Zero-dimensional Earth Energy Balance Model (EBM). Incoming solar radiation minus reflected albedo minus outgoing longwave radiation equals the rate of temperature change. Add CO₂ forcing, toggle ice-albedo feedback and discover climate tipping points where positive feedbacks overcome negative ones.
This is a zero-dimensional Energy Balance Model of Earth's climate. It treats the planet as a single point whose temperature changes when absorbed sunlight and emitted infrared radiation fall out of balance: a heat-capacity term integrates dT/dt = (Qin − Qout) / C year by year. Absorbed solar input is (1 − α)·S₀/4, outgoing longwave follows a greybody Stefan–Boltzmann law, and a logarithmic CO₂ forcing term shifts the balance, revealing feedbacks and tipping points.
The model steps surface temperature T forward in time. Absorbed sunlight is (1−α)·S₀/4 with S₀ = 1361 W/m², outgoing radiation is εσT⁴ (ε = 0.61), and CO₂ forcing is 5.35·ln(CO₂/280) W/m². Below 263 K an ice-albedo feedback raises reflectivity α toward 0.62, which can drive the planet into a self-sustaining Snowball state.
The CO₂ slider (150–2000 ppm) sets greenhouse forcing relative to a 280 ppm pre-industrial baseline. The Solar % slider (88–112%) scales the solar constant. The Ice-albedo checkbox toggles the reflectivity feedback. Reset returns to 288 K and year 0; Pause halts the integration. The telemetry panel reports T, ΔT, fluxes, forcing, albedo and simulated year.
Because the ice-albedo feedback is non-linear, an EBM can have two stable states at the same forcing: a warm Earth and a frozen Snowball Earth. Geological evidence suggests Earth truly entered near-global glaciation at least twice, roughly 700 million years ago.
It is the simplest climate model: it assumes the whole planet has one temperature and asks whether incoming absorbed sunlight matches outgoing infrared radiation. When they differ, the surface warms or cools. This zero-dimensional version captures the greenhouse effect, albedo and feedbacks without any geography or atmosphere layers.
Each step solves dT/dt = (Qin − Qout) / C. Qin is (1−α)·S₀/4 scaled by the solar slider, Qout is the greybody emission εσT⁴ minus the CO₂ forcing 5.35·ln(CO₂/280). The difference is divided by a large effective heat capacity (about 3×10⁸ J per m² per K) and multiplied by the timestep in seconds, so temperature relaxes gradually over simulated years.
The CO₂ slider changes the greenhouse forcing logarithmically around 280 ppm, so each doubling adds roughly 3.7 W/m². The Solar % slider scales the 1361 W/m² solar constant from 88 to 112 percent. The Ice-albedo checkbox switches the temperature-dependent reflectivity on or off, which is what allows the model to flip into a frozen state.
The equations are real and widely taught, but the model is deliberately simplified. It ignores clouds, ocean circulation, latitude and the time lag of deep oceans, and the heat capacity and emissivity are tuned to give a sensible pre-industrial 288 K. It illustrates the right qualitative behaviour, not precise forecasts of future warming.
The ice-albedo feedback is a positive feedback: cooling grows ice, ice reflects more sunlight, which causes more cooling. Once the surface drops past a threshold, this runs away until the whole planet is frozen and highly reflective. Escaping requires a very large forcing increase, so the system has two stable states and shows hysteresis, a classic tipping-point behaviour.
This interactive simulation implements a zero-dimensional Earth Energy Balance Model (EBM) that tracks how Earth's surface temperature evolves when incoming solar radiation and outgoing infrared radiation are out of balance. The core physics is the Stefan-Boltzmann law for longwave emission, a logarithmic CO2 greenhouse forcing term (5.35 ln(CO2/280) W/m2), and an optional ice-albedo feedback that raises planetary reflectivity as the surface cools below 263 K. Users can observe how the system seeks equilibrium, amplifies through positive feedbacks, and can tip into radically different stable states.
Energy balance models underpin modern climate science and have been used since the 1960s to estimate climate sensitivity, the expected global temperature rise per doubling of CO2. They are also central to understanding past climate extremes such as Snowball Earth glaciations and hot-house climates.
Earth's energy balance is the accounting of energy flowing into and out of the climate system. Sunlight (shortwave radiation) is absorbed at Earth's surface and in the atmosphere; the planet then re-emits this energy as longwave infrared radiation to space. When absorbed solar energy equals emitted infrared energy, global average temperature remains stable. Any imbalance causes warming or cooling until a new equilibrium is reached.
Use the CO2 slider (150 to 2000 ppm) to change greenhouse gas concentration relative to the pre-industrial 280 ppm baseline; each doubling adds about 3.7 W/m2 of forcing. The Solar % slider scales the solar constant between 88 and 112 percent of today's 1361 W/m2. Toggle the Ice-albedo checkbox to enable or disable the reflectivity feedback, then watch the temperature trace on the chart and the status label (STABLE, +2 degrees C THRESHOLD, SNOWBALL EARTH, etc.). Press Reset to return to 288 K.
Albedo is the fraction of sunlight a surface reflects. Ice and snow reflect roughly 60 to 80 percent of incoming solar radiation, while ocean and land reflect only about 10 to 30 percent. When temperatures fall below about 263 K (-10 degrees C) in this model, ice coverage grows and albedo rises, reflecting more sunlight and causing further cooling. This self-amplifying positive feedback can drive the planet into a fully glaciated Snowball Earth state from which escape requires a very large increase in CO2 or solar output.
The model integrates dT/dt = (Qin - Qout) / C, where Qin = (1 - alpha) * S0 / 4 is absorbed solar flux, Qout = epsilon * sigma * T^4 - F_CO2 is outgoing longwave minus greenhouse forcing, C is the effective heat capacity (~3x10^8 J m^-2 K^-1), and F_CO2 = 5.35 * ln(CO2/280) is the standard logarithmic radiative forcing formula derived from line-by-line radiative transfer calculations. The Stefan-Boltzmann constant sigma is 5.67x10^-8 W m^-2 K^-4 and emissivity epsilon is set to 0.61 to reproduce a 288 K pre-industrial equilibrium.
Equilibrium climate sensitivity (ECS) is the long-term global temperature rise after a doubling of atmospheric CO2. In this EBM without ice-albedo feedback, ECS is roughly 1.2 degrees C (the Planck response alone). With ice-albedo feedback enabled, sensitivity is amplified. The IPCC likely range for the real climate system is 2.5 to 4.0 degrees C, reflecting additional feedbacks from water vapour, clouds and lapse rate that this simple model omits. You can estimate ECS in the simulation by doubling CO2 from 280 to 560 ppm and reading the final equilibrium temperature change.
Yes. Geological evidence including glacial deposits (diamictites) found at tropical paleolatitudes, dropstone sediments, and cap carbonate formations indicates Earth experienced near-global or global glaciation at least twice: during the Sturtian (~717 million years ago) and Marinoan (~635 million years ago) glaciations of the Cryogenian period. The Snowball Earth hypothesis, developed primarily by Joseph Kirschvink and Paul Hoffman, proposes that a runaway ice-albedo feedback froze the oceans from pole to equator, with escape driven by volcanic CO2 accumulation over millions of years.
The zero-dimensional EBM framework was established independently by Mikhail Budyko and William Sellers in 1969, who both showed that ice-albedo feedback could produce multiple climate equilibria. Their work was among the first to use simple physics to quantify tipping points and climate sensitivity. Later, Gerard Roe, Eli Tziperman and others extended EBMs to explore hysteresis, bifurcations and the role of feedbacks in paleoclimate reconstructions.
The carbon cycle simulation on this site shows how CO2 moves between atmosphere, oceans and land, directly affecting the forcing term used here. Ocean acidification is a parallel consequence of rising CO2 that does not affect temperature directly but is driven by the same emissions. More complex climate models (General Circulation Models, GCMs) extend EBM physics to three dimensions with ocean currents, atmospheric dynamics and detailed cloud microphysics. The radiative transfer that sets the CO2 forcing coefficient is also studied in greenhouse gas spectroscopy and exoplanet atmosphere research.
EBM principles are used in spacecraft thermal design to ensure satellites maintain operating temperatures in orbit. Planetary scientists apply EBMs to other worlds, including Mars, Venus and exoplanets in habitable zones, to estimate whether liquid water could exist. Climate engineers proposing solar radiation management (stratospheric aerosol injection or space-based reflectors) use EBM calculations to estimate how much reduction in effective solar constant would offset a given CO2 forcing. Energy-industry projections of future temperatures also rely on simplified EBM-derived metrics such as transient climate response.
Satellites such as NASA's CERES instruments measure Earth's actual energy imbalance, currently estimated at about 0.9 W/m2, confirming ongoing heat accumulation. Researchers are refining estimates of effective radiative forcing for CO2, aerosols and clouds using high-resolution spectroscopy and machine learning emulators. Open questions include the role of cloud feedbacks (the largest source of uncertainty in ECS), the stability of marine ice sheets under warming, and whether any regional tipping elements (Amazon dieback, Atlantic Meridional Overturning Circulation) could trigger cascading global effects beyond what simple global EBMs predict.