NBSIR 75-822

STUDY OF ERRORS IN ABSOLUTE FLUX

DENSITY MEASUREMENTS OF CASSIOPEIA A

Motohisa Kanda

Electromagnetics Division

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado 80302

October 1975

Kanda, M., An error analysis for absolute

flux density .neasurements of Cassio;.3ia

h, IliES Trans. Instrum. Meas.^^25, %j

.

17>1g2 (SepL. 1976) T/T)-

?76

S'^rtie as

l\BSlK7^-b22 under

different title

Prepared for:

Advanced Concepts Office

United States Army Communications Command

Fort Huachuca, Arizona 85613

NBSIR 75-822

STUDY OF ERRORS IN ABSOLUTE FLUX

DENSITY MEASUREMENTS OF CASSIOPEIA A

Motohisa Kanda

Electromagnetics Division

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado 80302

October 1975

Prepared for:

Advanced Concepts Office

United States Army Communications Command

Fort Huachuca, Arizona 85613

U.S. DEPARTMENT OF COMMERCE, Rogers C. B. Morton, Secretary

James A. Baker, III. Under Secretary

Dr. Betsy Ancker-Johnson , Ass'stant Secretary for Science and Technology

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Acting Director

CONTENTS

Page

Abstract 1 I. Introduction 1 II. Theoretical Developments 2 III. Correction Factors and Their Accuracies 4 a. Star Shape Factor--k2 ^ b. Antenna Pointing Factor- -k^ 1 c. Atmospheric Transmission Factor- -k-j^ 8 d. Differential Sky Brightness Temperature Factor--k^ 3 e. Polarization Factor--k^ 9 f. Bandwidth Factor- -k^ H

g. System Response Factor--ky H

IV. Error Analysis 12

V. Conclusion 13

VI. Acknowledgment 13

References 14

iii
LIST OF FIGURES

Page

Figure 1. Block diagram o£ the measurement system 15

Figure 2. Relation o£ the polarization angle e and tilt angle t to

the polarization ellipse 15

Figure 3. Brightness temperature contour map of Cas A (coordinates

for Epoch AD 1950.0, after Rosenberg 1970b) 16

Figure 4. Reconstructed brightness temperature contour map of Cas A 17

Figure 5. Three-dimensional representation of Cas A brightness

temperature 18

Figure 6. Three dimensional representation of convolution integral of

Cas A with 2 minutes antenna half-power beamwidth 19

Figure 7. Three dimensional representation of convolution integral of

Cas A with 4 minutes antenna half-power beamwidth 20

Figure 8. Three dimensional representation of convolution integral of

Cas A with 6 minutes antenna half-power beamwidth 21

Figure 9. Star shape factor, k2 , versus antenna half-power beamdwidth- 22

Figure 10. Star shape factor error versus antenna half-power beamdwith- 23

Figure 11. Mean brightness temperature profile from the center of Cas A

outward (Epoch AD 1950.0, 5.0 GHz, after Rosenberg 1970b) 23

Figure 12. Antenna pointing factor, k^, versus antenna half-power

beamwidth 24

Figure 13. Antenna pointing factor, k^, versus a number of drift cuts across Cas A 25

Figure 14. The width of the Gaussian fitting for drift curves versus location of drift cut across Cas A 26

Figure 15. The width of the Gaussian fitting for drift curves versus antenna half-power beamwidth 27

Figure 16. Atmospheric transmission factor error versus elevation angle

(after Daywitt 1975) 28

Figure 17. Cosmic radio background around Cas A at 3.2 GHz (after

RaghavaRao et al. 1965) 29

Figure 18. Differential sky brightness temperature factor, k^ , versus half-power beamwidth 30

iv
STUDY OF ERRORS IN ABSOLUTE FLUX DENSITY MEASUREMENTS

OF CASSIOPEIA A

Motohisa Kanda

An error analysis for absolute flux density measurements of

Cassiopeia A is discussed. The lower-bound quadrature-accumulation

error for state-of-the-art measurements of the absolute flux density of Cas A around 7 GHz is estimated to be 1.71% for 3a limits.

The corresponding practicable error for the careful but not stateof-the-art measurement is estimated to be 4.46% for 3a limits.

Key words: Accuracy; antenna; calibration; Cassiopeia A; error

analysis; flux density, G/T [system gain/system noise temperature);

ground station; radio star.

I . Introduction

Strong cosmic "point" radio stars, such as Cassiopeia A, (Cas A), Cygnus

A (Cyg A), Taurus A (Tau A), etc., provide accurately positioned broadband

noise standards for G/T (system gain/system noise temperature) measurements

on large ground stations. Especially, the small angular size (approximately

4 arc-minutes) and the strong flux render Cas A useful for the measurements

of G (system gain) and G/T. The success of the measurement depends to a large extent on good knowledge and proper evaluation of the properties of

Cas A used as a standard noise source.

In order to measure the absolute values of G and G/T for a large ground

station, the absolute flux density of the radio star is required. Considerable

effort has been devoted to establish the absolute flux density of Cas A

[Allen et al . 1967; Baars 1973; Baars et al . 1972; 1965; Broten et al . 1960;

Findlay 1972; 1966; Findlay et al . 1965; Kellerman et al . 1969; Medd 1972;

Medd et al . 1965; Pawsey 1955; Penzias et al . 1965]. However, comparisons

among various sets of observations are often complicated by a lack of agreement on the scales of the absolute flux density, and, moreover, all the

observations have not been of comparable accuracy. Sometimes the problems

are compounded by the fact that the accuracy and the precision of the measurements are confused. In addition, a random error estimate which specifies the

precision of a measurement is not often clearly identified as to whether it

is for Ic, 3a or something else.

The purpose of this paper is to discuss the potential- accuracy of the

absolute flux density measurements of Cas A around 7 GHz. By use of a precise

brightness temperature contour map of Cas A, all the correction factors for

the absolute flux density measurements are analyzed in detail. This paper

will first discuss the theoretical development for the absolute flux density

measurements and then the corresponding detailed error analysis.
II. Theoretical Developments

In general, the absolute measurements of the flux density of Cas A con- sist of the following six steps (e.g., Allen et al. 1967; Broten et al. 1960;

Findlay et al . 1964; Medd 1972): 1) measuring the system output, when

the antenna is pointing at Cas A; 2) measuring the system output, 2* when

the antenna is pointing away from Cas A; 3) taking the difference, ^W^^^ ^,

between -j^ and 2*' 4) measuring the system outputs W^^-j^ ^

W^^^-j^ ^ when

the input of the radiometer is connected in succession to two thermal noise

standards with different noise temperatures; 5) taking the difference,

^^Q^^iy

between W^^^ ^ ^^'^ ^Cal 4' finally taking the ratio, Y, of AW^^^ ^ to ^^qq^i

The measurement scheme is shown in figure 1. Each measurement step is

discussed below in detail.

During the transit of Cas A, the available spectral power at the antenna

output port, W 1 (f ) in watts per hertz, is given by the antenna convolution

a , 1 integral

;

W

a

^-^(f,J^^) = kT^(f) + Mp(f)A^(f) / a(f ,f2)B(f ,fi)P^(f , (1)

where

^p(f) = I ^^[^05 2c^(f) cos 2e^(f) cos 2 [T^(f

)

-t^ (f ) ]

. + sin 2e^(f) sin 2e^(f)]

.

(2)

The symbols have the following meanings: k is Boltzmann's constant

- 2 3 -

1

(= 1.38x10 J K ); T-|^(f) is the system (minus receiver) noise temperature

with the antenna pointing at Cas A in kelvins; Mp(f) is the polarization mismatch factor; Ag(f) is the effective area of an antenna in square meters,

d(f) is the degree of polarization of the source, defined as the ratio of the

completely polarized power to the total power; e(f) and T(f) are referred to,

respectively, as the polarization angle and the tilt angle, and are specified

in figure 2; a(f,fi) is the atmospheric transmission coefficient; B(f,fi) is

- 2 - 1 - 2 the brightness o of Cas A in Wm Hz rad ;'no P (£,9,-0, ) is the normalized

antenna power pattern in which the difference "fi-f^^" reflects the fact that

the antenna may not be pointing towards the center, Q^, of Cas A. The subscripts w and a on £ and x denote factors associated with the wave and receiving antenna, respectively. By introducing the delivered power- to-available

power system gain, g(f), i.e.,

g(^£)

_ Delivered power at the system output port j-^-j Available power at the antenna output port

the delivered power from the output port of the receiver when the antenna is

pointing at Cas A is

2
^d,l = ^ / [T^^f) . T^^,(£)]g,(£) d£

+ / d£ Mp(£)Ag(£)g^(£) / dQ a (£ , f^) B (£ , fi) (£ , f^-f^^) dn

where Tg(£) is the e££ective input noise temperature o£ the radiometer.

When the antenna is pointing away £rom Cas A, the delivered power £rom

the output is given by

W^2 = ^ / [T2(£) + T^^2f£)] g2(£) d£ (5)

where T2C£) is the system (minus receiver) noise temperature with the antenna

pointing away £rom Cas A.

The delivered power contributed solely by Cas A is then determined by the

di££erence between and 2> i.e.,

Cas A d , 1 d ,

2

= k / [T^(£)g^(£) - T2(f)g2(£)] d£

+ k / [Tg^i(£)g;L(£) - Tg^2(f^g2'^^^]

+ / d£ Mp(£)A^(£)g^(£)

X / dfi a(£,f^)B(£,f2)P^(£,fi-fi^)

The calibration o£ the radiometer is accomplished, in principle, by use

o£ two standard noise sources. For example, the scheme shown in £igure 1 will calibrate the radiometer. When the input of the radiometer is disconnected £rom the antenna and is connected in succession to two thermal noise

standards with different noise temperatures, the delivered powers from the

output of the radiometer are \al,3 = k / [W,3(£) - d£, (7)

and

\al,4 = k / [T,,i,4(£) - T^,4(£)]g4(£) d£, (8)

where T^^-j^ is the available noise temperature of the thermal noise standard.

The difference between W^^-j^ ^ ^cal 4 ' AW-,hWt,-W^t. cal cal,3 cal,4

= k / [Teal,3(f^g3tf) ' ^cal , 4 ^ §4 ^

] (9)

+ k / [T^^3(f)g3(f) - Tg^4(f)g4(£)] df

.

3
CIO)

' By taking the ratio of AW^^^ ^ to AW^^-j^,

Y

= ^''Cas A

^\al

_ k / (T^g^-T,g^)d£ . k / (T^^ig.-T^^^g^) df

" ^ / [T^al,3^f)g3^f) - Tcal,4(f)g4(^^]

^ / df Mp(f)A^(f)g^(f) / a(f ,^^)B(f .^2)P^(f ,^^-^^)

+ k / [T^^3(f)g3(f) - T^^4(f)g4(f)] df

Using eq. (10), the total source flux density, S, of Cas A is given by

2k T -T

S = // B(n) dfi = — S^l^—cal^

Y. (11)

Ag k^ . . . k^

The k-j^ through k^ are correction factors which are listed below:

k-j^ = Atmospheric transmission factor,

k2 = Star shape factor,

kj = Bandwidth factor,

k^ = Differential sky brightness temperature factor,

k^ = Antenna pointing factor,

kg = Polarization factor,

kj = System response factor.

Using the precise brightness temperature contour map of Cas A, all the cor- rection factors and their accuracies are next analyzed in detail.

III. Correction Factors and Their Accuracies

The knowledge of the power gain and the corresponding effective area of

the standard gain antenna is basic to the problem. The measurement of the

power gain of the large standard antenna is one of the difficult tasks in

radio engineering. Accuracy considerations in the measurement of the antenna

power gain are discussed by the author (Kanda 1975; Wait et al . 1974). Uncertainties in the value of standard noise temperatures and in the ratio measurement, Y, are also important sources of error. These uncertainties are summarized in § IV. In this section, all the correction factors and their errors are analyzed by use of the precise brightness temperature contour map

of Cas A published by Rosenberg (1970b).

4
a. Star Shape Factor'-k^

The star shape factor k2 is given by

/ B(S2) P (fi) dQ

= ^ (12)

where B(fi) is the brightness o£ Cas A, and ^j^(^) is the normalized antenna

power pattern. Cas A is usually assumed to be a uniform disk with the resulting star shape correction factor approximated by (Baars et al . 1965)

x^

where

X = 5 (14)

1.2012 e,hp

and D is the angular diameter of Cas A and 3^^^ is the half -power -beamwidth of

the antenna. Hoivever, because Cas A is not a uniformly radiating disk, but

rather an anomalous source of radio waves where a large percentage of the

power is radiated from a circularly shaped ridge, a means must be devised for

estimating the magnitude of error caused by this assumption.

Figure 3 shows the brightness temperature contour map of Cas A measured

at epoch AD 1950.0 (Rosenberg 1970b), where the thick dashed curve represents

the zero contour relative to a cold area of the sky well separated from the

source, and where the contour interval is 200 kelvins . The brightness temperature is plotted in declination versus right ascension, and the highly

structured nature of Cas A is clearly evident. It is clear that the source

is well localized with a slight bulge towards the east (upper center) . The

high-resolution maps of Cas A show that all of the parameters of Cas A, flux

density, spectral index, degree of polarization and position angle, and

secular decay rate, which are considered to be constants in the broad antenna

beamwidth picture, do in fact vary across the face of the source in a highly

irregular manner (Rosenberg 1970a, 1970b) . The brightness temperature contour map of Cas A sho^oi in figure 3 is

reduced to numerical values by a superimposed grid with a value assigned for

each intersection of a horizontal and vertical line. This numerical representation is fed into a computer which reconstructs a comparable brightness

temperature contour map, and the result is shown in figure 4. From this

numerical representation, the computer plots out the relative brightness

temperature of Cas A in a three-dimensional representation which is shoim in

figure 5.

5
Using the computer-stored representation o£ Cas A it is possible to con- volute the star shape with the antenna power pattern to determine the magnitude

of the error caused by assuming a uniform disk for Cas A. In the convolution

integral, it is assumed that the normalized antenna power pattern can be very

closely approximated by the sine x function. Knowing that the function

. 2 sine X = given by

2 sine X = 0.5 when x = 1.392 radian, the normalized antenna power pattern is

P (e,(f)) = sine-

( 1 2.7846 2 . 7840 sinc^

- ^hp - *hp - (15)

where Q-^^ and (p-^^ represent the half -power-beamwidth of the antenna.

Figure 6 shows the three-dimensional representation of numerical convolution integral of Cas A with an antenna whose half -power-beamwidth is 2 minutes,

With this antenna, Cas A is still a discrete source with some structure with

maximum coupling observed toward the east. However, with an antenna of 4 minute half-power-beamwidth, the shape of the result for the numerical con- volution integral is very similar to that of the normalized antenna power

pattern itself as shown in figure 7 in three-dimensional presentation, and

Cas A can now be treated as a discrete source without structure. For further

reference, the three-dimensional representation of the numerical convolution

integral with an antenna of 6 minute half-power-beamwidth is shown in figure 8,

To determine the most appropriate equivalent disk size for Cas A, the

star shape factor using the actual brightness temperature contour map shown in

figure 4 and using a uniformly radiating 4.3 minute disk in eq. (13) are shown

in figure 9 as functions of antenna-half -power beamwidth. This 4.3 minute

uniformly radiating disk has been commonly used by radio astronomers (e.g.,

Baars et al . 1965). The discrepancies between the star shape correction factor using a precise brightness temperature contour map of Cas A and that using

a uniformly radiating disk distribution of various disk sizes are shown in

figure 10. It is found from this figure that, contrary to the 4.3 minute disk

which radio astronomers have been commonly using, the choice of 4.6 minutes

for the disk size gives less error in the star shape factor. For an antenna

of half-power-beamwidth larger than 4.6 minutes, the error by using a 4.6

minute disk distribution is at most 0.2%. It is also found from figure 10

that the star shape correction for an antenna with a half-power beamwidth less

than 4.6 minutes is very large (i.e., k2 is much less than one), and, therefore, the error associated with the correction factor is expected to be large.

From the explanation above, it is very natural to assume that the most

appropriate estimate for the size of Cas A is about 4.6 minute. This agrees

remarkably well with the mean radial profile shown in figure 11 (Rosenberg

1970b) . This mean profile of the map was obtained by averaging the emission

from annuli 5 arc seconds wide drawn from the center of the shell, and the

6
edge o£ the shell turns out to be approximately 140 seconds of arc from the

center, which corresponds to a 4.6 minute diameter for Cas A.

b. Antenna Pointing Factor--k

When the antenna is pointing slightly off the center of the radio star

by an amount Q , the system output will then be reduced by an amount

If the antenna pointing is highly accurate, the antenna can be pointed

at the center of the source (defined as the point of maximum antenna temperature) and is near unity. For example, convoluting the brightness temperature contour map of Cas A shown in figure 4 with a gaussian antenna power

pattern places the center of Cas A at 23^21^^20^ R--\ [o.) and 58°32'30" DEC (6).

These values are only slightly different from the optical center of the Cas A

nebulosities 23^21^^11^-4 and 58°32'18"-9 (Rosenberg 197 0b).

The antenna pointing factor for the measurement of the flux density of

Cas A is calculated by use of eq . (16) with the brightness temperature contour

map show^n in figure 5, and is shown in figure 12. It is found from this

figure that a typical antenna pointing accuracy of 51 in terms of half-power

beamwidth gives rise to 0.71 error in the measurement of the flux density.

When the antenna pointing accuracy is not high, k- can differ appreciably

from unity and is therefore difficult to calculate accurately. In this case star drift measurements can be used to circumvent the difficulty. Using a drift measurement technique, the antenna pointing factor, k- , is shoim in

figure 13 as a function of a number of drift cuts across Cas A. It is found

from this figure that, for an antenna i;ith half -power beamwidth of 6 minutes

or greater, five drift cuts across Cas A are minimum and optimum, and give

rise to a reasonable accuracy. More than five drift cuts will not provide

any significant advantages in determining the antenna pointing factor accu- rately. It is found from figure 13 that by taking five drift measurements

approximately one minute apart and fitting their peaks to a Gaussian curve, it

is possible to estimate the antenna pointing factor, k-, to within 0.5% for an antenna with a half-povrer of beamwidths 6 minutes or greater.

It should be pointed out that this drift measurement technique is also a very useful technique to determine the half -power beamwidth of an antenna.

Figure 14 shows how the width of the Gaussian fit for a drift curve changes

with different (in declination) cuts across Cas A. This figure indicates that,

for an antenna with a half -power-beamv/idth 4 minutes or less, the width of the

Gaussian fit for a drift curve does change with the location of the drift cut across Cas A. However, it is found from figure 15 that, for an antenna with

k. =

/ B(ro p^(S2) dQ

(16)

7
a half-power beamwidth o£ 6 minutes or greater, the width o£ the Gaussian fit

is constant regardless of where the location of the drift cut across Cas A

is and how large the half -power beamwidth of an antenna is, provided that the

main beam of the antenna power pattern is approximated by the Gaussian function.

c. Atmospheric Transmission Factor--k^

As electromagnetic waves from Cas A pass through the atmosphere to the

antenna, they are attenuated. The atmospheric transmission factor, k^,

accounts for the atmospheric reduction in the star flux density between the

upper reaches of the atmosphere and the antenna aperture at microwave frequencies, and is given by (Daywitt 1975)

h- h\K ^^^^

where k^, k^ , k^ are the transmission factors corresponding to atmospheric

diffusion, atmospheric gas absorption, and atmospheric refraction, respectively. To obtain the best accuracy it is assumed that measurements are performed

only at high elevation angles on clear days.

The details of the calculation and error analysis are discussed by

Daywitt (1975) . The atmospheric transmission factor error above 5 degrees at

7 GHz for an antenna with a half-power beamwidth of 10 minutes is shown in

figure 16.

d. Differential Sky Brightness Temperature Factor--k^

Since Cas A, Cyg A, and Tau A are all located close to the galactic plane,

special care must be taken with the cosmic background effect, particularly at

lower frequencies. In principle, the sky brightness temperature, T^^ , when the

antenna is pointing at Cas A and the sky brightness temperature, T2, when the

antenna is pointed off Cas A, are different. The difference between T^ and

T^ leads to the differential sky brightness temperature factor and is given by

k^ = 1 . ^

i ^ . (18)

It should be noted that in the drift measurements the noise temperature

difference T-j^ - T2 is due solely to the changing cosmic background as Cas A

passes through the antenna beam. To illustrate the magnitude of the difference

between T-j^ and T2 , a detailed radio emission contour map of the region around

Cas A is shown in figure 17 (RaghavaRao et al . 1965). The contour intervals

are designated by integers which are multiples of 0.87 kelvin above a uniform

background of 3 kelvin black-body radiation. Since the brightness for the

HII (ionized hydrogen) region is approximately flat above 1 GHz, T-|^-T2 can be

estimated by 0.9 kelvin per square frequency in GHz.

8
Assuming that:

1) the flux density SC£) o£ Cas A can be expressed as (e.g.. Wait et al.

1974)

S(£) = 3604x10"^^ £-0.875 ^^^^-2 ^^-1 ^g-j

where £ is a frequency in GHz, and

2) -the approximate relationship among the power gain, G, the effective area,

Ag, and the half-power -beamwidth , 9j^p> in minutes are given by

4ttA 9 . 578 X lO'

G = ^=~ — , (20)

hp

the differential sky brightness temperature factor, , is then approximately

given by

V - 1 + I- 52x10 ^ f^'^^ r?i^ ^4 " ^ ^ -14.7/0^ ^^^>

1 - e P

where f is the frequency in GHz and Gj^^ is the antenna half-power beamwidth

in minutes. The differential sky brightness temperature factor, k^ , is shown

in figure 18 as a function of half-power beamwidth of an antenna.

e. Polarization Factor--k^

D When the polarized component of the radiation from Cas A and the receiving

antenna polarization are not matched, it is necessary to introduce the polarization factor, k^, to account for the polarization mismatch. Assuming that

the degree of polarization, the polarization angle and the tilt angle do not

change appreciably with frequency within a small bandwidth and over the entire

solid angle subtended at the antenna by Cas A, the polarization correction

factor, k^ , is given approximately by

k^op= 2M = 1 + dTcos 2e cos 2e cos 2(t -t )

^ w a ^ w a (22)

+ sin 2e sin 2e 1

.

w a-" The polarization correction factor for special combinations of antennas and waves are compared in table 1.

9
Table 1 Polarization Correction Factor

Wave

Polarization

Antenna

Polari zation

Linear

Right

Circular

Left

Circular

(e =-45°) (e,=45°) ^ a ^ a

'

Linear

Left Circular

Right Circular (e =-45°)

(e =45°)

l+dcos2(T -T ) 1

^ w a-^

1 1 + d

1 1-d

1

1-d

1+d

The literature suggests that above 1 GHz the linear polarization of Cas A

is approximately \% and that below 1 GHz the polarization falls to zero. Table 2 is representative of the values reported.

Table 2 Degree of Linear Polarization of Cas A

Frequency

(GHz)

1

3

58.25

10

14. 5 15.25

15.75

19.4

Polarization

(^)

< 1

< 1 1.4 ± 0.5

0.5 ± 0.2

1.5

1.2 ± 0.5

1.9 ± 0.2

2.2 ± 0.3

1

Reference

(Baars 1973)

(Baars 1973)

(Rosenberg 1970b)

(Allen et al. 1967)

(Baars 1973)

(Boland et al. 1966)

(Allen et al . 1967)

(Allen et al . 1967)

(Mayer et al . 1968)

The degree of antenna polarization is commonly expressed in terms of

axial ratio, A, rather than polarization angle, e, well-known relation (Kraus et al . 1966)and one can arrive at the

cot e =

'I (23)

where and are, respectively, left and right circularly polarized field

components. For a non-ideally circularly polarized antenna with partially

linearly polarized waves from Cas A, the polarization correction factor, , is then given by

1+d cos 2(cot

10

-1

A) . (24)
Assuming that the axial ratio. A, o£ the standard gain antenna is 1.1 and the

linear polarization of Cas A above 1 GHz is 1%, the error in the polarization

correction factor is no more than 0.1%.

f. Bandwidth Factor-'k^

The frequency variation of the various parameters that comprise the elements of eq. (10) across the system bandwidth lead to the bandwidth correction. Assuming that the passband of a receiving system has a square window

and the power gain of the antenna is nominally 60 dB around 7 GHz which

corresponds to an effectiA

is given approximately by

2 corresponds to an effective area of 145 m , the bandwidth correction factor

kj = 1 + (Af/2f^)^ (25)

when Af is the bandwidth of a square pass band and f^ is its center frequency.

In the case of a 50 MHz bandwidth at a center frequency of 7 GHz, the bandwidth correction factor, k^ , is very close to unity.

g. System Response Factor--k-,

Instabilities due to receiver gain and its effective input noise temperature are often significant sources of error in the measurement of the flux

density of Cas A. In order to minimize the measurement errors, extremely good

receiver stability is required. Instabilities due to receiver gain and its

effective input noise temperature leads to the system response factor, k-

.

By assuming that the pass band of the receiving system has a square window,

and both the effective input noise temperature of the receiving system and

the noise temperature of the standard noise sources are independent of frequency within a small bandwidth, the system response factor is then given

approximately by

k^ = 1 + Ag AT * e Te

2kT T

e ^

e AS T -T ^e^ cal,3 cal,4j

(26)

where Ag/g is the fractional change in the system gain and AT^/T^ is the fractional change in the effective input noise temperature of the receiving system.

Assuming that (Ag/g) • (AT^/Tg) = 0.1°^ (3a), S = 656 x lO"^^ Wm"^ Hz"""- at 7 GHz,

A^ = 145 m^ at 7 GHz, T = 30 K , T ^ . , = 30 0 K , T . , =290 K, k^ is

e 'e 'cal,3 'cal,4 '7

approximately equal to 1± (0 .4%) (3a)

.

In addition to instability of gain and effective input noise temperature

of the receiving system, the system response factor, ky, should include a com- ponent due to resolution. The resolution component can be estimated by

(T Cas A "^sys-*^"^' ^^^^^

-^Cas A temperature rise due to Cas A, T^

11
is the system noise temperature, B is the bandwidth, and t is the time constant

of the receiving system. Assuming that T^^^ ^

= 36 K, T^^.^ = 150 K, B = 50

MHz, and t = 1 sec, the uncertainty due to resolution is approximately 0.1%.

. In the drift measurements the integrator of the radiometer distorts the

drift curve. The results of the distortion are a shift in the position of the

maximum with a decrease in its value and a broadening of the drift curve. The

amount of distortion depends on the integration time constant and the time it

takes the antenna to pass through Cas A. The detailed analysis on this subject is discussed by Kuz'min et al . (1966). When the time constant is on the

order of one second and the half-power transit time is on the order of two

minutes, the integration effect due to the time constant can be neglected.

IV . Error Analysis

An estimate of the limiting error for the calibration of the flux density

of Cas A around 7 GHz is summarized in table 3. Since all the contributing

sources of error are independent and, therefore, satisfy the condition for

quadrature error addition, the total error indicated in table 3 is a quadrature-accumulation error.

Table 3 The Error in Measuring the Absolute Flux Density of Cas A Around 7 GHz Error Contributing to Measuring

Uncertainty in Value of Quantity Flux Density of Cas A

Sources of Error

1. Antenna Power Gain

2. Temperature Calibration

Standard

3. Y Factor

4. Atmospheric Transmission

Factor k-j^

5. Star Size Factor k2

6. Bandwidth Factor kj

1 . Differential Sky Brightness Temperature Factor

8. Antenna Pointing Factor

9. Polarization Factor k.

10. System Response Factor

k,

Lower Bound Pract icable

60 0.06 dB O.IK

0.002 dB 0.99 0.005

0.9314 0.0044

1 0.0001

1 0.0006

1 0.005

1 0.001

1 0.005(3 )

60 0.18 dB

0.99 0.01

1 0.01

Lower Bound Practicable

1.39*

'0. 03%

0.05*

0. 51*

0.47% 0.01%

0.06%

0.51

0.1%

0.5%

4.23%

1.01%

0.5%

Quadrature-Accumulation Error 4.461

12
In the analysis of the error estimate, the following assumptions are made:

1) The standard gain antenna with a power gain of nominally 60 dB

around 7 GHz is calibrated to within a quadrature accumulation error of 0.06

dB for 3a limits (Kanda 1975; Wait et al . 1974). This lower bound estimate is

based on an ideal state-of-the-art measurement condition. The practicable

error for the standard gain antenna measurement which is based on careful but

not quite state-of-the-art conditions is estimated to be 0.18 dB for 3a limits

(Kanda 1975; Wait et al . 1974).

2) The measurement of power ratio Y in eqs . (10) and [11) utilizes

an automated measuring system [Boyle et al . 1972) developed around the most

accurate power measurement bridge known -- the XBS Type II self -balancing

bridge [Larsen et al . 1970). This bridge measures the ratio of stable noise

powers to an estimated error not greater than ± 0.002 dB [Wait et al. 1974).

3) The total uncertainty in the noise temperatures of standard

noise sources [e.g., 290 K and 300 K) is no greater than 0.1 K, i.e., 0.03%.

With these assumptions, it seems possible to calibrate the flux density

of Cas A around 7 GHz to within a quadrature-accumulation error of 1.71% for

3a limits. This is a good estimate of the lower -bound of the uncertainty

based on the state-of-the-art measurements. The corresponding practicable

error for the careful but not state-of-the-art measurements is estimated to

be 4.46% for 3c7 limits.

v. Conclusion

It has been found that the absolute flux density of Cas A around 7 GHz can be measured to within a quadrature -accumulation error of 1.71% for 3a

limits from the assumption that the standard gain antenna of nominally 60 dB

gain can be calibrated to within a quadrature accumulation error of 0.06 dB

for 3a limits. This procedure has established a lower-bound on the standard

error for the state-of-the-art measurement of the absolute flux density of

Cas A, in which a 3c random error estimate has been included along with the

various systematic errors . The practicable error for the standard gain an- tenna measurement which is based on careful but not quite state-of-the-art

conditions is estimated to be 0.18 dB for 3a limits. In this case, the practicable error for the measurements of the absolute flux density of Cas A

around 7 GHz is estimated to be 4.46% for 3a limits.

YI . Acknowledgment

The author wishes to thank B.K. Bender for her computer assistance, and

W.D. Daywitt and D.F. Wait for many stimulating discussions in the preparation

of this manuscript. Appreciation is also extended to C.K.S. Miller for his

continuing support in this subject.

13
References

Allen, R.J. and Barrett, A.H. (1967) Astrophysical Journal, 149, 1-13.

Baars, J.W.M. (1973) IEEE Trans. Antennas 5 Propagation, AP-21 , 461-474.

Baars , J.W.M. and Hartsuijker, A. P. (1972) Astronomy and Astrophysics, 17 , 172-181.

Baars, J.W.M., Mezger, P.G., and Wendker, H. (1965) Astrophysical Journal,

142 , 122-134.

Boland, J.W., Hollinger, J. P., Mayer, C.H. and McCullough T.P. (1966) Astrophysical Journal, 144 , 437-439.

Boyle, D.R., Clague, F.R., Reeve, G.R. , Wait, D.F. and Kanda , M. (1972) IEEE

Trans, on Instrumentation and Measurement, IM- 21 , 543-549.

Broten, N.W. and Medd, W.J. (1960) Astrophysical Journal, 132^, 279-285.

Daywitt, W.C. (1975) "Basic Error Equations for NBS G/T Measurements,"

NBSIR, to be published.

Findlay, J.W. (1966) Ann. Review Astronomy § Astrophysics, 4_, 77-93.

Findlay, J.W. (1972) Astrophysical Journal, 174, 527-528.

Findlay, J.W., Huatum, H. and Waltman, W.B. (1965) Astrophysical Journal, 141 , 873-884.

Kanda, M. (1975) IEEE Trans. Antennas § Propagation, AP-23 , 407-411,

Kellerman, K.I., Pauling -Toth , I.I.K. and Williams, P.J.S. (1969) Astrophysical

Journal, 157 , 1-34.

Kraus , J.D. and Tiuri, M.E. (1966) Radio Astronomy (McGraw-Hill Book Company,

New York) 108-130.

Kuz'min, A.D. and Salomonovich , A.E. (1966) Radioastronomical Methods of

Antenna Measurements (Academic Press, New York) 89-92.

Larson, N.T. and Clague, F.R. (1970) Proc. 1970 ISA Annual Conf . , 25, 712-770.

Mayer, C.H. and Hollinger, J. P. (1968) Astrophysical Journal, 151 , 53-63.

Medd, W.J. (1972) Astrophysical Journal, ITi, 41-50.

Medd, W.J. and Ramanov, K.V.V. (1965) Astrophysical Journal, 142^, 383-386.

Pawsey, J.L. (1955) Astrophysical Journal, 121 , 1-5.

Penzias, A. A. and Wilson, R.W. (1965) Astrophysical Journal, 1_42 , 1149-1155.

RaghavaRao, R., Medd, W.J., Higgs , L.A, and Broten, N.W. (1965) Royal Astronomical Society Monthly Notices, 129 , 159-168.

Rose.nberg, I. (1970a) Royal Astronomical Society Monthly Notices, 147 , 215-230.

Rosenberg, I. (1970b) Royal Astronomical Society Monthly Notices, 151 , 109-122.

Wait, D.F., Daywitt, W.C, Kanda, M. and Miller C.K.S. (1974) NBSIR 74-382.

14
ANTENNA

ca 1

CALIBRATED-j^

cal ,4

DIRECTIONAL

COUPLER RADIOMETER

RECEIVER

NOISE

SOURCE

Figure 1. Block diagram of the measurement system

MAJOR

AXIS

POLARIZATION

ELLIPSE

MINOR

AXIS

Figure 2. Relation of the polarization angle e and tilt angle x to the polarization ellipse

15
16
1 1 1 1 1 1 ! 1 1 r

DECLINATION

Figure 4. Reconstructed brightness temperature contour map o£

Cas A

17
Figure 5. Three-dimensional representation o£ Cas A brightness

temperature

18
gure 6. Three dimensional representation of convolution integral

of Cas A with 2 minutes antenna half-power beamwidth

19

Figure 8. Three dimensional representation of convolution integral

of Cas A with 6 minutes antenna half-power beamwidth

21
ANTENNA HALF-POWER BEAMWIDTH (MIN)

Figure 9. Star shape factor, k-^,, versus antenna half-power beamwidth

22
DIAMETER OF CAS A DISC (MIN)

ANTENNA HALF-POWER BEAMWIDTH (MIN)

Figure 10. Star shape factor error versus antenna hal£-pover beamwidth

Figure 11. Mean brightness temperature profile from the center of Cas A

outAs'ard (Epoch AD 1950 .0 , 5.0 GHz, after Rosenberg 1970b)

23
POINTING INACCURACY IN TERMS OF ANTENNA BEAMWIDTH {%)

Figure 12. Antenna pointing factor, , versus antenna half-power

beamwidth

24
Figure 13. Antenna pointing factor, k^, versus a number of drift

cuts across Cas A

25
ono

o

LlJ Q

1/5

(U>

V)>

u

f-1

<

•H nj

(/5 o

G Si

• H ct3

in (-»

rt u X -H M !-i t3 O "+HO

•M a

^3 O

•H -H

IS ^

0) o X o H r-H

CVI CM o

GO

•H

Ph

HiaiMWVBa VNNBiNV JO SWHJi NI HIQIM NVISSflVS

26
Figure 15. The width of the Gaussian fitting for drift curves versus antenna half-power beamwidth

27
I I I I _J I I I I I 0 1 0 20 30 40 50 60 70 80 90

ELEVATION (DEG)

(

Figure 16. Atmospheric transmission factor error versus elevation

angle (after Daywitt 1975)

28
RIGHT ASCENSION ( 1930 0)

Figure 17. Cosmic radio background around Cas A at 3.2 GHz (after

RaghavaRao et al . 1965)

29
30
U.5. DEPT. Ur L-UMM.

1 BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO.

NBSIR 75-822

2. Gov't Accession

No.

3. Recipient's Acces.sion No.

4. TITLE AND SUBTITLE

Study of Errors in Absolute Flux Density Measurements

of Cassiopeia A

5. Publication Date October 1975

6. Performing Organization Code

276 .06

7. AUTHORfS)

Motohisa Kanda

8. Performing Organ. Report No. NBSIR 75-822

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS , B 0 U 1 d G f LdbS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Project/Task/Work Unit No. 2766482

11. Contract/Grant No.

12. Sponsoring Organization Name and Complete Address (Street, City, State, ZIP)

U. S. Army Communications Command

Fort Huachuca, Arizona 85613

13. Type of Report & Period Covered

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-word or less factual summary of most si^ificant information. If document includes a significant

j bibliography or literature survey, mention it here.)

\

[ An error analysis for absolute flux density measurements of Cassiopeia A

i is discussed. The lower-bound quadrature-accumulation error for state-of-the-art

I measurements of the absolute flux density of Cas A around 7 GHz is estimated to

be 1.71% for 3a limits. The corresponding practicable error for the careful but

not state-of-the-art measurement is estimated to be 4.46% for 3a limits.

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name; separated by semicolons

)

Accuracy; antenna; calibration; Cassiopeia A; error analysis; flux density; G/T

(system gain/system noise temperature); ground station; radio star.

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