a

1

I




DEPARTMENT OF COMMERCE

BUREAU OF STANDARDS

[ISSUED MARCH )9IS1

TABLES AND GRAPHS FOR FACILITATING THE COMPUTATION OF

SPECTRAL ENERGY DISTRIBUTION BY PLANCK'S FORMULA

CONSISTING OF SEVEN SHEETS (FIVE CHARTS) AS FOLLOWS: TEXT (SB^T 1). GRAPHS: 1,000 TO 5,000° K. (SHEETS 2, 3, 4; CHARTS 1, 2, 3); S,000 TO 8,000° K. (SHEET 5; CHART 4); 8,000 TO 24..000°K. (SHEET 6; CHART 5). TABLES (SHEET 71.

MISCELLANEOUS PUBLICATION OF THE BUREAU OF STANDARDS No. SS SHEET I TEXT

By M. KATHERINE FREHAFER. Anaociaic Ph > i CHESTER L SNOW. DmfumW

Very frequently Jt hu betn necessity W compute the energy distribution ol a "black

IkkIv" in llie vMblC ipectnl ngim. lit vmic ptrtlcilUl temperature, from Planck's

formula;

Such computation consumes mticli time and |jl«r; in consciiucr

(linrt cuts in Llic way u[ (.iljlcs iitifl KMplit lime been devised. Using (lie equation In llic iurm

a table hiu been made ivitli values of M und correspond in f; values of gi- For any

Riven temptrulurc the Vravc lengths in the visible, nnd jinrlly in the ultra-violet and

itifni red regions, corresponding to these values o[ relative energy, can readily be

ubLiined (rum the products XJ. Hence the spectral energy curve can be plotted at

Tiro sets of tables llnvc also been made up, each (or -B temperatures, in which the

relative energy values computed directly from Planet's formula are recorded for the

ivavc lengths joo to 730 mti at intervals 0/ 10 millimicrons—one set energy Value equal to 100 nt 590 mil, the other equal to 100 at 560 mix. turc range is from 1,000 to 38,000° K, (accuracy, wiuiin 0.1s per cent). From one of these tables a scries of isochromatic curves has been

TJli ri IntiVE energy has been plotted (accuracy, within 0,33 per cent nbwluie tcinpcrntura, and one complete curve drawn for each of the 33 v From tliesc graphs, the energy distribution for the visible region can be obi

feu* minutes for any temperature whatever in the range above specified.

L INTRODUCTION

In connection with the work of the color laboratory, very

frequent use is made of the spectral energy distribution at various temperatures of "black body" radiation, as represented by

Planck's formula:

E, CD

Ek is the radiant energy for any wave length X, at any temperature

(degrees absolute); C, and C, are constants, and e is the base

of llu' natural logarithms. As a rule it is desired to know the

relative values of E at a given temperature for a range of 320 nni

througl l EEe visible spectrum; thatisj 400 to 'Jiomn, at inter- vnls of 10 nui. Moreover, these values must conform to a scale such that at some particular wave length E will always be the same for all temperatures, for instance, equal to 100 at wave

length 560 mu, J When computed directly from the formula, this proves to be a long and tedious process; therefore, certain short

cuts have been devised. Tables have been made and curves

plotted which greatly facilitate tlu's work. It was thought

worth while to publish these for the benefit of others who make

similar use of tin's formula, The following reference tables have

been found of great assistance: F. W. Newman and J. W. D.

Glaislier, Tables of Exponential Functions, C. E- VanOrstrand,

Tables of the Exponential Function.3

II. EXPLANATION OF TABLES

1. TABLE 1

Instead of using Planck's equation in the form given above, it was combined' with Wien's displacement law connecting the temperature with the wave length .it which the maximum energy

occurs; that is, \afi-A, a constant ( 2 ) This maximum energy is given by

EaSubstituting from (2) <\ a -j, equation (3) takes the form

<3)

Dividing (1) by (4) wc obtain

(5)

An inspection of this equation will show that the right-hand side

is B function of WJ alone. The values of the constants used in

lite computations are: A - 2,890 micron degrees

14,350 micron degrees.'

Table 1 has been arranged with a wide range of values of X0 and corresponding values of computed to four significant

figures.

> YnetuUMn < .

1 (.!,:,.. 1.. 11 ,,r, 1 Kr-eui Iw bb »bl* uuuacc with the CI Lu-klsh. VT>t wuili 11- P"lnt In uxAn'i energy tut lumliulir

J

1 1 mi Intl.. is, 1 .1 »).

' IMNW.-J In >h- Jour. W-*h, Ao.,LM,l,N'a ...Jno.i*. -f-j.

• WJmii. xl s Set i*ix- .*». '. 5n .' - aopwdbi

Use of Table j.—-From the pr ^ducts U the wave lengths can be readily computed for any temjj >erature 0. (These wave lengths

are in microns, and must be mul tiplied by 1,000 to be expressed

in millimicrons.) The values of relative energy at these wave

lengths are already known from t he table; hence the distribution curve for this temperature caul at once be dotted. The next

step is to transform the values ^

be equal to 100. This is acc curve the energy coordinate for 56

100) and multiplying the other

spectral energy curve is thus we

the required point at 560 niu.. It is frequently necessary to I beginning at 400 nui. These vali from the X0 computations, are re; tion curve mentioned above, and

0.2 to 0.6 per cent), provided the In order, therefore, to obtain

8 at every 10 nui, with a given

involves four distinct steps: (1) Cc

X0; (2) plotting of X against-^ gh

this curve the values of at it

plying these values by the reciproc product by 100. This process of computation, more rapidly performed than d

formula, still consumes a consid

over, without extreme care in accuracy obtainable may not be

use of Table 1 can not be recom

accuracy, under the conditions i:

it is very useful to give readily t tion curve and the relative valui conditions need not be met. Mc

the curve can be extended

so that at 560 mu the energy will

mplished by reading from the o mix, taking its reciprocal (times

coordinates by this factor. The

11 defined and mil pass through

mow the energy at every 10 m/i

tes, while not obtainable directly

idily obtained from the distribu- :an be read with accuracy (about

scale has been suitably chosen,

lergy values for any temperature due at 560 m/i, the use of Table 1 mputation of X from the products

*en in the table; (3) reading from

itervals of 10 nui; and (4) multial of the value at 560 mu, and the

vhile more easily and somewhat

rect calculation from Planck's

erable amount of time. More*

the plotting of the curves, the

greater than 0.6 per cent. The

mended for great speed or great nposed in the introduction ; but

he shape of the energy distribus of the energy when the above

reover, with the use of Table 1,

> the ultra-violet and infra-red

regions of the spectrum.

2, TABLES 2 A AND 2B These tables have been prepj Planck's formulas, with the ex(

1, 000 K., which were calculated^ formula was used for tcinperatui formula for 2,200 to 28,000° K.

these tables are all relative to t!

560 (or 590) mu the constant C, c Cj is taken equal to 14,350 micro

tables are considered correct to \ The temperatures for which the

tables are as follows: 1,000 to 2,000,

2,000 to 3,000, B 3,000 to 4,000, B 4,000 to 7,000, q 7,000 to 10,000, b' 10,000 to 28,000, c In Table 2 A the energy has bee each temperature; in Table 2B t 560 ma.

ared directly from Wien's and

;eption only of the values for directly from Table t. Wien's

-es 1,100 to 2,000° K., Planck's

Since the energy values given in

le value chosen for wave length loesnot enter into consideration, n degrees. The values in these

rithin 0.15 per cent.

i energy values are given in these

iy steps of ioo°.

iy steps of 200 . iy steps of 250°.

iy steps of 500°.

iy steps of i,ooo°.

iy steps of 2,000°. n made equal to 100 at 590 mu, for he value 100 has been chosen at

III. EXPLANATIC )N OF GRAPHS

The plotting of the graphs wa

Tables 2A and 2B. The energy dj known for some temperature inn

series of curves based on these t yield these data. This led to a s

for each wave length at intervals oj

1,000 to 28,000° K. The plotting of these curves u

following reasons: (1) It was de

plotting and of reading to 0.33 p vary from 0.007 (approximately

largest value is about 400,000 t necessitates several changes of st

(3) The isochromatics crowd

temperatures 4,000 to 7,000°. In view of these consideration!

plot the curves in sections on 4

series is given for which has tl other scale of values (100 at 59* these by the use of a suitable fj

is suggested by the existence of

istribution is often required to be

ermediate to those given, and a

ables should be very suitable to

eries of isochromatic curves, one

f lomu, from 400 to 720 nui. vr-l.,r» r1««r-.H ^tillr, tht. r-nirtet with some difficulties for the

•.sired to obtain an accuracy of «r cent. (2) The energy values

) to 3,000; in other words, the

times the smallest value. This

:ale for the energy coordinates. 1 very closely together between

i, it has been found necessary to o by 50 cm sheets. Only that

ae value of 100 at 560 m/i. The

i) can readily be obtained from

tctor. The accuracy mentioned

ntajW. 1: I' r i-.lir. Till. Utile ii ccamKible into

J tbc . a-j ~ henrenr. ojtc b«nfound to be -lightly

fdcoobi retion*. Xomogmnj pAnol raUfcrrcnmabove (0.33 per cent) should apply except for Ex less than 3.5 and

possibly also on very steep parts or near the line 0- 1,000° K. The temperature can be estimated to within i° for the range

1,000 to 8,ooo°, and to within 5° for the range 8,000 to 28,000°, The curves have been tested by comparing values furnished by

the graphs (for given temperatures other than those recorded in

Table 2B) with the values computed for these leinperaiurL's.

IV. PRACTICAL APPLICATION

It has been shown a that a number of the ordinary illttminants have a spectral energy distribution that approximates closely that

of a Planckian radiator. This property is valuable in that the unknown spectral energy distribution of an illuminant can very

readily be deternuned if it can be matched in color with a so-called "black body." The temperature to which the "black body" is brought in order to produce color match with the illuminant is defined as the color temperature of the latter. Since this quantity is not difficult to determine in laboratories having suitable equipment, the energy distribution in the visible region of such

illuminant can at once be recorded by reading from the charts the energy values for the given temperature.

V. SUMMARY

If it is required to know the spectral distribution of a Planckian

radiator at any temperature throughout the visible and into the

ultra-violet and infra-red regions, Table 1 affords a ready means of computing it with very little labor. But if the distribution curve

is required to pass through a certain point (for instance, 100 at 560 mu), and if the values of energy are desired for 10 mu intervals, further computation is necessary. Tables 2A and 2B have

been computed giving these values for 38 temperatures 111 the

interval 1,000 to 28,000° K.

In order to obtain directly the spectral distribution for any

temperature whatever in this interval, at wave lengths 10 mu

apart, and having the value 100 at 560 ma, a series of isochromatic curves has been constructed. From these curves, wliich arc Dlotted to a larce scale, the enerev values can be read to an a

usually within 0.33 per cent, the temperature values williin 0.1 per cent.

VI. APPENDIX

The question may arise as to what variation will be caused in the relative values of by the adoption of a value of C- different from 14,350; for example, Cobleutz's most recent value 10 for this constant is 14,320; the International Critical Tables has adopted

14,330; and further research may eventually cause further, though

probably very small, revision of these latest values.

C\~* An inspection of the equation, E^-*-^ shows that the -3-1

effect of a variation of C, with constant is the same as, but opposite in sense to, a variation of with C, constant. So far as the graphs are concerned, in which values of £x are plotted against values of for the various wave lengths, wc have,

therefore, a ready means of obtaining true values of £x (within the

precision already noted in the main part of the paper) for any

value of C, which it may be desired to use. The method is simply

this: Compute the percentage difference between 14,350 and the value to be used, Call this percentage difference p. Multiply llie temperature by 1 ±p/ioo and read values of from the graphs

at this corrected temperature. If C, is decreased, must be in- creased, and vice versa. Wliile true values of Ex may thus be obtained for any value of

C-, and as readily as for the value 14,350, the difference in Ek obtained by using 14,320 or 14,330 is so small as to be negligible for most work for which these tables and graphs will be used. In

the first place, the difference in temperature necessary to make the

correction on the graphs is about 0.2 ocr cent, the order of magnitude of differences in color temperature determinations by experi- enced observers. In the second place, the differences in values of £x relative to that at 560 nui caused by tlu's 0.2 per cent change in Cj is but a few tenths of 1 per cent, except at llie extreme wave

lengths for temperatures below about 1,500°, where they may be- come greater than 1 per cent. Spectral energy measurements can

scarcely be guaranteed mthin 1 per cent, while the discrepancies between spectral energy measurements and values of energy computed for the corresponding color temperature may be several per cent in the end regions of the visible spectrum.

While, therefore, true values of £t may be readily obtained from

the graphs for any value of C„ it is probable that for most work

the error involved in using values of £, based on C, -14,350^

that is, those given by the tables or read from the graphs without

the temperature correction—will be negligible.

• 111- tod PenrlBe. Tbc aulhy ol I isfat f

Ani£cul Light r - by t&* ' i c : I d! tt-CJ 7 b- juS..-. - i.ln . .. .11 . h1 • a 3. Set P-t-n No. **. Dnwabtr. *

umnnil ol ibt Cnlnr Irmtmu

DMptniaa. a S. So. Pu-it

(~1 duller- EttcM

For itlc by Uic Superintendent o! Document*, GoTcnin: t ent Printing; Office, Wubin(toa, D. C. Price 35 cenU.



DEPARTMENT OF COMMERCE

BUREAU OF STANDARDS

1500

ENERGY DISTRIBUTION BY PLANCK'S FORMULA

100 AT WAV E LENGTH 560 m|j.

2000 2500 5 3000 3500 4000

MISCELLANEOUS PUBLICATION No. 56 SHEET 2 CHART I

4500 l—

i

50

45

40

35

30

25

20

15

20

15

10

L

1000 1500 2000 2500 :booo 3500 4000 4500

TEMPERATURE K



DEPARTMENT OF COMMERCE

BUREAU OF STANDARDS

1000

300

o

in^ 290

J 280

y_ 270

o

cc 260

bJ

z

u

u

250

240

230 >

|— 220

<

_J 210

LjJ

CH 200

190

180

170

160

150

140

130

120

|

1 1 n !

100

90

80

70

60

1500

ENERGY DISTRIBUTION BY PLANCK'S FORMULA

100 AT WAVE LENGTH 560 m(x

2000 2500 :3000 3500 4000

MISCELLANEOUS PUBLICATION No. 56 SHEET 3 CHART 2

4500 5000

-no

no

Hoo

100

no

100

120

no

I 100

120

no

100

-4—i 90

H-i—(120

; no

100

90

00

90

80

70

60

1500 2000 2500 3000 3500 4000 4500

50

5000

TEMPERATURE ° K



DEPARTMENT OF COMMERCE

BUREAU OF STANDARDS

1000 ^ 2800 i o 111

m i £ 2700 H

f

^ 2600

2500

q£ 2400

. i 2300

2200

2100 Ld

>

|— 2000

<

—I 1900

Ld

CH 1800

1700

I DUU

1500

1400

1300

1200

I I 00

1000

900 |

800

700

600

500

400

300 \m- 1000

1200

ENERGY DISTRIBUTIC IN BY PLANCK'S FORMULA

100 AT WAV E LENGTH 560 mp.

1400 1600

MISCELLANEOUS PUBLICATION No. 56 SHEET 4 CHART 3

8000-:

7000

6000-

I 5000-ffl

I

3000-»

1000

m I 00 100

1200 1400 1600 1800

11 ^i^gtf^piiMijjM

2000 2200 2400

TEMPERATURE ° K



DEPARTMENT OF COMMERCE

BURCAU OF STANOARDS N

MISCELLANEOUS PUBLICATION No 56

5000 ^ 130 O

in

p)

n

(M o

120

5500

ENERGY DISTRIBUTIO

100 AT WAVE

6000 6500

BY PLANCK'S FORMULA

LENGTH 560 mp.

r000 7500 8000

SHEET 5 CHART 4

160

150

140

130

120

HI

100

90

80 80

70 70

60 :ii!-.ii:m--i^iiiu : : 5000 5500 6000 6500 ",'000 7500 8000

TEMPERATURE ° K



DEPARTMENT OF COMMERCE

BUREAU OF STANDARDS

8000 ~ 290

o liiilM inm 280

270

10000

>_ 260

O

Cr 250

. i Li-J -7 240

230

220 LU

>

h-210

<

-J 200

LU

Ql 190

180

i /u

160

150

140

130

120

I 10

100

90

80

70

60

50

40

8000

ENERGY DISTRIBUTION BY PLANCK'S FORMULA

IOO AT WAVE LENGTH 560 m>i

12000 14000 16000

-J LiJ I i _

MISCELLANEOUS PUBLICATION No. 56 SHEET 6 CHART 5

3 I 0i

300H

290 ' '-' 22000 24000 26000 28000B

18000 20000 22000 24000 26000 28000

1 1 1 Inn II

, I

. .h

i ^blrtr^rn^^u^hML^,^ 290

280

270

260

250

240

230

220

210

200

190

180

i"7n

I I w

160

150

140

130

120

|- I 10

1 100

I 90

I 80

10000 12000 14000 16000 18000 20000 22000 24000 26000 28000

TEMPERATURE ° K



M. Katherine Fr«h*f«r Cheater L. Snow TABLES AND GRAPHS FOR FACIl .ITATING THE COMPUTATION OF

SPECTRAL ENERGY DISTRIBU' TION BY PLANCK'S FORMULA

MIMTQJ^NEOLS PUBUCATION N* 55 SHEET 7 TABLES

i

!8r: I

II

i

i

9

I

i

i

i

I

1

t

ii

i

l

I

l

i

I;

f

I

I

1

il

1

i

1

li

i;

1

ii

1

Ii;

I

I

I

I

1

1

1

1

i

1111 1 I I I 1 1 I

III 1 1 i 1 1 1 I

is.-..:...::::.

• I I I I 1 11 11

f

:

1:::::::::=:: 1

1

I

li I1

I

1! 11

1

1

I

1

ii i 1! li 11 1! 1 i i' JP 1 1! 1

X Is oUUmlcral III I 1 1 II I I I

I If if I If II I 11

li as ii ii 11 II III 8b ::::::£ as SIS SI! 88 S3 fill 1 If I I I 1 I

I 1 li li 1 1 1

! II ii ii II II II 11 I II ii ii 1 |i II II

1

, —, —, --i

II

II M II II

II li 1 I

1

If

ii ii i ii i ii ii

If If I I If I I

1 1 1 li 1 1 1

II i II i ii 11 li

II I! II ii i 11 1

1

'II

SI i Si

1

I li 11

ifi i; m

nfrryaal

I T»it~j iu.i

| no.*

| ~iitTj

1 II

II

n

li

II

II

II

1

II II II II

II II II I II

1

I.:

1

i

ii

\ In Bimmioooi

Talon u EX lor lb

11 li 1 li il 1 li li

If 1 | li 1 1 il l

ii II 1 I! II 1 1 1

i 1 1 1 1 1 1 1

i 1 1 1 1 1 1 1

ii ii Ii II ii ii ii ii

1! ii 11 li ii ii ii ii