Free Blackbody Radiation Calculator

This blackbody radiation calculator computes the spectral radiance, total radiance, and peak emission characteristics of an ideal blackbody or a body that closely mimics one. You only need to supply the object's temperature and emissivity to obtain its radiance in either wave space (measured in watts) or photon space (measured in photons per second). In addition, you can pick a specific wavelength, frequency, or wavenumber to find the spectral radiance at that point. As a result, this tool serves multiple roles: a Planck's law calculator, a spectral radiance calculator, a Wien displacement calculator for peak wavelength, and a general blackbody spectrum calculator.

How to Use the Planck's Law Calculator

Using the calculator is straightforward:

1. Choose the spectral parameter you want to work with — wavelength, frequency, or wavenumber.
2. Decide whether you need the output in wave space (radiant energy) or photon space (photon count).
3. Enter the blackbody's temperature (in kelvin) and its emissivity (a value between 0 and 1).
4. The tool immediately returns the total radiance and the radiance emittance (exitance) in wave space; if photon space is selected, it also shows the total photon radiance and photon emittance.
5. It automatically calculates the peak spectral parameter (peak wavelength, peak frequency, or peak wavenumber) and the corresponding peak spectral radiance. For photon space, it reports the peak spectral photon radiance and its location.
6. To obtain the spectral radiance at a specific spectral coordinate, enter that value (e.g., a particular wavelength) after the initial calculation.

This workflow makes the calculator a convenient spectral radiance calculator across all spectral domains.

Understanding Blackbody Radiation

A blackbody is a theoretical object that absorbs every photon that strikes it, reflecting no electromagnetic radiation. Although perfect blackbodies do not exist in nature, many real objects — stars, heating elements, and cavity radiators — can be treated as close approximations. The radiation emitted by a blackbody in thermal equilibrium depends solely on its temperature, not on its shape or material.

At room temperature, the emitted spectrum lies mostly in the infrared, invisible to the human eye. When the body is heated above about 500 °C (930 °F), a portion of the emission shifts into the visible range, and the body begins to glow with a dull red color. More generally, the entire spectral shape shifts and scales with temperature, a behavior captured precisely by Planck's law.

Planck's Law: The Quantized Breakthrough

Classical electrodynamics predicted that the energy radiated at high frequencies (short wavelengths) would become infinite — the infamous "ultraviolet catastrophe." In 1901, Max Planck resolved this paradox by proposing that energy is emitted and absorbed in discrete packets, or quanta. This bold assumption gave birth to quantum physics and led to the correct distribution law now named after him.

Planck's law for spectral radiance per unit frequency is:

$$B_{\nu}(\nu, T) = \frac{2h\nu^{3}}{c^{2}} \frac{1}{e^{h\nu/(k_{\mathrm{B}}T)} - 1}$$

where:

• $B_{\nu}$ is the spectral radiance (power per unit area per unit solid angle per unit frequency),
• $\nu$ is the radiation frequency,
• $T$ is the absolute temperature of the blackbody,
• $h = 6.62607015 \times 10^{-34}\ \text{J·s}$ is Planck's constant,
• $c = 2.99792458 \times 10^{8}\ \text{m/s}$ is the speed of light in vacuum,
• $k_{\mathrm{B}} = 1.380649 \times 10^{-23}\ \text{J/K}$ is the Boltzmann constant.

The units of $B_{\nu}$ are $\text{W·sr}^{-1}\text{·m}^{-2}\text{·Hz}^{-1}$.

Integrating this spectral radiance over all frequencies gives the total radiance (also called radiance):

$$B = \int_{0}^{\infty} B_{\nu}\,d\nu = \frac{2\pi^{4} k_{\mathrm{B}}^{4} T^{4}}{15 h^{3} c^{2}}$$

with units $\text{W·sr}^{-1}\text{·m}^{-2}$. The total radiance emittance (or exitance), which is the power radiated per unit surface area, is numerically the same in this formulation:

$$M = \frac{2\pi^{4} k_{\mathrm{B}}^{4} T^{4}}{15 h^{3} c^{2}}$$

expressed in $\text{W/m}^{2}$. These two quantities differ in geometric meaning but share the same value for a Lambertian blackbody.

Spectral Radiance in Different Variables

Planck's law can be rewritten using wavelength $\lambda$ or wavenumber $\tilde{\nu}$ instead of frequency.

Wavelength-Dependent Form

$$B_{\lambda}(\lambda, T) = \frac{2hc^{2}}{\lambda^{5}} \frac{1}{e^{hc/(\lambda k_{\mathrm{B}}T)} - 1}$$

This form gives spectral radiance per unit wavelength. When $\lambda$ is expressed in micrometers, $B_{\lambda}$ has units $\text{W·sr}^{-1}\text{·m}^{-2}\text{·μm}^{-1}$.

Wavenumber-Dependent Form

$$B_{\tilde{\nu}}(\tilde{\nu}, T) = 2hc^{2}\tilde{\nu}^{3} \frac{1}{e^{hc\tilde{\nu}/(k_{\mathrm{B}}T)} - 1}$$

Here $\tilde{\nu}$ is the wavenumber (typically in $\text{cm}^{-1}$), and the spectral radiance units become $\text{W·sr}^{-1}\text{·m}^{-2}\text{·(cm}^{-1})^{-1}$.

These three versions are equivalent and can be converted from one to another using the relations $\nu = c/\lambda = c\tilde{\nu}$.

From Wave Space to Photon Space

Instead of radiant power, one can study the emission in terms of the number of photons leaving the surface each second. This is called photon space and is obtained by dividing each spectral radiance expression by the energy of a single photon.

• Frequency photon spectral radiance:

$$B_{\nu}^{P}(\nu, T) = \frac{B_{\nu}}{h\nu} = \frac{2\nu^{2}}{c^{2}} \frac{1}{e^{h\nu/(k_{\mathrm{B}}T)} - 1}$$

Units: $\text{photons/s/sr/m}^{2}\text{/Hz}$.

• Wavelength photon spectral radiance:

$$B_{\lambda}^{P}(\lambda, T) = \frac{B_{\lambda}}{hc/\lambda} = \frac{2c}{\lambda^{4}} \frac{1}{e^{hc/(\lambda k_{\mathrm{B}}T)} - 1}$$

Units: $\text{photons/s/sr/m}^{2}\text{/μm}$.

• Wavenumber photon spectral radiance:

$$B_{\tilde{\nu}}^{P}(\tilde{\nu}, T) = \frac{B_{\tilde{\nu}}}{hc\tilde{\nu}} = 2c\tilde{\nu}^{2} \frac{1}{e^{hc\tilde{\nu}/(k_{\mathrm{B}}T)} - 1}$$

Units: $\text{photons/s/sr/m}^{2}\text{/(cm}^{-1}$).

The total photon radiance (integrated over all frequencies) follows a similar pattern:

$$B^{P} = \int_{0}^{\infty} B_{\nu}^{P}\,d\nu = \frac{16\pi\,\zeta(3)\,k_{\mathrm{B}}^{3} T^{3}}{h^{3} c^{2}}$$

where $\zeta(3) \approx 1.202056903159594$ is the Riemann zeta function evaluated at 3. The photon exitance is then $M^{P} = \pi B^{P}$:

$$M^{P} = \frac{16\pi^{2}\,\zeta(3)\,k_{\mathrm{B}}^{3} T^{3}}{h^{3} c^{2}} \quad (\text{photons/s/m}^{2}).$$

Finding the Peak: Wien's Displacement and Beyond

The wavelength at which $B_{\lambda}$ reaches its maximum can be found by differentiating the expression with respect to $\lambda$ and setting the derivative to zero. The result is Wien's displacement law for an ideal blackbody:

$$\lambda_{\text{peak}} = \frac{hc}{a_{5}\,k_{\mathrm{B}} T}$$

where $a_{5} \approx 4.96511423174$. The corresponding peak spectral radiance is obtained by substituting $\lambda_{\text{peak}}$ back into $B_{\lambda}$.

Similarly, the peak frequency and peak wavenumber for spectral radiance are:

$$\nu_{\text{peak}} = \frac{a_{3}\,k_{\mathrm{B}} T}{h}, \qquad \tilde{\nu}_{\text{peak}} = \frac{a_{3}\,k_{\mathrm{B}} T}{h c}$$

with $a_{3} \approx 2.82143937212$.

In photon space, the maxima occur at different locations because the photon spectral radiances are different functions.

Domain	Peak location	Constant
Wavelength (wave space)	$\lambda_{\text{peak}} = \dfrac{hc}{a_{5}k_{\mathrm{B}}T}$	$a_{5} \approx 4.96511423174$
Frequency (wave space)	$\nu_{\text{peak}} = \dfrac{a_{3}k_{\mathrm{B}}T}{h}$	$a_{3} \approx 2.82143937212$
Wavenumber (wave space)	$\tilde{\nu}_{\text{peak}} = \dfrac{a_{3}k_{\mathrm{B}}T}{hc}$	$a_{3} \approx 2.82143937212$
Wavelength (photon space)	$\lambda_{\text{peak}}^{P} = \dfrac{hc}{a_{4}k_{\mathrm{B}}T}$	$a_{4} \approx 3.92069039487$
Frequency (photon space)	$\nu_{\text{peak}}^{P} = \dfrac{a_{2}k_{\mathrm{B}}T}{h}$	$a_{2} \approx 1.59362426004$
Wavenumber (photon space)	$\tilde{\nu}_{\text{peak}}^{P} = \dfrac{a_{2}k_{\mathrm{B}}T}{hc}$	$a_{2} \approx 1.59362426004$

The numbers $a_{2}, a_{3}, a_{4}, a_{5}$ are the dimensionless roots of certain transcendental equations that arise from setting the derivative of each spectral distribution to zero.

Accounting for Real Materials: Emissivity

No real object is a perfect blackbody, but many can be characterized by their emissivity $\varepsilon$, defined as the fraction of the blackbody power that the actual body emits. Emissivity ranges from 0 to 1, with $\varepsilon = 1$ for an ideal blackbody.

To compute the radiation from a real body, simply multiply the blackbody spectral radiance by $\varepsilon$. For example, the spectral radiance (per wavelength) of an actual surface is:

$$B_{\lambda}^{*}(\lambda, T) = \varepsilon\, B_{\lambda}(\lambda, T)$$

The same scaling applies to all the other measures: total radiance, photon radiance, and peak values. By inputting the emissivity into the calculator, you automatically obtain the realistic, reduced radiation levels for your material of interest.

Whether you are designing a thermal sensor, analyzing stellar spectra, or studying radiative heat transfer, this blackbody radiation calculator — functioning as a Planck's law calculator, spectral radiance calculator, Wien displacement calculator, peak wavelength calculator, and blackbody spectrum calculator — provides the core data you need in both wave and photon spaces.