SCIENCE · INVESTIGATE AND EXPLAIN

Compare greenhouse parameter sensitivities

Distinguish surface, layer and effective emitting temperatures across prescribed equilibria.

Learning outcome: Calculate emission and temperature ratios while separating parameter sensitivity from real climate sensitivity.

Starting recap

A fourth-root temperature response differs from a proportional flux response. Effective emitting temperature represents total radiation to space.

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Try

At fixed albedo, changing emissivity from 40% to 80% multiplies surface emission by 4/3 while outgoing to space remains Q.

Open the experiment

Predict

Can surface temperature increase while the equilibrium effective emitting temperature stays fixed?

Your prediction and reason

Experiment

Compare emissivities 0%, 40%, 80% and 100% at albedo 30%. Record flux multipliers and the distinct temperatures.

Trace a one-layer greenhouse energy balance

Worked teaching model · synthetic values · Global-mean radiative equilibrium, fixed sunlight 1361 W/m² and σ = 5.670374419 × 10⁻⁸ W m⁻² K⁻⁴. All unreflected sunlight reaches a blackbody surface. One uniform atmospheric layer is transparent to sunlight and has equal grey thermal absorptivity/emissivity ε. It emits upward and downward. No convection, spectral/vertical structure, heat capacity, time evolution, coupled feedback or CO₂-concentration mapping is included. Temperatures are solved model values, not measurements.

Atmospheric thermal emissivity: 80 % Planetary albedo: 30 %

Reset keeps your prediction, captures and practice history.

The atmospheric layer emits in both directions. Gross surface emission can exceed absorbed sunlight because downward radiation also enters the surface budget. The net radiative transfer from surface to layer is 158.783 W/m² toward the layer; downward emission alone is not the net direction of heat exchange.

Three radiative balances at the solved equilibrium (fluxes in W/m²)
Boundary	Input	Output
Surface	Sun 238.175 + air 158.783	Gross emission 396.958
Atmospheric layer	Absorbed surface IR 317.567	Up 158.783 + down 158.783
Whole system	Unreflected sunlight 238.175	Direct surface 79.392 + air 158.783

Surface, layer and effective temperatures have different meanings
Quantity	Value
Uniform modeled surface temperature	289.257 K
Uniform atmospheric-layer temperature	243.235 K
Effective emitting temperature of combined outgoing flux	254.578 K
No-layer surface reference at the same albedo	254.578 K
Surface difference from no-layer reference	34.678 K

Let Q = (1 − α)S/4 be absorbed solar power per area, F = σTₛ⁴ be gross surface emission, and D be the layer's emission in each direction. Layer balance gives 2D = εF; surface balance gives F = Q + D. Therefore F = Q/(1 − ε/2). Direct surface escape is (1 − ε)F, so radiation reaching space is (1 − ε)F + D = Q.

For ε greater than zero, atmospheric balance gives Tₐ = Tₛ/2^(1/4). At ε = 0, dividing by ε is invalid: the layer neither absorbs nor emits, and its temperature is unconstrained by these radiative equations. It is not zero kelvin. The surface then equals the no-layer reference.

The graph compares equilibrium solutions; it is not a warming timeline. Neither parameter responds to temperature. Atmospheric emissivity is a grey teaching parameter, not CO₂ concentration, and the plotted sensitivity is not real-world equilibrium climate sensitivity. Convection, atmospheric solar absorption, wavelength-dependent absorption and emission, clouds, vertical structure and recorded observations require additional models and evidence.

Modeled surface 289.257 K; effective emission 254.578 K. Modeled atmospheric layer 243.235 K.

Absorbed sunlight 238.175 W/m²; total radiation to space 238.175 W/m². Surface gross emission 396.958 W/m² includes energy returned from the layer. All three budgets balance; the comparison does not simulate adjustment through time or predict real climate sensitivity.

Observation to attach to a capture

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

Compare equilibria while keeping the solar supply fixed

Set planetary albedo to 30% and atmospheric thermal emissivity to zero. Record surface temperature, absorbed solar flux and total outgoing radiation. Now increase emissivity to 80%, leaving albedo unchanged. Predict whether total outgoing can remain balanced with the same solar supply even though the surface becomes warmer.

In this model it does. Surface temperature rises from about 254.5781 K in the no-atmosphere radiative reference to 289.2566 K at ε = 0.80. Absorbed solar flux and total outgoing both remain 238.175 W/m². The route by which thermal radiation reaches space changes while the external equilibrium balance remains intact.

The Penn State one-layer treatment distinguishes surface temperature from the equivalent temperature associated with radiation to space. The lab’s comparison uses that distinction to prevent “a warmer surface must mean a larger equilibrium planetary output” from becoming an unexamined assumption.

Derive a temperature ratio

Surface emission is F = Q/(1 − ε/2), where Q is the fixed absorbed solar flux. Let T₀ = (Q/σ) raised to the power 1/4 be the no-atmosphere reference temperature at the same albedo. Then Tₛ/T₀ = [1/(1 − ε/2)] raised to the power 1/4.

At ε = 0.80, the flux multiplier is 1/0.60, approximately 1.6667. The temperature multiplier is its fourth root, approximately 1.1362. A 66.7% larger surface emission therefore does not mean a 66.7% larger absolute temperature.

At ε = 1, surface emission is twice Q and surface temperature is 2 raised to the power 1/4 times T₀, about 302.7461 K for 30% albedo. That endpoint is the maximum thermal emissivity allowed in this particular single-layer model. It is not a universal upper limit on planetary surface temperatures.

Make a paper table for ε = 0, 0.40, 0.80 and 1. For each, predict the surface-flux multiplier before checking the lab: 1, 1.25, about 1.6667 and 2. Keep the fourth-root temperature multiplier in a separate column.

Track the temperature associated with space

The effective emitting temperature Tₑff is defined by σTₑff⁴ = total outgoing flux. At fixed albedo, equilibrium outgoing is Q, so Tₑff stays about 254.5781 K throughout the emissivity comparison. It does not become equal to Tₛ simply because both are temperatures.

For nonzero ε, the layer temperature is Tₐ = Tₛ divided by the fourth root of two. At ε = 1, all radiation reaching space comes from the layer, and Tₐ equals Tₑff. At partial emissivity, space receives a mixture of direct surface and layer contributions, so the layer temperature need not equal the effective emitting temperature.

At ε = 0, the layer temperature is unconstrained by radiative balance. Do not draw a continuous curve that interprets the missing value as a measured zero-temperature atmosphere. Its absence is an algebraic loss of constraint, not a numerical failure.

Keep parameter sensitivity separate from climate sensitivity

The slider changes a prescribed grey thermal parameter. It is not calibrated to carbon dioxide concentration, and a ten-percentage-point adjustment is not a CO₂ doubling. The model has no spectral absorption calculation, water-vapor response, cloud response or temperature-dependent albedo.

Consequently, the difference between two selected equilibrium temperatures is model parameter sensitivity. It is not a real-world equilibrium climate sensitivity estimate, feedback measurement or attribution result. Nor is it a warming trajectory: the lab calculates each equilibrium immediately without heat capacities or elapsed time.

Transfer to an unfamiliar emissivity

At fixed albedo, compare ε = 0.40 with ε = 0.80. What is the ratio of surface emissions? What is the ratio of surface absolute temperatures, and does total outgoing change?

Check the greenhouse-sensitivity transfer

The emission ratio is (1 − 0.40/2)/(1 − 0.80/2) = 0.80/0.60 = 4/3. The temperature ratio is the fourth root of 4/3, about 1.0746. Total outgoing remains Q because both cases are equilibrium solutions with the same absorbed solar input.

Apply

Public practice checks a specific claim. It does not establish independent mastery.

Explain which quantities remain fixed and why the slider is neither a CO₂ concentration nor a time-dependent feedback model.

Your explanation and limitations

Continue

Completion requires a saved prediction, at least one capture and a reflection. Presence checks do not grade your reasoning.

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Previous: Audit surface, atmosphere and space fluxes Next: Evaluate greenhouse evidence and model limits

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