Abstract

For many applications, a simple approximate equation for blackbody radiation can be more useful than is the Planck blackbody equation. An intermediate region exists where both the Rayleigh–Jeans equation and the Wien equation for blackbody radiation are inaccurate approximations to the Planck equation. Several approximate equations of simple mathematical form have been obtained that are more accurate in this intermediate region. These equations are applied to optical pyrometry to obtain formulas for brightness temperature and color temperature that can be used in the intermediate region.

© 1967 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |

  1. H. J. Kostkowski, R. D. Lee, in Temperature, Its Measurement and Control in Science and Industry (Reinhold Publishing Co., New York, 1962), Vol. 3, pp. 449–481.
  2. D. R. Lovejoy, in Temperature, Its Measurement and Control in Science and Industry (Reinhold Publishing Co., New York, 1962), Vol. 3, pp. 487–506.
  3. G. H. Dieke, in Temperature, Its Measurement and Control in Science and Industry (Reinhold Publishing Co., New York, 1955), Vol. 2, pp. 25–26.

Dieke, G. H.

G. H. Dieke, in Temperature, Its Measurement and Control in Science and Industry (Reinhold Publishing Co., New York, 1955), Vol. 2, pp. 25–26.

Kostkowski, H. J.

H. J. Kostkowski, R. D. Lee, in Temperature, Its Measurement and Control in Science and Industry (Reinhold Publishing Co., New York, 1962), Vol. 3, pp. 449–481.

Lee, R. D.

H. J. Kostkowski, R. D. Lee, in Temperature, Its Measurement and Control in Science and Industry (Reinhold Publishing Co., New York, 1962), Vol. 3, pp. 449–481.

Lovejoy, D. R.

D. R. Lovejoy, in Temperature, Its Measurement and Control in Science and Industry (Reinhold Publishing Co., New York, 1962), Vol. 3, pp. 487–506.

Other (3)

H. J. Kostkowski, R. D. Lee, in Temperature, Its Measurement and Control in Science and Industry (Reinhold Publishing Co., New York, 1962), Vol. 3, pp. 449–481.

D. R. Lovejoy, in Temperature, Its Measurement and Control in Science and Industry (Reinhold Publishing Co., New York, 1962), Vol. 3, pp. 487–506.

G. H. Dieke, in Temperature, Its Measurement and Control in Science and Industry (Reinhold Publishing Co., New York, 1955), Vol. 2, pp. 25–26.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (17)

Fig. 1
Fig. 1

The accuracy with which the Wien equation approximates the blackbody spectral radiance Nbλ plotted as a function of wavelength for selected values of temperature.

Fig. 2
Fig. 2

The accuracy with which the Rayleigh–Jeans equation approximates the blackbody spectral radiance Nbλ plotted as a function of wavelength for selected values of temperature.

Fig. 3
Fig. 3

The accuracy with which the indicated equations approximate the blackbody spectral radiance Nbλ plotted as a function of the product λT.

Fig. 4
Fig. 4

The indicated functions plotted in terms of the variable v (= c2T).

Fig. 5
Fig. 5

The accuracy with which the indicated functions ϕi approximate the Planck function, ϕp = (ev − 1)−1.

Fig. 6
Fig. 6

A plot of the function ϕ ˜ p(w). As w increases, ϕ ˜ pasymptotically approaches the straight line whose y intercept is −½ and whose slope is unity.

Fig. 7
Fig. 7

The values of the coefficients of the function ϕI(v) = b0 + (b1/v), for which ϕI(v) = ϕp(v).

Fig. 8
Fig. 8

The values of the coefficients of the function ϕII(v) = (d1/v) + (d2/v2), for which ϕII(v) = ϕp(v).

Fig. 9
Fig. 9

The values of the coefficients of the function ϕIII(v) = h0 + (h1/v) + (h2/v2), for which ϕIII(v) = ϕp(v).

Fig. 10
Fig. 10

The accuracy with which the Planck function ϕp is approximated by the function ϕI for a selected set of values of its coefficients b0 and b1, namely, those at v = 0, 1, 2, 3, 4, 5, 6, and 7.

Fig. 11
Fig. 11

The accuracy with which the Planck function ϕp is approximated by the function ϕII for a selected set of values of its coefficients d1 and d2, namely, those at v = 0, 1, 2, 3, 4, 5, 6, and 7.

Fig. 12
Fig. 12

The accuracy with which the Planck function, ϕp, is approximated by the function ϕIII for a selected set of values of its coefficients h0, h1, and h2, namely, those at v = 0, 1, 2, 3, 4, 5, 6, and 7.

Fig. 13
Fig. 13

A plot of the function Ψp(v). As v increases, Ψp asymptotically approaches the straight line of unit slope passing through the origin.

Fig. 14
Fig. 14

The values of the coefficients of the function ϕIV (v) = vβ0eβ1v, for which ϕIV(v) = ϕp(v).

Fig. 15
Fig. 15

The values of the coefficient of the function ϕV(v) = eγ1v, for which ϕV(v) = ϕp(v).

Fig. 16
Fig. 16

The accuracy with which the Planck function ϕp is approximated by the function ϕIV for a selected set of values of its coefficients β0 and β1, namely, those at v = 0, 1, 2, 2.3, 2.8, 3, 4, 5, and 6.

Fig. 17
Fig. 17

The accuracy with which the Planck function ϕp is approximated by the function ϕV for a selected set of values of its coefficient γ1, namely, those at v = 1, 2, 3, 4, 5, and 6.

Tables (1)

Tables Icon

Table I Tabulated Functions for Evaluating Various Coefficients

Equations (63)

Equations on this page are rendered with MathJax. Learn more.

N b λ = c 1 λ 5 { [ 1 + 1 1 ! c 2 λ T + 1 2 ! ( c 2 λ T ) 2 + ] - 1 } - 1 .
N b λ c 1 T [ c 2 λ 4 ( 1 + 1 2 c 2 λ T ) ] - 1 .
N b λ = c 1 λ - 5 e - c 2 / λ T ( 1 - e - c 2 / λ T ) - 1 .
N b λ c 1 λ - 5 e - c 2 / λ T ( 1 + e - c 2 / λ T ) .
ϕ ˜ p ( w ) i = 0 n a i w i ,
ϕ ( v ) = i = 0 n a i v i .
ϕ p ( v ) i = 0 n a i v i ,
ϕ ˜ I ( w ) = b 0 + b 1 w ,             for w 1 ϕ I I ( w ) = d 1 w + d 2 w 2 , for w 0.
ϕ ˜ I I I ( w ) = h 0 + h 1 w + h 2 w 2
ϕ I ( v ) = b 0 + b 1 / v ϕ I I ( v ) = d 1 / v + d 2 / v 2 ϕ I I I ( v ) = h 0 + h 1 / v + h 2 / v 2 .
b 0 ( v ) = ϕ p ( v ) + v ϕ p ( v ) b 1 ( v ) = - v 2 ϕ p ( v ) ,
d 1 ( v ) = v [ 2 ϕ p ( v ) + v ϕ p ( v ) ] d 2 ( v ) = - v 2 [ ϕ p ( v ) + v ϕ p ( v ) ] .
h 0 ( v ) = ϕ p ( v ) + 2 v ϕ p ( v ) + v 2 2 ϕ p ( v ) h 1 ( v ) = - v 2 [ 3 ϕ p ( v ) + v ϕ p ( v ) ] h 2 ( v ) = v 3 [ ϕ p ( v ) + v 2 ϕ p ( v ) ] .
d ϕ p ( v ) / d v = - e v ϕ p 2 ( v ) .
d ϕ p ( v ) / d v = - ψ p ( v ) [ ϕ p ( v ) / v ] ,
ψ ( v ) = i = 0 n α i v i
d ϕ p ( v ) / ϕ p ( v ) - ψ ( v ) [ d v / v ]
ϕ p ( v ) e - ( ψ ( v ) / v ) d v ,
β 0 ( v ) = ψ p ( v ) + ln ϕ p ( v ) 1 - ln v , β 1 ( v ) = - ψ p ( v ) ln v + ln ϕ p ( v ) v ( 1 - ln v ) ,
γ 1 ( v ) = - ln ϕ p ( v ) / v .
Δ ϕ I ϕ p = 1 ( 1 + b 1 b 0 1 v 0 ) Δ b 0 b 0 + 1 ( 1 + b 0 b 1 v 0 ) Δ b 1 b 1 .
N b λ c 1 λ 5 b 0 + c 1 c 2 T λ 4 b 1 ;
N b λ c 1 c 2 T λ 4 d 1 + c 1 c 2 2 T 2 λ 3 d 2 ;
N b λ c 1 λ 5 h 0 + c 1 c 2 T λ 4 h 1 + c 1 c 2 2 T 2 λ 3 h 2 ;
N b λ c 1 λ 5 ( λ T / c 2 ) β 0 e - β 1 c 2 / λ T ;
N b λ c 1 λ 5 e - γ 1 c 2 / λ T .
N b λ ( c 1 / c 2 ) ( T / λ 4 ) - ( c 2 / 2 λ 5 )
N b λ ( c 1 / c 2 ) ( T / λ 4 ) e - c 2 / 2 λ T
0 N b λ ( λ , T B B ) σ λ d λ = 0 N λ ( λ , T τ ) σ λ d λ .
N b λ ( λ e , T τ B r ) = N λ ( λ e , T τ ) .
0 N λ ( λ , T r ) σ λ d λ = 0 τ λ N λ ( λ , T s ) σ λ d λ ,
0 τ λ N λ ( λ , T s ) σ λ d λ = R 0 N b λ ( λ , T s B r ) σ λ d λ ;
0 N λ ( λ , T r ) σ λ d λ = 0 N b λ ( λ , T r B r ) σ λ d λ ,
0 N b λ ( λ , T r B r ) σ λ d λ = R 0 N b λ ( λ , T s B r ) σ λ d λ .
N b λ ( λ e m , T r B r ) = R N b λ ( λ e m , T s B r ) .
0 N b λ ( λ , T r B r ) σ λ d λ 0 N b λ ( λ , T s B r ) σ λ d λ = N b λ ( λ e m , T r B r ) N b λ ( λ e m , T s B r ) .
T s B r T r B r [ 1 + ( λ e m T r B r / c 2 ) ln R ] - 1 ,
[ λ e m ( T s B r , T r B r ) ] - 1 = ( 1 T r B r - 1 T s B r ) - 1 1 / T s B r 1 / T r B r 1 λ e ( T B r ) d ( 1 T B r ) ,
1 λ e ( T B r ) = 0 N b λ ( λ , T B r ) λ σ λ d λ 0 N b λ ( λ , T B r ) σ λ d λ .
[ λ e m ( T s B r , T r B r ) ] - 1 ½ { [ λ e ( T s B r ) ] - 1 + [ λ e ( T r B r ) ] - 1 } .
T s B r T r B r [ 1 + k c 2 1 T r B r ] - 1 .
T s B r ( 0 σ λ λ 4 d λ / 0 τ λ σ λ λ 4 d λ ) T r B r .
N b λ ( λ , T B B ) b 1 ( c 1 / c 2 ) ( T B B / λ 4 ) + b 0 ( c 1 / λ 5 )
T s B r b 1 r b 1 s T r B r R + c 2 b 1 s λ e m ( b 0 r R - b 0 s ) ,
λ e m = ( 0 σ λ λ 4 d λ / 0 σ λ λ 5 d λ ) .
T s B r T r B r R + c 2 2 λ e m ( 1 - 1 R ) .
T s B r γ 1 T r B r [ 1 + ( λ e m T r B r / c 2 ) ln R ] - 1 ,
1 λ e m ( T s B r γ 1 , T r B r ) = 1 ( 1 T r B r - γ 1 T s B r ) γ 1 / T s B r 1 / T r B r 1 λ e ( T B r ) d ( 1 T B r ) ,
1 λ e ( T B r ) = 0 N b λ ( λ , T B r ) λ σ λ d λ 0 N b λ ( λ , T B r ) σ λ d λ ,
1 λ e m ( T s B r / γ 1 , T r B r ) 1 2 ( 1 T r B r - T s B r ) [ γ 1 T r B r - T s B r λ e ( T s B r ) + ( 2 - γ 1 ) T r B r - T s B r λ e ( T r B r ) ] .
T s B r γ 1 T r B r [ 1 + ( k / c 2 ) ( 1 / T r B r ) ] - 1 .
ψ p ( v B r ) [ d T B r / T B r ] = d N b λ / N b λ ,
N b λ ( λ 1 , T c ) N b λ ( λ 2 , T c ) = N b λ ( λ 1 , T B r ( λ 1 ) ) N b λ ( λ 2 , T B r ( λ 2 ) ) .
T c ( 1 λ 1 - 1 λ 2 ) [ 1 λ 1 T B r ( λ 1 ) - 1 λ 2 T B r ( λ 2 ) ] - 1 .
( λ 2 / λ 1 ) 4 T c / T c ( λ 2 / λ 1 ) 4 T B r ( λ 1 ) / T B r ( λ 2 ) .
N b λ ( λ , T B B ) ( c 1 / c 2 ) ( T B B / λ 4 ) - ( c 1 / 2 λ 5 )
T c [ T B r ( λ 2 ) λ 1 - T B r ( λ 1 ) λ 2 ] { ( 1 λ 1 - 1 λ 2 ) + 2 c 2 [ T B r ( λ 2 ) - T B r ( λ 1 ) ] } - 1 .
N b λ ( λ , T B B ) b 1 c 1 T B B c 2 λ 4 + b 0 c 1 λ 5
T c c 2 [ b 0 ( v 2 c ) ϕ I ( v 1 B r ) - b 0 ( v 1 c ) ϕ I ( v 2 B r ) ] b 1 ( v 1 c ) λ 1 ϕ I ( v 2 B r ) - b 1 ( v 2 c ) λ 2 ϕ I ( v 1 B r ) ,
T c [ γ 1 ( v 2 c ) λ 2 - γ 1 ( v 1 c ) λ 1 ] [ γ 1 ( v 2 B r ) λ 1 T B r ( λ 1 ) - γ 1 ( v 1 B r ) λ 1 T B r ( λ 1 ) ] - 1 .
T c ( γ 1 ( v 2 c ) λ 2 - γ 1 ( v 1 c ) λ 1 ) [ 1 λ 2 T B r ( λ 2 ) - 1 λ 1 T B r ( λ 1 ) ] - 1 .
[ ψ p ( v 2 c ) - ψ p ( v 1 c ) ] ( d T c / T c ) = d ρ / ρ .
ϕ p ( v ) = - e v ϕ p 2 ( v ) ϕ p ( v ) = - e v ϕ p 2 ( v ) [ 1 - 2 e v ϕ p ( v ) ] ψ p ( v ) = v e v ϕ p ( v ) .

Metrics