Editing Wien's displacement law (section)

==Peak differs according to parameterization==
{| class="wikitable"
|+ Constants for different parameterizations of Wien's law
|-
! Parameterized by !! ''x''<math>_\mathrm{peak}</math> !! ''b'' (μm⋅K)
|-
| Wavelength, <math>\lambda</math> || {{val|4.965114231744276303|end=...}} || 2898
|-
| <math>\log\lambda</math> or <math>\log\nu</math> ||{{val|3.920690394872886343|end=...}}||3670
|-
| Frequency, <math>\nu</math> ||{{val|2.821439372122078893|end=...}}|| 5099
|}
{| class="wikitable"
|+ Other characterizations of spectrum
|-
! Parameterized by !! ''x'' !! ''b'' (μm⋅K)
|-
| Mean photon energy ||{{val|2.701|end=...}}|| 5327
|-
| 10% percentile ||{{val|6.553|end=...}}|| 2195
|-
| 25% percentile ||{{val|4.965|end=...}}|| 2898
|-
| 50% percentile ||{{val|3.503|end=...}}|| 4107
|-
| 70% percentile ||{{val|2.574|end=...}}|| 5590
|-
| 90% percentile ||{{val|1.534|end=...}}|| 9376
|}
The results in the tables above summarize results from other sections of this article. Percentiles are percentiles of the Planck blackbody spectrum.<ref>{{Cite journal
 |last1=Lowen |first1=A. N.
 |last2=Blanch |first2=G.
 |author2-link=Gertrude Blanch
 |year=1940
 |title=Tables of Planck's radiation and photon functions
 |journal=[[Journal of the Optical Society of America]]
 |volume=30 |issue=2 |page=70
 |doi=10.1364/JOSA.30.000070
 |bibcode=1940JOSA...30...70L
 }}</ref> Only 25 percent of the energy in the black-body spectrum is associated with wavelengths shorter than the value given by the peak-wavelength version of Wien's law.

[[File:230617T161650planckParam6000.svg|thumb|upright=1.45|Planck blackbody spectrum parameterized by wavelength, fractional bandwidth (log wavelength or log frequency), and frequency, for a temperature of 6000 K]]
Notice that for a given temperature, different parameterizations imply different maximal wavelengths. In particular, the curve of intensity per unit frequency peaks at a different wavelength than the curve of intensity per unit wavelength.<ref name="Marr2012"/>

For example, using {{nowrap|1=<math>T</math> = {{convert|6000|K}}}} and parameterization by wavelength, the wavelength for maximal spectral radiance is {{nowrap|1=<math>\lambda</math> = 482.962 nm}} with corresponding frequency {{nowrap|1=<math>\nu</math> = 620.737 THz}}.  For the same temperature, but parameterizing by frequency, the frequency for maximal spectral radiance is {{nowrap|1=<math>\nu</math> = 352.735 THz}} with corresponding wavelength {{nowrap|1=<math>\lambda</math> = 849.907 nm}}.

These functions are [[radiance]] ''density'' functions, which are [[Probability density function|probability ''density'' functions]] scaled to give units of radiance. The density function has different shapes for different parameterizations, depending on relative stretching or compression of the abscissa, which measures the change in probability density relative to a linear change in a given parameter. Since wavelength and frequency have a reciprocal relation, they represent significantly non-linear shifts in probability density relative to one another.

The total radiance is the integral of the distribution over all positive values, and that is [[Invariant (mathematics)|invariant]] for a given temperature under ''any'' parameterization.  Additionally, for a given temperature the radiance consisting of all photons between two wavelengths must be the same regardless of which distribution you use.  That is to say, integrating the wavelength distribution from <math>\lambda_1</math> to <math>\lambda_2</math> will result in the same value as integrating the frequency distribution between the two frequencies that correspond to <math>\lambda_1</math> and <math>\lambda_2</math>, namely from <math>c / \lambda_2</math> to <math>c / \lambda_1</math>.<ref>{{cite web |last1=King |first1=Frank |title=Probability 2003-04, Chapter 11, TRANSFORMING DENSITY FUNCTIONS |url=https://www.cl.cam.ac.uk/teaching/2003/Probability/ |publisher=University of Cambridge |date=2003}}</ref> However, the distribution ''shape'' depends on the parameterization, and for a different parameterization the distribution will typically have a different peak density, as these calculations demonstrate.<ref name="Marr2012"/>

The important point of Wien's law, however, is that ''any'' such wavelength marker, including the median wavelength (or, alternatively, the wavelength below which ''any'' specified percentage of the emission occurs) is proportional to the reciprocal of temperature.  That is, the shape of the distribution for a given parameterization scales with and translates according to temperature, and can be calculated once for a canonical temperature, then appropriately shifted and scaled to obtain the distribution for another temperature.  This is a consequence of the strong statement of Wien's law.