AUGUST, 1938

The Correlation of Color Temperatures Based on the Wien and the PlanckRadiation Formulas

ROGER S. ESTEYSpencer Lens Company, Buffalo, N. Y.

(Received January 14, 1938)

THE scientific specification of temperatureT colors has been developed in two differentfields, that of color temperature calculation andthat of clorimetric calculation. Since thesemethods do not in all cases yield equivalentresults, the correlation presented below will beof value in bringing these methods together.

Color temperature calculations are greatlyfacilitated by the use of reciprocal temperatureinstead of temperature itself. This was proposedby Priest' with the thought that this reciprocalscale would be of particular value in pyrometryand colorimetry not only because computationstake on simpler form, but also because on thereciprocal scale the least perceptible chromaticitydifference is a constant independent of tempera-ture and has the convenient approximate valueof one micro-reciprocal-degree (abbreviation"mired").

The I. C. I. standard observer and coordinatesystem2 is, of course, admirably adapted to thespecification of color stimuli in general. Itsusefulness for exact or approximate temperaturecolors is enhanced by the availability of iso-temperature lines3 by which color points slightlyoff the Planckian locus can be assigned anappropriate color temperature.

Unfortunately these two methods are not onthe same temperature scale because, while thecalorimetric calculation method is based ondistributions of spectral energy in accordancewith Planck's formula, the scale of reciprocaltemperature develops mathematically from theWien radiation formula as has been shown by

'I. G. Priest, "A Proposed Scale for Use in Specifyingthe Chromaticity of Incandescent Illuminants and VariousPhases of Daylight," J. 0. S. A. 23, 41 (1933).

2 D. B. Judd, "The 1931 I. C. I. Standard Observer andCoordinate System for Colorimetry," J. 0. S. A. 23, 359(1933).

3 D. B. Judd, "Estimation of Chromaticity Differencesand Nearest Color Temperature on the Standard 1931I. C. I. Colorimetric Coordinate System," J. 0. S. A. 26,421 (1936).

Gage.4 At low temperatures or short wave-lengths the Wien and Planck formulas yield thesame results but at larger values of temperatureor wave-length the differences become significant.

A conversion table has, therefore, been pre-pared by which reciprocal temperatures may beconverted from the Wien to the Planck basis foruse in clorimetric computations or for otherpurposes. The series of spectral energy distribu-tions which form the basis for the table could becomputed directly from the Wien formula butcan be more conveniently obtained from Planckenergy distribution data.

If the Planck and Wien formulas are expressedin the form

Ex c CIX-5(eC2XO -1)-'

Ex o: CX- 5 (eC2IX0)-land

(1)

(2)

respectively, it follows that

(E,)./ (Ex) p= (eC2/XO- 1)/eC21X0. (3)

This ratio permits values of spectral energy bythe Wien formula to be conveniently obtainedfrom readily available tabulated data5 based onthe Planck formula. Values of the ratio in Eq.(3) have been computed over a considerablerange of XO and are listed in Table I.

TABLE 1. Ratio of values of Wien to Planck formula spectralenergies for a range of X0.

Xxox NMICRON MICRONDEGREES (EX)w/(Ex)p DEGREES (EX)w/(Ex)p

2000 0.9992 7000 0.87 123000 0.9906 8000 0.83394000 0.9721 9000 0.79745000 0.9429 10000 0.76196000 0.9082 11000 0.7287

12000 0.6976

4 H. P. Gage, "Color Filters for Altering Color Tem-perature. Pyrometer Absorption and Daylite Glasses,"J. 0. S. A. 23, 46 (1933).

5 Frehafer and Snow, "Tables and Graphs for Facilitat-ing the Computation of Spectral Energy Distribution byPlanck's Formula," Nat. Bur. Stand. Misc. Pub. No. 56,1925.

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J. . S. A. VOLUME 2&

ROGER S. ESTEY

TM PRATUR_ C.OLO

0.20 C30 0.40 0.50 0.60

> .

FIG. 1. Mixture diagram showing Wien and Planck temperatureunified tristimulus coefficients.

A table of relative energies (Ex), was corm- and thereforeputed and used to obtain spectral distributions These point(Wien) for a range of temperatures. The tri- diagram andlinear color coordinates of these distributions thus givingwere then computed and the color points plotted. jective colorThese are shown on the mixture diagram, Fig. 1. temperaturesWhere the Wien distribution color points depart presented as cmaterially from the Planckian locus, interpola- against Wiention between iso-temperature lines was used is shown into obtain the corresponding Planckian color plotted on atemperatures.

It was found that the coordinates of the color gpoints corresponding to Planckian distributions TABLE II. Corren

derived from the Frehafer and Snow datadiffered slightly from those reported by Judd3

so that it was necessary to make sure that COLO PEAPE

differences in color temperature referred to measurements made on the same basis. These 4,000 4,005,000 5,02small variations (approximately one mired) 6,000 6,07

which are probably due to changes in the 7,000 7,158,000 8,29

radiation constants and the thermodynamic 9,000 9,51

temperature scale made since 1925, have no 10,000 10,8411,000 12,31effect on differences at high temperatures, but 12,000 13,95

become important at 4000 and 5000'K, where 13,000 15,7714,000 17,84the difference between the Wien and Planck 15,000 20,24

color temperature scales is so small. 16,000 22,88

The result of this work is to provide a series 18,000 28,77

of color points derived from the Wien formula

colors; x and

0.70

y are two of the

e having a Wien color temperature.s were plotted on the mixturePlanck color temperatures assigned,a correlation on the basis of sub-between Wien and Planck color

, The data are most convenientlycolor temperature differences plottedi color temperatures. Such a plotFig. 2 where the differences arelogarithmic scale because of the

A. From a large scale plot of the

lation of Wien and Planck color temperatures.

RECIPROCAL TEMPERATURE,RATURE, K MIREDS

DIFFER- 1/ 1/ DIFFER-K RNCR WIEN CK ENCE

6 6 250.0 249.6 0.48 28 200.0 198.9 1.15 75 166.7 164.6 2.18 158 142.9 139.7 3.23 293 125.0 120.6 4.42 512 111.1 105.1 6.00 840 100.0 92.2 7.80 1,310 90.9 81.2 9.70 1,950 83.3 71.7 11.60 2,770 76.9 63.4 13.55 3,845 71.4 56.0 15.40 5,240 66.7 49.4 17.30 6,880 62.5 43.7 18.80 8,770 58.8 38.8 20.00 10,740 55,6 34.8 20.8

294

CORRELATION OF COLOR TEMPERATURES

data a smooth curve was drawn to give thevalues listed below in Table II.

It is interesting to compare these data withthose computed by Gage4' 6 using as equivalent,Wien and Planck spectral energy distributionswhich had the same ratio of energy at two wellseparated wave-lengths, 450 and 650 m/.t Gage'sdata are indicated in Fig. 2, and show excellentagreement with the results obtained by this morerigorous method which was unavailable whenthe earlier data were obtained.

The estimation of the accuracy of results ofthis sort is exceedingly difficult. The deviationof computed points from the curve is small.The agreement with previous data is close.Probably the accuracy of this work is roughlysimilar to other colorimetric computations madeat 10 m intervals and carrying 4 significantfigures. In Table II the differences can probablybe relied upon within -t5 percent.

It is a pleasure to express appreciation to6 H. P. Gage and Norman Macbeth, "Filters for Artificial

Daylighting, Their Grading and Use," Trans. I. E. S. 31,995 (1936).

3z

0.z

LU ,00

3¢5III

'a

I2 00

00U

10

CORRELATION OF PLANCKANDWIEN

COLOR TE.PERAT.R.S

Of+-~~~~~

1000 8000 1E00- 6000I | WIEN COLOR TE-PEPA-UR I

FIG. 2. Correlation of Planck and Wien color temperatures.

Electrical Testing Laboratories and to theSpencer Lens Company for their kind coopera-tion in the preparation of this paper.

295