planck {celestial} R Documentation

Planck's Law and Related Functions

Description

Functions related to Planck's Law of thermal radiation.

Usage

```cosplanckLawRadFreq(nu,Temp=2.725)
cosplanckLawEnFreq(nu,Temp=2.725)
cosplanckLawEnWave(lambda,Temp=2.725)
cosplanckPeakFreq(Temp=2.725)
cosplanckPeakWave(Temp=2.725)
cosplanckSBLawEn(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) = (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.

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.

Aaron Robotham

References

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

`cosgrow`

Examples

```#Classic example for different temperature stars:

waveseq=10^seq(-7,-5,by=0.01)
log='x', type='l', xlab=expression(Wavelength / m),
legend('topright', legend=c('3000K','4000K','5000K'), col=c('red','green','blue'), lty=1)

#CMB now:

log='x', type='l', xlab=expression(Frequency / Hz),
abline(v=cosplanckPeakFreq(),lty=2)

log='x', type='l', xlab=expression(Wavelength / m),
abline(v=cosplanckPeakWave(),lty=2)

#CMB at surface of last scattering:

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

log='x', type='l', xlab=expression(Frequency / Hz),
abline(v=cosplanckPeakFreq(TempLastScat),lty=2)

log='x', type='l', xlab=expression(Wavelength / m),
abline(v=cosplanckPeakWave(TempLastScat),lty=2)

#Exact number of photons produced by black body:

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

#Stefan-Boltzmann Law: