About

A blackbody is a theoretical object that absorbs all incident electromagnetic radiation and re-emits energy solely as a function of its thermodynamic temperature T. The spectral distribution of this emission follows Planck's radiation law, derived by Max Planck in 1900 to resolve the ultraviolet catastrophe predicted by classical Rayleigh-Jeans theory. Errors in thermal radiation calculations propagate directly into material science, astrophysical modeling, and infrared sensor calibration. Misjudging peak emission wavelength by even 5% can misclassify a star's spectral type or invalidate a pyrometer reading.

This calculator computes the full spectral radiance curve B(λ, T) across the electromagnetic spectrum using Planck's law with CODATA 2018 constants. It derives peak wavelength λ_max via Wien's displacement law and total radiant exitance M via the Stefan-Boltzmann law. The visible spectrum region (380 - 780 nm) is highlighted on the spectral plot with approximate CIE chromaticity color rendering. Note: real surfaces deviate from ideal blackbody behavior by an emissivity factor ε < 1. This tool assumes ε = 1.

Formulas

The spectral radiance of a blackbody at temperature T and wavelength λ is given by Planck's radiation law:

B(λ, T) = 2hc²λ⁵ ⋅ 1e^hcλk_BT − 1

where h = 6.62607015 × 10⁻³⁴ J⋅s (Planck constant), c = 2.99792458 × 10⁸ m/s (speed of light), and k_B = 1.380649 × 10⁻²³ J/K (Boltzmann constant).

Wien's displacement law gives the peak emission wavelength:

λ_max = bT

where b = 2.897771955 × 10⁻³ m⋅K is Wien's displacement constant.

The total radiant exitance (power per unit area) integrated over all wavelengths follows the Stefan-Boltzmann law:

M = σT⁴

where σ = 5.670374419 × 10⁻⁸ W⋅m⁻²⋅K⁻⁴ is the Stefan-Boltzmann constant. For a sphere of radius R, total luminosity is L = 4πR²σT⁴.

Reference Data

Object / Source	Temperature (K)	λ_max (nm)	Peak Region	Radiant Exitance (W/m²)
Cosmic Microwave Background	2.725	1,063,000	Microwave	3.13 × 10⁻⁶
Liquid Nitrogen	77	37,600	Far Infrared	2.00
Dry Ice (CO₂)	195	14,860	Mid Infrared	82.1
Room Temperature	293	9,890	Mid Infrared	418
Human Body	310	9,350	Mid Infrared	524
Boiling Water	373	7,770	Mid Infrared	1,100
Candle Flame	1,800	1,610	Near Infrared	5.96 × 10⁴
Incandescent Bulb	2,500	1,160	Near Infrared	2.22 × 10⁵
Halogen Lamp	3,200	905	Near Infrared	5.96 × 10⁵
Sun (Photosphere)	5,778	501	Visible (Green)	6.32 × 10⁷
Sirius A	9,940	291	Ultraviolet	5.54 × 10⁸
Vega	9,602	302	Ultraviolet	4.82 × 10⁸
Blue Supergiant (Rigel)	12,100	240	Ultraviolet	1.21 × 10⁹
O-Type Star	40,000	72	Extreme UV	1.45 × 10¹¹
Lightning Bolt Channel	30,000	97	Extreme UV	4.59 × 10¹⁰
Nuclear Fireball (1 ms)	100,000	29	Soft X-ray	5.67 × 10¹²
Tokamak Plasma Edge	1,000,000	2.9	X-ray	5.67 × 10¹⁶

Frequently Asked Questions

The exponential term e^hc/(λk_BT) in the denominator of Planck's law controls the cutoff. As T increases, the exponent shrinks for shorter wavelengths, allowing them to contribute more radiance. Wien's law quantifies this: λ_max is inversely proportional to T. Doubling the temperature halves the peak wavelength. At 5778 K (solar photosphere), the peak is at 501 nm (green). At 3000 K (incandescent bulb), it shifts to 966 nm in the near infrared, which is why incandescent bulbs are inefficient visible-light sources.

Real surfaces have emissivity ε(λ, T) ranging from 0 to 1. A graybody has constant ε across all wavelengths. Polished aluminum has ε ≈ 0.05, oxidized steel ≈ 0.79, human skin ≈ 0.98. Multiply this calculator's spectral radiance by ε for realistic values. The Stefan-Boltzmann result becomes M = εσT⁴. Ignoring emissivity in pyrometry causes temperature reading errors that scale with the fourth root of the emissivity ratio.

Planck's law remains valid at all temperatures above absolute zero. At 2.725 K (cosmic microwave background), the peak wavelength is approximately 1.06 mm in the microwave band. The curve shape is preserved. However, at sub-kelvin temperatures, the total radiant exitance becomes extremely small (σT⁴ drops as the fourth power). Practical measurement requires bolometers with noise-equivalent power below 10⁻¹⁸ W/√Hz. The calculator handles temperatures down to 1 K but numerical precision limits appear below approximately 3 K due to floating-point exponent range.

Wien's peak at 501 nm falls in the green region, but human color perception integrates across the entire visible band (380 - 780 nm). The Sun's blackbody curve at 5778 K has significant radiance from violet through red. The roughly flat distribution across the visible spectrum stimulates all three cone types (S, M, L) approximately equally, producing a white percept. Atmospheric Rayleigh scattering removes short wavelengths, shifting ground-level perception toward yellow. No star appears green because any blackbody temperature that peaks in green also emits strongly in red and blue.

The Rayleigh-Jeans law B = 2ck_BT / λ⁴ is the classical limit of Planck's law, valid only when hc / (λk_BT) ≪ 1 (long wavelengths or high temperatures). It predicts radiance increasing without bound as λ → 0, the so-called ultraviolet catastrophe. For the Sun at 5778 K, the approximation diverges significantly below about 3000 nm. This calculator uses the exact Planck formula, so no divergence occurs.

Yes. The calculator numerically integrates Planck's function from 380 to 780 nm and divides by the total exitance (σT⁴ / π for hemispheric spectral radiance). For the Sun at 5778 K, approximately 37% of total energy is in the visible band. An incandescent bulb at 2500 K emits only about 5% in the visible range, confirming its low luminous efficiency. This visible fraction is displayed in the results panel.