IB Physics HLClimate & the Greenhouse EffectPaper 1 & 2Energy Balance~11 min read
Energy Balance Calculations
Now let’s put real numbers on all that heat coming in and going out. Energy balance is just careful bookkeeping: as long as the energy a planet absorbs equals the energy it radiates away, its temperature holds steady. Tip that balance and the planet warms or cools. This is the maths behind every climate model — and it’s friendlier than it looks.
📘 What you need to know
At equilibrium, power absorbed from the Sun = power radiated away, so the temperature stays constant
If absorbed > radiated the planet warms; if radiated > absorbed it cools
Sunlight is caught on a disc of area πr2 but radiated from the whole sphere 4πr2, so the average solar intensity is S/4
The absorbed solar intensity is S(1−a)/4, where a is the albedo
A planet with no atmosphere settles at σT4 = S(1−a)/4
The real Earth is warmer than this, because the atmosphere absorbs infrared and re-radiates some of it back down
In simple models we assume surfaces act as black bodies (surface emissivity e = 1) and stay at constant temperature
The core idea: energy in = energy out
Picture a set of kitchen scales. On one side is the power the planet soaks up from the Sun; on the other is the power it radiates back to space. When the two sides match, the beam sits level and the temperature doesn’t budge.
When absorbed solar power equals radiated power, the “scales” balance and the temperature stays constant. Absorb more than you radiate and the planet warms; radiate more than you absorb and it cools.
The whole topic in one line: write down “power in = power out”, put the numbers in, and solve for the temperature.
A planet with no atmosphere
Start with the simplest case — a bare planet, no air. Two things to balance.
Power in. Sunlight only hits the side facing the Sun, so the planet catches it on a flat disc of area πr2. But it re-radiates from its whole surface, a sphere of area 4πr2. Spread the incoming light over the whole sphere and you divide by 4. Some is also bounced straight back by the albedo a, so the intensity actually absorbed is:
Absorbed solar intensityIin = S(1 − a) ÷ 4
Power out. A planet at temperature T radiates like a black body, with intensity σT4. Set the two equal and you have the balance:
Energy balance (no atmosphere)σT4 = S(1 − a) ÷ 4
WE 1
Estimate the temperature Earth would sit at with no atmosphere. Take the solar constant S = 1360 W m−2 and albedo a = 0.30. (σ = 5.67 × 10−8 W m−2 K−4.)
Step 1 — absorbed intensity = S(1−a)/4= 1360 × (1 − 0.30) ÷ 4
= 238 W m⁻²
Step 2 — set equal to σT⁴ and solveT⁴ = 238 ÷ (5.67×10⁻⁸)
T = (238 ÷ 5.67×10⁻⁸)1/4T ≈ 255 K (−18 °C)Brrr! That’s well below freezing. The real surface sits at about 288 K (15 °C), so the greenhouse effect is worth roughly 33 K of warming.
This is the number that shows why the greenhouse effect matters. A bare Earth balances at about 255 K — a frozen ball. The 33 K gap up to the cosy 288 K we actually enjoy is the natural greenhouse effect doing its job. Learn this comparison; examiners love it.
Adding the atmosphere: the one-layer model
Now give the planet a thin atmosphere and treat it as a single layer sitting above the surface. Sunlight still comes down and warms the ground. But the atmosphere also absorbs infrared and re-radiates it — half heads up to space, half comes back down to the surface. So the surface now gets warmed by two things: sunlight, plus the atmosphere’s downward glow.
The atmosphere soaks up infrared and re-radiates it both ways. Sunlight (orange) plus the atmosphere’s downward glow (red) both warm the surface, which then radiates upward as a black body. Some solar energy is also reflected away first — that’s the albedo.
Any radiating body follows the same rule for the intensity it gives off:
Intensity radiated by a bodyI = eσT4
So to find the surface temperature, add up everything the surface absorbs and set it equal to what a black-body surface radiates:
Surface energy balanceσTs4 = (1 − a)I + eσTa4
WE 2
In a one-layer model, the solar intensity reaching the top of the atmosphere is I = 350 W m−2, the albedo is a = 0.30 and the atmosphere’s emissivity is e = 0.75. If the atmosphere warms to Ta = 248 K, estimate the new surface temperature and the rise from its present 288 K. Treat the surface as a black body.
Step 1 — sunlight reaching the surface = (1−a)I= 0.70 × 350 = 245 W m⁻²Step 2 — atmosphere’s downward glow = eσT₀⁴= 0.75 × 5.67×10⁻⁸ × 248⁴ ≈ 161 W m⁻²Step 3 — total absorbed by surface= 245 + 161 = 406 W m⁻²Step 4 — surface is a black body: σTₛ⁴ = 406
Tₛ = (406 ÷ 5.67×10⁻⁸)1/4Tₛ ≈ 291 KStep 5 — the riseΔT ≈ 291 − 288 = 3 KA warmer atmosphere sends more infrared back down, so the surface has to run hotter to balance its books.
🛠️ Doing an energy balance calculation
Write the rule: power in = power out (that’s what “balance” means).
Power in: start from the solar intensity, take off the albedo with (1−a), and divide by 4 if it’s the whole planet.
Add any downward glow from the atmosphere: eσTa4.
Power out: the surface radiates σT4 (black body, e = 1).
Solve for T — rearrange and take the fourth root. Keep temperatures in kelvin.
Absorbed solar power
+ atmosphere’s downward glow
Total power in
set equal to σT⁴ out
Solve for temperature
💡 Top tips
Balance = equal. Every question boils down to “power in = power out”.
Kelvin, and the fourth power. Never °C in Stefan–Boltzmann, and don’t forget to take the fourth root at the end.
Divide by 4 only for a whole planet (disc catches, sphere radiates); a per-square-metre surface balance doesn’t need it.
Surface = black body in these models, so its emissivity is 1 — but the atmosphere’s emissivity usually isn’t.
Absorbed = (1 − albedo) × incoming — take the reflected part off first.
⚠ Common mistakes
Leaving temperatures in °C — Stefan–Boltzmann needs kelvin
Forgetting the fourth root when solving for T (finding T4 and stopping)
Dividing by 4 in a per-unit-area surface balance where it doesn’t belong (or forgetting it for a whole planet)
Using e = 1 for the atmosphere — only the surface is treated as a black body
Forgetting to remove the reflected fraction: absorbed = (1 − a) × incoming
Quick recap: A planet’s temperature settles where power in = power out. With no atmosphere, σT4 = S(1−a)/4, giving Earth about 255 K. Add a one-layer atmosphere and the surface also catches the atmosphere’s downward infrared eσTa4, pushing the surface up towards its real 288 K. Add more greenhouse gas and the balance tips warmer.
And that wraps up Climate & the Greenhouse Effect! You can now start from raw sunlight, account for albedo and emissivity, follow the energy through the atmosphere, and calculate the temperature a planet settles at — the same core method real climate scientists build on. Bring this balancing trick and your Stefan–Boltzmann skills to any energy-balance question and you’ll be in great shape.
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