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An accurate beam model for the NRAO 91 meter radio telescope Picha, James Rothwell 1986-12-31

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AN A C C U R A T E B E A M M O D E L FOR T H E N R A O 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 S U B M I T T E D IN P A R T I A L F U L F I L L M E N T O F T H E REQUIREMENTS  FOR THE DEGREE OF  MASTER OF APPLIED  SCIENCE  in T H E F A C U L T Y O F G R A D U A T E STUDIES D E P A R T M E N T O F PHYSICS  We accept this thesis as conforming to the required standard  T H E UNIVERSITY O F BRITISH  COLUMBIA  September 1986 © James Rothwell Picha, 1986  In p r e s e n t i n g  t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of  requirements f o r an advanced degree a t the  the  University  of B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make it  f r e e l y a v a i l a b l e f o r reference  and  study.  I further  agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may department o r by h i s or her  be granted by the head of representatives.  my  It is  understood t h a t copying or p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l gain  s h a l l not be allowed without my  permission.  Department o f  P h  y  s i c s  The U n i v e r s i t y of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date  September 15, 1986  written  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  I. Introduction  x  :  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 ° - d r i f t Scan l ' North of Beam B  40  26. 0 ° - d r i f t Scan Residuals l ' North of B e a m 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 M o d e l  46  31. Theoretical Beam Model M a p  50  32. E m p i r i c a l B e a m Model M a p  51  33. 11° -  120'/min Scan of Empirical B e a m M o d e l  53  34. 11° -  120'/min Scan Residuals of Empirical Beam Model  54  35. New B e a m M a p  55  36. O l d B e a m M a p  56  37. 0 ° - 60'/min Residual Subscans  65  List of Symbols  a  Telescope reflector radius  6  Galactic longitude Theoretical beam model  B{0,<f>)  Complex electric field amplitude  ' E{9,4>) E{A)  Complex electric field amplitude of feed A  E(B)  Complex electric field amplitude of feed B  E  E plane electric field  v  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  HPBW  Half power beam width H P B W of gaussian convolved with the beam model  H 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) H plane electric field  H  p  Hp  H plane electric field (positive i-axis)  H~  H plane electric field (negative x-axis)  »  Complex number  J (u)) n  Integer order Bessel function  k  Wave number  K  Receiver constant  8  vii  I  Galactic longitude  L  Feed leg width  n  Number of scans  PRD  Peak ratio (observed data)  PRM  Peak ratio (beam model)  T  Radial integration  T  parameter  Point source vector  T  B e a m center vector  C  2-axis vector R  Distance to point source  RT  Feed horn rotation angle  SS  Separation between peak points (observed data) Separation between peak points (beam model)  SM  Observation integration time  t ILF  Equivalent integration time  T  Receiver system temperature  sys  x  M o d e l coordinate x-axis  y  M o d e l coordinate y-axis  z  M o d e l coordinate 2-axis  Z  Telescope latitude Complex conjugate Telescope right ascension  a ct  c  B e a m center right ascension  ct  2-axis right ascension  z  6  Telescope declination  6  C  B e a m center declination  S  z  2-axis declination  L\T  Receiver noise  TTns  viii  y shift parameter  Ay  Receiver high frequency bandwidth  &VHF  &<t>  Angle subtended by feed legs  «z  i-axis direction feed horn displacement  t  v  y-axis direction feed horn displacement  e  z  2-axis direction feed horn displacement Polar angle in model coordinates  0 0  Beam center polar angle in model coordinates  C  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 hours of time  h  m  ,min  minutes of time seconds of time  8  °  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 astronomical 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 reflector 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 wavelengths 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 exp(-tfc(2/ + R)) E(9,<f>) =• AixRf 2  l  exp {ikt cos 9')I(r)(l z  - cos 8')r dr  (l)  where I(r) = I Jo  (sin <f>'E + cos <f>'H ) exp (z'fcar sin 0cos [<f> - <f>')) 2  2  p  •exp [ik(t  x  p  sin 0'cos <j>' + e„ sin 0'sin 4>')) dip',  .=  COS>  °J£zJUl o r + 4/ 2  t =  2  (2)  ,  2  , 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, i is the x direction x  feed displacement, e  y  is the y direction feed displacement, i is the z direction feed z  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  E  p  is the complex far zone electric field amplitude in the E plane, and H  p  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 theoretical beam model E{9,<f>) of Imbriale et al by expanding E  and H  p  p  within I{r). T h e  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 characterized 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). T h u s by the addition of asymmetry and blockage the incident feed illumination is characterized as for  Ep,  for TT + A < £ < < / > ' < 2TT - Ar/>;  0,  elsewhere.  Jf+,  for  H',  for  £\<f><<!>' < f ; f<0'<ir-A&  for TT + Ar/» < <f>' <  H„ = < Hp, ff+  for  I 0, where A<f> = arcsin ( ^ ) .  &<f> < < £ ' < 7 r - A0;  E+,  ^-<4>'<  ^;  2TT - A<f>;  elsewhere.  T h e 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  °° +  J2m+iH(-l)  r  m=  /2cos(2m + l)A< > r  (E+-E;)sin(2m+l)P\-  m  0  cos(2m-(-3)A< > r  2m + 3  2m +  1  cos(2m - l)&<j>\ 2m -  1  /  -l) -sin(2m-rl)Ar> + ( i / + - # - ) c o s ( 2 m + l)/? ( * ± 1 2m + 1 m  +  (-l)  m + 1  - sin(2m + 3 ) A ^ 2m + 3  +  (-l)  m + 1  - s i ni ( 2 m - l ) A ^ J 2m  y  F i g u r e 1. Diagram of the telescope coordinate system (model coordinates). The origin corresponds to the focal point of the telescope.  5  X  Hp  +  0'  ^  Ep+  Ep  Hp  F i g u r e 2. Diagram of the E+, E~, # , and #~ +  p  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  +  E+  ^2nH(-l) CO 2n/3  + E;  + H+ + H~  (-sin  2nA<f>)  n  S  n  n=l  E+ + E7-H+-H-  +  / (s\n2{n+l)A<j> in2(n+l s  V  +  n+ 1  sin 2 ( n - ljAtA-sin 2 ( n - 1)(TT — A</>) 2(n - 1)  )]  w here ijj — fcsin ^'((ex + psin 0 cos <f>) + [t + P sin 0 sin <^) ) ^ , 2  2  1  2  v  tan /3 =  t + p sin sin 0 , ;— , and t + psin 9 c o s <f> y  p—  x  ar 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< ) • E * A  [ A )  - E< ) • E ' B  [ B )  where E< ) and E'B) are the far zone complex electric field amplitudes of feeds A and B A  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  A s shown previously, the theoretical beam model requires several parameters to specify the parabolic reflector dimensions, the feed illumination patterns, and the feed horn locations. A l l these parameters were derived from information provided by the National Radio Astronomy Observatory.  The 91 meter telescope, located at a latitude of 3 8 ° 25'46.3", is a meridian instrument.  transit  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 G a A s F e t receiver and sectorial feedsystem were located at the telescope focus. T h e 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 c m .  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. O n 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. 9  In  100 90 80 70 60 50 40 30 20 10 0 10 20 30 40 50 60 70 80 90 100 PLUS , . . . i I i AZIMUTH ANGLE (DEG.) M , N U S  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  F i g u r e 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  90-8070•60•50+ •40-  -20AZIMUTH, A N G L E IDEG.) r—1—1 1 I r— 0 80 7 6 ^ 6 MINUS  10 ' 4 0  3 0^  10  10  10-  20  I  I 40  I  t  I I I J ; I I I I 60 6 0 ^ 70 80 90 PLUS 1  20+  405060-  L_ PHASE A N G L E (DEG  30-  F i g u r e 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  90- 80- -  70-60- -  50-r•40•3(  AZIMUTH A »JGLjE (|DEG.) r— 1 1 i  90  80 70 60 50  k IU£  10  40  30  20  -4-  I  0-  10  I  10 20  10--  I  I .I  30  . . I I I I I t i l l 40 50 60 70 80 90 PLUS  20-50-40-  50+SUI  60+^-  z  < •  70  UI  80 90 J  <  X .  a I L  F i g u r e 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. T h i s 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 t  z  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. T h i r d there is an instrumental effect caused by the gravitational deformation telescope reflecting dish.  of the  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 t  x  = -.0470635 cos {RT)  t  y  -  x  = +.0470635 cos [RT)  - .0074(Z - 6)  - .0470635 sin (RT)  meters meters  for feed A and i  t  y  - .0074(2 - S)  = +.0470635 sin (RT)  meters 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 DECLINATION  Figure  30  35  40  45  50  (DEG.)  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 ascribed to each data point indicate the location of the differential beam center  (a ,6 ) C  C  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 , t 5 ) always correspond to the c  c  model coordinates (0 ,4> ) and vice versa. C  C  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 [a ,6 ) Z  Z  in the equatorial system are known. W i t h these  two assumptions the transformation from data coordinates [a,6) to model coordinates (M)  >s  sin 9 cos <j> — sin 6 cos 6 — sin 6 cos 6 cos ( a Z  Z  z  — a)  (1)  sin 9 sin <f> — cos 6 sin (a — a)  (2)  z  cos 9  where {a ,6 ) Z  Z  sin 6 sin 6 + cos 8 cos 6 cos [a — a)  —  Z  Z  (3)  z  are the z axis coordinates in the equatorial system.  Because the feed horns are displaced in the telescope focal plane, the coordinates [ zi^z) a  a r  e  not usually known.  However (a ,6 ) Z  Z  can be determined from the corre-  sponding beam center points (a ,c5 ) and (9 ,<f> ). Using the fact that (a, cS) corresponds c  c  c  c  to (0, <j>) when ( a , 8 ) is known in equations (l), (2), and (3) one sets (a, 6) — ( a , c5 ) and 2  c  Z  18  c  19  20  (9,<j>) — (9 ,<f>c) in these equations and then inverts the equations to solve for c  T h e solution to (a ,6 ) z  z  (a ,6 ). z  z  is  . / sin 9 sin <f> \ + arcsm I ) V cos b J cos 9 cos 6 cos (a — a ) + sin 6 sin 9 cos <f> coso = --5 c o s ( a — a ) c o s 6 + sin 6 sin 6 cos ^ - sin 9 cos ^ sin b cos o cos [a — a ) C  <*z — a  c  c  (4)  c  C  C  z  c  C  C  z  2  2  e  C  c  C  z  c  Now that the point (a ,6 ) z  to model coordinates  z  (5)  2  z  C  c  z  C  6  c  is determined, one can transform from data coordinates [ot,6)  (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). T h e telescope was driven in declination along its meridian at 1 2 0 ' / m t n ,  60'/ratn and O'/min (drift).  O f 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.  T h e theoretical beam model was  calibrated within the declination range 2 3 ° < 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 9 0 ° beam rotation to the telescope meridian and 0'/min declination drive rate are called 9 0 ° - d r 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.  A s 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 Name 3C165 3C287 1829+290 3C131 3C236 DA267 NGC7027 3C388 3C349 3C196 3C295 3C52 DA251 1358+624  ^1950  23°22'8" 25°24'37" 29°4'57" 31°24'32" 35°8'48" 39°15'24" 42°21'3" 45°30'22" 47°7'9" 48°22'7" 52°26'13" 53°17'46" 55°44'42" 62°25'8"  "1950  6 40 4.9 13 28 15.96 18 29 17.94* 4 50 10.55 fc  m  fc  s  ra  fc  s  m  fe  m  s  10 3 5.39 9 23 55.29 2l' 5 9.39 18 42 35.49 fc  m  ft  s  m  l  s  m  ft  s  m  s  16 58 5.06' 8 9 59.42 14 9 33.5 l 45 14.9 8 31 4.38 13 58 58.3* fc  m  /l  m  fe  fe  s  m  s  m  fc  s  m  fc  8  m  Flux (Jansky) 0.77+.03 3.26+.06 1.15+.04 0.86+.04 1.34+.08 7.57+.13 5.44+.05 1.77+.04 1.14+.04 4.36+.06 6.53+.08 1.48+.06 5.60+.06 1.77+.02  0° drift  90° drift  0° 60'/min  5 5  1 1  1  6 6  1 1  1 1  11° 120'/mm 4  4  3 5 5 6 5  1 1 1 1  1 1 1 4  6 4  1  1 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, AT , Tms  follows  the derivation of M . E . Tuiri (1964), AT where the receiver constant K  a  the receiver bandwidth  L\VHF  r m s  =  K1 T  s-  a  \/L\u  s  HF  1  aya  t  L F  — 2, the receiver system temperature T , = 70 K and ay  — 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. Conse23  quently for each receiver  A r  r  m  ,  —  16.3 mK for the driven data and AT  Tms  — 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 9 0 ° - d r i f t data is parallel to a line joining both beam A and beam B (called the main axis).  Consequently a typical 9 0 ° - d r 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, theoretical model peak ratio.  p^ff^,  is 5% greater than the  T h i s 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 9 0 ° - d r i f t data a feed separation factor EX the feed horn position equations. Originally the t  z  was added to  parameter was varied to account for  the peak point separation; however, this reduced the accuracy of the fit of the model to the data. W i t h the addition of A y and EX t  the feed horn positions equations became  = - .0470635 EX cos [RT)  - .0074 (Z - 6)  meters  € = -.0470635 EX sin (RT)  + Ay  meters  t  = +.0470635 £ " X c o s (RT)  - .0074 (Z - 6)  meters  = +.0470635 EX sin (RT)  + Ay  meters  x  v  for feed A a n d , x  t  y  for feed B .  Simple iteration methods were derived to determine the A y and EX  parameters for  the 9 0 ° - d r 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  \ ( PRM  2  - PRD  25  \  3  C  3  8  8  C  Figure 1 3 . 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 parameters, At/i  and A y , the observed data peak ratio, PRD, 2  model peak ratios, PRM\ the beam peak separation.  and PRM 2  The EX  and the two previous theoretical  parameter is directly proportional to  The new separation factor, EX2,  previous separation factor, EX , X  is determined from the  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 9 0 ° - d r i 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 9 0 ° - d r i f t data. Since the 9 0 ° - d r 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 ° - d r i f t scans were used independently to check the A y and EX derived from the 9 0 ° - d r i f t scans.  parameter values  The 0 ° - d r i f t scan track is perpendicular to the main  beam axis. T h u s a typical 0 ° - d r 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 ° - d r 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. T h e 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.  T h e 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 ° - d r 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 A y = —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° - 6 0 ' / m i n data showed that the data scans are much broader than the model even after allowing for the expected broadening due to the 0.2 integration time of the data. The s  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 H P B W s equal the observed data H P B W s , two previous estimates for H and the corresponding model beam H P B W s 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 A y , is  H  3  =  ( \  HAM \( HAM l  2  HBM -HBDY\  - HAD  HBM\  H A D ) \ HAMX~HAM2  where Hz is the new gaussian H P B W , Hi and H  2  the gaussian HPBWs, HAM\ HBM\  and HBM  2  {Hi-Hi)  2  and HAM  2  - HBM ) 2  J  2  f  H  2  are the two previous estimates of  are the two previous model A beam HPBWs,  are the two previous model B beam HPBWs, HAD is the observed  data A beam H P B W , and HBD is the observed data B beam H P B W . The EX parameter was iterated the same way as was done in the 90°-drift data, and the A y 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 29  D  A  2  6  7  A  S  V  a Oo  2  -I  0.0  16.0  '  1  96.0  54.0  72.0  90.0  Samples  T  162.0  180.0  F i g u r e 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  F i g u r e 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  F i g u r e 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  180.0  SCRN: DR267.R  0  23  45  68  90  SAMPLES  113  135  158  180  F i g u r e 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  162.0  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  180  SCRN:  DR267.RN  90  SAMPLES F i e u r e 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  B  D 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 F i g u r e 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 ° - d r i f t scan through the source DA267. T h e B beam center is at the source.  38  SCAN: DA267.B  F i g u r e 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  SAMPLES  113  135  156  180  F i g u r e 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  0  1  34  I  1  68  1  f  101  135  SAMPLES  1  169  1  203  1  236  1  270  Figure 28. Plot of the residuals between the final theoretical beam model and the 0 ° - 6 0 ' / m i n scan through the source 3 C 1 9 6 . The maximum residual is 9.4% of the beam peak.  44  o  3  C  5  2  SCAN: 3C52.NEU  F i g u r e 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 < 6 2 ° added three more parameters to the theoretical model. The y axis shift parameter 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 declination 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 9 0 ° -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 B e a m M o d e l Parameters  EX  Scan T y p e  Ay  90°-drift  - . 0 2 2 ± .003  1.01 ± .005  0°-drift  - . 0 2 2 ± .003  1.01 ± .005  - . 0 2 2 ± .003  1.01 ± .005  - . 0 2 2 ± .003  1.03 ± .005  (meters)  0° 11° -  60'/min 120'/min  48  H  Average  # Sources  (arcseconds)  Peak Residual  Observed  3.7%  9  2.8%  10  89 ± 8  7.1%  8  115 ± 6  3.1%  4  VI.  Comparison with Previous B e a m M o d e l  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). T h e 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 1 2 0 ' / m m driven data. W i t h 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.  T h e 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 m a x i m u m 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 A y = -0.022 meters, EX = 1.03 and H -  115" (see Figure 31). T h e older empirical beam  model above 10% of its peak is a modified gaussian whose parameters are determined from 0 ° - d r 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).  T h e same methods were used to compare scans as were used in the calibration process. B o t h beam models were shifted one to two seconds of time in right ascension until the model peak ratios equaled the observed data peak ratios. A s 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). T h e 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  _ 60.0 CD CD LU O  O UJ Q  59.8  11 58 60  12  0  RIGHT ASCENSION(HRS .MINS..SECS.)  F i g u r e 31. Contour map of the theoretical beam model for a declination of 59.9° The contour values are ± 80, 50, 30, 15, 7, 3, and 1% of the beam peak.  50  _ 60.0 • CD CD UJ O O UJ Q  UJ  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°. T h e contour values are ± 80, 50, 30, 15, 7, 3, and 1 % of the beam peak.  51  10:1.  Consequently the 11° -  1 2 0 ' / m m data show the theoretical beam to be 3 times  more accurate than the empirical beam model.  T h e 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. T h e region of sky from a = 23 to 6 —  fc  12  m  15 to a = 23 s  h  6 1 . 0 ° contains Sharpless objects 157 and 156.  surrounding it, and S156 is a compact H II  region.  19  m  0 and from 6 = 58.8° s  S157 has extended structure  T h e 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). 1978)  Next a maximum entropy deconvolution method (Gull et al  using the theoretical beam model and the empirical beam model produced two  full intensity sky maps of the area (using the software of B r a u n 1981). The noise and default level parameters were set to 18 m K and 15 m K 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 x  2  — 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 m a p . 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. F r o m 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. W i t h the assumption that the small sources surrounding S156 are artifacts of the beam, the ratio of the S156 52  3  C  5  2  F i g u r e 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  F i g u r e 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  RIGHT ASCENSION(HRS..MINS..SECS.)  F i g u r e 35. Contour map  29  of the test area deconvolved with the theoretical beam model.  Contour values are 500, 300, 150, 70, 50, and 30  55  mK.  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  T w o 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 c m ; 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 ° - 6 0 ' / 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, 6 0 ' / m i n and 1 2 0 ' / m m scans H equaled 0", 89", and 115" respectively. T h e cause of beam broadening is not understood, but high frequency oscillations are the suspected cause. T h e 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. A l 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  A b r a m o w i t z , M . , and Stegun, I.A. (Eds.).(1972). New York:  B r a u n , Robert.  Handbook of Mathematical Functions.  Dover.  (1981).  G109.1-1.0.  Radio Continuum Observations of the Supernova Remnant  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 B a n k , West Virginia.  Gregory, P . C . , and Taylor, A.R.(1981).  "Radio Patrol of the Northern M i l k y Way: A  Survey for Variable Sources". The Astrophysical Journal, 248, 596-605.  Gregory, P . C . , and Taylor, A.R.(1986). Catalog of Sources II".  "Radio Patrol of the Northern Milky Way. A  The Astronomical Journal, 92, 371-411.  G u l 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 Propagation, AP-22, 742-745.  Israel, F . P . (1976).  "Aperture Synthesis Observations of Galactic H II Regions ". As-  tronomy and Astrophysics, 59, 27-41.  J o r d a n , E . C . and B a l m a i n , 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, N o . 11.  K r a u s , 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: M c G r a w - H i 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. U n 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  and Propagation,  AP-12, N o . 7.  61  Transactions on Antennas  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 position is r\ —  (XI,J/I,2I)  and the minimum peak position is r~2  beam center position f — (x ,y ,z ) c  c  c  (x2> J/2> ^2)5  —  then the  is constrained by  c  r"  c  x  r~i = r"  2  f  X  (1)  c  Reduction of vector equation ( l ) gives x  g  Vc  _  Xl + X  _  J/l + J/2  2  z  c  Zi +  ^ \  Z  2  Assuming that the beam center lies on a unit sphere (x . + y% + z\ = l ) , the beam center 2  point f is uniquely determined to be x — c  R  —  , y = + , Vl  c  {{xi + x ) 2  2  + (j/i +  c  2  1  2  (O ,<f>c) is to be determined then c  9 — arccos (z ) C  c  4> — arctan ( — I . c  \x / c  62  and z = c  Z  '^  Z ;  where  Furthermore if the beam center point  j / 2 ) + (21 + 2 2 ) ) ^ 2  y2  Appendix B - Oscillations  Scalar field theory was used to develop an accurate beam model for the N R A O 91 meter 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. N o 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.  W i t h this in m i n d , 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.  T o 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 Name DA267 3C196 3C295 DA251  "i Hz .65 .59 .76  v  2  Hz .64 .60 .81 .60  v  3  Hz .64 .65 .70 .50  v  4  Hz .61 .51 .76 .68  Average # of points 17 17 17 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  F i g u r e 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 9 0 ° rotation angles. In addition 1 2 0 ' / m m declination driven scans were obtained with a beam rotation angle of 11° to the scan track.  N o significant oscillations were found in the residuals of the theoretical beam  model and the data driven in declination at O'/min and 1 2 0 ' / m m .  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.  T h i s suggests that at the 1 2 0 ' / m i n 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. T h e 60'/min data provides direct evidence for a low frequency oscillation of 0.65 H z , and the broadening of the 60'/min and the 1 2 0 ' / m t n 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 f o r CCid=VRSS 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58  C C C C C C C  THIS PROGRAM GENERATES A 64X64 GRIDED BEAM MODEL SUITABLE FOR RUNNING ON MEMXXX. ALSO THE BEAM MAYBE VIEWED ON GRIDSHOW. INPUTS - UNIT 2 FEED ILLUMINATION DATA OUTPUTS - UNIT 11 MODEL BEAM HEADER FILE OUTPUTS - UNIT 12 MODEL BEAM GRID FILE COMPLEX*16 EH(2,10,40) REAL*8 DFLOAT,H,EX REAL*8 RA,DE,ROT,X0,Y0,Z0,SP,CP,HB,IR,ID,DRA,DDE REAL*8 R0(40),KS(40),PI,BM,RT,L,X(2),Y(2) REAL*8 SRA,ERA,SDE,EDE,SDC,SAM,SAS,RAS,DES REAL*4 RB,BEAM(64,64),PK,OUT(4096),CVBEAM(64,64),G(65) INTEGER*4 I,J,K,N,NSTP,D,HD,WD COMMON /ABLOCK/ RO,KS,EH,X,Y,L,NSTP COMMON /BBLOCK/ X0,Y0,ZO,SP,CP,RAS CALL FWRITE(6,'Enter the D e c l i n a t i o n of Map: ') CALL FWRITE(6,'(degs,arcmins,arcsecs): ') CALL FREAD(5,'3(R*8): ',SDC,SAM,SAS) CALL FWRITE(6,'<R*8> <R*8> <R*8>: ,SDC,SAM,SAS) CALL FWRITE(6,'Enter the Beam r o t a t i o n angle (degs): ') CALL FREAD(5, R*8: ',ROT) CALL FWRITE(6,'<R*8>: ',ROT) CALL FWRITE(6,'Enter the Dec. broadening ( a r c s e c s ) : ') CALL FWRITE(6,'(DO NOT ENTER HB EQUAL TO ZERO!): * ) CALL FREAD(5,'R*8: ',HB) CALL FWRITE(6,'<R*8>: ',HB) CALL FWRITE(6,'Enter s e p a r t a t i o n f a c t o r : ') CALL FREAD(5,'R*8: ',EX) CALL FWRITE(6,'<R*8>: ',EX) CALL FWRITE(6,'Enter R.A. d i r e c t i o n s p a c i n g ( a r c m i n s ) : ') CALL FREAD(5,'R*8: ',DRA) CALL FWRITE(6,'<R*8>: ',DRA) CALL FWRITE(6,'Enter Dec. d i r e c t i o n s p a c i n g ( a r c m i n s ) : ') CALL FREAD(5,'R*8: ',DDE) CALL FWRITE(6,'<R*8>: ',DDE) 1  1  C C C  INITIALIZE CONSTANTS PI=3.141592654D0 RAS=PI DES=PI*(SDC+(SAM+SAS/60.DO)/60.DO)/180.DO RT=-PI*R0T/180.DO IR=DRA*Pl/(10800.D0*DCOS(DES)) ID=DDE*PI/10800.D0 SRA=PI-32.D0*IR SDE=DES-33.D0*ID N=64  C C INITIALIZE ARRAYS C CALL COEFS1 CALL COEFS2(EX,RT,DES) H=HB*PI/648000.DO CALL CONVOL(lD,H,WD,HD,G) C C GENERATE BEAM MODEL C 68  L i s t i n g o f ASPG:COBRA.FAST a t 16:14:23 o n AUG 28, 1986 f o r C C i d = V R S S 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 1 02 103 104 1 05 106 107 108 109 110 111 112 113 114 115 116  DO 10 J = 1 , 6 4 DE=SDE+ID*DFLOAT(J) DO 2 0 1 = 1 , 6 4 RA=SRA+IR*DFLOAT(I) C A L L CENTBM(RA,DE,BM) BEAM(I,J)=SNGL(BM) 20 CONTINUE 10 C O N T I N U E  C C CONVOLVE BEAM MODEL WITH T H E G A U S S I A N C PK=0. DO 3 0 1=1,64 DO 34 J = 1 , 6 4 C V B E A M ( I , J ) = 0. DO 3 6 K=1,WD D=J-HD+K-1 I F ( ( D . L T . 1 ) . O R . ( D . G T . 6 4 ) ) GOTO 36 CVBEAM(I,J)=CVBEAM(I,J)+G(K)*BEAM(I,D) 36 CONTINUE RB=ABS(CVBEAM(I,J)) I F ( R B . L T . P K ) GOTO 34 PK=RB 34 C O N T I N U E 30 CONTINUE C C P U T NORMALIZED BEAM I N OUTPUT ARRAY C DO 4 0 J = 1 , 6 4 DO 50 1 = 1 , 6 4 K=(J-1)*64+I OUT(K)=CVBEAM(65-1,65-J)/PK 50 C O N T I N U E 40 CONTINUE C C W R I T E G R I D S T Y P E H E A D E R T O U N I T 11 C W R I T E G R I D S T Y P E G R I D T O U N I T 12 C WRITE(11,100)N,N,N,SRA,SDE,IR,ID CALL WFILE(12,0,OUT,0,4096) 100 F O R M A T ( 3 I 6 , 4 F 1 0 . 6 ) STOP •END C 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 C A T A P O I N T I N MODEL COORDINATES. C SUBROUTINE BEAM(S,BM) COMPLEX*16 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 REAL*8 R0(40),KS(40),X(2),Y(2),S(2),BM,DREAL,L INTEGER*4 J,C,M,NSTP,K COMMON / A B L O C K / R 0 , K S , E H , X , Y , L , N S T P C C I N I T I A L I Z E ARRAYS C DO 4 0 M=1,2 DO 4 0 C = 1 , 2 I(M,C)=DCMPLX(0.DO,0.DO) 69  L i s t i n g of ASPG:COBRA.FAST at 16:14:23 on AUG 28, 1986 f o r CCid=VRSS 117 118  119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 1 39 140 141 142 143 144 145 146 147 148 149 1 50 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174  40 CONTINUE C C CALCULATE INTEGRALS NUMERICALLY BY SIMPSON'S RULE. C C=2 K=NSTP-1 DO 20 J=1 ,K C=3-C CALL INTEG(S,J,B) DO 20 M=1,2 I(M,C)=I(M,C)+B(M) 20 CONTINUE C=NSTP CALL INTEG(S,C,B) DO 30 M=1,2 V(M)=B(M)+4.D0*I(M,1)+2.D0*I(M,2) 30 CONTINUE C C FIND DIFFERENTIAL BEAM INTENSITY C BM=DREAL(V(1)*DCONJG(V(1))-V(2)*DCONJG(V(2))) RETURN END C C THIS SUBROUTINE SEARCHES THE BEAM TO FIND C THE MAXIMUM AND MINIMUM PEAK POINTS. C SUBROUTINE SEARCH(SC,SS) REAL*8 SC(2),SS(2),B(5),AN(2),ST,BM,SN REAL*8 D(2,5)/0.DO,0.DO,1.DO,0.DO,0.DO, + 1.DO,-1.DO,O.DO,O.DO,-1.DO/ INTEGER*4 M,I,J,JM SN=-1.D0 DO 10 M=1 ,2 SN=-SN AN(1)=SC(M) AN(2)=SS(M) CALL BEAM(AN,B(1)) B(1)=SN*B(1) DO 20 1=3,9 ST=(.1D0)**(I) 25 JM=1 BM=0.D0 DO 30 J=2,5 AN(1)=SC(M)+D(1,J)*ST AN(2)=SS(M)+D(2,J)*ST CALL BEAM(AN,B(J)) B(J)=SN*B(J) IF ((B(J)-B(1)).LT.BM) GOTO 30 JM=J BM=B(J)-B(1) 30 CONTINUE IF (JM.EQ.1) GOTO 20 SC(M)=SC(M)+D(1,JM)*ST SS(M)=SS(M)+D(2,JM)*ST B(1)=B(JM) GOTO 25 20 CONTINUE 70  L i s t i n g of ASPG:COBRA.FAST at 16:14:23 on AUG 28, 1986 f o r CCid=VRSS 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 1 99 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 ' 230 231 232  10  CONTINUE RETURN END  C C THIS SUBROUTINE CALCULATES THE COMPLEX DATA COEFICIENTS C FROM THE FEED ILLUMINATION DATA. READ FROM UNIT 2. C SUBROUTINE COEFS1 C0MPLEX*16 CD(2,4,40),EH(2,10,40),CDEXP,CANG,DCMPLX COMPLEX*16 CF3,CF4,CF7,CF8 REAL*8 RAS,RO(40),KS(40),X(2),Y(2),X0,YO,Z0,SP,CP REAL*8 PD(4,40),AD(2,4,40),F,A,PI,P2,P4 REAL*8 SH,R,ANG,L,SPH,DPH,CPH,S2H,DARSIN,DCOS REAL*8 DSIN,DSQRT,F1,F2,F3,F4,F5,F6,F7,F8,F9,F10 INTEGER*4 NSTP,J,I,M COMMON /ABLOCK/ R0,KS,EH,X,Y,L,NSTP COMMON /BBLOCK/ X0,Y0,Z0,SP,CP,RAS C C READS UNIT 2 FOR THE FEED ILLUMINATION DATA. C READ(2,150) NSTP,L READ(2,200)((PD(J,I),1=1,NSTP),J=1,4) READ(2,200)(((AD(M,J,I),I=1,NSTP),J=1,4),M=1,2) . 150 F0RMAT(I4,F6.3) 200 FORMAT(10D17. 1 0) C C INTIALIZE CONSTANTS C F=38.735D0 A=45.72D0 . PI=3.141592654D0 P2=PI/2.D0 P4=P2/2.D0 SH=(A-L)/DFLOAT(NSTP) C C FILL ARRAYS TO BE USED BY SUBROUTINE INTEG. C DO 5 1=1,NSTP R=SH*DFLOAT(I)+L R0(I)=(R*R+4.D0*F*F)/(4.D0*F) DO 5 J=1,4 ANG=PD(J,I) CANG=DCMPLX(0.D0,ANG) DO 5 M=1,2 CD(M,J,I)=AD(M,J,I)*CDEXP(CANG) 5 CONTINUE DO 50 1=1,NSTP R=SH*DFLOAT(I)+L SPH=L/R DPH=DARSIN(SPH) CPH=DSQRT((1.D0+SPH)*(1.D0-SPH)) S2H=SPH*SPH F1=P2-DPH+SPH*CPH F2=P2-DPH-SPH*CPH F3=4.DO*CPH*(2.DO+S2H)/3.D0 F4=8.D0/3.D0-4.D0*SPH*(1.D0-S2H/3.DO) F5=P4-DPH/2.D0+SPH*CPH*(.5D0+S2H) F6=DPH/2.D0-P4+SPH*CPH*(1.5D0+S2H) 71  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 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290  F7=CPH*(8.D0+S2H*(4.D0+48.D0*S2H))/l5.D0 F8 = SPH*(4.D0-S2H*(20.D0/3.D0-16.D0*S2H/5.D0))-8.DO/1 F9=2.D0*SPH*S2H*CPH*(4.D0*S2H-1.D0)/3.D0 F10=SPH*CPH*((14.D0-8.D0*S2H)*S2H/3.D0-2.D0) CF3=DCMPLX(0-.D0,F3) CF4=DCMPLX(0.D0,F4) CF7=DCMPLX(0.D0,F7) CF8=DCMPLX(0.D0,F8) DO 52 M=1,2 EH(M,1,1)=(CD(M,1,I)+CD(M,2,I))*F1 EH(M,2,I)=(CD(M,3,I)+CD(M,4,I))*F2 EH(M,3,I)=(CD(M,1,1)-CD(M,2,I))*CF3 EH(M,4,I)=(CD(M,3,I)-CD(M,4,I))*CF4 EH(M,5,I)=(CD(M,1,I)+CD(M,2,1))*F5 EH(M,6,I)=(CD(M,3,I)+CD(M,4,I))*F6 EH(M,7,I)=(CD(M,1,1)-CD(M,2,I))*CF7 EH(M,8,I)=(CD(M,3,I)-CD(M,4,I))*CF8 EH(M,9,I)=(CD(M,1,1)+CD(M,2,I))*F9 EH(M,10,1)=(CD(M,3,I)+CD(M,4,I))*F10 52 CONTINUE 50 CONTINUE RETURN END  C C THIS SUBROUTINE CALCULATES THE RECEIVER POSITION C AND THEN DETERMINES THE BEAM CENTER. C SUBROUTINE COEFS2(EX,RT,DES) COMPLEX*16 EH(2,10,40) REAL*8 W,L,R0(40),KS(40),X(2),Y(2),X0,Y0,Z0,SP,CP REAL*8 SH,DFLOAT,R,SC(2),SS(2),C(2),DSQRT,R1,RT REAL*8 RAS,PI,K,A,DES,EX,Z INTEGER*4 NSTP,I COMMON /ABLOCK/ R0,KS,EH,X,Y,L,NSTP COMMON /BBLOCK/ X0,Y0,Z0,SP,CP,RAS C C INITIALIZE CONSTANTS. C W=0.0632D0 Z=0.6707217897D0 PI=3.14259265D0 K=2.D0*PI/W A=45.72D0 SH=(A-L)/DFLOAT(NSTP) C C CALCULATE RECEIVER POSITIONS. C X(1)=-.047625D0*EX*DCOS(RT)-.4220407119D0*(Z-DES) Y(1)=-.047625D0*EX*DSIN(RT)-.022D0 X(2)=+.047625D0*EX*DCOS(RT)-.4220407119D0*(Z-DES) Y(2)=+.047 625D0*EX*DSIN(RT)-.022D0 SP=DSIN(DES) CP=DCOS(DES) DO 10 1=1,NSTP R=SH*DFLOAT(l)+L KS(I)=K*R/R0(I) 10 CONTINUE CALL INIT(SC,SS) 72  5.DO  Listing 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348  of ASPG:COBRA.FAST at 16:14:23 on AUG 28, 1986 f o r CCid=VRSS CALL SEARCH(SC,SS) DO 20 1=1,2 C(I)=DSQRT(1.DO-SC(I)*SC(I)-SS(I)*SS(I)) 20 CONTINUE R1=((SC(1)+SC(2))**2+(SS(1)+SS(2))**2+(C(1)+C(2))**2) R1=DSQRT(R1) X0=(SC(1)+SC(2))/R1 Y0=(SS(1)+SS(2))/R1 Z0=(C(1)+C(2))/R1 RETURN END  C C THIS SUBROUTINE CALCULATES THE BEAM IN CELESTIAL C SPHERE COORDINATES C SUBROUTINE CENTBM(RA,DE,B) REAL*8 RA,DE,B,ZRA,CZ,SZ,X0,Y0,Z0,SP,CP,S(2),BT REAL*8 DARSIN,DCOS,DSIN,RAS COMMON /BBLOCK/ X0,Y0,Z0,SP,CP,RAS ZRA=RA+DARSIN(Y0/DCOS(DE)) CZ=Z0*DCOS(DE)*DCOS(ZRA-RA)+X0*DSIN(DE) CZ=CZ/((DCOS(DE)*DCOS(ZRA-RA))* * 2 +DSIN(DE)* * 2) SZ=DSIN(DE)*CZ-X0 SZ=SZ/(DCOS(DE)*DCOS(ZRA-RA)) S(1)=SP*CZ-SZ*CP*DCOS(ZRA-RAS) S(2)=CP*DSIN(ZRA-RAS) CALL BEAM(S,BT) B=BT RETURN END C C THIS SUBROUTINE CALCULATES THE BEAM INTEGRAND TO BE C USED BY BEAM. IT IS A SUM OF BESSEL FUNCTIONS. C SUBROUTINE INTEG(S,N,B) COMPLEX*16 EH(2,10,40),B(2) REAL*8 JB(5),R0(40),KS(40),S(2),X(2),Y(2),U,V REAL*8 W,R,DSQRT,L INTEGER*4 NC,N,M,NSTP COMMON /ABLOCK/ R0,KS,EH,X,Y,L,NSTP DO 10 M=1,2 U=R0(N)*S(1)+X(M) V=R0(N)*S(2)+Y(M) R=DSQRT(U*U+V*V) W=KS(N)*R CALL DBSJIN(W,5,JB,NC) B(M)=(EH(M,1,N)+EH(M,2,N))*JB(1) B(M)=B(M)+(EH(M,3,N)*V+EH(M,4,N)*U)*JB(2)/R B(M)=B(M)+(EH(M,5,N)+EH(M,6,N))*(U*U-V*V)*JB(3)/(R*R) B(M)=B(M)+(EH(M,7,N)*V*(3.D0*U*U-V*V)+ + EH(M,8,N)*U*(U*U-3.D0*V*V))*JB(4)/(R*R*R) B(M)=B(M)+(EH(M,9,N)+EH(M,10,N))* + (V**4-6.D0*V*V*U*U+U**4)*JB(5)/(R**4) B(M)=KS(N)*B(M) 10 CONTINUE RETURN END C 73  Listing 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406  o f ASPG:COBRA.FAST a t 16:14:23 on AUG 2 8 , 1986 f o r CCid=VRSS C THIS SUBROUTINE GUESSES THE VALUES FOR THE MAXIMUM C AND MINIMUM BEAM PEAK LOCATIONS TO BE USED BY SEARCH. C THE SUBROUTINE USES MODEL COORDINATES. C SUBROUTINE I N I T ( S C , S S ) COMPLEX*16 E H ( 2 , 1 0 , 4 0 ) REAL*8 X ( 2 ) , Y ( 2 ) , S C ( 2 ) , S S ( 2 ) , R 0 ( 4 0 ) , K S ( 4 0 ) REAL*8 K,KM,CDABS,L INTEGER*4 I,IM,NSTP,M COMMON /ABLOCK/ R0,KS,EH,X,Y,L,NSTP DO 20 M=1,2 KM=0.D0 DO 10 1=1,NSTP K=CDABS(EH(M,1,1)+EH(M,2,I))*KS(l) I F (K.LT.KM) GOTO 10 IM=I KM=K 10 CONTINUE SC(M)=-X(M)/R0(IM) SS(M)=-Y(M)/R0(IM) 20 CONTINUE RETURN END C C THIS SUBROUTINE GENERATES THE ARRAY TO BE USED C FOR THE GAUSSIAN CONVOLUTION. C SUBROUTINE CONVOL(ID,H,WD,HD,G) REAL*8 ID,H REAL*4 G ( 6 5 ) , A , X , M INTEGER*4 WD,HD,I,WE A=4.*ALOG(2.)/(SNGL(H)**2) X=SQRT(ALOG(1000.)/A) HD=IFIX(X/SNGL(ID)) WD=2*HD+1 DO 10 1=1,WD X=FLOAT(l-HD-1)*SNGL(ID). G(I)=EXP(-A*X**2) 10 CONTINUE M=2. WE=WD-1 DO 20 1=2,WE M=6.-M G(I)=M*G(I) 20 CONTINUE RETURN END C C THIS SUBROUTINE WRITES DATA TO UNIT I U I N GRIDS FORMAT. C SUBROUTINE W F I L E ( I U , I D , X , J B , L N ) INTEGER*2 LENGTH DIMENSION X ( L N ) DATA MOD/2/ LENGTH=LN*4 LINE=(JB/32+1)*1000 CALL WRITE(X,LENGTH,MOD,LINE,IU,£100) RETURN 74  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 407 408 409 410  100 101  WRITE(6,101) FORMAT('I/O e r r o r occured i n WFILE') STOP END  75  

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