N A S A TECHNICAL NOTE N A S A TN D-2750 e /

-L. --

.? . .

RADIATION GEOMETRY FACTOR BETWEEN THE EARTH AND A SATELLITE

.ai ~

. I ,/' < - : ,q

by Tommy C. Banmiter

Huntsville, A Za. George C. Marshall Space FZigbt Center .. ,:.*: .

N A T I O N A L A E R O N A U T I C S A N D SPACE A D M I N I S T R A T I O N W A S H I N G T O N , D. C. J U L Y 1965

https://ntrs.nasa.gov/search.jsp?R=19650019693 2020-02-13T16:56:12+00:00Z

TECH LIBRARY KAFB. NM

0079693

NASA T N D-2750

RADIATION GEOMETRY FACTOR BETWEEN THE EARTH

AND A SATELLITE

By Tommy C . Bann i s t e r

George C . M a r s h a l l Space Fl ight Cen te r Huntsvi l le , Ala.

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION

For sale by the Clearinghouse for Federal Scientif ic and Technical Information Springfield, Virginia 22151 - Price $2.00

TABLE O F CONTENTS

Page

SUMMARY ................................................ i

INTRODUCTION ............................................ i

PROCEDURE .............................................. 2

A. F for Earth's IR to Cylinder ............................ 2

B. F for Earth's IR to Flate Plate .......................... 4

C. Albedo-F for Both Cylinder and Flate Plate .................. 4

CONCLUSION .............................................. 5

REFERENCES ............................................. 27

iii

LIST OF ILLUSTRATIONS

Figure

I.

2.

3.

4.

5.

6 .

7.

8.

9.

io .

11.

12.

13.

Title Page

Geometry for Planetary Thermal Radiation to a Cylinder ....... 6

Geometric Factor for Earth Thermal Radiation Incident to a Cylinder vs. Altitude, with Attitude Angle as a Parameter . . . . . . 7

Geometry Factor vs. Altitude for IR to a Cylinder. . . . . . . . . . . . 8

Geometry for Planetary Thermal Radiation to a Flat Plate . . . . . . 9

Geometric Factor for Earth Thermal Radiation Incident to a Flat Plate vs. Altitude with Attitude Angle as a Parameter . . . . . . 10

Geometry Factor vs. Altitude for IR tQ a Flat Plate . . . . . . . . . . 11

Geometry for Planetary Albedo to a Cylinder . . . . . . . . . . . . . . . 12

Geometry for Planetary Albedo to a Flat Plate . . . . . . . . . . . . . . 13

Geometric Factor for Albedo to a Cylinder vs. Altitude with Zenith Distance Between Cylinder and Sun as a Parameter y = O-J, @c = 0 9 - 1 8 0 " . ............................. 14

Geometric Factor for Albedo to a Cylinder vs. Altitude with Zenith Distance Between Cylinder and Sun as Parameter y = 6 0 ° , @ = O " .................................. 15

C

Geometric Factor for Albedo to a Cylinder vs. Altitude with Zenith Distance Between Cylinder and Sun as a Parameter y = 6 0 " , @ c = 9 0 " ................................. 16

Geometric Factor for Albedo to a Cylinder vs. Altitude with Zenith Distance Between Cylinder and Sun as a Parameter y = 6 0 " , @c =180" ................................ 17

Geometric Factor for Albedo to One Side of a Flat Plate vs. Altitude with Zenith Distance Between Flat Plate and Sun as Parameter y = O " , 4c = 0" - 180". .................... 18

iv

LIST OF ILLUSTRATIONS (Concluded )

Figure Title Page

14. Geometric Factor for Albedo to One Side of a Flat Plate vs. Altitude with Zenith Distance Between Flat Plate and Sun as Parameter y = 30", GC = 0" .......................... 19

15.

16.

17.

18.

Table

I

II

111

w

Geometric Factor for AlEedo to One Side of a Flat Plate vs. Altitude with Zenith Distance Between Flat Plate and Sun as Parameter y = , 3 0 ° , GC = 90" .......................... 20

Geometric Factor for Albedo to One Side of a Flat Plate vs. Altitude with Zenith Distance Between Flat Plate and Sun as Parameter y = 3 0 ° , GC = 180" ....................... . 2 1

Geometry Factor vs. Altitude fo r Albedo to a Cylinder ( y = 6 0 " ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2

Geometry Factor YS. Altitude for Albedo to a Flat Plate ( y = 3 0 ° ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3

LIST OF TABLES

Title Page

B.. For Cylinder .................................. 24

A. For Cylinder ................................. 25

B.. For Flat Plate ................................. A F o r a F l a t P l a t e ............................... 26

11

ik

11

ik

26

V

RADIATION GEOMETRY FACTOR BETWEEN THE EARTH AND A SATELLITE

SUMMARY

In solving thermal problems of satellites, the geometry factors must be known. A brief discussion of geometry factors, an analytical expression, and computer procedure is given.

Methods for computing these geometry factors from basic relations require lengthy computer programs with many inputs. A method w a s devised in the Space Thermo- dynamics Branch of the Research Projects Laboratory of MSFC for obtaining these factors for space thermal calculations from a subroutine similar to the common trigono- metric subroutines. It requires only 600 octal locations. of the total memory of the IBM 7094 Mod. 11 used by the Computation Laboratory. Max- imum computer time required per factor obtained is .0015 seconds on the IBM 7094 Mod. 11. This subroutine has been used for over two years without any apparent problems.

This is less than 10 percent

INTRODUCTION

The geometry factor, F, between two bodies of various geometrical configurations must be known in order to solve most heat problems encountered in space. A brief de- scription of geometry factors follows.

The Stefan-Boltzmann radiation law (fourth power) relates the amount of radiative flux emitted from a black body to its absolute temperature. A non-black body radiates a fraction of the amount of energy flux radiated by a black body at the same temperature. This fraction is known as the total hemispherical emittance ( o r emissivity) of the non- black body at the given temperature.

Assuming opaque bodies and Lambert's cosine law, the angular distribution of the flux radiated from an elemental area on a body, (a) , can be calculated. If another body, (b) , is placed in the vicinity of (a ) , each elemental area on (b) has incident upon it some

fTaction of the flux from each elemental area on (a). It follows that the amount of incident flux absorbed on (b) can be calculited by a double integration. This amount is a fraction of the total flux emitted, and this fraction is called the geometry factor, F.

Reflected parallel radiation from an outside source, o r albedo, also .enters into orbital problems. Assuming diffuse reflection, an albedo-F can be obtained by a similar double integration. Except for the simplest geometrical configurations, the integrations for F and albedo-F must be carr ied out by numerical techniques.

A fact which is often overlooked is that, by using the reciprosity theorem, the (the geometry factor from a body, A, to another body, B) can be calculated if the

(the geometry factor from the body, B, to the body, A) is known. Obviously, the F ~ - + ~

F ~ 4 ~ simplest route in computation should always be chosen. For example, the F is much more easily calculated between the earth and a smaller object, such as a satellite, by thinking in terms of the satellite radiating to the earth instead of vice versa,. although in practical problems the F is used to compute earth radiation to a satellite.

Convair Corporation performed the necessary integrations, previously mentioned, for several geometrical configurations. Their results were published in graphical form [ iA.

Presently, the Space Thermodynamics Branch of Research Projects Division is using these results for two of these geometrical configurations for both albedo and the so- called IR (earth's infrared) , which is the earth's Stefan-Boltzmann radiation. The two configurations are: ( i) a sphere (earth) and a flat plate (on a satellite) , and (2) a sphere (earth) and a cylinder (on a satellite).

Since most thermal problems of orbiters require many tedious arithmetic opera- tions, the IBM 7090 computer in the Computation Division has been utilized. An analyti- cal expression for F as a function of the lmown geometrical parameters (such a s attitude, etc. ) is imperative for computing purposes. Calculating the factors from basic assump- tions by numerical integration is too lengthy to use for a subroutine as visualized by R- RP-T. Since polynomials a r e very adaptive to computers in that they require a minimum of memory storage, minimum computing time, and are easily programmed, an empe'rical polynomial was generated with a "sum of the least squares polynomial curve fitting" routine a t the Computation Laboratory.

PROCEDURE

A. F FOR EARTH'S IR TO CYLINDER

The following parameters are geometrically defined in figure i for IR be- tween a planet and a cylinder:

2

r = altitude (km)

y = attitude angle (deg)

Figure 2 shows the F versus altitude for a cylinder a s plotted in [ 21 for several representative attitudes. Figure 3 shows the'same curves a s figure 2 on Cartesian coordinates over the range 182 km s r5 3500 km for y = 0" , 20" , 40" , 6 0 ° , and 90".

Letting F be a function of h for each curve in figure 3 , a least square curve f i t equation was obtained of degrees 2 through 9. to be the smallest degree acceptable. Thus, for each y value ( 0" , 20" , 40" , 60" , 90") , a 5th degree polynomial equation was obtained:

The 5th degree equations were determined

i 5 ( F ) ~ = ~ . . r

1J i = O

where ,

j = 0 when y = O",

j = I when y = Z O O ,

j = 2 when y = 409

j = 3 when y = 60°,

and j = 4 when y = 90" . The B.. 's a r e listed in table I.

1J

Other least square curve fit processes were used to obtain the B.. a s a function of 1J

y. Of these, the 4th degree polynomial equations were best:

with the A.. given in table II. Hence, 1J

i r k 5 4

i=O k=O F = A i k Y Y

3

for a cylinder 0" 5 yz 90" , since the cylinder is symmetrical about its axis. The above equations are not valid outside the l imits 0" y 5 90" and 182 km 5 r 5 3500 km.

B. F FOR EARTH'S IR TO FLAT PLATE

Similar figures (4 , 5 &. 6) and tables (I11 &. IV) are shown for a flat plate radiating on one side. The resulting equation for the F of a flat plate is:

i r k 4

= AikY i = O k=O

Since y can range from 0" to 180" for a flat plate, and the case 110" to 180" is not covered in the polynomial, most of the F's in this range a r e zero for all practical pur- poses. The non-zero F's a r e obtained by a linear interpolation included in the subroutine for F. The interpolation routine is not as adequate as desired, due to the fact that F is not a linear function of y o r h and F - 0 at different h values for a given y . it necessary to interpolate differently over specified increments of altitude. a discontinuity in interpolative values of F at the value of h where any two interpretative groups a r e separated. crossing one of the boundaries.

This fact makes This causes

A discontinuity (less than * 3010 Fmax) in computation occurs upon It should be noted that this occurs when 0 < F < . 2 .

C. ALBEDO-F FOR BOTH CYLINDER AND FLAT PLATE

Figures 7 and 8 show a graphical definition of the parameters necessary to compute the albedo-F.

The albedo-F is not only a function of parameters y and h, but of B S and @c also. These angles did not enter into the IR case because of symmetry. Figures 9 through 18 (taken from [ 21 ) a r e typical examples of graphs for albedo-F.

Intuitively, the dependence of the albedo-F upon 0, is thought to be approximately proportional to cos 0, and that F s albedo-F when 8, = 0. Also , albedo-F is a week function of c$~. These assumptions a r e illustrated in figures 17 and 18 where the solid line curves a r e typical integrated curves and the dotted line curves are the corresponding curves obtained by using the above assumption. Hence,

albedo-F E F cos0 if cos 0 > 0 S'

and albedo-F = 0, if cos 0 5 0 .

This subroutine is coded to calculate F for the flat plate or the cylinder. Only F is in the output; if the albedo-F is desired, one must multiply by cos O s .

4

CONCLUSION

A s far as the IR is concerned, there is one basic type of e r r o r (neglecting the interpolation routine). Assuming the data from the Convair reports to be valid, this is the F obtained for values of y not used in the curve fitting process. Since the Bij and A i j are continuous, the e r r o r is estimated to be within &270 of Fmax. Over an entire orbit, it is believed that these e r r o r s will usually have a cancelling effect. For the albedo-F the same e r r o r s occur as in IR, plus an e r ro r , brought about by using the assumption albedo- FrF ,cos 8 less than + 570 of Fmax.

when cos 8, > 0 and albedo-F = 0, when cos 8, 5 0, which is estimated to be S’

Methods for computing these geometry factors from basic relations require lengthy computer programs with many inputs. A method w a s devised in R-RP-T for obtaining these factors for space thermal calculations from a subroutine similar to the common trigonometric subroutines. It requires only 600 octal locations. This is less than 10 percent of the total memory of the IBM 7094 Mod. I1 used by the Computation Laboratory. Maximum computer time required per factor obtained is .0015 seconds on the IBM 7094 Mod. 11. This subroutine has been used for over two years without any apparent problems.

In making typical computations of the radiation, it must be remembered that the factors are defined such that one uses the actual a rea of one side of the plate and the projected area (length x diameter) of the cylinder.

5

FIGURE I. GEOMETRY FOR PLANETARY THERMAL RADIATION TO A CYLINDER

6

a

I .o

0.1

.01

ALTITUDE -nautical miles FIGURE 2. - GEOMETRIC FACTOR FOR EARTH THERMAL RADIATION INCIDENT TO A

CYLINDER VS. ALTITUDE, WITH ATTITUDE ANGLE AS A PARAMETER

I .4

I .2

a 0 1.0 I-

.8

.6

.4

.2 0 I O 0 0 2000

y=90° I 1

3000 ALTITUDE - kilometers

FIGURE 3. GEOMETRY FACTOR VS. ALTITUDE FOR IR TO A CYLINDER

FLAT PLATE

. . . . . . . ____ .-

FIGURE 4. GEOMETRY FOR PLANETARY THERMAL RADIATION TO A FLAT PLATE

9

1'0

a

FIGURE 5. GEOMETRIC FACTOR FOR EARTH THERMAL RADIATION INCIDENT TO A FLAT PLATE vs. ALTITUDE WITH A ~ T I T U D E ANGLE AS A PARAMETER

FIGURE 6 . GEOMETRY FACTOR VS. ALTITUDE FOR IR TO A FLAT PLATE

TERMINATOR

~

FIGURE 7. GEOMETRY FOR PLANETARY ALBEDO TO A CYLINDER

12

13

I

ps 0 I- u i5

I 1.00

.50 E P-

F .IO 0 - a + w

0 . W (3

= 05F .01 L

100

m

500

I I I a f 1 L f I t T

1 1 1000

ALTITUDE - nautical miles

5000

FIGURE 9. GEOMETRIC FACTOR FOR ALBEDO TO A CYLINDER VS. ALTITUDE WITH ZENITH DISTANCE BETWEEN CYLINDER AND SUN AS A PARAMETER y = O " , $ c = O o - 180" . -

14

a 0 I- O

E 0 a t- W s 0 W (3

I .oo

.so

.I 0

.05

.a i / -

I O 0

1 1 e== 0" s,

i i

-1 I u II 500 1000

A LT I T U D E - n aut i ca I m i I es 5000

FIGURE 10. GEOMETRIC FACTOR FOR ALBEDO TO A CYLINDER VS. ALTITUDE WITH ZENITH DISTANCE BETWEEN CYLINDER AND SUN AS PARAMETER y = 6 0 ° , @c = 0"

15

I O 0 500 1000 5000

ALTITUDE- nautical miles

FIGURE 11. GEOMETRIC FACTOR FOR ALBEDO T O A CYLINDER VS. ALTITUDE WITH ZENITH DISTANCE BETWEEN CYLINDER AND SUN AS A PARAMETER y = 6 0 ° , = 90"

16

I

I

a LI

-ALTITUDE - nautical miles FIGURE 12. GEOMETRIC FACTQR FOR ALBEDO T O A CYLINDER VS. ALTITUDE WITH

ZENITH DISTANCE BETWEEN CYLINDER AND SUN AS A PARAMETER y = 60", +c = 180"

17

I .oo

.50

aL 0 I- o i2 - .IO a k w z 0 .05 w (3

0 D

60" I I 1

70" 4

T ~ --

80" -b

! I 85" -

1

00 500 1000 ALTITUDE- nautical miles

! - 5000

FIGURE 13. GEOMETRIC FACTOR FOR ALBEDO TO ONE SIDE O F A FLAT PLATE VS. ALTITUDE WITH ZENITH DISTANCE BETWEEN FLAT PLATE AND SUN AS PARAMETER y = 0", GC = 0" - 180"

IO0

- 1 i - e,= 04

d

t ..

- -a rmk

- 30"

50"-

-70"-

-1 I-- 80"

LI 4 - 850 -

I

t I I

;/c 9( D

500 1000 5000 ALTITUDE-nautical miles

FIGURE 14. GEOMETRIC FACTOR FOR ALBEDO TO ONE SIDE O F A FLAT PLATE VS. ALTITUDE WITH ZENITH DISTANCE BETWEEN FLAT PLATE AND SUN AS PARAMETER y = 30°, @c = 0"

19

I .00.

.50

a 0 I- O .Q: L L

O - . IO Ikz. I- W t s .05 (3

-

I O 0 500 1000 5000 ALTITUDE -naut ica l miles

FIGURE 15. GEOMETRIC FACTOR FOR ALBEDO TO ONE SIDE O F A FLAT PLATE VS. ALTITUDE WITH ZENITH DISTANCE BETWEEN FLAT PLATE AND SUN AS PARAMETER y = 3 0 ° , @c = 90" ,

20

I .oo

-50

. IO

.05

500 1000 5000 ALTITUDE - nautical miles

FIGURE 16. GEOMETRIC FACTOR FOR ALBEDO TO ONE SIDE OF A FLAT PLATE VS. ALTITUDE WITH ZENITH DISTANCE BETWEEN FLAT PLATE AND SUN AS PARAMETER y = 30", $c = 180"

21

1.2 - I I I

I I I

1.0 -

. .8

.6

.4

.2 TIMES COS 90" ~ #c = Oo 4; = 180"

I \ I - I\ - I 1 -.

I \ I I ---- O0- IO00 2000 3000

ALTITUDE - kilometers

FIGURE 17. GEOMETRY FACTOR VS. ALTITUDE FOR ALBEDO TO A CYLINDER ( y = 606)

0 I- o

> I- W z 0 W a

E a

.8

.6

0 1000 2000 3000 ALTITUDE - kilometers

FIGURE 18. GEOMETRY FACTOR VS. ALTITUDE FOR ALBEDO TO A FLAT PLATE ( y = 30")

TABLE I

j=O ( y = 0")

i = O +. 12912x10'

j = i j=2 j=3 j =4

+. 1 2 9 8 8 ~ 1 0 ~ +. 1 3 3 0 3 ~ 1 0 ~ +. 13444x10' +. 1 4 0 8 6 ~ 1 0 ~

( y = 20") ( y = 40") ( y = 60") ( y = 90")

1 i = i 1 -. 1 1 5 4 6 ~ 1 0 - ~ 1 -. 1 2 1 2 9 ~ 1 0 - ~ 1 -. 1 1 5 8 4 ~ 1 0 - ~ 1 -. 9 5 3 3 5 ~ 1 0 - ~ 1 -. 8 4 1 9 1 ~ 1 0 - ~ 1 i = 2 1 +.69623~10-~ + .81600~10-~ + .79503~10-~ 1 +.58425~10-~ - 1 . +.44063x1011

-9 i = 3 -. 2 9 2 3 ' / ~ 1 0 - ~ -. 3 3 9 8 6 ~ 1 0 - ~ -. 3362 9x1 O-' -. 22681x10 -. 15707~10 I i = 4 +. 5 3 0 8 3 ~ 1 0 - ~ ~ 1 +. 7 3 9 1 1 ~ 1 0 - ~ 1 +. 7 3 5 0 0 ~ 1 0 - ~ ~ 1 +. 46419x10~ :~ I +. 3 0 8 3 9 ~ 1 0 - ~ ~

i = 5 -. 43402~10-~ ' -. 63521~10-~ ' -. 63232~10-~ ' -. 37997x10 -. 24892~10-~ '

TABLE I1

i = 5 1 - . 4 3 4 0 3 ~ i O - ~ ~ I -. 1 0 2 6 7 ~ 1 0 - ~ ~

Aik FOR CYLINDER

-. 1 8 0 9 1 ~ i O - ~ ~ +. 11289~10-~ ' -. 8 6 1 9 6 ~ i O - ~ ~

k = O k = i k = 2 k = 3 k = 4

i = O +. 12912x10' -. 17242xiO-' +. 1 5 6 6 5 ~ 1 0 - ~ -. 2 9 1 6 6 ~ 1 0 - ~ +. 17221xiO-'

2 i = i -. i i 5 4 6 ~ 1 0 - .22624~10-~ -. 1 6 6 9 9 ~ 1 0 - ~ +. 7 9 0 2 3 ~ 1 0 - ~ ~ -. 59320~10- '~

1 i = 2 + .69695~10-~ ~ +.53858~iO-~ +. 1s843xio-9 -. 92692~10-" ~ +. 68441~10- '~

1 i = 3 1 -.26237xiO-' 1 -.37223x10-" ~ - .91473~10- '~ , +. 49288~ iO- I~ I -. 36760~10- '~

I i = 4 I +, 53083~10- '~ I +. 1 0 4 0 2 ~ i O - ~ ~ ~ +. 20723~iO- '~ j -. i 2 1 7 i ~ i O - ' ~ I +. 91996~10- '~

I

TABLE III

k = O

B.. FOR FLAT PLATE 1J

-..- j=3 j=4 j=5 j=6

( y = 50") ( y = 70") ( y = 90") ( y = 110")

0240x10 +. 77200x10 +. 58590x10 +.41410x10 +. 2375x10

-. 3 2 5 2 0 ~ 6 ~ -. 38502~iO-~ -. 3 7 3 3 4 ~ 1 0 ~ ~ -. 3615OxiO-' -. 26265~iO-~

.91757xiO-' +. 15112~10-~ +. 17036~10-~ +. 18877~10-~ +. 14473~10-~

-lo -. 34398~10- '~ -. 444i6~10- '~ -. 51021x10~10 -. 39877~iO-~O

k = i k = 2

- 14 - 14 1 i = 4 1 +.21995xi0-l4 I +. 22752~10- '~ 1 +. 12802xiO I +. 3 1 4 0 8 ~ 1 0 - ~ ~ 1 + . 4 6 7 7 3 ~ 1 0 - ~ ~ ~ + . 53038x10 I + . 4 2 3 0 2 ~ i O - ~ ~ I

k = 4

TABLE IV

Aik FOR A FLAT PLATE

k = 5 k = 6 k = 7 1 I i = 0 I +. 1 0 5 0 7 ~ 1 0 ~ I+. 74328xiO-' I -. 1 3 3 9 7 ~ 1 0 - ~

i = i

i = 2 F i = 3

I i = 4 ~ +.21995x10-l4 ~+. 8 3 3 4 1 ~ i O - ~ ~ ' -. 92378x;O-'

k = 3 - +. 58132~10-~

-. 14765~10-~

+. 12167~10-~

+. 76986xiO-"

+. 37480~10- '~

-. 12748~10-~ I+. 14561~ iO-~ I -. 82937xiO-io( +. 1 8 6 1 5 ~ 1 0 - ~ ~ I

-. 25507~10- '~ 1+.28049~10-'~ I -. 15570~10-~~1+. 3437i~iO- '~ I -. 83013~10- '~ I-. 83013~10-~ ' 1 +. 45444~10-'~1-. 99389xiO-" 1 -. 74129~10- '~ I+. 77824xIO-'' - . 4 1 7 3 7 ~ 1 0 - ~ ~ (+. 89956~10- '~ 1

REFERENCES

I. Ballinger, J. C., Elizalda, J. C., Garcia-Varela, R. M., and Christensen, E. H.: Environmental Control Study of Space Vehicles, Part I1 ( Thermal.Environment of Space) , Convair (Astronautics) Division, General Dynamics Corporation, November 1960.

2. Ballinger, J. C. and Christensen, E. H. : Environmental Control Study of Space Vehicles, Part 11, Supplement A. Convair (Astronautics) Division, General Dynamics Corporation, January 1961.

NASA-Langley, 1965 M408

I I

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