A bare rock at Earth's distance from the Sun would sit near −18 °C. Earth's surface sits near +15 °C. The 33-degree gap is settled radiative physics: sunlight in, infrared out, and an absorbing atmosphere that re-emits downward as well as up. Down the center, data flows: the star's flux goes in, the energy balance closes, the surface temperature comes out. The blue team builds and defends it; the red team tries to break it.
source S. Arrhenius, On the Influence of Carbonic Acid in the Air upon the Temperature of the Ground, Phil. Mag. Ser. 5, 41(251), 237–276 (1896) — doi:10.1080/14786449608620846; building on Fourier (1824). Rendered, not quoted.
A planet warms until what it radiates away equals what it absorbs. It intercepts sunlight over a disk of area πr² but radiates from its whole sphere 4πr² — a geometric factor of 4.
Balance: S(1−a)/4 = σT⁴, giving the effective temperature T_eff = (S(1−a)/4σ)^¼. A single absorbing atmospheric layer re-emits half its infrared back down, so the surface must radiate more — lifted by exactly 2^¼ ≈ 1.19.
Live values for the current sliders:
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The premise of this sphere is Stefan–Boltzmann: a body radiates as σT⁴. Feed the Sun's flux through it and you get T_eff ≈ 255 K. Add one absorbing slab and the surface climbs above it.
Arrhenius (1896) was first to quantify that lift for carbon dioxide. The same balance is the next sphere's premise — a lapse rate carries the surface warmth up through the troposphere. Each sphere is the next one's premise.
The blue team's live check: recompute Earth's balance from scratch (S≈1361, a≈0.30) and confirm T_eff≈255 K, the slab surface≈303 K, and the 33 K gap. If red tampers, this badge is where it shows.
Three numbers enter: the solar constant S (W/m² at the planet's orbit), the albedo a (fraction of sunlight reflected straight back), and the Stefan–Boltzmann constant σ = 5.67×10⁻⁸. Earth: S ≈ 1361, a ≈ 0.30.
Of the arriving beam, a fraction a is reflected; the rest, (1−a), is absorbed — then spread over four times the area it was caught on. That absorbed flux is what you feed the panel below.
One absorbing layer: it re-emits infrared up and down, so the surface runs warmer than the effective temperature.
Move any slider — every temperature is computed from S(1−a)/4 = σT⁴ on the spot, never looked up.
What the machine produces, proven: for Earth's inputs the bare effective temperature is ≈255 K (−18 °C); one absorbing slab lifts the surface to ≈303 K; and the observed 288 K surface sits ≈33 K above the effective temperature — the real greenhouse warming. Full climate projections are richer AMBER; this engine is the settled radiative core.
The blue team's witness (left) confirms these numbers live; the red team (right) tries to make them wrong.
And the 303 K slab number over-shoots the observed 288 K precisely because one opaque grey layer traps more than Earth's partly-transparent air does. "Effective temperature" is exact; "surface temperature" needs the real absorption spectrum — which is why the panel marks projections AMBER.
"Greenhouse gas traps heat like glass in a greenhouse." Kept, corrected. A real greenhouse mostly stops convection; the atmosphere works by re-radiating infrared downward — same warming, different mechanism.
"The effect violates energy conservation — you can't warm a surface with its own emission." Cut. Energy still balances at the top of the atmosphere; the slab only slows the escape, raising the surface temperature until σT⁴ from the top again equals S(1−a)/4.
"Without greenhouse gases Earth would be about 0 °C." Cut. It computes to ≈255 K = −18 °C — a full 33 K colder than today, not near freezing.
The red team's move: drop the (1−a) factor so the planet absorbs all incoming sunlight. T_eff comes out too high and the 255 K / 33 K gap collapses. The blue team's witness (window 7) is watching.
Absorb every photon and Earth's T_eff jumps toward 278 K, so the gap to the 288 K surface shrinks from 33 K to ~10 K — wrong. The witness recomputes, disagrees with the known balance, and turns red. Nothing is faked; the attack is real and it is caught.