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UBC Theses and Dissertations

An accurate beam model for the NRAO 91 meter radio telescope 1986

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A N A C C U R A T E B E A M M O D E L FOR T H E NRAO 91 M E T E R RADIO T E L E S C O P E by J A M E S R O T H W E L L PICHA B. A. Sc., The University of British Columbia, 1984 A THESIS SUBMITTED IN PARTIAL F U L F I L L M E N T OF T H E REQUIREMENTS FOR T H E D E G R E E OF M A S T E R OF APPLIED SCIENCE in T H E F A C U L T Y OF G R A D U A T E STUDIES D E P A R T M E N T OF PHYSICS We accept this thesis as conforming to the required standard T H E UNIVERSITY OF BRITISH C O L U M B I A September 1986 © James Rothwell Picha, 1986 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. I t i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of P h y s i c s The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date September 15, 1986 Abstract Scalar field theory has been used to develop an accurate beam model for use with the National Radio Astronomy Observatory 91 meter radio telescope and the 6 cm dual feed system. The theoretical beam model was calibrated, to an accuracy of 3% of the beam peak, with a small sample of radio point sources within the declination range 23° < S < 62°. The new beam model is shown to be effective in deconvolving differential beam maps, to a dynamic range of 30:1, by a maximum entropy deconvolution method. ii Table of Contents Page Abstract ii List of Tables iv List of Figures v List of Symbols vii Acknowledgements x I. Introduction : 1 II. Model Theory 3 III. Model Parameters • 9 IV. Coordinates 18 V. Calibration 22 VI. Comparison with Previous Beam Model 49 VII. Conclusions 58 References 60 Appendix A - Beam Center 62 Appendix B - Oscillations 63 Appendix C - Software 67 iii List of Tables Table Page I. Number of Scans per Point Source 23 II. Summary of the Theoretical Beam Model Parameters 48 III. Frequencies of Residual Subscans 64 iv List of Figures Figure Page 1. Model Coordinates 5 2. Electric Field Illumination 6 3. Incident Electric Field Blockage and Asymmetry 8 4. E Plane Electric Field Amplitude for Feed A 10 5. H Plane Electric Field Amplitude for Feed A 11 6. E Plane Electric Field Amplitude for Feed B 12 7. H Plane Electric Field Amplitude for Feed B 13 8. E Plane Electric Field Phase 14 9. H Plane Electric Field Phase 15 10. North-South Feed Horn Displacement * 17 11. Beam in Model Coordinates 19 12. Beam in Data Coordinates 20 13. 90°-drift Scan 26 14. 90°-drift Scan Residuals 27 15. 0°-drift Scan l ' South of Beam A 30 16. 0°-drift Scan Residuals l ' South of Beam A 31 17. 0°-drift Scan of Beam A 32 18. 0°-drift Scan Residuals of Beam A 33 19. 0°-drift Scan 1' North of Beam A 34 20. 0°-drift Scan Residuals 1' North of Beam A 35 21. 0°-drift Scan 1' South of Beam B 36 22. 0°-drift Scan Residuals 1' South of Beam B 37 23. 0°-drift Scan of Beam B 38 24. 0°-drift Scan Residuals of Beam B 39 v 25. 0°-dr i f t Scan l ' North of Beam B 40 26. 0°-dr i f t Scan Residuals l ' North of Beam B 41 27. 0° - 6 0 ' / m m Scan , . -43 28. 0° - 60'/mi'n Scan Residuals 44 29. 0° - 120'/min Scan of Theoretical Beam Model 45 30. 0° - 120'/min Scan Residuals of Theoretical Beam Model 46 31. Theoretical Beam Model Map 50 32. Empirical Beam Model Map 51 33. 11° - 120'/min Scan of Empirical Beam Model 53 34. 11° - 120'/min Scan Residuals of Empirical Beam Model 54 35. New Beam Map 55 36. Old Beam Map 56 37. 0° - 60'/min Residual Subscans 65 List of Symbols a Telescope reflector radius 6 Galactic longitude B{0,<f>) Theoretical beam model ' E{9,4>) Complex electric field amplitude E{A) Complex electric field amplitude of feed A E(B) Complex electric field amplitude of feed B Ev E plane electric field Ep E plane electric field (positive y-axis) E~ E plane electric field (negative y-axis) EX Separation parameter / Telescope focus H II Ionized hydrogen nebulae H P B W Half power beam width H H P B W of gaussian convolved with the beam model HAD H P B W of beam A (observed data) HAM H P B W of beam A (beam model) HBD H P B W of beam B (observed data) HBM : H P B W of beam B (beam model) Hp H plane electric field Hp H plane electric field (positive i-axis) H~ H plane electric field (negative x-axis) » Complex number Jn(u)) Integer order Bessel function k Wave number K8 Receiver constant v i i I Galactic longitude L Feed leg width n Number of scans PRD Peak ratio (observed data) PRM Peak ratio (beam model) T Radial integration parameter T Point source vector TC Beam center vector 2-axis vector R Distance to point source RT Feed horn rotation angle SS Separation between peak points (observed data) SM Separation between peak points (beam model) t Observation integration time ILF Equivalent integration time Tsys Receiver system temperature x Model coordinate x-axis y Model coordinate y-axis z Model coordinate 2-axis Z Telescope latitude Complex conjugate a Telescope right ascension ctc Beam center right ascension ctz 2-axis right ascension 6 Telescope declination 6C Beam center declination Sz 2-axis declination L\TTTns Receiver noise viii Ay y shift parameter &VHF Receiver high frequency bandwidth &<t> Angle subtended by feed legs « z i-axis direction feed horn displacement t v y-axis direction feed horn displacement ez 2-axis direction feed horn displacement 0 Polar angle in model coordinates 0C Beam center polar angle in model coordinates 9' Polar angle integration parameter A Wavelength 4> • Azimuthal angle in model coordinates <f>c Beam center azimuthal angle in model coordinates <t>' Azimuthal angle integration parameter Units cm centimeters GHz, MHz, Hz gigaHertz, megaHertz, Hertz mK, K milliKelvin, Kelvin h hours of time m,min minutes of time 8 seconds of time ° degrees ' arcminutes " arcseconds ix Acknowledgements First 1 would like to thank my supervisor Dr. P. C. Gregory for his assistance and guidance throughout this work. Also I wish to thank Dr. J . R. Fisher for providing information about the N R A O 91 meter telescope. I am very grateful to M . A . Potts and A . Reid for their advice on computing. Finally, I would like to thank my family for their help in typing and proof reading this manuscript. x I. Introduction With the Galactic Radio Patrol project, Gregory and Taylor created a unique as- tronomical data base. In their search for compact source variability, they repeatedly mapped a large portion of the galactic plane (/ = 40° to / = 220° and b — —2° to b — + 2 ° ) , using the National Radio Astronomy Observatory (NRAO) 91 meter telescope and a 6 cm dual feed receiver . A full intensity atlas of this region can be created by deconvolving the differential beam pattern from the data, but previous attempts to do this with an empirically derived beam model and a maximum entropy deconvolution method have met with limited success. The major problem seems to have been that the empirical beam model is accurate only to 10% of the beam peak. This work derives a more accurate telescope beam model. The most direct method of deriving an accurate telescope beam model is to map a large number of point sources spread out over the declination range of the observations. Because the 91 meter telescope is a transit instrument, this mapping requires a large amount of observing time, approximately 21 days for the dual beam system, leaving very little time for the Galactic Ra\iio Patrol observations. To reduce the time spent observing calibration sources, a theoretical telescope beam model has been developed and calibrated with a small number of point sources. It is anticipated that the theoretical beam model will be especially useful for the proposed second phase of the Galactic Radio Patrol project which will use a new seven feed system. This thesis discusses the theoretical beam used to model the differential beam of the N R A O 91 meter telescope at a wavelength of 6 cm. Specifically the theory used to construct the beam model and the methods used to calibrate the beam model with a small number of point sources are presented in sections II and V. In section VI, the theoretical beam model is compared to the old empirical beam model developed by Taylor (1982) and Braun (I98l),and the maximum entropy deconvolution method is used to do 1 dynamic range tests of the theoretical beam model. Also, in appendix B , as a by-product of this work, the instrumental effects of the N R A O 91 meter telescope are quantified, and possible causes of the instrumental effects are discussed. 2 II. Model Theory The present theoretical model is based on the work of Imbriale et al (1974). They derived a scalar approximation for the far zone electric field pattern of a parabolic reflec- tor system in which the receiver feed position may undergo large lateral displacements. This scalar approximation requires that the E and H plane feed illumination patterns be symmetric, and that the incident radiation not be blocked from the telescope reflector. However, the N R A O 6 cm dual feed system illumination pattern is asymmetric and the 91 meter radio telescope feed legs partially block incident radiation from the reflector. To account for the asymmetric feed illumination and the blockage, the work of Imbriale et al is developed further in this thesis. Using the approximations [that the feed illumination pattern is fixed with respect to the reflector coordinates, that the feed displacement is accounted for only in the electric field phase, and that the observation point {9,<f>) is within approximately eight wave- lengths of the boresight (see Figure 1)], Imbriale et al derived the scalar approximation for the far zone electric field E[9,<j>). Thus -t'fca 2exp(-tfc(2/ + R)) E(9,<f>) =• AixRf l exp {iktz cos 9')I(r)(l - cos 8')r dr (l) where I(r) = I (sin 2<f>'Ep + cos 2<f>'Hp) exp (z'fcar sin 0cos [<f> - <f>')) Jo •exp [ik(tx sin 0'cos <j>' + e„ sin 0'sin 4>')) dip', COS>.= °J£zJUl (2), o 2 r 2 + 4 / 2 t = , fc = 27r/A, A is the electric field wavelength, / is the telescope focal length, R is the distance to the point {9,<j>), a is the telescope radius, ix is the x direction feed displacement, ey is the y direction feed displacement, iz is the z direction feed displacement, r is a dimensionless radial integration parameter defined by equation 2, <f>' is the azimuthal angle integration parameter, 9' is the polar angle integration parameter, 3 Ep is the complex far zone electric field amplitude in the E plane, and Hp is the complex far zone electric field amplitude in the H plane (see Figure 2). In this thesis the feed illumination asymmetry and blockage are added to the theo- retical beam model E{9,<f>) of Imbriale et al by expanding Ep and Hp within I{r). The feed illumination asymmetry in the E and H planes are represented as Ep and E~, and as Hp and H~ respectively. Furthermore the feed illumination blockage is simply char- acterized by zeroing the incident field wherever it is blocked. In the case of the 91 meter telescope the feed legs lie in the y = 0 plane, hence the blockage is characterized by L, the feed leg width (see Figure 3). Thus by the addition of asymmetry and blockage the incident feed illumination is characterized as E+, for &<f> < < £ ' < 7 r - A0; Ep, for TT + A <£<</>'< 2TT - Ar/>; 0, elsewhere. J f+ , for £\<f><<!>' < f ; H', for f < 0 ' < i r - A & H„ = < Hp, for TT + Ar/» < <f>' < ^; ff+ for ^-<4>'< 2TT - A<f>; I 0, elsewhere. where A<f> = arcsin (^) . The solution to 7(r) (determined by inserting the incident feed illumination and then integrating) is an infinite sum of integer order Bessel functions. The complete solution is °° r / + J2m+iH(-l)m (E+-E;)sin(2m+l)P\- m = 0 2cos(2m + l)A<r> 2m + 1 cos(2m-(-3)A<r> cos(2m - l)&<j>\ 2m + 3 2m - 1 / + ( i / + - # - ) c o s ( 2 m + l)/? ( * ± 1 - l ) m - s i n ( 2 m - r l ) A r > 2m + 1 + ( - l ) m + 1 - sin(2m + 3 ) A ^ ( - l ) m + 1 - s i n 2m + 3 + 2m i ( 2 m - l ) A ^ J y Figure 1. Diagram of the telescope coordinate system (model coordinates). The origin corresponds to the focal point of the telescope. 5 X Hp+ 0' ^ Ep Hp Ep+ F i g u r e 2. Diagram of the E+, E~, # p +, and #~ electric field components in the model coordinate system. The 9' coordinate, which describes the angular dependence of E+, E~, H + , and i f " , is perpendicular to the <f>' coordinate. 6 + ^ 2 n H(-l) nCO S 2 n / 3 n = l E+ + E; + H+ + H~ (-sin 2nA<f>) n + E+ + E7-H+-H- (s\n2{n+l)A<j> sin 2(n- ljAtA-sin 2(n- 1)(TT — A</>) + 2(n - 1) / s i ( l V n + 1 )] w here ijj — fcsin ̂ '((ex + psin 0 cos <f>)2+ [tv + P sin 0 sin <^)2)1^2, ty + p sin sin 0 , ar tan /3 = ;— , and p — tx + psin 9 c o s <f> sin 9' Fortunately the I{r) series converges rapidly so that it is approximated by truncating the series after the fourth order Bessel function (accurate to 0.1% of the peak). Inserting 7(r) into equation ( l ) reduces E(9,<f>) to a one dimension integral. Because this integral is intractable it is solved by numerical integration. Thus E(9,4>) represents the scalar approximation of the far zone electric field at point (9,<f>) for one receiver feed. The differential beam of the 6 cm dual feed system is modeled by B{9,4>) = E<A) • E * [ A ) - E<B) • E ' [ B ) where E<A) and E'B) are the far zone complex electric field amplitudes of feeds A and B respectively. Thus the theoretical beam model B{9,<f>) accounts for the asymmetric feed illumination of the 6 cm dual feed system and the incident radiation blockage of the 91 meter radio telescope. 7  III. Model Parameters As shown previously, the theoretical beam model requires several parameters to spec- ify the parabolic reflector dimensions, the feed illumination patterns, and the feed horn locations. Al l these parameters were derived from information provided by the National Radio Astronomy Observatory. The 91 meter telescope, located at a latitude of 38° 25'46.3", is a meridian transit instrument. The physical characteristics used in the theoretical model to describe the parabolic reflector are the reflector radius o, the focal length / , and the feed leg width L which are 45.72 meters, 38.735 meters, and 2.13 meters respectively. Although the feed legs are a lattice structure, in this analysis they are assumed to block incident radiation completely For the phase I Galactic Radio Patrol work the dual channel 6 cm cooled GaAsFet receiver and sectorial feedsystem were located at the telescope focus. The 6 cm receiver was operated at a center frequency of 4.75 G H z with a bandwidth of 580 M H z . Thus the theoretical model assumes that the observational wavelength A is 6.32 cm. The feed horn consists of two sectorial horns fixed together. Measurements of the feed horn electric field amplitude and phase patterns were obtained from Dr. J . R. Fisher of N R A O (see Figures 4, 5, 6, 7, 8, and 9). The E and H plane electric field amplitude patterns of both feeds were measured at the receiver center frequency. Unfortunately the E and H plane electric field phase patterns had not been measured for the dual 6 cm feed system; instead, the E and H plane phase patterns had been measured on a geometrically scaled version of the 6 cm feed horn. On the assumption that the scaled feed horn electric field phase patterns are not significantly different from the 6 cm phase patterns, the scaled feed horn phase patterns were used. The most significant theoretical model parameters are the feed horn positions. In 9 100 90 80 70 60 50 40 30 20 10 0 , M , N U S . . . i I i AZIMUTH ANGLE (DEG.) 10 20 30 40 50 60 70 80 90 100 PLUS L i Figure 4. Diagram of feed A E plane electric field amplitude pattern measured at 4.75 GHz. The azimuth angle axis corresponds to 0' the polar angle in this analysis (J. R. Fisher private communication). 10 Figure 5. Diagram of feed A H plane electric field amplitude pattern measured at 4.75 GHz. The azimuth angle axis corresponds to 9' the polar angle in this analysis (J. R. Fisher private communication). 11 Figure 6. Diagram of feed B E plane electric field amplitude pattern measured at 4.75 GHz. The azimuth angle axis corresponds to 0' the polar angle in this analysis (J. R. Fisher private communication). 12 Figure 7. Diagram of feed B H plane electric field amplitude pattern measured at 4.75 GHz. The azimuth angle axis corresponds to 9' the polar angle in this analysis (J. R. Fisher private communication). 13 AZIMUTH, A N G L E I DEG.) r—1—1 1 I r— 0 8 0 7 6 ^ 6 ' 4 0 3 0^ 10 M I N U S 9 0 - - 80- 70- •60- •50+ •40- -20- 10 t 10- 20+ 30- 40- 50- 60- 10 2 0 I I I I I I J ; 1 I I I I 4 0 6 0 6 0 ^ 7 0 8 0 9 0 P L U S L_  P H A S E A N G L E (D EG  L_  P H A S E A N G L E (D EG  L_  P H A S E A N G L E (D EG  L_  P H A S E A N G L E (D EG  L_  P H A S E A N G L E (D EG  L_  P H A S E A N G L E (D EG  Figure 8. Diagram of the E plane electric field phase pattern measured at the center frequency on a scaled feed horn. The azimuth angle axis corresponds to 9' the polar angle in this analysis (J. R. Fisher private communication). 14 AZIMUTH 1 1 i A r— »JGLjE (|DEG.) - 4 - 90 80 70 60 50 40 30 90- - 80- - 70-- 60- - 50-r- •40- •3( 20 10 10 0- I I I I I k IU£ 10-- 20-- 50-- 40- 50+S- 10 20 I I I I I 70 80 90 J I L UI 60+^-z < • UI < X . a . . . t i l l 30 40 50 60 70 80 90 PLUS Figure 9. Diagram of the H plane electric field phase pattern measured at the center frequency on a scaled feed horn. The azimuth angle axis corresponds to 0' the polar angle in this analysis (J. R. Fisher private communication). 15 particular the theoretical model seems to be more sensitive to the feed horn displacement within the telescope focal plane than to the displacement perpendicular to the focal plane. This result corresponds to the findings of the work done to determine the best receiver focus position of the 91 meter telescope (Fisher and Payne 1982). For these reasons the theoretical model assumes that the feed horn z displacement tz is zero. The feed horn position in the focal plane is determined by three factors. First the feed horns are a fixed 9.413 cm apart. Second the feed horns are mounted on a turntable that rotates, accurate to one degree, about the center point between the two feeds. Third there is an instrumental effect caused by the gravitational deformation of the telescope reflecting dish. Specifically J . R. Fisher and H. E . Payne discovered, after observing several sources over a range of declinations and feed horn positions, that the best reflector focus position changed in the north-south direction at a rate of .74 cm per degree from the telescope zenith (see Figure 10). Conversely this work assumes that the feed horn positions change while the best telescope focus point remains fixed. With the above constraints on the feed horn dimensions and movements the feed horn positions used by the theoretical model are tx = -.0470635 cos {RT) - .0074(Z - 6) meters ty - - .0470635 sin (RT) meters for feed A and ix = +.0470635 cos [RT) - .0074(2 - S) meters ty = +.0470635 sin (RT) meters for feed B , where RT is the feed horn system rotation angle, 6 is the telescope declination, and Z is the telescope latitude. It should be noted that the parameters derived here are not obtained from the calibration data. Rather the model calibration section discusses the adjustments made to these model parameters to improve the fit of the theoretical beam model to the calibration data. 16 471 25A : t P — to <-> O tu Be ul 2 -12.7 £ Si ' to S -254L 10 15 20 25 30 35 40 45 50 DECLINATION (DEG.) F i g u r e 10. Plot of the best reflector focus position versus the declination of the tele- scope. The points correspond to the observed data and the line indicates a slope of .74 cm per degree from the telescope zenith (J. R. Fisher private communication). 17 IV. Coordinates Before the theoretical model can be calibrated by the observed data or used to deconvolve the observed data, one must be able to transform a point in the equatorial coordinate system (a, 6) to a point in the theoretical model coordinate system (9,<f>) (see Figures 11 and 12). A l l points in the equatorial system are called data coordinates, and all points in the theoretical model system are called model coordinates. The differential beam center point is defined as the point equally spaced between the two main lobes of the differential beam (see Figure 11). The telescope is calibrated so that the coordinates (aC,6C) ascribed to each data point indicate the location of the differential beam center when the data point is measured. In the model coordinates the beam center point is (^o^c) (see Appendix A) . Thus the data coordinates (a c , t5 c ) always correspond to the model coordinates (0C,4>C) and vice versa. The transformation from data coordinates to model coordinates relies on the two assumptions that the telescope y axis points due west in the equatorial coordinate system, and that the z axis coordinates [aZ,6Z) in the equatorial system are known. With these two assumptions the transformation from data coordinates [a,6) to model coordinates ( M ) >s sin 9 cos <j> — sin 6 cos 6Z — sin 6Z cos 6 cos ( a z — a) (1) sin 9 sin <f> — cos 6 sin (az — a) (2) cos 9 — sin 6Z sin 6 + cos 8Z cos 6 cos [az — a) (3) where {aZ,6Z) are the z axis coordinates in the equatorial system. Because the feed horns are displaced in the telescope focal plane, the coordinates [azi^z) a r e not usually known. However (aZ,6Z) can be determined from the corre- sponding beam center points (a c ,c5 c ) and (9c,<f>c). Using the fact that (a, cS) corresponds to (0, <j>) when ( a 2 , 8Z) is known in equations (l), (2), and (3) one sets (a, 6) — ( a c , c5c) and 18 19 20 (9,<j>) — (9c,<f>c) in these equations and then inverts the equations to solve for (az,6z). The solution to (az,6z) is . / sin 9C sin <f>c \ <*z — a c + arcsm I ) (4) V cos bc J cos 9C cos 6C cos (az — ac) + sin 6C sin 9C cos <f>c cosoz = - - 5 (5) c o s 2 ( a 2 — a e ) cos 2 6C + sin 6C sin 6C cos ^ z - sin 9C cos ^ c sin bz - 6 cos oc cos [az — ac) Now that the point (az,6z) is determined, one can transform from data coordinates [ot,6) to model coordinates (9,<f>) using equations (l) , (2), and (3). Thus the observed data can be reliably compared to the theoretical beam model. 21 V . Calibration After the information from the N R A O had been used to estimate the theoretical beam model parameters, point source data were used to calibrate the model further. In fact, to improve the accuracy of the beam model, three more parameters had to be added to the model. Also, during the model calibration, significant differences between the observed beam and the theoretical beam indicate that there are instrumental effects reducing the telescope performance. Possible causes of these problems were analyzed with the hope that the telescope performance would be improved, or at least be quantified. Fourteen point sources were observed at three declination drive rates and three beam rotation angles (see Table I). The telescope was driven in declination along its meridian at 120' /mtn, 60'/ratn and O'/min (drift). Of course since the telescope is a transit instrument, the 15'cos 6/min rotation of the earth was added to the telescope's motion. Also the beam was rotated so that different parts of the beam were observed by each scan. In total four types of calibration sources were used. The scans observed at 120'/min declination drive rate were observed during the nighttime, and the scans observed at the other drive rates were observed during the daytime. The theoretical beam model was calibrated within the declination range 23° < 6 < 6 2 ° , since outside this range few strong point sources were observed and the data and the model differed by more than 3% of the beam peak. Throughout this work, data observed at 90° beam rotation to the telescope meridian and 0'/min declination drive rate are called 90°-dr i f t data, data observed at 0° beam rotation to the telescope meridian and O'/min declination drive rate are called 0° - drift data, data observed at 0° beam rotation to the scan track and 60'/min declination drive rate are called 0° — 60'/min driven data, and data observed at 11° beam rotation to the scan track and 120'/min declination drive rate are called 11° — 120'/mtn driven data. As one of the main reasons for this work was to derive a telescope beam model 22 Table 1. Numbers of Scans per Point Source Source ^1950 "1950 Flux 0° 90° 0° 11° Name (Jansky) drift drift 60'/min 120'/mm 3C165 23°22'8" 6 f c40m4.9 s 0.77+.03 4 3C287 25°24'37" 13 f c28ra 15.96s 3.26+.06 5 1 1829+290 29°4'57" 18fc29m17.94* 1.15+.04 5 1 1 3C131 31°24'32" 4 f e50m10.55 s 0.86+.04 4 3C236 35°8'48" 10 f c3m5.39 s 1.34+.08 6 1 1 DA267 39°15'24" 9 f t23m55.29 s 7.57+.13 6 1 1 NGC7027 42°21'3" 2l' l5m9.39 s 5.44+.05 3 3C388 45°30'22" 18 f t42m35.49 s 1.77+.04 5 1 1 3C349 47°7'9" 16 f c58m5.06' 1.14+.04 5 1 3C196 48°22'7" 8 / l9m59.42 s 4.36+.06 6 1 1 3C295 52°26'13" 14 f e9m33.5 s 6.53+.08 5 1 1 3C52 53°17'46" l f e45m14.9 s 1.48+.06 4 DA251 55°44'42" 8 f c31m4.38 8 5.60+.06 6 1 1 1358+624 62°25'8" 13fc58m58.3* 1.77+.02 4 1 accurate to 3% of the beam peak, the sources chosen to calibrate the theoretical model had to have a high flux density so that the receiver noise was less than 1% of the peak of the scan. In general, sources with flux density greater than 1.0 Jansky were selected. However two point sources with a flux density less than 1.0 Jansky, observed at 120'/mtn drive rate, were used because there were so few scans at this drive rate. Their noise was reduced by the averaging of repeated observations. The receiver noise, ATTms, follows the derivation of M . E . Tuiri (1964), KaTs A T r m s = 1 s-1 aya \/L\uHF t L F where the receiver constant Ka — 2, the receiver system temperature Tay, = 70 K and the receiver bandwidth L\VHF — 580MHz. The equivalent integration time t^p is related to the integration time t by the relation t 1.57 The 11° - 120'/min and 0° — 60'/mtn driven data have an integration time of .2 seconds whereas the 0°-drift and 90°-drift data have an integration time of 1.0 seconds. Conse- 23 quently for each receiver A r r m , — 16.3 mK for the driven data and ATTms — 7.3 mK for the drift data. Each scan is the average of two receivers so that the noise is reduced by \ /2. Furthermore the 11° - 120'/mt" n driven data were observed n times reducing the noise further by i/n. Analysis of the receiver noise showed it to be less than 1% of the peak of each scan. Finally it should be noted that pointing corrections, based on the 0° - 60'/min driven data and the 90°-drift data (Taylor 1982), were added to the coordinates of each data scan. Two processes are involved in calibrating the theoretical beam model with the point source data. First iterative methods were used to find the beam model parameters which cause the model best to fit the observed data. The iterative methods used for each type of scan are discussed in detail below. The second process quantifies the fit of the beam model to the observed data by directly comparing the model to the data. The method used to compare the theoretical beam model to the point source data was the same for all types of scan data. Because the beam model needs only to predict the relative intensity and positioning of the beam, both the observed data and the theoretical model were normalized by their peak point value, and the theoretical model was shifted until its peak point position coincided with the observed data peak point position. It is important to note that because of noise at the data beam peaks, the peak point position was determined as the average of the half power beam width positions from either side of the beam. After the model and data scans were normalized and the model peak position shifted, graphs of the theoretical beam superimposed on the observed beam and graphs of the residual difference between the theoretical beam model and the observed beam were plotted. These graphs provided information on the accuracy of the theoretical beam model. The most accurate calibration data are the drift scan data. First it is necessary to consider the calibration of the model to the 90°-drift scan data. The scan track in 24 the 90° -dr i f t data is parallel to a line joining both beam A and beam B (called the main axis). Consequently a typical 90°-dr i f t scan has a positive lobe and a negative lobe corresponding to feed A and feed B (see Figure 13). Initial comparison of the theoretical model to the 90° drift data showed that the model did not fit the data. Detailed analysis shows that the observed data peak ratio, p f̂f̂ , is 5% greater than the theoretical model peak ratio. This suggested that the model feed position parameters were wrong. The theoretical model peak ratio is most sensitive to a y direction change in the feed horn position. Therefore a y shift parameter A y was added to the feed horn position equations. Another problem with the theoretical model is that the observed data separation between peak points is greater than the theoretical model separation between peak points. To fit the 90°-dr i f t data a feed separation factor EX was added to the feed horn position equations. Originally the tz parameter was varied to account for the peak point separation; however, this reduced the accuracy of the fit of the model to the data. With the addition of A y and EX the feed horn positions equations became tx = - .0470635 EX cos [RT) - .0074 (Z - 6) meters € v = -.0470635 EX sin (RT) + A y meters for feed A and, tx = +.0470635 £"Xcos (RT) - .0074 (Z - 6) meters ty = +.0470635 EX sin (RT) + A y meters for feed B. Simple iteration methods were derived to determine the A y and EX parameters for the 90°-dr i f t data. For small values the A y parameter is assumed to be linearly related to the beam peak ratio. Also, since the A y parameter is determined once the model beam peak ratio equals the observed data peak ratio, the two previous estimates for A y and the corresponding model beam peak ratios are used to linearly interpolate a new A y value. It can be shown that / PRMi \ ( PRM2 - PRD \ 25 3 C 3 8 8 C Figure 13. Plot of the final theoretical beam model (dashed lines) superimposed on a 90°-drift scan through the source 3C388. 26 SCAN: 3C388 C Figure 14. Plot of the residuals between the final theoretical beam model and the 90°-drift scan through the source 3 C 3 8 8 . The maximum residual is 2 . 5 % of the beam peak. 27 where the new y shift parameter, At/3, is determined from the two previous shift parame- ters, At/i and A y 2 , the observed data peak ratio, PRD, and the two previous theoretical model peak ratios, PRM\ and PRM2- The EX parameter is directly proportional to the beam peak separation. The new separation factor, EX2, is determined from the previous separation factor, EXX, the previous theoretical model peak separation, 5 M l 5 and the observed data peak separation, 5 5 , by the relation When the A y parameter changes by less than 1% (after approximately 6 iterations), the A y and EX parameters have converged. The average peak residual of all the 90°-dri f t scans is 3.7%. Because the parameters derived for each source are relatively constant, the average parameter values A y = -0.022 meters and EX — 1.01 are used by the beam model to fit the 90°-dr i f t data. Since the 90°-dr i f t scan data are its most sensitive measurement, the A y parameter is fixed at -0.022 meters for the rest of this work. The final theoretical model fit to the data for the source 3C388 is shown in figure 13 and the residuals are shown in figure 14. 0°-dr i f t scans were used independently to check the A y and EX parameter values derived from the 90°-dr i f t scans. The 0°-dr i f t scan track is perpendicular to the main beam axis. Thus a typical 0°-dr i f t scan shows either a positive or a negative main lobe due either to feed A or feed B respectively (see Figure 15). Between four and six separate 0°-dr i f t scans, offset by one arcminute, were observed for each source. Residuals between the data and the model were found for each 0° -drift scan. The peak residuals for each source were averaged to give the average peak residual for a source. These values were averaged again to give the average peak residual for the 0 ° - drift data which is 2.8% of the beam peak. The final theoretical model fits to the data for the source DA267 are shown in figures 15, 17, 19, 21, 23, and 25 and the residuals are shown in figures 16, 18, 20, 22, 24, and 26. Thus the 0° -dr i f t data, which consist of several offset scans through 28 the same source, provide good independent confirmation that the theoretical beam model using Ay = —0.022 meters and EX — 1.01 describes the observed drift scan beam to the 3% level of the beam peak. Next the theoretical beam model was calibrated by the driven declination point source data. The 0° — 60'/min data scan track was parallel to the beam main axis. Thus a typical 0° — 60'/min data scan has a positive and negative lobe corresponding to feed A and feed B (see Figure 27). The initial work calibrating the theoretical model with the 0° -60' /min data showed that the data scans are much broader than the model even after allowing for the expected broadening due to the 0.2s integration time of the data. The theoretical model parameters could not account for the 0° — bO'/min data broadening. It seemed that the best way to fit the theoretical beam model to the 0° - 60'/min data was to convolve the theoretical model with a 1 dimensional gaussian of half power beamwidth H in the declination direction, using the assumption that the broadening is caused by the declination drive motor shaking the receiver feeds at a high frequency. Thus a third parameter H, the gaussian half power beamwidth, was added to the theoretical beam model. Because the H parameter is determined once the model beam HPBWs equal the observed data HPBWs, two previous estimates for H and the corresponding model beam HPBWs are used to linearly interpolate a new H value. The iteration formula used to find H, which is similar to the formula used to determine Ay, is where Hz is the new gaussian H P B W , Hi and H2 are the two previous estimates of the gaussian HPBWs, HAM\ and HAM2 are the two previous model A beam HPBWs, HBM\ and HBM2 are the two previous model B beam HPBWs, HAD is the observed data A beam HPBW, and HBD is the observed data B beam HPBW. The EX parameter was iterated the same way as was done in the 90°-drift data, and the Ay parameter was set equal to -0.022 meters. Because the parameters derived for each source are relatively constant, the average parameter values EX =1.01 and H = 89" were used by the beam H3 = \ H A D ) \ ( HAMl\( HAM2 - HAD HAMX~HAM2 HBM2-HBDY\ {Hi-Hi) HBM\ - HBM2) J 2 f H2 29 D A 2 6 7 A S V a O o 2 0.0 -I ' 16.0 96.0 54.0 1 72.0 90.0 Samples T 162.0 180.0 Figure 15. Plot of the final theoretical beam model (dashed lines) superimposed on a 0°-drift scan through the source DA267. The A beam center is l ' south of the source. 30 SCAN: DR267.RS ft Figure 16. Plot of the residuals between the final theoretical model and the 0°- scan through the source DA267. The A beam center is 1' south of the source, maximum residual is 1.9% of the beam peak. 31 D A 2 6 7 A 162.0 180.0 Figure 17. Plot of the final theoretical beam model (dashed lines) superimposed 0°-drift scan through source DA267. The A beam center is at the source. 32 SCRN: DR267.R 0 23 45 68 90 113 135 158 180 SAMPLES Figure 18. Plot of the residuals between the final theoretical beam model and the 0°- drift scan through the source DA267. The A beam center is at the source. The maximum residual is 2.8% of the beam peak. 33 D A 2 6 7 A N 1 6 2 . 0 1 8 0 Figure 19. Plot of the final theoretical beam model (dashed lines) superimposed 0°-drift scan through source DA267. The A beam center is 1' north of the source. 34 SCRN: DR267.RN 90 SAMPLES Fieure 20. Plot of the residuals between the final theoretical beam model and the 0 - drift scan through the source DA267. The A beam center is 1' north of the source. The maximum residual is 1.5% of the beam peak. 35 D B Samples 72.0 n.o Figure 2 1 . Plot of the final theoretical beam model (dashed lines) superimposed on a 0°-drift scan through the source DA267. The B beam center is 1' south of the source. 36 SCAN: DR267.BS 0 23 45 68 80 113 133 158 180 SAMPLES Figure 22. Plot of the residuals between the final theoretical beam model and the 0°- drift scan through the source DA267. The B beam center is 1' south of the source. The maximum residual is 1.7% of the beam peak. 37 D A 2 6 7 B Samples F i g u r e 23. Plot of the final theoretical beam model (dashed lines) superimposed on a 0° -dr i f t scan through the source DA267. The B beam center is at the source. 38 SCAN: DA267.B Figure 24. Plot of the residuals between the final theoretical beam model and the 0°- drift scan through the source DA267. The B beam center is at the source. The maximum residual is 1.5% of the beam peak. 39 Figure 25. Plot of the final theoretical beam model (dashed lines) superimposed on a 0°-drift scan through the source DA267. The B beam center is l' north of the source. 40 SCAN: DA267.BN 0 23 45 68 90 113 135 156 180 SAMPLES Figure 2 6 . Plot of the residuals between the final theoretical beam model and the 0 ° - drift scan through the source DA267. The B beam center is 1 ' north of the source. The maximum residual is 1.9% of the beam peak. 41 model to fit the 0° - 60'/min driven data. However the residuals of the theoretical model and the observed data are very large (see Figure 28). Further analysis showed that the large residuals are caused by an oscillation in the 0° — 60'/min driven data at a frequency of 0.65 Hz (see Appendix B). More importantly, comparison of the 0° — 60'/min data with the 11° — 120'/min data indicates that the broadening of the telescope beam pattern is dependent on the declination drive rate. The 11°— 120'/mtn data are the most important data used to calibrate the theoretical model since the Galactic Radio Patrol data base was observed at this drive rate. Each point source observed at 11° — 120'/m»n has three or four repeats of the same scan. The typical 11° - 120'/mtn driven scan has a positive and negative lobe corresponding to feed A and feed B respectively, as does the typical 0° — 60'/min scans. Initially, calibrating the theoretical beam model with the 11° — 120'/mm data using the same iteration methods as were used for the 0° — 60'/min data was not successful since the 11° — 120'/mm scan peak ratios were all greater than the theoretical beam model peak ratios. This suggested that the scan coordinates were wrong. Furthermore a qualitative analysis of all the 11° — 120'/mm scans showed that a positive shift in the right ascension coordinate would increase the theoretical beam model peak ratio. Also independent of this work, Dr. N. Duric (personal communication) has found that there is a positive systematic one to two seconds of time difference between point source coordinates derived from observations at the V L A and point source coordinates derived from the Galactic Radio Patrol data base. Consequently to calibrate the theoretical model to the 11° — 120'/min driven data, it is necessary to shift the right ascension coordinates until the model and data peak ratios are equal. The iteration method used to determine the shift in the right ascension coordinate was the same as the method used to determine the Ay parameter, with the assumption that the shift in the right ascension coordinate is linearly related to the beam peak ratio. As expected, the theoretical model peak ratio equals the 11° — 120'/mtn data peak ratio when the scan coordinates are shifted 42 3 C 1 9 6 F i g u r e 27. Plot of the final theoretical beam model (dashed lines) superimposed on the 0° - 6 0 ' / m m scan through the source 3C196. 43 SCRN: 3C196 I i 1 1 f 1 1 1 1 1 0 34 68 101 135 169 203 236 270 SAMPLES Figure 28. Plot of the residuals between the final theoretical beam model and the 0 ° - 6 0 ' / m i n scan through the source 3C196 . The maximum residual is 9.4% of the beam peak. 4 4 o 3 C 5 2 SCAN: 3C52.NEU Figure 30. Plot of the residuals between the final theoretical beam model and the 11° - 120'/mm scan through the source 3C52. The maximum residual is 2.6% of the beam peak. 46 one to two seconds of time. The EX and H parameters used to calibrate the theoretical model to the 11° - 120'/min driven data were determined concurrently. Because the parameters derived for each source are relatively constant, the average parameter values EX =1.03 and H = 115" were used to fit the beam model to the 11° - 120'/mm driven data. The average peak residual between the theoretical model and the 11° — 120'/mm data is 3.1%. The final theoretical model fit to the data for the source 3C52 is shown in figure 29 and the residuals are shown in figure 30. Calibration of the theoretical beam model by the 11° - 120'/mtn data shows that there is a positive systematic one to two seconds of time scan coordinate error in the Galactic Radio Patrol data base, and that the observed telescope beam pattern depends on the declination drive rate. Calibration of the theoretical model by the point source observations, between 23° < 8 < 62° added three more parameters to the theoretical model. The y axis shift param- eter was constrained by the 90° drift scans to be —0.022 meters. The feed separation parameter EX and the gaussian half power width parameter H were found to depend on the declination drive rate. Table II summarizes the Ay, EX, and H parameter values for the 90°-drift, 0°-drift, 0° - SO'/mm, and 11° - 120'/rom data scans. The broad- ening of the beam between the 90°-drift data and the 0° — 60'/min driven data was thought possibly to be a daytime versus nighttime effect. This idea is contradicted by the fact that both the 90°-drift scans and the 0° - 60'/min scans were observed during the daytime. Unfortunately there are no driven scans which would illustrate the declina- tion driven telescope beam pattern perpendicular to the scan direction. Several parallel driven scans, offset one to two arcminutes, must be used to determine the telescope beam perpendicular to the scan direction, since at the high declination drive rates of 60'/min and 120'/min the scan direction is within 14° and 7° of the telescope meridian. Instead it was assumed that there was no declination drive rate broadening in the right ascension direction. The good fit of the model to the 0°-drift scans after the model was calibrated by the 90° -drift scans is consistent with this assumption. The addition of the y shift 47 parameter A y , the feed separation factor EX, and the gaussian half power beam width parameter H to the theoretical beam model enabled the beam model to be calibrated by the data observed at the three different declination drive rates. Table II. Summary of the Theoretical Beam Model Parameters Scan Type A y EX H Average # Sources (meters) (arcseconds) Peak Residual Observed 90°-dr i f t - .022 ± .003 1.01 ± .005 3.7% 9 0° -dr i f t - .022 ± .003 1.01 ± .005 2.8% 10 0° - 60'/min - .022 ± .003 1.01 ± .005 89 ± 8 7.1% 8 11° - 120'/min - .022 ± .003 1.03 ± .005 115 ± 6 3.1% 4 48 VI. Comparison with Previous Beam Model Now that the theoretical beam model has been calibrated it is possible to compare its performance to that of the older empirical beam model (Braun 1981). The objective is to choose the beam model which best represents the actual beam. Since the calibration work shows that the beam pattern is dependent on the telescope drive rate, and the Galactic Radio Patrol data are 120'/min declination driven scans, the only useful data to evaluate the two beam models are the 120' /mm driven data. With this in mind two separate methods were used to evaluate the beam models. One method derived from the calibration work finds the average peak residual between the beam models and the 120'/min driven scans. The second method compares two separate full intensity sky maps generated using the two beams, a section of the Galactic Radio Patrol data base, and a maximum entropy deconvolution algorithm. A n estimate of each beam model's dynamic range was made using these two comparison methods. Throughout this comparison the parameters used by the theoretical model are Ay = -0.022 meters, EX = 1.03 and H - 115" (see Figure 31). The older empirical beam model above 10% of its peak is a modified gaussian whose parameters are determined from 0° -dr i f t scans and 0° — 60'/min scans. A gaussian is fitted to the empirical beam model below the 10% peak level (see Figure 32). The same methods were used to compare scans as were used in the calibration process. Both beam models were shifted one to two seconds of time in right ascension until the model peak ratios equaled the observed data peak ratios. As mentioned earlier, there is only one scan track through each source. For all four sources the theoretical model fits the data much better than the empirical model (see Figures 29, 30, 33, and 34). The average peak residual of the theoretical model is 3.1%, and the average peak residual of the empirical model is 10.6%. Thus the theoretical beam model should have a dynamic range of 32:1 whereas the empirical beam model should have a dynamic range of only 49 _ 6 0 . 0 - CD CD L U O O U J Q 59.8 11 58 60 RIGHT ASCENSION(HRS 12 0 .MINS..SECS.) Figure 31. Contour map of the theoretical beam model for a declination of 59.9° contour values are ± 80, 50, 30, 15, 7, 3, and 1% of the beam peak. The 50 _ 6 0 . 0 • CD CD U J O O U J Q U J 59.8 11 58 60 12 0 60 RIGHT flSCENSION(HRS..MINS..SECS.) Figure 32. Contour map of the previous empirical beam model for a declination of 59.9°. The contour values are ± 80, 50, 30, 15, 7, 3, and 1% of the beam peak. 51 10:1. Consequently the 11° - 120 ' /mm data show the theoretical beam to be 3 times more accurate than the empirical beam model. The second method used to evaluate the accuracy of the beam models is to deconvolve the differential beam from a small area of the Galactic Radio Patrol data base. The area of sky was chosen so that it had both extended structure and an unresolved point structure. The region of sky from a = 23 f c 1 2 m 15 s to a = 23h 1 9 m 0 s and from 6 = 58.8° to 6 — 61.0° contains Sharpless objects 157 and 156. S157 has extended structure surrounding it, and S156 is a compact H II region. The 11° — 120'/rat'n scan data of this region were placed in a 256x256 point grid with the space between grid points being 0.5' (Braun 1981). Next a maximum entropy deconvolution method (Gull et al 1978) using the theoretical beam model and the empirical beam model produced two full intensity sky maps of the area (using the software of Braun 1981). The noise and default level parameters were set to 18 m K and 15 mK respectively. After 40 iterations the map generated using the theoretical beam model (new beam map) converged with X 2 = 1.25707, and the map generated using the empirical beam model (old beam map) converged with x2 — 1.13413 (see Figures 35 and 36). Generally the two maps have the same structure; however, detailed analysis shows that all the peak values of the new beam map are greater than the peak values of the old beam map. Furthermore the unresolved sources on the new map are more circular than the unresolved sources on the old beam map. In fact the empirical beam model seems to produce triangular shaped point sources. The compact H II region S156 was used to determine the dynamic range of each beam. From the high resolution observations done by F. P. Israel (1977) S156 is shown to be a relatively isolated radio source with an angular extent of approximately 15". Since the telescope beam has an angular extent of approximately 3', S156 is essentially a point source compared to the N R A O 91 meter telescope beam. With the assumption that the small sources surrounding S156 are artifacts of the beam, the ratio of the S156 52 3 C 5 2 Figure 33. Plot of the previous empirical beam model (dashed lines) superimpoi the 11° - 120'/min scan through the source 3C52. 53 SCAN: 3C52.0LD 0 10 2 0 2 9 3 9 4 9 5 9 6 8 7 8 SAMPLES Figure 34. Plot of the residuals between the previous empirical beam model and the 11° - 120'/mm scan through the source 3C52. The maximum residual is 13.4% of the beam peak. 54 23 18 0 23 16 0 23 14 0 29 12 0 23 10 I) 2318 0 23 6 0 29 RIGHT ASCENSION(HRS..MINS..SECS.) F i g u r e 35. Contour map of the test area deconvolved with the theoretical beam model. Contour values are 500, 300, 150, 70, 50, and 30 mK. 55 Figure 36. Contour map of the test area deconvolved with the empirical beam model. Contour values are 500, 300, 150, 70, 50, and 30 mK. 56 peak value to the peak artifact value gives an estimate of the dynamic range of each beam model. On the new beam map the peak value of S156 is 11,130 mK, and the peak value of the sources surrounding S156 is 372 mK; and on the old beam map the peak value of S156 is 7,561 mK, and the peak value of the sources surrounding S156 is 443 mK. Thus the new beam map shows that the theoretical beam model has a dynamic range of 30:1 whereas the old beam map shows that the empirical beam model has a dynamic range of much less than 17:1 since the triangular shaped point sources are not believable. Both the single scan data and the maximum entropy method full-intensity maps show that the theoretical beam model is more accurate than the empirical beam model. 57 VII. Conclusions Two major results have come from this work: a theoretical beam model has been developed accurate to 3% of the peak to describe the N R A O 91 meter telescope beam at 6 cm; and new instrumental effects of the N R A O 91 meter telescope have been observed and discussed. The main achievement of this work was the development of the theoretical beam model which, compared with the previous empirical beam model, was shown in two separate tests to be more useful. A comparison of the models for 120'/min declination driven scans shows that the theoretical model is accurate to 3% of the beam peak (for 23° < S < 6 2 ° ) , whereas the empirical model is accurate to only 10% of the beam peak. Furthermore, a full intensity map generated from the data by a maximum entropy deconvolution method and the theoretical beam model achieved a dynamic range of approximately 30:1. It is expected that the theoretical beam model developed in this work will greatly simplify the calibration of the new seven feed receiver planned for the N R A O 91 meter telescope. Three new instrumental effects of the N R A O 91 meter telescope were discovered: the beam shape broadening, the oscillations in the 0° - 60 '/ m i n data, and the beam peak point separation. The beam shape broadening was modeled by the convolution of a one-dimensional declination direction gaussian of half power beam width H with the theoretical model. For the drift, 60 '/ m i n and 120' /mm scans H equaled 0", 89", and 115" respectively. The cause of beam broadening is not understood, but high frequency oscillations are the suspected cause. The residuals of 0 ° - 6 0 ' / r a m data and the theoretical model showed that there is an oscillation of 0.65 Hz in the data. This oscillation increases the noise in the data to approximately 10% of the scan peak. Fortunately oscillations were not observed in the drift or 120'/ratn declination driven scans. The comparison of the theoretical model to the observed data showed that the beam peak point separation 58 depends on the declination drive rate, as both the drift and 60'/min scans had a feed separation factor of 1.01, and the 120'/mtn scans had a feed separation factor of 1.03. Al l these results indicate that work should be done to improve the declination drive mechanism of the N R A O 91 meter telescope; however, while the instrumental effects of the telescope persist, 60'/min drive rate scans should not be used for observations requiring high dynamic range mapping, and scans at different drive rates should not be compared directly. 59 References Abramowitz, M. , and Stegun, I.A. (Eds.).(1972). Handbook of Mathematical Functions. New York: Dover. Braun , Robert. (1981). Radio Continuum Observations of the Supernova Remnant G109.1-1.0. Unpublished master's thesis, University of British Columbia, Van- couver, B . C . Fisher, J .R. , and Payne, H.E.(1982). Measurements of the North-South Focal Point Motion and Astigmatism of the 300 Foot Telescope. Engineering Memo No.148, N R A O Green Bank, West Virginia. Gregory, P .C. , and Taylor, A.R.(1981). "Radio Patrol of the Northern Milky Way: A Survey for Variable Sources". The Astrophysical Journal, 248, 596-605. Gregory, P .C. , and Taylor, A.R.(1986). "Radio Patrol of the Northern Milky Way. A Catalog of Sources II". The Astronomical Journal, 92, 371-411. Gul l , S .F . , and Daniell, G.J.(1978). "Image reconstruction from incomplete and noisy data ". Nature, 272, 686-690. Imbriale, W . A . , Ingerson, P .G . , and Wong, W.C.(1984). "Large Lateral Feed Displace- ments in a Parabolic Reflector ". IEEE Transactions on Antennas and Propaga- tion, AP-22, 742-745. Israel, F .P. (1976). "Aperture Synthesis Observations of Galactic H II Regions ". As- tronomy and Astrophysics, 59, 27-41. Jordan, E . C . and Balmain, K . G . (1968). Electromagnetic Waves and Radiating Systems (2nd ed.). New Jersey: Prentice Hall. 60 Kay, S .M . , and Marple, S.L. Jr.(1981). "Spectrum Analysis: A Modern Perspective ". Proceedings of the IEEE, 69, No. 11. Kraus, J.D.(1986). Radio Astronomy (2nd ed.). Ohio: Cygnus-Quasar Books. National Radio Astronomy Observatory (I983,rev. ed). S00 foot Telescope Observer's Manual. Green Bank, West Virginia. Silver, S. (1944). Microwave Antenna Theory and Design. New York: McGraw-Hi l l . Smart, W . M . (1977). Textbook on Spherical Astronomy (6th ed. rev. by R . M . Green). Cambridge: Cambridge University Press. Taylor, A . R . (1982). A Survey of the Galactic Plane for Variable Radio Emission. Un- published doctoral dissertation, University of British Columbia, Vancouver. Taylor, A . R . and Gregory, P.C. (1983). "Radio Patrol of the Northern Milky Way: A Catalog of Sources. I ". The Astronomical Journal, 88, 1784-1809. Tuirri, M . E . (1964). "Radio Astronomy Receivers ". IEEE Transactions on Antennas and Propagation, AP-12, No. 7. 61 Appendix A - Beam Center The beam center of the theoretical model is defined to be the point equally spaced between the maximum and minimum peak points. Given that the maximum peak po- sition is r\ — ( X I , J / I , 2 I ) and the minimum peak position is r~2 — (x2> J/2> ^2)5 then the beam center position fc — (xc,yc,zc) is constrained by r"c x r~i = r"2 X fc (1) Reduction of vector equation ( l ) gives x g _ Vc _ z c ^ \ Xl + X2 J/l + J/2 Zi + Z2 Assuming that the beam center lies on a unit sphere (x2. + y% + z\ = l ) , the beam center point fc is uniquely determined to be xc — , yc = Vl+y2, and zc = Z ' ^ Z ; where R — {{xi + x2)2 + (j/i + j/2) 2 + (21 + 2 2 ) 2 ) 1 ^ 2 - Furthermore if the beam center point (Oc,<f>c) is to be determined then 9C — arccos (zc) 4>c — arctan ( — I . \xc/ 62 Appendix B - Oscillations Scalar field theory was used to develop an accurate beam model for the N R A O 91 me- ter telescope. Physical characterstics of the N R A O 91 meter telescope and observations of point sources were used to constrain and calibrate the theoretical beam model. During the calibration of the theoretical beam model, significant oscillations were found in the residuals between the theoretical beam model and the 0° beam rotation 60'/min declination driven data (called 0° - 60'/min driven data). Figure 28 shows a plot of the typical residuals between the theoretical beam model and the 0° — 60'/min driven data. The 0° — 60'/min residual data were analyzed and a common frequency was found. No mechanism for the oscillation was determined, but resonances in the feed support stimulated by the drive motor were suspected. A method was derived to determine the oscillation frequency and amplitude of the 0° — 60'/min residuals by analyzing the results of a simulation of the oscillations in residuals. A n analysis of the residuals showed that the amplitude was smallest at the beam peaks and below the 5% peak level beam, and greatest at the inflection points of the beam. In the case of the 0° — 6 0 ' / m t » driven data, there are two main lobes to the differential beam, and each beam lobe has two inflection points. Therefore the 0° — bO'/min driven data residuals consist of four packets of oscillations at the same frequency, with each packet centered on one inflection point of the differential beam. With this in mind, the 0° - 60'/min driven data residuals were divided into four residual subscans. The endpoints of the four residual subscans were the two peak points and the four 5% level points of the 0° - 60'/min driven scans. Some of the residual subscans had gradient baselines. To remove the gradient baseline, the mean, the least squares fit line, and the least squares fit parabola of the residual subscan were subtracted from each residual subscan (see Figure 37). Next a baseline processed residual subscan was discarded if 63 one oscillation amplitude was less than three times the receiver noise of the original 0° — 60'/min driven scan. This reduced the number of residual subscans by half to the residual subscans from the four strongest sources. Next the spectral density of each residual subscan was determined. It was decided that since each subscan had only approximately 17 points, a maximum entropy method should be used to determine the spectral density. The maximum entropy method (Kay et al 1981) requires that the model order parameter of the data to be analyzed be specified. The model order parameter, which is the order of the autoregressive process used to model the data, is 2 for the residual subscan data. Table III shows the peak frequencies of the residual subscans determined by the maximum entropy method. These results show that the average oscillation frequency in the 0° - 60'/min driven data is 0.65 Hz. Table III. Frequencies of Residual Subscans Source " i v2 v3 v4 Average Name Hz Hz Hz Hz # of points DA267 .65 .64 .64 .61 17 3C196 .59 .60 .65 .51 17 3C295 .76 .81 .70 .76 17 DA251 .60 .50 .68 18 A crude estimate of the oscillation amplitude, which is thought to be the angular distance the feed legs move, was determined from the 0° — 60'/min residual subscan amplitudes which are the peak residual values. The average 0° — 60'/min residual subscan amplitude was approximately 10% of the beam peak. The simulated residuals show that the oscillation amplitude must be at least 4.7" for a residual subscan amplitude of 10%. Therefore the oscillation amplitude in the 0° — 60'/min driven data is estimated to be 4.7" or greater. After determination of the oscillation frequency and amplitude in the 0° - 60'/min driven data residuals, the same analysis was used to look for oscillations in the residuals 64 SCAN: DA267 Figure 37. Plots of four baseline processed residual subscans extracted 60'/min scan through the source DA267. 65 of the theoretical beam model and the data driven at other declination rates. Drift scans (O'/min) were observed at both 0° and 90° rotation angles. In addition 120' /mm declination driven scans were obtained with a beam rotation angle of 11° to the scan track. No significant oscillations were found in the residuals of the theoretical beam model and the data driven in declination at O'/min and 120 ' /mm. The absence of the 0.65 Hz oscillations in the residuals of the beam model and the 120'/mtn data was not surprising since the telescope was driven so fast that the residual subscans were too short to measure the 0.65 Hz frequency. However, the measured telescope beamwidth was found to be larger for the 120'/rotn data than for the 0° - 6 0 ' / m m driven data. This suggests that at the 120'/min drive rate the feed supports are oscillating at a higher frequency which is causing the telescope beam to appear broadened at a sampling rate of 5Hz. The 60'/min data provides direct evidence for a low frequency oscillation of 0.65 Hz, and the broadening of the 60'/min and the 120'/mtn data provides indirect evidence for high frequency oscillations. Therefore it is possible that the 0.65 Hz oscillation is one of many oscillation modes in the movement of the feeds supports. 66 Appendix C - Software This appendix is a listing of the Fortran code used to generate the theoretical beam model. 67 L i s t i n g of ASPG:COBRA.FAST at 16:14:23 on AUG 28, 1986 for CCid=VRSS 1 C THIS PROGRAM GENERATES A 64X64 GRIDED BEAM MODEL 2 C SUITABLE FOR RUNNING ON MEMXXX. ALSO THE BEAM MAYBE 3 C VIEWED ON GRIDSHOW. 4 C INPUTS - UNIT 2 FEED ILLUMINATION DATA 5 C OUTPUTS - UNIT 11 MODEL BEAM HEADER FILE 6 C OUTPUTS - UNIT 12 MODEL BEAM GRID FILE 7 C 8 COMPLEX*16 EH(2,10,40) 9 REAL*8 DFLOAT,H,EX 10 REAL*8 RA,DE,ROT,X0,Y0,Z0,SP,CP,HB,IR,ID,DRA,DDE 11 REAL*8 R0(40),KS(40),PI,BM,RT,L,X(2),Y(2) 12 REAL*8 SRA,ERA,SDE,EDE,SDC,SAM,SAS,RAS,DES 13 REAL*4 RB,BEAM(64,64),PK,OUT(4096),CVBEAM(64,64),G(65) 14 INTEGER*4 I,J,K,N,NSTP,D,HD,WD 15 COMMON /ABLOCK/ RO,KS,EH,X,Y,L,NSTP 16 COMMON /BBLOCK/ X0,Y0,ZO,SP,CP,RAS 17 CALL FWRITE(6,'Enter the Declination of Map: ') 18 CALL FWRITE(6,'(degs,arcmins,arcsecs): ') 19 CALL FREAD(5,'3(R*8): ',SDC,SAM,SAS) 20 CALL FWRITE(6,'<R*8> <R*8> <R*8>: 1,SDC,SAM,SAS) 21 CALL FWRITE(6,'Enter the Beam rotation angle (degs): ') 22 CALL FREAD(5,1R*8: ',ROT) 23 CALL FWRITE(6,'<R*8>: ',ROT) 24 CALL FWRITE(6,'Enter the Dec. broadening (arcsecs): ') 25 CALL FWRITE(6,'(DO NOT ENTER HB EQUAL TO ZERO!): * ) 26 CALL FREAD(5,'R*8: ',HB) 27 CALL FWRITE(6,'<R*8>: ',HB) 28 CALL FWRITE(6,'Enter separtation factor: ') 29 CALL FREAD(5,'R*8: ',EX) 30 CALL FWRITE(6,'<R*8>: ',EX) 31 CALL FWRITE(6,'Enter R.A. d i r e c t i o n spacing(arcmins): ') 32 CALL FREAD(5,'R*8: ',DRA) 33 CALL FWRITE(6,'<R*8>: ',DRA) 34 CALL FWRITE(6,'Enter Dec. d i r e c t i o n spacing(arcmins): ') 35 CALL FREAD(5,'R*8: ',DDE) 36 CALL FWRITE(6,'<R*8>: ',DDE) 37 C 38 C INITIALIZE CONSTANTS 39 C 40 PI=3.141592654D0 41 RAS=PI 42 DES=PI*(SDC+(SAM+SAS/60.DO)/60.DO)/180.DO 43 RT=-PI*R0T/180.DO 44 IR=DRA*Pl/(10800.D0*DCOS(DES)) 45 ID=DDE*PI/10800.D0 46 SRA=PI-32.D0*IR 47 SDE=DES-33.D0*ID 48 N=64 49 C 50 C INITIALIZE ARRAYS 51 C 52 CALL COEFS1 53 CALL COEFS2(EX,RT,DES) 54 H=HB*PI/648000.DO 55 CALL CONVOL(lD,H,WD,HD,G) 56 C 57 C GENERATE BEAM MODEL 58 C 68 L i s t i n g o f A S P G : C O B R A . F A S T a t 1 6 : 1 4 : 2 3 o n AUG 2 8 , 1 9 8 6 f o r C C i d = V R S S 59 DO 10 J = 1 , 6 4 60 D E = S D E + I D * D F L O A T ( J ) 61 DO 20 1=1,64 62 R A = S R A + I R * D F L O A T ( I ) 63 C A L L C E N T B M ( R A , D E , B M ) 64 B E A M ( I , J ) = S N G L ( B M ) 65 20 C O N T I N U E 66 10 C O N T I N U E 67 C 68 C C O N V O L V E BEAM MODEL WITH T H E G A U S S I A N 69 C 70 P K = 0 . 71 DO 30 1=1,64 72 DO 34 J = 1 , 6 4 73 C V B E A M ( I , J ) = 0. 74 DO 36 K=1,WD 75 D=J-HD+K-1 76 I F ( ( D . L T . 1 ) . O R . ( D . G T . 6 4 ) ) GOTO 36 77 C V B E A M ( I , J ) = C V B E A M ( I , J ) + G ( K ) * B E A M ( I , D ) 78 36 C O N T I N U E 79 R B = A B S ( C V B E A M ( I , J ) ) 80 I F ( R B . L T . P K ) GOTO 34 81 PK=RB 82 34 C O N T I N U E 83 30 C O N T I N U E 84 C 85 C PUT N O R M A L I Z E D BEAM I N O U T P U T ARRAY 86 C 87 DO 40 J = 1 , 6 4 88 DO 50 1=1,64 89 K = ( J - 1 ) * 6 4 + I 90 O U T ( K ) = C V B E A M ( 6 5 - 1 , 6 5 - J ) / P K 91 50 C O N T I N U E 92 40 C O N T I N U E 93 C 94 C W R I T E G R I D S T Y P E HEADER T O U N I T 11 95 C W R I T E G R I D S T Y P E G R I D TO U N I T 12 96 C 97 W R I T E ( 1 1 , 1 0 0 ) N , N , N , S R A , S D E , I R , I D 98 C A L L W F I L E ( 1 2 , 0 , O U T , 0 , 4 0 9 6 ) 99 100 F O R M A T ( 3 I 6 , 4 F 1 0 . 6 ) 100 S T O P 101 •END 1 02 C 103 C T H I S S U B R O U T I N E C A L C U L A T E S T H E MODEL BEAM I N T E N S I T Y 104 C A T A P O I N T I N MODEL C O O R D I N A T E S . 1 05 C 106 S U B R O U T I N E B E A M ( S , B M ) 107 C O M P L E X * 1 6 E H ( 2 , 1 0 , 4 0 ) , 1 ( 2 , 2 ) , B ( 2 ) , V ( 2 ) , D C O N J G , D C M P L X 108 R E A L * 8 R 0 ( 4 0 ) , K S ( 4 0 ) , X ( 2 ) , Y ( 2 ) , S ( 2 ) , B M , D R E A L , L 109 I N T E G E R * 4 J , C , M , N S T P , K 110 COMMON / A B L O C K / R 0 , K S , E H , X , Y , L , N S T P 111 C 112 C I N I T I A L I Z E A R RAYS 113 C 114 DO 40 M=1,2 115 DO 40 C=1,2 116 I ( M , C ) = D C M P L X ( 0 . D O , 0 . D O ) 69 L i s t i n g of ASPG:COBRA.FAST at 16:14:23 on AUG 28, 1986 for CCid=VRSS 117 40 CONTINUE 118 C 119 C CALCULATE INTEGRALS NUMERICALLY BY SIMPSON'S RULE. 120 C 121 C=2 122 K=NSTP-1 123 DO 20 J=1 ,K 124 C=3-C 125 CALL INTEG(S,J,B) 126 DO 20 M=1,2 127 I(M,C)=I(M,C)+B(M) 128 20 CONTINUE 129 C=NSTP 130 CALL INTEG(S,C,B) 131 DO 30 M=1,2 132 V(M)=B(M)+4.D0*I(M,1)+2.D0*I(M,2) 133 30 CONTINUE 134 C 135 C FIND DIFFERENTIAL BEAM INTENSITY 136 C 137 BM=DREAL(V(1)*DCONJG(V(1))-V(2)*DCONJG(V(2))) 138 RETURN 1 3 9 END 140 C 141 C THIS SUBROUTINE SEARCHES THE BEAM TO FIND 142 C THE MAXIMUM AND MINIMUM PEAK POINTS. 143 C 144 SUBROUTINE SEARCH(SC,SS) 145 REAL*8 SC(2),SS(2),B(5),AN(2),ST,BM,SN 146 REAL*8 D(2,5)/0.DO,0.DO,1.DO,0.DO,0.DO, 147 + 1.DO,-1.DO,O.DO,O.DO,-1.DO/ 148 INTEGER*4 M,I,J,JM 149 SN=-1.D0 1 50 DO 10 M=1 ,2 151 SN=-SN 152 AN(1)=SC(M) 153 AN(2)=SS(M) 154 CALL BEAM(AN,B(1)) 155 B(1)=SN*B(1) 156 DO 20 1=3,9 157 ST=(.1D0)**(I) 158 25 JM=1 159 BM=0.D0 160 DO 30 J=2,5 161 AN(1)=SC(M)+D(1,J)*ST 162 AN(2)=SS(M)+D(2,J)*ST 163 CALL BEAM(AN,B(J)) 164 B(J)=SN*B(J) 165 IF ((B(J)-B(1)).LT.BM) GOTO 30 166 JM=J 167 BM=B(J)-B(1) 168 30 CONTINUE 169 IF (JM.EQ.1) GOTO 20 170 SC(M)=SC(M)+D(1,JM)*ST 171 SS(M)=SS(M)+D(2,JM)*ST 172 B(1)=B(JM) 173 GOTO 25 174 20 CONTINUE 70 L i s t i n g of ASPG:COBRA.FAST at 16:14:23 on AUG 28, 1986 for CCid=VRSS 175 10 CONTINUE 176 RETURN 177 END 178 C 179 C THIS SUBROUTINE CALCULATES THE COMPLEX DATA COEFICIENTS 180 C FROM THE FEED ILLUMINATION DATA. READ FROM UNIT 2. 181 C 182 SUBROUTINE COEFS1 183 C0MPLEX*16 CD(2,4,40),EH(2,10,40),CDEXP,CANG,DCMPLX 184 COMPLEX*16 CF3,CF4,CF7,CF8 185 REAL*8 RAS,RO(40),KS(40),X(2),Y(2),X0,YO,Z0,SP,CP 186 REAL*8 PD(4,40),AD(2,4,40),F,A,PI,P2,P4 187 REAL*8 SH,R,ANG,L,SPH,DPH,CPH,S2H,DARSIN,DCOS 188 REAL*8 DSIN,DSQRT,F1,F2,F3,F4,F5,F6,F7,F8,F9,F10 189 INTEGER*4 NSTP,J,I,M 190 COMMON /ABLOCK/ R0,KS,EH,X,Y,L,NSTP 191 COMMON /BBLOCK/ X0,Y0,Z0,SP,CP,RAS 192 C 193 C READS UNIT 2 FOR THE FEED ILLUMINATION DATA. 194 C 195 READ(2,150) NSTP,L 196 READ(2,200)((PD(J,I),1=1,NSTP),J=1,4) 197 READ(2,200)(((AD(M,J,I),I=1,NSTP),J=1,4),M=1,2) 198 . 150 F0RMAT(I4,F6.3) 1 99 200 FORMAT(10D17. 1 0) 200 C 201 C INTIALIZE CONSTANTS 202 C 203 F=38.735D0 204 A=45.72D0 205 . PI=3.141592654D0 206 P2=PI/2.D0 207 P4=P2/2.D0 208 SH=(A-L)/DFLOAT(NSTP) 209 C 210 C FILL ARRAYS TO BE USED BY SUBROUTINE INTEG. 211 C 212 DO 5 1=1,NSTP 213 R=SH*DFLOAT(I)+L 214 R0(I)=(R*R+4.D0*F*F)/(4.D0*F) 215 DO 5 J=1,4 216 ANG=PD(J,I) 217 CANG=DCMPLX(0.D0,ANG) 218 DO 5 M=1,2 219 CD(M,J,I)=AD(M,J,I)*CDEXP(CANG) 220 5 CONTINUE 221 DO 50 1=1,NSTP 222 R=SH*DFLOAT(I)+L 223 SPH=L/R 224 DPH=DARSIN(SPH) 225 CPH=DSQRT((1.D0+SPH)*(1.D0-SPH)) 226 S2H=SPH*SPH 227 F1=P2-DPH+SPH*CPH 228 F2=P2-DPH-SPH*CPH 229 ' F3=4.DO*CPH*(2.DO+S2H)/3.D0 230 F4=8.D0/3.D0-4.D0*SPH*(1.D0-S2H/3.DO) 231 F5=P4-DPH/2.D0+SPH*CPH*(.5D0+S2H) 232 F6=DPH/2.D0-P4+SPH*CPH*(1.5D0+S2H) 71 L i s t i n g of ASPG:COBRA.FAST at 16:14:23 on AUG 28, 1986 for CCid=VRSS 233 F7=CPH*(8.D0+S2H*(4.D0+48.D0*S2H))/l5.D0 234 F8 = SPH*(4.D0-S2H*(20.D0/3.D0-16.D0*S2H/5.D0))-8.DO/1 5.DO 235 F9=2.D0*SPH*S2H*CPH*(4.D0*S2H-1.D0)/3.D0 236 F10=SPH*CPH*((14.D0-8.D0*S2H)*S2H/3.D0-2.D0) 237 CF3=DCMPLX(0-.D0,F3) 238 CF4=DCMPLX(0.D0,F4) 239 CF7=DCMPLX(0.D0,F7) 240 CF8=DCMPLX(0.D0,F8) 241 DO 52 M=1,2 242 EH(M,1,1)=(CD(M,1,I)+CD(M,2,I))*F1 243 EH(M,2,I)=(CD(M,3,I)+CD(M,4,I))*F2 244 EH(M,3,I)=(CD(M,1,1)-CD(M,2,I))*CF3 245 EH(M,4,I)=(CD(M,3,I)-CD(M,4,I))*CF4 246 EH(M,5,I)=(CD(M,1,I)+CD(M,2,1))*F5 247 EH(M,6,I)=(CD(M,3,I)+CD(M,4,I))*F6 248 EH(M,7,I)=(CD(M,1,1)-CD(M,2,I))*CF7 249 EH(M,8,I)=(CD(M,3,I)-CD(M,4,I))*CF8 250 EH(M,9,I)=(CD(M,1,1)+CD(M,2,I))*F9 251 EH(M,10,1)=(CD(M,3,I)+CD(M,4,I))*F10 252 52 CONTINUE 253 50 CONTINUE 254 RETURN 255 END 256 C 257 C THIS SUBROUTINE CALCULATES THE RECEIVER POSITION 258 C AND THEN DETERMINES THE BEAM CENTER. 259 C 260 SUBROUTINE COEFS2(EX,RT,DES) 261 COMPLEX*16 EH(2,10,40) 262 REAL*8 W,L,R0(40),KS(40),X(2),Y(2),X0,Y0,Z0,SP,CP 263 REAL*8 SH,DFLOAT,R,SC(2),SS(2),C(2),DSQRT,R1,RT 264 REAL*8 RAS,PI,K,A,DES,EX,Z 265 INTEGER*4 NSTP,I 266 COMMON /ABLOCK/ R0,KS,EH,X,Y,L,NSTP 267 COMMON /BBLOCK/ X0,Y0,Z0,SP,CP,RAS 268 C 269 C INITIALIZE CONSTANTS. 270 C 271 W=0.0632D0 272 Z=0.6707217897D0 273 PI=3.14259265D0 274 K=2.D0*PI/W 275 A=45.72D0 276 SH=(A-L)/DFLOAT(NSTP) 277 C 278 C CALCULATE RECEIVER POSITIONS. 279 C 280 X(1)=-.047625D0*EX*DCOS(RT)-.4220407119D0*(Z-DES) 281 Y(1)=-.047625D0*EX*DSIN(RT)-.022D0 282 X(2)=+.047625D0*EX*DCOS(RT)-.4220407119D0*(Z-DES) 283 Y(2)=+.047 625D0*EX*DSIN(RT)-.022D0 284 SP=DSIN(DES) 285 CP=DCOS(DES) 286 DO 10 1=1,NSTP 287 R=SH*DFLOAT(l)+L 288 KS(I)=K*R/R0(I) 289 10 CONTINUE 290 CALL INIT(SC,SS) 72 L i s t i n g of ASPG:COBRA.FAST at 16:14:23 on AUG 28, 1986 for CCid=VRSS 291 CALL SEARCH(SC,SS) 292 DO 20 1=1,2 293 C(I)=DSQRT(1.DO-SC(I)*SC(I)-SS(I)*SS(I)) 294 20 CONTINUE 295 R1=((SC(1)+SC(2))**2+(SS(1)+SS(2))**2+(C(1)+C(2))**2) 296 R1=DSQRT(R1) 297 X0=(SC(1)+SC(2))/R1 298 Y0=(SS(1)+SS(2))/R1 299 Z0=(C(1)+C(2))/R1 300 RETURN 301 END 302 C 303 C THIS SUBROUTINE CALCULATES THE BEAM IN CELESTIAL 304 C SPHERE COORDINATES 305 C 306 SUBROUTINE CENTBM(RA,DE,B) 307 REAL*8 RA,DE,B,ZRA,CZ,SZ,X0,Y0,Z0,SP,CP,S(2),BT 308 REAL*8 DARSIN,DCOS,DSIN,RAS 309 COMMON /BBLOCK/ X0,Y0,Z0,SP,CP,RAS 310 ZRA=RA+DARSIN(Y0/DCOS(DE)) 311 CZ=Z0*DCOS(DE)*DCOS(ZRA-RA)+X0*DSIN(DE) 312 CZ=CZ/((DCOS(DE)*DCOS(ZRA-RA))* * 2 +DSIN(DE)* * 2) 313 SZ=DSIN(DE)*CZ-X0 314 SZ=SZ/(DCOS(DE)*DCOS(ZRA-RA)) 315 S(1)=SP*CZ-SZ*CP*DCOS(ZRA-RAS) 316 S(2)=CP*DSIN(ZRA-RAS) 317 CALL BEAM(S,BT) 318 B=BT 3 1 9 RETURN 320 END 321 C 322 C THIS SUBROUTINE CALCULATES THE BEAM INTEGRAND TO BE 323 C USED BY BEAM. IT IS A SUM OF BESSEL FUNCTIONS. 324 C 325 SUBROUTINE INTEG(S,N,B) 326 COMPLEX*16 EH(2,10,40),B(2) 327 REAL*8 JB(5),R0(40),KS(40),S(2),X(2),Y(2),U,V 328 REAL*8 W,R,DSQRT,L 329 INTEGER*4 NC,N,M,NSTP 330 COMMON /ABLOCK/ R0,KS,EH,X,Y,L,NSTP 331 DO 10 M=1,2 332 U=R0(N)*S(1)+X(M) 333 V=R0(N)*S(2)+Y(M) 334 R=DSQRT(U*U+V*V) 335 W=KS(N)*R 336 CALL DBSJIN(W,5,JB,NC) 337 B(M)=(EH(M,1,N)+EH(M,2,N))*JB(1) 338 B(M)=B(M)+(EH(M,3,N)*V+EH(M,4,N)*U)*JB(2)/R 339 B(M)=B(M)+(EH(M,5,N)+EH(M,6,N))*(U*U-V*V)*JB(3)/(R*R) 340 B(M)=B(M)+(EH(M,7,N)*V*(3.D0*U*U-V*V)+ 341 + EH(M,8,N)*U*(U*U-3.D0*V*V))*JB(4)/(R*R*R) 342 B(M)=B(M)+(EH(M,9,N)+EH(M,10,N))* 343 + (V**4-6.D0*V*V*U*U+U**4)*JB(5)/(R**4) 344 B(M)=KS(N)*B(M) 345 10 CONTINUE 346 RETURN 347 END 348 C 73 L i s t i n g of ASPG:COBRA.FAST a t 16:14:23 on AUG 28, 1986 f o r CCid=VRSS 349 C THIS SUBROUTINE GUESSES THE VALUES FOR THE MAXIMUM 350 C AND MINIMUM BEAM PEAK LOCATIONS TO BE USED BY SEARCH. 351 C THE SUBROUTINE USES MODEL COORDINATES. 352 C 353 SUBROUTINE INIT(SC,SS) 354 COMPLEX*16 EH(2,10,40) 355 REAL*8 X ( 2 ) , Y ( 2 ) , S C ( 2 ) , S S ( 2 ) , R 0 ( 4 0 ) , K S ( 4 0 ) 356 REAL*8 K,KM,CDABS,L 357 INTEGER*4 I,IM,NSTP,M 358 COMMON /ABLOCK/ R0,KS,EH,X,Y,L,NSTP 359 DO 20 M=1,2 360 KM=0.D0 361 DO 10 1=1,NSTP 362 K=CDABS(EH(M,1 , 1)+EH(M,2,I))*KS(l) 363 IF (K.LT.KM) GOTO 10 364 IM=I 365 KM=K 366 10 CONTINUE 367 SC(M)=-X(M)/R0(IM) 368 SS(M)=-Y(M)/R0(IM) 369 20 CONTINUE 370 RETURN 371 END 372 C 373 C THIS SUBROUTINE GENERATES THE ARRAY TO BE USED 374 C FOR THE GAUSSIAN CONVOLUTION. 375 C 376 SUBROUTINE CONVOL(ID,H,WD,HD,G) 377 REAL*8 ID,H 378 REAL*4 G(65),A,X,M 379 INTEGER*4 WD,HD,I,WE 380 A=4.*ALOG(2.)/(SNGL(H)**2) 381 X=SQRT(ALOG(1000.)/A) 382 HD=IFIX(X/SNGL(ID)) 383 WD=2*HD+1 384 DO 10 1=1,WD 385 X=FLOAT(l-HD-1)*SNGL(ID). 386 G(I)=EXP(-A*X**2) 387 10 CONTINUE 388 M=2. 389 WE=WD-1 390 DO 20 1=2,WE 391 M=6.-M 392 G(I)=M*G(I) 393 20 CONTINUE 394 RETURN 395 END 396 C 397 C THIS SUBROUTINE WRITES DATA TO UNIT IU IN GRIDS FORMAT. 398 C 399 SUBROUTINE WFILE(IU,ID,X,JB,LN) 400 INTEGER*2 LENGTH 401 DIMENSION X(LN) 402 DATA MOD/2/ 403 LENGTH=LN*4 404 LINE=(JB/32+1)*1000 405 CALL WRITE(X,LENGTH,MOD,LINE,IU,£100) 406 RETURN 74 L i s t i n g of ASPG:COBRA.FAST at 16:14:23 on AUG 28, 1986 for CCid=VRSS 407 100 WRITE(6,101) 408 101 FORMAT('I/O error occured in WFILE') 409 STOP 410 END 75

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