IB Physics HLClimate & the Greenhouse EffectPaper 1 & 2Solar Radiation~10 min read
The Solar Constant
Every second, the Sun floods the Earth with energy — and almost all life depends on how much of it arrives. To do climate physics we need to pin that amount down with a single number: the solar constant. It tells us the power landing on each square metre at the top of our atmosphere, and it’s the starting point for every energy-balance calculation about our planet.
📘 What you need to know
The solar constantS is the Sun’s radiation intensity arriving perpendicular to the top of Earth’s atmosphere, at the Earth’s mean distance from the Sun
Its average value is S = 1.36 × 103 W m−2
It is calculated by spreading the Sun’s power over a sphere: S = P ÷ 4πd2
It varies slightly with the Earth’s elliptical orbit and the Sun’s 11-year sunspot cycle (~0.1%)
Because of the inverse-square law, other planets receive different intensities depending on distance
Averaged over the whole rotating planet, the mean incoming intensity is S ÷ 4
What the solar constant is
Life on Earth runs on sunlight, so it’s worth having one clean figure for how much solar energy reaches us. That figure is the solar constant, and it’s defined carefully:
The solar constant — definition
The intensity of the Sun’s radiation arriving perpendicular to the top of the Earth’s atmosphere, when the Earth is at its mean distance from the Sun
Every word earns its place. “Intensity” means power per unit area (W m−2). “Perpendicular” means the radiation falls straight onto a flat surface facing the Sun. And “top of the atmosphere” is crucial — it’s measured above the air, before clouds and gases scatter and absorb any of it, not at the ground. Its accepted average value is:
Average valueS = 1.36 × 103 W m−2
The word “constant” is a little cheeky — it’s not perfectly fixed. Two things nudge it. The Earth’s orbit is a slight ellipse, so we’re a bit closer to the Sun in January and a bit further in July. And the Sun’s own output flickers by about 0.1% over its 11-year sunspot cycle. For exam purposes you treat S as constant, but it’s good to know why it wobbles.
Calculating the solar constant
Where does that number come from? Picture the Sun pouring out its total power P in all directions. By the time that energy reaches us, it has spread evenly across the surface of an enormous imaginary sphere, with the Sun at the centre and a radius equal to the Earth-Sun distance d.
The Sun’s total power P spreads over a sphere of radius d (the Earth-Sun distance). Dividing by the sphere’s surface area gives the intensity that reaches us.
Since the surface area of that sphere is 4πd2, the intensity landing on each square metre — the solar constant — is simply the total power divided by that area:
Solar constant from the Sun’s powerS = P ÷ 4πd2
Here P is the Sun’s total power output (its luminosity, in W), d is the Earth-Sun distance (m), and S comes out in W m−2. This is really just the inverse-square law in disguise: double the distance and the same power spreads over four times the area, so the intensity drops to a quarter.
WE 1
The Sun radiates a total power of 3.8 × 1026 W. The mean Earth-Sun distance is 1.5 × 1011 m. Calculate the solar constant.
Step 1 — the Sun’s power spreads over a sphere
S = P ÷ 4πd²
Step 2 — substituteS = (3.8×10²⁶) ÷ [4π × (1.5×10¹¹)²]S = (3.8×10²⁶) ÷ (2.83×10²³)S = 1340 W m⁻² (~1.3 kW m⁻²)Reassuringly close to the accepted 1.36×10³ W m⁻² — the small gap is just rounded input data.
WE 2
The solar constant is measured as 1360 W m−2 at a distance of 1.5 × 1011 m. Use this to estimate the Sun’s total power output.
Step 1 — rearrange S = P ÷ 4πd²
P = S × 4πd²
Step 2 — substituteP = 1360 × 4π × (1.5×10¹¹)²P = 1360 × 2.83×10²³P = 3.8 × 10²⁶ WThe same equation runs backwards to weigh the Sun’s output from a single ground-level measurement — neat.
Other planets, other intensities
Because it all comes from the inverse-square law, the solar constant is really “Earth’s number.” Any planet gets its own value depending on how far it sits from the Sun. A planet closer in is blasted with far more intensity; one further out gets a feeble trickle.
Earth receives a solar constant of 1360 W m−2 at a distance of 1.00 AU. Venus orbits at 0.72 AU. Estimate the solar intensity arriving at the top of Venus’s atmosphere.
Step 1 — use the inverse-square ratio
SV = SE × (dE ÷ dV)²
Step 2 — substitute (AU cancels)SV = 1360 × (1.00 ÷ 0.72)² = 1360 × 1.93SV = 2620 W m⁻²Venus sits closer in, so it’s hit with nearly double Earth’s intensity — one reason it’s such a scorching world.
Spreading it over the whole planet
There’s one subtlety that trips people up. The solar constant is the intensity on a surface facing the Sun head-on. But the Earth is a rotating sphere: only the daylit side faces the Sun, and even there the ground curves away from the beam. So the average intensity over the whole planet is much less than S.
The Earth intercepts sunlight across its circular shadow — a disc of area πr2. But it then spreads that captured energy over its entire spherical surface, 4πr2. The ratio of these two areas is exactly one quarter:
The planet catches sunlight across its disc (πr2) but re-radiates over its whole surface (4πr2). The quarter ratio gives an average incoming intensity of S ÷ 4.
Average intensity over the whole planetS × (πr2 ÷ 4πr2) = S ÷ 4
WE 4
Taking the solar constant as 1360 W m−2, calculate the average intensity of solar radiation arriving per square metre of the Earth, averaged over the whole planet.
Step 1 — whole-planet average is S/4
average = S ÷ 4
Step 2 — substituteaverage = 1360 ÷ 4average = 340 W m⁻²This S/4 figure is the number you actually feed into Earth’s energy-balance models — you’ll meet it again soon.
Sun’s power P
÷ 4πd²
Solar constant S
÷ 4
Planet average S/4
🛠️ Solving a solar constant problem
Sun’s power given? Use S = P ÷ 4πd2 with d the Earth-Sun distance.
Need the Sun’s power? Rearrange to P = S × 4πd2.
Different planet? Scale with the inverse-square ratio S2 = S1(d1/d2)2.
Whole-planet average? Divide the solar constant by 4.
Check the units — intensity is always W m−2.
💡 Top tips
Top of the atmosphere, not the ground. The definition is deliberately above the air.
It’s an inverse-square law.S = P/4πd2 is just power spread over a sphere.
Don’t confuse S with S/4. The bare constant faces the Sun; the quarter is the planet-wide average.
“Constant” is approximate — orbit and sunspot cycle shift it by around 0.1%.
Distance is Sun-to-planet, not the Sun’s radius — use the orbital distance.
Quick recap: The solar constant S = 1.36 × 103 W m−2 is the Sun’s intensity at the top of Earth’s atmosphere, found from S = P/4πd2. It changes with distance (inverse square), and averaged over the whole rotating Earth becomes S/4.
⚠ Common mistakes
Quoting the solar constant at the ground — it’s defined at the top of the atmosphere
Forgetting the factor of 4 when finding the whole-planet average (S/4)
Using the Sun’s radius instead of the Earth-Sun distance in 4πd2
Dropping the square on the distance in the inverse-square scaling
Assuming S is identical for every planet — it depends on distance from the Sun
You’ve now got the incoming side of Earth’s energy books — how much solar power actually turns up. But arriving isn’t the same as staying: some of that sunlight bounces straight back off clouds, ice and land, and the rest gets absorbed. Next we’ll quantify that split with albedo and emissivity, then use the whole picture to balance Earth’s energy budget and see how the greenhouse effect tips it.
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