planck: Planck's Law and Related Functions

In asgr/celestial: Collection of Common Astronomical Conversion Routines and Functions

Planck's Law and Related Functions

Description

Functions related to Planck's Law of thermal radiation.

Usage

cosplanckLawRadFreq(nu,Temp=2.725)
cosplanckLawRadWave(lambda,Temp=2.725)
cosplanckLawEnFreq(nu,Temp=2.725)
cosplanckLawEnWave(lambda,Temp=2.725)
cosplanckLawRadFreqN(nu,Temp=2.725)
cosplanckLawRadWaveN(lambda,Temp=2.725)
cosplanckPeakFreq(Temp=2.725)
cosplanckPeakWave(Temp=2.725)
cosplanckSBLawRad(Temp=2.725)
cosplanckSBLawRad_sr(Temp=2.725)
cosplanckSBLawEn(Temp=2.725)
cosplanckLawRadPhotEnAv(Temp=2.725)
cosplanckLawRadPhotN(Temp=2.725)
cosplanckCMBTemp(z,Temp=2.725)

Arguments

nu	The frequency of radiation in Hertz (Hz).
lambda	The wavelength of radiation in metres (m).
Temp	The absolute temperature of the system in Kelvin (K).
z	Redshift, where z must be > -1 (can be a vector).

Details

The functions with Rad in the name are related the spectral radiance form of Planck's Law (typically designated I or B), whilst those with En are related to the spectral energy density form of Planck's Law (u), where u=4\pi I/c.

To calculate the number of photons in a mode we simply use E=h\nu=h c / \lambda.

Below h is the Planck constant, k_B is the Boltzmann constant, c is the speed-of-light in a vacuum and \sigma is the Stefan-Boltzmann constant.

cosplanckLawRadFreq is the spectral radiance per unit frequency version of Planck's Law, defined as:

B_\nu(\nu,T) = I_\nu(\nu,T) = \frac{2 h \nu^3}{c^2} \frac{1}{e^{h \nu / k_B T}-1}

cosplanckLawRadWave is the spectral radiance per unit wavelength version of Planck's Law, defined as:

B_\lambda(\lambda,T) = I_\lambda(\lambda,T) = \frac{2 h c^2}{\lambda^5} \frac{1}{e^{h c / \lambda k_B T}-1}

cosplanckLawRadFreqN is the number of photons per unit frequency, defined as:

B_\nu(\nu,T) = I_\nu(\nu,T) = \frac{2 \nu^2}{c^2} \frac{1}{e^{h \nu / k_B T}-1}

cosplanckLawRadWaveN is the number of photons per unit wavelength, defined as:

B_\lambda(\lambda,T) = I_\lambda(\lambda,T) = \frac{2 c}{\lambda^4} \frac{1}{e^{h c / \lambda k_B T}-1}

cosplanckLawEnFreq is the spectral energy density per unit frequency version of Planck's Law, defined as:

u_\nu(\nu,T) = \frac{8 \pi h \nu^3}{c^3} \frac{1}{e^{h\nu/k_B T}-1}

cosplanckLawEnWave is the spectral energy density per unit wavelength version of Planck's Law, defined as:

u_\lambda(\lambda,T) = \frac{8 \pi h c}{\lambda^5} \frac{1}{e^{h c / \lambda k_B T}-1}

cosplanckPeakFreq gives the location in frequency of the peak of I_\nu(\nu,T), defined as:

\nu_{peak} = 2.821 k_B T

cosplanckPeakWave gives the location in wavelength of the peak of I_\lambda(\lambda,T), defined as:

\lambda_{peak} = 4.965 k_B T

cosplanckSBLawRad gives the emissive power (or radiant exitance) version of the Stefan-Boltzmann Law, defined as:

j^* = \sigma T^4

cosplanckSBLawRad_sr gives the spectral radiance version of the Stefan-Boltzmann Law, defined as:

L = \sigma T^4/\pi

cosplanckSBLawEn gives the energy density version of the Stefan-Boltzmann Law, defined as:

\epsilon = 4 \sigma T^4 / c

Notice that j^* and L merely differ by a factor of \pi, i.e. L is per steradian.

cosplanckLawRadPhotEnAv gives the average energy of the emitted black body photon, defined as:

<E_{phot}> = 3.729282 \times 10^{-23} T

cosplanckLawRadPhotN gives the total number of photons produced by black body per metre squared per second per steradian, defined as:

N_{phot} = 1.5205 \times 10^{15} T^3 / \pi

Various confidence building sanity checks of how to use these functions are given in the Examples below.

Value

Planck's Law in terms of spectral radiance:

cosplanckLawRadFreq	The power per steradian per metre squared per unit frequency for a black body (W.sr^{-1}.m^{-2}.Hz^{-1}).
cosplanckLawRadWave	The power per steradian per metre squared per unit wavelength for a black body (W.sr^{-1}.m^{-2}.m^{-1}).

Planck's Law in terms of spectral energy density:

cosplanckLawEnFreq	The energy per metre cubed per unit frequency for a black body (J.m^{-3}.Hz^{-1}).
cosplanckLawEnWave	The energy per metre cubed per unit wavelength for a black body (J.m^{-3}.m^{-1}).

Photon counts:

cosplanckLawRadFreqN	The number of photons per steradian per metre squared per second per unit frequency for a black body (photons.sr^{-1}.m^{-2}.s^{-1}.Hz^{-1}).
cosplanckLawRadWaveN	The number of photonsper steradian per metre squared per second per unit wavelength for a black body (photons.sr^{-1}.m^{-2}.s^{-1}.m^{-1}).

Peak locations (via Wien's displacement law):

cosplanckPeakFreq	The frequency location of the radiation peak for a black body as found in cosplanckLawRadFreq.
cosplanckPeakWave	The wavelength location of the radiation peak for a black body as found in cosplanckLawRadWave.

Stefan-Boltzmann Law:

cosplanckSBLawRad	Total energy radiated per metre squared per second across all wavelengths for a black body (W.m^{-2}). This is the emissive power version of the Stefan-Boltzmann Law.
cosplanckSBLawRad_sr	Total energy radiated per metre squared per second per steradian across all wavelengths for a black body (W.m^{-2}.sr^{-1}). This is the radiance version of the Stefan-Boltzmann Law.
cosplanckSBLawEn	Total energy per metre cubed across all wavelengths for a black body (J.m^{-3}). This is the energy density version of the Stefan-Boltzmann Law.

Photon properties:

cosplanckLawRadPhotEnAv	Average black body photon energy (J).
cosplanckLawRadPhotN	Total number of photons produced by black body per metre squared per second per steradian (m^{-2}.s^{-1}.sr^{-1}).

Cosmic Microwave Background:

cosplanckCMBTemp

The temperaure of the CMB at redshift z.

Author(s)

Aaron Robotham

References

Marr J.M., Wilkin F.P., 2012, AmJPh, 80, 399

See Also

cosgrow

Examples

#Classic example for different temperature stars:

waveseq=10^seq(-7,-5,by=0.01)
plot(waveseq, cosplanckLawRadWave(waveseq,5000),
log='x', type='l', xlab=expression(Wavelength / m),
ylab=expression('Spectral Radiance' / W*sr^{-1}*m^{-2}*m^{-1}), col='blue')
lines(waveseq, cosplanckLawRadWave(waveseq,4000), col='green')
lines(waveseq, cosplanckLawRadWave(waveseq,3000), col='red')
legend('topright', legend=c('3000K','4000K','5000K'), col=c('red','green','blue'), lty=1)

#CMB now:

plot(10^seq(9,12,by=0.01), cosplanckLawRadFreq(10^seq(9,12,by=0.01)),
log='x', type='l', xlab=expression(Frequency / Hz),
ylab=expression('Spectral Radiance' / W*sr^{-1}*m^{-2}*Hz^{-1}))
abline(v=cosplanckPeakFreq(),lty=2)

plot(10^seq(-4,-1,by=0.01), cosplanckLawRadWave(10^seq(-4,-1,by=0.01)),
log='x', type='l', xlab=expression(Wavelength / m),
ylab=expression('Spectral Radiance' / W*sr^{-1}*m^{-2}*m^{-1}))
abline(v=cosplanckPeakWave(),lty=2)

#CMB at surface of last scattering:

TempLastScat=cosplanckCMBTemp(1100) #Note this is still much cooler than our Sun!

plot(10^seq(12,15,by=0.01), cosplanckLawRadFreq(10^seq(12,15,by=0.01),TempLastScat),
log='x', type='l', xlab=expression(Frequency / Hz),
ylab=expression('Spectral Radiance' / W*sr^{-1}*m^{-2}*Hz^{-1}))
abline(v=cosplanckPeakFreq(TempLastScat),lty=2)

plot(10^seq(-7,-4,by=0.01), cosplanckLawRadWave(10^seq(-7,-4,by=0.01),TempLastScat),
log='x', type='l', xlab=expression(Wavelength / m),
ylab=expression('Spectral Radiance' / W*sr^{-1}*m^{-2}*m^{-1}))
abline(v=cosplanckPeakWave(TempLastScat),lty=2)

#Exact number of photons produced by black body:

cosplanckLawRadPhotN()

#We can get pretty much the correct answer through direct integration, i.e.:

integrate(cosplanckLawRadFreqN,1e8,1e12)
integrate(cosplanckLawRadWaveN,1e-4,1e-1)

#Stefan-Boltzmann Law:

cosplanckSBLawRad_sr()

#We can get (almost, some rounding is off) the same answer by multiplying
#the total number of photons produced by a black body per metre squared per
#second per steradian by the average photon energy:

cosplanckLawRadPhotEnAv()*cosplanckLawRadPhotN()

asgr/celestial documentation built on Dec. 12, 2023, 10:06 a.m.