Blackbody Radiation Explorer

Adjust temperature to see how thermal emission changes shape and peak wavelength.

Worked example — Blackbody Radiation Explorer

Wien peak wavelength	502.04 nm
Peak spectral band	Visible

These are the values the calculator shows when the page opens. Change any input above to recompute them.

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Lesson

The theory — Blackbody Radiation Explorer

Planck's law gives the brightness a perfect thermal emitter produces at each wavelength, from one number: its temperature. Getting it right forced physics to accept that energy comes in discrete packets — this curve is where the quantum began.

What each symbol means

B(λ, T): the spectral radiance — brightness per unit wavelength. The plot on this page shows its shape only; the readout gives no radiance value.
λ_max: the peak wavelength, the one number this page prints: 502.04 nm at 5772 K, 241.48 nm at 12000 K, 9659.24 nm at 300 K.
h: Planck's constant. Set it to zero and the law collapses back to the classical prediction that failed.
k_B T: the thermal energy scale. A wavelength is bright when its photon costs less than this, and dark when it costs much more.

Where the formula comes from

Count the ways light can vibrate inside a warm box. Short wavelengths fit in far more ways than long ones — the count grows as 1/λ⁴. Classical physics then hands every one of them the same average energy, k_B T.
That prediction is a disaster. With no shortest wavelength, the total radiated energy is infinite — every warm object, including you, would flood the room with ultraviolet. This is the ultraviolet catastrophe, and it was not a rounding error but a flat contradiction with the fact that warm objects sit there quietly.
Planck's repair: a vibration at frequency ν may hold energy only in whole multiples of hν. A short wavelength means a large hν, so when hν greatly exceeds k_B T there is not enough thermal energy to buy even one packet, and that mode stays dark.
That is the e^(hc/λk_B T) − 1 sitting in the denominator of the law. It suppresses short wavelengths exponentially, which is what makes the curve turn over instead of climbing forever — and a curve that turns over has a peak. Differentiate to find it and you get the wavelength this page reports.

How to read what you see

The curve is drawn normalised to its own maximum, so its height carries no information — every temperature reaches the top of the frame. Only the shape and the position of the peak are meaningful here. In reality a hotter body is brighter at every wavelength, not merely bluer, and that difference is invisible on this plot by design.

Assumes: A body in thermal equilibrium at one definite temperature, opaque, absorbing everything that lands on it and reflecting nothing. Real surfaces fall short of that by a factor called emissivity, which scales the whole curve down without changing its shape or its peak.
Breaks when: Planck's law describes thermal emission only. A fluorescent tube, a neon sign and a laser all produce light without being hot, and their spectra are narrow lines that this curve does not approximate at any temperature. The page shows a quieter limit too: load the room-temperature preset and the peak moves to 9659.24 nm, far off the right-hand edge, so the plot draws no peak marker at all — the physics is unchanged, the window is simply too narrow to contain it.

The Sun's peak is green 🖖

Load the Sun preset: the tool reports a peak at 502 nm and paints its colour chip green, because 502 nm is green light. Yet sunlight is white. A peak is one wavelength, while your eye sums the entire curve, and a 5772 K Planck spectrum delivers ample red and blue on either side of that maximum — they average to white. The chip is honest about the peak and misleading about the star. Worth keeping in view: of the four presets only the Sun's peak lands in the visible band at all, with the filament out at 1035 nm in the infrared and the 12000 K star down at 241 nm in the ultraviolet.

Why everything warm glows 🖖

Anything above absolute zero radiates light, and this tool shows the one rule behind it: the shape of a blackbody's spectrum depends only on its temperature, never on what it is made of. A lump of iron and a distant star at the same temperature emit exactly the same curve. Color is a thermometer — slide from red-hot to blue-white and you are reading temperature straight off the glow.

The universe is the perfect blackbody 🖖

The most flawless blackbody ever measured sits in no laboratory — it is the whole sky. The cosmic microwave background, left over from about 380,000 years after the Big Bang, matches a Planck spectrum at T = 2.725 K to better than 0.01%, peaking near 1.06×10⁻³ m in the microwave range. NASA's FIRAS instrument found no deviation larger than a few parts in 10⁵, making cosmology's oldest light also physics' cleanest thermal curve.

Problem solved in full

Wien's constant from Planck's law for a 5772 K surface 6 steps
Sunlight comes from a 5772 K surface. Derive Wien's constant from Planck's law rather than looking it up, find the peak wavelength — then get the Sun's total power output from the same two numbers.
1. Planck's law gives the intensity per unit wavelength. The peak is where its derivative vanishes, so differentiate — and the result will not be an ordinary equation.
  
  $B_{λ} (T) = \frac{2 h c ^{2}}{λ ^{5}} \frac{1}{e ^{h c / λk T} - 1}, \frac{\partial B _{λ}}{\partial λ} = 0$
2. Substituting the dimensionless group x = hc/λkT collapses the derivative into a transcendental equation with no closed-form root. Iterating from x = 5 converges in a handful of passes.
  
  $x = \frac{h c}{λk T} : x = 5 (1 - e^{- x}) ⟹ x = 4.9651$
3. Because x is a pure number, the product λT is a constant made only of h, c and k. This is why Wien's displacement constant exists at all: nothing about the object survives the algebra.
  
  $b = \frac{h c}{4.9651 k} = \frac{1.9864 \times 1 0 ^{- 25}}{6.8555 \times 1 0 ^{- 23}} = 2.8978 \times 1 0^{- 3} m K$
4. Divide by the temperature. 502 nm is green, which is worth pausing on — the Sun peaks in green and looks white, because the eye integrates the whole curve rather than reporting its maximum.
  
  $λ_{m a x} = \frac{b}{T} = \frac{2.8978 \times 1 0 ^{- 3}}{5772} = 5.0204 \times 1 0^{- 7} m = 502.04 nm$
5. The area under the same Planck curve is Stefan's law, and it is the fourth power that makes stars so unforgiving: 10% hotter is 46% brighter.
  
  $F = σ T^{4} = 5.6704 \times 1 0^{- 8} \times 577 2^{4} = 6.294 \times 1 0^{7} W/ m^{2}$
6. Multiply by the surface area. This is the solar luminosity, and it agrees with the measured value to the digits given.
  
  $L = 4 π R_{⊙}^{2} F = 6.082 \times 1 0^{18} \times 6.294 \times 1 0^{7} = 3.828 \times 1 0^{26} W$
Answer

502.04 nm, and 3.83 × 10²⁶ W. The second number is the striking one: the colour of sunlight and the Sun's radius are enough to recover its luminosity to four significant figures, with no measurement of brightness anywhere in the chain. That is the whole reason stellar astrophysics works. A star is too far away to weigh and too bright to approach, but its spectrum gives a temperature, an angular size gives a radius, and Stefan's law converts the pair into an energy budget — which is where a main-sequence lifetime, and eventually a supernova, come from.

$λ_{m a x} = \frac{b}{T} = 502.04 nm; 4 π R_{⊙}^{2} σ T^{4} = 3.828 \times 1 0^{26} W$

References (3)

The distribution the curve is drawn from: M. Planck, "Ueber das Gesetz der Energieverteilung im Normalspectrum." Annalen der Physik 309(3), 553–563, 1901.
The 5772 K solar effective temperature the Sun preset uses: A. Prša et al., "Nominal Values for Selected Solar and Planetary Quantities: IAU 2015 Resolution B3." The Astronomical Journal 152(2), 41, 2016.
The Wien displacement constant used to place the peak marker: E. Tiesinga, P. J. Mohr, D. B. Newell and B. N. Taylor, "CODATA recommended values of the fundamental physical constants: 2018." Reviews of Modern Physics 93(2), 025010, 2021.

By the 42Math team · Updated 2026-07-11

Reviewed against published sources. Corrections welcome. Editorial policy · Corrections

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Example problems

Room temperature - 300 K peaks at 9659 nm - 9.7 micrometres, deep in the infrared and squarely in the band thermal cameras are built for.
Sun-like photosphere - 5772 K peaks at 502 nm, the only preset here whose peak lands in the visible band at all. The tool paints that chip green; sunlight is white.
Hot O/B star - 12000 K peaks at 241 nm, in the ultraviolet. The star still looks blue-white, because what reaches your eye is the tail of the curve rather than its maximum.
Incandescent filament - 2800 K peaks at 1035 nm, past the red end of vision. A filament lamp is mostly an infrared heater that happens to glow.