Wien's peak: λ_max = 2.898×10⁶ nm·K / T Stefan-Boltzmann: P = σT⁴ (σ = 5.67×10⁻⁸ W/m²K⁴)
About Blackbody Radiation & Planck's Law
A blackbody is an idealised object that absorbs all incoming electromagnetic radiation and re-emits it purely as a function of its temperature. Max Planck solved the "ultraviolet catastrophe" in 1900 by proposing that electromagnetic energy is quantised, leading to his famous formula: B(λ,T) = (2hc²/λ⁵) · 1/(e^(hc/λkT) − 1). This single equation correctly predicts the complete spectral shape of radiation from the Sun, stars, incandescent light bulbs, and even the Cosmic Microwave Background at 2.73 K.
Drag the temperature slider from 500 K to 30 000 K and watch the spectral radiance curve shift. The yellow marker tracks Wien's displacement law peak (λ_max = 2898 μm·K / T); the colour swatch and star presets show you how stellar colour relates to surface temperature. Toggle the logarithmic y-axis, Wien's approximation overlay, or normalised view to explore different aspects of the distribution.
Frequently Asked Questions
What is Wien's displacement law?
Wien's law states that the peak wavelength of a blackbody spectrum shifts inversely with temperature: λ_max = b/T, where b = 2.898 × 10⁻³ m·K. At 5778 K (the Sun's photosphere) this gives λ_max ≈ 501 nm — green light, right in the middle of the visible band. Our eyes evolved to be most sensitive near this wavelength, which is no coincidence.
Why does total radiated power increase so steeply with temperature?
The Stefan-Boltzmann law says total power per unit area P = σT⁴, where σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴. The fourth-power dependence is dramatic: doubling the temperature increases radiated power by a factor of 16. The Sun emits about 63 MW/m² at its surface, while a red dwarf star at 3000 K emits only around 4.6 MW/m².
What was the "ultraviolet catastrophe"?
Before Planck's 1900 breakthrough, classical physics (the Rayleigh-Jeans law) predicted that a hot object should radiate infinite energy at short wavelengths — an obviously wrong result called the ultraviolet catastrophe. Planck resolved it by assuming energy is emitted in discrete quanta E = hν, which suppresses the high-frequency tail and perfectly matches measured spectra. This quantisation assumption launched the entire field of quantum mechanics.
How does Wien's approximation differ from Planck's law?
Wien's approximation replaces the Planck factor 1/(e^x − 1) with e^(−x), valid when x = hc/λkT ≫ 1 (i.e., at short wavelengths or low temperatures). It underestimates radiance at long wavelengths but is simpler and was historically the first successful formula fitting part of the blackbody curve. Toggle the "Show Wien's approximation" option to compare the two curves directly.
What colour is the Sun when viewed from space?
Although Wien's peak for the Sun falls at ~501 nm (green), the Sun appears white from space because it emits substantial energy across the entire visible spectrum from 380–750 nm. The roughly flat spectral radiance across the visible band produces white light. At ground level, Rayleigh scattering removes short wavelengths, making the Sun appear yellow-white and the sky blue.
Why do incandescent bulbs produce so much heat?
A standard incandescent filament runs at about 2856 K. At this temperature, Wien's peak is at ~1015 nm — deep in the infrared. Only about 5–10% of the emitted energy falls in the visible range; the rest is wasted as heat. Energy-efficient LEDs produce light through a fundamentally different (non-thermal) electroluminescent process, which is why they are far more efficient.
What is the Cosmic Microwave Background and why is it a blackbody?
The CMB is the thermal afterglow of the Big Bang, now cooled to 2.725 K. It has the most perfect blackbody spectrum ever measured — deviations are less than 50 parts per million. At this temperature, Wien's peak falls at about 1.06 mm (microwave). Its near-perfect blackbody shape tells us the early universe was in thermal equilibrium for hundreds of thousands of years before recombination.
How do astronomers use blackbody radiation to measure stellar temperatures?
By comparing a star's colour index — the ratio of brightness in two wavelength bands (e.g., B and V filters) — astronomers can fit a Planck curve and read off the effective surface temperature. Hot O-type stars (>30 000 K) appear blue-white, while cool M-type dwarfs (~3000 K) appear deep red. This method, called photometric classification, works even for stars too faint for spectroscopy.
Can real objects be perfect blackbodies?
No real object is a perfect blackbody — the emissivity ε (ranging from 0 to 1) scales the Planck spectrum: M = εσT⁴. Polished metals have ε ≈ 0.02–0.1 (poor emitters), whilst matte black paint or human skin have ε ≈ 0.95–0.98 (near-perfect emitters). Laboratory blackbody standards use a cavity with a tiny aperture — radiation entering is trapped by multiple reflections and almost never escapes, making ε effectively 1.
What happens at very high temperatures like 30 000 K?
At 30 000 K (an O-type star), Wien's peak falls at ~97 nm — well into the far-ultraviolet. Most of the star's radiated energy is invisible to human eyes and is absorbed by Earth's atmosphere. Such stars ionise the gas around them, creating HII regions (emission nebulae) visible in telescope images as colourful clouds of glowing hydrogen.