Login Name: 

Forgot Password?
GATS, Inc.
Gas-Cell Simulator |  Atmospheric Paths |  My Spectra |  Line List Browser |  Blackbody Calculator |  Atmosphere Browser |  Solar Calculator |  Unit Converter
Calculating Blackbody Radiance

Blackbody Calculator

Calculation of Blackbody Radiance

What is a Blackbody?

A blackbody is a hypothetical object that absorbs all incident electromagnetic radiation while maintaining thermal equilibrium.  No light is reflected from or passes through a blackbody, but radiation is emitted, and is called blackbody radiation.  The prefix black is used because at room temperature such an object would emit almost no visible light, appearing black to an observer.

No physical object exactly fits this definition, but most behave at least in part as blackbodies.  Calculation of the radiometric quantities associated with blackbody radiation is extremely important in physics, chemistry, optics, engineering, astronomy and many other areas.

History of blackbody theory

In 1900, Max Planck developed the modern theory describing the radiation field of a blackbody.  At the time, there were two distinct models for blackbody radiation: the Rayleigh-Jeans law, which fit the measurements well at low frequencies, and Wien’s law, which worked well at high frequencies, but neither worked everywhere.  Planck, by making the ingenious assumption that the energy of the modes of the electromagnetic field must be quantized, developed the theory that fits observations at all parts of the spectrum.  This leap marked the birth of quantum mechanics and modern physics.

Radiometric systems of units

There are many choices of units when dealing with radiometric quantities, and each discipline has its preferred units.  Spectroscopists traditionally prefer wavenumber, infrared engineers use wavelength, and physicists typically deal with frequency.   Thermal calculations generally involve radiated/received power, but many systems, including the human eye, operate as efficient quantum detectors, and photon flux is the appropriate measure.  The choice of units is not trivial, as the functional forms differ.  For example, the power emitted per unit area of a blackbody at temperature T is proportional to T 4, but the photon flux is proportional to T 3.

References containing the basic formulas abound, but it is difficult to find any single source with formulas given in each system of units.  Here we collect a comprehensive set of radiometric formulas in all the common units. We consider spectral units of frequency (Hz), wavelength (μm) and wavenumber (cm-1).  For each, we derive the basic blackbody formulas in terms of both power (W) and photon flux.  Beginning with the Planck blackbody function in units of W m-2 sr-1 Hz-1, all other functions are derived.   We also derive useful formulas for computing integrated band radiance, and present sample C++ computer codes in Appendix A.  Appendix B describes the Doppler effect on the observed blackbody radiation spectrum of moving sources.  Finally, all significant formulae are summarized in Appendix C for quick reference. 


Calculation of a Blackbody Radiance
Units of Frequency
Units of Wavelength
Units of Wavenumbers
Radiance: Integrating the Planck Equation
In-band Radiance: Integrating the Planck Equation over a Finite Range
Appendix A: Algorithms for Computing In-band Radiance
Appendix B: The Doppler Effect
Appendix C: Summary of Formulas
Blackbody Calculator
Print Version


Gas-cell Simulator
Microsoft Internet Explorer 6.0 and above
Netscape 6.2 and above
Konqueror 3.0 and above
Safari 2.0 and above
Opera 8.5 and above
Firefox 1.5 and above
Other browsers may work but are not fully tested or supported.
Transmittance: ratio of received radiation intensity, I, to incident light intensity, I0

Transmittance: ratio of received radiation intensity, I, to incident light intensity, I0

The information provided will not be shared, sold or used in any way other than to contact users to announce new features.
Radiance: radiant flux radiated per unit area, per unit solid angle, per wavenumber

light with wavenumber between σ and σ + dσ
Radiance: radiant flux radiated per unit area, per unit solid angle, per wavenumber

light with wavenumber between σ and σ + dσ
Isotopes are forms of an element whose nuclei have the same atomic number, the number of protons in the nucleus,but different atomic masses because they contain different numbers of neutrons.
Cell: model the transmission/radiance of a gas cell. Specify it's length, temperature and pressure, and the vmrs of the absorbing gases.

Wavenumber cm-1: the number of
wavelengths of light per centimeter

LINEPAK: The GATS spectral radiance and transmission software library. Performs detailed and accurate line-by-line modeling of molecular absorption. Efficient and flexible, LINEPAK is at the heart of analysis systems for many major atmospheric remote sensing missions, including HALOE, SABER, LIMS, SOFIE, CRISTA, and CLAES.
Tangent Path: Model the transmission or radiance of a ray that passes completely through the Earth's atmosphere but does not intersect the Earth. The path is specified by the tangent height, the height at the point of closest approach to the surface. The pressure, temperature and vmrs of absorbing gases at each altitude are chosen from a database of atmospheric states.
Slant path: Model the transmission or radiance of a ray between two arbitrary points in the Earth's atmosphere. The points are specified by their heights and the zenith angle from one to the other.
VMR: volume mixing ratio. The fractional number of molecules of a species in a volume.

Individual vmrs and their sum must be between 0 and 1.

If the vmrs sum to less than 1, the rest of the gas in the cell is assumed transparent.(Lineshapes for molecules with vmr less than 1 are air-broadened.)
Clicking this will display the data as text in a new browser window. Right-clicking will download the data file to your computer (recommended). These files can be extremely large depending on the spectrum simulated.
Clicking this will open a new browser window suitable for printing.
You have exceeded your daily limit.

To help ensure the availability of our servers, public use is limited to 50 calculations per day. Subscribe now for uninterrupted service. Subscribers also have access to advanced features such as large wavebands, multiple gases, choice of units, radiance spectra, logo-free high-resolution graphics, ascii data files,
full tech support and much more.


Spectroscopy and remote sensing tools for researchers, teachers, and students

Subscribe now for full access to the Spectral Calculator tools.

Get priority use of advanced, state-of-the-art radiative transfer algorithms--the same ones used by NASA for many remote sensing missions. Subscribers gain access to large wavebands, multiple gases and cells, choice of units, radiance spectra, logo-free high-resolution graphics, data files, full tech support, and much more.


Temperature Offset: The model atmosphere (US_Standard, Tropical, etc.) determines the temperature, pressure and gas concentrations at each height in the atmosphere. To adjust the temperature from the model value, enter a temperature offset (from -50 to 50 K). The Atmosphere Browser tool displays the temperature profiles for the model atmospheres.
Atmosphere: An atmosphere contains profiles of temperature and gas concentrations at all altitudes. There are six system-supplied atmospheres for Earth and one for Mars. Custom atmospheres can be uploaded from the Atmosphere Browser.
Scale Factor for Gas Concentrations: The model atmosphere (US_Standard, Tropical, etc.) determines the gas concentrations at each altitude. To adjust a gas concentration, choose a scale factor, from 0 to 1000. For example, to simulate an atmosphere with 20% more water vapor than the model, enter a scale factor of 1.2 for H2O. Note: while the model atmospheres are physically realistic, using large scale factors can produce unphysical situations where the gas abundance exceeds 100%. If this occurs, an error message will be displayed.
The atmosphere model (US_Standard, Tropical, etc.) determines the temperature, pressure and gas concentrations at each height in the atmosphere. To adjust a gas concentration, choose a scale factor other than 1 (from 0 to 1000). For example, to simulate a path with 20% more water vapor, use a scale factor of 1.2 for H2O. The Atmosphere Browser tool displays the temperature, pressure and gas mixing ratios for the model atmospheres..
Below is a list of valid characters:
A-Z      a-z      0-9   _      -      .

  (      )       [      ]       :       ;       ,

Create Atmosphere
Modify Atmosphere
Delete Atmosphere
Plot/Extract Species
Atmosphere List
Add/Replace Species
Duplicate Atmosphere
Delete Species