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79

TA B L E S 52-57.-THE BLACKBODY AND ITS RADIANT ENERGY

TA B L E 5 2 . 4 Y M B O L S A N D D E F IN I N G EX PR ES SI ON S F OR R A D I A N TENERGY m

Radiant energy is energy traveling in the form of electromagnetic waves. It is measuredin units of energy such as ergs, joules, calories, and kilowatt hours. Some units, symbols,and abbreviations used in discussing radiant energy are as follows :

Designation

Radiant energy .......Spectral radiant energy.

Radiant energy density.

Radiant flux ..........Radiant flux density.

Radiant intensity of asource ..............

Spectral radiant intensity

Radiant flux density ofa source per unit solidangle ...............

Radiant intensity of asource per unit area..

Radiant flux per unitarea ...............

Symbol anddefining

expression

UdU

dUd Vd U

dW=--d A

d w

dJdx

U,= -

+ P) = 7

u =

J=*

J A =

d WB , N )do

dJd A

d A

B = -

E = - d

Unit

......

. . . . . .

erg/cm

watt, erg/sec

watt/cma

watt/steradian

watt/steradian

watt/(steradian cm)

watt/(s teradian cm)

......

Proposedterm 01.

Radiant energy

Spectral radiant

Radiant energy density

energy

Radiant flux (radi-ance *)

Radiant flux density(radiancy *)

Radiant intensity

Spectral radiant inten-sity

Steradiancy *

Steradiancy *

Irradiancy

The standard radiator is the blackbody, which may be defined as a body that absorbsall the radiation that falls upon it, i.e., i t neither reflects nor transmits any of the incidentradiation. From this simple definition and some very plausible assumptions it can be shownthat the blackbody radiates more energy than any other temperature radiator when bothare at the same temperature. The total amount of energy (i.e., for all wavelengths)radiated by a blackbody depends upon the temperature raised to the fourth power and a

constantu

that had to be measured: W = uT4If a blackbody is radiating to another blackbody it will a t the same time receive radiation

from the second blackbody and, under the proper geometrical conditions, the net radiationlost by the first blackbody is

The spectral distribution of this radiation is given by the Planck equation :

For values of the product AT less than 3OOOp deg, the Wien equation

gives values that are correct to better than 1 percent.The values of a number of the radiation constants have been selected from Table 26 andare given in Table 53. AAI the blackbody calculations given were made with these constants.Some calculated results for the total radiation W for a series of temperatures and of J Xfor a range of temperatures and for wavelengths have been calculated and are given inTables 54-56.

W = U Ti4 - a

J X Ad/[exp(cr/AT) - 1 t

J X= cAd/[exp (cr/AT)l

a Rev. Sci. Instr., vol. 7. p. 322, 1936. These terms apply only to a source. The term radianceis not recommended as a substitute for radiant f lux; however, if a s ingle term is desired to express theradiant flux from a source, the word radiance is suggested as the most logical. t See footnote 5a.

ad For a more extensive list of va lues of J , reference should be made to two papers by Parry Moon:Journ. Math. and Phys., vol. 16, p. 133. 1937 ; Publ. Electr. En g., Massachusetts Institut e of Tech-

nology, 1947.SMITHSONIAN PHYSICAL TABLES

p. 7.

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SO T A B L E B S .-RA D IATIO N C O N S TA N T S

Velocity of light.. ............................ c = 2.99776 x 10 cm sec-'Planck's constant ............................. h = 6.6242 x erg-secBoltzmann's constant k = 1.3805X lo-'' er g deg-'Stefan-Boltzmann constant*Wien's displacement law.......................

u = 5.673 X lo- e r g cm deg-' sec-'= Ac1A4F AT)

The principal corollaries are: L T = 6J m- -

AT6 - I

T h e fir st corollary is sometimes given as the W ien's displacement law , and6 as thedisplacement constant.Wien d isp lacement constan t . .. . . . . . . . . . . . . . . . . .First radiation constantt

All lengths in cm, d A = 1 c m . .A re a cm', in p , d A = 0 . 0 1 ~ ...............

Second radiation constant......................

6 = 0.2897 cm deg

c1 = 3.740 X

c z = 1.4380cm deg

e rg sec-' cm2CI = 3.740 >( 10' er g sec cm2

T he un it of energy chosen for the above values is the erg. Any oth er unitof energy (orpower) may be used if the proper conversion factor is used (Table7).

Va l u e s of c z u s e d at d if fe re n t times.-This second radiation constant has been de-termined many times in the last40 years. Sho wn below ar e the values used at differenttimes. [ A new determ ination of the value ofcz by G. A. W. Rutgers (Physica, vol. 15,p. 985, 1949) gives two values : 14325. 20 and 14310. 20 p deg.1

National Bureau NelaDate of Standards Park

1911.. 14.500~K

1915..1917 143501922. . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . 14320g1925. . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . 1432061936. . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . 14320 I/1944 143201949.. 14380

14500p K

144601435014350143201432014320-

For 27r solid angle. For the general case, c1 may be written in the following symbolic form:(wavelength unit) x power unit

area x wavelength interval x solid angle1 = numeric

This form shows that the value of the numeric depends upon the several uni ts used-in thi s case 5 .I f I , is the normal intensi ty, i.e., per un it solid angle perpendicular to the surface, s JAo gives the

radia2ion per 2 8 solid angle. Th e energy radiated within a unit solid angle around the normal, is 0.92 J oThe above values are for a plane blackbody; for a spherical blackbody the radiation for 2a solid angleequals 2aJ0.

For calculations the use of the radiation constants u and c2 as given follows directly and causes butlittle trouhle. Th e numeric for c2 must be expressed in the unit of wavelength times the absolute tempera-ture. If t he wavelength is expressed in 11 the numeric hecomes 14380.

When Planck's equation is used fo r calculation s, it may be writ ten as follows for blackbody of a re a A:

J,dX = ( A c i P / [ exp ( c 2 / X T )- I)dXwhere dX is the wavelength interval fo r which t he radiation is to he calculated. Th e first value of c1given in the table is for all dimensions in centimeters-a condition almost never met in practice. Th esecond value is for the wavelength expressed in microns and d X = 0.01fl.

I f this second value of c2 be used in calculation with Planck's equation and summed step by step,the results will be the total energy per second, per 2a solid angle, per unit area for the wavelengthinterval covered X expressed in u .

t I . , G Priest: in January 1932, used cp= 14350 in his work on color temperature. I J. F. Skog-land, in 1929, used c t = 14330 in his tables of s pectral energy d istr ihuti on of a blackbody. D. U.Judd, in 1933, used c I = 14350 in his calculations related to the T C I tandard observer.

SYITHSONIAN PHYSICAL TABLES

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QIi5BD

4DW

vrn

TA B L E 5 4 . - R A D I AT I O N IN E R G S ( W X 10 ) A N D G R A M -C A L O R IE S( W ' X 10 ') P E R C M ' P E R S EC , F O R 2 a S O L I D A N G L E ,F R OM A P E R F E C T R A D I AT O R AT t o F R O M -2 7O oC T O + 56 C A N D F O R T F R O M 300K T O 5500K

Tzmp.C

-270-250-200-190-180-160-150-140-130-120-110-100

9000

- 0- 0

000

- 10- 8- 6- 4- 2 0

2

erg cm-2 sec-1-5.6561.6321.6252.7134.2729.3011.3051.7832.3823.1214.0205.1006.3837.8969.6621.1711.4071.6761.9832.3302.7202.8042.8902.9773.0673.1583.252

n

-313333444444444555555555555

cal cm-z sec-1Temp .

5 r - 2 C

u = 5.672 x lo-' erg cmP deg-4 sec-

1.3513.8993.8836.4821.0212.2223.1184.2615.6937.4589.6051.2191.5251.8872.3092.7983.3614.0064.7385.5676.5006.7006.9047.1147.3277.5467.769

-10 4- 7 6- 5 8- 5 10- 4 12- 4 14- 4 16

4 18- 4 20- 4 22- 4 24- 3 26- 3 28- 3 30- 3 32- 3 34- 3 36- 3 38- 3 40- 3 42- 3 44- 3 46- 3 48- 3 50- 3 52- 3 54- 3 56

erg cm-2 sec-1

57-----3.3473.4453.5453.6463.7513.8573.9654.0764.1894.3054.4234.5434.6664.7914.9195.0495.1825.3175.4555.5965.7395.8856.0346.1866.3416.4986.658

n

555555555555555555555555555

W'

7.9988.2318.4708.7138.9629.2169.4759.7401.0011.0291.0571.0861.1151.1451.1751.2061.2381.2711.3041.3371.3711.4061.4421.4781.5151.5531.591

cal cm-2 sec-* erg cm-2 sec-1 cal cm-2 sec-

~~

Temp .---7 K-3 30 0* 4.5944-3 373.16 1.0998-3 400 1.4520-3 500 3.5450-3 600 7.3509

700 1.36193-3 800 2.3233-3 900 3.7214-2 1 0 0 5.6720_-2 1500 2.8715-2 2000 9.0752-2 2500 2.2156

5 1.0978 -26 2.6280 -26 3.4700 -26 8.4707 -26 1.7565 -1~ .7 3.2542 -17 5.5515 -17 8.8922 -17 1.3553 08 6.8614 08 2.1685 19 5.2942 1

-2 3500 8.5115-2-2-2-2 ,.

L

-2-2-2-2-2-2-2-2-2

~ . . . 9 2.0338 2

5500 5.1902 10 1.2402 34500 2.3259 10 5.5577 2

Energy radiated from OO( K ir be obtained from the value for th is temp rature )J mu l t ip ly in g it )J 10. Likewi e 01 th :I p ratu : th t are 1 jmes thevalues g iven in the table.

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54

x.102 0.30

.40.45

.50

.55

.60

.65

.70

.75

.so

.901.001.50

2.002.503.004.005.00

10.0050.00

100.00

x.102 0.30.40.45

.so

.55

.60

.65.70

.7s

.so

.901 001.50

2.002.503.004.00

5.0010.0050.00

100.00

TA B L E 55. -CALCULATED SPECTRA L INTENSlTl i ES JA F O R A RANGE O FWA V E L E N G T H S FOR A BL A CK BO D Y O F U N I T A R E A F OR A R A N G E

O F T E MPE RA T U RE S FRO M 5 0 K TO 25,000'K (concluded)

3000 3200' 3500 4000'

A J A n J A 1 J, n

-5.698 -13 1.1396 -11 5.3650 -10 9.1199 -84.562 2.0402 1.3998 1.8251 -11.7710 4.807 - 1.7358 0 9.616 02.2819 0 4.825 0 1.2640 1 4.565 14.795 0 9.330 0 2.1962 1 6.877 1

8.215 0 1.4957 1 3.2321 11.2195 1 2.1028 1 4.237 11.6321 1 2.6895 1 5.113 12.0227 1 3.2083 1 5.807 12.3657 1 3.6313 1 6.303 1

2.6465 1 3.94912.8600 1 4.1643.0957 1 4.3273.1245 1 4.2282.1026 1 2.5919

1.1703 1 1.38186.600 0 7.607 0 9.178 0

3.9044 0 4.432 0 5.247 01.5780 0 1.7598 0 2.0369 07.442 8.217 9.391

1 6.611 1I 6.754 11 6.662 11 6.248 11 3.4032 1

1 1.7181 1

6.081 6.5931.1897 1.27321.9581 8.130

6000 8000-i n A

1.4597 -2 5.840 07.302 1 1.4607 35.223 2 3.8565 39.152 2 4.129 39.906 2 3.8032 3

- 2 7.361- 4 1.3978- 6 8.926

10 ,000 15 ,000~-.1268 2 2.5671 4

8.820 3 9,763 41.2857 4 6.571 41.0313 4 3.6571 48.653 3 2.7323 4

J h ?I A

9.032 11.0789 21.2052 21.2825 21.3168 2

1.3165 21.2903 21.1884 21.0561 24.9320 1

2.3215 11.1922 1

6.650 02.5076 01.1372 0

8.657 -21.6064 -41.0219 5

2 0 , 0 0 0 ~

1.2094 -46.647 01.0561 2

2.7563 23.4034 2

3.8137 24.004 24.018 23.9080 23.7175 2

3.48103.22272.7040 22.2338 28.487 1

3.6384 11.7735 1

9.570 03.4705 01.5393 0

1.1225 -12.0216 -41.2808 5

25,000

J A n2.8224 53.3001 51.5411 57.255 45.1415 4

I, I I

1.1917 66.981 52.6523 51.1370 57.825 4

9.9983 2 3.3793 3 7.148 3 2.0625 4 3.7257 49.6424 2 2.9415 3 5.869 3 1.5762 4 2.7563 49.024 2 2.5311 3 4.816 3 1.2201 4 2.0780 48.279 2 2.1601 3 3.9614 3 9.563 3 1.5936 47.496 2 1.8485 3 3.2718 3 7.586 3 1.2456 4

6.728 2 1.5780 3 2.7160 3 6.084 3 9.800 3 1.3667 46.007 2 1.3494 3 2.2670 3 4.931 3 7.836 3 1.0845 44.748 2 9.945 2 1.6067 3 3.3311 3 5.178 3 7.078 33.7449 2 7.429 2 1.1643 3 2.3256 3 3.5536 3 4.810 31.2494 2 2.1278 2 3.0625 2 5.505 2 8.008 2 1.0537 3

1.1106 2 1.9004 2 2.7017 2.049 1 8.024 12.3814 1 3.6391 1 4.926 1 8.194 1 1.1494 2

2.5026 1 4.088 1 5.684 1.2584 1 1.8756 18.443 0 1.3487 1 1.8549 1.451 0 6.438 0

1.9460 0 2.7665 0 3.5918 0 5.6640

7.741 0

5.542 44.026 42.9907 42.2654 41.7461 4

3.5077 21.4804 27.280 12.3625 1

9.818 02.4184 -1 3.7177 -1 5.020 -1.3811 -1 1.8994 -1

2.4375 -4 3.2700 4 4.0 99 -4 6.1 85 -4 8.254 -31.5 391 -5 2.0 663 -5 2.5 793 -5 3.8 748 -5 5.194 -5

6.318 -11.0318 -36.448 -5

SMITHSONIAN PHYSICAL TABLES

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TA B L E 56.-B L A CK B OD Y S P E C T R A LINTENSITIES 85Auxiliary table for a short method of calculating JX for any temperature. (Menzel, Harvard Uni-

versity.)Let J o = intensity for To= 10,000 OK; or another temperature T K :

For ease of calculation To was taken as 10,000 OK. = abular JA X 10 watts, for cm* for 2* solidangle per 0 . 1 ~ . Choose A = k To / T; then JX= Jo T/T0) . A s an example find J Xfor 0 . 5 ~ nd 6oOO Kfrom value of JA or 0 . 3 ~iven in Table 55. 0 . 5 ~= 0 . 3 ~ 0,000/6000. J X for 0 . 3 ~ 1.2857 X lo'. J Afor A = 0 . 5 ~= 1.2857 X lo4 X (6,000/10,000)6= 9.998 X 10'.

J / J o = CAf(exp ( @ d o ) - )I /[A6(exp ( a /AT) - 11

10,000~

A J ,OLOO 1.3224.0150 1.1427.0200 6.949.0250 4.005.0300 2.3444

.0350 1.0214

.0400 8.906.0450 2.6833

.0500 3.8700

.0550 3.2828

.0600 1.8773

.0650 7.950

.07OO 2.6652

.0750 7.427

.0800 1.7823

.0850 3.7891

.Om 7.288

.0950 1.2894.lo00 2.1269

.lo50 3.3049

.1100 4.881

.1150 6.899

.1200 9.391

.1250 1.2365

.1300 1.5819

.1350 1.9732

.1400 2.4062

n A

-49 .1450-29 .15m-20 .1600-14 . 700-10 .1800

- 7 .1900- 6 .2000- 4 .21#- 3 . 2 m- 2 .2300- 1 .2400

1 .29001 .30002 .32002 .34002 .3600

2 .38002 .4OOo2 .42003 .44003 .4600

3 .48003 so00

J ,

2.87763.38064.4585.5866.716

7.8058.8209.7351.05361.12151.17691.22041.25241.27391.2859

1.28951.28571.26011.21631.1606

1.09771.03139.6408.9778.335

7.7247.148

U

33333

33344

44444

44444

44333

33

A

.5500

.6000

.6500

.7000

.7500

.8000

.8500.9000

.95001.000

1.1001.2001.3001.4001.500

1.6001.7001.8001.9002.000

2.2002.4002.6002.8003.000

3.5004.000

J ,

5.869 34.816 33.9614 33.2718 32.7160 3

2.2670 31.9031 31.6067 31.3641 31.1643 3

8.613 26.494 24.980 23.8782 23.0625 2

2.4487 21.9805 21.6183 21.3348 21.1106 2

7.867 15.724 14.262 13.2372 12.5026 1

1.4015 18.443 0

A

4.5005.0006.0007.0008.000

9.00010.0012.0014.0016.00

18.0020.0025.0030.0035.00

40.0045.0050.0055.0060.00

65.0070.0080.0090.00

100.00

J ,

5.3833.59181.77619.7565.797

3.65482.41841.18076.4333.7904

2.37901.56676.46923.13461.6954

9.9796.2364.0992.80421.9793

1.43901.06986.3063.93402.5793

n

000

-1-1

-1-1-1-2-2

-2-2-3-3-3

-4-4-4-4-4

-4-4-5-5-5

SMITHSONIAN PHYSICAL TABLES

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86 T A B L E 5 7.-C HA N GE S DUE T O A C H A N G E IN c2

The adoption of a new value for cz changes the calculated values forJ A by an amounttha t varies indirectly' with both the w avelength and the tem peratu re for value s ofAT