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Radio continuum observations of the supernova remnant G109,1-1.0 Braun, Robert 1981

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RADIO CONTINUUM OBSERVATIONS OF THE SUPERNOVA REMNANT G109.1-1.0  by  • ROBERT BRAUN B.Sc,  The U n i v e r s i t y  of B r i t i s h Columbia, 1980  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE  in THE DEPARTMENT OF PHYSICS  We accept t h i s  t h e s i s as conforming  to the r e q u i r e d  standard  THE UNIVERSITY OF BRITISH COLUMBIA September 1981  © Robert Braun, 1981  In p r e s e n t i n g requirements  this thesis f o r an  of  British  it  freely available  agree for  that  for  that  for reference  and  study.  I  for extensive be  her  copying or shall  DE-6  (2/79)  by  the  publication  not  be  of  further this  this  It  thesis my  is  thesis  a l l o w e d w i t h o u t my  Columbia  make  head o f  representatives.  of  The U n i v e r s i t y o f B r i t i s h 2075 Wesbrook P l a c e V a n c o u v e r , Canada V6T 1W5  copying of  granted  the  University shall  permission.  Department  the  Library  h i s or  f i n a n c i a l gain  degree at the  p u r p o s e s may by  f u l f i l m e n t of  I agree that  permission  department or understood  advanced  Columbia,  scholarly  in partial  written  i i  Abstract The  supernova  remnant  G109.1-1.0  d i f f e r e n t i a l l y at 6 cm wavelength telescope  of  6 cm d i f f e r e n t i a l the  the  91  observed m  transit  the NRAO at Green Bank, West V i r g i n i a .  development of a complete  to  using  was  d e c l i n a t i o n dependant model of the  response, the o b s e r v a t i o n s  equivalent  After  total  were  restored  i n t e n s i t y o b s e r v a t i o n s u s i n g the  method d e s c r i b e d by D.T. Emerson et a l (1979) as w e l l as the maximum entropy method of Daniell  1978).  image  Both  reconstruction  approaches  gave  (Gull  successful  r e c o n s t r u c t i o n s of the f i e l d at s i m i l a r computational The maximum entropy method i s shown  to  and  provide  costs.  effective  r e c o n s t r u c t i o n even under the s p a r s e r sampling c o n d i t i o n s of the  Gregory-Taylor  application  of  differentially  this  (1981) work  recorded  survey; to survey  o b s e r v a t i o n s of G109.1-1.0 i n d i c a t e structure  which  i s similar  allowing  the  the  majority  data  base.  an  extended  direct  of The  the 6  cm  half-disc  i n both shape and extent to the  X-ray emission (Gregory and Fahlman 1980) from the remnant. An unusual f e a t u r e has a l s o become .apparent  which  extends  from the northern edge of the remnant f o r 35 arcmin  radially  away from the center t o the north-west. circular  shape  The g e n e r a l l y  semi-  of the remnant, as i n d i c a t e d by both the X-  ray and r a d i o o b s e r v a t i o n s , can be understood i n t e r a c t i o n with an a d j a c e n t molecular c l o u d .  i n terms of an  iii  Table of Contents  Abstract  i i  Table of Contents  i i i  L i s t of F i g u r e s and I l l u s t r a t i o n s  iv  Acknowledgements  vi  1.  Introduction  ....1  2.  Observations  3  3.  Calibrations  9  4.  Reduction  20  4.1 B a s e l i n e Removal  20  4.2 G r i d d i n g the Data  21  4.3 Theory of the Emerson Method  23  4.4 Implementation of the Emerson Method  31  4.5 Emerson-Clean  34  Hybrid  4.6 Theory of the Maximum Entropy Method  35  4.7 Implementation of the Maximum Entropy Method  37-  5.  Comparison  6.  Other O b s e r v a t i o n s of the F i e l d  61  7.  Summary and C o n c l u s i o n s  67  References  of R e s u l t s  49  70  iv  L i s t of F i g u r e s and I l l u s t r a t i o n s  Figure 1)  Feed and Receiver  System Schematic  6  2)  Scanning Sequence and Beam Geometry  7  3)  L o c a t i o n of a l l A v a i l a b l e Data Scans  8  4)  RA and Dec P o i n t i n g C o r r e c t i o n s  14  5)  Beam S e p a r a t i o n ,  15  6)  T y p i c a l Model F i t s t o Half P r o f i l e s  7)  Beam Parameters  .17  8)  Beam Model Output  19  9)  Raw D i f f e r e n t i a l Scans  42  10)  I l l u s t r a t i o n of Data G r i d d i n g  43  11)  I n t e g r a t i o n Path f o r Transform I n v e r s i o n  44  12)  Emerson Method C o n v o l u t i o n  45  13)  A r t i f a c t s of R e s t o r a t i o n  46  14)  Convolved Beam Model  47  15)  L o c a t i o n of Data Scans  16)  Emerson Method Restored Map  53  17)  C r o s s - s e c t i o n s through S152 i n Emerson Map  54  18)  Maximum Entropy Restored Map  55  19)  C r o s s - s e c t i o n s through S152 i n Maximum Entropy Map..56  20)  Maximum Entropy Map smoothed t o r e s o l u t i o n of  R e l a t i v e Gain and S e n s i t i v i t y  16  Function  f o r Sparse Data Test  48  Emerson  Method Map  57  21)  Maximum Entropy Restored Map from Sparse Data  58  22)  Cross-sections  through  Maximum Entropy Map  S152  in  Sparse  Data 59  V  23)  Sparse  Data  Maximum  Entropy  Map  smoothed  r e s o l u t i o n of Emerson Method Map 24)  Maximum  Entropy  Map  60  smoothed  to  10  Resolution 25)  Overlay  arcmin 64  of  Maximum  Entropy  Map  and  Observations 26)  to  Cross-sections Entropy Map  X-ray 65  through  Central  Peak  in  Maximum 66  vi  Acknowledgements I. would f i r s t  of a l l l i k e t o thank my s u p e r v i s o r ,  P.C. Gregory  for h i s assistance  undertaking.  Of  discussions A.R. Bank  with  Taylor.  M.A. P o t t s  Dr.  help  as  guidance  well,  were  W.L.H. Shuter, Dr.  Thanks a l s o go t o the s t a f f  for their  the observing  great  and  of  Finally,  f o r her a s s i s t a n c e  I  would  i n the software  this  stimulating  S.F. G u l l  h o s p i t a l i t y and a s s i s t a n c e  program.  in  Dr.  NRAO,  and Green  i n c a r r y i n g out like  to  thank  development.  1  1. I n t r o d u c t i o n The v a r i a b l e r a d i o source survey begun by and  A.R.  T a y l o r i n 1977,  while s t i l l  P.C.  under way,  Gregory  has a l r e a d y  begun to make s i g n i f i c a n t c o n t r i b u t i o n s to our knowledge the was  radio  sky  (Gregory and T a y l o r 1981).  designed to  explore  e x t e n s i v e sky coverage recognition brightness  source  of  which  remnants.  variability,  One  recorded  and Fahlman (1980) vicinity.  are  probably  associated  to  6 cm  emission  in  survey scans prompted  obtain  X-ray  The very unusual X-ray  map  shell.  I t was  with  such source, G109.1-1.0, showing a  observations  the  Gregory in  its  s t r u c t u r e showed evidence  of a j e t - l i k e extension from a compact source w i t h i n a circular  its  f e a t u r e s of low s u r f a c e  s h e l l - l i k e s t r u c t u r e and c e n t r a l l y peaked differentially  the survey  and high s e n s i t i v i t y have allowed the  of a number of extended many  supernova  compact  While  decided to produce  a total  semi-  intensity  of the r e g i o n surrounding G109.1-1.0 at a wavelength  6 cm  to  of  allow a d e t a i l e d comparison  of the X-ray and  of radio  morphology. Total 6 cm due  i n t e n s i t y measurements  made  at  wavelengths  and s h o r t e r can be s u b j e c t to c o n s i d e r a b l e degradation to atmospheric  method  of  water  beam-switching  vapour.  For  this  reason,  atmospheric  emission when mapping sources of small angular, s i z e the  beam  spacing.  the  has long been employed i n s i n g l e  d i s h o b s e r v a t i o n s to c a n c e l the f l u c t u a t i o n s i n  to  of  Limits  relative  on u s e f u l beam spacing are  imposed however, by the requirement  that there be  sufficient  2  n e a r - f i e l d beam o v e r l a p to p r o v i d e e f f e c t i v e The  method  since  it  was was  differential  not a p p l i e d to the mapping of l a r g e thought  response  that was  the not  which  allows  the  practical recently  analytic  e q u i v a l e n t s i n g l e beam response s e v e r a l beam spacings.  unravelling  a  However, Emerson et a l (1979) have method  cancellation.  In  of  deconvolution  illustrated  reconstruction  a  of the  f o r sources which extend f o r  addition,  problems  the  undertaking.  the  maximum  entropy  method of image r e c o n s t r u c t i o n has been s u c c e s s f u l l y to  fields  ( G u l l and D a n i e l l ,  approach o f f e r s an a l t e r n a t e method f o r  applied  1978).  This  the r e c o n s t r u c t i o n  of d i f f e r e n t i a l o b s e r v a t i o n s . With was  both of these r e d u c t i o n p o s s i b i l i t i e s at hand, i t  decided  measurement  to  take  technique  advantage in  mapping  of  the  undertaken  fact  that  the  i n t h i s manner.  differential  the source G109.1-1.0.  A d d i t i o n a l m o t i v a t i o n f o r t h i s method from  the  of  measurement  came  e n t i r e Gregory-Taylor survey The development of an  was  effective  method of r e s t o r i n g d i f f e r e n t i a l measurements made under the same  conditions  that  technique  possible  as to  the t o t a l  the this  survey would allow a p p l i c a t i o n of much  larger  data  base,  i n t e n s i t y mapping of the e n t i r e  plane between 1 =40° to 220° and b =-2° to 2° with 3 resolution  making galactic arcmin  and an improvement i n s e n s i t i v i t y of at l e a s t  times over p r e s e n t l y a v a i l a b l e r e s u l t s f o r the the survey a r e a .  majority  50 of  3  2. Observations Observations of  were  c a r r i e d out i n August and September  1980 using the 91 meter  transit  t e l e s c o p e of the N a t i o n a l  Radio Astronomy  Observatory at Green Bank,  The  r e c e i v e r c o n f i g u r a t i o n used was  feed  and  West  that used f o r the survey of Gregory and T a y l o r Taylor  i d e n t i c a l to (Gregory  1981) and i s shown s c h e m a t i c a l l y i n F i g u r e  The  feed  system  consists  of  a  rotatable  mount  centered  on  projected  beams  of  separated  by  amplifier  receivers  about  7  about  arcmin.  were  used;  3  of  arcmin  the  and  are  cooled  parametric  each  with a t o t a l  system  The  two  were switched i n a n t i p h a s e between the horns at a  frequency of 50 Hz; p r o v i d i n g two of  6 cm  Two  temperature of 70 K and 3 db bandwidth of 580 MHz. receivers  fixed  the t e l e s c o p e a x i s of  At a wavelength  FWHM  1.  These are  symmetry at the prime focus. have  and  two horns s e n s i t i v e to  r i g h t hand c i r c u l a r l y p o l a r i z e d r a d i a t i o n . to  Virginia.  independent  measurements  the d i f f e r e n t i a l output of the feeds. In  order  to  G109.1-1.0, a scan region  between  map  the  field  sequence  RA( 1950)  was  22h 54m  surronding designed  the remnant  to  to 23h 04m  sample  the  and D e c d 950)  58.0° to 59.6°.  To o b t a i n maximum sky coverage i n  days  f o r these o b s e r v a t i o n s , a d e c l i n a t i o n  drive  r a t e of 120 arcmin/min and an i n c l i n a t i o n of the feeds  away  from  available  the  scan  observations southbound  direction  consisted scans  across  by  of  11°  were  alternating the  field,  used.  the  ten  Each day's  northbound  staggered to g i v e  and 1.6  4  arcmin  s p a c i n g of p a r a l l e l  scans a f t e r  10 days of  mapping.  A scanning sequence and the beam geometry are i l l u s t r a t e d i n F i g u r e 2. The  i n c l i n a t i o n of the feeds from the scan d i r e c t i o n by  11° p r o v i d e s about  1.5 atcmin  s e p a r a t i o n of the beam c e n t e r s  p e r p e n d i c u l a r to the scan d i r e c t i o n . analysis  somewhat,  e f f e c t i v e coverage 1/2 FWHM  and  this  technique  c o m p l i c a t i n g the  provides  increased  area f o r scan spacings g r e a t e r than about  corresponds  Gregory-Taylor  While  exactly  to  that  used  i n the  survey.  Immediately preceding and f o l l o w i n g each day's 8 minute scan  sequence the t e l e s c o p e was h e l d s t a t i o n a r y  firing  of  calibration  a  stable of  noise  the  tube  data.  to  f o r the 12s  provide  temperature  The rms r e c e i v e r gain v a r i a t i o n  was 5%. A sample time of 0.2 seconds was chosen as used Gregory-Taylor  survey.  the 3 db c u t o f f appropriate  The frequency,  f =2.4 Hz was used as  f o r the p o s t - d e t e c t i o n low pass  integration  time  1/(2f(1.57))=0.13 sec ( T i u r i  At  i n the  is  filter.  then  The  given  by  1966) a l l o w i n g one t o c a l c u l a t e  the t h e o r e t i c a l rms r e c e i v e r n o i s e f l u c t u a t i o n s  (Tiuri  1966)  from AT  r m s  =K • T s  s y s  /VSuSt:  (2.1)  given the constant K =2 f o r a beam switched r e c e i v e r s  This  gives  f a c t o r of output  T  r m s  =l6  mK,  by combining  system.  which can be f u r t h e r reduced the  two  receiver  outputs.  by a The  of both r e c e i v e r s together with p o i n t i n g i n f o r m a t i o n  5  was w r i t t e n onto magnetic tape a t the t e l e s c o p e s i t e .  These  q u a n t i t i e s were a l s o monitored i n  strip  chart  recorder  real  time  and the p o s i t i o n a l d i s p l a y .  suspect due  to  conditions  or  faulty  telescope  interference  via a  Data which was  tracking,  was  noted  poor  at  weather  this  time.  U n f o r t u n a t e l y some of the p r e c e d i n g reasons r e s u l t e d i n the loss  of  one  day's complete scanning sequence as w e l l as a  number of a d d i t i o n a l scans. scans i s i l l u s t r a t e d The continuum  data  The l o c a t i o n of a v a i l a b l e  i n F i g u r e 3.  received  initial  r e d u c t i o n program  Virginia.  This  program  processing  CONDARE uses  3  was  the t e l e s c o p e written  University  onto of  British  the noise  tube f i r i n g s t o  After visual  tape  data  recorded  The program output  for transport  to  the  Columbia where the remainder of the  data p r o c e s s i n g was c a r r i e d computer.  the NRAO  Charlottesville,  was not on t r a c k .  magnetic  by  at  provide temperature c a l i b r a t i o n and d i s c a r d s while  data  out  using  an  AMDAHL  470 V8  i n s p e c t i o n , the two r e c e i v e r outputs  were averaged f o r the map scans.  B  V  V  1  TO F R O N T END  'B' PARAMP  SWITCHES A  'A' PARAMP  NOISE  CAL.  CAL  TRANSISTOR  AMP  AMP  4.5 to 5.1 GHz  A.5 to 5.1 GHz  FILTER  FILTER  A  SWITCH  SWITCH DRIVE OSC.  F R O N T END SWITCHES  NOISE TRANSISTOR  TO  MIXER  MIXER  I F AMP  OSC. IF LOAD  SQ. LAW DETECT.  DETECT.  LOW PASS F I LTER B' OUT  AMP  COLD  SQ. LAW  SYNC. DETECT.  DRIVE  x2  L . 0. AMP  L.O.  SYNC. DETECT. LOW PASS F I LTER A  OUT  7  54' 0  RIGHT  A S C E N S I O N ( H R S . . M I N S . . S E C S . )  F i g u r e 2 . Scanning Sequence and Beam Geometry. One day's complete scanning sequence, showing alignment.  the beam  8  Figure 3. L o c a t i o n of a l l A v a i l a b l e Data Scans. T y p i c a l spacing of p a r a l l e l scans i s 1.6 arcmin.  9  3. C a l i b r a t i o n s Calibration approximately  scans  were  45 unresolved  obtained  sources  through  a  set of  of known f l u x d e n s i t y at  d e c l i n a t i o n s between -10° and 70° i n a d d i t i o n t o the mapping scans.  These  were  parameters necessary carried  out  in  obtained  to  determine  for reduction  conjunction  of  with  the  the  the t e l e s c o p e data  and  survey  were  program of  Gregory and T a y l o r . Drift direction set  to  scans  with  (type  1) were f i r s t  determine  RA  d e c l i n a t i o n dependence.  the  beams  aligned  obtained  pointing  with  the  scan  f o r the c a l i b r a t i o n  corrections  and  their  (See F i g u r e 4.) Scans d r i v e n to the  north or south at 60 arcmin/min with RA p o i n t i n g c o r r e c t i o n s applied  and  beams a l i g n e d with the scan d i r e c t i o n  (type 2)  were next c a r r i e d out t o determine Dec p o i n t i n g c o r r e c t i o n s . (See F i g u r e 5.) pointing  These type 2 scans were repeated  corrections  applied  (type  3)  s e p a r a t i o n of the two beams as a f u n c t i o n (See and  F i g u r e 5.)  aligned  passing and  of  both  obtain  the  declination.  F i n a l l y using the Dec p o i n t i n g c o r r e c t i o n s  beam s e p a r a t i o n data, d r i f t  beams  to  with  perpendicular  scans were obtained with the to  the  scan  direction  through only the upper or lower beam center  5 ) , to o b t a i n beam shape i n f o r m a t i o n i n t h i s Both the maximum entropy  techniques  and  Emerson  and  (type 4  direction.  data  r e q u i r e a knowledge'of the i n s t r u m e n t a l  reduction profile.  To  determine the beam shape and i t s d e c l i n a t i o n dependence,  one  c o u l d c o n s i d e r the complete d i f f e r e n t i a l  mapping  of  a  10  large  number  of  unresolved sources.  r e q u i r e d f o r such an undertaking however.  The  method  is  adopted  The o b s e r v a t i o n time somewhat  here was t o make use of the  i n f o r m a t i o n p r o v i d e d by the approximately through  prohibitive,  perpendicular cuts  the two beams p r o v i d e d by the scans of type 3, 4 and  5 above.  The p r o f i l e of each beam i n the  directions  could  N,  S,  E  and  W  be combined with beam s e p a r a t i o n and gain  r a t i o data t o give an approximate  model  of  the  dual-beam  response p a t t e r n . A f o r t r a n computer program to c a r r y out a l e a s t fit  to  a  developed  particular  p r o f i l e model had a l r e a d y been  by A.R. T a y l o r , who k i n d l y allowed i t s use f o r the  following  work.  asymmetry  about  fit  beam  squares  Since their  the  two  beams  show  significant  peak v a l u e s , the model f u n c t i o n was  s e p a r a t e l y to e i t h e r h a l f of  each  beam  profile.  The  model f u n c t i o n i s a m o d i f i e d gaussian of the form f(x)=exp{-2.7726[1+c •(x/H)•(x/H-1/2)](x/H) }  (3.1)  2  in which H i s the FWHM which corresponds in  the same u n i t s as x, and c i s  the  the  to the h a l f  parameter  (affecting  r e l a t i v e width of the f u n c t i o n ' s c e n t r a l p l a t e a u ) which  a l l o w s a b e t t e r f i t than a  simple  takes  with  raw  data  scans  gaussian. only  The  initial  program  temperature  c a l i b r a t i o n , e s t a b l i s h e s s u i t a b l e baseline regions criterion of  profile  of  a l l o w i n g d i f f e r e n c e s i n adjacent data  only 20 mK) and removes a l i n e a r b a s e l i n e .  and/or  negative  quadratically  (by the  peaks  are  first  The  normalized  i n t e r p o l a t e d peak temperature.  Each  samples positive  to  their  side  of  11  the  peak  is  then s e p a r a t e l y f i t by a f u n c t i o n of the  shown i n equation 3.1.  T h i s i s done  h a l f power p o i n t , H d i r e c t l y is  then  varied  those data  to  values  first  from the data.  obtain above  by  a least  10%  of  the  fixing  The  squares  form the  parameter c  f i t using only  peak  temperature.  T y p i c a l model f i t s to h a l f p r o f i l e s are shown i n F i g u r e 6. Scans  driven  to  the  north  analysed s e p a r a t e l y to determine were  affected  differences  by  driving  and  to  the  whether the beam parameters  direction.  No  i n parameter values were observed  v a r i a t i o n of beam parameters with d e c l i n a t i o n F i g u r e 7. fit  In each case the l e a s t  i s shown that was The  function  used  in  level.  however. is  to model the 3.1  beam  c  was  found to be q u i t e down  15% l e v e l  at low  to  the  10%  was  used to extend  amplitude  equation  the model  i n the f o l l o w i n g manner. (3.2)  2  Rewriting  To  l e v e l s , a gaussian  g(x)=A-exp{-a(x/H) } 3.1  i n terms of z= x/H  and a l l o w i n g an  C gives, f(z)=C-exp{-b[1+C-Z(Z-1/2)]z }  (3.3)  2  f u n c t i o n f ( z ) reaches a value k'C  (k<l) when  k.C=C-exp{-b[1+c-z (Z -1/2)]z } 2  0  .  in  polynomial  give r i s e to o s c i l l a t i o n s of the f u n c t i o n .  matched slope (equation 3.2)  below the  or  shown  variation.  shapes  b e t t e r represent the beam response  The  The  However, f o r l a r g e arguments, negative v a l u e s of the  parameter  of  significant  squares q u a r t i c  equation  adequate i n d e s c r i b i n g the  south were  q  ln(k)=-b[1+c-z (z -1/2)]z Q  Q  0  2 Q  12  allowing a solution  for z  from the q u a r t i c  Q  equation,  -z "-z /2+z /c+ln(k)/b-c=0 3  o  (  2  o  o  A matched f u n c t i o n a l value at z=z  (3.5)  2  0  D=-b-C[4c'Z -3c•z /2 + 3  g i v e s the c o n d i t i o n ,  Q  2z ,]exp{-b[1+c•z (z -1/2)]z }  2  0  )  g i v e s the c o n d i t i o n ,  0  k-C=A'exp{-a•z } while matching s l o p e s at z=z  3 > 4  2  0  0  Q  Q  Q  =-2a'A'Z -exp{-a-z }  (3.6)  2  0  0  D i v i d i n g equation 3.6 by equation 3.5 g i v e s , D/(k.C)=-2a-z  or  0  S u b s t i t u t i o n of equation  a=-D/(2k-C•z )  (3.7)  0  3.7 i n t o equation 3.5 then g i v e s ,  A=k-C/(exp{D-z /2k.C})  (3.8)  0  Equations  3.7  and  3.8  then provide the parameters of the  gaussian extension f o r z ^ z . Q  In c o n s t r u c t i n g the two dimensional beam model, i t was assumed that the two c u t s a v a i l a b l e through each beam center (from  scans  of  type 3, 4 and 5) represented p e r p e n d i c u l a r  c u t s i n the RA and Dec d i r e c t i o n s . G109.1-1.0 rotation  the  two  provides  l5«cos(Dec)  85°.  a  Away  are  right  arcmin/min.  the Gregory-Taylor to  cuts  For the o b s e r v a t i o n s  i n c l i n e d by 83° s i n c e e a r t h  ascension  For the f u l l  drive  rate  declination  survey, the i n c l i n a t i o n v a r i e s from  m u l t i p l i c a t i v e combination  the  of  RA  and  Dec  of  range of from  76°  directions,  a  of the form,  f(z,,z )=C-exp{-b([1+c,z,(z,-1/2)]z, + 2  2  [1+c z (z -1/2)]z )}  (3.9.)  2  2  2  2  2  was assumed to apply above the 15% l e v e l , using of c , c , H 1  2  1  and H  2  a p p r o p r i a t e f o r each quadrant.  the  values  To extend  13  this  model  in  the manner d e s c r i b e d above, equation  first  transformed  to  cylindrical  coordinates ( r , e )  polar  such that, z , =r (cos©) and z = r ( s i n e ) 2  3.9 i s  giving,  f ( r , e ) = C ' e x p { - b [ r - ( r / 2 ) . (c ^ o s ^ + c z s i n © ) 2  3  3  +r"(cTcos^e+Czsin"©)]} At  any  angle  and  the parameters of the r a d i a l  © , one can determine the matching r a d i u s r gaussian  manner analogous to that employed above. parameters rather  are  than  extended  determined  f o r each  individual  at  one  beam  extension  grid  i n t e r v a l s of © location.  beam model output  other  beam  center.  The  the two beams  respect  RA  to  =  An  only from  had  constant,  a  scans  small  i t is  only  expected  i n a s i m i l a r way.  investigation  the  f e e d / d i s h response that  l e d to with such  obtained  of  with  The  with to be  present  r e c o g n i t i o n t h a t the d u a l -  alignment the  f o r which the  inclination  a p p l i c a b l e t o data c o l l e c t e d has  example  at Dec = 58.8° i s shown i n F i g u r e 8.  line joining  constant.  The  p a t t e r n was t h e r e f o r e formed by simply adding the  Since t h i s model was generated  from  these  models t y p i c a l l y c o n t r i b u t e l e s s  f u n c t i o n a l models of the i n d i v i d u a l beams. the  Q  in a  In p r a c t i c e ,  degree  rectangular  than 0.5% a t the l o c a t i o n of the dual-beam  (3.10)  feeds  i s very aligned  different with  Dec =  14  RA Pointing Corrections (Cat.-Obs.)  ~  .2F-  £  >1F-\  0 •30  Dec Pointing Corrections (Cat.-Obs.)  •20 •10 <_>  in  Q  O  -10 -20 -30  0  10  20  30 40 50 Declination ( ° )  60  70  F i g u r e 4. RA and Dec P o i n t i n g C o r r e c t i o n s . The t e l e s c o p e p o i n t i n g c o r r e c t i o n s determined from scans of types 1 and 2 (see t e x t ) through the c a l i b r a t i o n sources.  15  Beam A  0  10  20  Sensitivity  30 40 50 D e c l i n a t i o n (°)  60  70  F i g u r e 5. Beam S e p a r a t i o n , R e l a t i v e Gain and S e n s i t i v i t y . Telescope beam parameters determined from scans of type 3, 4 and 5 (see t e x t ) through the c a l i b r a t i o n s o u r c e s .  16  I  i  i  i  0.0  i  i  i  i  1.0  i  i  i  i I  2.0  3.0  ARCM1N  I 0-0  1  1  i  I  I  I  1.0  i  '  2.0  '  •  •  '  3-0  ARCMIN  F i g u r e 6. T y p i c a l Model F i t s t o H a l f P r o f i l e s . The dots i n d i c a t e the normalized data, while the s o l i d represents the model f i t t o the north and south p r o f i l e s of beam A f o r DA251.  line half  17  F i g u r e 7a. Beam Parameters. The model parameters f o r the four orthogonal from the center of beam A.  radial  cuts  18  B (North)  B(South)  Declination (°)  F i g u r e 7b. Beam Parameters. The model parameters f o r the from the center of beam B.  Declination (°)  four  orthogonal  radial  cuts  19  AN  Beam ' B '  W  Beam 'A'  F i g u r e 8. Beam Model Output. The two dimensional dual beam model at Dec=58.8° c o n s t r u c t e d from equation 3.9 and the parameters of f i g u r e 7 with a matched radial gaussian extension below the 15% l e v e l determined at 1° i n t e r v a l s . Contours a r e at±70, 50, 30, 20, 10, 7, 5, 3, 2, 1, 0.7, 0.5, 0.3, 0.2, and±0.1% of the beam A peak.  20  4.  Reduction Before meaningful  i n t e r p r e t a t i o n of the o b s e r v a t i o n s i s  p o s s i b l e , c o n s i d e r a b l e data preliminary  reduction  gridding  the  of  reduction  phases  data  are  of  is  necessary.  baseline  described  The  removal  and  These  are  below.  followed by a d e s c r i p t i o n of the methods employed to recover total  intensity  i n f o r m a t i o n from the raw d i f f e r e n t i a l  data  scans shown i n F i g u r e 9.  4.1  B a s e l i n e Removal The  r a p i d l y d r i v e n scans  operation  are  subject  used  here  to non-constant  for  seen  telescope  zenith  angle.  generally  have  sources  by  complicate the definition.  the  choice However,  two  Scans very  of  a  G109.1-1.0 was region  low  little  level  mapped.  scan  allowing  direction,  a  more  The  adds  the  calibration  extended  s t r u c t u r e to  to  for  the  baseline  structure in problem. northwest  to the edge  differential  the  spillover  f u n c t i o n of the  serious  extension  due  nature  In of  of  the  of  the  beams not a l i g n e d with  further  complication  of  both p o s i t i v e and negative responses which are not  n e c e s s a r i l y of equal Scans  a  regions  found to continue almost  being  the  the mapping of extended  measurements, e s p e c i a l l y with the two the  as  through  suitable  the g a l a c t i c plane can pose particular,  horns  mapping  baseline levels  p r i m a r i l y to v a r i a t i o n s i n the d i f f e r e n c e of radiation  the  from  the  amplitude. Gregory-Taylor  survey  which  passed  21  through  the  r e g i o n being mapped were used to a s s i s t  r e d u c t i o n at t h i s stage. degrees  centered on the g a l a c t i c  i n e x a c t l y the same above. of  a  way  baseline  measurements were made. common  the  mapping  scans  out  described  relative  profile  curvature,  a  obtained of , 2  this linear  Plotted baseline f i t s the  choice  of  A program was and  specified  allow  and unsmoothed map  the  convolving  arcmin  the  FWHM.  these  with  a  Since the smoothed  vicinity  did  not  baseline  was  s u b t r a c t e d from them.  show  significant  to the survey scans were then used  to  b a s e l i n e l e v e l in the G109.1-1.0  map  w r i t t e n to d i s p l a y each  interactive  b a s e l i n e was  by  in  which  and  were  To a s s i s t  to  low l e v e l f e a t u r e s , smoothed v e r s i o n s of the map  survey scans i n  scans.  level  of  gaussian  4.2  plane and were c a r r i e d  4  recognition  survey scans  scan  as  approximately  T h e i r g r e a t e r l e n g t h a l l o w s more a c c u r a t e d e f i n i t i o n reasonable  guide  These scans span  i n the  baseline  smoothed  map  specification.  The  then removed from both  the  smoothed  scans.  G r i d d i n g the Data Two  regularly  dimensional  data  processing  sampled r e c t a n g u l a r g r i d .  normally r e q u i r e s a  When  one's  data  been o b t a i n e d along d i r e c t i o n s other than those d e f i n i n g processing  array,  the  with noncommensurate sample s p a c i n g , the  method of p l a c i n g t h i s i n f o r m a t i o n onto a important  has  grid  c o n s i d e r a t i o n . The method which was  t h i s o p e r a t i o n i s o u t l i n e d below.  becomes  an  employed f o r  22  Consider For  each  data  locations Figure  a s e t of data point,  (Pj ,  Pj i  k  10.  +  p o i n t s Dj a t p o s i t i o n s  there /  k  p  will  jM  >  One c a n d i s t r i b u t e  be  j+1k*1  p  the data  four )  a  s  (XJ ,y  nearest  {  ). grid  illustrated in  v a l u e Dj among  these  four p o i n t s according t o the weights, Wj  =(1-Ax)(1-Ay)  kj  Hki  =AX(1-Ay)  V i i  =(1"Ax)Ay  W  W  such If  that  (W  one t h e n  jki  j-1k-1i  ) +( W  accumulates  =AxAy ) + (W  H k i  ) +(Wj.  j k+l j  a t each g r i d  )= 1  l k # 1 i  l o c a t i o n the  jk=*i (Wjki) D i  D  and  normalizes  one  obtains  relevant  a  cells  close.  which  Given calculate  jk  i jki w  with  no  resolution  data  2  cells of  2  k  i s normalized  0.5  arcmin  w  by d i v i s i o n w a  assign  c e l l w i d t h and  sample Dj , one c a n  >«i  j k i  .  were  FWHM.  2  with  jk-<iiW  values  sufficiently  a s s o c i a t e d with  2  of  s i z e and  k  T h i s i s g i v e n by, « jk=£i< jki  and  cell  f o r w h i c h Wj >0.3  of data  v a r i a n c e o f <y j  2  to  average  sample h a s come  of 3 arcmin  t h e v a r i a n c e <s \  k  weighted  one may n o t w i s h  for grids  value D j .  a  D e p e n d i n g on t h e g r i d  end, o n l y g r i d  values  the  = j :  filled  resolution,  to  instrumental  w  points.  To t h i s  assigned  data  grid  data  instrumental to  each with  )  2  also  the gridded  23  4.3 Theory of the Emerson Technique Under many circumstances  i t i s possible to a n a l y t i c a l l y  r e s t o r e a s e t of d i f f e r e n t i a l o b s e r v a t i o n s to the e q u i v a l e n t single  beam  observations,  cancellation  of  while  atmospheric  d i f f e r e n t i a l method.  benefiting  fluctuations  Consider  from  the  p r o v i d e d by the  the b r i g h t n e s s  distribution,  P due t o a t e l e s c o p e ' s s i n g l e beam response B convolved  with  the sky b r i g h t n e s s d i s t r i b u t i o n S. P=S * B If  the  dual  represented function  D,  beam by  response  the  then  (4.1)  of  the  convolution  the  dual  telescope  B * D  beam  for  response  can  be  some s u i t a b l e to  the  sky  d i s t r i b u t i o n would be 0=S * B * D Taking  the f o u r i e r transform of equations  gives  p=s«b  and  o=S'b'd  One  (4.2)  could  distribution  in  principal  then  obtain  4.1 and 4.2  the  single  beam  f u n c t i o n P from, P=F" {o/d} 1  or e q u i v a l e n t l y (and more convolving  the  dual  convenient  computationally)  by  beam o b s e r v a t i o n s with the f u n c t i o n T  given by, T=F- {1/d} 1  so that  P=0 * T  Let us assume f o r the moment that the dual beam f u n c t i o n , can be w r i t t e n as  D  24  D(x)=6(x+X /2)-A6(x-X /2) 0  for  the beam spacing k .  Now l e t us s o l v e f o r the f u n c t i o n  Q  T(x).  (4.3)  0  The f o u r i e r transform of D(x) can be w r i t t e n as d(k) = J ° e x p { - 2 j r i k x } [ 6 (x + X. /2 ) - A 6 ( x - X / 2 ) ] d x Q  G  = exp{irikX. }-A'exp{-irikX. } 0  The  0  f o u r i e r transform of T(x) would then be t(k) = l/d(k) = l / ( e x p { i r i k x  so that T(x)  0  i s given by  J e x p { 2 7 r i k x } / ( e x p { i r i k X . }-A• exp{-trikX. })  T(x)=  The  }-A^exp{-irikk })  c  0  dk  0  i n t e g r a n d has simple p o l e s i n the complex k plane where exp{irikX. }=A«exp{-jrikX. } 0  0  frikX. = l n ( A ) - i r i k X  or  0  k=-iln(A)/(2irX  giving  When  |k|  approaches  expression  0  0  )+n/X  0  infinity,  f o r T(x) above  the  the  that  f o r x>0 the i n t e g r a l of equation this  contour  t r u e i n the lower given  by equation  integrand zero.  of r a d i u s R  (or  along  half  n=0,±1,±2,...  approaches  c o n s i d e r s a s e m i c i r c u l a r contour lower)  n = 0 , ± 1 , ±2 , . . .  + 2n7ri  the  the  upper  lemma s t a t e s  4 . 5 approaches  as R approaches i n f i n i t y .  h a l f plane  in  I f one then  in  complex plane; Jordan's  (4.5)  for x<0.  The  zero  The same i s simple  poles  4 . 5 w i l l be above, below or on the r e a l k  a x i s depending upon whether A i s l e s s than, g r e a t e r then  or  equal to one. First  consider  i n t e g r a t i o n path f o r contour  the x>0  case are  of shown  A=1.  The  i n Figure  poles  and  11.  The  i s c l o s e d i n the upper h a l f plane, and the path i s  indented around each of the p o l e s on  the  real  axis.  The  25  Cauchy  principal  obtained for  by  the  value  letting  upper  respectively.  of  the  R approach  half This  plane  desired  infinity  contour  and  and  i s then  € approach  pole  indent  zero .  contours  gives Residues = 0  T(x)-jriE  T ( x ) = rriE Res [ exp{ 2rr i kx} / (exp{ ir i kX  or  integral  }-A • e x p { - i r i kX  Q  }) ]  0  = TriE( exp{27rikx}/7riX. ( e x p { r r i k X }+A«exp{-irikX } ) 0  <  Q  = E exp{ 2 n i r i x / X }/X n  Q  The  series  strength  X  Eexp{2nirix/X }  n  alternating 4.6  x=mX  instead gives 0  rise  T(X)=1/2 For lower  x<0,  half  the  the  plane  (-1 )  G  to d e l t a so  Combining for  the  equation  strength X  c  for  contour  w o u l d be  (4.7)  closed in  the  give 0  E 6(x+(2m+1)k /2) 4.7  4.8  and  }) ]  Q  (x<0)  0  m  }-A• e x p { - i r i k X  f o r the  (4.8)  complete  solution  A=1,  In  the  above the  while  of  with  (x>0)  G  equations  T(X)=1/2  the  series  f u n c t i o n s of  T(x)=-iriE Res[exp{2rrikx}/(exp{?rikX =-1/2  However,  the  E 6(x-(2m+1 ) X / 2 )  integration  f u n c t i o n s of  that,  m  to  (4.6)  n  to d e l t a  denominator  (m=0,1,2,...),  x=(2m+1)X /2  }/2X  (m=0 , 1 , 2, . . .) .  0  p h a s e of  0  corresponds  0  for  0  (exp{ n?r i }+exp{-nir i })  Q  = Eexp{2nirix/X n  0  upper  case  real  0  of A<1,  k axis.  half  for x<0,  (4.9)  E[6(x-(2m+1)X /2)-6(x+2m+1)X /2)]  m  plane  T(x)=0.  Q  the  poles  C l o s i n g the for For  of  equation  integration  x>0 encloses x>0  then,  a l l of  4 . 4 a l ll i e contour the  in  poles,  26  T(x)=2irii: Res[exp{2irikx}/(exp{trikX } -A • exp{ -IT i kX } ) ] 0  = 2/X  0  Iexp{2irikx}/iriX (exp{irikX }+A«exp{-irikX }) ]  0  0  K  0  0  T(x)=2/X Iexp{x• ln.(A)/X +2njrix/X } n Q  0  0  /[exp{ln(A)/2+riTri }+A«exp{-ln (A)/2-niri } ] T(x)=A In  ( 2 x  " o A  ) / 2 X o  E 6(x-(2m+1 )X /2) m  (m=0 , 1 , 2 , . . . )  c  the case of A>1 , the p o l e s of equation  the r e a l  k axis.  (4.10)  4.5 a l l l i e below  One now o b t a i n s the c o n t r i b u t i o n from  the  p o l e s only f o r x<0. T(x)=-2iriE Res [ exp{ 2tr ikx}/(exp{ ir i k X }-A. exp{ - IT i kX }) ] D  =  The  _ (2x-A )/2> A  0  solution  0  E  6  m  ( +(2m+1 )X /2) x  formal  (m=0 ,1 ,2 , .. . )  Q  f o r T(x) i n equations  shown i n F i g u r e 12.  The  Q  (4.11)  4.9, 4.10 and 4.11 are  discontinuous  behaviour  s o l u t i o n when one approaches A=1 suggests  of the  the use of  the more p h y s i c a l form, T(x) = ( 1/2) - A  ( 2 x  - ° X  ) / 2 A  o  l[6(x-(2m+1 )X /2)-6(x+(2m+1 )X /2) ] G  c  m (m=0,1,2,...) for  A  near  illustrated While represents it  1  and  the an  The  restored t o t a l similar  to  of  order  X .  This  0  i s also  i n F i g u r e 12. restoration infinite  need o n l y extend  restored.  |x|  (4.12)  over  expected  function  sum of d e l t a  equation  rms  noise  2.1.  can  field  associated  be  Since  4.12  functions; in practise  twice the s i z e of the  i n t e n s i t y maps  equation  of  written  one  being  with in  a  i s combining  the form the  i n f o r m a t i o n from both horns, K =1, and s i n c e the c o n v o l u t i o n s  of the d i f f e r e n t i a l data with equation  4.12 corresponds  summation over n independant  spaced  samples  by  to a  the beam  27  spacing k , the noise w i l l be Vrf times  larger giving  a  AT f o r each r e c e i v e r .  r m s  =Vn-T  /VAuAt  s y s  (4.13)  As one might expect,  with the s i z e of f i e l d being r e s t o r e d . for comparison with s i n g l e beam absence  of  s i z e 4X. .  noise  beam  measurements  fluctuations  measurements  increases  The break even p o i n t made  occurs  In the presence of atmospheric  0  single  atmospheric  the noise  i n the  f o r f i e l d s of  f l u c t u a t i o n s the  w i l l have a s u b s t a n t i a l l y  higher  level. The  dual  beam  function  given  in  equation  4.3  is  a p p l i c a b l e t o d i f f e r e n t i a l measurements made u s i n g two beams of  identical  shape with at most some v a r i a t i o n  in relative  amplitude.  Even with beams of  identical  the  poor  estimation  of  or beam s e p a r a t i o n w i l l  give  r i s e to c h a r a c t e r i s t i c p a t t e r n s of a r t i f a c t s along the  beam  inclination spacing. It  direction  i s conceivable  beam  function.  multiples  to  of  the  assumed beam  i n F i g u r e 13.  accomodate  One  could  even  f u n c t i o n of the form given  allow  direction  at  more  serious  one  d i f f e r e n c e s between the beams by a more complex  dimensional would  gain  These are i l l u s t r a t e d  dimensional dual  relative  shape,  i n equation  a  two  4.3 which  f o r the d i f f e r e n t p r o f i l e s of the beams i n the  perpendicular  complexities  imagine  of  the  to  the  associated  beam.  alignment.  analysis  seem  The  somewhat  f o r b i d d i n g however. A more s t r a i g h t f o r w a r d method of d e a l i n g with d i f f e r e m t beam shapes i s the f o l l o w i n g .  Suppose one has  an  accurate  28  representation observed  of the t e l e s c o p e dual beam response,  brightness  distribution,  0  due  to  C.  The  the  sky  brightness d i s t r i b u t i o n S i s 0=S * C One can form a m o d i f i e d d i s t r i b u t i o n 0' from 0'=S * C * C* where  C'(x)=-C(-x).  written  (4.14)  The m o d i f i e d d i s t r i b u t i o n can then be  i n the form 0'=S * B'* D  in  which  D=6 (x + X. /2 )-6 (x-X /2 ) f o r a m o d i f i e d s i n g l e beam Q  c  response given by B'= F" {c-c'/d} 1  in  which  lower  quantities.  case  letters  indicate fourier  One can then d e r i v e the  modified  transformed single  beam  b r i g h t n e s s d i s t r i b u t i o n P' from P'= 0'* T  (4.15)  where T(x) i s given p r e c i s e l y by equation approach its  allows  major  4.9.  While  this  the a p p l i c a t i o n of s t r a i g h t f o r w a r d theory,  drawback  is  the  degradation  of  instrumental  r e s o l u t i o n by a f a c t o r of about V T . It the  i s p o s s i b l e to o b t a i n a s i g n i f i c a n t  standard  Emerson  improvement  over  c o n v o l u t i o n procedure by imposing  a d d d i t i o n a l c o n s t r a i n t on the r e s u l t .  Consider  the  an  dual-  beam f u n c t i o n f o r i d e n t i c a l beams, D(x)=6(x+X /2)-6(x-X. /2) 0  The  complete  obtained  from  differential  0  observations  of  any  stucture  a c o n v o l u t i o n with such a f u n c t i o n must have  29  s t r i c t l y balanced  positive  and  negative  response.  This  i m p l i e s that the sum of a l l samples along  the beam alignment  direction  the more s t r i n g e n t  must  be  zero,  as  well  as  requirement that the sum of of a s e t of samples direction zero.  along  this  which are spaced by the beam spacing must a l s o be  I f one then  writes  the dual-beam  function  of an  asymmetric beam p a i r as D' (x) =6 (x + X./2 )-6 (x-X.o/2)+ € (x ) 0  it  i sclear  rise  to  that the asymmetry embodied i n  an  effectively recover  imbalance  remove t h i s  total  convolution  i n the  response.  f u n c t i o n T(x)  information  by  given by equation  does  not  data,  i t s d e t a i l e d removal i s not t r i v i a l .  use  know  along  the beam  give  one  could  one  could  applying  4.9.  Since  the one  the form of the imbalance i n the  the p r o p e r t i e s of a balanced  t h i s undertaking.  If  imbalance from the data,  intensity  generally  e(x) w i l l  data  One can however,  set t o  guide  one i n  By r e q u i r i n g that the sum of data  samples  alignment d i r e c t i o n and spaced by the beam  spacing be zero, one can o b t a i n a t l e a s t p a r t i a l  removal  of  € ( X ) .  We can form t h i s sum, Z(x) by the c o n v o l u t i o n Z = 0 *N for and  N(x) = E[6(x-(2m+1 )X /2)+6(x+(2m+1 )X. /2) ] m 0  0,  the a c t u a l  brightness. equal  to  dual  Q  beam  I f the beams were zero  f o r a l l x.  a c t u a l l y c o n t r i b u t e t o Z(x)  observations balanced  then  m=0,1,2,... of Z  the sky would  be  Suppose that n data values Dj  f o r some x so that  30  Z(x)=Ej Dj The  most  straightforward  way  of a l t e r i n g  the data so that  Z(x)=0 i s by forming the m o d i f i e d data values Dj ' from Dj '=Dj -Z/n where each data sample  (4.16)  i s m o d i f i e d by a f i x e d amount and f o r  which Ij Dj '=Ej (Dj -Z/n)=0 . Consider now  the r e s t o r e d s i n g l e beam o b s e r v a t i o n s  from  obtained  P=0 * T  with T(x) = l/2Z[6(x-(2m+1 )X. /2)-6(x+(2m+1 )X /2) ] m=0,1,2,... m Q  as given i n equation 4.9.  0  For a p a r t i c u l a r value of x, P(x)  can be w r i t t e n as n  n.  +  P ( X ) = 1/2 •E Dj -1/2  D  1  for  the n  +  (4.17)  k  p o s t i t i v e l y c o n t r i b u t i n g data samples and the n_  which c o n t r i b u t e n e g a t i v e l y .  These q u a n t i t i e s  to those d e f i n i n g Z above by n=n  n  samples  {Dj } = {Dj }U{D } k  ie the n samples i n the set {Dj } are s p l i t in  related  + n.  +  and  are  the  into  +  set {Dj} and n_ samples i n the set {D }. k  m o d i f i e d data  values  of  equation  4.16  one  m o d i f i e d r e s t o r e d d i s t r i b u t i o n by s u b s t i t u t i o n  Using the  obtains  the  i n t o equation  4.17 P' (X) = 1/2  Sj (Dj-Z/n)-1/2 ? " (D -Z/n) k  =P(x) - n «Z/2n +  k  + n_«Z/2n  P ' ( x ) = P ( x ) + Z-(n. - n ) / 2 n  (4.18)  +  The m o d i f i e d r e s t o r e d d i s t r i b u t i o n standard  restored  distribution  is given  thus by  given  by  equation  the 4.17  31  m o d i f i e d by an a d d i t i o n a l term which attempts the  unbalanced  response  in  the data.  to  eliminate  An i d e a l data base  w i l l of course not be a f f e c t e d by t h i s procedure  since  the  f u n c t i o n Z(x) w i l l a l r e a d y be i d e n t i c a l l y z e r o .  However, an  a c t u a l . data base which s u f f e r s from unbalanced  response due  to beam asymmetry as w e l l as inadequate noise  fluctuations  approach.  will  benefit  considerably  The p r a c t i c a l e f f e c t i v e n e s s  affected  by  the  extent  of  representation  the  of  from  this  region  method  data m o d i f i c a t i o n by a l a r g e number of This  was  the  approach  used  to  r e s t o r e d map d e s c r i b e d i n the next  4.4 Implementation  of the Emerson  direction,  a  scan  by  the d i l u t i o n of the unrelated  obtain  samples.  the s u c c e s s f u l l y  section.  Technique  beam  observed. carried  out  of  the  however,  were  with the beams i n c l i n e d by 11° t o the scanning  application  of  a  c o n v o l u t i o n s would then r e q u i r e rotation  here  (The q u e s t i o n of beam s i m i l a r i t y  The  provide  d i s t r i b u t i o n which would have been  The o b s e r v a t i o n s undertaken  direction. later.)  brightness  the  scan c o n v o l u t i o n with the  f u n c t i o n T(x) as given i n equation 4.12 should single  is  response  Given data c o l l e c t e d by s i m i l a r beams a l i g n e d with scanning  this  being r e s t o r e d .  Whenever p o s s i b l e , areas with d i s t i n c t d i f f e r e n t i a l should be r e s t o r e d s e p a r a t e l y to prevent  and  the  the beam alignment,  series initial  i s dealt  with  of one dimensional interpolation  and  data base to provide one a x i s p a r a l l e l to followed  by  another  rotation  to the  32  d i s p l a y c o o r d i n a t e s of RA and Dec. perform, a  two. dimensional  data base gridded i n  the  A l t e r n a t i v e l y , , one c o u l d  c o n v o l u t i o n on an i n t e r p o l a t e d  final  coordinates.  The  latter  o p t i o n was chosen here. The first  raw  data scans with l i n e a r b a s e l i n e s removed were  smoothed by c o n v o l v i n g each with a gaussian p r o f i l e of  2 arcmin FWHM to reduce  the l e v e l of  high  frequency  noise  fluctuations.  The 2 arcmin  gaussian h a l f w i d t h was chosen as  a  to  adequate smoothing without  comprimise  broadening  of  provide  the  instrumental  independent  data  sets  consisting  southbound  scans  were  then  (RA,Dec)  1950  grids  binwidths ARA and functions  of  by  ADec  T(x)  were  method chosen  the  angle  that  gridded  allow  the  e x a c t l y on other g r i d  the  beams  two  onto  of s e c t i o n 4.2. to  upon a p p l i c a t i o n of T(x) t o any g r i d since  The  of s o l e l y northbound or  individually  the  to f a l l  response.  undue  point. make  The delta  locations  Specifically,  with  respect  to  RA=constant i s e=11°-tan" (15cos(Dec)/120) 1  (rounded  t o the nearest i n t e g e r degree),  i t i s r e q u i r e d that  (nADec) + (mARA) = ( X / 2 ) 2  2  2  Q  and a l s o that The  (mARA)/(nADec)=tane  integers  resolution.  n  and  m  were  chosen  to  p r o v i d e adequate  The values used were n=8 and m=1  and ADec about 0.5 arcmin  which g i v e ARA  f o r a wide range of d e c l i n a t i o n s .  A f t e r g r i d d i n g , i t i s necessary  to f i l l  l o c a t i o n s with i n t e r p o l a t e d values i n  order  unassigned to  grid  provide  a  33  smooth  fully  sampled  interpolation points  in  data base.  technique, which uses  each  of  they  define.  the  two dimensional  nearest  two  two p e r p e n d i c u l a r d i r e c t i o n s ,  a l l o w the d e f i n i t i o n of range  A linear  values  outside  of  data  does not  the  intensity  Since the data scans make only a small  angle with respect t o RA=constant, one a l r e a d y has v i r t u a l l y complete f i l l i n g  in this direcion.  For these two reasons i t  was decided to perform one dimensional Dec=constant  direction  polynomial.  Since  discontinuities  one  spliced quadratic  is  then  filtering  corresponding  circular  a  to  in  in  two fact  significantly  this  different.  a r e furthermore  are  different  transform  1 arcmin FWHM  a  beam  s t r o n g f u n c t i o n s of  by more than  that  shapes  shapes.  The  and  declination. two  15%, while the i n t e g r a t e d same  effective  owing  t o the  use of the Emerson  method u s i n g the c o n v o l v i n g f u n c t i o n of equation thus  somewhat  beam  shapes  limited  from  would r e s u l t  the  constraint  the  outset.  The  i n some combination of  the a r t i f a c t s i l l u s t r a t e d i n F i g u r e 13. effects,  were then  of G109.1-1.0, the peak gains of the  different  beam  restoration  different  grids  minor  investigation  The  gains of the beams a r e very n e a r l y the  was  with  6 cm f a r - f i e l d beams of the NRAO 9 1 m t e l e s c o p e are  At the d e c l i n a t i o n  4.12  Lagrangian  fourier  by  i n the  gaussian.  strengths  beams  the  convolution  I t was shown i n chapter 3 of the  left  from row t o row, the f i l l e d  smoothed by gaussian plane,  by  interpolation  To  minimize  of balanced response,  such  described in  34  the p r e v i o u s s e c t i o n was a p p l i e d to the time  at  the  as the c o n v o l u t i o n f u n c t i o n of equation 4 . 9 .  to prevent the  data  the propagation of any remaining  strong  In order  artifacts  from  compact source S152 i n t o the extended s t r u c t u r e  of G109.1-1.0, the region surrounding S152/153 and (with  same  distinct  differential  response),  S148/149  r e c e i v e d separate  a p p l i c a t i o n of the c o n v o l u t i o n f u n c t i o n of equation  4.9  as  m o d i f i e d by equation 4.18. The averaged  northbound and southbound r e s t o r e d g r i d s were then together t o reduce  artifacts,  except  in  the  the l e v e l area  of  the  containing  uncorrelated the S h a r p l e s s  regions where more complete i n f o r m a t i o n was a v a i l a b l e i n the northbound data s e t .  The expected  map, o b t a i n e d from equation 4.13  rms noise i n the r e s t o r e d for a  13X.  field,  0  when  using both r e c e i v e r s and data s e t s i s 15 mK.  4.5 Emerson-Clean Hybrid A  commonly  used  method  f o r improving  the r e s u l t s of  a s t r o n o m i c a l measurements made with a p o s i t i v e primary which  has  known  (Hogbom 1974). more  sidelobe  structure  The method i s based  complicated  structure  i s that c a l l e d CLEAN  on the  can  be  assumption simulated  a p p r o p r i a t e summation of p o i n t responses.  In  data  the  base  is  searched  largest positive  iteratively  for  (or negative) response.  some f r a c t i o n of the i n s t r u m e n t a l response the  data  base  while  the  beam  fraction  that by  practise,  At t h i s  a  p o s i t i o n of position,  i s subracted and  an  location  from are  35  accumulated.  A f t e r source removal i s complete down to  level,  collection  the  of  weighted  delta  convolved with a " c l e a n beam" and added  some  functions  back  to  the  is data  base. In  the case of an asymmetric d i f f e r e n t i a l  response,  after convolution  with  function  T(x)  with a p o s i t i v e primary  one  is left  with known s i d e l o b e s t r u c t u r e . chapter obtains  the  instrumental  Emerson  the  of  3 at DEC=58.8° and assuming equal gains f o r T(x)  one  pattern i l l u s t r a t e d  beam  reponse  model  the response  Using  restoration  i n F i g u r e 14.  should i n p r i n c i p a l then be able to CLEAN  the  One  individually  r e s t o r e d g r i d s of t h i s type of s i d e l o b e s t r u c t u r e . A  fortran  o p e r a t i o n was "observed" by T(x) beam  written.  program to c a r r y out the c l e a n i n g  Simple  test  sources were  (of equation 4.9) model  shown  in  and  cleaned  Figure  with  14.  the  The  Emerson  r e d u c t i o n of s i d e l o b e l e v e l from  about  convolved showed 10%  "observed"  made on the a c t u a l r e s t o r e d maps (of the  procedure)  intensity  results  However, t e s t s made with only p a r t i a l l y  and t r i a l s  completely  with the beam model, r e s t o r e d to t o t a l  significant 2%.  computer  d i d not show s i g n i f i c a n t  to data  standard  improvement by  the c l e a n i n g o p e r a t i o n .  4.6  Theory of the Maximum Entropy Method The maximum entropy approach  attempts is  most  to f i n d the unique uniform  while  to  image  reconstruction  sky b r i g h t n e s s d i s t r i b u t i o n that  consistent  with  the i n s t r u m e n t a l  36  p r o f i l e and data t h a t one uniformity the  available.  The  degree  and  configurational reasonable  is  obtained  entropy,  form  S  by  of  maximizing  the  for this quantity  image.  (Frieden  the  The  1972,  is  where ,pj i s the normalized map  the map  The  most  Skilling  S=-Ej pj l o g ( p j )  for  of  i s d e f i n e d as the minimum i n f o r m a t i o n content of  picture  1981)  has  intensity  function  S  has  (4.19)  intensity  in p i x e l  statistic. gaussian  with  the  of  n.  a g l o b a l maximum where a l l the fj are  data  is  measured  the  image.  by some s u i t a b l e  For c o n v o l u t i o n problems, i n which errors  given by  f; summed over the number of p i x e l s  e q u a l , c o r r e s p o n d i n g to no s t r u c t u r e at a l l i n Consistency  j  one  assumes  known v a r i a n c e f o r the i n d i v i d u a l  data  samples, an a p p r o p r i a t e c h o i c e i s the normalized c h i - s q u a r e d statistic  ( G u l l and D a n i e l l x ^ 2  i n which' D  k  deviation  for  (F -D ) /(r* 2  k  k  2 k  )  i s the data value i n p i x e l of  normalization pixel  1978)  that  data  f a c t o r and F  sample, i s the  k  (4.20) k, <r  i s the standard  k  y  is  derived  an  data  arbitrary value  k from  B  the i n s t r u m e n t a l response  to p i x e l  j.  Choosing  a v a l u e of x = 1 +3.29/VFf r e p r e s e n t s the 99% c o n f i d e n c e 2  in  in  the  f i t to d a t a .  l a r g e r than t h i s  (Abramowitz and Stegun 1964).  indicate  a  poor  f i t , while  very  y=N, level  Values small  37  v a l u e s imply the u n d e s i r a b l e f i t t i n g The  solution  constraint  i s found by maximizing x =1  that  t o n o i s e i n the d a t a .  2  .  Using  the  S s u b j e c t to the  method  of  Lagrange  m u l t i p l i e r s , one wishes t o maximize, Q=S-X.x  (4.21)  2  with X. assuming a s u i t a b l e value to s a t i s f y the c o n s t r a i n t .  4.7 Implementation  of the Maximum Entropy Method  Since a s t r o n o m i c a l o b s e r v a t i o n s t y p i c a l l y or  more  data  difficulties  locations,  one  can  easily  involved in obtaining a solution  involve imagine to  l i n e a r o p t i m i z a t i o n problem of equation 4.21 with limits  on  solution  computation i s that  J. Skilling. John S k i l l i n g  time.  developed  non-  reasonable for i t s  G.J. D a n i e l l  and  The a l g o r i t h i m was most r e c e n t l y d e s c r i b e d by (1981).  The maximum  entropy  subroutine  was  used  i n c o n j u n c t i o n with a computer program w r i t t e n by S.F.  Gull  in July  application.  1981 to accomodate t h i s p a r t i c u l a r A  slightly  c o n f i g u r a t i o n a l entropy  i s used  modified  astronomical  form  of  allows  the  (4.21)  s p e c i f i c a t i o n of a d e f a u l t l e v e l , DEF t o  which the p i c t u r e w i l l the c o n t r a r y .  the  i n the subroutine  S=-Ej fj [ l o g ( f j /DEF)-1 ] which  2  the  the  The method used here S.F. G u l l ,  128  tend i n the absence of i n f o r m a t i o n to  The program begins with a  proceeds  iteratively  toward  monitored  by the c u r r e n t v a l u e s of x  TEST=(1/2)•(grad(S)/|grad(S)|  flat  a solution.  -  2  picture  and  I t ' s progress i s  and the q u a n t i t y TEST  grad(x )/|grad(x )|) 2  2  2  38  which  measures  the  simultaneous  satisfaction  c r i t e r i a which d e f i n e the s o l u t i o n . the s p e c i f i e d value of x  obtain  In  one  can  practice,  a f t e r on the order of 10 i t e r a t i o n s provided a  two  (=1.00000) while TEST<0.01  2  specified  of the  that  one  reasonable .instrumental p r o f i l e Bj^ and  has  sample  variance e ^ . 2  A s p e c i a l f e a t u r e of the present availability  of two  the northbound and at d i f f e r e n t data  set  D|'  (assuming  <  etc.)  s o l u t i o n simply by with  the  data s e t s (corresponding to  One  can  use  an  to  constrain  appropriate  the  maximum  instrumental  for  data/resonse  pairs.  Details  program  available  in  the  specification concerning  the  the  program  of maximum entropy  by  B  use of  "MEM  images r e q u i r e s , to  unsmoothed  the method of s e c t i o n 4.2.  data. The  These eventual  twice  that  of the measurements, with s i g n a l to noise of about 100. reason  for the g r i d s . instrumental and  a cell One  s i z e of ARA=ADec=0.75 arcmin a l s o r e q u i r e s the  profile(s).  this  1981).  r e s o l u t i o n o b t a i n a b l e with t h i s method i s about  this  '-  multiple  manual,  The  provided  of equation  of  Program" (R.Braun and P.C.Gregory  begin with, the g r i d d e d s e t ( s ) of  entropy  response  Deconvolution  were  feeds  additional  i n c o r p o r a t i n g i t i n t o the sum  production  the  i t r e p r e s e n t s the same wavelength,  G u l l ' s program a l l o w s  are  is  southbound scans) measured with the  r o t a t i o n angles.  polarization,  4.20  independent  observations  measured  The model developed  r o t a t e d to the a p p r o p r i a t e i n c l i n a t i o n  was or  For  chosen modeled  i n chapter 3  angle  was  used  39 for  this  in the  purpose.  F i n a l l y , one  individual  required  by  normalized  the  data  must estimate the v a r i a n c e  samples.  maximum  entropy  2  K  each  quantity  subroutine  actually  i s twice  the  inverse variance, E =2/(y*  at  The  grid  location.  The  k  )  value E = 0  i s a s s i g n e d to any  K  unmeasured g r i d l o c a t i o n s so that these do not c o n t r i b u t e t o x . 2  As  installed,  specification  of  r e p r e s e n t i n g the rms data  sample.  the  calling  a  program  fixed  noise  noise  allowed  figure  fluctuations  Experimentation  with  of  a  around  regions  of  suggested  that more  necessary  to p r o v i d e a more uniform  to the  elaborate  measured figure heavily  emission.  estimation  This  might  be  (and hence c r e d i b l e )  fit  data.  The  v a r i o u s types of nonrepeatable  that might be expected 1)  noise  each  f i t were  intense  the  fff=constant,  f i x e d noise  showed that the r e s i d u a l s of the program concentrated  for  short  time  signal  fluctuations  i n the data are the f o l l o w i n g :  scale  0.2  (about  sec)  receiver  noise  fluctuations 2)  intermediate  time  scale  (mins.)  receiver  gain  fluctuations 3) longer time  scale  (min.  to  days)  telescope  pointing  fluctuations Receiver  noise f l u c t u a t i o n s can be d e s c r i b e d by a f i x e d  n o i s e f i g u r e , «^ , a p p l y i n g to a l l data samples.  The  rms  vlaue  40  Cf =0.015  K  Receiver  was  found  gain and  important  appropriate  telescope  since  the  data  for  pointing was  the  data  fluctuations  collected  gain f l u c t u a t i o n s w i l l g i v e r i s e  scale l i n e a r l y From was  (to f i r s t  consideration expected to  to  at  the  5%  level  intensity  to  uncertainty  i s changing most r a p i d l y .  variations  of  the  tfp=0.1 arcmin, the sample  the  NRAO  is  above.  not  Since  as  (<*g=0.05).  the  observed  While the rms  pointing  are thought to be  this  effect  into  straightforward  intensity  effect  c o n t r i b u t e most  where  telescope  i n c o r p o r a t i o n of  variances  mentioned  91m  which  intensity.  of noise tube c a l i b r a t i o n s , t h i s  P o i n t i n g v a r i a t i o n s , on the other hand, w i l l significantly  10 days.  errors  order) with the observed  contribute  become  during 8 minute  i n t e r v a l s of d r i v e n t e l e s c o p e o p e r a t i o n on each of Receiver  here.  gradient  the  as  those  information  perpendicular  to the scan d i r e c t i o n  access,  s i n c e the steepest g r a d i e n t s occur  between  the  hence approximately  track,  the  two  and  beams and  f o l l o w i n g scheme was Inverse  on  i s somewhat d i f f i c u l t  the  scan  to  adopted.  variance  scans  were  generated  from smoothed  scans from E '=2/ [<r k  f  in  which  for  the maximum entropy  local  U D * ) + U -grad(D ' )) ] 2  g  k  2  p  k  the t o t a l number of g r i d d e d data p o i n t s used  intensity  K/arcmin. the  r=N,  +  2  r  r e c o n s t r u c t i o n and  gradient  along  the  These scans were then g r i d d e d  unsmoothed  data  scans  to p r o v i d e  grad(D ') i s  scan  the  k  in  units  i n the same way the q u a n t i t y E  k  of as for  41  each g r i d d e d data sample D^. above  for  both  Using the standard  the northbound  deviations  and southbound data s e t s , a  maximum entropy s o l u t i o n with x =0.99994 and TEST=0.0009 was 2  o b t a i n e d i n 12 i t e r a t i o n s .  gf  to  0.020  resolution  result,  asymptotically with l i t t l e While  gave  V a r y i n g the f i x e d  overly while  slow  rapid the  convergence  use  hope of a s a t i s f a c t o r y  a  low  led  to  result.  the Gregory-Taylor survey data was  coarsely  to  <tf =0.010  of  figure  progress a f t e r more than 20 i t e r a t i o n s  the same way as that f o r t h i s f i e l d , more  noise  spaced.  To  in  the scans are g e n e r a l l y  assess  producing r e l i a b l e maps with l e s s  collected  the  complete  possibility data  of  sampling,  approximately h a l f of the a v a i l a b l e scans of G109.1-1.0 were removed from the data s e t . shown  in  F i g u r e 15.  The remaining data l o c a t i o n s are  The t y p i c a l spacing of p a r a l l e l  scans  was 3.2 arcmin, although there were some gaps of 4.8  arcimn.  This  entropy  data  processing  was in  gridded the  same  and  received  manner  d e s c r i b e d above.  s o l u t i o n was obtained i n 12 i t e r a t i o n s with TEST=0.0010.  maximum  Again a  x =1.00003 2  and  DK (1950)  59°  D»c(1950)  F i g u r e 9. Raw D i f f e r e n t i a l Scans. The unsmoothed differential scans with o n l y b a s e l i n e s removed are d i s p l a y e d with intensity r e p r e s e n t e d as a vertical deflection from the scan path. The dashed l i n e s mark b = 0 ° and b = - 1 ° .  43  P.  d k Ak r  1  1  1  <  A  j  ^  AX=  uJ  Aj  P  > *•  /dj  Ay = A k / d k  Figure 1 0 .  I l l u s t r a t i o n of Data G r i d d i n g .  44  F i g u r e 1 1 . I n t e g r a t i o n Path f o r Transform I n v e r s i o n . The i n t e g r a t i o n path i n the complex k plane f o r determining the Emerson c o n v o l u t i o n f u n c t i o n when A = 1 .  45  a) 0.5-  1 1  1  1  l | I  X  1 *>•  -  b) 0.5-  l  X  0 0.5-  X  ^  F i g u r e 12. Emerson Method C o n v o l u t i o n F u n c t i o n , a) The s o l u t i o n f o r the c o n v o l u t i o n f u n c t i o n when A=1. b) The s o l u t i o n f o r A<1, shown f o r A=0.5. c) The s o l u t i o n f o r A>1, shown f o r A=2. d) A more continuous s o l u t i o n near A=1 suggested f o r use with f i n i t e x and shown f o r A=0.95.  46  b)  °1\ 1  4A/2  V  T7—^7~x*  t  F i g u r e 13. A r t i f a c t s of R e s t o r a t i o n . a) The unbalanced reponse due t o a point source. b) The resulting artifacts i n the r e s t o r e d map when equal gains were assumed, c) The response t o a p o i n t source with beam separation X +AX. d) The r e s u l t i n g a r t i f a c t s when a beam s e p a r a t i o n X i s assumed. C  0  47  Figure 14. Convolved Beam Model. The beam model of f i g u r e 8 r o t a t e d to one of the measuring configurations and convolved with the f u n c t i o n of equation 4.9.  48  F i g u r e 15. L o c a t i o n of Data Scans f o r Sparse Data T y p i c a l spacing of p a r a l l e l scans i s 3.2 arcmin.  Test.  49  5. Comparison of R e s u l t s The 6 cm t o t a l from  the  Emerson  intensity  map  convolution  c o n s t r a i n t of balanced response i n F i g u r e 16. visible S152.  Decaying  along  G109.1-1.0  method  obtained  and m o d i f i e d by the  (see s e c t i o n 4.3)  is  shown  s i d e l o b e response a t the 5% l e v e l i s  the beam alignment d i r e c t i o n to the south of  The extended  free  of  s t r u c t u r e of the remnant appears  to  be  of such a r t i f a c t s of the r e c o n s t r u c t i o n method down to  the 15 mK l e v e l . t h i s map, through  To g i v e an i n d i c a t i o n of the r e s o l u t i o n of  cross-sections the  peak  of  at  fixed  S152  are  RA  and  shown  at  fixed  Dec  i n F i g u r e 17.  The  h a l f w i d t h s of the response to t h i s source a r e 4.35 and arcmin  respectively.  The  expected  halfwidths  3.46  of  the  t e l e s c o p e beam i n these d i r e c t i o n s , on the b a s i s of the beam model and the data smoothing, are 4.12 and 3.00 arcmin. d i f f e r e n c e i n these v a l u e s i s l i k e l y  due  to  the  The  extended  s t r u c t u r e of S153 c e n t e r e d only 5 arcmin to the SE of S152. Recent  VLA o b s e r v a t i o n s of S152 at 6 cm wavelength  arcsec resolution  s i g n i f i c a n t beam broadening  of  standard  The  testing  method seemed to i n d i c a t e an inadequacy  while  more  improvement  Emerson r e s u l t was o b t a i n e d by a p p l y i n g  CLEAN to the r e s t o r e d maps.  procedure.  observing  here.  As i n d i c a t e d i n s e c t i o n 4.5, no s i g n i f i c a n t the  3.7  (Gregory, p e r s o n a l communication) i n d i c a t e  a FWHM of 20 arcsec r u l i n g out the p o s s i b l i t y  over  with  of  this  hybrid  i n the i n t e r p o l a t i o n  A one d i m e n s i o n a l , s p l i c e d q u a d r a t i c Lagrangian, effective  than a l i n e a r or c u b i c i n t e r p o l a t i o n  50  scheme, was not able distribution  to  to  represent  better  data spacing of 1.6  than  the  spaced  by 1.6 arcmin), of  data  about 7% given the a v a i l a b l e  image of the same f i e l d ,  a v a i l a b l e data,  inspection  differential  arcmin.  The maximum entropy of  the  this  using a l l  (two c r o s s e d s e t s of p a r a l l e l  is  shown  image  in  Figure  indicates  18.  A  brief  substantially  higher  r e s o l u t i o n than the Emerson c o n v o l u t i o n r e s u l t . hydrogen regions S152 and  S153  are  scans  clearly  The  ionized  resolved  and  f i n e r s t r u c t u r e i s v i s i b l e w i t h i n the s e m i c i r c u l a r remnant. Cross-sections  through  the peak of S152 show 1.88 and  arcmin  FWHM f o r the constant RA and constant Dec c u t s .  Figure  19.) The r e s o l u t i o n of the map  readily to  apparent  however.  This w i l l  in  general,  1.65 (See  i s not  depend upon the s i g n a l  n o i s e r a t i o of the data used which v a r i e s over the f i e l d .  To determine Emerson  the  maps  at  agreement  of  the  a more s i m i l a r  maximum  entropy  and  r e s o l u t i o n , the former  was  convolved  with an e l l i p t i c a l gaussian with h a l f w i d t h s of 3.9  and  arcmin  3.0  directions. the  in  the  The r e s u l t  constant  RA  and  constant  i s shown i n F i g u r e 20.  T h i s map and  Emerson r e s u l t have very c o n s i s t e n t temperatures  r e g i o n s surronding S152/153, S148/149 and the the  NW,  difference  down  to  almost  the  i n the r e s u l t s , l i e s  15 mK  level.  Dec  i n the  extension The  i n the l e v e l of the  major central  p o r t i o n of the remnant, being perhaps 30 mK higher over entire  r e g i o n f o r the Emerson r e u l t .  the 50 mK  higher  temperature  to  this  T h i s i s c o n t r a s t e d to  obtained  for  the  southern  51  remnant  hot  spot  These d i f f e r e n c e s  i n the smoothed maximum entropy may  be  due  to  insufficient  a p p l i e d to t h i s p o r t i o n of the maximum entropy The  two  approaches  i n t e r p r e t a t i o n s f o r another missing  scans  southbound  (see  data  structure.  in  the  be  reason.  Figure  sets  Since  may  3) the  gaps  in  introduced  resulting  number  of  i n both the northbound  and  vicinity  of  the  would  result.  sparser  resolution.  of  the  A  interpolate  across  quadratic  interpolation  data  of  sampling  Perpendicular  The  solution.  remnant,  data  (two  c r o s s e d s e t s of  arcmin) i s shown i n F i g u r e results  in  somewhat  and  1.89  arcmin  are most apparent especially  along  FWHM.  of  Figure  are The  i n the  extended  its  northern  r e s t o r e d compact s t r u c t u r e i s remarkably  to that of the complete data map  21. lower  c r o s s - s e c t i o n s through S152  show 2.11  undersampling  s t r u c t u r e of the  than  image of t h i s same f i e l d using only  available  shown i n F i g u r e 22 and  extended  use both data  tend to smooth over such gaps i n the data,  p a r a l l e l scans spaced by 3.2  edge.  extended  i n a g e n e r a l l y higher deduced emission l e v e l  half  effects  the  Emerson method cannot  The maximum entropy  The  divergent  There are a  the more f u l l y c o n s t r a i n e d maximum entropy  about  result.  the i n d i v i d u a l data s e t s , some u n c e r t a i n t y i s  into  procedure  smoothing  giving  s e t s s i m u l t a n e o u s l y , and one must f i r s t any  result.  18.  similar  Even  the  s t r u c t u r e i s represented s u f f i c i e n t l y w e l l to allow  recognition  of major f e a t u r e s .  to the r e s o l u t i o n  of  the  A f t e r smoothing t h i s  Emerson  map  (Figure  result  23),  the  52  extended  s t r u c t u r e i s r e p r e s e n t e d very n e a r l y as w e l l as the  similarly  smoothed f u l l y  makes t h i s  possible  northbound  and  i s the  southbound  c o n s t r a i n the s o l u t i o n . technique  one  sampled maximum entropy map.  does  and  4.8  data  sets  use to  of  both  convolution  not have t h i s advantage s i n c e the maps alignments  must  be  restored  The r e q u i r e d i n t e r p o l a t i o n a c r o s s gaps of 3.2  arcmin  (with  the  present beam FWHM of 3 arcmin)  would i n t r o d u c e severe u n c e r t a i n t i e s i n the i n d i v i d u a l sets  which  frequency  the  simultaneously  When using the Emerson  p e r t a i n i n g t o d i f f e r e n t beam individually.  effective  What  would not be c o r r e l a t e d with a s p e c i f i c  response.  No e f f e c t i v e means f o r removing  u n c e r t a i n t i e s would thus be a v a i l a b l e when combining r e s t o r e d maps.  data  spatial these the two  53  F i g u r e 16. Emerson Method Restored Map. Contours are at 15, 30, 50, 70, 100, 150, 200, 300, 500 and 700 mK. Negative contours a t -30 and -15 mK and v a l l e y s are hatched.  F i g u r e 17.  C r o s s - s e c t i o n s through S152  i n Emerson  Map.  55  F i g u r e 18. Maximum E n t r o p y C o n t o u r s a r e a t 15, 30, 50, 700, 1000 and 2000 mK.  Restored 70, 100,  Map. 120,  200,  300,  500,  Figure Map.  19.  C r o s s - s e c t i o n s through S152  i n Maximum Entropy  57  4  0  23  RIGHT  2  0  23  0  0  22 58  0  22 5  A S C E N S I O N ( H R S . . M I N S . . S E C S . )  F i g u r e 20. Maximum Entropy Map smoothed to r e s o l u t i o n of Emerson Method Map. The map of f i g u r e 18 convolved with an e l l i p t i c a l gaussian to o b t a i n approximately the r e s o l u t i o n of the map of figure 16.  58  l  —  i  1  1 — — — i  1  1  ^  1  1  r  F i g u r e 21. Maximum Entropy Restored Map from Sparse Data. The map made with only those data l o c a t i o n s shown i n f i g u r e 15. Contours are at 15, 30, 50, 70, 100, 120, 200, 300, 500, 700, and 1000 mK.  Figure 22. Cross-sections Maximum E n t r o p y Map.  through  S152  in  Sparse  Data  60  F i g u r e 23. Sparse Data Maximum Entropy Map smoothed to r e s o l u t i o n of Emerson Method Map. The map of f i g u r e 21 convolved with an e l l i p t i c a l gaussian to obtain approximately the r e s o l u t i o n of the map of figure 16.  61  6. Other Observations of the Emission  from  the  Field  vicinity  of  G109.1-1.0  has been  d e t e c t e d i n a number of surveys of the g a l a c t i c p l a n e . survey of Wilson and Bolton (i960) at 960 MHz beam  detected  emission from t h i s v i c i n i t y  i n t e g r a t e d f l u x of 75 Jy, RaghavaRao  et  f l u x of 40 J y .  Churchwell  (1972) at 1400  first  map  structural  with  map  allow  wavelengths.  form in  survey  by  in  appears At 6 cm,  The  beam  the SW  only one hot spot apparent  at  the NE. extend  and  to examine some 6 cm  total  the maximum entropy method and i s shown i n  S152/153 the  two  and maps,  Figure  24  S148/149 the  to have  structure  s i g n i f i c a n t l y d i f f e r e n t at these the smoothed s t r u c t u r e i s i n  similar  the  intensity  the NE and SW p o r t i o n s of the remnant's h a l f d i s c .  in  of  provided  of a broad p l a t e a u with hot spots of s i m i l a r  emission  not  MHz  beam e s t a b l i s h e d an  10 arcmin  the 21 cm continuum emission shows a  in  3200  remnant.  While  G109.1-1.0  (CTB109) with an  resolution  resolution  c o i n c i d e n t peak emission  two  with  the  produced  comparison.  within  MHz  of  smoothed to 10 arcmin  arcmin  The more recent work of F e l l i  adequate  features  intensity  the  a l (1965) with 38 arcmin  integrated  the  while  with 48  The  While  plateau,  the  i s c o n c e n t r a t e d at a s m a l l e r r a d i u s and  (with  twice  the  a p o s i t i o n 7 arcmin  plateau  intensity)  south of the 6 cm hot  U n f o r t u n a t e l y the F e l l i  and Churchwell  map  is spot does  north of DEC=58°52' so that comparison of the  extension v i s i b l e  i n the 6 cm maps i s not  Information about  the  environment  NW  possible. of  G109.1-1.0  is  62  provided  by  the  carbon  monoxide  (1980) and H e y d a r i - M a l a y e r i the  presence  et a l  of a molecular  observations (1981).  of  These  Israel  indicate  c l o u d of V(LSR)= -53km/s w i t h i n  which the S h a r p l e s s r e g i o n s S147-S153 are  contained.  This  c l o u d appears to butt a g a i n s t the western face of the  source  with map  a  minor  at  RA=22h 59m,  (prominent al  extension  in  the  i n t o the southern  Dec=58.5° 13  CO  the NW  extension of the 6 cm  molecular 5 arcmin  gulf  Another molecular  continues  in  this  a  major  as  far  i n t e n s i t y begins  direction.  The  these  RA=23h  to  densest  00m,  fade  and  p a r t of  this  e x t r e m i t y , w i t h i n about  of the major hot spot of the 6 cm of  as  et  c l o u d becomes prominent where  c l o u d i s near i t s southern  correlation  extension  o b s e r v a t i o n s of H e y d a r i - M a l a y e r i  (1981)) i n t o the northern  Dec=58.7°.  and  g u l f of the 6 cm  observations  extension.  suggests  a  i n t e r a c t i o n , or at l e a s t a c o n s i d e r a b l e degree s i g h t a b s o r p t i o n between the molecular  The  physical  of  line  of  c l o u d complex and  the  c e n t r a l f e a t u r e and extension of G109.1-1.0. The  X-ray  observations  of Gregory and Fahlman  are s c h e m a t i c a l l y superimposed on the 6 cm map and  show  the  unusual  bifurcated  eastward and northeastward (Fahlman and Gregory  1981).  feature  which  maximum  entropy  r e s t o r e d 6 cm map  for a f e a t u r e at a p o s i t i o n coordinates.  The  radio  in  1E2259+586 source,  Dec=58°36'38".  a l s o shows evidence  agreement  coordinates  25  extends  c o o r d i n a t e s of t h i s  a c c u r a t e to 5 a r c s e c , are RA=22h59m03.4s, and The  in Figure  from the X-ray p u l s a r The  (1980)  with taken  the from  X-ray the  63  p e r p e n d i c u l a r c r o s s - s e c t i o n s of F i g u r e 26,  to  which  known  p o i n t i n g c o r r e c t i o n s have been a p p l i e d , are RA=22h59m01s and Dec=58°36'55". to  0.5  These c o o r d i n a t e s a r e thought  arcmin.  The  e q u i v a l e n t p o i n t source f l u x of t h i s  peak, r e l a t i v e to the r i d g e 25 mJy.  Recent  to be a c c u r a t e  in  which  i t i s embedded  VLA o b s e r v a t i o n s at 21 cm (Gregory  is  private  communication) show no evidence f o r a compact source at t h i s p o s i t i o n down to the 0.5 mJy l e v e l , while a is  observed  about  marginal evidence prominent  X-ray  c e n t r a l source. RA=23h00m,  2  arcmin  i s seen  west  for a radio  The  Dec=58.6°  feature.  the  pulsar.  counter-part  6 cm  map  which  shows extends  The  X-ray  and  a  weak  enhanced  emission  with the notable RA=22h59m,  Only  to the  into  a  spur  in  exception  Dec=58.3°.  near  region  of  i n p o s i t i o n with the  radio  X-  data do show o v e r a l l  agreement on the extent of emission as w e l l as some of  source  f e a t u r e which curves n o r t h e a s t e r l y from the  otherwise weak e m i s s i o n , and agrees ray  of  compact  regions  the southern h a l f of the source, of  the  radio  hot  spot  near  64  Figure 24. Maximum Entropy Map smoothed to 10 arcmin Resolution. The map of f i g u r e 18 convolved with an elliptical gaussian to o b t a i n the r e s o l u t i o n of the 21 cm continuum map of F e l l i and.Churchwell (1972).  DECLINATION(DEC. DEGS.) Figure 25. Overlay of Maximum Entropy Map and X-ray Observations. Four relative intensity levels of the X-ray emission (Gregory and Fahlman 1980) are o v e r l a i d on the 6 cm continuum map of f i g u r e 1 8 . The dashed l i n e marks the outer boundary of the X-ray e m i s s i o n .  66  RA=22 59 01 h  m  s  0.8  *  U  0.4-  0.0K  15  CMIN  45  60  Figure 26. C r o s s - s e c t i o n s through C e n t r a l Peak i n Maximum Entropy Map. Perpendicular c r o s s - s e c t i o n s through the 6 cm peak which i s c o i n c i d e n t i n p o s i t i o n with the compact X-ray source and has an e q u i v a l e n t p o i n t source f l u x of 25 mJy.  67  7.  Summary and The  Conclusion  production  structure  from  of  total  beam-switched  p r a c t i c a l undertaking. switching without  intensity  technique  One to  observations  can  et  measurements  al  has  be  showed  how  analytically  restored  T h i s method  here  successfully  to o b s e r v a t i o n s  applied  l e s s than i d e a l circumstances. beams were known to be of were,  scanning  in  r e s t o r e d map expectations  of  15 mK (See  maximum entropy  developed  extended made under  case,  different  significantly  these  applied  the  section  to  the  shape. from the  complications,  the  processing.  Within  processing  one  of  other  allows  instrumental  (as also  reconstructing  methods phase  need f o r i n t e r p o l a t i o n and  r e s o l u t i o n enhancement.  of  T h i s approach o f f e r s a number of over  noise,  with  4.3.)  problem  advantages  high frequency  two  with v i r t u a l l y no evidence  significant  the  the  method of image r e c o n s t r u c t i o n  d i f f e r e n t i a l measurements.  undesirable  was  to  by S . F . G u l l , G . J . D a n i e l l and J . S k i l l i n g ) was  successfully  eliminates  beam  shows a noise l e v e l c o n s i s t e n t  about  processing a r t i f a c t s . The  rotated  Despite  (Figure 16)  In the present  significantly  addition,  direction.  the  a  differential  s i n g l e beam response.  They  become  information.  equivalent and  extended  e l i m i n a t e atmospheric f l u c t u a t i o n s  (1979)  could  of  take advantage of  losing large scale s t r u c t u r a l  Emerson  maps  for  this  of  data method  pre-filtering  deconvolution  from  of an  response and as a bonus, p r o v i d e s T h i s i s accomplished with  moderate  68  computational c o s t s , which were i n t h i s case l e s s than of  the  Emerson  method.  The  resulting  image ( F i g u r e 18)  shows r e s o l u t i o n enhancement by a f a c t o r of all  of  the  justified the of  indicated  areas  by the data.  field  discrepancy  to  within  lies  about  in  the  20 mK.  The  only  deduced  emission  significant level  in  maximum entropy map.  of the  30 mK  lower  This difference i s l i k e l y  due  the more e f f e c t i v e use of a l l a v a i l a b l e data to c o n s t r a i n  the maximum entropy s o l u t i o n . The  maximum  effective 23 was  with  result  entropy  3.2  obtained  method  was  arcmin, and  also The  shown  to  seems  be  image i n F i g u r e  i s remarkably  with more complete  return  5.)  i n t e r l e a v e d s e t s of p a r a l l e l  t h i s study, t h i s approach  information  (See chapter  sparser data sampling.  o b t a i n e d from two  spaced by about  of  of  images show agreement over much  h a l f - d i s c of the remnant; being about  to  with  resolution  extended the  2,  emission appearing to be  When smoothed to the  Emerson r e s u l t , the two the  of  those  scans  s i m i l a r to the  sampling.  On the b a s i s  to  the  offer  optimum  when a p p l i e d to completely or p a r t i a l l y  sampled d a t a . The present work can now  be  directly  r e d u c t i o n of the Gregory-Taylor survey. the  survey  data  base  p a r a l l e l northbound as 2.4 has  and  will  spacings  p r o d u c t i o n of images  to  As of October multiple  of 3.2 with  More than 85% of t h i s arcmin or l e s s . greater  the 1981,  repeats of  southbound scans spaced by as  arcmin at Dec=60°.  scan  contain  applied  little  data  base  T h i s a l l o w s the  q u a l i t y • than  that  of  69  Figure  23 (due to s u b s t a n t i a l l y lower noise) t o be made f o r  the m a j o r i t y of the survey a r e a . The  past  indicate  a  and  observations  G109.1-1.0  1981).  with  The  the  X-ray  estimated  There i s no obvious X-ray  pulsar  6 cm  r e l a t i v e to the r i d g e i n which i t i s  (Gregory  flux  of t h i s  embedded  is  source  25 mJy.  j e t , although  the extent of emission and  6 cm o b s e r v a t i o n s have a l s o r e v e a l e d an unusual  similar. feature  which  extends  arcmin  r a d i a l l y away from the c e n t e r to the north-west.  from the northern edge of the remnant f o r 35  g e n e r a l l y s e m i - c i r c u l a r shape of the remnant, by  and  r a d i o c o u n t e r p a r t at the p o s i t i o n of the  h a l f - d i s c appearance of the remnant are remarkably The  with  A peak i n the 6 cm emission was found at a  position coincident  prominent  of  complex i n t e r n a l s t r u c t u r e and i n t e r a c t i o n  i t s environment.  Fahlman  present  as  The  indicated  both the X-ray and r a d i o o b s e r v a t i o n s , can be understood  in terms of an i n t e r a c t i o n with an adjacent molecular c l o u d . A  thorough  unpublished  discussion  of  these  and  o b s e r v a t i o n s i s being prepared  with P.C. Gregory  and G.G. Fahlman.  more  recent  in collaboration  70  References.  1) Gregory,P.C.  and Taylor,A.R.,  2) Gregory,P.C.  and Fahlman,G.G., 1980, Nature,  3) Emerson,D.T., K l e i n , U .  1981, Ap.J.  ( i n press) 287, p805  and Haslam,C.G.F., 1979, A s t r o n .  Astrophys., 76, p92 4) G u l l , S . F .  and D a n i e l l , G . J . , 1978, Nature,  5) T i u r i , M.E.,  1966,  in  'Radio  272, p686  Astronomy',  Kraus,J.D.,  McGraw-Hill 6) Hogbom,J.A., 1974, A s t r o n . 7) Frieden,B.R., 8) S k i l l i n g , J . , 9)  1981, ( p r e p r i n t ) and  Stegun,I.A.,  1964,  and Bolton,J.G.,  of  1960, PASP, 72, p331  11) RaghavaRao,R., Medd,W.J., Higgs,L.A.,  12)  'Handbook  F u n c t i o n s ' , Dover, New York  10) Wilson,R.W.  1965,  Suppl. , 1_5, p4l7  1972, J.opt.Soc.Am., 62, p511  Abramowitz,M.  Mathematical  Astrophys.  and  Broten,N.W.,  MNRAS, 129, pl59 Felli,M.  and Churhwell,E.,  1972, A s t r o n .  Astrophys.  Suppl., 5, p369 13) I s r a e l , F . P . , 14)  1980, A s t r o n .  Heydari-Malayeri,M.,  J . , 85, pi 612  Kahane,C.  and  Lucas,R.,  1981,  (preprint) 15) Fahlman,G.G.  and Gregory,P.C., 1981, Nature,  ( i n press)  

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