planck: Planck's Law and Related Functions
In celestial: Collection of Common Astronomical Conversion Routines and Functions

Description Usage Arguments Details Value Author(s) References See Also Examples

Functions related to Planck's Law of thermal radiation.

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)

`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).

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) = (2.h.nu^3/c^2).(1/(exp(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) = (2.h.c^2/lambda^5).(1/(exp(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) = (2.nu^2/c^2).(1/(exp(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) = (2.c/lambda^4).(1/(exp(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) = (8.pi.h.nu^3/c^3).(1/(exp(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) = (8.pi.h.c/lambda^5).(1/(exp(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]> = 3.729282e-23 T

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

N[phot] = 1.5205e+15.T^3/pi

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

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.

Aaron Robotham

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

cosgrow

#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()