TECHNICAL NOTE

Revised equivalent temperature formula

H. Z. Sar-EI

To save having to repeat computations of radiances of several thermal sources with different tempera­tures that are radiating simultaneously or, alternatively, to compute the total radiance of a single source composed of several patches with different areas and temperatures, a concept of area-weighted temperature has been defined and used [U.S. Army Electronics Command Rep. 7043 (U.S. Army Electronics Command, Fort Monmouth, N.J., 1975); Opt. Eng. 15, 525-530 (1976)]. This equivalent temperature was said to be substituted in Planck function to obtain the total spectral radiance. Theoretical and computational examinations of the concept of area-averaged temperature reveal the need for its revision. To keep the advantage of a short-cut computation of the total radiance in the above-mentioned cases, a revised formula for the equivalent temperature is suggested.

To compute the total radiance of a radiating source composed of several patches, each of which has its particular area, spectral emissivity, and temperature, Ratches and coworkers1,2 proposed the use of an equivalent temperature, TAVG, given by the relation

where Ai and Ti are the area and temperature of the ith patch of the source, respectively. Ratches and coworkers had referred to TAVG as an approximate temperature of the whole source or as an area-weighted temperature. Equation (1) can be adopted as a good approximation for sources or patches of equal spectral emissivities and rather high tempera­tures. For instance the temperature of each source or patch that is included in Eq. (1) should be at least 2000 K for a wavelength of approximately λ = 10 μm and hc < λkT, where h, k, and c are the Planck constant, Boltzmann constant, and light velocity, respectively. This is easily determined from the Planck blackbody formula when one writes the de­nominator of the formula in an approximate form subject to known mathematical rules. Ratches and co-workers1-2 also implicitly ascribed equal spectral

The author is with Department 6053, El-Op Electro-Optic Industries Company, P.O. Box 1165, Rehovot 76110, Israel.

Received 24 May 1993; revised manuscript received 26 October 1993.

0003-6935/94/091749-02$06.00/0. © 1994 Optical Society of America.

emissivities to all radiative participants. Conse­quently a revised approach to calculate more accu­rately the approximate equivalent temperature of a thermally nonhomogeneous source is suggested. Let Ti, ελ., and Ai be the temperature, spectral emissiv­ity, and area of the ith patch, respectively. Then, the whole spectral radiant intensity of the patched source, Iλ, is given by the Planck function3,4

where

λ is the wavelength of the radiation, and c1 and c2 are constants.4 On the other hand, Iλ of the source as a whole should conform to the Planck function, so that

where

Teq is the searched equivalent temperature and ελ is the effective spectral emissivity of the whole source. Equation (5) is trivial, but so far we have no knowl­edge about ελ. Because Eqs. (2) and (4) are physically equal, we have

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From Eq. (7) we can deduce both ελ and Teq as follows. Because the validity of Eq. (7) is independent of its parameter values, it surely holds for the simple case of uniform thermal distribution source, too. In this trivial case, Eq. (7) reduces to the relation

which provides ελ. Extraction of Teq from Eqs. (6) and (7) yields

It is easy to find from Eq. (9), at once, the conditions by which Eq. (1) was brought about and to see the physical limits of Eq. (1). As can be seen from Eq. (9), Teq is λ dependent. Examination of the extent of this dependence for the two practical spectral regions, 3-5 μm and 8-12 μm, as well as for their subregions and other spectral regions, indicated that for a differ­ence of less than 1% between the computed radiances, according to Eqs. (2) and (4), a substitution of a value for λ in Eq. (9) that equals the arithmetic mean of the spectral region was good enough. Trials to reduce

differences between computed results from Eqs. (2) and (4) to less than 0.01% came out with very minor variations in Teq values. For instance, to improve approximation to the true radiance given by Eq. (2) from less than 1% to better than 0.02%, several different cases had been arbitrarily chosen and exam­ined. The differences between the computed values of the equivalent temperature among the various cases, amounted to only approximately 0.2 K. The associated Teq values changed around 500 K in all the cases examined.

The author is grateful to Yoram Liran for stimulat­ing him to investigate the subject and to Uri Mahlab for his fruitful comments on the manuscript.

References 1. J. A. Ratches, W. R. Lawson, L. P. Obert, R. J. Bergmann, T. W.

Cassidy, and J. M. Swenson, "Night vision laboratory static performance model for thermal viewing systems," U.S. Army Electronics Command Rep. 7043 (U.S. Army Electronics Com­mand, Fort Monmouth, N.J., 1975).

2. J. A. Ratches, "Static performance model for thermal imaging systems," Opt. Eng. 15, 525-530 (1976).

3. W. L. Wolfe, "Radiation theory," The Infrared Handbook, W. L. Wolfe and G. J. Zissis, eds. (Environmental Research Institute of Michigan, Ann Arbor, Mich., 1989), Chap. 1, pp. 1-21.

4. Ref. 3, pp. 1-2.

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