SCIENCE · INVESTIGATE AND EXPLAIN

Close a planet’s two-way radiative exchange

A spherical planet intercepts starlight over a disk but emits over a sphere. Uniform global redistribution therefore gives absorbed mean flux S(1−A)/4, not S itself. Albedo A describes reflected incoming radiation; longwave emissivity describes thermal absorption and emission and is a different property.

Learning outcome: Retain separate surface, layer and top-boundary balances. Change emissivity at fixed incoming radiation; explain why temperature can change while top-boundary equilibrium output stays fixed.

Starting recap

Keep interior, thermal, surface-history, orbital and detection assumptions distinct before drawing a planetary inference.

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Try

At S=1200 W/m², absorbed mean input is 210 W/m². With ε=1, surface emission is 420 W/m² and the atmosphere emits 210 upward and 210 downward. The top boundary still loses exactly 210 W/m²; surface temperature is 2^(1/4) times effective emission temperature.

Open the experiment

Predict

Does downward atmospheric emission count as a new external energy source?

Your prediction and reason

Experiment

Retain separate surface, layer and top-boundary balances. Change emissivity at fixed incoming radiation; explain why temperature can change while top-boundary equilibrium output stays fixed.

Close shortwave input and two-way longwave exchange

Worked teaching model · synthetic values · Original global-mean equilibrium: albedo 0.3, uniform redistribution, surface black in longwave; one layer transparent to shortwave, absorbing/emitting fraction epsilon in longwave both upward/downward. Te⁴=S(1−A)/(4sigma), Ts⁴=Te⁴/(1−epsilon/2). For epsilon>0 Ta⁴=Ts⁴/2; at epsilon 0 layer temperature is unconstrained and omitted. No convection, cloud feedback or real weather.

Incoming stellar irradiance: 1200 W/m² Gray atmospheric longwave emissivity: 0.5 fraction

Reset keeps your prediction, captures and practice history.

An infrared-active layer changes surface exchange while top-of-atmosphere energy still closes.

The layer emits in both directions and the surface absorbs stellar plus downward infrared energy. The simple equilibrium omits convection, clouds, spectral windows and dynamics, so it is not a planetary climate forecast.

Original conditional model ledger
Quantity	Value	Unit
Absorbed stellar global-mean input	210	W/m²
Effective emission temperature	246.6902	K
Surface equilibrium temperature	265.0859	K
Surface upward longwave	280.0000	W/m²
Layer downward longwave	70.00000	W/m²
Top-of-atmosphere outgoing longwave	210.0000	W/m²
Layer equilibrium temperature	222.9098	K

Original supplied models, not actual planet observations or a unique interior, atmosphere or discovery claim.

An infrared-active layer changes surface exchange while top-of-atmosphere energy still closes.

Observation to attach to a capture

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

Understand

1. Define the planetary evidence

2. Connect mechanism and inference

The original model fixes albedo at 0.3, takes a black longwave surface and uses one atmosphere layer transparent to incoming shortwave but absorbing fraction ε of surface longwave. At equilibrium the layer emits equally upward and downward. For ε>0 its temperature satisfies Ta⁴=Ts⁴/2, and the surface balance gives σTs⁴=S(1−A)/[4(1−ε/2)].

At the top boundary, direct transmitted surface emission (1−ε)σTs⁴ plus upward atmospheric emission εσTa⁴ equals absorbed stellar input. Downward atmospheric emission warms the equilibrium surface without creating energy; it is an internal exchange that must not be added again as external input.

When ε=0, the layer is radiatively disconnected in this model, so its temperature is unconstrained rather than uniquely zero. Actual atmospheres involve spectral windows, convection, clouds, circulation and chemistry. This gray equilibrium comparison is not a forecast for Earth, Venus or an exoplanet, and radiative temperature alone cannot establish liquid water or life.

3. Work the supplied case

4. Predict and investigate

Does downward atmospheric emission count as a new external energy source?

Retain separate surface, layer and top-boundary balances. Change emissivity at fixed incoming radiation; explain why temperature can change while top-boundary equilibrium output stays fixed.

Apply the evidence to a changed case

At S=1200 and ε=0.5, find surface emission, upward atmospheric emission and direct transmitted emission.

Retain the evidence frame, calculations and assumptions before opening public feedback.

Compare a worked response

Surface emission is 210/0.75=280 W/m². Atmospheric emission in each direction is ε×280/2=70; direct transmission is 140. Their top-boundary sum is 210, while surface output 280 equals external absorption 210 plus downward 70.

Source and boundaries

Primary source. Checked 31 October 2026. Original diagrams and supplied model records are not actual planetary observations. No direct solar viewing, instrument operation or physical experiment is requested.

Apply

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

At S=1200 and ε=0.5, find surface emission, upward atmospheric emission and direct transmitted emission.

Your explanation and limitations

Continue

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

Skipping retains experiments and awards no mastery.

Previous: Separate atmospheric thickness from an escape claim Next: Separate a condensation threshold from a formation history

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