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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 University 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 th i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September 1981 © Robert Braun, 1981 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 the requirements f o r an advanced degree at the U n i v e r s i t y o f 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 i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r 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 o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head o f my department o r by h i s o r her r e p r e s e n t a t i v e s . I t i s understood t h a t copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department of The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 DE-6 (2/79) i i Abstract The supernova remnant G109.1-1.0 was observed d i f f e r e n t i a l l y at 6 cm wavelength using the 91 m t r a n s i t telescope of the NRAO at Green Bank, West V i r g i n i a . After development of a complete declination dependant model of the 6 cm d i f f e r e n t i a l response, the observations were restored to the equivalent t o t a l intensity observations using the method described by D.T. Emerson et a l (1979) as well as the maximum entropy method of image reconstruction (Gull and Daniell 1978). Both approaches gave successful reconstructions of the f i e l d at similar computational costs. The maximum entropy method i s shown to provide e f f e c t i v e reconstruction even under the sparser sampling conditions of the Gregory-Taylor (1981) survey; allowing the direct application of thi s work to the majority of the d i f f e r e n t i a l l y recorded survey data base. The 6 cm observations of G109.1-1.0 indicate an extended h a l f - d i s c structure which i s similar in both shape and extent to the X-ray emission (Gregory and Fahlman 1980) from the remnant. An unusual feature has also become .apparent which extends from the northern edge of the remnant for 35 arcmin r a d i a l l y away from the center to the north-west. The generally semi-c i r c u l a r shape of the remnant, as indicated by both the X-ray and radio observations, can be understood in terms of an interaction with an adjacent molecular cloud. i i i Table of Contents Abstract i i Table of Contents i i i L i s t of Figures and I l l u s t r a t i o n s iv Acknowledgements v i 1. Introduction ....1 2. Observations 3 3. Calibrations 9 4. Reduction 20 4.1 Baseline Removal 20 4.2 Gridding the Data 21 4.3 Theory of the Emerson Method 23 4.4 Implementation of the Emerson Method 31 4.5 Emerson-Clean Hybrid 34 4.6 Theory of the Maximum Entropy Method 35 4.7 Implementation of the Maximum Entropy Method 37-5. Comparison of Results 49 6. Other Observations of the F i e l d 61 7. Summary and Conclusions 67 References 70 i v L i s t of Figures 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) Location of a l l Available Data Scans 8 4) RA and Dec Pointing Corrections 14 5) Beam Separation, Relative Gain and S e n s i t i v i t y 15 6) Typical Model F i t s to Half P r o f i l e s 16 7) Beam Parameters .17 8) Beam Model Output 1 9 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 Gridding 43 11) Integration Path for Transform Inversion 44 12) Emerson Method Convolution Function 45 13) A r t i f a c t s of Restoration 46 14) Convolved Beam Model 47 15) Location of Data Scans for Sparse Data Test 48 16) Emerson Method Restored Map 53 17) Cross-sections through S152 in Emerson Map 54 18) Maximum Entropy Restored Map 55 19) Cross-sections through S152 in Maximum Entropy Map..56 20) Maximum Entropy Map smoothed to resolution of Emerson Method Map 57 21) Maximum Entropy Restored Map from Sparse Data 58 22) Cross-sections through S152 in Sparse Data Maximum Entropy Map 59 V 23) Sparse Data Maximum Entropy Map smoothed to resolution of Emerson Method Map 60 24) Maximum Entropy Map smoothed to 10 arcmin Resolution 64 25) Overlay of Maximum Entropy Map and X-ray Observations 65 26) Cross-sections through Central Peak in Maximum Entropy Map 66 v i Acknowledgements I. would f i r s t of a l l l i k e to thank my supervisor, Dr. P.C. Gregory for his assistance and guidance in th i s undertaking. Of great help as well, were stimulating discussions with Dr. W.L.H. Shuter, Dr. S.F. Gull and A.R. Taylor. Thanks also go to the staff of NRAO, Green Bank for their h o s p i t a l i t y and assistance in carrying out the observing program. F i n a l l y , I would l i k e to thank M.A. Potts for her assistance in the software development. 1 1. Introduction The variable radio source survey begun by P.C. Gregory and A.R. Taylor in 1977, while s t i l l under way, has already begun to make s i g n i f i c a n t contributions to our knowledge of the radio sky (Gregory and Taylor 1981). While the survey was designed to explore compact source v a r i a b i l i t y , i t s extensive sky coverage and high s e n s i t i v i t y have allowed the recognition of a number of extended features of low surface brightness many of which are probably associated with supernova remnants. One such source, G109.1-1.0, showing a s h e l l - l i k e structure and c e n t r a l l y peaked emission in the d i f f e r e n t i a l l y recorded 6 cm survey scans prompted Gregory and Fahlman (1980) to obtain X-ray observations in i t s v i c i n i t y . The very unusual X-ray structure showed evidence of a j e t - l i k e extension from a compact source within a semi-c i r c u l a r s h e l l . It was decided to produce a t o t a l i ntensity map of the region surrounding G109.1-1.0 at a wavelength of 6 cm to allow a detailed comparison of the X-ray and radio morphology. Total intensity measurements made at wavelengths of 6 cm and shorter can be subject to considerable degradation due to atmospheric water vapour. For t h i s reason, the method of beam-switching has long been employed in single dish observations to cancel the fluctuations in atmospheric emission when mapping sources of small angular, size r e l a t i v e to the beam spacing. Limits on useful beam spacing are imposed however, by the requirement that there be s u f f i c i e n t 2 ne a r - f i e l d beam overlap to provide e f f e c t i v e c a n c e l l a t i o n . The method was not applied to the mapping of large f i e l d s since i t was thought that the unravelling of the d i f f e r e n t i a l response was not a p r a c t i c a l undertaking. However, Emerson et a l (1979) have recently i l l u s t r a t e d a method which allows the analytic reconstruction of the equivalent single beam response for sources which extend for several beam spacings. In addition, the maximum entropy method of image reconstruction has been successfully applied to deconvolution problems (Gull and D a n i e l l , 1978). This approach o f f e r s an alternate method for the reconstruction of d i f f e r e n t i a l observations. With both of these reduction p o s s i b i l i t i e s at hand, i t was decided to take advantage of the d i f f e r e n t i a l measurement technique in mapping the source G109.1-1.0. Additional motivation for this method of measurement came from the fact that the entire Gregory-Taylor survey was undertaken in th i s manner. The development of an e f f e c t i v e method of restoring d i f f e r e n t i a l measurements made under the same conditions as the survey would allow application of that technique to th i s much larger data base, making possible the t o t a l intensity mapping of the entire g a l a c t i c plane between 1 =40° to 220° and b =-2° to 2° with 3 arcmin resolution and an improvement in s e n s i t i v i t y of at least 50 times over presently available results for the majority of the survey area. 3 2. Observations Observations were c a r r i e d out in August and September of 1980 using the 91 meter t r a n s i t telescope of the National Radio Astronomy Observatory at Green Bank, West V i r g i n i a . The feed and receiver configuration used was i d e n t i c a l to that used for the survey of Gregory and Taylor (Gregory and Taylor 1981) and i s shown schematically in Figure 1. The feed system consists of two horns sensitive to right hand c i r c u l a r l y polarized radiation. These are fixed to a rotatable mount centered on the telescope axis of symmetry at the prime focus. At a wavelength of 6 cm the projected beams have FWHM of about 3 arcmin and are separated by about 7 arcmin. Two cooled parametric amplifier receivers were used; each with a t o t a l system temperature of 70 K and 3 db bandwidth of 580 MHz. The two receivers were switched in antiphase between the horns at a frequency of 50 Hz; providing two independent measurements of the d i f f e r e n t i a l output of the feeds. In order to map the f i e l d surronding the remnant G109.1-1.0, a scan sequence was designed to sample the region between RA( 1950) 22h 54m to 23h 04m and Decd 950) 58.0° to 59.6°. To obtain maximum sky coverage in the ten days available for these observations, a declination drive rate of 120 arcmin/min and an i n c l i n a t i o n of the feeds away from the scan d i r e c t i o n by 11° were used. Each day's observations consisted of alternating northbound and southbound scans across the f i e l d , staggered to give 1.6 4 arcmin spacing of p a r a l l e l scans after 10 days of mapping. A scanning sequence and the beam geometry are i l l u s t r a t e d in Figure 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° provides about 1.5 atcmin separation of the beam centers perpendicular to the scan d i r e c t i o n . While complicating the analysis somewhat, thi s technique provides increased e f f e c t i v e coverage area for scan spacings greater than about 1/2 FWHM and corresponds exactly to that used in the Gregory-Taylor survey. Immediately preceding and following each day's 8 minute scan sequence the telescope was held stationary for the 12s f i r i n g of a stable noise tube to provide temperature c a l i b r a t i o n of the data. The rms receiver gain variation was 5%. A sample time of 0.2 seconds was chosen as used in the Gregory-Taylor survey. The frequency, f =2.4 Hz was used as the 3 db cutoff for the post-detection low pass f i l t e r . The appropriate integration time At is then given by 1/(2f(1.57))=0.13 sec ( T i u r i 1966) allowing one to calculate the t h e o r e t i c a l rms receiver noise fluctuations ( T i u r i 1966) from AT r m s=K s • T s y s /VSuSt: (2.1) given the constant Ks=2 for a beam switched receiver system. This gives T r m s = l 6 mK, which can be further reduced by a factor of by combining the two receiver outputs. The output of both receivers together with pointing information 5 was written onto magnetic tape at the telescope s i t e . These quantities were also monitored in real time v i a a s t r i p chart recorder and the po s i t i o n a l display. Data which was suspect due to faulty telescope tracking, poor weather conditions or interference was noted at t h i s time. Unfortunately some of the preceding reasons resulted in the loss of one day's complete scanning sequence as well as a number of additional scans. The location of available data scans i s i l l u s t r a t e d in Figure 3. The data received i n i t i a l processing by the NRAO continuum reduction program CONDARE 3 at C h a r l o t t e s v i l l e , V i r g i n i a . This program uses the noise tube f i r i n g s to provide temperature c a l i b r a t i o n and discards data recorded while the telescope was not on track. The program output was written onto magnetic tape for transport to the University of B r i t i s h Columbia where the remainder of the data processing was carr i e d out using an AMDAHL 470 V8 computer. After v i s u a l inspection, the two receiver outputs were averaged for the map scans. TO FRONT END SWITCHES A SWITCH DRIVE OSC. 'B' PARAMP TRANSISTOR AMP 4.5 to 5.1 GHz FILTER MIXER I F AMP SQ. LAW DETECT. SYNC. DETECT. LOW PASS F I LTER B' OUT B V V NOISE CAL. 1 NOISE C A L COLD LOAD x2 L . 0. AMP L.O. 'A' PARAMP TRANSISTOR AMP A.5 to 5.1 GHz FILTER MIXER IF AMP SQ. LAW DETECT. SYNC. DETECT. LOW PASS F I LTER A OUT TO FRONT END SWITCHES A SWITCH DRIVE OSC. 7 54' 0 R I G H T A S C E N S I O N ( H R S . . M I N S . . S E C S . ) Figure 2 . Scanning Sequence and Beam Geometry. One day's complete scanning sequence, showing the beam alignment. 8 Figure 3. Location of a l l Available Data Scans. Typical spacing of p a r a l l e l scans i s 1.6 arcmin. 9 3. Calibrations Ca l i b r a t i o n scans were obtained through a set of approximately 45 unresolved sources of known flux density at declinations between -10° and 70° in addition to the mapping scans. These were obtained to determine the telescope parameters necessary for reduction of the data and were ca r r i e d out in conjunction with the survey program of Gregory and Taylor. D r i f t scans with the beams aligned with the scan d i r e c t i o n (type 1) were f i r s t obtained for the c a l i b r a t i o n set to determine RA pointing corrections and their declination dependence. (See Figure 4.) Scans driven to the north or south at 60 arcmin/min with RA pointing corrections applied and beams aligned with the scan d i r e c t i o n (type 2) were next ca r r i e d out to determine Dec pointing corrections. (See Figure 5.) These type 2 scans were repeated with both pointing corrections applied (type 3) to obtain the separation of the two beams as a function of declination. (See Figure 5.) F i n a l l y using the Dec pointing corrections and beam separation data, d r i f t scans were obtained with the beams aligned perpendicular to the scan d i r e c t i o n and passing through only the upper or lower beam center (type 4 and 5), to obtain beam shape information in thi s d i r e c t i o n . Both the maximum entropy and Emerson data reduction techniques require a knowledge'of the instrumental p r o f i l e . To determine the beam shape and i t s declination dependence, one could consider the complete d i f f e r e n t i a l mapping of a 10 large number of unresolved sources. The observation time required for such an undertaking is somewhat p r o h i b i t i v e , however. The method adopted here was to make use of the information provided by the approximately perpendicular cuts through the two beams provided by the scans of type 3, 4 and 5 above. The p r o f i l e of each beam in the N, S, E and W directions could be combined with beam separation and gain r a t i o data to give an approximate model of the dual-beam response pattern. A fortran computer program to carry out a least squares f i t to a pa r t i c u l a r beam p r o f i l e model had already been developed by A.R. Taylor, who kindly allowed i t s use for the following work. Since the two beams show s i g n i f i c a n t asymmetry about their peak values, the model function was f i t separately to either half of each beam p r o f i l e . The model function i s a modified gaussian of the form f(x)=exp{-2.7726[1+c •(x/H)•(x/H-1/2)](x/H) 2} (3.1) in which H i s the FWHM which corresponds to the half p r o f i l e in the same units as x, and c i s the parameter (affecting the r e l a t i v e width of the function's central plateau) which allows a better f i t than a simple gaussian. The program takes raw data scans with only i n i t i a l temperature c a l i b r a t i o n , establishes suitable baseline regions (by the c r i t e r i o n of allowing differences in adjacent data samples of only 20 mK) and removes a lin e a r baseline. The posit i v e and/or negative peaks are f i r s t normalized to their quadratically interpolated peak temperature. Each side of 11 the peak i s then separately f i t by a function of the form shown in equation 3.1. This i s done by f i r s t f i x i n g the half power point, H d i r e c t l y from the data. The parameter c is then varied to obtain a least squares f i t using only those data values above 10% of the peak temperature. Typical model f i t s to half p r o f i l e s are shown in Figure 6. Scans driven to the north and to the south were analysed separately to determine whether the beam parameters were affected by dr i v i n g d i r e c t i o n . No s i g n i f i c a n t differences in parameter values were observed however. The variation of beam parameters with declination i s shown in Figure 7. In each case the least squares quartic polynomial f i t i s shown that was used to model the v a r i a t i o n . The function in equation 3.1 was found to be quite adequate in describing the beam shapes down to the 10% l e v e l . However, for large arguments, negative values of the parameter c give r i s e to o s c i l l a t i o n s of the function. To better represent the beam response at low l e v e l s , a gaussian of matched slope (equation 3.2) was used to extend the model below the 15% l e v e l in the following manner. g(x)=A-exp{-a(x/H)2} (3.2) Rewriting equation 3.1 in terms of z= x/H and allowing an amplitude C gives, f(z)=C-exp{-b[1+C-Z(Z-1/2)]z 2} (3.3) The function f(z) reaches a value k'C (k<l) when k.C=C-exp{-b[1+c-z 0(Z q-1/2)]z 0 2} or . ln(k)=-b[1+c-z Q(z Q-1/2)]z Q 2 12 allowing a solution for z Q from the quartic equation, -z o"-z o 3/2+z o 2/c+ln(k)/b-c=0 ( 3 > 4 ) A matched functional value at z=z 0 gives the condition, k-C=A'exp{-a•z02} (3.5) while matching slopes at z=z Q gives the condition, D=-b-C[4c ' Z 0 3-3c•z 0 2/2 + 2z 0,]exp{-b[1+c•z Q(z Q-1/2)]z Q 2} =-2a'A 'Z 0-exp{-a-z 0 2} (3.6) Dividing equation 3.6 by equation 3.5 gives, D/(k.C)=-2a-z0 or a=-D/(2k-C•z0) (3.7) Substitution of equation 3.7 into equation 3.5 then gives, A=k-C/(exp{D-z0/2k.C}) (3.8) Equations 3.7 and 3.8 then provide the parameters of the gaussian extension for z^z Q. In constructing the two dimensional beam model, i t was assumed that the two cuts available through each beam center (from scans of type 3, 4 and 5) represented perpendicular cuts in the RA and Dec di r e c t i o n s . For the observations of G109.1-1.0 the two cuts are incl i n e d by 83° since earth rotation provides a right ascension drive rate of l5«cos(Dec) arcmin/min. For the f u l l declination range of the Gregory-Taylor survey, the i n c l i n a t i o n varies from 76° to 85°. Away from the RA and Dec di r e c t i o n s , a m u l t i p l i c a t i v e combination of the form, f(z,,z 2)=C-exp{-b([1+c,z,(z,-1/2)]z, 2+ [ 1 + c 2 z 2 ( z 2 - 1 / 2 ) ] z 2 2 ) } (3.9.) was assumed to apply above the 15% l e v e l , using the values of c 1 , c 2 , H 1 and H 2 appropriate for each quadrant. To extend 13 t h i s model in the manner described above, equation 3.9 i s f i r s t transformed to c y l i n d r i c a l polar coordinates ( r , e ) such that, z , =r (cos©) and z 2=r(sine) giving, f (r , e)=C'exp{-b[r 2-(r 3/2). (c ^ o s^+czsin 3©) +r"(cTcos^e+Czsin"©)]} (3.10) At any angle © , one can determine the matching radius r Q and the parameters of the r a d i a l gaussian extension in a manner analogous to that employed above. In practice, these parameters are determined at one degree inter v a l s of © rather than for each rectangular gr i d l o c a t i o n . The extended individual beam models t y p i c a l l y contribute less than 0.5% at the location of the other beam center. The dual-beam pattern was therefore formed by simply adding the functional models of the individual beams. An example of the beam model output at Dec = 58.8° i s shown in Figure 8. Since t h i s model was generated only from scans for which the l i n e joining the two beams had a small i n c l i n a t i o n with respect to RA = constant, i t i s only expected to be applicable to data c o l l e c t e d in a similar way. The present investigation has led to the recognition that the dual-feed/dish response with such alignment i s very d i f f e r e n t from that obtained with the feeds aligned with Dec = constant. 14 ~ .2F-£ >1F-\ 0 •30 •20 •10 <_> O in Q -10 -20 -30 RA Pointing Corrections (Cat.-Obs.) Dec Pointing Corrections (Cat.-Obs.) 0 10 20 30 40 50 60 70 Declination ( ° ) Figure 4. RA and Dec Pointing Corrections. The telescope pointing corrections determined from scans of types 1 and 2 (see text) through the c a l i b r a t i o n sources. 15 B e a m A Sensitivity 0 10 20 30 40 50 60 70 Decl inat ion (°) Figure 5. Beam Separation, Relative 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 text) through the c a l i b r a t i o n sources. 16 I i i i i i i i i i i i I 0.0 1.0 2.0 3.0 A R C M 1 N I 1 1 i I I I i ' ' • • ' 0-0 1.0 2.0 3-0 ARCMIN Figure 6. Typical Model F i t s to Half P r o f i l e s . The dots indicate the normalized data, while the s o l i d l i n e represents the model f i t to the north and south half p r o f i l e s of beam A for DA251. 17 Figure 7a. Beam Parameters. The model parameters for the four orthogonal r a d i a l cuts from the center of beam A. 18 B (North) B(South) Declination (°) Declination (°) Figure 7b. Beam Parameters. The model parameters for the four orthogonal r a d i a l cuts from the center of beam B. 19 A N Beam ' B ' Beam 'A' W Figure 8. Beam Model Output. The two dimensional dual beam model at Dec=58.8° constructed from equation 3.9 and the parameters of figure 7 with a matched r a d i a l gaussian extension below the 15% le v e l determined at 1° in t e r v a l s . Contours are 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 interpretation of the observations i s possible, considerable data reduction i s necessary. The preliminary reduction phases of baseline removal and gridding of the data are described below. These are followed by a description of the methods employed to recover t o t a l intensity information from the raw d i f f e r e n t i a l data scans shown in Figure 9. 4.1 Baseline Removal The rapidly driven scans used here for the mapping operation are subject to non-constant baseline levels due primarily to variations in the difference of the s p i l l o v e r radiation seen by the two horns as a function of the telescope zenith angle. Scans through the c a l i b r a t i o n sources generally have very l i t t l e extended structure to complicate the choice of suitable regions for baseline d e f i n i t i o n . However, the mapping of extended structure in the gala c t i c plane can pose a more serious problem. In p a r t i c u l a r , a low l e v e l extension to the northwest of G109.1-1.0 was found to continue almost to the edge of the region being mapped. The d i f f e r e n t i a l nature of the measurements, es p e c i a l l y with the two beams not aligned with the scan d i r e c t i o n , adds the further complication of allowing both posi t i v e and negative responses which are not necessarily of equal amplitude. Scans from the Gregory-Taylor survey which passed 21 through the region being mapped were used to a s s i s t in the reduction at t h i s stage. These scans span approximately 4 degrees centered on the g a l a c t i c plane and were ca r r i e d out in exactly the same way as the mapping scans described above. Their greater length allows more accurate d e f i n i t i o n of a reasonable baseline l e v e l r e l a t i v e to which the measurements were made. To a s s i s t in the recognition of common low l e v e l features, smoothed versions of the map and survey scans were obtained by convolving these with a gaussian p r o f i l e of , 2 arcmin FWHM. Since the smoothed survey scans in t h i s v i c i n i t y did not show s i g n i f i c a n t curvature, a linear baseline was subtracted from them. Plotted baseline f i t s to the survey scans were then used to guide the choice of baseline l e v e l in the G109.1-1.0 map scans. A program was written to display each smoothed map scan and allow interactive baseline s p e c i f i c a t i o n . The s p e c i f i e d baseline was then removed from both the smoothed and unsmoothed map scans. 4.2 Gridding the Data Two dimensional data processing normally requires a regularly sampled rectangular g r i d . When one's data has been obtained along directions other than those defining the processing array, with noncommensurate sample spacing, the method of placing t h i s information onto a g r i d becomes an important consideration. The method which was employed for th i s operation i s outlined below. 22 Consider a set of data p o i n t s Dj at p o s i t i o n s (XJ ,y{ ). For each data p o i n t , there w i l l be four nearest g r i d l o c a t i o n s ( P j k , P j + i k / pjM > pj+1k*1 ) a s i l l u s t r a t e d i n F i g u r e 10. One can d i s t r i b u t e the data value Dj among these four p o i n t s a c c o r d i n g to the weights, Wj k j =(1-Ax)(1-Ay) WHki =AX(1-Ay) V i i =(1"Ax)Ay Wj-1k-1i =AxAy such t h a t (W j k i ) + ( W H k i ) + (W j k +l j ) + ( W j . l k # 1 i ) = 1 If one then accumulates at each g r i d l o c a t i o n the Djk=*i (Wjki ) D i and normalizes each with w j k = j : i w j k i one o b t a i n s a g r i d f i l l e d with a weighted average of r e l e v a n t data p o i n t s . Depending on the g r i d c e l l s i z e and ins t r u m e n t a l r e s o l u t i o n , one may not wish to a s s i g n values to c e l l s to which no data sample has come s u f f i c i e n t l y c l o s e . To t h i s end, only g r i d c e l l s f o r which Wj k>0.3 were assign e d v a l u e s f o r g r i d s of 0.5 arcmin c e l l w i d t h and ins t r u m e n t a l r e s o l u t i o n of 3 arcmin FWHM. Given the v a r i a n c e <s2\ of data sample Dj , one can a l s o c a l c u l a t e the v a r i a n c e of <y2jk a s s o c i a t e d with the grid d e d data value D j k . T h i s i s given by, « 2jk=£i< wjki >2«2i and i s normalized by d i v i s i o n with w a j k - < i i W j k i . ) 2 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 restore a set of d i f f e r e n t i a l observations to the equivalent single beam observations, while benefiting from the cancellation of atmospheric fluctuations provided by the d i f f e r e n t i a l method. Consider the brightness d i s t r i b u t i o n , P due to a telescope's single beam response B convolved with the sky brightness d i s t r i b u t i o n S. P=S * B (4.1) If the dual beam response of the telescope can be represented by the convolution B * D for some suitable function D, then the dual beam response to the sky d i s t r i b u t i o n would be 0=S * B * D (4.2) Taking the fourier transform of equations 4.1 and 4.2 gives p=s«b and o=S'b'd One could in p r i n c i p a l then obtain the single beam d i s t r i b u t i o n function P from, P=F"1{o/d} or equivalently (and more convenient computationally) by convolving the dual beam observations with the function T given by, T=F-1{1/d} so that P=0 * T Let us assume for the moment that the dual beam function, D can be written as 24 D ( x ) = 6 ( x + X 0 / 2 ) - A 6 ( x - X 0 / 2 ) ( 4 . 3 ) for the beam spacing kQ. Now l e t us solve for the function T(x). The fourier transform of D(x) can be written as d(k) = J°exp { - 2 j r i k x } [ 6 (x + X.G/2 ) - A 6 (x-X Q / 2 ) ] d x = exp { i r i k X . 0 }-A'exp { - i r ikX. 0 } The fourier transform of T(x) would then be t(k) = l/d(k) = l/(exp{irikx c }-A^exp{-irikk 0}) so that T(x) i s given by T(x)= J exp { 2 7 r i k x}/(exp { i r i k X . 0 }-A• exp{-trikX.0 }) dk The integrand has simple poles in the complex k plane where exp { i r i kX. 0}=A«exp { - j r i kX . 0 } or f r i k X . 0 = l n ( A ) - i r i k X 0 + 2n7 r i n = 0 , ± 1 , ± 2 , . . . giving k = - i l n ( A ) / ( 2 i r X 0 )+n/X0 n = 0 , ± 1 , ± 2 , . . . ( 4 . 5 ) When |k| approaches i n f i n i t y , the integrand in the expression for T(x) above approaches zero. If one then considers a semicircular contour of radius R in the upper (or the lower) half complex plane; Jordan's lemma states that for x>0 the integral of equation 4 . 5 approaches zero along t h i s contour as R approaches i n f i n i t y . The same i s true in the lower half plane for x<0. The simple poles given by equation 4 . 5 w i l l be above, below or on the real k axis depending upon whether A i s less than, greater then or equal to one. F i r s t consider the case of A= 1 . The poles and integration path for x>0 are shown in Figure 1 1 . The contour i s closed in the upper half plane, and the path i s indented around each of the poles on the real axis. The 2 5 Cauchy p r i n c i p a l value of the d e s i r e d i n t e g r a l i s then obtained by l e t t i n g R approach i n f i n i t y and € approach zero . f o r the upper h a l f plane contour and pole indent contours r e s p e c t i v e l y . T h i s g i v e s T ( x ) - j r i E Residues = 0 or T(x) = rriE Res [ exp{ 2rr i kx} / (exp{ ir i kX Q }-A • exp{ - i r i kX 0 }) ] = TriE( <exp{27rikx}/7riX. 0 (exp{rrikX Q }+A«exp{-irikX 0 } ) = E n exp{ 2n i r i x / X Q }/XQ (exp{ n?r i }+exp{-nir i }) = Eexp{2nirix/X 0 }/2XG (-1 ) n ( 4 . 6 ) n The s e r i e s Eexp{2nirix/X 0} corresponds to d e l t a f u n c t i o n s of n s t r e n g t h X 0 f o r x=mX0 (m=0 , 1 , 2, . . .) . However, with the a l t e r n a t i n g phase of the denominator the s e r i e s of equation 4 . 6 i n s t e a d g i v e s r i s e to d e l t a f u n c t i o n s of s t r e n g t h X c f o r x=(2m+1)X 0/2 (m=0 ,1,2,...), so t h a t , T(X)=1/2 E 6(x-(2m+1 )X G/2) ( x > 0 ) ( 4 . 7 ) m For x < 0 , the i n t e g r a t i o n contour would be c l o s e d i n the lower h a l f p l a ne to give T(x)=-iriE Res[exp{2rrikx}/(exp{?rikX 0 }-A• exp{ - i rikX Q } ) ] =-1/2 E 6(x+(2m+1)k 0 /2) ( x < 0 ) ( 4 . 8 ) m Combining equations 4 . 7 and 4 . 8 f o r the complete s o l u t i o n f o r A=1, T(X)=1/2 E[6(x-(2m+1)X 0/2)-6(x+2m+1)X Q/2)] ( 4 . 9 ) m In the case of A<1, the p o l e s of equation 4 . 4 a l l l i e above the r e a l k a x i s . C l o s i n g the i n t e g r a t i o n contour i n the upper h a l f plane f o r x> 0 e n c l o s e s a l l of the p o l e s , while f o r x < 0 , T ( x ) = 0 . For x> 0 then, 26 T(x)=2irii: Res[exp{2irikx}/(exp{trikX 0 } -A • exp{ -IT i kX0 } ) ] = 2/X0 Iexp{2irikx}/iriX 0 (exp{irikX 0 }+A«exp{-irikX0 }) ] K T(x)=2/X QIexp{x• ln.(A)/X 0+2njrix/X 0 } n /[exp{ln(A)/2+riTri }+A«exp{-ln (A)/2-niri } ] T(x)=A ( 2 x" Ao ) / 2 X oE 6(x-(2m+1 )X c/2) (m=0 , 1 , 2 , . . . ) (4.10) m In the case of A>1 , the poles of equation 4.5 a l l l i e below the real k axis. One now obtains the contribution from the poles only for x<0. T(x)=-2iriE Res [ exp{ 2tr ikx}/(exp{ ir ikX D }-A. exp{ - IT i kXQ }) ] =_A(2x-A0)/2>0 E 6(x+(2m+1 )X Q/2) (m=0 ,1 ,2 , .. . ) (4.11) m The solution for T(x) in equations 4.9, 4.10 and 4.11 are shown in Figure 12. The discontinuous behaviour of the formal solution when one approaches A=1 suggests the use of the more physical form, T(x) = ( 1/2) - A ( 2 x- X° ) / 2 Ao l[6(x-(2m+1 )XG/2)-6(x+(2m+1 )X c/2) ] m (m=0,1,2,...) (4.12) for A near 1 and |x| of order X 0. This i s also i l l u s t r a t e d in Figure 12. While the restoration function of equation 4.12 represents an i n f i n i t e sum of delta functions; in practise i t need only extend over twice the size of the f i e l d being restored. The expected rms noise associated with the restored t o t a l intensity maps can be written in a form similar to equation 2.1. Since one i s combining the information from both horns, Ks=1, and since the convolution of the d i f f e r e n t i a l data with equation 4.12 corresponds to a summation over n independant samples spaced by the beam 27 spacing ka , the noise w i l l be Vrf times larger giving AT r m s=Vn-T s y s /VAuAt (4.13) for each receiver. As one might expect, the noise increases with the size of f i e l d being restored. The break even point for comparison with single beam measurements made in the absence of atmospheric fluctuations occurs for f i e l d s of size 4X.0 . In the presence of atmospheric fluctuations the single beam measurements w i l l have a substantially higher noise l e v e l . The dual beam function given in equation 4.3 i s applicable to d i f f e r e n t i a l measurements made using two beams of i d e n t i c a l shape with at most some variation in r e l a t i v e amplitude. Even with beams of i d e n t i c a l shape, the poor estimation of r e l a t i v e gain or beam separation w i l l give r i s e to c h a r a c t e r i s t i c patterns of a r t i f a c t s along the beam i n c l i n a t i o n d i r e c t i o n at multiples of the assumed beam spacing. These are i l l u s t r a t e d in Figure 13. It i s conceivable to accomodate more serious one dimensional differences between the beams by a more complex dual beam function. One could even imagine a two dimensional function of the form given in equation 4.3 which would allow for the d i f f e r e n t p r o f i l e s of the beams in the di r e c t i o n perpendicular to the beam. alignment. The complexities of the associated analysis seem somewhat forbidding however. A more straightforward method of dealing with differemt beam shapes i s the following. Suppose one has an accurate 28 representation of the telescope dual beam response, C. The observed brightness d i s t r i b u t i o n , 0 due to 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 modified d i s t r i b u t i o n 0' from 0'=S * C * C* (4.14) where C'(x)=-C(-x). The modified d i s t r i b u t i o n can then be written in the form 0'=S * B'* D in which D=6 (x + X.Q/2 )-6 (x-X c/2 ) for a modified single beam response given by B'= F" 1{c-c'/d} in which lower case l e t t e r s indicate fourier transformed quantities. One can then derive the modified single beam brightness 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 precisely by equation 4.9. While th i s approach allows the application of straightforward theory, i t s major drawback i s the degradation of instrumental resolution by a factor of about VT . It i s possible to obtain a s i g n i f i c a n t improvement over the standard Emerson convolution procedure by imposing an addditional constraint on the r e s u l t . Consider the dual-beam function for i d e n t i c a l beams, D(x)=6(x+X0/2)-6(x-X.0/2) The complete d i f f e r e n t i a l observations of any stucture obtained from a convolution with such a function must have 29 s t r i c t l y balanced pos i t i v e and negative response. This implies that the sum of a l l samples along the beam alignment d i r e c t i o n must be zero, as well as the more stringent requirement that the sum of of a set of samples along t h i s d i r e c t i o n which are spaced by the beam spacing must also be zero. If one then writes the dual-beam function of an asymmetric beam pair as D' (x) =6 (x + X.0/2 )-6 (x-X.o/2)+ € (x ) i t i s clear that the asymmetry embodied in e(x) w i l l give r i s e to an imbalance in the response. If one could e f f e c t i v e l y remove t h i s imbalance from the data, one could recover t o t a l intensity information by applying the convolution function T(x) given by equation 4.9. Since one does not generally know the form of the imbalance in the data, i t s detailed removal i s not t r i v i a l . One can however, use the properties of a balanced data set to guide one in thi s undertaking. By requiring that the sum of data samples along the beam alignment d i r e c t i o n and spaced by the beam spacing be zero, one can obtain at least p a r t i a l removal of € ( X ) . We can form th i s sum, Z(x) by the convolution Z = 0 * N for N(x) = E[6(x-(2m+1 )X0/2)+6(x+(2m+1 )X.Q/2) ] m=0,1,2,... m and 0, the actual dual beam observations of the sky brightness. If the beams were balanced then Z would be equal to zero for a l l x. Suppose that n data values Dj actually contribute to Z(x) for 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 modified data values Dj ' from Dj '=Dj -Z/n (4.16) where each data sample i s modified by a fixed amount and for which Ij Dj '=Ej (Dj -Z/n)=0 . Consider now the restored single beam observations obtained from P=0 * T with T(x) = l/2Z[6(x-(2m+1 )X.Q/2)-6(x+(2m+1 )X 0/2) ] m=0,1,2,... m as given in equation 4.9. For a p a r t i c u l a r value of x, P(x) can be written as n+ n. P(X) = 1/2 •E1 Dj -1/2 D k (4.17) for the n + p o s t i t i v e l y contributing data samples and the n_ which contribute negatively. These quantities are related to those defining Z above by n=n+ + n. and {Dj } = {Dj }U{Dk} ie the n samples in the set {Dj } are s p l i t into n + samples in the set {Dj} and n_ samples in the set {Dk}. Using the modified data values of equation 4.16 one obtains the modified restored d i s t r i b u t i o n by substitution into equation 4.17 P' (X) = 1/2 Sj (Dj-Z/n)-1/2 k?" (Dk-Z/n) =P(x) - n +«Z/2n + n_«Z/2n P'(x)=P(x) + Z-(n. - n +)/2n (4.18) The modified restored d i s t r i b u t i o n i s thus given by the standard restored d i s t r i b u t i o n given by equation 4.17 31 modified by an additional term which attempts to eliminate the unbalanced response in the data. An ideal data base w i l l of course not be affected by thi s procedure since the function Z(x) w i l l already be i d e n t i c a l l y zero. However, an actual . data base which suffers from unbalanced response due to beam asymmetry as well as inadequate representation and noise fluctuations w i l l benefit considerably from this approach. The p r a c t i c a l effectiveness of thi s method i s affected by the extent of the region being restored. Whenever possible, areas with d i s t i n c t d i f f e r e n t i a l response should be restored separately to prevent the d i l u t i o n of the data modification by a large number of unrelated samples. This was the approach used to obtain the successfully restored map described in the next section. 4.4 Implementation of the Emerson Technique Given data c o l l e c t e d by similar beams aligned with the scanning d i r e c t i o n , a scan by scan convolution with the function T(x) as given in equation 4.12 should provide the single beam brightness d i s t r i b u t i o n which would have been observed. The observations undertaken here however, were ca r r i e d out with the beams inclined by 11° to the scanning d i r e c t i o n . (The question of beam s i m i l a r i t y i s dealt with l a t e r . ) The application of a series of one dimensional convolutions would then require i n i t i a l interpolation and rotation of the data base to provide one axis p a r a l l e l to the beam alignment, followed by another rotation to the 32 display coordinates of RA and Dec. Alternatively,, one could perform, a two. dimensional convolution on an interpolated data base gridded in the f i n a l coordinates. The l a t t e r option was chosen here. The raw data scans with linear baselines removed were f i r s t smoothed by convolving 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 halfwidth was chosen as a comprimise to provide adequate smoothing without undue broadening of the instrumental response. The two independent data sets consisting of solely northbound or southbound scans were then i n d i v i d u a l l y gridded onto (RA,Dec) 1950 grids by the method of section 4.2. The binwidths ARA and ADec were chosen to allow the delta functions of T(x) to f a l l exactly on other g r i d locations upon application of T(x) to any grid point. S p e c i f i c a l l y , since the angle that the beams make with respect to RA=constant i s e=11°-tan" 1(15cos(Dec)/120) (rounded to the nearest integer degree), i t is required that (nADec) 2 + (mARA)2 = (X Q/2) 2 and also that (mARA)/(nADec)=tane The integers n and m were chosen to provide adequate resolution. The values used were n=8 and m=1 which give ARA and ADec about 0.5 arcmin for a wide range of declinations. After gridding, i t i s necessary to f i l l unassigned gri d locations with interpolated values in order to provide a 33 smooth f u l l y sampled data base. A linear two dimensional interpolation technique, which uses the nearest two data points in each of two perpendicular di r e c t i o n s , does not allow the d e f i n i t i o n of values outside of the intensity range they define. Since the data scans make only a small angle with respect to RA=constant, one already has v i r t u a l l y complete f i l l i n g in th i s d i r e c i o n . For these two reasons i t was decided to perform one dimensional interpolation in the Dec=constant d i r e c t i o n by a spliced quadratic Lagrangian polynomial. Since one i s then l e f t with minor d i s c o n t i n u i t i e s from row to row, the f i l l e d grids were then smoothed by gaussian f i l t e r i n g in the fourier transform plane, corresponding to convolution by a 1 arcmin FWHM c i r c u l a r gaussian. It was shown in chapter 3 of this investigation that the two 6 cm f a r - f i e l d beams of the NRAO 9 1 m telescope are in fact s i g n i f i c a n t l y d i f f e r e n t . The beam shapes and strengths are furthermore strong functions of declination. At the declination of G109.1-1.0, the peak gains of the two beams are d i f f e r e n t by more than 15%, while the integrated gains of the beams are very nearly the same owing to the di f f e r e n t beam shapes. The e f f e c t i v e use of the Emerson restoration method using the convolving function of equation 4.12 was thus somewhat limited from the outset. The d i f f e r e n t beam shapes would result in some combination of the a r t i f a c t s i l l u s t r a t e d in Figure 13. To minimize such e f f e c t s , the constraint of balanced response, described in 34 the previous section was applied to the data at the same time as the convolution function of equation 4 . 9 . In order to prevent the propagation of any remaining a r t i f a c t s from the strong compact source S152 into the extended structure of G109.1-1.0, the region surrounding S152/153 and S148/149 (with d i s t i n c t d i f f e r e n t i a l response), received separate application of the convolution function of equation 4.9 as modified by equation 4.18. The northbound and southbound restored grids were then averaged together to reduce the l e v e l of the uncorrelated a r t i f a c t s , except in the area containing the Sharpless regions where more complete information was available in the northbound data set. The expected rms noise in the restored map, obtained from equation 4.13 for a 13X.0 f i e l d , when using both receivers and data sets i s 15 mK. 4.5 Emerson-Clean Hybrid A commonly used method for improving the results of astronomical measurements made with a po s i t i v e primary beam which has known sidelobe structure i s that c a l l e d CLEAN (Hogbom 1974). The method i s based on the assumption that more complicated structure can be simulated by an appropriate summation of point responses. In practise, a data base i s searched i t e r a t i v e l y for the position of largest p o s i t i v e (or negative) response. At t h i s p o s i t i o n , some fra c t i o n of the instrumental response i s subracted from the data base while the fraction and location are 35 accumulated. After source removal i s complete down to some l e v e l , the c o l l e c t i o n of weighted delta functions i s convolved with a "clean beam" and added back to the data base. In the case of an asymmetric d i f f e r e n t i a l instrumental response, after convolution with the Emerson restoration function T(x) one i s l e f t with a positive primary reponse with known sidelobe structure. Using the beam model of chapter 3 at DEC=58.8° and assuming equal gains for T(x) one obtains the response pattern i l l u s t r a t e d in Figure 14. One should in p r i n c i p a l then be able to CLEAN the i n d i v i d u a l l y restored grids of t h i s type of sidelobe structure. A fortran computer program to carry out the cleaning operation was written. Simple test sources were completely "observed" with the beam model, restored to t o t a l intensity by T(x) (of equation 4.9) and cleaned with the convolved beam model shown in Figure 14. The re s u l t s showed s i g n i f i c a n t reduction of sidelobe l e v e l from about 10% to 2%. However, tests made with only p a r t i a l l y "observed" data and t r i a l s made on the actual restored maps (of the standard Emerson procedure) did not show s i g n i f i c a n t improvement by the cleaning operation. 4.6 Theory of the Maximum Entropy Method The maximum entropy approach to image reconstruction attempts to find the unique sky brightness d i s t r i b u t i o n that i s most uniform while consistent with the instrumental 36 p r o f i l e and data that one has av a i l a b l e . The degree of uniformity i s defined as the minimum information content of the picture and i s obtained by maximizing the configurational entropy, S of the image. The most reasonable form for t h i s quantity (Frieden 1972, S k i l l i n g 1981) i s S=-Ej pj l o g(pj ) (4.19) where ,pj i s the normalized map intensity in pi x e l j given by for the map intensity f; summed over the number of pixels n. The function S has a global maximum where a l l the fj are equal, corresponding to no structure at a l l in the image. Consistency with the data i s measured by some suitable s t a t i s t i c . For convolution problems, in which one assumes gaussian errors of known variance for the individual data samples, an appropriate choice i s the normalized chi-squared s t a t i s t i c (Gull and Daniell 1978) x 2 ^ ( F k - D k ) 2 / ( r * 2 k ) (4.20) in which' D k i s the data value in p i x e l k, <rk i s the standard deviation of that data sample, y i s an a r b i t r a r y normalization factor and F k i s the derived data value in p i x e l k from for B the instrumental response to pi x e l j . Choosing y=N, a value of x 2 = 1 +3.29/VFf represents the 99% confidence l e v e l in the f i t to data. (Abramowitz and Stegun 1964). Values larger than t h i s indicate a poor f i t , while very small 37 values imply the undesirable f i t t i n g to noise in the data. The solution i s found by maximizing S subject to the constraint that x 2 = 1 . Using the method of Lagrange m u l t i p l i e r s , one wishes to maximize, Q=S-X.x2 (4.21) with X. assuming a suitable value to s a t i s f y the constraint. 4.7 Implementation of the Maximum Entropy Method Since astronomical observations t y p i c a l l y involve 1282 or more data locations, one can e a s i l y imagine the d i f f i c u l t i e s involved in obtaining a solution to the non-linear optimization problem of equation 4.21 with reasonable l i m i t s on computation time. The method used here for i t s solution i s that developed S.F. G u l l , G.J. Daniell and J. S k i l l i n g . The algorithim was most recently described by John S k i l l i n g (1981). The maximum entropy subroutine was used in conjunction with a computer program written by S.F. Gull in July 1981 to accomodate this p a r t i c u l a r astronomical a p p l i c a t i o n . A s l i g h t l y modified form of the configurational entropy i s used in the subroutine S=-Ej fj [log(fj /DEF)-1 ] (4.21) which allows the s p e c i f i c a t i o n of a default l e v e l , DEF to which the picture w i l l tend in the absence of information to the contrary. The program begins with a f l a t picture and proceeds i t e r a t i v e l y toward a solution. It's progress i s monitored by the current values of x 2 and the quantity TEST TEST=(1/2)•(grad(S)/|grad(S)| - g r a d ( x 2 ) / | g r a d ( x 2 ) | ) 2 38 which measures the simultaneous s a t i s f a c t i o n of the two c r i t e r i a which define the solution. In practice, one can obtain the s p e c i f i e d value of x2 (=1.00000) while TEST<0.01 after on the order of 10 i t e r a t i o n s provided that one has s p e c i f i e d a reasonable .instrumental p r o f i l e Bj^ and sample variance e2^ . A special feature of the present observations i s the a v a i l a b i l i t y of two independent data sets (corresponding to the northbound and southbound scans) measured with the feeds at d i f f e r e n t rotation angles. One can use an additional data set D|<' (assuming i t represents the same wavelength, po l a r i z a t i o n , etc.) to constrain the maximum entropy solution simply by incorporating i t into the sum of equation 4.20 with the appropriate instrumental response B '-Gull's program allows for the s p e c i f i c a t i o n of multiple data/resonse pa i r s . Details concerning the use of t h i s program are available in the program manual, "MEM Deconvolution Program" (R.Braun and P.C.Gregory 1981). The production of maximum entropy images requires, to begin with, the gridded set(s) of unsmoothed data. These were provided by the method of section 4.2. The eventual resolution obtainable with t h i s method i s about twice that of the measurements, with signal to noise of about 100. For t h i s reason a c e l l size of ARA=ADec=0.75 arcmin was chosen for the grids. One also requires the measured or modeled instrumental p r o f i l e ( s ) . The model developed in chapter 3 and rotated to the appropriate i n c l i n a t i o n angle was used 39 for t h i s purpose. F i n a l l y , one must estimate the variance in the individual data samples. The quantity a c t u a l l y required by the maximum entropy subroutine i s twice the normalized inverse variance, E K = 2 / ( y * 2 k ) at each grid location. The value E K = 0 i s assigned to any unmeasured grid locations so that these do not contribute to x 2. As i n s t a l l e d , the c a l l i n g program allowed for the s p e c i f i c a t i o n of a fixed noise figure fff=constant, representing the rms noise fluctuations of each measured data sample. Experimentation with a fixed noise figure showed that the residuals of the program f i t were heavily concentrated around regions of intense emission. This suggested that more elaborate noise estimation might be necessary to provide a more uniform (and hence credible) f i t to the data. The various types of nonrepeatable signal fluctuations that might be expected in the data are the following: 1 ) short time scale (about 0 . 2 sec) receiver noise fluctuations 2 ) intermediate time scale (mins.) receiver gain fluctuations 3) longer time scale (min. to days) telescope pointing fluctuations Receiver noise fluctuations can be described by a fixed rms noise figure, «^ , applying to a l l data samples. The vlaue 40 Cf =0.015 K was found appropriate for the data here. Receiver gain and telescope pointing fluctuations become important since the data was c o l l e c t e d during 8 minute intervals of driven telescope operation on each of 10 days. Receiver gain fluctuations w i l l give r i s e to errors which scale l i n e a r l y (to f i r s t order) with the observed i n t e n s i t y . From consideration of noise tube c a l i b r a t i o n s , t h i s e f f e c t was expected to contribute at the 5% l e v e l (<*g=0.05). Pointing variations, on the other hand, w i l l contribute most s i g n i f i c a n t l y to the uncertainty where the observed intensity i s changing most rapidly. While the rms pointing variations of the NRAO 91m telescope are thought to be tfp=0.1 arcmin, the incorporation of t h i s effect into the sample variances is not as straightforward as those mentioned above. Since intensity gradient information perpendicular to the scan d i r e c t i o n i s somewhat d i f f i c u l t to access, and since the steepest gradients occur between the two beams and hence approximately on the scan track, the following scheme was adopted. Inverse variance scans were generated from smoothed scans from Ek'=2/r[<r2f + U g D k * ) 2 + U p - g r a d ( D k ' ) ) 2 ] in which r=N, the t o t a l number of gridded data points used for the maximum entropy reconstruction and grad(D k') i s the l o c a l intensity gradient along the scan in units of K/arcmin. These scans were then gridded in the same way as the unsmoothed data scans to provide the quantity E k for 41 each gridded data sample D^. Using the standard deviations above for both the northbound and southbound data sets, a maximum entropy solution with x2=0.99994 and TEST=0.0009 was obtained in 12 i t e r a t i o n s . Varying the fixed noise figure gf to 0.020 gave overly rapid convergence to a low resolution result, while the use of <tf =0.010 led to asymptotically slow progress after more than 20 ite r a t i o n s with l i t t l e hope of a sa t i s f a c t o r y r e s u l t . While the Gregory-Taylor survey data was c o l l e c t e d in the same way as that for th i s f i e l d , the scans are generally more coarsely spaced. To assess the p o s s i b i l i t y of producing r e l i a b l e maps with less complete data sampling, approximately half of the available scans of G109.1-1.0 were removed from the data set. The remaining data locations are shown in Figure 15. 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 data was gridded and received maximum entropy processing in the same manner described above. Again a solution was obtained in 12 i t e r a t i o n s with x2=1.00003 and TEST=0.0010. 5 9 ° DK (1950) D » c ( 1 9 5 0 ) Figure 9. Raw D i f f e r e n t i a l Scans. The unsmoothed d i f f e r e n t i a l scans with only baselines removed are displayed with intensity represented as a v e r t i c a l d e f l e c t i o n from the scan path. The dashed l i n e s mark b = 0 ° and b = - 1 ° . 43 P. d k A k r 1 1 1 < A j > P ^ u J *• A X = Aj / d j 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 Figure 1 1 . Integration Path for Transform Inversion. The integration path in the complex k plane for determining the Emerson convolution function when A=1. 45 a) 0.5- l | I 1 *>• 1 1 1 1 - X b) 0.5-l X 0 0.5- ^ X Figure 12. Emerson Method Convolution Function, a) The solution for the convolution function when A=1. b) The solution for A<1, shown for A=0.5. c) The solution for A>1, shown for A=2. d) A more continuous solution near A=1 suggested for use with f i n i t e x and shown for A=0.95. 4 6 b) °1\ 1 4 A / 2 V t T 7 — ^ 7 ~ x * Figure 13. A r t i f a c t s of Restoration. a) The unbalanced reponse due to a point source. b) The resulting a r t i f a c t s in the restored map when equal gains were assumed, c) The response to a point source with beam separation XC+AX. d) The resulting a r t i f a c t s when a beam separation X 0 i s assumed. 47 Figure 14. Convolved Beam Model. The beam model of figure 8 rotated to one of the measuring configurations and convolved with the function of equation 4.9. 48 Figure 15. Location of Data Scans for Sparse Data Test. Typical spacing of p a r a l l e l scans i s 3.2 arcmin. 49 5. Comparison of Results The 6 cm t o t a l intensity map of G109.1-1.0 obtained from the Emerson convolution method and modified by the constraint of balanced response (see section 4.3) i s shown in Figure 16. Decaying sidelobe response at the 5% l e v e l i s v i s i b l e along the beam alignment d i r e c t i o n to the south of S152. The extended structure of the remnant appears to be free of such a r t i f a c t s of the reconstruction method down to the 15 mK l e v e l . To give an indication of the resolution of th i s map, cross-sections at fixed RA and at fixed Dec through the peak of S152 are shown in Figure 17. The halfwidths of the response to t h i s source are 4.35 and 3.46 arcmin respectively. The expected halfwidths of the telescope beam in these di r e c t i o n s , on the basis of the beam model and the data smoothing, are 4.12 and 3.00 arcmin. The difference in these values i s l i k e l y due to the extended structure of S153 centered only 5 arcmin to the SE of S152. Recent VLA observations of S152 at 6 cm wavelength with 3.7 arcsec resolution (Gregory, personal communication) indicate a FWHM of 20 arcsec r u l i n g out the p o s s i b l i t y of observing s i g n i f i c a n t beam broadening here. As indicated in section 4.5, no s i g n i f i c a n t improvement over the standard Emerson result was obtained by applying CLEAN to the restored maps. The testing of th i s hybrid method seemed to indicate an inadequacy in the interpolation procedure. A one dimensional, spliced quadratic Lagrangian, while more e f f e c t i v e than a linear or cubic interpolation 50 scheme, was not able to represent the d i f f e r e n t i a l data d i s t r i b u t i o n to better than about 7% given the available data spacing of 1.6 arcmin. The maximum entropy image of the same f i e l d , using a l l of the available data, (two crossed sets of p a r a l l e l scans spaced by 1.6 arcmin), i s shown in Figure 18. A brief inspection of th i s image indicates substantially higher resolution than the Emerson convolution r e s u l t . The ionized hydrogen regions S152 and S153 are c l e a r l y resolved and finer structure i s v i s i b l e within the semicircular remnant. Cross-sections through the peak of S152 show 1.88 and 1.65 arcmin FWHM for the constant RA and constant Dec cuts. (See Figure 19.) The resolution of the map in general, i s not readily apparent however. This w i l l depend upon the signal to noise r a t i o of the data used which varies over the f i e l d . To determine the agreement of the maximum entropy and Emerson maps at a more similar resolution, the former was convolved with an e l l i p t i c a l gaussian with halfwidths of 3.9 and 3.0 arcmin in the constant RA and constant Dec d i r e c t i o n s . The result i s shown in Figure 20. This map and the Emerson result have very consistent temperatures in the regions surronding S152/153, S148/149 and the extension to the NW, down to almost the 15 mK l e v e l . The major difference in the r e s u l t s , l i e s in the l e v e l of the central portion of the remnant, being perhaps 30 mK higher over this entire region for the Emerson r e u l t . This i s contrasted to the 50 mK higher temperature obtained for the southern 51 remnant hot spot in the smoothed maximum entropy r e s u l t . These differences may be due to i n s u f f i c i e n t smoothing applied to t h i s portion of the maximum entropy r e s u l t . The two approaches may be giving divergent interpretations for another reason. There are a number of missing scans (see Figure 3) in both the northbound and southbound data sets in the v i c i n i t y of the extended structure. Since the Emerson method cannot use both data sets simultaneously, and one must f i r s t interpolate across any gaps in the individual data sets, some uncertainty i s introduced into the r e s u l t . A quadratic interpolation procedure would tend to smooth over such gaps in the data, r e s u l t i n g in a generally higher deduced emission l e v e l than the more f u l l y constrained maximum entropy solution. The maximum entropy image of t h i s same f i e l d using only about half of the available data (two crossed sets of p a r a l l e l scans spaced by 3.2 arcmin) is shown in Figure 21. The sparser data sampling results in somewhat lower resolution. Perpendicular cross-sections through S152 are shown in Figure 22 and show 2.11 and 1.89 arcmin FWHM. The e f f e c t s of undersampling are most apparent in the extended structure of the remnant, es p e c i a l l y along i t s northern edge. The restored compact structure i s remarkably similar to that of the complete data map of Figure 18. Even the extended structure i s represented s u f f i c i e n t l y well to allow recognition of major features. After smoothing t h i s result to the resolution of the Emerson map (Figure 23), the 52 extended structure i s represented very nearly as well as the s i m i l a r l y smoothed f u l l y sampled maximum entropy map. What makes t h i s possible i s the ef f e c t i v e use of both the northbound and southbound data sets to simultaneously constrain the solution. When using the Emerson convolution technique one does not have th i s advantage since the maps pertaining to d i f f e r e n t beam alignments must be restored i n d i v i d u a l l y . The required interpolation across gaps of 3.2 and 4.8 arcmin (with the present beam FWHM of 3 arcmin) would introduce severe uncertainties in the individual data sets which would not be correlated with a s p e c i f i c s p a t i a l frequency response. No e f f e c t i v e means for removing these uncertainties would thus be available when combining the two restored maps. 53 Figure 16. Emerson Method Restored Map. Contours are at 15, 30, 50, 70, 100, 150, 200, 300, 500 and 700 mK. Negative contours at -30 and -15 mK and valleys are hatched. Figure 17. Cross-sections through S152 i n Emerson Map. 5 5 F i g u r e 18. Maximum Entropy Restored Map. Contours are at 15, 30, 50, 70, 100, 120, 200, 300, 500, 700, 1000 and 2000 mK. Figure 19. Cross-sections through S152 i n Maximum Entropy Map. 5 7 4 0 23 2 0 23 0 0 22 58 0 22 5 R I G H T A S C E N S I O N ( H R S . . M I N S . . S E C S . ) Figure 20. Maximum Entropy Map smoothed to resolution of Emerson Method Map. The map of figure 18 convolved with an e l l i p t i c a l gaussian to obtain approximately the resolution of the map of figure 16. 58 l — i 1 1 — — — i 1 1 ^ 1 1 r Figure 21. Maximum Entropy Restored Map from Sparse Data. The map made with only those data locations shown in figure 15. Contours are at 15, 30, 50, 70, 100, 120, 200, 300, 500, 700, and 1000 mK. Figure 22. Cross-sections through S152 i n Sparse Data Maximum Entropy Map. 60 Figure 23. Sparse Data Maximum Entropy Map smoothed to resolution of Emerson Method Map. The map of figure 21 convolved with an e l l i p t i c a l gaussian to obtain approximately the resolution of the map of figure 16. 61 6. Other Observations of the F i e l d Emission from the v i c i n i t y of G109.1-1.0 has been detected in a number of surveys of the g a l a c t i c plane. The survey of Wilson and Bolton (i960) at 960 MHz with 48 arcmin beam detected emission from t h i s v i c i n i t y (CTB109) with an integrated flux of 75 Jy, while the 3200 MHz survey of RaghavaRao et a l (1965) with 38 arcmin beam established an integrated flux of 40 Jy. The more recent work of F e l l i and Churchwell (1972) at 1400 MHz with 10 arcmin beam provided the f i r s t map with adequate resolution to examine some stru c t u r a l features of the remnant. The 6 cm t o t a l intensity map produced by the maximum entropy method and smoothed to 10 arcmin resolution i s shown in Figure 24 to allow comparison. While S152/153 and S148/149 have coincident peak emission in the two maps, the structure within G109.1-1.0 appears s i g n i f i c a n t l y d i f f e r e n t at these two wavelengths. At 6 cm, the smoothed structure i s in the form of a broad plateau with hot spots of similar intensity in the NE and SW portions of the remnant's half d i s c . While the 21 cm continuum emission shows a similar plateau, the emission in the SW i s concentrated at a smaller radius and only one hot spot (with twice the plateau intensity) i s apparent at a position 7 arcmin south of the 6 cm hot spot in the NE. Unfortunately the F e l l i and Churchwell map does not extend north of DEC=58°52' so that comparison of the NW extension v i s i b l e in the 6 cm maps i s not possible. Information about the environment of G109.1-1.0 i s 62 provided by the carbon monoxide observations of I s r a e l (1980) and Heydari-Malayeri et a l (1981). These indicate the presence of a molecular cloud of V(LSR)= -53km/s within which the Sharpless regions S147-S153 are contained. This cloud appears to butt against the western face of the source with a minor extension into the southern gulf of the 6 cm map at RA=22h 59m, Dec=58.5° and a major extension (prominent in the 1 3CO observations of Heydari-Malayeri et a l (1981)) into the northern gulf as far as RA=23h 00m, Dec=58.7°. Another molecular cloud becomes prominent where the NW extension of the 6 cm intensity begins to fade and continues in t h i s d i r e c t i o n . The densest part of t h i s molecular cloud i s near i t s southern extremity, within about 5 arcmin of the major hot spot of the 6 cm extension. The c o r r e l a t i o n of these observations suggests a physical interaction, or at least a considerable degree of l i n e of sight absorption between the molecular cloud complex and the central feature and extension of G109.1-1.0. The X-ray observations of Gregory and Fahlman (1980) are schematically superimposed on the 6 cm map in Figure 25 and show the unusual bifurcated feature which extends eastward and northeastward from the X-ray pulsar 1E2259+586 (Fahlman and Gregory 1981). The coordinates of t h i s source, accurate to 5 arcsec, are RA=22h59m03.4s, and Dec=58°36'38". The maximum entropy restored 6 cm map also shows evidence for a feature at a position in agreement with the X-ray coordinates. The radio coordinates taken from the 63 perpendicular cross-sections of Figure 26, to which known pointing corrections have been applied, are RA=22h59m01s and Dec=58°36'55". These coordinates are thought to be accurate to 0.5 arcmin. The equivalent point source flux of thi s peak, r e l a t i v e to the ridge in which i t i s embedded i s 25 mJy. Recent VLA observations at 21 cm (Gregory private communication) show no evidence for a compact source at thi s position down to the 0.5 mJy l e v e l , while a compact source is observed about 2 arcmin west of the pulsar. Only marginal evidence i s seen for a radio counter-part to the prominent X-ray feature which curves northeasterly from the central source. The 6 cm map shows a weak spur near RA=23h00m, Dec=58.6° which extends into a region of otherwise weak emission, and agrees in position with the X-ray feature. The X-ray and radio data do show ove r a l l agreement on the extent of emission as well as some regions of enhanced emission in the southern half of the source, with the notable exception of the radio hot spot near RA=22h59m, Dec=58.3°. 64 Figure 24. Maximum Entropy Map smoothed to 10 arcmin Resolution. The map of figure 18 convolved with an e l l i p t i c a l gaussian to obtain the resolution 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 r e l a t i v e intensity levels of the X-ray emission (Gregory and Fahlman 1980) are overlaid on the 6 cm continuum map of figure 1 8 . The dashed l i n e marks the outer boundary of the X-ray emission. 66 0.8 * U 0.4-0 . 0 K R A = 2 2 h 5 9 m 0 1 s 15 CMIN 45 60 Figure 26. Cross-sections through Central Peak in Maximum Entropy Map. Perpendicular cross-sections through the 6 cm peak which i s coincident in position with the compact X-ray source and has an equivalent point source flux of 25 mJy. 67 7. Summary and Conclusion The production of t o t a l intensity maps of extended structure from beam-switched observations has become a p r a c t i c a l undertaking. One can take advantage of the beam switching technique to eliminate atmospheric fluctuations without losing large scale s t r u c t u r a l information. Emerson et a l (1979) showed how d i f f e r e n t i a l measurements could be a n a l y t i c a l l y restored to the equivalent single beam response. This method was extended here and successfully applied to observations made under less than ideal circumstances. In the present case, the two beams were known to be of s i g n i f i c a n t l y d i f f e r e n t shape. They were, in addition, rotated s i g n i f i c a n t l y from the scanning d i r e c t i o n . Despite these complications, the restored map (Figure 16) shows a noise l e v e l consistent with expectations of about 15 mK with v i r t u a l l y no evidence of processing a r t i f a c t s . (See section 4.3.) The maximum entropy method of image reconstruction (as developed by S.F.Gull, G.J.Daniell and J . S k i l l i n g ) was also successfully applied to the problem of reconstructing d i f f e r e n t i a l measurements. This approach o f f e r s a number of s i g n i f i c a n t advantages over other methods of data processing. Within one processing phase t h i s method eliminates the need for interpolation and p r e - f i l t e r i n g of high frequency noise, allows for deconvolution from an undesirable instrumental response and as a bonus, provides resolution enhancement. This i s accomplished with moderate 68 computational costs, which were in th i s case less than those of the Emerson method. The resulting image (Figure 18) shows resolution enhancement by a factor of about 2, with a l l of the indicated areas of emission appearing to be j u s t i f i e d by the data. When smoothed to the resolution of the Emerson res u l t , the two images show agreement over much of the f i e l d to within 20 mK. The only s i g n i f i c a n t discrepancy l i e s in the deduced emission l e v e l of the extended h a l f - d i s c of the remnant; being about 30 mK lower in the maximum entropy map. This difference i s l i k e l y due to the more e f f e c t i v e use of a l l available data to constrain the maximum entropy solution. (See chapter 5.) The maximum entropy method was also shown to be e f f e c t i v e with sparser data sampling. The image in Figure 23 was obtained from two interleaved sets of p a r a l l e l scans spaced by about 3.2 arcmin, and i s remarkably similar to the result obtained with more complete sampling. On the basis of t h i s study, t h i s approach seems to offer the optimum information return when applied to completely or p a r t i a l l y sampled data. The present work can now be d i r e c t l y applied to the reduction of the Gregory-Taylor survey. As of October 1981, the survey data base w i l l contain multiple repeats of p a r a l l e l northbound and southbound scans spaced by as l i t t l e as 2.4 arcmin at Dec=60°. More than 85% of t h i s data base has scan spacings of 3.2 arcmin or l e s s . This allows the production of images with greater q u a l i t y • than that of 69 Figure 23 (due to subst a n t i a l l y lower noise) to be made for the majority of the survey area. The past and present observations of G109.1-1.0 indicate a complex internal structure and interaction with i t s environment. A peak in the 6 cm emission was found at a position coincident with the X-ray pulsar (Gregory and Fahlman 1981). The estimated 6 cm flux of t h i s source r e l a t i v e to the ridge in which i t i s embedded i s 25 mJy. There i s no obvious radio counterpart at the position of the prominent X-ray j e t , although the extent of emission and h a l f - d i s c appearance of the remnant are remarkably s i m i l a r . The 6 cm observations have also revealed an unusual feature which extends from the northern edge of the remnant for 35 arcmin r a d i a l l y away from the center to the north-west. The generally semi-circular shape of the remnant, as indicated by both the X-ray and radio observations, can be understood in terms of an interaction with an adjacent molecular cloud. A thorough discussion of these and more recent unpublished observations is being prepared in collaboration with P.C. Gregory and G.G. Fahlman. 70 References. 1) Gregory,P.C. and Taylor,A.R., 1981, Ap.J. (in press) 2) Gregory,P.C. and Fahlman,G.G., 1980, Nature, 287, p805 3) Emerson,D.T., Klein,U. and Haslam,C.G.F., 1979, Astron. Astrophys., 76, p92 4) Gull,S.F. and Daniell,G.J., 1978, Nature, 272, p686 5) T i u r i , M.E., 1966, in 'Radio Astronomy', Kraus,J.D., McGraw-Hill 6) Hogbom,J.A., 1974, Astron. Astrophys. Suppl. , 1_5, p4l7 7) Frieden,B.R., 1972, J.opt.Soc.Am., 62, p511 8) S k i l l i n g , J . , 1981, (preprint) 9) Abramowitz,M. and Stegun,I.A., 1964, 'Handbook of Mathematical Functions', Dover, New York 10) Wilson,R.W. and Bolton,J.G., 1960, PASP, 72, p331 11) RaghavaRao,R., Medd,W.J., Higgs,L.A., and Broten,N.W., 1965, MNRAS, 129, pl59 12) F e l l i , M . and Churhwell,E., 1972, Astron. Astrophys. Suppl., 5, p369 13) Israel,F.P., 1980, Astron. J., 85, pi 612 14) Heydari-Malayeri,M., Kahane,C. and Lucas,R., 1981, (preprint) 15) Fahlman,G.G. and Gregory,P.C., 1981, Nature, (in press) 

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