Vol. Science · Astronomy · Insolation

Habitable zone
flux and temperature

Starlight thins with distance. At orbital distance d in AU, the flux relative to Earth’s is S/S⊕ = (L/L☉) / d². Watts per square metre use the NASA Earth Fact Sheet solar irradiance, 1361.0 W/m². Equilibrium temperature assumes that light is shared over the whole planet and radiated as a blackbody: T_eq = [F (1−A) / (4σ)]^¼, the same statement as T_★ √(R_★ / 2a) (1−A)^¼. A is the Bond albedo. The strip is the habitable zone from Kopparapu et al. 2014, conservative and optimistic, for a 1 Earth-mass planet. Luminosity and temperature are the quantities on the blackbody desk and the HR diagram. Spectral type is the stars desk. The rough rings on exoplanets are not this calculation. Proxima is on nearest stars.

Classroom model. Full heat redistribution, emissivity 1, no atmosphere, no clouds, a circular orbit, and the star’s light only. A point inside the strip is not a habitable world, and a point outside is not a proof against liquid water under some other atmosphere. Mars is inside the conservative strip on this page. Venus is not. Earth’s no-greenhouse temperature is not its 288 K surface. The polynomial is for 2600 K to 7200 K and is not used outside that range.

Interactive · Flux, temperature, and the zone

Distance, logarithmic · drag the marker · conservative inside optimistic

Optimistic
Conservative
Planet

Cited systems · the star, the orbit, and a Bond albedo

Bond albedo · fact-sheet values, not a guess for an exoplanet

L / L☉ L☉

Teff K

Distance AU

Albedo A

The sliders need JavaScript. The cited anchors are still below. S/S⊕ = (L/L☉) / d². The solar irradiance is 1361.0 W/m² (NASA Earth Fact Sheet). With Bond albedo 0.294 at Earth’s semimajor axis 1.00000011 AU, this full-redistribution formula gives 255.1 K. Venus, Bond albedo 0.77 at 0.72333199 AU, gives 226.6 K. Mars, Bond albedo 0.250 at 1.52366231 AU, gives 209.8 K. Earth’s average surface temperature on the same sheet is 288 K. Kopparapu et al. 2014 give the habitable-zone polynomial for 2600 K to 7200 K. TRAPPIST-1 at 2566 K is outside that range.

Not a verdict

Inside the strip is a flux limit. It is not a living world.

The conservative edges are the runaway greenhouse and the maximum greenhouse. The optimistic edges are recent Venus and early Mars. Kopparapu et al. 2014 print those four for a 1 Earth-mass planet, and only for effective temperatures from 2600 K to 7200 K. The runaway-greenhouse row depends on mass. This page uses the 1 Earth-mass coefficients. Recent Venus, maximum greenhouse, and early Mars are the same at every mass in that table.

Mars sits inside the conservative strip and the fact sheet’s average temperature is about 214 K, under a thin atmosphere. Venus sits closer than the recent-Venus distance, and its average temperature is 737 K. Earth’s no-greenhouse result here is 255.1 K. The same fact sheet lists the average surface temperature as 288 K (15 °C). The gap is greenhouse warming, which this model leaves out.

Equilibrium temperature assumes the absorbed starlight is shared over the whole surface and radiated with emissivity 1. A planet that keeps the heat on the dayside runs hotter. Clouds, oceans, and a different gas mixture move the real edges. Being in the strip is not a detection of liquid water, and missing the strip is not a proof against it.

Literacy · the numbers on the readout

S / S⊕: Stellar flux at the planet divided by the flux at 1 AU from a star with L = L☉. The formula is (L/L☉) / d² with d in AU. It does not depend on albedo. Earth’s preset is 1.000 because the fact-sheet semimajor axis is 1.00000011 AU.
Watts per square metre: S/S⊕ times 1361.0 W/m², the solar irradiance on the NASA Earth Fact Sheet. Spreading the IAU nominal luminosity, 3.828 × 10²⁶ W, over a sphere of radius 1 au (149,597,870,700 m) gives 1361.17 W/m². This desk multiplies by the fact-sheet figure.
Equilibrium temperature: T_eq = [F (1−A) / (4σ)]^¼. The 4 is the whole surface emitting what a cross-section absorbs: full heat redistribution. σ is the CODATA Stefan–Boltzmann constant, the same computation as the blackbody desk. Emissivity is 1. There is no greenhouse.
The radius form: T_eq = T_★ √(R_★ / 2a) (1−A)^¼. R_★ here is the radius implied by L = 4πR²σT⁴, using L = (L/L☉) × 3.828 × 10²⁶ W. That form and the flux form differ only because 1361.0 W/m² is not exactly the nominal luminosity spread over 1 au. At the Earth preset the gap is 0.008 K, inside the tenth this page prints.
Bond albedo: The fraction of incident starlight the planet reflects, over all wavelengths and all angles. Earth 0.294, Venus 0.77, Mars 0.250, from the NASA fact sheets. Geometric albedo is a different number and is not used here. Exoplanet Bond albedos are not measured on this page.
Greenhouse: An atmosphere can let starlight in and hold infrared on the way out. Earth’s surface is warmer than T_eq. Venus is much warmer. This desk does not add that term. The fact-sheet surface temperatures are printed beside the model so the gap stays visible.
The four limits: Optimistic inner edge: recent Venus. Conservative inner edge: runaway greenhouse. Conservative outer edge: maximum greenhouse. Optimistic outer edge: early Mars. S_eff = S_eff☉ + aT_★ + bT_★² + cT_★³ + dT_★⁴, with T_★ = T_eff − 5780 K. Distance is √((L/L☉) / S_eff) AU. The zero of the polynomial is 5780 K, the solar temperature in that paper, not the IAU nominal 5772 K.
Outside 2600–7200 K: Kopparapu et al. 2014 state the coefficients for 2600 K ≤ T_eff ≤ 7200 K. Below or above that, this page still computes flux and T_eq, and it does not draw a zone. TRAPPIST-1’s 2566 K is the preset that shows the refusal. The endpoints 2600 K and 7200 K are inside the window.

Sources · fact sheets, IAU, CODATA, and the papers

Cited, not invented

NASA NSSDCA — Earth Fact Sheet — Bond albedo 0.294, solar irradiance 1361.0 W/m², black-body temperature 254.0 K, average temperature 288 K (15 °C), J2000 semimajor axis 1.00000011 AU. This desk’s formula with 1361.0 and 0.294 returns 255.1 K, not the sheet’s 254.0 K.
NASA NSSDCA — Venus Fact Sheet — Bond albedo 0.77, solar irradiance 2601.3 W/m², black-body temperature 226.6 K, average temperature 737 K, J2000 semimajor axis 0.72333199 AU.
NASA NSSDCA — Mars Fact Sheet — Bond albedo 0.250, solar irradiance 586.2 W/m², black-body temperature 209.8 K, average temperature ~214 K, J2000 semimajor axis 1.52366231 AU.
IAU 2015 Resolution B3 (Prša et al. 2016, AJ 152, 41) — nominal solar luminosity 3.828 × 10²⁶ W, nominal solar effective temperature 5772 K, nominal solar radius 6.957 × 10⁸ m. Conversion constants, not a measurement of the Sun. The resolution text is also at the IAU archive.
IAU 2012 Resolution B2 — 1 au = 149,597,870,700 m exactly. Used when the radius form of T_eq is compared with the flux form.
NIST CODATA — Stefan–Boltzmann constant — σ = 5.670374419… × 10⁻⁸ W m⁻² K⁻⁴, exact, from the defined h, c, and k. This desk computes σ = 2π⁵k⁴ / (15h³c²) using h = 6.62607015 × 10⁻³⁴ J·s, c = 299,792,458 m/s, and k = 1.380649 × 10⁻²³ J/K.
Kopparapu et al. 2014, ApJL 787, L29 — habitable zones for 2600 K to 7200 K. Equation (4): S_eff = S_eff☉ + aT_★ + bT_★² + cT_★³ + dT_★⁴, T_★ = T_eff − 5780 K. Equation (5): d = √((L/L☉) / S_eff) AU. Table 1, 1 M⊕ row, as printed: recent Venus 1.776, 2.136×10⁻⁴, 2.533×10⁻⁸, −1.332×10⁻¹¹, −3.097×10⁻¹⁵; runaway greenhouse 1.107, 1.332×10⁻⁴, 1.58×10⁻⁸, −8.308×10⁻¹², −1.931×10⁻¹⁵; maximum greenhouse 0.356, 6.171×10⁻⁵, 1.698×10⁻⁹, −3.198×10⁻¹², −5.575×10⁻¹⁶; early Mars 0.32, 5.547×10⁻⁵, 1.526×10⁻⁹, −2.874×10⁻¹², −5.011×10⁻¹⁶. The 2013 paper is Kopparapu et al. 2013, ApJ 765, 131.
Ribas et al. 2017, A&A 603, A58 — Proxima Centauri: T_eff = 2980 ± 80 K, L = 0.00151 ± 0.00008 L☉. The same central values as the blackbody desk.
Anglada-Escudé et al. 2016, Nature 536, 437 — Proxima b: semimajor axis 0.0485 AU, credible interval 0.0434–0.0526 AU. That paper’s star was T_eff = 3050 K and L = 0.00155 L☉. This preset keeps the later stellar parameters and the discovery orbit.
Agol et al. 2021, PSJ 2, 1 — TRAPPIST-1: T_eff = 2566 ± 26 K, planet e semimajor axis 0.02925 ± 0.00025 AU. Luminosity (5.53 ± 0.19) × 10⁻⁴ L☉ adopted from Ducrot et al. 2020. 2566 K is below 2600 K, so the zone is not drawn.
Jenkins et al. 2015, AJ 150, 56 — Kepler-452b: T_eff = 5757 ± 85 K, semimajor axis 1.046 AU (+0.019, −0.015), insolation 1.10 S⊕ (+0.29, −0.22), equilibrium temperature 265 K (+15, −13) assuming Bond albedo 0.3 and complete redistribution. L/L☉ on this preset is 1.10 × (1.046)². They compared the planet with Kopparapu et al. 2013 and called it slightly outside the conservative zone.

Classroom picture of insolation and a 1-D habitable zone. Full redistribution, no greenhouse. Mars can sit in the strip without a liquid-water surface. Not a catalog and not a habitability verdict. Earthrise preview: NASA / Bill Anders (public domain).

Keep going

Curve

Axis	Orbital distance in AU, logarithmic. Drag the marker, or use the distance slider.
Optimistic	Recent Venus to early Mars. The wider band.
Conservative	Runaway greenhouse to maximum greenhouse. Drawn inside the optimistic band.
Outside the window	If Teff is below 2600 K or above 7200 K, the bands are not drawn. Flux and T_eq still update. TRAPPIST-1 e is the preset that shows this.

Systems	Earth, Venus, Mars, Proxima b, TRAPPIST-1 e, Kepler-452b. Each sets luminosity, temperature, distance, and a starting albedo.
Albedo	Earth 0.294, Venus 0.77, Mars 0.250, from the NASA fact sheets. An exoplanet preset starts on Earth’s value and says so. It is not a measurement of that planet.
Sliders	Luminosity and distance are logarithmic. Temperature and albedo are linear. Typing a number leaves the preset.

S / S⊕	(L/L☉) / d². Albedo does not enter.
Flux	That ratio times 1361.0 W/m².
T_eq	[F (1−A) / (4σ)]^¼, in kelvin and in °C. Full redistribution. No greenhouse.
Implied radius	The radius that satisfies L = 4πR²σT⁴ for the luminosity and temperature you set.
Zone	Where the distance sits among the four limits, for the central values only. Uncertainties are in the note, not in the label.

Strip	The canvas takes focus. Left and Right nudge the distance. Shift is a coarser step.
Buttons	Enter or Space selects. The pressed button is aria-pressed. Targets are at least 44 pixels on a side.
This dialog	The question mark, or ?, opens it. Escape closes it. Tab stays inside until you close it. Click the dim page behind the card to close it.

Habitable zone
flux and temperature

Inside the strip is a flux limit. It is not a living world.

Cited, not invented

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Habitable zoneflux and temperature

Inside the strip is a flux limit. It is not a living world.

Cited, not invented

Blackbody

HR diagram

Stars

Exoplanets

Nearest stars

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Habitable zone
flux and temperature