celestial (version 1.4.6)

planck: 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).

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.

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.

References

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

See Also

cosgrow

Examples

# NOT RUN {
#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()

# }