Radiance
Physical size  

Surname  Radiance  
Formula symbol  

The radiation density or radiation density L (also specific intensity , English radiance ) provides detailed information about the location and direction dependency of the radiation emitted by a transmitting surface.
definition
Introduction and consideration of limit values
Consider a body (for example an incandescent lamp , a light emitting diode ) which emits radiation (measured for example in watts ) into its surroundings. As a rule, each point on the body will emit different amounts of radiation in different directions. If this characteristic is to be described in detail, the concept of radiance is necessary.
It is not possible to state how many watts emanate from an infinitely small point on the surface of the body, since the finite radiation power is distributed over an infinite number of such points and therefore zero watts are allotted to a single surface point. Instead, one looks at a small area around the point in question, sets the (finite) radiation power emanating from this area in relation to its (finite) area and lets the area shrink to zero. Although the radiated power and the radiating area both approach zero, the ratio of both tends towards a finite limit value ( differential quotient ), the area output or specific radiation of the point, measured in watts per square meter.
In the same way, it is not possible to specify which power is emitted in a certain direction, since the finite radiation power is distributed over an infinite number of possible directions and therefore zero watts are allotted to each individual direction. Instead, one considers a small solid angle surrounding the desired direction , sets the (finite) power output in this solid angle in relation to the (finite) size of the solid angle and lets the solid angle shrink to zero. Again, both the solid angle and the radiated power contained therein tend towards zero, but their ratio towards a finite limit value, the radiant intensity emitted in the relevant direction , measured in watts per steradian .
The radiation density combines both and in this way describes both the location and the directional dependence of the radiation emitted by an infinitely small surface element.
Radiance
The radiation density indicates which radiation power is emitted from a given point of the radiation source in the direction given by the polar angle and the azimuth angle per projected surface element and per solid angle element .
is the angle between the radiation direction and the surface normal .
The definition of the radiance has the special feature that the emitted radiant power is not related to the radiating surface element , as usual , but to the surface element projected in the direction of emission . The radiation power emitted in a certain direction depends on the one hand on the (possibly directiondependent) physical radiation properties of the surface and, on the other hand, purely geometrically on the projection of the radiating surface element effective in the radiation direction. The second effect has the effect that the radiant power emitted at the polar angle is lower by a factor than the power emitted perpendicularly. The division by the factor calculates this geometric effect, so that only a possible physical directional dependency due to the surface properties remains.
Surfaces that no longer show any directional dependence of the radiance after the factor has been calculated are called diffuse emitters or Lambertian emitters . A Lambertian surface element emits the same radiance in all directions:
The radiation power emitted by it in a certain direction only varies with the cosine of the radiation angle; Such emitters are therefore particularly easy to handle mathematically:
In particular, when integrating over the solid angle, the now angleindependent radiance can be drawn as a constant in front of the integral (see below).
For the definition of the radiation density, it is irrelevant whether the radiation emitted by the surface element is (thermal or nonthermal) selfemission , transmitted or reflected radiation, or a combination thereof.
Radiance is defined at every point in space where radiation is present. Instead of a radiating surface element, think of a fictitious radiating surface element in space.
The photometric equivalent of the radiance is the luminance , which can therefore be used for illustration: The luminance is a measure of the brightness with which a surface is perceived. If you look at a diffusely luminous surface, e.g. B. a sheet of paper, from different directions, the perceived luminance of the area remains constant, while the total amount of light reaching the viewer depends on the projected area and therefore varies with the cosine of the viewing angle. Similarly, the radiance of a diffuse radiator is the same in all directions, but the radiant power emitted in a certain direction also depends on the beam area projected in the relevant direction.
Spectral radiance
The spectral radiance (engl. Spectral radiance ) (unit: W · m ^{2} · Hz ^{1} · sr ^{1} ) of a body indicates which radiation power of the body at the frequency in the by the polar angle and the azimuth angle of projected given direction per Area, per solid angle and per frequency width. ^{}^{}^{}
The spectral radiance is also given as (unit: W · m ^{−3} · sr ^{−1} ) related to the unit wavelength interval . ^{}^{}
The spectral radiance provides the most detailed representation of the radiation properties of a radiator. It explicitly describes the directional dependence and the frequency (or wavelength) dependence of the emitted radiation. The other radiation quantities can be derived from the spectral radiance by integrating them over the directions and / or frequencies. Integration over the relevant frequency or wavelength interval in particular again provides the radiance, which is therefore also called the total radiance if it has to be distinguished from the spectral radiance.
Black body
A black body is an idealized body that completely absorbs all electromagnetic radiation that hits it. For thermodynamic reasons, the thermal radiation emitted by such a body has a universal spectrum and it must necessarily be a Lambertian radiator. Real emitters never fully achieve these ideal properties, but can come close to them. The radiation properties of a black body can therefore often be used as a good approximation for a real body.
The deviation of a real radiator from the black ideal can be recorded by an emissivity . Since a real emitter cannot radiate more strongly than a black body at the same temperature at a given wavelength, the emissivity must always be less than 1. The emissivity can depend on the wavelength and, if the real emitter is not a Lambertian emitter, also depending on the direction. The emissivities are determined by comparing the radiance or the spectral radiance of the real and black body.
Spectral radiance of a blackbody
Derivation
According to Planck, the following applies to the spectral energy density of a black body:
This gives the spectral radiance:
The factor can be understood in such a way that the radiation spreads with the speed in the entire solid angle .
Inference
For the spectral radiance of a black body the absolute temperature applies according to Planck
in the frequency display:
With
:  spectral radiance of the blackbody,  W m ^{−2} Hz ^{−1} sr ^{−1}  
:  Frequency,  Hz 
and in the wavelength display:
With
:  spectral radiance of the blackbody,  W m ^{−2} μm ^{−1} sr ^{−1}  
:  Wavelength,  m, µm  
:  absolute temperature,  K  
,  :  Planck's quantum of action ,  Js 
:  Speed of light ,  m / s  
:  Boltzmann constant ,  J / K 
is the radiant power from the surface element in the frequency range and in that between the azimuth angles and and the polar angles and space spanned angle element is radiated. The directional dependency of this radiation power is only due to the geometric factor; the spectral radiance itself is independent of direction.
When converting between frequency and wavelength representation, it should be noted that because of
applies:
The ratio of the spectral radiance of a surface element of a given radiator emitted in a certain direction and observed at a certain wavelength to the spectral radiance of a black body of the same temperature observed at the same wavelength is the directional spectral emissivity of the surface element.
If one integrates the spectral radiance of the black body over all directions of the halfspace in which the surface element radiates, one obtains the spectral specific radiation of the black body. The integral provides an additional factor . For the formula see the article " Planck's law of radiation ".
Total radiance of a blackbody
If the spectral radiance is integrated over all frequencies or wavelengths, the total radiance is calculated :
The evaluation of the integral yields because of :
With
 : Total radiance of the black body, W m ^{−2} sr ^{−1} .
is the radiant power that is radiated from the surface element at all frequencies into the solid angle element in that direction .
The ratio of the total radiation density emitted in a certain direction of a surface element of a given radiator to the total radiation density of a black body of the same temperature is the directed total emissivity of the surface element.
If one integrates the total radiation density of the black body over all directions of the halfspace in which the surface element radiates, one obtains the specific radiation of the black body. The integral provides an additional factor . For the formula, see the article “ Stefan Boltzmann Law ”.
application
Changing the definition equation for the radiance provides the radiant power that is radiated from the surface element into the solid angle element , which lies in the direction described by the angles and :
If the radiation of a finitely large radiating surface is to be determined in a finitely large solid angle , then integrate via and :
The representation of the solid angle element in spherical coordinates was used:
Since it can generally depend on the location on the beam surface and the directions swept over, a very complicated integral may result. A significant simplification occurs when the radiant surface is a Lambertian radiator (the radiance is independent of direction) with constant surface properties (the radiance is independent of location). Then the radiance is a constant number and can be drawn in front of the integral:
The integral now only depends on the shape and position of the solid angle and can be solved independently of . In this way, only by the sender and receiver geometry dependent general view factors are determined.
If, for example, the radiation in the entire halfspace overlooked by the beam surface is considered, the value for the integral results and the radiation of a Lambertian radiator of the surface in the entire halfspace is simple:
 (Radiant power of a Lambertian radiator in the half space)
If the radiation surface is a black body of temperature , the required radiation density can be calculated immediately using Planck's law of radiation (see formulas above). If it is a gray body , the Planck radiance has to be reduced in order to reduce the emissivity . A possible location and direction dependency of the emissivity as well as possible reflections can make the integration more difficult.
Basic photometric law
The basic photometric law states that the luminance remains unchanged on the way from the light source to the illuminated surface. In radiometry, this applies analogously: The radiance at the location of the transmitter in the direction of the receiver is equal to the radiation density at the location of the receiver from the direction of the transmitter. For a detailed description see Luminance # Photometric Basic Law .
Relation to other radiometric quantities and to photometry
radiometric quantity  Symbol ^{a)}  SI unit  description  photometric equivalent ^{b)}  symbol  SI unit 
Radiant flux radiant power, radiant flux, radiant power 
W ( watt ) 
Radiant energy through time 
Luminous flux luminous flux, luminous power 
lm ( lumens ) 

Radiant intensity irradiance, radiant intensity 
W / sr  Radiation flux through solid angles 
Luminous intensity luminous intensity 
cd = lm / sr ( candela ) 

Irradiance irradiance 
W / m ^{2}  Radiation flux through the receiver surface 
Illuminance illuminance 
lx = lm / m ^{2} ( lux ) 

Specific radiation emission current density, radiant exitance 
W / m ^{2}  Radiation flux through the transmitter surface 
Specific light emission luminous exitance 
lm / m ^{2}  
Radiance radiance, radiance, radiance 
W / m ^{2} sr  Radiant intensity through effective transmitter area 
Luminance luminance 
cd / m ^{2}  
Radiant energy amount of radiation, radiant energy 
J ( joules ) 
by radiation transmitted energy 
Amount of light luminous energy, quantity of light 
lm · s  
Irradiation irradiation, radiant exposure 
J / m ^{2}  Radiant energy through the receiver surface 
Exposure luminous exposure 
lx s  
Radiation yield radiant efficiency 
1  Radiation flux through absorbed (mostly electrical) power 
Luminous efficiency (overall) luminous efficacy 
lm / W 
literature
 HD Baehr, K. Stephan: Heat and mass transfer. 5th edition. Springer, Berlin 2006, ISBN 9783540323341 , chap. 5: thermal radiation.
Individual evidence
 ↑ ^{a } ^{b} electropedia , International Electrotechnical Dictionary (IEV) of the International Electrotechnical Commission : Entry 8450130 (area of lighting) has the translation: radiance = "radiation density"
 ↑ DIN EN ISO 9288: Heat transfer by radiation  physical quantities and definitions. Beuth Verlag, August 1996