Blackbody Radiation Calculator — Planck's Law, Radiance, Exitance & Peak Spectrum

Blackbody Radiation Calculator — Planck's Law, Radiance & Peak Spectrum

Our blackbody radiation calculator will help you calculate the radiation spectrum of a blackbody, or of any body that can be closely approximated as one. All you need to provide is the body's temperature and emissivity to determine its radiance and peak spectral radiance. You can even specify a particular wave's wavelength, frequency, or wavenumber to find the spectral radiance at that point. You can also work in wave space (watts) or in photon space (photons per second).

Blackbody radiation is the fundamental concept that led us toward quantum physics. The sections below introduce Planck's law and show how to calculate blackbody radiation from temperature. If terms like wavelength, frequency, and wavenumber are new to you, our wavelength calculator is a good place to start.

How to use this blackbody radiation calculator

This blackbody radiation spectrum calculator is a versatile tool:

1. Select the spectral parameter used for the calculation — wavelength, frequency, or wavenumber. The chosen parameter controls which peak is reported and which optional spectral value you can enter.
2. Select the calculation space — wave space (results in watts) or photon space (results in photons per second).
3. Choose a unit system — metric (W·m⁻²) or American (BTU·h⁻¹·ft⁻²) for the power-flux outputs. Temperature can be entered in K, °C, or °F.
4. Enter the characteristics of the body — its temperature and emissivity (ε between 0 and 1).
5. The calculator instantly returns the total radiance and radiant emittance (exitance) in wave space. In photon space it also returns the total photon radiance and photon exitance.
6. It automatically computes the peak spectral parameter (peak wavelength, frequency, or wavenumber) and the corresponding peak spectral radiance. With the photon-space option, it also reports the peak in photon space.
7. To get the spectral radiance at a specific wave, enter that wavelength, frequency, or wavenumber in the optional field.

Blackbody radiation

A blackbody is an ideal body that absorbs all electromagnetic radiation incident on it, reflecting nothing back. A perfect blackbody doesn't exist in nature, but many objects approximate one. Blackbody radiation is the electromagnetic radiation emitted spontaneously by a blackbody in thermal equilibrium with its surroundings. The spectrum is continuous and depends only on the body's temperature.

At room temperature most of the spectrum lies in the infrared region, invisible to the naked eye. If the body were hotter than about 500 °C (930 °F), part of the emission would fall in the visible region and the body would glow dull red. Although they are not perfect blackbodies, the thermal radiation of many objects can be approximated as blackbody radiation, especially as a first approximation.

Planck's law for blackbody radiation

Classical theory could not explain blackbody radiation because it assumed energy is continuous; it predicted that the radiated energy would tend to infinity at higher frequencies — the so-called ultraviolet catastrophe. In 1901, Max Planck proposed that energy is quantized, increasing or decreasing only in small increments. This successfully explained blackbody radiation and paved the way for quantum physics.

According to Planck's law, the spectral radiance of a blackbody per unit wavelength is:

B_λ(λ, T) = 2hc² / [ λ⁵ ( e^{hc / λk_BT} − 1 ) ]

• B_λ — spectral radiance per unit wavelength (W·sr⁻¹·m⁻³)
• λ — wavelength (m), T — absolute temperature (K)
• h = 6.62607015 × 10⁻³⁴ J·s — Planck constant
• c = 2.99792458 × 10⁸ m/s — speed of light
• k_B = 1.380649 × 10⁻²³ J/K — Boltzmann constant

Calculating blackbody spectrum radiation with wavelength or wavenumber

Planck's law can be written in terms of frequency or wavenumber instead of wavelength:

• Per frequency: B_ν(ν, T) = 2hν³ / [ c² ( e^{hν / k_BT} − 1 ) ]  (W·sr⁻¹·m⁻²·Hz⁻¹)
• Per wavenumber: B_ν̃(ν̃, T) = 2hc²ν̃³ / ( e^{hcν̃ / k_BT} − 1 )  (W·sr⁻¹·m⁻¹, with ν̃ = 1/λ)

Because each form has a different mathematical shape, the peak you find in wavelength space does not correspond to the peak in frequency or wavenumber space. That's why this calculator lets you pick the spectral parameter explicitly.

How to calculate blackbody radiation in photon space

In wave space we measure energy (watts). In photon space we count photons per second by dividing the spectral radiance by the energy of a single photon (E = hν = hc/λ). The total photon exitance of a blackbody scales with the cube of temperature:

M_q = ε × 1.5204 × 10¹⁵ × T³   (photons·s⁻¹·m⁻²)

The corresponding photon radiance is M_q / π. Selecting the photon-space option also reports the peak of the photon distribution, which sits at a different position than the wave-space peak.

Calculating peak wavelength of blackbody radiation

The peak of the wavelength distribution follows Wien's displacement law:

λ_max = b / T,   b = 2.8977720 × 10⁻³ m·K

The total radiance and radiant emittance come from the Stefan–Boltzmann law: M = εσT⁴ and L = M / π, where σ = 5.670374 × 10⁻⁸ W·m⁻²·K⁻⁴.

Emissivity of a real body

A perfect blackbody has emissivity ε = 1. Real objects are gray bodies with ε < 1, so they emit a fraction ε of the radiation a perfect blackbody would emit at the same temperature. Multiply the blackbody radiance, exitance, and spectral radiance by ε to obtain the values for a real body. Typical emissivities: polished metal ≈ 0.05, oxidized steel ≈ 0.8, human skin ≈ 0.98, soot ≈ 0.95.

Metric and American (US) units

• Temperature: Kelvin (K, scientific standard), Celsius (°C, metric), Fahrenheit (°F, American).
• Power flux (metric): radiant emittance in W·m⁻², radiance in W·sr⁻¹·m⁻².
• Power flux (American): radiant emittance in BTU·h⁻¹·ft⁻², radiance in BTU·h⁻¹·sr⁻¹·ft⁻²  (1 W·m⁻² ≈ 0.317 BTU·h⁻¹·ft⁻²).

Example: the Sun's surface

1. Temperature ≈ 5,778 K, emissivity ≈ 1.
2. Radiant emittance: M = σT⁴ ≈ 6.32 × 10⁷ W·m⁻².
3. Peak wavelength: λ_max = b / T ≈ 501.5 nm (green-yellow, visible).

FAQs

What is a blackbody?

A blackbody is an idealized object that absorbs all incident electromagnetic radiation and re-emits energy that depends only on its temperature. It is the perfect emitter and the perfect absorber.

Why doesn't the peak wavelength match the peak frequency?

The spectral radiance has a different mathematical form per unit wavelength, per unit frequency, and per unit wavenumber. Maximizing each gives a different location on the spectrum, so the peaks do not correspond to the same point. Use the spectral-parameter selector to choose which peak you want.

What is the difference between wave space and photon space?

Wave space measures radiated energy in watts. Photon space counts the number of photons emitted per second, obtained by dividing the energy by the per-photon energy hν. The two distributions peak at different positions.

How does emissivity affect the result?

Emissivity ε scales every radiated quantity linearly. A real body with ε = 0.8 emits 80% of the radiance, exitance, and spectral radiance of a perfect blackbody at the same temperature.

What is the radiant emittance (exitance)?

Radiant emittance, or exitance, is the total power radiated per unit surface area into the hemisphere above the surface (W·m⁻²). It equals εσT⁴. The radiance is the exitance divided by π and is measured per steradian (W·sr⁻¹·m⁻²).