Blackbody Radiation Simulator

What is Planck’s blackbody radiation law?

Planck’s law describes the spectral radiance of electromagnetic radiation emitted by a blackbody in thermal equilibrium at absolute temperature T. As temperature increases, the total power radiated per unit area rises in proportion to T⁴ (Stefan-Boltzmann law: P/A = σT⁴), while peak emission wavelength shifts towards shorter wavelengths following Wien’s displacement law: λ_max · T = 2.898 × 10⁻³ m·K.

A blackbody is an object that absorbs all the light that falls on it, at every wavelength. Because it absorbs perfectly it also emits perfectly, and the light it gives off depends on nothing but its temperature: not on what it is made of, nor on its shape. Max Planck found the formula for that light in 1900, and it is the curve this simulator draws. A lump of iron, a bulb filament and a star at the same temperature glow with nearly the same colour, apart from how far each falls short of the ideal.

Nothing is exactly a blackbody, but a great deal comes close. The inside of a furnace seen through a small hole is the textbook case. Stars are close, apart from the absorption lines their atmospheres cut into their spectra. Your skin is close in the infrared, where its emissivity is about 0.98. And the cosmic microwave background, the afterglow of the early universe, is a blackbody at 2.72548 K.

Reading the simulator

The first view sweeps the temperature from 1,500 K up to 12,000 K and back. Any touch stops it where it is; the Sweep button starts it again. Then set a temperature in any of these ways:

• Drag along the colour strip at the foot of the scene, which runs from 1,000 to 40,000 K on a log scale, with the stellar classes under it.
• Drag the peak of Planck’s curve sideways. The temperature follows from Wien’s law, T = b / λmax, so the chart becomes a Wien’s law calculator.
• Type a temperature in kelvin, Celsius or Fahrenheit, from 1 K to 100,000 K, or choose one of the named temperatures: lava, lamps, the Sun and other stars, you, and the cosmic microwave background.
• Use the keys on the focused scene: the arrows change the temperature by 1 percent, Page Up and Page Down by 10 percent, and Home and End go to 1,000 and 40,000 K.

The glow in the dark panel is the colour a blackbody at that temperature shows on your screen, with the photons it gives off: a streak in its own colour for each visible photon, a faint ring for infrared, which you feel as warmth but cannot see, a violet ring for ultraviolet, and a wider ring for a microwave, longer than 1 mm, as 91 percent of the cosmic microwave background’s photons are. The chart can switch to log-log axes, from 10 nm to 10 mm, where every peak fits, and below 1,000 K it does so on its own, so the peaks of a body at room temperature and of the cosmic microwave background can be seen.

Planck’s curve and its two peaks

Planck’s law gives the spectral radiance, the power a surface sends out per square metre, per steradian and per unit of wavelength: B(λ, T) = (2hc² / λ⁵) × 1 / (e^(hc/λkT) − 1), where h is Planck’s constant, c the speed of light and k Boltzmann’s constant. At the Sun’s 5,772 K the curve climbs steeply out of the ultraviolet, peaks at 502 nm and falls away slowly through the infrared. That peak per unit wavelength is where Wien’s law puts it.

The same light drawn per unit of frequency peaks somewhere else: at 339 THz, which is a wavelength of 883 nm, in the infrared. Neither peak is wrong. The two curves cut the same power into different slices: equal steps of frequency are narrow in wavelength at the blue end and wide at the red end, so the frequency curve’s peak sits further to the red. The Peak per frequency readout gives it at every temperature, as νmax = 2.821 kT / h.

So why is the Sun not green? Because the peak is only the top of a very broad curve. Sunlight carries plenty of every visible colour, with 46.5 percent of a 5,772 K blackbody’s power between 380 and 780 nm, and together they look white. Drawn here, a 5,772 K blackbody is #fff0ea, a white with the faintest warmth in it. The Electromagnetic Spectrum Explorer puts that peak on the whole spectrum, from radio waves to gamma rays.

Worked example: Wien’s law

The peak wavelength is Wien’s constant over the temperature in kelvin, λmax = b / T, with b = 2.898 × 10⁻³ m K.

1. The Sun. 2.898 × 10⁻³ ÷ 5,772 = 5.021 × 10⁻⁷ m, which is 502 nm, in the green.
2. You. First turn 37 °C into kelvin: 37 + 273.15 = 310.15 K. Then 2.898 × 10⁻³ ÷ 310.15 = 9.34 × 10⁻⁶ m, which is 9.34 µm, in the thermal infrared. Thermal cameras look at 8 to 14 µm, the band where bodies near room temperature shine brightest and the air lets their light through.
3. The cosmic microwave background. 2.898 × 10⁻³ ÷ 2.72548 = 1.063 × 10⁻³ m, which is 1.06 mm, in the microwaves.

Both constants are on the Physical Constants page, derived as this simulator derives them: σ and b come from h, c and k, which have been exact since 2019.

Worked example: the Stefan-Boltzmann law

Each square metre of a blackbody radiates M = σT⁴, with σ = 5.6704 × 10⁻⁸ W m⁻² K⁻⁴, at every wavelength together.

1. The Sun’s surface. 5.6704 × 10⁻⁸ × 5,772⁴ = 6.294 × 10⁷ W/m², which is 62.9 MW from every square metre.
2. The whole Sun. Its surface is 4πR², with the IAU’s nominal radius of 695,700 km, so it radiates 6.294 × 10⁷ × 4π × (6.957 × 10⁸)² = 3.83 × 10²⁶ W, the IAU’s nominal solar luminosity of 3.828 × 10²⁶ W.
3. Twice as hot. σ(2T)⁴ = 16σT⁴, so doubling the temperature multiplies the power by 16. That fourth power is why a small rise in temperature makes such a difference to the power.
4. You. A blackbody at 37 °C radiates 5.6704 × 10⁻⁸ × 310.15⁴ = 525 W/m². A room at 20 °C, 293.15 K, radiates 418.8 W/m² back, and with skin’s emissivity of 0.98 the net loss is 0.98 × (524.7 − 418.8) = 104 W/m². Bare skin is usually a few degrees cooler than the inside of you, so the real figure is smaller.

Colour temperature in lighting and photography

A lamp’s colour temperature is the temperature of the blackbody whose colour it matches. For a tungsten filament, which glows because it is hot, that is close to the filament’s own temperature, and the CIE’s standard for household tungsten light, illuminant A, is defined as a Planckian radiator at about 2,856 K.

An LED or a fluorescent tube makes its light another way, and its spectrum is nothing like Planck’s curve: a white LED’s is usually a blue peak and a broad yellow hump. So its rating is a correlated colour temperature, the temperature of the blackbody whose colour it is nearest. ANSI C78.377 sorts LED lamps into nominal categories, 2,700, 3,000, 4,000, 5,000 and 6,500 K among them, and lets a lamp in each sit a little above or below the blackbody line, which is why two 3,000 K lamps can look slightly different. 2,700 K and 3,000 K are the warm whites sold for homes, here #ffad58 and #ffb86d; 4,000 K is the trade’s neutral white, #ffd3a5; and 5,000 K and 6,500 K are the cool daylight whites. Gas discharge lamps go further still, with light in a few sharp lines, like the hydrogen lines in the Bohr Model and Hydrogen Spectrum Simulator.

Screens are set to a daylight standard, CIE illuminant D65, whose correlated colour temperature is about 6,504 K. Daylight is not a blackbody: D65 sits 0.0032 above the blackbody line in CIE 1960 uv, towards green. That is why the 6,500 K blackbody here, #fff8fe, is a faint pink beside your screen’s white.

Your eyes adapt to the light around you, and a camera’s white balance does the same job, scaling red, green and blue until a white card is white. A room lit at 2,700 K looks orange in a photograph set for daylight and white once your eyes have adjusted. The two white cards in the simulator show the light both ways. Photographers call 3,200 K tungsten, the light of studio tungsten-halogen lamps and the light tungsten-balanced film is made for.

Star colours and classes

A star’s light is close to a blackbody’s at its surface temperature, so its colour follows its temperature. Betelgeuse, at 3,600 K, peaks at 805 nm, just beyond the red, and looks orange. The Sun peaks in the green and looks white. Rigel, at 12,000 K, peaks in the ultraviolet, at 241 nm, and looks blue-white. Astronomers sort stars into spectral classes by the lines in their spectra, and the lines follow the temperature, so the classes run from cool to hot in this order.

The names in the table are tradition: astronomers call the Sun a yellow star, though its blackbody colour is white, and the yellow is partly the sky’s doing, since the atmosphere scatters blue light out of the direct beam.

Why hot bulbs waste energy

At 2,700 K a blackbody puts 8.4 percent of its power into visible light and almost all the rest into infrared. Weighting each wavelength by how sensitive the eye is to it gives the luminous efficacy of the radiation: about 12 lm/W at 2,700 K, against the 683 lm/W of pure green light at 555 nm, the most any light can have. Hotter is better, up to a point. The efficacy climbs to a maximum of about 95 lm/W near 6,600 K and falls beyond it as the light moves into the ultraviolet. A tungsten filament cannot get near that: tungsten melts at 3,422 °C, so filaments run far cooler, and most of what they radiate is heat.

These efficacies use an analytic fit to the CIE’s eye sensitivity, which is weakest in the far red, so below about 2,000 K, where almost all the visible light is deep red, they are rough.

Named temperatures

Each photon in these spectra carries an energy E = hc / λ, and the idea that light’s energy comes in such packets grew out of Planck’s curve. The Photoelectric Effect Simulator shows where it led, and the Photon Energy Calculator gives the energy of a photon of any wavelength.

What the colours on your screen can and cannot show

Every colour here is computed, not chosen. The temperature gives a chromaticity on the Planckian locus, from Krystek’s 1985 rational fit up to 15,000 K and from Planck’s curve summed against an analytic fit to the CIE 1931 colour matching functions above that; the chromaticity becomes XYZ, and XYZ becomes sRGB by the matrix in the W3C’s CSS Color specification, the same one the spectrum explorer’s rainbow uses.

• Below about 1,900 K the true colour is more saturated than any sRGB screen can show, so white is added until it fits, and the swatch is paler than the light itself. The Colour readout says so.
• Brightness is shown apart from colour. Every swatch is drawn at full brightness, since no screen can be as bright as the Sun, so the meter under the sphere shows brightness on a log scale: a 2,700 K surface is 0.0064 times as bright as the Sun’s.
• Below 1,000 K nothing is drawn. Things first glow a dull red, seen only in the dark, near 798 K, the Draper point.
• Your screen changes them. The colours are worked out for an sRGB screen viewed beside daylight. A wide-gamut screen, a night mode or a phone’s colour profile shifts what you see.

Common mistakes

• Leaving Celsius in Wien’s law. Wien’s and Stefan’s laws need kelvin. For a body at 37 °C, 2.898 × 10⁻³ ÷ 37 gives 78.3 µm, more than eight times the true 9.34 µm. The field here takes °C and °F and shows the kelvin it uses.
• Reading the peak as the colour. The peak is not the colour you see. The Sun peaks in the green and is white; a 4,000 K lamp peaks at 724 nm, in the deep red, and is a neutral white.
• Calling an LED a blackbody. A 3,000 K LED is not at 3,000 K and its spectrum is not Planck’s. Its rating only says which blackbody its colour is closest to.
• Thinking power grows in step with temperature. It grows as the fourth power: a star twice as hot radiates 16 times as much from each square metre.
• Thinking cool things do not radiate. Everything above absolute zero does. You radiate hundreds of watts per square metre, all of it infrared, and a thermal camera sees it.

Constants used: Speed of light in vacuum (c), Planck constant (h), Boltzmann constant (k), Stefan-Boltzmann constant (σ) and Wien’s displacement constant (b).