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Analysis of radio recombination lines Perley, Richard Alan 1970

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v AN ANALYSIS OF RADIO RECOMBINATION LINES by R. A. PERLEY B . S c , Un ive r s i ty of B r i t i s h Columbia, 1968 A THESIS SUBMITTED IN PARTIAL PULFILMENT OE THE REQUIREMENTS FOR THE DEGREE OF . MASTER OF SCIENCE i n the Department of PHYSICS We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1970 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that 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 fo r reference and study. I f u r t h e r agree t h a t permiss ion fo r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood that 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 ga in s h a l l not be a l lowed without my w r i t t e n p e r m i s s i o n . Department of PHY&CS The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada Date SEPT 2 9 19?d ABSTRACT The formation of spect ra l l i ne s i n the radio region from ga lac t i c nebulae i s s tudied. Equations describing the strengths of these l i ne s are developed for both the l o c a l thermodynamic equi l ibr ium case and the s t a t i s t i c a l equi l ibr ium case. An experiment i n which radio l i n e s from ga lac t i c nebulae were detected i s described. The reduction of the observations to obtain estimates of the temperature and density of the ga lac t i c gas i s performed, and comments on the v a l i d i t y of the analys is are included. - i i i -TABLE OF CONTENTS Chapter 1 INTRODUCTION Page 1 His tory and Background 1 2 Purpose of the Experiment 4 Chapter. 2 GENERAL THEORY 1 Background 5 2 Def in i t ions . . 13 3 Equation of Transfer 17 4 The b n Problem 30 Chapter 3 THE EXPERIMENT 1 The Receiving Equipment 35 2 The Observing Program 36 3 Data Reduction 37 4 Continuum Observations • • • • 40 5 Results 41 6 Elec t ron Temperatures 45 7 Concluding Remarks 52 Appendix A ELECTRON SCATTERING A l - i v -ACKNOWLEDGEMENTS . I g ra te fu l ly acknowledge the assistance and helpful d iscus-sions of Dr. W. L . H. Shuter. His guidance was a major factor i n the preparation of th i s thes i s . I would also l i k e to record my thanks to both Dr. Wi l l i am McCuteheon and my colleague, Mr. "M. J . Mahoney, for he lpfu l and enlightening d iscuss ions , both re la ted and unrelated to th i s t hes i s . Many thanks are due to the s ta f f of the Dominion Radio Ast rophysica l Observatory for generously providing the i r f a c i l i t i e s . F i n a l l y , I wish to acknowledge the Nat ional Research Council for t h e i r f i n a n c i a l assis tance. - 1 -•-Chapter 1 INTRODUCTION. 1-1) In 1929, I . Langmuir gave the name plasma to those re -l a t i v e l y f i e l d - f r e e regions of a highly ionized gas where pos i t i ve and negative space charges are nearly balanced. Plasmas are of considerable as t rophysical in teres t because of the known existence of ionized gases both i n and near the s ta r s , and are eas i ly de-tected because of the i r cha rac t e r i s t i c emission of r a d i a t i o n . There are two important emission mechanisms which contribute i n the radio region. One mechanism i s known as free-free emission, -responsible for the "continuum" or "background" r ad ia t ion of H I I regions - those regions of ionized hydrogen that surround hot s ta rs . Free-free emission i s caused by the def lec t ion of electrons passing close by,an ionized atom, and var ies slowly wi th frequency. The other -mechanism i s that of bound-bound emission, where an e lect ron i n some quantum l e v e l n characterized by energy E changes to l e v e l n ' and energy E ' . Accompanying t h i s change of state w i l l be an emitted photon whose frequency i s given by the f ami l i a r r e l a t i o n (1-D -where-h i s P lanck ' s constant. I f E - E ' i s l a rge , the frequency of the emitted r ad ia t ion w i l l be high and w i l l be observed i n the o p t i c a l range. I f E - E ' i s smal l , (say 10 "'"''erg), then the emitted r ad ia t ion w i l l be i n the radio range. -These two-emission mechanisms are-the only s i gn i f i c an t mech-anisms i n the radio range (10 GHz.to 100 MHz). Others w i l l not be discussed. - 2 -The fundamental differences between the two mentioned emission mechanisms are eas i ly seen. Because the free-free emission i s the r e su l t of myriads of encounters of d i f ferent v e l o c i t i e s and impact parameters, the emitted r ad ia t ion i s smeared out and var ies l i t t l e with ei ther frequency or temperature. In the bound-bound emission, however, the frequency i s nearly always the same for any pa r t i cu l a r t r a n s i t i o n . F u r t h e r , i t s i n t ens i ty i s dependent upon the l e v e l population which i n turn i s a sens i t ive function of temperature. Hence the bound-bound emission takes the form of a peak of emitted power and i s ca l l ed a "recombination l i n e " since i t i s part of the process where a free e lectron recombines with an ionized atom and eventually descends back to the ground s ta te . These notions are more f u l l y developed i n Chapter 2. Since the main constituentr.t of ga lac t i c plasmas i s hydrogen, and since the emissions of t h i s element were those observed i n th i s experiment, we consider the bound-bound l i n e frequencies of t h i s element. For the large values of n involved i n radio l i n e s , the l i n e frequencies are given to su f f i c i en t accuracy by the Rydberg formula Here, Z i s the e f fec t ive nuclear charge, c the v e l o c i t y of l i g h t , and R the appropriate Rydberg constant. Typica l values of n for a t r ans i t ions ( An =1 ) are from 90 to 200. The notat ion used for radio recombination l i n e s i s s i m i l a r to that used for o p t i c a l l i n e s . The chemical symbol of the element i s wr i t t en f i r s t , followed by the Greek l e t t e r i nd ica t ing the change of n , i . e . a, $, y» e represent'"An = 1,2,3,A,5 respec t ive ly . Hence Hl68a denotes -3-the t r a n s i t i o n of hydrogen from n = 169 to n = 168. Extensive tables of radio-frequency recombination l i n e s have been recent ly published. Because of the very small populations of the high quantum leve l s involved , radio recombination l i n e s are expected to be very weak. In the o p t i c a l range, however, the l i ne s are very strong due to the low quantum l eve l s involved, and the l i n e s of hydrogen, helium, oxygen, ni trogen and others have been in tens ive ly observed for many years. The h i s to ry of the radio l i n e s i s quite short , due to the e a r l i e r mentioned weakness and a suspicion that the l i ne s would be so extensively broadened by pressure broadening that they would become ind i s t ingu ishab le from the continuum. A further obstacle to observa-t i o n was the lack of su i tab le receivers and spectrometers to resolve the l i n e i 2 Nevertheless, i n 1959, N . S . Kardashev , using a homogeneous model of a nebula, and neglecting a l l pressure broadening ef fec ts , -predicted that the r a t i o of l i n e to continuum i n t e n s i t i e s for hydrogen i n l o c a l thermodynamic equi l ibr ium would be 4 v D l _ <_.gx/0~'2_±L T c T e . d-3) where A~d0 i - s t n e Doppler (or f u l l 1/e) width, equal to 0.60 times the f u l l width at ha l f power. T_ and ~T_ are the observed l i n e and continuum temperatures respec t ive ly , T e i s the free e lec t ron temperature. Kardashev further predicted that by using a 22m parabolic antenna, and modern rece iv ing equipment, one should be able to detect l i n e s of frequency as low as 1 GHz from nebulae whose emission measure (=Jn_n_dl) i s greater than lO^pc cm ^ . 1 A . E . L i l l ey , P: Palmer, Astrophys. J . Supp. 16^ , 143,(1968) 2 N . S . Kardashev, Sov. Astron. A J , 3, 813, (I960) - 4 -The f i r s t reported observations of radio l i ne s were by two 1 Russian groups i s 1964. Their resu l t s were, however, not very con-v i n c i n g . The f i r s t North American detect ion was the Hl09a l i n e i n 1965. Since then,observations of hydrogen, helium, and another emitter have been made over a wide range of frequencies. I t i s in te res t ing to note that l i t t l e , i f any, evidence of pressure broadening has been found. 1-2) Many in te res t ing and useful resu l t s can be gained from ob-servations of these l i n e s . Included are: the e lectron temperature, the mean turbulent v e l o c i t y , the r e l a t i v e r a d i a l v e l o c i t y of the neb-u l a , the helium to hydrogen abundance r a t i o , the structure of i n d i v i d -u a l nebulae, and the s p i r a l s tructure of our galaxy. Combining two or more re su l t s at d i f ferent frequencies w i l l give information about the existence of thermodynamic equi l ib r ium. In t h i s experiment, the purpose was: 1) To derive electron temperatures for the nebulae Orion, Omega, and W43 from observations of the Hl68a*Hl67a t r an s i t i ons , and 2) To combine these observations wi th other observations made at d i f fe ren t frequencies to evaluate better, estimates of temperature, dens i ty , and emission measure using a non-equil ibrium analys is des-cribed i n Chapter 2, and . -3) To lay the groundwork for a proposed future experiment which w i l l measure the important parameter by measuring r a t i o s of a and 8 l i n e s from the same upper l e v e l , using theory developed i n Chapter 2. A l l observations for part: 1 were taken i n November and December 1969 at the Dominion Radio Ast rophysica l Observatory near Pent ic ton . 1 Z . V . D . Dravskikh and A . F . Dravskikh, Ast ron. Z i r k . , 282, 2, (1964) 2 B, Hoglund, P .G . Mezger, Science, 150, 339, (1965) Chapter 2 GENERAL THEORY 2-1) Telescopic examination of the sky reveals the existence of d i f fuse , cloudy patches, ca l l ed nebulae by the ear ly observers. I t was l a t e r r ea l i zed that there are two d i s t i n c t types of nebulae-external ga lax ies , and ga lac t i c nebulae. The former are large groups of s t a r s , s i m i l a r to our own galaxy, which look diffuse from thei r extreme dis tance. The l a t t e r are members of our own galaxy, and are composed of gas and dust. Galac t ic nebulae form two d i s t i n c t subgroups: planetary nebulae, which appear smooth and sphe r i ca l , looking rather l i k e •^-planets .,-.and. diffuse nebulae, of-various forms and much la rger . Diffuse --.-nebulae shine by two di f ferent mechanisms. One mechanism i s the re -f l e c t i o n of s t a r l i g h t from a nearby s ta r . This gives a continuous spectrum with absorption l i n e s s imi l a r to those of the s ta r . These "nebulae are ca l l ed r e f l e c t i o n nebulae. The other means i s of f l u o r -escence. A hot star ionizes the surrounding gas and the subsequent recombinations give r i s e to an emission l i n e spectrum with a back-ground continuum which i s not necessar i ly due to scat ter ing of s ta r -l i g h t . These nebulae are known commonly as gaseous nebulae and are the subject of t h i s thes i s . Observations of gaseous nebulae show that they are d i s t r ibu ted mainly i n the plane of the galaxy and are associated wi th 0 and B s ta r s . Their dimensions are va r i ed ; most have a diameter between 1 and 10 pc, and a mean diameter of 3.6 pc. Their shapes are extremely v a r i e d , showing de f in i t e filamentary extensions and -^dark lanes. Their contours can.be e i ther sharp of b lur red . Spec-. t roscopic analys is .shows an abundance of spec t ra l l i n e s . Eas i l y - 6 -recognized are the Balmer ser ies of hydrogen and l i n e s of neutral helium. In addi t ion are numerous very strong l ines which for some time were un iden t i f i ed . They were l a te r shown to be forbidden l ines of O i l , OIII , NT,01, N i l , plus others. F i n a l l y , permitted l ines of the above elements are also found, along with l ines of Ne, Fe, A, Ca, CI , and S. About 90%. of the atoms i n nebulae are hydrogen, v i r t u a l l y a l l the rest i s helium. The others are a l l found i n trace amounts. The phys ica l p ic ture we adopt i s the fo l lowing : the plane of the galaxy i s r i c h i n gas and dust. I f a s u f f i c i e n t l y hot star (0 or B) i s near enough to a cloud of gas, we have a gaseous nebula. Those regions too far removed to be ionized by the s t e l l a r r ad ia t ion w i l l remain neu t r a l . The l a t t e r regions are ca l l ed HI regions. The former are ca l l ed HII regions. Of in teres t to us now i s where an HII region ends and where an HI region begins. In other words, how b ig can an HII region be? The d i s t inguish ing feature between the two i s the i o n i z a t i o n of hydrogen. The i o n i z a t i o n po ten t i a l of th i s element ss 13.61 eV, so only those photons more energetic. than th i s (corresponding to v>3.29xlO i 5 Hz) w i l l ion ize hydrogen. Appl ica t ion of the Planck rad ia t ion law shows that only stars hotter than T e 20,000 K can be expected to maintain s i g n i f i c a n t HII regions. A simple development of the s i ze of an HII region fo l lows: Consider a steady state cloud of ionized hydrogen. The number of ion iza t ions per second must equal the number of recombinations per second. Not a l l of the ion iza t ions w i l l be due to photons emitted by the s ta r , since recombinations to the ground state w i l l r e su l t i n emission of a Lyman continuum photon. However, i f we assume that a l l Lyman con-tinuum photons are consumed i n the nebula, the fol lowing i s t rue: the number of i on iza t ions -7-due to photons emitted by the star are equal to the number of recom-binat ions to a l l l eve l s above the f i r s t , ( i . e . a l l n>2) Thus, l e t Su(0)= the number of photons capable of i o n i z i n g hydrogen emitted by the star per second, a (2)_ the recombination coef f ic ien t to a l l l eve l s except the f i r s t of hydrogen, n^= the number density of protons, and r s = the radius of the HII region. The number of ion iza t ions per •second throughout the whole nebula i s S u(0)./ The number of recombin-at ions per sec per cm3 i s n enpCx (2) , through the whole nebula i t i s " 3 i r r s 3 n e n p a ^ • Thus, subject to the previous r e s t r i c t i o n s , S u ( ° ) ~ f T r r s ' ^ p o c " ^ (2 - i ) More exact der ivat ions give the same r e s u l t . This equation - i s i l l u s t r a t e d i n Table (2-1), taken from Spi tze r .* TABLE 2-1 < -RADII OF HII REGIONS : t r a l Type T R/R© s_C°> r s n p ^ / 3 05 56,000 K 7.0 31 * 10hB s ec - 1 100 pc cm' 6 44,000 6.7 9.0 66 7 36,000 6.3 2.7 \44 8 30,000 6.0 0.77 29 9 25,000 5.6 0.19 18 BO 21,000 . 5.3 0.041 11 1 18,000 5.0 .- 0.008 6.4 In the above t ab le , R /R© i s the radius i n solar u n i t s , T c i s the "temperature of a black body whose f lux shortwards of the Lyman l i m i t 1 L. S p i t z e r . J r . , Diffuse Matter i n Space, Interscience Publishers , 1968; Table 4.5, p 117 - 8 -equals that of the s ta r . More exact treatments show that wi th in the boundary, i on i za t i on i s v i r t u a l l y complete, but that a very th in (compared to the s i ze of the nebula) t r a n s i t i o n region separates the HI region from the HII region. I t i s known that very l i t t l e , i f any, of the rad ia t ion short-wards of the Lyman l i m i t escapes from the nebula. This fact leads one to ask where a l l th i s energy goes. Since the majori ty of the energy emitted by the star i s i n th i s region, the energy transferred i s enormous. Closer examination of the phys ica l processes gives the answer. As previously mentioned, there i s a steady state between ion iza t ions and recombinations. An e lec t ron , recombining with a proton can end up i n any of the energy l e v e l s . I f the e lectron f a l l s to the ground state d i r e c t l y , a Lyman continuum photon i s emitted which w i l l immediately ion ize another another neut ra l hydrogen atom. Thus i n t h i s case there i s no net change. I f , however, the e lectron f a l l s in to any other l e v e l , a photon of another series i s emitted which immediately excapes from the nebula. The electron . w i l l then cascade down the l e v e l s , emitting other photons u n t i l i t eventually reaches the ground s ta te . Since nearly a l l neut ra l atoms are i n the ground s ta te , ion iza t ions from excited states can be ignored. The net r e su l t of the ionization-capture-cascade sequence i s that a photon shortwards of the Lyman l i m i t i s converted ( i . e . degraded) in to a number of other photons of longer wavelength, plus one Lyman ser ies photon which i t s e l f exci tes a neutra l atom. The Lyman ser ies photon w i l l eventually end up as a Lyman a photon plus others. Thus - we have: L y c o n t i n u u m Ly a•+ others -9-What happens to the Lyman a photons i s not f u l l y known. E v i -dently the Ly a density w i l l increase u n t i l equi l ibr ium i s reached with some process(es) which al low these photons to excape. In any case, the density w i l l be very h igh . We conclude that the energy i n the Lyman continuum i s degraded to lower frequencies. What, then, i s the temperature of HI1 regions? In a pure hydrogen HII region, the only important cool ing mechanism i s by the emission of the Balmer, Paschen, and other hydrogen ser ies l i n e s , since i t i s e a s i l y shown that the cooling from free-free emission i s n e g l i g i b l e . Theoret ical c a l c u l a t i o n show that th i s would lead to a k i n e t i c temperature of the order of 25,000 or 30,000 degrees K e l v i n , much i n excess of observations. As mentioned e a r l i e r , other very strong l i n e s are observed, coming from the trace elements. The in t ens i ty wi th which these atoms radia te i s too great to be explained by the capture-cascade mechanism as described for hydrogen. Rather, these atomic l eve l s are mainly populated by c o l l i s i o n with e lectrons . A l l the forbidden l i n e s come from states w i th in 3.3 eV of the ground s ta te , w i th in the energy range of electrons at a k i n e t i c temperature of some 10,000 K. Hence, the proposed mechanism i s that fast electrons c o l l i d i n g r w i t h a n a t o m . o f : o x y g e n , nitrogen, neon, or other type w i l l cause an exc i t a t i on of the atom, t ransferr ing energy from the e lec -tron to the atom. The subsequent r ad ia t ive de-exc i ta t ion of these atoms causes a photon to escape from the nebula, since the density of the trace elements i s so very low. This mechanism i s a very e f f i c i e n t one, and i s the major cool ing mechanism operating; i n HII regions. Theore t ica l ca lcu la t ions show that the e lectron temperature of nebulae wi th impur i t ies should be around 7000 K, and th i s i s close to the observed temperatures. One other phys ica l process needs to be mentioned. This i s the - 1 0 -free-free emission due to e lec t ron- ion encounters. The accelera t ion of the e lect ron by the ion causes emission of a photon and accounts for the background for both o p t i c a l and radio regions. I t i s espec ia l ly important at radio frequencies since emission by the star at radio wavelengths i s completely n e g l i g i b l e . This important process i s w e l l understood and i s t reated, for example, i n Karzas and Latter^ and by Oster^. The question of thermodynamic equi l ibr ium i s a c r i t i c a l one. I f a gas i s enclosed ins ide a black body as defined i n thermodynamics, the energy of the gas equally divided among the various degrees of free-dom, both k i n e t i c and i n t e r n a l . The state of the gas i s described by a s ing le macroscopic parameter, the thermodynamic temperature T. In t h i s s ta te , the d i s t r i b u t i o n of atomic and e lect ron v e l o c i t i e s Is described by the Maxwell d i s t r i b u t i o n : The d i s t r i b u t i o n of atoms among the i r i n t e rna l energy states i s given by the Boltzmann equation: O n f = ^ e (2-3) The d i s t r i b u t i o n of atoms over the i r d i f ferent stages of ion-i z a t i o n i s given by the Saha equation: • y , * = 2^2rJ* T > f , fe (2-4) The spec t ra l d i s t r i b u t i o n of r ad ia t ion in t ens i ty i s given by the Planck funct ion: 1 W.J . Karzas and R. L a t t e r , Astrophys. J . Suppl. 6_. 167, (1961) 2 L . Oster, Astrophys. J . 137, 332, (1963) -.11-I „ _ E U T ) = z ^ ^ d z r <2"5) In the previous equations, grj- i s the s t a t i s t i c a l weight of the j 1 " * 1 energy l e v e l of the r t n ionized species, ~ 2J ^ r j ^3-P(~£ rj/$T) , x3/z ' i s the p a r t i t i o n funct ion, ,the p a r t i t i o n function for free e lect rons . The other symbols have the i r usual meanings. "The as te r i sk means thermodynamic equi l ibr ium values . A gas which i s i n the s tate previously described i s said to be i n thermodynamic ^equil ibrium, abbreviated T .E . This state i s never phys i ca l ly attained . in nature due to the-absence of true blackbodies. Implied i n the d e f i n i t i o n i s that no photons or pa r t i c l e s may excape from the gas, and hence there can be no net f lux w i t h i n a gas i n T .E . This i s -equivalent to asking for i n f i n i t e o p t i c a l depth i n the r e a l world (a ro le played by the wal ls of a,,block body enclosure), a s i t ua t i on .which i s not found. Close approximations of th i s s i t ua t i on are found i n s t e l l a r i n t e r i o r s , however. Not wishing to lose the s i m p l i c i t y of the equations(2-2) to (2-5), one makes the approximation of " l o c a l thermodynamic equ i l ib r ium" , -abbreviated L . T . E . Here one assumes the v a l i d i t y of the equations (2-2) to (2-5), but allows s p a t i a l (and poss ib ly temporal) v a r i a t i o n of the parameter T. Iri other words, we assume the conditions of T . E . to hold for a small region about the point i n question, al lowing the temperature to vary from place to p lace . This allows a f lux of r ad i a t ion to move through the gas. The ca l cu l a t i on of th i s f l ux requires knowledge of the composition of the.gas and of the relevant cross sections of the consituents. The question immediately ar i ses as to whether the as-.•assumption of L . T . E . _ i s v a l i d - i n a given - s i tua t ion . The -answer to ..this depends upon the f l u x ; i s the in te rac t ion of the emerging rad ia t ion (which or ig inates i n the hotter regions) su f f i c i en t to s i g n i f i c a n t l y a l t e r the l e v e l populations of the cooler regions, the d i s t r i b u t i o n of which has been assumed by eq(2-3)? What i t b o i l s down to i s that i f the photoelect r ic t r a n s i t i o n rate i s small compared to the corres-ponding c o l l i s i o n a l t r a n s i t i o n ra te , or i f the photoelectr ic t r a n s i -t i o n rates balance i n d e t a i l , then the assumption of L . T . E . i s v a l i d . In gaseous nebulae, the extremely low dens i t ies reduce the c o l l i s i o n ra te to a point where i t becomes very l i k e l y that the assumptions of L . T . E . are not v a l i d . What, then, can we assume? A l l fol lowing work i n th i s thesis i s based upon the assumption of s t a t i s t i c a l equ i l ib r ium. The only assumption t h i s en ta i l s i s that of equi l ibr ium i n time-we assume that the l e v e l populations are constant i n time. From th i s we must c a l cu l a t e , with a knowledge of a l l relevant parameters of the gas, the d i s t r i b u t i o n of s ta tes , emergent f l u x , r ad i a t ion i n t e n s i t y , e t c . We can recover two of our equations, namely (2-3) and (2-4) by de-f i n i n g a factor such that i f nj = the number density of the j t n l e v e l of the r*-*1 stage of i o n i z a t i o n , and n<| = the number density of the same species i n the "equivalent thermodynamic system", then (2-6) The "equivalent thermodynamic system" i s defined as a system i n thermodynamic equi l ibr ium i n which the temperature equals the ac tual k i n e t i c temperature, i n which the e lectron density i s equal to i t s ac tual va lue , and i n which the density of r+1 ionized atoms TI*+\ equals the actual value n r + l » T h i s d e f i n i t i o n allows us to use the Boltzman and Saha equations. The other two equation, however, are of doubtful y a l i d i t y , and these quant i t ies must be calculated from - 1 3 -f i r s t p r i n c i p l e s . F i n a l l y , a word about dens i t i e s . In th i s thes i s , the customary assumption about constant density i s adhered to. Th is , of course, i s not the case. There i s considerable evidence that nebulae are ex-tremely clumpy. However, i n defence of the assumption, one can say that the emissive properties to be considered are not s trongly dependent upon density and that the assumption of homogeneity at leas t i l l u s t r a t e s the basic propert ies w e l l . Recently, considerable work has been done on the in te rpre ta t ion of r e a l nebulae i n terms of homogeneous nebulae. I t appears that applying the theore t i ca l r esu l t s from homogeneous nebulae to observations of r e a l nebulae w i l l r e su l t i n reasonably and meaningful answers provided the values of temperature and density obtained are in te rpre-ted i n the sense of an average weighted with respect to the emission measure. This point w i l l be elaborated l a t e r . 2-2) Def in i t ions We define here those fundamental quant i t ies which are used extensively i n the theory of r a d i a t i o n . Of fundamen-t a l importance i s the a) In tens i ty ^Jnu Consider a point P on surface do. L i s a f ixed l i n e at angle 0 from the normal N to surface do. d_ i s an elementary cone of s o l i d angle.. The aggregate of a l l such cones over surface da defines a s emi - in f in i t e truncated cone wi th da as the f i n i t e end. Radiat ion i s being t rans-mitted through da. Denote by dE v the energy passing through da i n time dt between frequencies v and v+dv i n the d i r e c t i o n L . Theory and 1 R.H.Hjellming.M.H.Andrews, T . J . Sejnowski, Astrophys. J . 157,573,(1969) -14-experiment then f ind that the r a t i o Cose d-'cJ^dowcJt (2-7) tends to a p o s i t i v e de f in i t e l i m i t as da, dvy dt , dto,-> 0 i n any manner. This l i m i t we c a l l the spec i f i c i n t e n s i t y , and denote by I v I t i s of some in teres t to note that the spec i f i c in t ens i ty i s constant along the path of any ray i n free space. Proof of th i s state ment i s found i n M i l n e . ^ " From the d e f i n i t i o n (2-7), the t o t a l energy passing through a surface i s given by (2-8) where the i n t eg ra l i s taken over the ha l f sphere. The f lux F^ i s defined as the energy transfer per un i t area do per uni t time dt per un i t frequency i n t e r v a l dv. In terms of the spec i f i c i n t e n s i t y , i t i s (2 - 9 ) F v — ^ Coo 6 cJ 4ir We define the energy density u v as the amount of energy per un i t volume i n the frequency i n t e r v a l v to v+dv i n the neighborhood of the given poin t . I t i s re la ted to the i n t ens i ty by Mo= I^du; (2-10) b) The Absorption Coeff ic ient Radia t ion , t raversing matter, i s found to be attenuated. Consider a beam of r ad ia t ion of in t ens i ty I v t ravers ing a slab of mater ia l of thickness ds. Upon emergence 1 E .A . Milne Thermodynamics of th Stars , Handbuch der Astrophysik, V o l 3 , Part 1, Ch 2 , p72, 1930. -15 -the beam Intensi ty i s found to be I v + d l v » The absorption i s found to be proport ional to the i n t e n s i t y . We define < v , the l inea r ab-sorpt ion coef f ic ien t by the r e l a t i o n _ L , = - * . I , J S ( ? , ! ) The absorption of photons from a beam i s comprised of two causes. One i s true absorption by atoms, where the photon i s absorbed to r a i se the energy of the atom. The photon w i l l be re-emitted l a t e r , not necessar i ly at the same frequency. The. other mechanism i s that of sca t ter ing by e lectrons , atoms, or p a r t i c l e s , where the photon i s merely re-directed into another s o l i d angle wi th no change i n frequency. We define a mass absorption coef f ic ien t k v by V v = p A (2-12) where p i s the mass densi ty . c ) The Emission Coeff ic ient Consider a volume element dV which emits energy d E v i n the frequency range v to v+dv into s o l i d angle dco i n time d t . We define the emission coef f ic ien t A by A mass emission coef f ic ien t j v i s defined by £V= pj» (2-14) where p i s again the mass densi ty . d) Opt ica l Depth This useful quantity i s defined as d f'O — Vv d S (2-15) or 7X> = $*vds (2-16) in A small o p t i c a l depth ( x v « 1 ) indicates that a photonAthe region has only a small' p robab i l i t y of in te rac t ion wi th the gas. In f ac t ? -16-the p robab i l i t y of in te rac t ion i s given by l - e x p ( - x v ) . e) The Source Function The source funct ion, denoted S v i s defined by the r a t i o * ~~ (2-17) For a gas i n thermodynamic equi l ib r ium, the fol lowing r e l a t i o n , known as K i r c h o f f ' s Law, holds 3 I C i £hv/*~r _ / (2-18) f) The Equation of Transfer Consider a cy l inder of length ds and cross sect ion da. As e/w -Radiation of i n t ens i ty I v ( s ) enters one end, and ex i t s wi th in t ens i ty I v (s+ds) . By eq.(2-7) . the energy change w i t h i n the cyl inder i s dEs, = f l v ^ + JsJ - I^^JJ j^Jtdtu (2-19) The energy change from absorption i s , from (2-7) and (2-11) d £ ^ _ = - Kvl^dsd<rdv<_tdw (2-20) The energy gain from emission i s , by (2-13) dE - ^ d ^ d c c d t d c r d s (2-21) Equating gains and losses , and d iv id ing by dadvdtdio I „ ( S + _ s ) - J V ^ S ) = - V v l ^ d s •+ f v J s (2-22) Div id ing through by icvds and introducing (2-15) and (2-17), we f ind ' •dL. = - I . + S . J r . This equation i s the equation of t ransfer . In terms of l i nea r d i s --17-tance s, the equation i s . . . • =1 - K l ^ i- £ o (2-24) As Note that t h i s equation holds only for plane p a r a l l e l atmospheres, as no account of curvature has been taken. 2-3) The Coeff ic ients and Solut ion of the Equation of Transfer Equation (2-24) may eas i ly be al tered to apply to a l i n e form-ing gas. For th i s case, the t o t a l emission coe f f i c i en t , e v , i s composed of two par ts , a continuum coef f i c ien t e c , and a l i n e coef-f i c i e n t , et .• S i m i l a r l y for the absorption coe f f i c i en t . Thus we make the subst i tu t ions V-u — Ve •+ V_ (2-25) £ , , , = _ c + £ s (2-26) Equation (2-24) then becomes 41? - -C*. + * , ) T V + ( _ « * _ > J (2-27) As I f we define the o p t i c a l depth to be that of the continuum A 7c = Vc As (2-28> then we can a l t e r (2-27) to read where 5c = V v c (2-30) ^ = \ (2-31) AC x (2-32) (2-33) -18-This equation i s of the form + P ( a ) c j f o ) = Q(<x) (2_34) (2-35) The general so lu t ion i s o Applying th i s to (2-30) gives 0 where we have assumed that there i s no incident r ad ia t ion on the ne-bu la , and that zero o p t i c a l depth occurs on the far s ide of the nebula. App l i ca t i on of the so lu t ion (2-36) depends upon the four r e l e -vant c o e f f i c i e n t s , ic, , K c , e j , , e c . We deal wi th the l i n e coef f ic ien ts f i r s t , s t a r t ing wi th the emission c o e f f i c i e n t , z_g. This quantity w i l l obviously be propor t ional to the number of atoms i n the j 1 - * 1 l e v e l . Applying Boltzmann's equation, eq.(2-3) , and (2-6) g ives , 7)rj =. ^ <___.V-> S (2-37) The ra t io : 22c may be eliminated with the help of eq (2-4) to give , # - Bri/*T sr\Ti = bj 3 r j ^ g ^ r „ e (2-38) R e s t r i c t i n g the problem to the hydrogen atom, we have g r j = 2 n 2 , where n i s the p r i n c i p a l quantum number of l e v e l j , n r + ^ = n_, the proton densi ty , f r + i = 1 for ionized hydrogen, and f e = 4 .83*10^ 5 T 3 / 2 In taking f r + 2 = 1, we have i m p l i c i t l y assumed that the zero energy corresponds to a free e lectron at r e s t . Hence E r j i n eq (2-38) now becomes the binding energy, rather than the exc i t a t ion energy. - 1 9 -Thus, P r . = r C ' . - -Zd8jJ<? e r 9 (2-39) & /ft* Now, the t o t a l energy output from a small element of gas dV i n some pa r t i cu l a r l i n e w i l l be the product of the number of atoms able to make the t r a n s i t i o n (given by (2-38)) times the volume times the p r o b a b i l i t y of t r a n s i t i o n , times the energy per t r a n s i t i o n . I f we denote the lower l e v e l by j , the upper l e v e l by k, the p robab i l -i t y of a t r a n s i t i o n , per second, i s given by the E in s t e in coe f f i c i en t for spontaneous emission, j , and the energy per t r a n s i t i o n i s hv j^ . The t o t a l energy output, per un i t time over the ent i re l i n e i s 4 £,„ = ^ * Hi* h V i * *f V ^ * (2-40) Equation (2-40) has no s p a t i a l dependence since spontaneous emission occurs with equal p r o b a b i l i t y i n any d i r e c t i o n . The emission co-e f f i c i e n t , integrated over the l i n e i s then, by comparison wi th eq (2-13) i _] \foA-0 = ^ ^ H * i (2-41) 'in* I f we introduce an emission l i n e p r o f i l e <J>(Av) such that £, = € <t>(A^>) - (2-42) ans subst i tu te in to (2 -41) , then we f ind £=^/l-^/n*t f* a - (2-43) provided that the p r o f i l e i s normalized : ty(AJ)d* = 1 (2-44) Hence, we have A s i m i l a r de r iva t ion gives the l i n e absorption c o e f f i c i e n t . The - 2 0 -net f lux of photons, per un i t area into a slab of thickness dx i s _k_L , and the net f lux of photons coming out w i l l be +e"^)-£ , so that the change of f lux i s -<_____ . Now, the rate of photon ab-sorpt ion i s , per atom}U-yBj-^, so the t o t a l absorption i s njUyBj^dx, where B j ^ i s the E ins t e in coef f ic ien t for absorption. Subtracted from th i s are those photons emitted by stimulated emission njU v Bkjdx, so that we have • .. ~M c = K 3ik - ^ 8*j) U, d * (2-46) Hence, ... Sec v 7 Now, d i f f e r e n t i a t i n g eq (2-10) with respect to :x gives - - l ( <__!> A Co (2-48) dec C J doc ec or (2-49) d a where we have used equation (2-11). By comparing (2-47) and (2-49), remembering that the energy absorp-t i o n i n the former refers to the whole l i n e , we have, \ * f ^ = ^ K siK - e>j) ( 2 - 5 0 ) "By def ining an absorption p r o f i l e s imi l a r to the emission p r o f i l e , we get \MS =• ^ * ( ^ j Si*-"** 6*i) </>(&•») (2-51) By employing the w e l l known re la t ionsh ip 3 i 8 * = fc&j = r - ^ S * . f l " . (2-52) --and-using equation X2-38), we express (2-45) and (2-51) i n more —convenient form, after some algebra -21-- ^K^^h-OjH ft HI e k T _ 5 3 ) V e _ b j f r W f r h ^ g , , - C B j / k T (l - £ ^ U^V) (2-54) f«f„c ** / r Of great importance i s the l i n e source funct ion, S ,which i s , by d e f i n i t i o n . . „ n - (e"~Ei ' (2-55) S _ - = 2 ^ 3 / where we have again used eq (2-52), and have assumed that the l i n e p r o f i l e s of emission and absorption are i d e n t i c a l . Note that i n thermal equ i l ib r ium, Sj = B v , s ince i n that case, bj = b^ = 1. A spec i a l case, important for radio l i n e s , i s when the l i n e p r o f i l e <j>(Av) i s Gaussian i n shape. Then we wr i t e ^ = A e (2-56) where Av i s the frequency sh i f t from l i n e center, A V Q i s the frequency sh i f t of the l i n e centre from the true center frequency, r e s u l t i n g from any net motion of the gas c loud. I f we can assume that the l i n e of s ight through the cloud contains a large number (>10) of turbulent c e l l s , then z W o = ^A/ISTT <2-57> where A\>n i s the Doppler width,-m^ the mass of the emitter (proton), and go i s the mean turbulent speed. The Doppler width, Avn, i s re la ted to the half-power width, Av^, by A - J - ' -. A-VL = 0-6006 A v L A ° ~ ZUT£ (2-58) -22-By def ining the o s c i l l a t o r strength, f b y •R. - JLl£ f . ^ J * ~ h-tf i H (2-59) and using (2-52) again, the l i n e coef f ic ien ts e_ and Kf can be f i n a l l y expressed as, at l i n e center, A - t > 0 T " £ (2-60) VAT\ (2-61) We have assumed Q —/ , since E^ , the binding energy i s much smaller than kT, the average thermal energy. Here, m represents the lower l e v e l , n , the upper l e v e l . We turn now to the continuum coe f f i c i en t s . There are a number of processes which could conceivably contribute to the continuum opacity of nebulae. They are: a) Bound-freeJabsorption by atoms, K^f b) Free-free absorption by ions , K j f c) Photon scat ter ing from free e lectrons , K _ S d) Photon scat ter ing from dust and other s o l i d p a r t i c l e s , K _ _ Of these, K£,f« Kff » due to the very low density of atoms compared! to ions(mainly protons). , A l s o , we expect K s s to be very smal l , s ince; i t i s known that the i n t e r s t e l l a r medium i s quite transparent to radio frequencies. However, comparatively l i t t l e i s known of the absorptive and emissive propert ies of grains. Corresponding to the two remaining c o e f f i c i e n t s , are the emission c o e f f i c i e n t s : a) Free-free emission(bremsstrahlung), e^f b) Photon emission by electron sca t t e r ing , e e s -23-The emission coef f ic ien t for free-free t r ans i t ions for electrons m t h 1 a Maxwellian v e l o c i t y d i s t r i b u t i o n i s 4fT 2 Exact analys is has shown that the assumption of a Maxwellian v e l o c i t y d i s t r i b u t i o n i s v a l i d to an excel lent approximation for a l l p a r t i c l e s i n both HI and HII regions. This fact allows us to use equations (2-18) and (2-19) to obtain K f f o r i s the Gaunt factor for free-free emission "Equation (2-64) i s an approximation v a l i d for radio frequencies, Y i s E u l e r ' s constant, equal to 0.577. Typica l values for gff i n the radio region are of the order of 5. App l i ca t i on of the radio-frequency approximation i n (2-63) and (2-62) allows us to wr i te the coef f i c ien t s as e _ 5". 44 x IO " 3 9 <j+f 22£^fc ( 2 - 6 5 ) X-w = 1-17 XlO~ Z ^ff /^e^/» (2-66) T" L . Spi tzer J r . , Diffuse Matter i n Space, Interscience Publ i shers , 1968. eq. (2-46) 2 L . H . A l l e r , D. Bohm, Astrophyg. J . , 105, 131, (1947) -24-Th e ro l e of electron scat ter ing may be important. An analysis of th i s effect i s done i n Appendix A. The resu l t s indicate that e lect ron sca t ter ing can be neglected at the lower frequencies con-sidered i n th i s thesis ( v - 10 s Hz) , but may become important at higher frequencies. We may now return to the so lu t ion of the equation of transfer for l i n e r a d i a t i o n , equation (2-36), which i s T s o w [rc . c v 5 u + - u V r dr/ (2-67) We now make the fundamental assumption of homogeneity, that i s , the coef f ic ien t s n v, S_, and Sx are not dependent upon T C . Equation (2-67) can be eas i ly integrated i n th i s case to read I V (T C ) = i s _ t J 2 r k (/ - e ^ * ) (2-68) A> O p t i c a l l y Thin Case At the frequencies of in teres t (1.4 Ghz), most nebulae are o p t i c a l l y t h i n , so we may expand the exponential i n (2-68). For reasons which w i l l be c lear l a t e r , we expand to second order e == l-dtnJZ t f (2-69) so that = ( t/», 5,) ( £ - C / ^ v ) r V 2 ) (2-70) By solv ing the equation for the continuum only , wie obtain IcCr t)= S c f ? c - T//Z\ ; (2-71) The quantity which we are interested i n i s the r a t i o of l i n e i n t ens i ty to continuum i n t e n s i t y : ( I v - I c ) / I c . Using (2-70) and (2-71), -25-I^ZJS . £ ) _ (I (2-72)' We now apply eq(2-72) to the two cases on hand, namely, L . T . E . and non-L .T .E . 1): L . T . E . Nebula In t h i s case, S ; , given by (2-55), becomes = & X T ) (2-73) and, S c = G^CT) (2-74) The previous expression i s by assumption, since the free electrons are i n thermodynamic equi l ib r ium. From eq (2-72), then, we have i ^ L c = ^ ( / - T c ) (2-75) Ic which, to f i r s t order i s simply r * = ^ v = (2-76) where r* stands for the l ine-to-continuum r a t i o i n L . T . E . The coef f ic ien t s K and « c are given by (2-61) and (2-66) respec t ive ly . We f i n d , upon evaluat ion, that at l i n e center, r* = t-68•xio* <ssL£±ss ( -L—\ ( 2 - 7 7 ) Tne term[7^—77775^-^is included in(2-77) to account for that part of NuT*) ' the continuum produced by s ing ly ionized helium. The cont r ibut ion from doubly ionized helium and other elements i s n e g l i g i b l e . A simple r e l a t i o n between quantum.number and frequency, derived from .the Rydberg formula, and accurate enough,for most purposes, i s V 2= 2 n (2-78) -26-where R i s the Rydberg constant, equal to 1.097x10^ cm ^. Thus, = 6 5$xlo'5 m'* ^AC~' (2-79) For a - l i n e s , the o s c i l l a t o r strength i s given by^ " h-,,^ = O-IVI <n (2-80) For B - l i n e s , TV-*,.* = 0-OZ6 5 n (2-81) Subst i tu t ion into (2-77) y i e ld s r / = 2.IIXlO-"^l±— ( - — ^ ] (2-82) The r a t i o of a to B l i n e s v i s of i n t e re s t . Noting that Av^ i s pro-ffer. **>e tome offer level por t iona l to v, we f ind f 1^ li/lw1 (2"83) When approximate values are suppl ied, we have r y * = oS9T r«* (2-84) Another r a t i o i s of in t e re s t , that of a and 8 l i n e s whose frequencies are nearly the same. In th i s case, eq (2-77) supplies r * r * • el As and example, consider the 156a and 1976 l i n e s . Then (2-85) gives  rfi* = O 230 f « 6 * . (2-86) 2) Non-L.T .E . Nebula Here we use the complete equation (2-72)/ The r a t i o occurs twice. Upon applying (2-17), we f ind that Sc - Le ' ' (2-87) Sc £ c Hence we rewri te (2-72) as r ~ ec t ' ^) i l ' + & J ( 2 _ 8 8 ) 1 H.C. Goldwire, J r . , Astrophys. J . Supp. 17, 445, (1968) -27-where r stands for the non-L. 'T.E. l ine-to-continuum r a t i o . Noting that Ti-vht = ^ , we can rewri te (2-88) as (2-89) The various coef f i c ien t s are (at l i n e center) Is = i-tf no' 26 ^W^3iiK / / <7j„ = I-69x10 ^ V T _ (1 + *%$/ ^-9°) w^^_i= . . r / _ _v_e" f e7f / 7 ( 2_ 9 1 ) S r _ * T - / — = b", — (2-92) In the radio approximation, hv « kT, so j_c _ ___ ^ ___T / (2-93) t Since n i s large (n > 100) for radio l i n e s , we can make the approximation (2-94) AH S° *L - _ _ a L - i i X a n A^km (2-95) Applying the same reduction to (2—91) gives OQv - b m r * & ( - 4A_L) 4n\ (2-96) *?T I ^-v J n / Depending on the values of kT/hv and '^f , n and -r^- can be ei ther negative, p o s i t i v e , or even zero. From comparison, i t i s obvious that r i v i - s n e g l i g i b l e compared to j>c/S_f , hence we w i l l neglect i t , g iv ing (2-97) -28-The equations (2-89) and (2-97) are i n exact agreement with Gold-berg 's equations which use a d i f ferent method of so lu t ion^ . We define the enhancement r a t i o R by R ~ ^r* (2-98) Evident ly which properly goes to one as b n goes to one. The r a t i o bm/bjj i s always very close to one, so we can rewri te R to be Thus i t i s evident that for t r ans i t ions whose energy change i s much less than kT, small deviat ions of l e v e l populations from thermal eq-u i l i b r i u m values can cause large changes i n emitted l i n e strengths. The consequence (of th i s i s that l i n e s emitted from regions not i n L . T . E . may be much stronger than the L . T . E . theory would expect, which, by eq.(2~77) would g ive an electron temperature lower than the r e a l va lue . B) O p t i c a l l y Thick Nebulae At low frequencies, the nebulae become o p t i c a l l y t h i ck , and i n th i s case, eq (2-68), for T C > > 1 reduces to (2-101) By def ining r again as the r a t i o of l i n e to continuum i n t e n s i t i e s , we f ind i:that "H-v ( - S c \ > (2-102) c 1 L . Goldberg, I n t e r s t e l l a r Ionized Hydrogen, W. Benjamen, 1967. -29-In the case of L . T . E . , S = S c , so r* = O (2-103) Thus there no l i n e s formed at a l l , and the nebula radiates l i k e a blackbody. For n o n - L . T . E . , we can neglect r)v wi th respect to one i n the denominator. Then „ r = %(l-where we have used (2-87). Using (2-95) i n (2-104), we f ind kT Since ^ >> 1, we can wr i te (2-105) as # > — -KL A£r\b^ An , r - €c d^ C) The General Case Graphs of the emitted f lux from HII regions show that most nebulae become o p t i c a l l y th ick near 1-2 GHz, so that neither of the two previous approximations w i l l be correc t . I i i the in te rp re ta t ion of the observed l i n e s , the general so lu t ion i s used. That i s , from (2-68) / t / ^ (2-107) and Ic (To) = 5 c ( / - e T t ) (2-108) so -30-where the coef f ic ien t s are defined as before: /vfrf) /V«Y*) 7". = / ' 7 7 X / g - z y f 2-4) The b n Problem. Measuring d ^ n ^ Obviously the cha rac te r i s t i c s of l i n e emission i n radio frequencies i s s trongly dependent upon any deviat ions i n atomic l e v e l populations. From a theore t i ca l standpoint, the values of b n can be calculated provided a l l the var ious rate coef f ic ien t s are known. This problem has been solved"^ by making use of the postulate of s t a t i s t i c a l equ i l ib r ium. By considering any l e v e l n , ca lcu la t ing the rate of population to and depopulation from other l eve l s and the continuum by a l l relevant r ad i a t i ve and c o l l i s i o n a l processes, and comparing the r e su l t to that expected from Boltzmann's equation, the values of b n and may be ca lcu la ted . The resu l t s indicate that b n approaches 1 asymptot ical ly as n--> °°. Typica l values of b n and < ^ ~ n at 1.4 GHz and 10,000*K are 0.97 and 1.0x10" 3 r e spec t ive ly . The important factor l - ^ A n ^ ~ * can be quite la rge , as much as -350 for the 158a hv i n l i n e . The values can be measured a l so . The equation for the l i n e to continuum r a t i o i s 1 T . J . Sejnowski, Robert M. Hje l lming, A s t r o p h y s . J . , 156, 915, (1969) - 3 1 -The r a t i o i s dependent upon T p , b ^ / a n d Al^n . By making obser-vations of an a and 8 l i n e from the same upper l e v e l , one of the 3 var iab les may be el iminated. Then, using "the well-known values for b n , unambiguous values may be obtained for T e and . From (2-113), By d i v i d i n g the equations, w e f ind _ •rgegr tCcpt t - - ' K ^ ^ C M V ic^/ (2-116) which, making the approximations x c a , T cg « 1, becomes • — ' v ,^ e (2-117) By subs t i tu t ing (2-117) into (2-114), we can wr i te =• £ effi£c^y<P - r> <f* £cP y« (2-118) This can be rearranged to a s l i g h t l y more convenient form (2-119) From previous work i n th i s chapter, we know that the coef f ic ien ts are: fe* = C, f = C, bm-ifvji**.* (2-120) -32-Sc(i = Cz <frffl (2-121) - Cj (2-122) where 2 Tw * V 1 NCH*-)J (2 = 5 - 4 7 X 1 0 U BJ*- I I + NIHjf 3 yltz I /V^Oy Here, E .M. stands for emission measure £. V). =. j/n^ry, _7 and i s expressed i n uni t s of cm~^pc. Thus eq (2-119) becomes, upon subs t i tu t ion (2-123) (2-124) fi^ -'-n •f«»-i,. 1 5^ s* / ( 2 r - ^ M H ^ „ - r ^ ^ W l H f f l g g _ Subs t i tu t ion of (2-119) to (2-122) in to (2-117) gives -.(2-125) ;2-126) %«f\2r_i,.^ Al** - rprrr? f^ .,,^  Ai/0/3 J i - ' T A more informative method of wr i t i ng the l a s t two equations i s accomplished by the fol lowing transformations: Let T a be the tem-perature derived from the a - l i ne data alone using the L . T . E . theory, and define Tg i n a s i m i l a r manner. Then, 0 (2-127) (2-128) -33-Then, by subs t i t u t ing , 2 V. 2 ^ v ; T , - ^ V T - / (2-129) D / N ..- T T T — = r — r = r 1 ( 2 " 1 3 0 ) The two preceding equations are nearly i d e n t i c a l to those f i r s t derived by Palmer."^ We see that measurements of a and 3 l i ne s from the same upper l e v e l w i l l a l low us to evaluate unambiguous values of T_ and 4Jn2ik^n } providing only that we know b n and the emission measure, both of which are known w i t h some confidence. Two important points concerning the value of emission measure to be used need, to be mentioned. The value of emission measure usua l ly quoted i s that for the center of the nebula, whereas the the value needed should be an average over the area of the nebula sampled by the telescope beam. Secondly, i n der iv ing the emission measure from continuum measurements, an assumed temperature must be used. This i s e a s i l y shown. For a homogeneous model, the br ight ness at frequency v i s , for the o p t i c a l l y t h i n case, = 2***Te^JJ (2.131) The received f l u x , when the source i s much smaller than the antenna beam i s given by * 2*i£~k-/2s\C5ff^2!£^ (2-132) ^T1 Thus, - 3 4 -—T— 2, £. y). CA — i (2-133) where C Is a constant. I f we denote by E Q the emission measure derived by assuming temper-ature To, then the true value E needed i n eq(2-130) w i l l be: E = f o (-^]'Z 3'^tX v) (2-134) where v i s the frequency at which these observations were made. -35-Chapter 3 THE EXPERIMENT 3-1) The Receiving Equipment A l l observations were made at the Dominion Radio Ast rophysica l Observatory near Pent icton B .C . using the i r 25.6 m parabaloid antenna. Figure 3-1 i s a s imp l i f i ed block diagram of the receiver system and spectrometer. Power col lec ted by the horn i s fed to a low noise parametric r e f l e c t i o n ampl i f ier Xparamp) . A gas discharge tube i s coupled in to th i s l i n e by a 20 dB coupler and i s used for c a l i b r a t i o n . The output from the paramp i s then amplifed by a 1-2 GHz t rans i s to r ampl i f i e r . Following th i s i s a tunable f i l t e r , used to ensure s ing le sideband reception from the mixer , which follows the f i l t e r and heterodynes the high frequency s igna l to 35 MHz by using a l o c a l o s c i l l a t o r (L.O.) set 35 MHz below ( the observing frequency f s . This l o c a l o s c i l l a t o r i s a phase locked backward wave o s c i l l a t o r (B.W.O.), and has reso lu t ion and accuracy of 1 part i n 1 0 9 . The 35 MHz I . F . i s then amplifed and transmitted to the cont ro l room by cable, where i t i s further amplifed, converted t o 10.7 MHz, and amplifed again. The s igna l then passes through a 6 MHz bandwidth f i l t e r , a f i n a l ampl i f i e r , and i s passed to the spectrom e t e r . An automatic gain cont ro l (A.G.C.) loop i s f i t t e d here to ensure constant s igna l l e v e l to the spectrometer. The spectrometer i s a 100 channel instrument. The s igna l from the receiver i s divided by 100 contiguous f i l t e r s , each 10 KHz wide, and centered on 10.7 MHz, so the coverage i s from 10.2 to 11.2 MHz, which t ranslates to a 1 MHz bandwidth at the observing frequency. Each f i l t e r leads to a diode detector after which there are two RC in tegra tors , l abe l l ed A and B. The sequence o f operation i s as fo l lows : For one minute, the input switches are on A, so that a l l the A net--36-works are rece iv ing information. At the end of one minute, the -input switches to B bank. Meanwhile, the voltages on A bank are being read i n sequence by a d i g i t a l voltmeter, after which the capa-c i t o r s are discharged to ready the bank for further in tegra t ion . At the end of the second minute, the input switches back to A, and bank B i s read. In th i s way, continuous receiving of the s igna l i s assured. A l l voltages read are stored by a computer, the means and usage of which w i l l be explained l a t e r . 3-2) The Observing Program The sources chosen for observation were the three br ightest sources i n D i e t e r ' s list"^ which are obser-vable from our l a t i t u d e . They were: (1) The Omega Nebula, W38, a = 18 h18 m44?3, <5 = -16 11 '45"; (2) The Orion Nebula, W10; a = 05 h 33 m 48? l , <5 = -05 24'29";^and (3) W43 complex; a = 18 h46m02?6, 6 = -01 58'45". The coordinates are precessed to 1969.9. The t r ans i t ions chosen were the 168a (v = 1374.601 MHz), and the 167a (v = 1399.368 MHz). The quoted frequencies are laboratory frequencies. A l l sources were observed i n a period from November 25 to December 6, 1969. The observing program was as fo l lows : The spectrometer runs -on a two minute c y c l e . The on-l ine computer (A D i g i t a l PDP-9) was ^programmed to accept 5 cycles a t ' a time i n two sets , so.a computer _„-cycle..was ,20 ..minutes... .The. f i r s t .set of 5 two-minute cycles was spent "observing a nearby "(displaced by about 1 degree) region where there —was no l ine- r ad i a t ion expected. The second set was spent observing the source. At the end of the 20 minute c y c l e , the computer per-formed the fol lowing ca l cu l a t i on for each channel of each bank and - punched the r e su l t i ng number on paper tape: 1. N. H- Oieicr } (Qstro^ys. X l5Q. 435, /I967) The on-off mode of observations i s adopted to el iminate any slowly changing gain of the ampl i f i e r s . 3-3) Data Reduction of the UBC IBM 360-67 Computer data reduction was done on the U . B . C . campus. Two undesirable effects confuse the output from the PDP-9 on-l i n e computer at the observatory. The f i r s t , and most ser ious , i s due to the spectrometer. I t was found that the 200 detectors and RC integrators (2 banks of 100 each) are not balanced with respect to each other. That i s , they w i l l not a l l have the same output when fed an i d e n t i c a l input . With a broadband input , the output i s ex-tremely ragged, so much so that the weak s igna ls we were attempting to detect would be l o s t . However, i t was found that the value of the output for nearly a l l channels was l i nea r wi th the input . • This suggested that a correc t ion factor could be defined to el iminate this e f fec t . This was done i n the fol lowing way: At least 2 or 3 times per day, a c a l i b r a t i o n check was done on the spectrometer by feeding i n a known amount of broadband power for a s ing le computer cyc l e , using the cold sky as the reference. Of the 100 channels of output, about 50 or 60 showed l i t t l e or no devia t ion from the average. These channels were deemed to be normal, and for each c a l i b r a t i o n run, a leas t squared f i t parabola was f i t t e d to these points only (by the computer), ignoring the r e s t . Then, for each' channel on both banks, the fol lowing r a t i o was computed A l l remaining (3-2) -38-where YF(I) i s the f i t t e d value of the I t h channel, and Y ( l ) i s the spectrometer value of the I1-*1 channel. This r a t i o was found to agree to w i th in 1% for a l l c a l i b r a t i o n runs for about 80% of the channels, and to w i th in 2% for 15% of the channels. The remaining 5% were e r r a t i c , due probably to fau l ty switches. This excel lent agreement between runs compares to an expected noise f luc tua t ion of about 1% at the l e v e l at which the ca l ib ra t ions were made. High confidence i n th i s procedure was gained from the excel lent consistency of the r a t i o s . There are, however, dangers i n the procedure which are nearly impossible to check. The c a l i b r a t i o n power input i s some 400 times larger than the s igna l l e v e l . This was necessary to make the output much larger than the noise f luc tua t ions . A test was run wi th a ser ies of successively smaller power inputs to check the l i n e a r i t y , but one can only reduce the power input to about ha l f the normal c a l i -b ra t ion l e v e l before the noise f luctuat ions begin to confuse the output. Thus, one has to assume the effects shown at high input l eve l s are the same as those at low input l e v e l s , (e .g. the recombination l i n e s ) . Once the cor rec t ion factor T(I ) had been decided upon, each number on every tape was m u l t i p l i e d by the factor to give a new 'cor-rected output. In most cases, the cor rec t ion was less than 5%. After t h i s step, the second obscuring effect was removed. This effect i s that of bandpass shape, which i s caused by the ampl i f iers amplifying more at one end of the bandpass than the other. The output looks l i k e a shallow parabola when plot ted as a function of channel number. To el iminate t h i s , a b e s t - f i t second or thirdrlorder (de-pending on which was bet ter) polynomial was f i t t e d to those channels where no l i n e was expected, and a new value for each channel was de--39-fined by NY (I) = Y(I) - YF(I) (3-2) where NY(I) i s the corrected value of the spectrometer output, and the other symbols have the i r previous meanings. The channels from 40 to 75 were blanked out at th i s stage to prevent the computer from being influenced by these points during the f i t t i n g process. Neglect of t h i s precaution leads for some very unusual r e s u l t s . After t h i s stage, i t was found that some runs were s t i l l very unusual i n appearance, and a number of further tests were done to e l im-inate those which would degrade the average more than could be recovered by the other tapes. F i r s t , i f any channel was more than three times the average f luc tua t ion more or less than the previous channel, i t s value was replaced by the average of the channels before and a f te r . .This was done to el iminate those e r r a t i c channels caused by bad switches i n the spectrometer. Next, i f any run had more than a set number of consecutive negative numbers, (usually set at 8 or 10), then that run was rejected from the averaging process. F i n a l l y , i f the sum of the channels i n which the l i n e appears (40 to 70) was less than a set number (usually 0 or -10) , then that run was deleted from the average. The procedure was to ' t ry d i f ferent combinations of the three parameters: Order of baseline f i t (2 or 3 ) , r e j ec t ion number (8 or 10), and center sum(-10 to 10) to the same data to f ind three or more combinations which used the largest number of data to give the same r e s u l t s . The f i n a l steps i n the reduction program were to add and average a l l tapes r e fe r r ing to the same source and frequency (except those rejected by the previous c r i t e r i a ) , and then to f i t a least squares f i t Gaussian p r o f i l e to the data. The l a s t rout ine (known as LQF i n -40-th e U . B . C . Computing Center l i b r a r y routines) returns the three parameters: area, center, and standard dev ia t ion , a l l of which were used i n the data in te rp re ta t ion . 3-4) The Continuum Observations Since the important quantity i n recombination l i n e work i s the r a t i o of l i n e in t ens i ty to continuum i n t e n s i t y , observations were made of the continuum as w e l l . This was achieved by making numerous scans through the sources i n both r igh t ascension and d e c l i n a t i o n . (For th i s part of the experiment, the A . G . C . loop must be disconnected.) These scans were recorded on chart paper, and l a t e r averaged graphica l ly by hand. Direct comparison of the source in t ens i ty to known values wasunade poss ib le by observation of known standard sources, and by using the noise tube c a l i b r a t o r . An added benefit of these observations was the measurement of ha l f power beamwidth (HPBW) of the antenna, obtained d i r e c t l y from the scans due to the source s o l i d angle Q g i s very much smaller than the antenna main beam s o l i d angle fi^. Table 3-1 presents the resu l t s from the c a l i b r a t i o n sources, while Table 3-2 presents the observations from the three nebulae. . TABLE 3-1 OBSERVATIONS OF CALIBRATION SOURCES SOURCE FLUX 6 a 6 6 R CASSIOPEIA A 2450±50 39+2' 3 9 ± 2 " 1.50±.02 CYGNUS A 1650+100 39±2' 39±2' .1.001.03 ID"" The f lux i s measured i n uni ts of^watts m""2Hz~ . The quant i t ies 0 a and are the ha l f power beam widths i s r igh t ascension and dec l ina t ion respec t ive ly . R i s the r a t i o of the maximum pen de--41-f l e c t i o n of the source to that of the noise tube c a l i b r a t o r , and i s the important r e su l t to be used l a t e r . The quoted source f lux i s taken from graphs of absolute measurements made by a number of other observers. TABLE 3-2 CONTINUUM OBSERVATIONS OF THREE NEBULAE SOURCE R J L R 2 F v OMEGA 0.425+.01 0.43±.02 700+20 -ORION 0.23 +.02 0.241.04 380±20 W43 0.16 +.03 0.181.06 270+50 The fluxes F v were again 'taken from other observations, and R^vfche : fcheore t ica l r ra t io rof "the :.source in t ens i ty to the noise tube c a l i b r a t o r , was computed using these fluxes and the resu l t s -of Table 3-1. R 2 , the measured r a t i o of source to c a l i b r a t i o n , was taken of the chart records. Table 3-2 shows the very good agreement between our observations and the expected value, su rp r i s -i n g l y good since we were using an unswitched rece iver . 3-5) Results of Line Observations The resu l t s of the data r e -duct ion scheme described i n the previous sections are as fo l lows : TABLE 3-3 DERIVED LINE PARAMETERS -SOURCE FREQUENCY TRANSITION T t T c T / T c A V .Av£ V L S R (MHz) C K °K 7, (kHz) (km/sec) W43 1374.601 168a 0.214 19±8 1.111.20 145±8 1.611,30 99.5±1.0 1399.368 167 a 0 . 2 1 4 •- 1.111.20 148113 1.64+.30 99.6±1.0 ORION 1374.601 1 6 8 a 0.173 2716 0.651.07 194+10 1.261.15 -2.711.0 1399.368 167« 0.192 0.721.10 197117 1.421.21 - 3 . 4 H . 0 OMEGA 1374.601 168« 0.314 47+6 0.671.06 177+7 1.181.10 20.711.0 1399.368 167o(0.327 0.701.07 187+10 1.311.14 17.911.0 -42-The temperatures quoted are antenna temperatures, a quantity de-fined i n the fol lowing manner: If the antenna i s replaced by a matched res is tance , the temperature at which the r e s i s t o r supplies the same noise power as the observed source i s ca l l ed the antenna temperature for that source. This quantity was measured from ob-servations of the c a l i b r a t i o n sources and"of the noise tube. The power received from a source whose s o l i d ; a n g l e tts i s much smaller than the antenna main lobe i s : W / d v =r i r9eT-o-/2scJ-0 (3-3) where A e i s the ef fec t ive antenna area and the factor \ i s present because an antenna can receive only one mode of p o l a r i z a t i o n . The power supplied by an i d e a l l y matched r e s i s to r i s The in tens i ty I v and the f lux F v are re la ted by: (for ns«n^) J v = LL (3-5) where ft^ i s the antenna s o l i d angle. By a fundamental theorem from antenna theory, -0.*f)e - ^ (3-6) Combining (3-6), (3-5), and (3-3) gives From the d e f i n i t i o n of antenna temperature, the above expression i s equal to so that I f , however, we have a c a l i b r a t i o n power source ins ide the rpcpiMpr which WP nan fnrn on. we can obtain T& without having  - 4 3 -to know the other parameters, because i f our c a l i b r a t i o n noise source i s at some temperature T R , and i s per fec t ly matched, then eq ](3-4) applies d i r e c t l y : W v c k > = k T « oln> ( 3 - 1 0 ) I f t h i s c a l i b r a to r gives a de f lec t ion on the output which i s C times the de f l ec t ion caused by the observed source, then, from ( 3 - 8 ) and ( 3 - 1 0 ) , we have T R = CTA ( 3 - l D The only problem l e f t unresolved i s the measurement of T^. In our case, the ca l ib ra to r was a plasma noise tube which operates at a quoted f igure of 1 0 , 0 0 0 K . Remembering that the device i s coupled into the receiver by a 2 0 dB coupler, the ef fec t ive tem-—perature i s , at f i r s t glance, 'TK = lOO°k- ( 3 - 1 2 ) However, a l l i s not so simple. Other than the poss ible errors i n the operating temperature, and the coupling at tenuation, no account has been made of losses from mis-matching and cables. Thus the e f fec t ive temperature w i l l be c loser to : TK = no °i< ( 3 - 1 3 ) Remembering that Cygnus A gave the same de f l ec t ion as the ca l ib ra to r allows us to state TA^- ~i lO ± lO'K (3 -14) The continuum antenna temperatures i n column 5 were then taken d i r e c t l y from eq (3 -14) and Table 3 - 2 . The l i n e to continuum r a t i o was obtained from the r a t i o of the spectrometer output for observations on source (as processed by computer), and from the c a l i b r a t i o n runs using the noise tube as reference. The l a t t e r values were reduced to continuum values by -44-use of Table 3-2. The l i n e antenna temperatures were then obtained by mul t ip ly ing eolumns 5 and 6. The f u l l width at half-power, Av^, was obtained direatlypfromg the computer f i t t e d gaussian. Column 9 i s the value of the nebula v e l o c i t y wi th respect to the l o c a l standard of rest ( L . S . R ) , which i s defined as the centroid of motion of the nearby s ta rs . The program to reduce the observed l i n e sh i f t s to L . S . R . v e l o c i t i e s was k ind ly supplied by Mr. M . J . Mahoney. For comparison, Table 3-4 presents other observations of these sources at nearby frequencies. TABLE. 3-4 OTHER LINE OBSERVATIONS SOURCE FREQUENCY TRANSITION Av L A v t f c V LSR REF W43 1683.20 157 * 1.51+-.11 169+7 2.62±0.19 95.0+2 6 ORION 1424.73' 166 cc 0.66 212±15 1.40+0.2 1 1374.60 168<* 1.39±0.19 - 3 . 4 ± 1 . 3 2 1424.73 166* 0.8 215+10 1.72 0.0 3 OMEGA 1651 158 * 1.15+.17 209+60 2.4010.81 21.0+5 4 1424.73 166* 0.83±.17 183+60 1.52±0.56 20.1+7 5 1424.73 166« 1.05 203±20 2.13 20 3 Where d i r ec t comparison i s poss ib le , our r e su l t s are i n good agreement wi th previously published work. Better comparison can be made when a l l ava i lab le data i s p lo t t ed . Graphs 3-1 to 3-3 give 1 J.A.DeBoer, A . D . H i n , U.J .Schwartz , H.Van Woerden, B u l l . Astron. Ins t . Netherlands, 19, 460,(1968) 2 T.K.Menon, J.Payne, Astrophys. Le t t e r s , 3_, 25, (1969) 3 R.X.McGee, F.F.Gardner, Nature, 213, 579, (1967) £ A . E . L i l l y , D.H.Menzel .H.Penfie ld , B.Zuckerman, Nature 209, 468, (1966) 5 P,Palmer, B.Zuckerman, Nature "209,1118, (1966) 6 E.Churchwell , J . E d r i c h , Ast ron. and Astrophys, , 6, 261, (1970) -45-T • a l l ava i l ab le data of &VT— • Graphs 3-4 to 3-6 show the observations T of — , while graphs 3-7 to 3-9 are p lo ts of Av L. In a l l these graphs, the points p lot ted with c i r c l e s indica te our data. 3-6) Der iva t ion of Elec t ron Temperatures. a) L . T . E . Temperatures Using the assumption of L . T . E . , we extract the e lectron temperatures from eq (2-77): The fo l lowing subst i tu t ions are now employed . -r- O-IS -OIO V J J The approximation for the gaunt factor i s due to Altenhoff et al."* and i s accurate to better than 2% for these frequencies. The r a t i o of ionized helium to hydrogen i s from many observations, and i s an approximate factor which w i l l be applied to a l l nebulae. See, for 2 instance, Palmer. Applying (3-16) into (3-15) gives where v i±s i n GHz, and Av L i s i n kHz. Eq. (3-17) was applied to the observations, and the derived L . T . E . temperatures are displayed i n Table 3-5. 1 Altenhoff , Mezger, S t r a s s l , Wendker and Westerhout,Veroeff. Sternwarte, Bonn, No. 59, (1960) 2 P.Palmer, B.Zuckerman, H.Penf ie ld , A . E . L i l l e y , Astrophys. J . 156, 887, (1969) -46-TABLE 3-5' LTE TEMPERATURES SOURCE v(MHz) T(°K) ORION 1399 7900 1374 8490 OMEGA 1399 9280 1374 8980 TO3 1399 6970 1374 6860 The temperatures obtained for a l l three sources are s i g n i f i c a n t l y higher than other radio measurements made at shorter wavelengths, e spec i a l ly for the Omega nebula. The explanation for t h i s i s quite simple. At the longer wavelengths, the nebula becomes o p t i c a l l y t h i ck , which degrades the l i n e emission due to se l f -absorpt ion . This r e su l t s i n a lower A v j ^ being measured, which gives a higher c value of T when eq (3-17) i s appl ied . _ ' Hence, we must use the exact analysis s o l u t i o n , eq (2-109), which takes in to account both increasing o p t i c a l depth and non-L.T.E. e f fec ts . A recent paper by Brocklehurst^ gives the best evaluations C-& b_ and cfAik-p y e t der ived. The method of determining T, n P , and 1 1 dm 2 E .M. i s that developed by Hje l lming, Andrews, and Sejnowski , and 3 employed by Hjellming and Davies , and i s as fo l lows : 1 M. Brocklehurst , Mon. Not. R. Astron. Soc. (1970), 148, 417 2 R.M. Hje l lming, M.H. Andrews, T . J . Sejnowski, Astrophys. J . , 157, 573, (1969) 3 R.M. Hjel lming,R.D.Davies , Astron & Astrophys. , j[, 53, (1970) 47-Eq. (2-109) i s solved for d i f ferent values of T e , N e , E , and v, with the set of the above var iab les which gives the so lu t ion which best f i t s the observed data being adopted as the so lu t i on . I t i s important to note that two observations d i f f e r i ng great ly i n f r e -quency (at least by the order of 4) are required to obtain unique values of T e , N e , and E. A s ing le observation w i l l f i t a large number of so lu t ions , and at best w i l l only determine a l i m i t to each parameter.. The errors on each derived parameter i s the v a r i a t i o n allowed i n the theore t i ca l so lu t ion which s t i l l allows a f i t to the errors on the data. At t h i s point a fundamental question must be r a i sed . What relevance i s the derived values of T e , N e , and E to the nebula, since the equation (2-109) i s derived for a homogeneous nebula, while the ga l ac t i c nebulae are very clumpy? This question i s answered by Hje l lming , Andrews, and Sejnowski"*" on the basis of a theore t i ca l study of l i n e formation i n models of nebulae which contained var ia t ions o f ' N and T„. Their conclusion was that one obtains consistent and e e meaningful r e su l t s i f the values of T , and N e derived by the above method are interpreted as averages weighted wi th respect to the emission measure. That i s , the derived values of T £ and N e are ac tua l l y averages over the nebula of the form: (3-18) (3-19) where P j^ i s the main beam s o l i d angle 1 i b i d . -48-The in te rp re ta t ion of the derived value of the emission measure i s more d i f f i c u l t . Since the majority of l i n e r ad ia t ion i s emitted from the denser regions, the simple average emission measure {{N&ZJSJJZ £ = (3-20) w i l l probably be too smal l . The relevant value should be weighted wi th respect to e lect ron densi ty . This point has not yet been c l a r i f i e d . Eq (2-109) reads: r = £ ( l ^ ' l - ^ e - ^ ) + e r-U-^~ T') 0 - 2 1 ) Using (3-16), the coef f ic ien ts f iven by (2-110), (2-111), and (2-112) can be rewri t ten i n the p r a c t i c a l form E.M. fl . N(HJ) 1 -1-36^2;l> N c H t ) J (3-24) y ~ 8-235X 10-r In the above equations, E i s i n pc cm" 6 , v i n GHz, and Av-^  i n kHz. Since the b^ and ^ - ^ T factors are tabulated as functions of n , i t 11 dsn i s more convenient to express the coef f ic ien ts as functions of n rather than v. Thus, we f ind = 4-67X1018 k TT~w^ I (3"25) NtH+) -49-where we have used the fol lowing r e l a t i o n ^ s 4 £ S ^ ' (3-28) F i n a l l y , we need an expression for A v T . I f the broadening i s purely Doppler, then we may use eq (2-57). A v t = 0-60<> f 2 ' (3-29) c V srr\h 5o However, observations show that Av^ i s not l i n e a r l y re la ted to v, but ra ther , L\^L - (3-30) The reason for . th i s effect i s probably a combination of two ef fec ts . The f i r s t i s that Stark broadening may be having an effect at lower frequencies. The other effect i s that the lower frequency measurements sample larger regions of the nebula, due to the larger main beam s o l i d angle. For each nebula, the, we must use the observed data to evaluate the coef f ic ien t s i n eq (3-30d and (3-28) i n order to evaluate (3-21). From published data, and from our own observations, we f ind TABLE 3-6 C p ORION 109 0.951.01 OMEGA 138 0.941.01 W43 106 1.001.02 Graphs 3-7 to 3-9, from which the above Table was made, show that eq (3-30) holds very w e l l for a l l frequencies yet observed -50-for Omega and W43, but only for v>3GHz for Orion.'i . For frequencies less than 1.7 GHz, A V L seems to be constant near 200 kHz. This in t e res t ing devia t ion from the power law (3-30) i s due to an addi t iona l •broadening mechanisnijpossibly that due to Stark broadening by e lect rons , an effect described by Griem.* From h is paper, one expects the graph to r i s e wi th decreasing frequency witih a slope of As ye t , no concrete evidence has been obtained to show th is e f fec t . I t appears that there i s a large disagreement between den-s i t i e s measured by the non-L.T.E. analys is on one hand, and densi t ies -predicted by the absence of l i n e broadening on the other. The broadening i h the Orion source i s more l i k e l y to be due to a sampling ef fec t . The Orion nebula, i s known to be extremely turbulent and clumpy. At lower frequencies, the telescope half-power beam width ®HPBW becomes large i n accordance with the re la t ionsh ip OHPB^ - ~- (3-31) and w i l l thus sample larger areas of the nebula. . I f the edges of ...the .nebula are turbulent , as they are. i n Orion, then a broadened l i n e p r o f i l e w i l l r e s u l t . -Each sou rce -wi l l now be discussed-separately, a) Orion A """"" ""This nebula has a rather high e lect ron temperature and - ione of ^the largest emission measures of any-HII region. Because of " i t s high emission measure, there i s considerable enhanced emission, which -by the L . T . E . theory, returns a. temperature much lower than the actual temperature. High reso lu t ion studies of th i s nebula reveal that i t i s very h ighly clumped and turbulent , probably more so than any 1 H.R.Griem,.Astrophys. J . , 148, 547, (1967) - 5 1 -other nebula known. Graphs(3-13) shows the best f i t t heore t i ca l so lu t ion and some of the better data points . The best f i t l i n e i s the so lu t ion to eq (3-21) with the parameters T e = 10,000'K N e = 2x10k cm" 3 E = 2 x l 0 7 pc c m - 6 The errors on each parameter are of the order of 20%, except 10% for T e . This error w i l l be reduced wi th more careful f i t t i n g . b) Omega Nebula This nebula, which i s the br ightest of a l l nebulae, i s a considerably cooler nebula than Orion. The best f i t graph, graph(3-14) gives the fol lowing parameters T e = 7500 K N e = 4X102* c m - 3 E = 1.0*10 7 pc cm" e Also included are three graphs (3"10) to (3-12) which i l l u s t r a t e the f i t t i n g procedure and show the s e n s i t i v i t y of the f i t when the three dependent parameters are changed. The f i t t i n g of the curves to t h i s nebula was more ca re fu l ly attended to than ei ther of the others, which resul ted i n a lower er ror , namely, 10%, 15%, and 20% respec t ive ly . The resu l t s for both nebulae agree w e l l wi th the result! of Hjel lming and Davies.^ c) WA3 Much less can be said about th i s nebula, which has far less observational information. Hjellming and Davies could only give upper l i m i t s to the three parameters due to the lack of low frequency information. Since the pub l i ca t ion of the i r work, three low frequency points have been measured-the twoi in th i s thes i s , and the Hl57a l i n e 1 i b i d -52-of Churchwell and Edr ich . In the f i t t i n g of the theore t i ca l so lu t i on , an obscuring effect which emanates from the so lu t ion made exact f i t t i n g rather d i f f i c u l t . The re su l t for W43 i s . T e =7000 K N e = 5*103 c m - 3 E = 1.0x10 s pc cm - G These parameters are the f i r s t derived for t h i s nebula by th is method. The errors are rather la rge , about 15%, 20%, and 30% respec t ive ly . 3-7) Concluding Remarks The main point to be emphasized i s that a non-L .T .E . analysis must be used i n in te rpre t ing radio l i n e data from ga l ac t i c nebulae. I t i s quite naive to bel ieve that the nebulae are i n L . T . E . when theore t i ca l inves t iga t ions point p l a i n l y to the contrary. I t was evident from the e a r l i e s t inves t iga t ions that something was wrong wi th the L . T . E . theory. Attempts to explain the d i f ferent temperatures derived by th i s theory at d i f ferent frequencies without removing the assumption of L . T . E . included sampling areas, and e lec t ron temperature f luc tua t ions . As has been pointed out, a l l l i n e data from any one source w i l l y i e l d unique and unambiguous values of temperature, densi ty , and emission measure wi th excel lent accuracy. S t i l l , . m a n y problems remain, and much more observation i s necessary. S p e c i f i c a l l y ; 1) While analyzing the W43 data, a pecul ia r effect occurred whenever the l i n e absorption coef f ic ien t became equal and opposite the continuum absorption coe f f i c i en t . This effect occurred for values of n around 100 to 120, and emission measure less than 101* 1 i b i d . -53-pc cm -k. The r e su l t was that the l i n e to continuum r a t i o was considerably reduced on the high frequency side of the anomaly, becoming negative for one or two values of n , then rose and became greater than normal on the low frequency s ide . The o s c i l l a t i o n s were very steep and whole range of th i s behavior i s quite smal l . More work needs to be done on th is ef fec t , which was not noted by Hjellming i n h i s work, nor has been observed yet.Of course, the lack of any observations of th i s resu l t do not mean that i t does not e x i s t , since the ava i lab le data on th i s source i s very skimpy. 2) The whole problem of c o l l i s i o n broadening i s i n an unsatisfac~ tory s ta te . I f Griem's theory of broadening of hydrogenic l i n e s at h igh quantum leve l s i s used, then the dens i t ies of the Orion and Omega nebulae must be at least 100 times lower than the measured v a l u e s . i n order for no broadening to have occurred as i s observed. Since an error of t h i s sort i s hardly l i k e l y considering the number of d i f f e r i n g ways th i s quantity has been measured, i t would seem that the theory i t s e l f i s quite i n jeopardy. Much needed are ob-servations of the nebulae at frequencies under 1 GHz to search for broadening. A re-examination of the theory i s also i n order. A search for spec t ra l l i n e s at low frequency would be a rather d i f f i -c u l t and tedious experiment (because the nebulae are o p t i c a l l y th ick) But should be car r ied out. 3) There are two other methods of measuring the temperature using non-L .T .E . ana lys i s . One has already been dealt w i t h - the observing of na and ( n - l ) B l i n e s . The advantage of t h i s experiment i s that unambiguous values of T e and can be obtained. The major disadvantage i s that since the observing frequencies of the two l i n e s are a factor of 2 apart," the beam areas of the telescope w i l l d i f f e r by a factor of 4, thus making i t necessary to measure the beta l i n e i n a number of d i f ferent areas, and averaging l a t e r . Another method i s to observe the a ,g ,y ,6 , . . . . l i n e s which occur at nearly the same frequency. Since the enhancement of the higher order l i n e s i s much reduced( because the outer quantum leve l s are more c lo se ly i n L . T . E . ) temperatures derived by the elementary L . T . E . theory should converge on the true temperature as the order of the l i n e i s increased. This method i s known as the asymptotic method. I t s major disadvantage i s that i t requires the measurement of very weak l i n e s . I t has the decided advantage that a l l l i n e s are at the same frequency. A) A low-frequency survey of HII regions i s necessary to comple-ment the extensive highrfrequency survey of Reifenste in e t . a l . ^ Once th i s has been completed, then a comprehensive non-L.T.E. analys is of these l ine-forming regions can be performed s i m i l a r ! to that done i s t h i s thes i s . 1 E .C.Rei fens te in I I I , T .L .Wi l son , B.F.Burke,P.G.Mezger, and W. J .Al tenhoff , Astron. & Astrophys. , 3 5 7 , ( 1 9 7 0 ) -55-FIGURE CAPTIONS Figure 3-1 i s the block diagram of the receiver and spectrometer as described i n page 35. Figures 3-2, 3-3, and 3-4 are the p lo ts of the data after a l l analysis has been completed. Each dot represents one channel out-put of the spectrometer. The smooth curve i s the computer f i t t e d leas t squares f i t gaussian. Graphs 3-1, 3-2, and 3-3 are p lo ts of a l l ava i l ab le data of A^ Jk- for the three sources of in t e res t . The l i n e i s an eye-f i t ted best f i t to the data. This l i n e has not been attempted for W43 due to a lack of data i n the intermediate range from n=110 to n=155. Graphs 3-4, 3-5, and 3-6 are p lo ts of a l l ava i l ab le data of T L / T c for the three sources. The l i n e i s again eye - f i t t ed . Graphs 3-7, 3-8, and 3-9 show the v a r i a t i o n of l i n e width &~0L with frequency, and the absence of any c o l l i s i o n broadening. Graphs 3-10, 3-11, 3-12 are included to i l l u s t r a t e the method of non-L.T.E. analysis described i n the text . Each graph shows the v a r i a t i o n of the r a t i o T L / T c when one parameter i s allowed to vary , keeping the other two f i x e d . Graphs 3-13, 3-14, and 3-15 show the f i t of the best so lu t ion of the theore t i ca l equation for T L / T C wi th the parameters given i n the tex t . RECEIVER O c m L.O. g hi L.O. zo AS kOKN cot/nan 4S.7 35 MHt _ PR£/**lP I 35 I.F~ nous ruge FILTER ltTntxcK PAW, FILTERS <_ _ ' OVTSI0J _ I PrWrTrTp ~ — — - CONTROLS SPECTROMETER P R O M t-AM r -• Q — P K ^ » — / V W — - f f i — It MP ChUHO co»PVT£R TO *,P£CTRoNieT£lt fftPBR OOTPOl 03 I -o o g > o > CO O M E G A NEBULA •AO -35 -— I 1 ; 1 1 . 1 L_ 1 I L • ' t too -so -bo - 4 0 . -20 -o zo 40 6 0 eo too V i S R (km / s e c ) X I I I I ! I I I ( l_ -too -80 -bo -AO -Zo O Bo 4o 60 80 V.5ff(km/sec) OMEGA .<5£|5f.f/,. 3z4 CYCLES X !• O I V I S I U N S r t . K mun p ro 10 _LL _l_L M M ; [ ! I I ! -LLL - U . ORION HALF WIDTHS l-o i o 3 o Bo t-o 10 20 3 o so -O(OHz) W43 H A L F WIDTHS ESffifittt GRAPH 3 - 9 looo 1 0 3 o 4o 6<? -0(0 H: 10 o ZOO 4o T! - A l -APPENDIX A THE IMPORTANCE OF ELECTRON SCATTERING The coef f i c ien t of absorption for th i s effect is'*' Of importance i s the r a t i o Ev iden t ly , we can neglect e lect ron scat ter ing as an important process i f 2-66 xto" Vd*£ > > 4 . (A_3) T y p i c a l l y i n HII regions, tip - 103, T - 104. Using g f f = 5, the c r i t e r i o n becomes /3-3 x to" -v-2 y7 l (A-4) This indicates that for frequencies less than 1 0 1 0 Hz, '(10 GHz), free-free absorption w i l l dominate e lect ron sca t te r ing . For high frequencies, and/or high temperatures and/or low dens i t i e s , the reverse i s t rue . Because the scat ter ing by free electrons i s comparable to free-free absorption, we must invest igate scat ter ing as a source of photons. The energy scattered in to s o l i d angle oriented i n tiie d i r e c t i o n 9,<J> from s o l i d angle oriented at B 1 , ^ 1 i s where Iv(e',<j>') i s the spec i f i c i n t ens i ty i n d i r e c t i o n %' ,<j>' , 1 D. Clayton, P r i n c i p l e s of S t e l l a r Evolut ion and Nucleosynthesis, McGraw-Hi l l , -A2-p(0,<f>,e',<j)') i s the scat ter ing phase funct ion, which gives the angular d i s t r i b u t i o n of sca t te r ing . Obviously, the t o t a l energy emission i s then tr Zn (A-6) O 0 Tor Thompson scat ter ing from free e lectrons , ' f>(e,%&,<?') = | (/ t ^ x ) (A-7) where i s the angle between incident and scattered rays , and i s e a s i l y found to be The geometry i s i l l u s t r a t e d below. \ dA'. Figure A - l Photons are scattered from s o l i d angle d^ ' into s o l i d angle dJ3. The z-axis i s oriented i n the d i r e c t i o n of the net f l u x . The phase function i s normalized such.that 4TT (A-9) - A 3 -Now, the basic equation of t ransfer , eq(2-24) reads «i 5 (A-10) I f we d i s t i ngu i sh between scat ter ing and true emission and true absorption, then ( A - l l ) ( A - 1 2 ) ( A - 1 3 ) But, using equation (A-6) , J ^pte, «*» ©; ^ J l ^ <?) -^e'deUf ( A - i 4) ds 4TT, This form of the equation of transfer has in te res t ing consequences. a) I so t ropic r a d i a t i o n : I v(0 ,,<j> ,) = I v . •Using eq. ( A - 9 ) , we f ind dJy ^ ~ V* I „ + <fa ( A _ 1 5 ) d 5 This r e s u l t i s of only academic use, since one never r ea l i z e s i s -otropy except i n thermal equ i l ib r ium. b) In a medium s u f f i c i e n t l y dense ( i . e . o p t i c a l l y t h i c k ) , the i n t ens i ty w i l l approach a form 1 . 0 < ? ) - I0LO,r) <-1 \*ce' (A-16) Using eq , ( A - 8 ) , we obta in , from ( A - 1 4 ) , « - V f t 1* -f U - V * X , ceo 9' (A_17) d S which reduces to eq ( A - 1 5 ) i f | K S ' I , | « | i c A I V | c) Result (b), however, i s not at a l l v a l i d for o p t i c a l l y th in nebulae. I f we assume a spher i ca l , homogeneous nebula for a s t a r t , the angular dependence of r ad ia t ion w i l l be i n d i r ec t proportion to path' length through the nebula. * y Figure A-2 Path Lengths i n an i d e a l HII region Using the law of cosines- the path length s i s S ~ - rocx>9 t J Rz- rlAW-&' By al lowing R to be the scale fac tor , and defining 'Jk = V* (A-18) (A-19) (A-20) then l9 (A-21) (A-22) which allows us to wr i te Integrat ion of (A-14) using (A-21) and (A-8) i s long and tedious, and y i e l d s a very complicated r e s u l t . App l i ca t ion of these resu l t s i s complicated,by the factLthat the angular va r i ab le 9 and r a d i a l . v a r i a b l e k are not separable. We assume, then, using eq (A-4), and the r e su l t s of spec ia l cases (a) and (b) as a guide, that sca t ter ing can be neglected i n HII regions at these low frequencies. I t i s worth noting that the only other way to f ind the r e l a t i v e importance of scat ter ing v s . true absorption would be to attempt and exact so lu t ion of eq (A14), which should be w r i t t e n as -A5-o o -a rather formidable problem, which becomes worse i f the r e s t r i c t i o n of homogeneity i s removed, al lowing the coef f ic ien ts to become de-pendent upon r , 6, and <j>. 

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