Temperature measurements of black body most closely fitting solar extraterrestrial spectru

Fussell, W. B.

Schutt, J. B.

English

NASA Technical Reports Server

NASA TECHNICAL NOTE N A S A  TN D-1845 
~ -
TEMPERATURE OF A GRAY BODY 
MOST CLOSELY FITTING THE 
SOLAR EXTRATERRESTRIAL SPECTRUM 
by William B. Fztssell and John B. Schzttt 
Goddard Space Flight Center 
Greenbe14 M d .  
- =  L 
NATIONAL AERONAUTICS A N D  SPACE A D M I N I S T R A T I O N  WASHINGTON,  D. C. AUGUST 1 9 6 4  
https://ntrs.nasa.gov/search.jsp?R=19640017747 2020-03-24T06:27:47+00:00Z
TECH LIBRARY KAFB, NM 
0154315 
TEMPERATUREOFAGRAYBODY 
MOST CLOSELY FITTING 
THE SOLAR EXTRATERRESTRIAL SPECTRUM 
By William B. Fussell and John B. Schutt 
Goddard Space Flight Center 
Greenbelt, Md. 
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION 
For sale by the Off ice of Technical  Services, Department of Commerce, 
Washington, D.C. 20230 -- Price $0.50 
TEMPERATUREOFAGRAYBODY 
MOST CLOSELY FITTING 
THE SOLAR EXTRATERRESTRIAL SPECTRUM 
by 
William B. Fussell and John B. Schutt 
Goddard Space Flight Center 
SUMMARY 
A definition of "closeness" of f i t  of a gray body curve to the 
sun's extraterrestrial  spectrum is presented. F. S. Johnson's 
data are used for the solar spectrum. For the wavelength inter- 
val 0.3 to 2.4 microns (pertinent to thermal design), the magni- 
tude of C, calculated for use in Planck's black body formula is 
8.6504 X 1O-I' watt-cm2 at a temperature of 5742* 0.5 O K .  
i 

CONTENTS 
Summary .................................. i 
INTRODUCTTON ............................. 1 
METHOD .................................. 1 
RESULTS .................................. 3 
DISCUSSION ................................ 4 
iii 
. ... - ~ -. ..... .. . , , , 
TEMPERATUREOFAGRAYBODY 
MOST CLOSELY FITTING 
THE SOLAR EXTRATERRESTRIAL SPECTRUM 
by 
William B. Fussell* and John B. Schutt 
Goddard Space Flight Center 
INTRODUCTION 
The problem of simulating the extraterrestrial solar illumination for testing the thermal design 
of satellites has recently become of wide interest. If the materials for the satellite surface have 
been selected, the problem reduces to one of matching the absorbed energy from the simulating 
source-for each material composing the surface-to the absorbed solar energy for that material. 
However, many materials have spectral absorptivity curves which vary slowly with wavelength be- 
tween wavelength limits including most of the sun's energy. A solar simulator which matches the 
sun's spectrum fairly well on a finely resolved basis, and which matches the total energy exactly, 
will automatically match the solar absorptivities and the simulated absorptivities closely for these 
materials. To this end, the black body function was used in constructing a gray body curve for 
which a solar f i t  was obtained by parametrically varying the temperature. 
METHOD 
In establishing a closeness of f i t  criterion for approximating the sun's extraterrestrial spectrum 
by that of a black body, it seems pertinent at this point t o  recall the geometric interpretation of the 
quantities in the black body equation which can be treated as parameters. Consequently, any crite- 
rion which is established by means of parametric variations can only be defined as ultimate when it 
is the most valid one from geometric considerations. 
From the black body definition 
C. 
where A is the wavelength in cm, T is the temperature in OK, and c, and c, a r e  constants (or 
parameters). Any of the quantities T, cl, and c, may be chosen as a parameter. First consider 
*Mr. Fussel l  is now working for the National Bureau of Standards, Washington, D. C. 
1 
II Ill l l 1 l l 1 l 1 l l  I I I I 
variations in C, . It is clear that any change in the magnitude of C, shifts the spectral distribution 
up or down along the ordinate depending on the sign of the chosen change for c,. Since the parameters 
C, and T a r e  geometrically equivalent (in a reciprocal manner), only one of them need be considered; 
T was chosen because of its experimental accessibility. Variations in T affect the shape of the 
black body distribution and, of course, the wavelength at which the maximum spectral intensity occurs. 
Therefore, any criterion which is to be established for fitting the sun's extraterrestrial spectral dis- 
tribution to Equation 1 can be determined by means of parametric variations in T and c,, or a 
gray body f i t .  
'I'he most straightforward approach is to find the temperature at which the black body maximum 
matches the sun's maximum, 0.475 micron. This temperature is readily found to be 6060°K. But 
this method gives a black body curve which contains a greater integrated energy than the sun's inte- 
grated spectral distribution, since by definition the black body distribution gives an upper limit to 
the emissive ability of any body at a given temperature. Consequently, an upper bound on the re- 
quired temperature has been established from this matching process. Next, an estimate of the lower 
bound can be found. To accomplish this, c, and T may be varied in an effort to minimize the 
residud 
where HA and the vertical lines represent the sun's spectral distribution and absolute magnitude, 
respectively. From Equation 2 the lower bound is found to be approximately 5738°K. It is apparent 
that Equation 2 alone cannot give the required closeness of f i t  t o  the desired accuracy because one 
criterion cannot be used to minimize a bivariant distribution. The final criterion, the one defined as 
geometrically ultimate and, consequently, most pertinent from the thermal design viewpoint, is the 
system: 
and 
P c 
J J (4) 
The addition of Equation 4, to establish the most meaningful criterion, allows a temperature to be 
found by using matched a reas  as the basis. The absolute magnitude in Equation 3 is necessary in 
order to insure that the total deviation between the two curves is minimized. From Equation 1 it is 
clear that c, can be eliminated in Equation 3 by means of Equation 4, which gives a single relation 
for the determination of T . These equations give a temperature of 5742 f 0.5 O K  and a value of 
8.65040 x 1 O - l '  watt-cm2 for C, for the 0.3 to 2.4 micron ( p )  region. 
2 
It remains to show that Equation 2 should give results identical to Equations 3 and 4, indicating 
therefore that 5742°K is the desired temperature. If the spectral region of interest is broken into 
intervals of J ~ ( T ;  cl ) > H~ and Jh(T; C, ) < H*, where the intervals of the first kind a r e  designated by 
A ~ ,  and those of the second kind by 4 , then Equations 2 and 3 may be rewritten: 
where the first subscript of A refers to the interval type and the second is an index which enumer- 
ates subintervals. When the derivatives of the right-hand side of Equation 5 (ignore Equation 4) are 
taken with respect to C, and T in an effort to obtain simultaneous equations for C, and T, the 
following results are obtained: t 
and 
Consequently, Equation 4 must be included in order to define C, and obtain the desired results. This 
implies the equivalence of Equation 2 and the system of Equations 3 and 4. By taking the derivative 
of the right-hand side of Equation 5 with respect to T and equating the result  to zero in order to ob- 
tain a minimum (make use of Equation 4), Equation 5 may be rewritten: 
where E is a number characterizing the f i t  of the black body curve to the sun's distribution and 
Al ,  become the sets {A,,  i }  and {Az,  j }  , respectively, with 1 5 i 5 N and 1 5 j 5 M. 
If the f i t  is exact, E = 0 and the equality sign holds; however, if the f i t  is not exact, the left-hand 
side of Equation 5 gives the best fit,  which means that Equations 3 and 4 give the desired temperature. 
and Az, 
RESULTS 
The results given after each method of calculation described in the previous section are for the 
wavelength range of 0.3-2.4 p .  Calculations were also made for the intervals 0.4-1.2 and 0.220-7.0~; 
3 
the former region is of interest for purposes of solar cell illumination, and the latter is a complete 
cover of Johnson's data.* All results are given in Table 1. The auxiliary range of 0 .3-2.0~ is in- 
cluded for comparison purposes. 
C, (watt-cm2) J HAdA(watt/cm2) 1 -  
6.76111 x 9.86137 x 
8.72019 x 1.29419 x lo- '  
8.65040 x 10-17 1.32629 x l o - '  
8.80334 x lo-'  1.39424 x 10 -l 
It is apparent that the best f i t  (2.97 percent) was obtained for the 0.4-1.2~ region, which indi- 
cates that the adjusted black body curve conforms reasonably well to Johnson's data throughout the 
visible and into the near infrared. Less  satisfactory agreement (6.08 percent) was obtained for the 
region of 0 .3-2.0~ because a single black body formula cannot be made to f i t  the steep rise in emis- 
siveness characteristic of the sun's radiation in the region of 0.34-0.4~. Agreement is enhanced, 
however, when the region is extended to 2 . 4 ~  because the additional 2.0-2.4~ region, where the f i t  
is good, makes the subvisible region (0.34-0.4~) of somewhat less weight. The f i t  for the complete 
range covered by Johnson's data, 0.220-7.0~ , is least satisfactory. The black body curve (with only 
two adjustable parameters) cannot be made to f i t  the steep rise in the 0.220-0.4~ region as well as 
the tail of the sun's spectral distribution in the infrared. 
J I JA - HXldA(watt/cm2) 
2.92699 x 10-3 
7.86669 x 
7.88560 x 
1.03376 x 1 O-' 
In order to clarify the f i t  to the solar extraterrestrial spectrum by the above gray body matching 
technique, a gray body curve was calculated for the 0.3-2.4~ region. The gray body curve of this 
region for a temperature of 5742°K is compared with the solar data in Figure 1. According to col- 
umn 6 of Table 1 the areal mismatch of these curves is 5.94 percent when computed relative to the 
solar constant. It can be concluded that a gray body curve agrees well with the solar data curve, 
except for the irregular region of 0.41-0.55P. What has happened is that virtually exact fits occurred 
at 0.43 and 0.52p, and the intermediate irregularities could not be f i t  by a single gray body curve. 
1 
Range 
(microns) 
0.4 - 1.2 
0.3 -2.0 
0.3 -2.4 
0.220 -7.0 
T ( O K )  
6142 f 0.5 
5730 f 0.5 
5742 f 0.5 
5693 f 0.5 
~~ 
8 H i d  A 
2.97 
6.08 
5.94 
7.41 
DISCUSSION 
If the black body approximation to the solar spectrum is to be useful from the thermal design 
viewpoint, the average absorptance of a spacecraft coating material, as calculated from the black 
body spectrum, should be fairly close in value to the average solar absorptance of the material. 
Mathematically expressed, it is desirable that zBB = zs , where 
*Johnson, F. S., "The Solar Constant," J .  Meteorology I l ( 6 ) :  431-439, December 1954. 
4 
0.22 
- 0.20 
0.18 
0.16 
e 
Y. 
- 
I 
0.14 
I 
5 
Y) + 
0 
& 0.12 
w 
V 
Z 
5 
0 0.10 
5 
d 
d 
k3 
_I 
+ 0.08 
0- cn
0.06 
0.04 
0.02 
0 
and 
I111 
0.2 0.4 0.6 0.8 1 .o  1.2 1.4 1.6 1.8 2.0 2.2 2.4 
WAVELENGTH (P) 
Figure 1-Cornparisan o f  a 5742°K gray body curve with the solar extraterrestrial spectrum. 
with 
zBB = the average absorptance for a black body, 
zs = the average solar absorptance, 
HA = the solar spectral intensity per unit wavelength bandwidth, as given by Johnson, 
JA(T) = the Planck black body spectral intensity for  absolute temperture T, 
ah = the absorptance of the material at wavelength A. 
For simplicity, it is desirable to normalize HA and JA to H'A andJ i  : 
Equations 7 and 8 now become 
and 
Arbitrarily, a 10 percent proportional difference inaBB and zs will be chosen as the criterion 
of usefulness; that is, if 
the black body approximation will be considered useful. Equation 11 is actually I 
JO 
6 
It can be seen that the results given in column 5, Table 1 for the quantity 
do not indicate the magnitude of EP (Equation 11). Consider now the following two cases: 
1. If ah is approximately constant, then the proportional e r r o r  becomes 
and, since 
it is clear that 
lom [ J i (T)  - H i ]  d h  = 0 
andEP = 0.  
2. If aA can be represented by a power ser ies  in A ,  that is, 
then 
in which the e r ro r  is now weighted by the slope. 
NASA-Langley, 1964 7 
II I II I l l  I II 
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Temperature of a gray body most closely fit- ting the solar extraterrestrial spectrum

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