intensity it emits — under those conditions, its temperature stays constant.
If absorbed and emitted intensity are not equal, the temperature will rise or fall until a new balance is reached.
The simplest climate model treats the atmosphere and the surface as two separate bodies, each radiating according to I = eσT4, each held at its own constant temperature.
In this one-layer model, the surface absorbs radiation from two sources at once: solar radiation that reaches it directly, and infrared radiation re-emitted downward by the atmosphere above it.
Unless told otherwise, the surface is usually modelled as a black body (e = 1), while the atmosphere is given its own, lower emissivity.
What “Energy Balance” Actually Means
Every worked problem in this topic reduces to the same underlying statement: at a stable temperature, a body radiates away exactly as much energy as it takes in. Push more energy in — by increasing solar intensity, lowering albedo, or thickening a greenhouse gas layer — and the temperature has to rise until outgoing radiation catches back up.
Condition for a stable temperature
intensity absorbed = intensity emitted
The Simplest Case — A Planet With No Atmosphere
Strip everything back to a bare planet in space. It absorbs a fraction (1 − a) of the mean solar intensity S⁄4 falling on it, and radiates according to the Stefan–Boltzmann law. Setting these equal and solving for temperature gives a genuinely useful formula:
Equilibrium temperature (no atmosphere)T = 4√[ (1 − a)S ÷ (4eσ) ]
Adding an Atmosphere — The One-Layer Model
Real planets aren’t bare — an atmosphere layer sits above the surface, absorbing some outgoing infrared and sending part of it straight back down. The surface then has to balance two incoming contributions against its own emission: the solar radiation that makes it through the atmosphere, plus the radiation the atmosphere itself re-emits downward.
The surface absorbs both the solar intensity that reaches it directly and the infrared intensity the atmosphere radiates back down — both contributions have to be added before solving for the new surface temperature.
🧮 Solving an energy-balance problem
List the knowns — temperatures, emissivities, albedo and the incoming solar intensity.
Find the solar intensity reaching the surface directly, usually using the atmosphere’s emissivity as the fraction that gets through.
Use I = eσT4 to find the intensity the atmosphere itself radiates back down.
Add the two contributions together to get the total intensity absorbed at the surface.
Solve I = σT4 for the surface, treating it as a black body (e = 1), to find its new temperature.
Subtract the original surface temperature to get ΔT.
Quick recap: at equilibrium, absorbed intensity equals emitted intensity. In a one-layer model, add the solar contribution and the atmospheric contribution before solving for the surface’s new temperature.
WE 1
Model a planet with no atmosphere. Its albedo is 0.30 and it behaves as a black body. The solar constant at its orbit is 1360 W m⁻². Calculate its equilibrium surface temperature.
Step 1 — Find the mean absorbed intensityabsorbed = (1 − 0.30) × 1360 ÷ 4 = 238 W m⁻²Step 2 — Apply the Stefan–Boltzmann law (e = 1)238 = (5.67 × 10⁻⁸) × T⁴T⁴ = 4.20 × 10⁹T ≈ 255 KThat’s roughly 33 K colder than Earth’s actual average surface temperature of 288 K — the difference is exactly the warming the greenhouse effect provides.
WE 2
In a one-layer model, the mean solar intensity above the atmosphere is 340 W m⁻² and the atmosphere’s emissivity is 0.75. The atmosphere’s temperature is 255 K, and the surface — treated as a black body currently at 288 K — absorbs both the solar radiation reaching it directly and the radiation the atmosphere emits downward. Estimate the surface’s new equilibrium temperature and the resulting increase, ΔT.
Step 1 — Solar intensity reaching the surface directlyI_s(solar) = 0.75 × 340 = 255 W m⁻²Step 2 — Intensity radiated downward by the atmosphereI_atm = 0.75 × (5.67 × 10⁻⁸) × 255⁴ ≈ 180 W m⁻²Step 3 — Total intensity absorbed at the surfaceI_total = 255 + 180 = 435 W m⁻²Step 4 — Solve for the new surface temperature (e = 1)435 = (5.67 × 10⁻⁸) × T_s⁴ → T_s ≈ 296 KStep 5 — Find ΔTΔT = 296 − 288ΔT ≈ 8 KSmall changes in atmospheric temperature and emissivity translate into a real shift in surface temperature — this is exactly the kind of reasoning behind climate-model predictions.
💡 Top tips
Unless a question says otherwise, treat Earth’s surface as a black body (e = 1) — the atmosphere is usually the layer with the lower emissivity.
Work in intensity (W m⁻²), not total power — area cancels out of almost every one of these problems, so there’s rarely any need to bring it in.
Keep a clear label on every temperature and intensity you calculate — with two separate bodies in the model, it’s easy to substitute the wrong one by accident.
⚠ Common mistakes
Solving for the surface’s new temperature using only one of the two incoming contributions — both the direct solar intensity and the atmosphere’s downward radiation need to be added first.
Applying the atmosphere’s emissivity to the surface’s own emission — the surface is normally assumed to radiate as a black body.
Rounding intermediate values too early — because temperature depends on the fourth root of intensity, small early rounding errors can shift the final answer by several kelvin.
That’s Topic B.2 complete — albedo, emissivity, the solar constant, greenhouse gases and full energy-balance models, all working together. Next up, we’ll move into a new topic building on these same black-body ideas.
Want this to actually stick before the exam?
Book a free session and we’ll work through energy-balance problems, step by step, until they’re second nature.
