Page updated: April 9, 2020
Author: Curtis Mobley
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Blackbody Radiation

The birth date of modern physics can be regarded as December 14, 1900, when Max Planck presented his derivation of the spectral distribution of radiant energy in thermodynamic equilibrium with matter at a given temperature. The derivation of this distribution is both conceptually and mathematically subtle, and Max well deserved his subsequent Nobel Prize. Planck’s function is commonly called the blackbody radiation spectrum.

As derived in most physics books (e.g., Liboff (1980), Chapter 2; or Eisberg and Resnick (1985), Chapter 1), Planck’s function is expressed as a spectral energy density:

UE(ν) = 8πhν3 c3 1 ehνkT 1, (1)

where ν is frequency in s1, h = 6.6261 1034Js is Planck’s constant, c = 2.9979 108ms1 is the speed of light in vacuo, k = 1.3806 1023JK1 is Boltzmann’s constant, and T is the temperature in Kelvin. UE(ν) thus has units of J(m3s1) = J(m3Hz), or energy per unit volume per unit frequency interval (with frequency measured in Hertz = 1 cycle per second).

For ease of comparison with the sun’s irradiance, or with the irradiance measured at the entrance of a black body cavity, Eq. (1) can be converted to spectral plane irradiance as a function of wavelength. The energy contained in a unit frequency interval dν must equal the energy contained in the corresponding wavelength interval dλ, i.e.,

UE(ν)|dν| = UE(λ)|dλ|.

Recalling that ν = cλ gives dν = (cλ2)dλ, and Eq. (1) becomes

UE(λ) = UE(ν) dν dλ = 8πhc λ5 1 ehcλkT 1, (2)

which has units of has units of J(m3m), or energy per unit volume per unit wavelength interval (with wavelength measured in meters).

The scalar irradiance Eo is related to the energy density by Eo = Uc. One way to see this is to think of the many photons making up the energy density. How many photons ”hit” a small spherical detector per unit time, there to be recorded as scalar irradiance, equals how many photons there are times how fast they are moving, i.e. Eo = Uc. Radiation in thermodynamic equilibrium is isotropic and unpolarized. For isotropic radiance, Eo = 4Ed, where Ed is the plane irradiance. Thus Eq. (2) can be converted to spectral plane irradiance by a factor of c4:

Ed(λ) = c 4UE(λ) = 2πhc2 λ5 1 ehcλkT 1. (3)

This is the form of Planck’s law seen, for example, in Leighton (1959), page 65.

Two final transformations of Eq. (3) are needed for comparison with the sun’s solar irradiance as measured at the top of the earth’s atmosphere, as seen in Figs. (1) and (2) of the next page on light from the sun. First, in accordance with the r2 law for irradiance, the irradiance emitted at the sun’s surface (presumed to be a blackbody in the present discussion) is reduced by a factor of (RsunRearth)2 to obtain the irradiance at the mean distance of the earth’s orbit. Here Rearth = 1.496 108km is the radius of the earth’s orbit, and Rsun = 6.95 105km is the sun’s radius. Finally, a factor of 109 is applied to Eq. (3) to convert the wavelength spectral interval from meters to nanometers. The resulting equation is

Ed(λ) = Rsun Rearth 22πhc2 λ5 1 ehcλkT 1109, (4)

where Ed(λ) is now in Wm2nm1, although the wavelength is still measured in meters on the right-hand side of the equation for consistency with the SI units for h,c and k.

Integrating Eq. (3) over all wavelengths gives the total plane irradiance emitted by a black body:

Ed = σT4, (5)

where σ = (2π5k4)(15h3c2) = 5.6703 108Wm2K4 is the Stefan-Boltzmann constant. The sun’s total (over all wavelengths) irradiance as measured at the top of the atmosphere is 1368Wm2. Carrying this value back to the sun’s surface via a factor of (RearthRsun)2 and inserting the result into Eq. (5) gives a corresponding black body temperature of T = 5,782K. That is, a black body at this temperature emits the same total irradiance as does the sun. This temperature is then used in Eq. (4) to generate the black body spectrum seen in the figures of the solar energy page.

Other forms of the blackbody spectrum are sometimes useful. As already noted, blackbody radiation is isotropic. For isotropic radiance Lo, Ed = πLo, where π has units of steradian. Thus formula (3) for plane irradiance can be converted to a formula for blackbody radiance LBB by dividing by π:

LBB(λ) = 2hc2 λ5 1 ehcλkT 1. (6)

For some applications it is useful to know the photon density or photon irradiance. The photon density UQ is obtained from the energy density by dividing the energy density UE by the energy hν of a single photon. Thus Eq. (1) gives

UQ(ν) = 8πν2 c3 1 ehνkT 1, (7)

where UQ has units of photons(m3Hz). Similarly, Eq. (3) can be divided by the energy per photon in wavelength units, hcλ, to obtain the photon plane irradiance

Qd(λ) = 2πc λ4 1 ehcλkT 1, (8)

where Qd has units of photons(sm2m). Integrating this equation over all wavelengths gives the total number of photons emitted per second per unit area by a blackbody:

Qd = σQT3, (9)

where σQ = (4.808πk3)(h3c2) = 1.520 1015photonss1m2K3 is the photon equivalent of the Stefan-Boltzmann constant. Thus the total energy emitted by a blackbody is proportional to T4, but the total number of photons emitted is proportional to T3. As the temperature increases, the blackbody spectrum shifts toward the blue, and relatively fewer more-energetic short-wavelength photons are needed to keep up with the increasing energy output.

It is also common to use wavenumber ν̃ = 1λ as the spectral variable. A change of variables based on UE(ν)|dν̃| = UE(λ)|dλ| and dλdν̃ = λ2 then gives

UE(ν̃) = UE(λ) dλ dν̃ = 8πhcν̃3 1 ehcν̃kT 1,

which has units of J(m3m1), or energy per unit volume per unit wavenumber interval (with wavenumber measured in 1/meters). Other formulas in terms of wavenumber are obtained as before.

Table 1 summarizes various formulas for blackbody radiation. These cover everything needed for optical oceanography. However, the Spectral Calculations website has much additional information about blackbody radiation, including such esoterica as how the spectrum shifts if the blackbody source is moving at relativistic speeds.

Figure 1 shows the energy and photon densities, and energy and photon irradiances, for a temperature of T = 5782K, corresponding approximately to the sun’s surface temperature. These curves were computed using the first four formulas in Table 1. It should be noted that the energy spectra have their maxima at about 500 nm for this temperature, whereas the photon spectra have their maxima at about 635 nm. That is, where the sun’s output is a maximum depends on what measure of the output is used, as well as on which variable is used for the spectral density. This important matter is discussed further on the page on a common misconception about energy spectra.






Quantity Spectral Variable Units Formula




Energy density wavelength J m3mUE(λ) = 8πhc λ5 1 ehcλkT1
    
Photon density wavelength photons m3m UQ(λ) = 8π λ4 1 ehcλkT1
    
Energy irradiancewavelength W m2mEd(λ) = 2πhc2 λ5 1 ehcλkT1
    
Photon irradiancewavelength photons sm2m Qd(λ) = 2πc λ4 1 ehcλkT1
    
Energy density frequency J m3HzUE(ν) = 8πhν3 c3 1 ehνkT1
    
Photon density frequency photons m3Hz UQ(ν) = 8πν2 c3 1 ehνkT1
    
Energy irradiancefrequency W m2HzEd(ν) = 2πhν3 c2 1 ehνkT1
    
Photon irradiancefrequency photons sm2Hz Qd(ν) = 2πν2 c2 1 ehνkT1
    
Energy density wavenumber J m3m1UE(ν̃) = 8πhcν̃3 1 ehcν̃kT1
    
Photon density wavenumber photons m3m1 UQ(ν̃) = 8πν2 1 ehcν̃kT1
    
Energy irradiancewavenumber W m2m1Ed(ν̃) = 2πhc2ν̃3 1 ehcν̃kT1
    
Photon irradiancewavenumber photons sm2m1Qd(ν̃) = 2πcν̃2 1 ehcν̃kT1




    

Table 1: 1. Blackbody radiation formulas for energy and photon density and for energy and photon plane irradiance, in spectral units of wavelength λ, frequency ν, and wavenumber ν̃. Formulas require wavelength in meters and wavenumber in 1/meters. Divide the Ed and Qd formulas by π to obtain formulas for blackbody radiances.


PIC

Figure 1: 1. Blackbody spectra for energy and photon densities, and for energy and photon irradiances, for a temperature of 5782 K. The inset values give the totals over all wavelengths.

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