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*"»fAU ov *S NBS TECHNICAL NOTE 1073

U.S. DEPARTMENT OF COMMERCE / National Bureau of Standards
NATIONAL BUREAU OF STANDARDS

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NATIONAL BUREAU OF STANDARDS

LI3HAI Y c

An Error Analysis for the Use of Presently

7

\%

Available Lunar Radio Flux Data in

]C

Broadbeam Antenna-System

Measurements

William C. Daywitt

Electromagnetic Fields Division

National Engineering Laboratory

National Bureau of Standards

U.S. Department of Commerce

Boulder, Colorado 80303

/ V %

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Issued February 1984
National Bureau of Standards Technical Note 1073

Natl. Bur. Stand. (U.S.), Tech Note 1073, 35 pages (Feb. 1984)

CODEN: NBTNAE

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1984

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, DC 20402
CONTENTS

Page

1. Introduction 1

2. Lunar Flux Density and Average Brightness Temperature 3

3. Shape Factor 8

4. Conclusions. 9

5 Acknowl edgments 9

6 . References 10

Appendix A 13

Appendix B 18

Appendix C 20

Appendix D 31

An Error Analysis for the Use of Presently Available Lunar Radio Flux

Data in Broadbeara Antenna-System Measurements

W. C Day witt

National Bureau of Standards

Boulder, Colorado 80303

Simple, precise expressions for lunar diameter., average

brightness temperature, flux density, and shape factor are presented. An analysis of the relationship between these

parameters and corresponding errors are included. For broadbeam

(HPBW>d) antennas, results show that flux density and shape factor

can be determined with errors less than 13 percent and 0.4 percent

respectively at frequencies below 10 GHz. Extension of the

analysis to higher frequencies is indicated.

Key words: antenna temperature; error analysis; G/T: lunar flux

density; shape factor.

Introduction

The antenna-system parameter G/T [1] can be directly measured by using

the incoherent radiation from a radio star [2,3], thus circumventing difficulties associated with measuring G and T separately. Since the extraterrestrial

sources used in the direct measurements function as noise power transfer

standards, their flux densities and shape factors must be amenable to precise

calibration. The angular diameter of the moon and its predictable output make

it an appropriate source for the measurement of some antenna systems, and the

purpose of this paper is to present results of an investigation into the

potential accuracy of using the moon for these measurements.

The National Bureau of Standards (NBS) has been involved in G/T measure- ments since 1972 when it began a series [4,5] of studies concerning the use of some of the "brighter" sources, primarily Cassiopeia A, as precise transfer

standards. Since that time it has developed [3,6,7,8] an earth-termi nalmeasurement-system for direct measurement of G/T and Satellite EIRP that

operates off the IF patch panel of the receiving earth terminal. The measurement algorithms were developed for use on large (~ 18 m) antenna systems in

the frequency range from 1 to 10 GHz, where a source the size of Cassiopeia A

is appropriate. Recently NBS has performed successful measurements on a small
(~ 6 m) antenna system using the moon as a transfer standard for which the

analysis presented here was completed.

Application of the lunar output to antenna and antenna-system measurements is not new. It has been used by radio astronomers [9.10] and communications engineers [11,12] alike, and the thermal radiation from its central

region has been proposed as a radiometric standard for narrowbeam microwave

and infrared observations [13]. The following presentation adds a detailed

error analysis concerning the use of the radio output from the entire lunar

surface as a standard for "broadbeam" (half-power beamwidth > lunar diameter)

antennas that has previously been missing from the literature The analysis

was designed specifically for 1 to 10 GHz, although it will be clear from the

presentation where it can be expanded outside this region. The main body of

the text is a collection of simple but precise algorithms for calculating the

lunar output and an accompanying summary of the error. For clarity, the more

analytical aspects of the presentation are located in the appendices.

The receiver of an antenna system responds to the delivered power from

its antenna waveguide terminal. This power is proportional to the antenna

temperature [14] and is related to the radiation from a radio source in the

antenna mainbeam by an equation of the form

T

a

= C k

2 S (1)

where T

a is the antenna temperature, S is the flux density from the radio

source, k 2 is a shape factor that accounts for variation of the source flux across the antenna beam pattern, and C is a factor containing system parameters unrelated to the lunar output. The equation indicates that an accurate

calculation of Ta can only be made if both S and k

2 are accurately known.

Consequently, in determining the usefulness of a particular radio source the

accuracy with which the shape factor can be calculated must also be considered. Therefore, an analysis of the shape factor is included along with

that of the flux density.
2. Lunar Flux Density and Average Brightness Temperature

An expression for the antenna temperature arising from a radio source in

the antenna mainbeam is the starting point for a precise error analysis. From

this expression follows those specifications (like frequency range or the

ratio of half -power beamwidth to source diameter) that dictate under what

conditions use of the chosen radio source will provide accurate results.

Because of its variation with lunar phase, the lunar flux density is best

calculated as a function of the apparent lunar diameter and average brightness

temperature. This function is derived in Appendix A and can be conveniently

expressed as

S = 7.349 f

2 T d2 (2)

where S is the flux density in Janskys (10~ 26 watts/Hz/meter2

), f is the

measurement frequency in GHz, T is the average lunar brightness temperature in

kelvins, and d is the angular diameter of the moon in degrees subtended at the

antenna site. This diameter can be easily calculated from eq (B2) in Appendix

B with an error less than ±0.1 percent.

Disregarding errors in f, T, and d, eq (2) is an approximation (Appendix

A) which is accurate to ±4.34 percent. It appears that this error could be

reduced to ±1 percent if the yearly variation in the solar insolation at the

lunar surface due to the earth's eccentric orbit were taken into account.

An expression for the average brightness temperature that contains

effects of lunar phase can be written in the form [9]

T = T 1 . (_L) cos(.

T

*) (3)

where T , (T,/T ), and the phase lag Y are functions of frequency that can be

determined from radio measurements. <j> is the lunar phase angle U =0° at new

moon) which is related [15] to the phase (fraction F of the apparent lunar

disk illuminated by the sun) through the equation
o) = arccosd - 2F). (4)

F can be obtained from The Astronomical Almanac [16] to an accuracy of

+2 percent by a simple parabolic fit over a three-day period. This leads to a maximum error in cos<t> of 0.04 and a negligible error in T. Disregarding

errors in the parameters appearing in eq (3), the error (Appendix D) in T due

to retaining only the fundamental o) on the right-hand side of the equation is

+0.18 percent.

The value of eq (3) lies in the fact that with it the average brightness

temperature T can be calculated for any lunar phase, leaving only three

parameters, T , (T\/T ), ?, to be accurately measured at a few frequencies and

extrapolated in between. This process will be illustrated in the remainder of

the section for 1 to 10 GHz with data taken from the literature [9] and

reproduced in table 1.

Table 1. Parameters for the average lunar brightness temperature

Frequency (GHz) T (kelvins) VT

o ¥(d egree) VT

0.6 240 -- — -- 0.86 235 -- -- --

1.5 225 0.00667 44 0.00800

3.13 218 0.0183 42 0.0204

9.375 210 0.0619 40 0.0681

18.75 207 0.155 35 0.151

37.5 205 0.195 32 0.217

75 203 0.266 24 0.326

The values in table 1 for T are quoted with accuracies from 5 to 10

percent, although with more recent measurement techniques [5,17,18] they could

all be determined to approximately 5 percent. The accuracies of the ratios

T,/T and Tj_/T are quoted to 2 percent, and the phase lag ¥ to 10° for frequencies below 15 GHz, and 5° for frequencies above. Figures 1, 2, 3, and 4 were plotted from these data.
240

230

T (K) 220

210 -

200

T = 207.7 + 24.43/f

ID = 97.6%

T

1

10 20 30

FREQUENCY (GHz)

I

40

Figure 1. Constant term versus frequency for the average

brightness temperature.

10

-1

10

-2

10

.-3

I

/

' •

»

• ©

J /

y/ •1 1 224 • •=- = 0.004212 f'-"* / T« /

'0

ID = 99.8%

5 10 50 100

FREQUENCY (GHz)

Figure 2. First-harmonic ratio versus frequency for the

average brightness temperature.
50 n

PHASE LAG

(DEGREE)

43.83

1+0.0109f - - ¥ = arctan(ytt-)

$ = -^-0080

10 20 30 40 50 60

FREQUENCY (GHz)

70 •» 80

Figure 3. Phase lag versus frequency for the first harmonic of the

average brightness temperature.

10r1

10

10

-3

I

'-'. A

/

/ I -. /

*

•

f

- Ti /

I — = 0.005192 f f Tn

_ in — qo o«*

5 10 50 100

FREQUENCY (GHz)

Figure 4. First-harmonic ratio versus frequency for the

subterrestial brightness temperature.
Figure 1 shows the constant term, T , of the average brightness temperature with 10 percent error bars plotted as a function of frequency. Assuming

a radio absorption coefficient proportional to frequency [13] and independent

of depth into the lunar soil, T is linear in inverse frequency. This

functional form is least-squares fit to the data from 0.86 to 18.75 GHz with

the resulting formula shown in the figure. The index of determination [19],

denoted by ID in the figure, indicates that 97.6 percent of the variance shown

in the data is explained by the form of T . The largest difference between

the curve and the fitted data from 1 to 10 GHz is 1.4 percent. Combining this

with a 5 percent achievable error (not the 10 percent error bars shown) in the

data indicates that it is possible to determine T to +6.4 percent.

The data for T,/T from column 3 of table 1 is plotted in figure 2. The

points from 1.5 to 18 75 GHz were least-squares fit with a power-law curve,

resulting in a 99.8 percent index of determination. A maximum error of 4.6

percent between these points and the curve is obtained. Combining this error

with the 2 percent quoted for the ratio leads to an error of 6.6 percent when

using the formula shown in the figure. This gives a maximum error of +0.46

percent for T in the 1 to 10 GHz range.

Figure 3 is a plot of the phase lag with the error limits associated with

table 1. The "arctan" curve is used in Appendix C and is of no concern now. The least-squares curve that is calculated from the formula shown has an index

of determination of 99.1 percent. It is within 2° of the data from 1 to 10

GHz, and combined with the error bars leads to an error of 12° in that region.

This leads to a maximum error of +1.22 percent for T.

The previous systematic error components are collected in table 2. It

should be remembered that the 5 percent error for T represents an achievable

error, and should not be applied in practice to all of the data in column 2 of

the table. Table 2 shows a maximum 12.8 percent linear sum of errors which is

acceptable for G/T measurements.
Table 2, Summary of systematic flux density errors from 1 to 10 GHz

Source of Error

Variation in Solar

Insolation (Appendix B)

Neglect of Higher Harmonics

in Eq (D3) (Appendix D)

cos 4>

d

Quadrature Sum

Magnitude

3.34% + 1%

0.18%

5% + 1.4%

6.6%

0.04

12°

0.1%

Resulting % Error in S

4.34

0.18

6.4

0.46

0.00

1.22

0.2

±7.8% (0.33 dB)

Linear Sum ±12.8% (0.52 dB)

3. Shape Factor

The shape factor ^ in eq (1) approaches unity as the ratio of the

antenna half-power beamwidth (HPBW) to lunar diameter increases without limit.

For finite ratios it may be calculated from [5]

.2

1 - e (5)

where

x

2

= 0.6441(d/9

H

) 2

(6)

and where d and 9 H are the apparent (or optical) angular diameter of the moon

(Appendix B) and antenna HPBW respectively. Disregarding errors in d or

H

, the approximations leading to eq (5) (Appendix C) generate an error no larger

than ±0.32 percent in k 2 for 8 H /d greater than unity and a frequency less than

than 10 GHz.
Taking the total differential of k

2 will give an estimate of the errors

in k

2 due to errors in d and 0^. From Appendix B the error in d is ±0.1

percent. The error due to 0^ depends upon the scheme used in viewing the

moon. In at least one scheme [3] the systematic error vanishes, and hence

will be taken to be zero here. Performing the differential leads to a ±0.06

percent error in k

2 due to the ±0.1 percent lunar diameter error. Thus, the

total error in k

2 from using eq (5) is ±0.38 percent (0.32 + 0.06) for H /d

greater than unity and frequencies less than 10 GHz.

4. Conclusions

The lunar flux density and shape factor can be easily calculated for any

lunar phase and earth-moon separation distance using both the equations

presented and The Astronomi cal Al manac [16]. The maximum systematic errors

for the flux density and shape factor are ±13 percent and ±0.4 percent

respectively for broadbeam (0^> d) antennas in the 1 to 10 GHz frequency

range. This flux-density error could be reduced from ±13 percent to ±10

percent by taking the variable solar insolation into account. The regression

equations shown in figures 1, 2, and 3 are of course somewhat arbitrary and

could be replaced by other forms.

The small value (±0.18 percent) of the "higher-harmonic" error in table 2 indicates that the analysis can readily be extended to higher frequencies, but

a frequency will ultimately be reached where the simple form of eq (3) will

have to be replaced with one containing harmonics of the fundamental <j>. Unfortunately, data like that in table 1 does not exist for these harmonics.

5. Acknowledgments

The author wishes to thank M. Kanda, D. F. Wait, and a number of other of

his colleagues in the Electromagnetic Fields Division of NBS for many helpful

and stimulating discussions.
>. References

[1] Altman, F. J.; Burgess, J. L. ; Driscoll, C. R. : McAllister, N. F.;

Selz, G. 0.; Tunstall, B.; Young, H. M. Satellite communications

reference data handbook. Defense Communications Agency. Report No. AD746165, distributed by NTIS, U.S. Department of Commerce; 1972 July.

[2] Kreutel , Jr., R. W.; Pacholder, A. 0. The measurement of gain and noise

temperature of a satellite communications earth station. Microwave

Journal. 12(10); 1969 October.

[3] Wait, D. F. Satellite earth terminal G/T measurements. Microwave

Journal. 20(4); 1977 April.

[4] Wait, D. F.; Daywitt, W. C; Kanda, M.; Miller, C. K. S. A study of the

measurement of G/T using Cassiopeia A. Nat. Bur. Stand. (U.S.) NBSIR

74-382; 1974 June.

[5] Kanda, M. An error analysis for absolute flux density measurements of

Cassiopeia A. IEEE Trans. Instr. Meas. IM-25(3): 173; 1976 September.

[6] Daywitt, W. C. Error equations use in the NBS precision G/T measurement

system. Nat. Bur. Stand. (U.S.) NBSIR 76-842; 1976 September.

[7] Daywitt, W. C. An error analysis for the measurement of satellite EIRP

using a calibrated radio star. IEEE Trans. Instr. Meas. I M- 2 7 ( 3 ) : 253;

1978 September.

[8] Daywitt, W. C. Atmospheric propagation equations used in the NBS earth

terminal measurement system. Nat. Bur. Stand. (U.S.) NBSIR 78-883;

1978 April.

[9] Kuz'min, A D. ; Salomonovich, A. E. Radioastronomical methods of

antenna measurements. New York, London: Academic Press; 1966.

10
[10] Baars, J. M. W. The measurement of large antennas with cosmic radio

sources. IEEE Trans. A. P. AP-2K4): 461; 1973 July.

[11] Johannsen, K. G.; Koury, A. The moon as a source for G/T

measurements. IEEE Trans. A. ES-: 718; 1974 September.

[12] Saam, T. J. Small computers revolutionize G/T tests. Microwaves. 37;

1980 August.

[13] Linsky, J. L. The moon as a proposed radiometric standard for microwave

and infrared observations of extended sources. The Astrophysical

Journal. Supp. Ser. 25(216): 163; 1973 February.

[14] Kraus, J. D.; Tiuri, M. E. Radio astronomy. New York: McGraw-Hill;

1966.

[15] Explanatory supplement to the astronomical ephemeris and the American

ephemeris and nautical almanac. London: H. M. Nautical Almanac Office,

Her Majesty's Stationery Office; 1961.

[16] The astronomical almanac for the year 1981. Nautical Almanac Office of

the U.S. Naval Observatory; 1981.

[17] Kanda, M. Accuracy considerations in the measurement of the power gain

of a large microwave antenna. IEEE Trans. A. P. 407; 1975 May.

[18] Kerns, D. M. Plane-wave scattering-matrix theory of antennas and

antenna-antenna interactions. Nat. Bur. Stand. (U.S.) Monogr. 162;

1981 June.

[19] Crow, E. L.; Davis, F. A.; Maxfield, M- W. Statistics manual. U.S.

Naval Ordnance Test Station; 1955.

[20] Ko, H. C. Antenna temperature and the temperature of electromagnetic

radiation. IEEE Trans. A. P. 126; 1964 January.

11
[21] Soboleva, N. S. Measurement of the polarization of lunar radio emission

on a wavelength of 3.2 cm. Soviet Astronomy-AJ. 6(6): 873, 1963 May.

[22] Krotikov, V. D.; Troitskii, V. S. Radio emission and nature of the

moon. Soviet Physics Uspekhi. 6(6): 841; 1964 May.

[23] Allen, C. W. Astrophysical quantities, Univ. of London: The Athlone

Press; 1976.

[24] Labs, D.; Neckel , H. The solar constant. Solar Physics. 19: 3; 1971.

[25] Baars, J. W. M.; Mezger, P.G.; Wendker, H. The spectra of the strongest

non-thermal radio sources in the centimeter-wavelength range. The

Astrophysical Journal. 142(1); 1965 July.

[26] Linsky, J. F. Models of the lunar surface including temperaturedependent thermal properties. ICARUS 5: 606; 1966.

[27] Muhleman, D. 0. Microwave emission from the moon. Lucus, J. W., ed.

Thermal characteristics of the moon. Progress in Astronautics and

Aeronautics Ser. 28; 1972.

[28] Salomonovich, A. E. Thermal radio emission of the moon and certain

characteristics of its surface layer. Kopal , Z.* Mikhailov, Z. K. ed.

The moon. London, New York: Academic Press; 1962.

12
APPENDIX A

DERIVATION OF A USEFUL RELATIONSHIP BETWEEN

LUNAR FLUX DENSITY AND AVERAGE BRIGHTNESS TEW>ERATURE

The antenna temperature [14] corresponding to the power available from

the antenna waveguide port is measured by the antenna system. Consequently,

it is with this quantity that an analysis must start, relating other measure- ment variables to it. The antenna temperature T

a corresponding to lunar

radiation in the antenna mainbeam can be written as

T

a

= T

a (a,<Kg) = --J-

/MTP

n dQ (Al)

where [14,20]

( r + r )

(V--^) C0! cos 2e

a cos 2(t -T

fl

) (A2)

and

t' + T" T = -L

-f-- • (A3)

The antenna temperature is a function of the relative earth-moon geometry

through the solid angle Q subtended by the moon at the antenna, the lunar

phase angle <\>, and the solar mean anomaly g [15]. The latter dependance

results from variable solar insolation caused by the eccentricity of the

earth's orbit. r\, Q

a and P

n are the antenna radiation efficiency, solid

angle, and normalized power pattern, respectively. M is a factor describing

the mismatch between the moon's radial polarization [21] and the antenna

polarization, with the angles e a

, T

fl and t defined on the Pointeare* sphere

13
[14]. dQ is the differential solid angle subtended by a small portion of the

lunar surface as seen from the antenna position on the earth. T' and T" are

radial and azimuthal (with respect to the subterrestial point on the lunar

surface) brightness temperatures of the lunar surface, with the radial

componant T

1 being the principle or strongest of the two. The quantity

(T

1 - T")/(T' + T") is the degree of polarization and varies from zero at the

subterrestial point to a maximum at the lunar limbs.

Depending on the frequency and elevation angle, atmospheric attenuation

can significantly reduce the magnitude of T

a

. Nevertheless, this effect has

been deliberately left out of eq (Al) in order to focus attention upon the

flux density and shape factor calculations.

Equation (Al) can be reduced to a more convenient form in a strictly

formal way, resulting in the equation

a

with the definitions

/TP

n dQ

k

2

= k

2

(Q s <t>,g,e H

) = ~jtfg- (A5)

/MTP dQ

k

6

= k

6 (Q,*, gi e H

) ^j^-^- (A6)

n

Q = total lunar solid angle = —*— . (A7)

d is the lunar angular diameter (Appendix B), k£, and kg are the shape and

polarization factors [5] respectively, and e H is the antenna half-power

beamwidth. The reason for defining these factors stems from the fact that

under certain conditions they can be easily and accurately approximated, and

help reduce the equation for T

a to a usable form. The shape factor k 2 is

discussed in Appendix C. The last ratio in eq (A4) is a function of Q, <t>, g

14
and represents an average brightness temperature T(Q,<hg). However, defining

this ratio as the lunar brightness temperature incorporates the earth-moon

geometry (through the presence of Q) in the definition, an undesirable feature

that can be eliminated by continuing the formal development a little bit

further:

^- = T(Q,<t>,g) = at' , (A8)

where

5 = S(Q,*,g) .liMlii (A9)

TU,g)

and

f e fU,g) e ^ . (A10)

T' is the average brightness temperature, and is a function of lunar phase and

solar mean anomaly. Its usefulness results from the fact that a is close to

unity. A model for T' has been developed [22] that reasonably fits the

experimental data [9] as a function of frequency and lunar phase <t>, although

no attempt was made in the literature to account for the variable solar insolation indicated by the presence of g in eq (A10) . In accounting for this

effect it is convenient to make one more formal manipulation:

?U,u/2) = aU,g)T' (All)

where

a(g) -zlii'^l (A12)

f(<D,g)

-n;/2 (or 3-rc/2) is the solar mean anomaly where the earth-sun separation

distance is equal to 1 astronomical unit (1 Au = the average earth-sun

separation distance [23]). Consequently, TU,it/2) is T' normalized to 1 Au.

The normalization factor a is not strongly dependent upon lunar phase and,

consequently the phase doesn't appear in the argument of a. It will be seen shortly that the error in T' from neglecting the variable solar flux can be as

15
high as ±3.34 percent, so it would be of value to use the normalized brightness temperature in future spectral characterizations such as the one in

table 1.

The earth-sun separation distance R in astronomical units is given by

[16]

R = 1 - 0.0167 cos g (A13)

where the quantity (0.0167) is the eccentricity of the earth's orbit about the

sun. The mean anomaly g in degrees is easily calculated from [16]

g

= 357.0 + 0.9856D (A14)

where D is the yearly day number [16], to an accuracy sufficient for eq (A13)

and those to follow. The solar insolation at the lunar surface is proportional to R

2

, hence the temperature of the lunar soil varies throughout the

year. Since the variation is slow relative to a synodic period, it appears

that the perturbing effect on the linear approximations of the lunar heat and

radiation equations can be treated quasi -statically, leading to an average

temperature T of the form

T n

= TnJ 1 + 0-0334 cos g) (A15)

where T 00 is the average temperature T at 1 Au. Equation (A15) leads to a normalizing factor of the form

cr(g) = 1 - 0.0334 cos g . (A16)

The preceding development of a assumes a constant solar output. However,

according to available data possible variations in the solar constant with

amplitudes up to ±1 percent cannot be ruled out [24]. Furthermore, no final

conclusions can be drawn at this time about possible correlations to other

solar phenomena [24], making easy detection of a change in the solar constant

difficult.

16
The parameter 5 in eq (A9) was estimated using the brightness temperature

of eq (C4) with the small variable term removed, and yielded a value equal to

the quantity (1 + 0.013d). With a lunar diameter d of 0.5° (0.0087 radians), 5 is only 0.01 percent greater than unity, implying that for the earth-moon

separation distance the lunar output is well represented by an average

brightness that is a property only of the lunar surface and not the relative

earth-moon geometry.

The brightness temperature given by eq (C4) was used to estimate the

polarization factor kg. The largest deviation from unity for any lunar phase

or antenna polarization over the 1 to 10 GHz range was found to be 0.1

percent, showing that kg can be dropped with little error.

Equation (A4) can now be written as

= ^a U) T

a ^- I—J k

2

S ± 0.1% + 0.01% (A17)

where the first error (±0.1 percent) comes from setting kg equal to unity and

the second (±0.01 percent) from setting 5 equal to unity, k is Boltzmann's

constant and \ is the measurement wavelength. The flux density S in eq (A17)

becomes

S = ml^r-\ T'Q ± 1% (A18)

2k \-

TQ ± 1% ± 3.34% (A18 1 m )

where the first error (±1 percent) comes from a possible variation in the

solar constant and the second (±3.34 percent) from the variable solar insolation due to the earth's eccentric orbit. The f appearing in (A18

1 ) and eq (2)

of Section 2 corresponds to that of eq (A8), thus leading to the additional

±3.34 percent error. Equation (A18 1 ) is the starting point for Section 2.

17
APPENDIX B

ANGULAR LUNAR DIAMETER

The apparent lunar diameter as seen by an observer on the earth's surface

can be deduced from the earth-moon geometry shown in figure 5, where d is the

average lunar angular diameter, rmis the average lunar radius, r

Q is the

geocentric earth-moon separation, a is the equatorial radius of the earth, LAT

is the latitude of the observer, re is the geocentric radius to the observer's

"sea level" and is a function of his latitude [15], z is the observer's

altitude above sea level, and el is the elevation angle to the lunar center.

The angular diameter is related to the other parameters by the equation

rm

d = 2 arc sin

2 2 2 W?

[r - (r + z) cos el] - (r + z) sin el

o e e

(Bl)

After dividing the numerator and denominator of this expression by the

earth's equatorial radius and performing a number of approximations, the

following equation for d in degrees is obtained:

d

_ ____

0-5182

(r /a)/60.268 - 0.0166 sin el o

The angular diameter is now only a function of the earth-moon distance and the

elevation angle at the time of observation. The quantity "60.268" is the mean

earth-moon distance in units of the earth's equatorial radius. The ratio r Q/a can be obtained from The Astronomical Almanac [16] at the beginning (midnight)

of each day of the year to an accuracy of 0.002 percent, and a simple parabolic extrapolation including the values for three consecutive days gives a result accurate to 0.02 percent. The approximations leading to eq (B2) add up

to a total error of 0.1 percent in calculating the lunar diameter from this

equation.

18
Figure 5. Relative earth-moon geometry

19
APPENDIX C

A DISK MODEL FOR THE LUNAR SHAPE FACTOR

The effect of an antenna power pattern upon the antenna temperature T

a is

contained in the convolution integral of eq (A5). Under favorable conditions,

the resulting shape factor k 2 can be accurately reduced to a simple expression

containing only the ratio d

e /eH

, where d

e is an effective lunar diameter and

e H is the antenna HPBW. Radio astronomers [25] first used a simple disk model

with a constant brightness temperature to describe the output of Casseopia A,

which was later refined [5] by choosing the effective diameter to minimize the

error from using such a simple model.

The shape factor described by eq (A5) can be formally written as

/TP

n dQ (l-e*

2

\

k

2

E ^w- =

I --3- -I l < C1 >

where

x

2 e (in2) (d

e /eH

) 2

(C2)

and

/TP dQ

I = I(Q,4»,g,e u) =(—A \_"_ (C3) (r? H

' ~ I 2 1 /TdQ

The first factor in eq (CI) is the shape factor for a disk of constant

brightness. If I were deleted from this equation, the ratio (1 - I)/I becomes

the relative error in using a disk model to estimate k 2

- The effective

diameter will be chosen to minimize this error for e H /d greater than unity.

20
The integrals in eq (C3) require some knowledge of the normalized antenna

power pattern P

n and the lunar surface brightness temperature. If these quantities were known with sufficient accuracy, T

a could be calculated directly

from eq (Al). Although this is not the case, it is often possible to approximate the power pattern inside the HPBW by a Gaussian function—sometimes even

down to the -lOdB point on the pattern. Defining the pattern width at the

-lOdB point to be the maximum usable source diameter d leads to a specification of 0.55d as the narrowest HPBW that should be measured using the source.

A model of the surface brightness temperature T that is quite adequate

for estimating the error in setting I equal to unity can be obtained from the

literature [22, 26, 27]. The model used in the calculations is

T = T(a,j3,<!>) = e T 1 - v2 cos

u

<\> (C4)

where

e = e(e

Q

) = e„(9

o

) + ^(9^ (C5)

2e„ = 1 -

2 V?

cos e - (e' - sin 9 ) c o o

2 V? cos 9 + (e

1 - sin 9 ) L o o

(C6)

2e = 1 - 1

2 V?

e' COS 9 - (e

1 - sin 9 \

L

~~

2 Vo

e

1 cos e + (e

1 - sin 9 ) L O

(C7)

7 = Y(9) = arc tan (y-^-j) (C8)

and

6 = 6(9) = 6

1 cos (C9)

21
With reference to figure 6, T is the surface brightness temperature with

selenographic coordinates A<j> and <\> which correspond to the coordinates a and p in the observer's coordinate frame on the earth's surface. $ is the lunar

phase angle for which I is calculated. The integrals in eq (C3) are performed

over the entire apparent solid angle of the surface with dQ as the

corresponding differential, e is the surface emissivity and is composed of a part e

M associated with radiation parallel to the plane of incidence defined

by the points e, m, and the surface point, and a part e perpendicular to that

plane, e' is an average dielectric constant set equal to 1.64 [9] for the

calculations. The angles, 9

Q and 9, are related to each other by Snell's law

and to a and p through the geometry shown in the figure. T is a constant

thermal temperature resulting from solar heating, and T^ is the corresponding

fundamental harmonic amplitude. ¥ is the phase lag between initial solar

heating and subsequent radio emission. 6-j_ is the ratio of the thermal

absorption coefficient to the radio absorption coefficient.

In the calculation of I, it can be seen from the ratio of the integrals

in eq (C3) that the constant T in eq (C4) will drop out and need not be

considered. This leaves the ratio (T^/T ) and the parameter 6j_ to be

determined. The data [9] plotted in figure 4 was least-squares fit between 1 and 20 GHz with a straight line, resulting in the formula for (Tj_/T ) shown in

the figure. The value of 6j that was used to calculate I was obtained by

making it proportional to inverse frequency, and fitting the resulting phase

lag ¥ to the data in figure 3.

As a check on the model for T, calculations from the model were compared

to measurements of the surface brightness temperature at 9.375 GHz [28].

Figures 7 and 8 show the results for the two lunar phases 168° and 262°

respectively. An equatorial and a polar profile of the brightness are shown

in both figures with the experimental data plotted with solid lines. The

dashed lines, resulting from the model calculations, were shifted upwards for

clarity.

Figure 9 shows the error (1 - I)/I at 9.375 GHz and 168° for different

values of the effective diameter d

e

. Both the half-power beamwidth and the

effective diameter are normalized by the apparent lunar diameter d. The

22
SUBSURFACE RAY

dQ =

EXTERNAL RAY

SUBTERRESTRIAL POINT

Sin 6 = e

1/2 Sin 6 = (Hq.^$j/ //

e^-OBSERVER'S POSITION

ON EARTH'S SURFACE

Figure 6. Pictorial relationship between observer and lunar coordinates.

23
EQUATORIAL

PROFILES

-1 -0.5 0.5 1 E

S

<*/amax

0/An«x

W

N

Figure 7. Comparison at 168° and 9.375 GHz between calculated and

measured profiles of the brightness temperature.

24
EQUATORIAL

PROFILES

POLAR

PROFILES

250

in

HI

LLI

UJ

200

150

100

1 MODEL " .

1 - RATA OT

1 DATA (/) - UJ - Z _ -50

I " CD - C£ - CD .

i i i i i i i i i i i i I i i i i - 1 -0.5 0.5 1

E a/amax W

c

> 9/Pmax N

Figure 8. Comparison at 262° and 9.375 GHz between calculated and

measured profiles of the brightness temperature.

25
-5«-

Figure 9. Error due to modeling k£ as a disk versus 9,,/d at 168° and

9.375 GHz for various effective diameters.

26
figure shows that the error is quite sensitive to d

e

. The object in choosing

d

e is to minimize this error for 9^/d greater than unity.

Figure 10 shows the error at 10 GHz for various lunar phases. Since the error for 45° is the largest, the d e minimizing the error for this phase will

automatically minimize the error at other phases. Figure 11 shows the results

where a d

e of 0.965d produces the least error for e^/d above unity.

The above procedure was repeated at 5.5 and 1 GHz. An average of the

three resulting diameters was taken as the effective diameter for the 1 to 10

GHz frequency range. The result of using this average diameter for the three

frequencies is shown in figure 12. It is clear from the figure that the

average diameter produces an error of less than 0.07 percent for the desired

range of 9 H /d. This diameter, 0.964d, is then used in calculating the lunar

shape factor with the disk model and leads to a value of x

2 given by

x

2

= 0.6441 (d/9

H

) 2

. (C10)

An attempt was made to determine the error at 9.375 GHz from using the

model contours (eq (C4)) instead of the actual contours from the literature

[28]. The two contours differ mostly towards the lunar limb. To make the

error calculations simpler, both the model and actual contours were approximated by a circularly symmetric fitting function of the form (cos y') n

- Y

1 is

the angle between the earth-moon axis and the lunar latitude (see figure 6).

When the model and actual contours were fit by this function, there resulted

exponents (n) with values of 0.118 and 0.151, respectively. The error was

then defined as (k 2

- ko')/^ 1 , where k 2 and k

2

' are the shape factors calculated from the second factor in eq (C3) for the model and actual contours,

respectively. The result, as a function of the ratio of HPBW to apparant

lunar diameter, is shown in figure 13 with a maximum 0.25 percent for the

ratio greater than unity.

27
1-1 %

PHASE

Figure 10. Error due to modeling l<2 as a disk versus 9,,/d at 10 GHz for

an effective diameter of 0.965d and various phase angles.

0.5iFigure 11. Error due to modeling l<2 as a disk versus 6,,/d at

45° and 10 GHz for various effective diameters.

28
0.5 r

Figure 12. Error due to modeling k

2 as a disk versus e./d for an effective diameter of 0.964d for various frequencies and

phase angles.

1.0,-

0.75 -

-TT-£ ]% 0.5

0.25

Figure 13. Model contour error versus 9 /d for k

2 at 9.375 GHz

29
Combining the errors due to I and the model contours results in

k

2 = g ± °- 07% * °- 25% < C11 )

where the second and third terms are the maximum errors in k

2 due to I and the

model contours respectively.

30
APPENDIX D

AN EXPRESSION FOR THE AVERAGE BRIGHTNESS TEIf>ERATURE

Equation (C4) in Appendix C for the brightness temperature of the lunar

surface is an approximation in which terms containing higher harmonics in the

phase <$> have been dropped. These deleted terms lead to higher terms in the

average brightness temperature of eq (A10) which, for three harmonics, may be

written as [22]

T = T - T cos U - <i> ) + To ol 12 c °s (24>

+ T cos (3< (Dl)

The maximum relative error from dropping the second and third harmonics is

A T

(D2)

This error increases with frequency, ultimately limiting the accuracy of the

flux calculation. The maximum frequency of interest in the NBS system is 10

GHz and the error will be estimated at this frequency. From figure 2, the

ratio T^Tq is 0.07 at 10 GHz. At 75 GHz [22], the ratio ?

2 /T

1 and f

3 /f1 are 0.08 and 0.01 respectively. Assuming that these last two ratios can be scaled

in frequency like the center-of-the-moon values shown in figure 4, the ratios

at 10 GHz decrease to 0.022 and 0.003 respectively. Inserting these values

into eq (D2) shows a total error of 0.18 percent for dropping the higher

harmonic terms. Thus, below 10 GHz, eq (Dl) can be written as [22, 27]

T = T + 0.18% (D3)

31
NBS-114A (rev. 2-eo

U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR

REPORT NO.

NBS TN-1073

2. Performing Organ. Report No. 3. Publication Date February 1984

4. TITLE AND SUBTITLE

An Error Analysis for the Use of Presently Available Lunar Radio Flux Data

in Broadbeam Antenna-System Measurements

5. AUTHOR(S)

William C. Daywitt

6. PERFORMING ORGANIZATION (If joint or other than NBS, see instructions)

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Grant No.

8. Type of Report & Period Covered

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. State, ZIP)

10. SUPPLEMENTARY NOTES

]] Document describes a computer program; SF-185, FIPS Software Summary, is attached.

11. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here)

Simple, precise expressions for lunar diameter, average brightness temperature, flux density, and shape factor are presented. An analysis of the

relationship between these parameters and corresponding errors are included. For

broadbeam (HPBW>d) antennas, results show that flux density and shape factor can

be determined with errors less than 13 percent and 0.4 percent respectively at

frequencies below 10 GHz. Extension of the analysis to higher frequencies is

indicated.

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolons)

antenna temperature; error analysis; G/T; lunar flux density; shape factor

13. AVAILABILITY

3 Unlimited

| | For Official Distribution. Do Not Release to NTIS

[~X~| Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C.

20402.

^\ Order From National Technical Information Service (NTIS), Springfield, VA. 22161

14. NO. OF

PRINTED PAGES

35

15. Price

USCOMM-DC 6043-P80

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