# Earth Energy Balance Simulator - Greenhouse Effect, CO₂ Forcing, Climate Sensitivity

> Planetary energy balance lab: effective temperature, N-layer greenhouse model, global energy budget diagram, CO₂ forcing, two-box warming and spectra.

- URL: https://qilingtools.com/en/physics/climate-energy
- Category: Physics Tools (https://qilingtools.com/en/physics)
- Tags: [Physics](https://qilingtools.com/en/tags/physics), [Education](https://qilingtools.com/en/tags/education)
- Last updated: 2026-10-29
- Price: Free, no sign-up
- Privacy: Runs locally in the browser; files are never uploaded

A teaching lab for planetary energy balance. Adjust the solar constant, albedo and atmosphere model to see the effective and surface temperatures, and compare with observations for Mercury, Venus, Earth, the Moon, Mars and Titan. You can also edit the global energy budget and check whether it balances, follow warming over time with the CO₂ forcing formula and a two-box ocean model, and see on a spectrum why incoming sunlight and outgoing infrared barely overlap.

## How to use

1. In Zero-D balance, pick a planet or set the solar constant, albedo and atmosphere model (none, single layer or N layers)
2. In Energy budget, edit the fluxes and watch the net balance at the top of the atmosphere, in the atmosphere and at the surface
3. In CO₂ forcing & warming, drag the CO₂ level and climate sensitivity, then pick a scenario for the two-box model
4. In Spectra, switch between normalised and absolute axes, change the surface temperature and compare with the gas absorption bands

## Features

- Effective temperature from S(1−A)/4 = σT⁴, single-layer and N-layer greenhouse models with a flux diagram
- Six planet and moon presets showing predicted vs observed temperature, the equivalent number of layers and why they differ
- Editable global energy budget diagram with arrow widths proportional to flux and automatic balance checks
- CO₂ radiative forcing 5.35 ln(C/C₀) and equilibrium warming, with adjustable ECS and the IPCC AR6 range marked
- Two-box (upper ocean + deep ocean) energy balance model integrated with RK4, reporting TCR, realised fraction and time constants
- Solar (5772 K) and Earth blackbody spectra with schematic H₂O, CO₂, O₃ and CH₄ absorption bands

## FAQ

### Why would Earth be only about 255 K (−18 °C) without a greenhouse effect?

Earth intercepts sunlight over its cross-section πR² but radiates from its whole surface 4πR², so the average absorbed flux is S(1−A)/4 ≈ 1361 × 0.7 / 4 ≈ 238 W/m². Setting that equal to σT⁴ gives T ≈ 255 K. The observed global mean surface temperature is about 288 K. The extra 33 K or so is the greenhouse effect: water vapour, CO₂ and other gases absorb infrared from the surface and send part of it back down, so the surface must be warmer to get the same energy out to space.

### Where does T_s = T_e (N+1)^(1/4) come from in the N-layer model?

Assume each layer is transparent to sunlight, fully absorbs infrared, and radiates σT⁴ both up and down. Writing the energy balance from the top down, the top layer must send σT_e⁴ to space, and each layer below radiates one more σT_e⁴ than the one above, so the surface emits (N+1)σT_e⁴. It is highly idealised, but it shows that a more opaque atmosphere means a warmer surface. Venus needs roughly 100 such layers to explain its 737 K.

### Why is CO₂ forcing logarithmic, ΔF = 5.35 ln(C/C₀)?

The centre of CO₂'s main absorption band near 15 µm is already saturated. Adding more CO₂ mostly widens the band's wings and raises the effective emission height, so the forcing grows roughly with the logarithm of concentration: each doubling adds about 3.7 W/m² regardless of the starting point. The 5.35 coefficient comes from radiative transfer calculations by Myhre et al. (1998) and works well from a few hundred to a couple of thousand ppm.

### What is the difference between ECS and TCR?

Equilibrium climate sensitivity (ECS) is the warming after CO₂ doubles and the whole system, including the deep ocean, has reached a new balance. IPCC AR6 gives a best estimate of 3 K with a likely range of 2.5–4 K. The transient climate response (TCR) is the warming at the moment of doubling when CO₂ rises 1% per year, around year 70. Because the deep ocean is still taking up heat, TCR is smaller (AR6 best estimate about 1.8 K). With this tool's default two-box parameters TCR is about 1.9 K; change the ocean parameters to see how it shifts.

### Can I use this tool to predict future temperatures?

No. These are zero-dimensional and two-box teaching models with CO₂ as the only forcing. They leave out aerosols, other greenhouse gases, the carbon cycle, the geography of cloud and ice feedbacks, and regional differences. Their purpose is to show the basic physics of energy conservation, radiative forcing and ocean heat inertia. Real climate projections rely on full Earth system models constrained by observations.

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