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Radio spectral line studies of the interstellar medium in the galactic plane

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Title:
Radio spectral line studies of the interstellar medium in the galactic plane
Creator:
Seacord, Andrew Wilkin, 1941-
Copyright Date:
1975
Language:
English
Physical Description:
viii, 235 leaves : ill. ; 28cm.

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Subjects / Keywords:
Absorption spectra ( jstor )
Emission spectra ( jstor )
Geometric lines ( jstor )
Line spectra ( jstor )
Noise spectra ( jstor )
Power lines ( jstor )
Radio observatories ( jstor )
Radio spectrum ( jstor )
Spectral index ( jstor )
Spectral line width ( jstor )
Astronomy thesis Ph. D ( lcsh )
Dissertations, Academic -- Astronomy -- UF ( lcsh )
Interstellar matter ( lcsh )
Radio sources (Astronomy) ( lcsh )
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bibliography ( marcgt )
non-fiction ( marcgt )

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Thesis:
Thesis--University of Florida.
Bibliography:
Bibliography: leaves 226-233.
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by Andrew Wilkin Seacord, II.

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RADIO SPECTRAL LINE STUDIES OF THE INTERSTELLAR MEDIUM
IN THE GALACTIC PLANE











By

ANDREW WILKIN SEACORD, II


A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY



UNIVERSITY OF FLORIDA

1975
































To My PaAentA

With GwatUude

Louise DeLano Peck Seacotd

and

WiLkin Henry Seacord















ACKNOWLEDGEMENTS


I wish to express my appreciation to Dr. Stephen

Gottesman who introduced me to the intriguing problem of

the galactic ridge and who has provided many comments and

suggestions during the course of my study of it. His en-

couragement and cooperation during the final stages of

this project are greatly appreciated.

The spectra observed as a part of this study were

obtained with the 43 meter telescope at the National

Radio Astronomy Observatory in Green Bank, West Virginia.

I appreciate the opportunity to have used these facilities.

I am grateful to the NRAO staff, particularly the opera-

tors and engineers whose support made the observations suc-

cessful.

I also wish to thank the other members of my doc-

toral committee whose comments and suggestions have been

most helpful. Actually, many people within the Department

of Physics and Astronomy deserve a word of thanks for

their general support, for providing travel funds, and for

making computer time available to me.

Finally, I owe many words of gratitude to my wife,

Penny, who has been patient with me and often provided me










with much-needed words of encouragement and to my daughter,

Becky, who has provided many interesting moments during

the course of my work here.
















TABLE OF CONTENTS


Page

ACKNOWLEDGEMENTS...................................... iii

ABSTRACT .................. .................. ......... vii


CHAPTERS

I INTRODUCTION...... ........................... 1

The Galactic Ridge......................... 1
Objective, Method and Scope of
This Study.............................. 4

II INTERSTELLAR MEDIUM ASTROPHYSICS ................ 8

Introduction............................... 8
The Transfer Equation and a General
Solution of It.......................... 8
Thermodynamic Equilibrium.................. 12
The Thermal Continuum............... ...... 14
Radio Recombination Lines.................. 18
Thermodynamic Temperatures and Level
Populations.......................... 19
Population and Depopulation Mechan-
isms of High n-Levels ................ 23
Collision Transition Probalities
and Cross-sections..................... 26
Radiative and Dielectronic Recombina-
tion Coefficients..................... 31
Departure Parameters bn and 6mn......... 34
Radio Recombination Line Absorption
Coefficient .......................... 39
Line Profiles........................... 43
Non-LTE Solution of the Transfer
Equation and Line Enhancement......... 49
The Radiation Field....................... 53
Ionization Rates .......................... 56
The 21 cm Hydrogen Line................. ... 58
Pulsar Dispersion Measure.................. 62










Page

III THE TELESCOPE SYSTEM SENSITIVITY AND THE
OBSERVING SCHEME .... ................ .......... 64

System Sensitivity .............. .......... 65
The Observing Scheme .............. ......... 67

IV THE RADIO RECOMBINATION LINE OBSERVATIONS...... 70

Introduction................................ 70
Observing Scheme and Data Reduction........ 75
The Spectra ................................ 83

V MODELS FOR THE INTERSTELLAR MEDIUM IN THE
DIRECTION 3C391 ............................... 107

3C391: A Summary of Previous Observa-
tions.................... .. ............ 107
The Models................................. 119
The D-Method Models ...................... 119
Isolated Cold Region Models .............. 147
The Hot Spot Models ...................... 156
H II Nebula Models ...................... 175

VI FURTHER CONSIDERATIONS AND APPLICATIONS OF
THE MODELS.................................... 182

Discussion of the Models ................... 182
The Pulsar 1858+03 ........... ............ 200
The 3C396 and COMP Spectra................. 206
Distribution of the Recombination Line
Sources Within the Ridge ................ 210

VII SUGGESTIONS FOR FURTHER RESEARCH ............... 216


APPENDIX. .............. .................... 222

BIBLIOGRAPHY.................................. 226

BIOGRAPHICAL SKETCH..................... ...... 234










Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy


RADIO SPECTRAL LINE STUDIES OF THE INTERSTELLAR MEDIUM
IN THE GALACTIC PLANE


By

Andrew Wilkin Seacord, II

August, 1975


Chairman: Stephen T. Gottesman
Major Department: Astronomy


This study develops a model for the source of radio

recombination line emission from the galactic plane. The

model is based mainly on recombination line spectra ob-

served in the direction of the supernova remnant 3C391.

The H157a recombination line was observed and used with the

H92a spectrum observed by Cesarsky and Cesarsky (1973) to

obtain a model consistent with both spectra. Other

criteria--such as the 80 MHz optical depth, the contri-

bution to the thermal continuum temperature, and ionization

rates--were also considered.

Four models were studied: a cold region illumi-

nated by 3C391; an isolated cold region; a group of hot,

dense H II regions; and a low density H II nebula. Of

these, the best model is a small (x85 pc diameter) H II

region with a temperature of 75000 K and an average elec-
-3 -3
tron density of 10 cm











The H157a spectrum in the direction of the pulsar

1858+03 was also observed. It was found that it is prob-

able that one of the recombination line nebulae lies along

the line of sight to the pulsar. In this case the pulsar

would be at a distance of 2.6 kpc in order for its dis-

persion measure to agree with the measured value of 402
-3
pc cm

The large scale distribution of the H II nebulae

was also studied. The results of available radio recom-

bination line spectra surveys from the galactic plane sug-

gest that they exist within an annular region,which ex-

tends from 4 to about 8.7 kpc from the galactic center.


v i i i
















CHAPTER I

INTRODUCTION



The Galactic Ridge



As a radio telescope is scanned across the plane

of the Milky Way at some galactic longitude,1 it is seen

to be a source of radio emission. For radio frequencies

of 100 MHz or greater, the intensity increases as the

galactic equator is approached and peaks at the equator

itself. Scans through the galactic plane made at several

different frequencies reveal that the emission has two

components, thermal and nonthermal. At frequencies below

a few GHz, the emission is predominantly nonthermal with

a spectral index between -2.7 and -3.0 (Westerhout, 1958;

Matthews, Pedlar and Davies, 1973). At 85 MHz the radi-

ation is almost completely nonthermal and the intensity




1 II
In this work, the revised galactic latitude, b ,
and longitude, 111, as adopted by the IAU (Blaauw et al.,
1960) will be used throughout. The superscript II, how-
ever, will be dropped from the notation.










distribution across the galactic plane has a total half-in-

tensity width of about 50 at longitude greater than 120

from either side of the galactic center (Hill, Slee, and

Mills,1958). As the frequency increases, the overall in-

tensity decreases and the width of the emission becomes

smaller, approaching a thin ridge at centimeter wavelengths.

At 1390 MHz the half-intensity width is generally about

195. The thickness of the nonthermal emission region in-

creases somewhat near the galactic center. This thin con-

tinuum emission region surrounding the galactic equator

has been named the "galactic ridge."

The galactic ridge has been recently mapped with

an antenna resolution of 11 arc minutes at frequencies of

1414, 2695, and 5000 MHz by Altenhoff et al. (1970).

Matthews, Pedlar, and Davies (1973) have used these maps

to obtain the latitude distribution of the continuum in-

tensities at three longitudes where no resolved sources

are present. They plotted the distribution with the in-

tensity scale at each of the frequencies adjusted so that,

if the emission were entirely nonthermal, the three dis-

tribution curves would be congruent. The plots are con-

gruent at latitudes greater than about 025, but inside this

range, the intensity increases markedly with frequency due

to thermal emission. These curves illustrate the two com-

ponents of the galactic ridge continuum emission.










It is known that hydrogen is concentrated within

a thin region around the galactic equator (cf. Westerhout,

1958; Baker and Burton,1975). The thermal emission comes

from the free-free process between electrons and protons

from ionized hydrogen.

At frequencies below about 30 MHz, a scan cutting

through the ridge shows an intensity profile with two

peaks, one on each side of the equator, The decrease of

intensity along the equator results from absorption of

the strong nonthermal emission by the free-free process

within the thermal emission region. The map by Shain,

Komesaroff, and Higgins (1961) at 19.7 MHz shows the ab-

sorption lane prominently.

The galactic ridge took on an interesting aspect

when Gottesman and Gordon (1970) observed low intensity

H157c radio recombination line spectra in two directions

in which the Altenhoff et al. (1970) maps show no resolved

sources. Gordon, Brown, and Gottesman (1972) found that

at a longitude of 330, the recombination line emission is

limited to a latitude of less than O05 above the equator.

In the region toward the galactic center, recombination

line emission has been observed at a latitude of -0.62

(Lockman and Gordon,1973). Previous to Gottesman and

Gordon's observations, radio recombination lines had only

been observed from known galactic H II regions. They have

not been detected from the interstellar medium in the solar

neighborhood.










A number of workers have studied the galactic

ridge in order to determine the nature of the radio recom-

bination line sources and how they relate to the thermal

continuum and 21 cm spectral line emission. The analysis

of Cesarsky and Cesarsky (1971, 1973b) and Lockman and

Gordon (1973) find that the electron temperature of the

radio recombination line emission regions is close to 20 K.

Other workers derive temperatures ranging from 103 to 104

K (Gottesman and Gordon,1970; Gordon and Gottesman,1971;

Matthews, Pedlar, and Davies,1973; Jackson and Kerr, 1971;

Chaisson,1974a; and Downes and Wilson,1974). Very recently

two papers have appeared,which conclude that these recom-

bination lines originate from low density H II regions

(Jackson and Kerr, 1975; Pankonin,1975).



Objective, Method, and Scope of This Study


The problem to which this study is directed is that

of determining the type of region responsible for the recom-

bination line emission in the galactic ridge. The first

step will be to observe recombination line emission from

regions in the ridge in which there are no H II regions

visible on the Altenhoff et al. (1970) maps and from which

other data have previously been obtained. The recombination

line spectrum will provide one piece of datum, the power

contained in the spectral line. However, there are several










parameters which define the type of region: electron tem-

perature, electron, neutral hydrogen, ionized hydrogen

number densities, size of region, and distribution of the

region, or regions, within the galactic ridge. The latter

may imply a relation between recombination line emission

and other phenomena such as low frequency radio absorption

and galactic structure.

Since the observations made as a part of this study

can only supply one bit of information, the identity of

the recombination line source can be obtained only after

other observations are utilized. Even so, there are still

more parameters whose values are to be found than there

are observed quantities. For that reason, the values of

some parameters must be chosen at the onset of model com-

putation. A type of model is hypothesized and theory or

the conclusions from previous observation suggest what

these values are. The second step will be to obtain, from

one region in the ridge, a model solution which is consis-

tent between two similar data. The ideal case for this

is the region toward the supernova remnant 3C391 from which

Cesarsky and Cesarsky (1973b) have observed an H92a recom-

bination line spectrum. This region has undergone a con-

siderable amount of scrutiny, particularly by those who

have attempted to measure the 80 MHz optical depth (Cas-

well et al., 1971; Dulk and Slee, 1972; Goss, 1972).










The third step in the process will be to assume

that this model applies in the mean sense to the entire

ridge and use the remaining data obtained for this study

to check the model. The method is to calculate a parame-

ter which depends on the model, to see if it agrees with

what is observed. Finally, recombination line surveys

will be combined in such a way as to examine the large

scale structure of the recombination line ridge.

The theoretical basis of this study will be devel-

oped in Chapter II. All equations used in the analysis

will be derived and approximations and assumptions will be

discussed. The third chapter will discuss theory behind

the observing procedure. The method and conditions under

which the observations were made and the display of the

processed spectra will occupy Chapter IV. Chapters II,

III, and IV provide a basis for the subsequent work.

Four types of regions on which to base the models

will be selected. Three of them result from previous obser-

vation of 3C391, a summary of which begins Chapter V. The

Hot Spot model is based on unpublished observation of Alten-

hoff et al. (1973). The remainder of Chapter V develops

the four models. Chapter VI is involved with the selection

of the most probable model and tests this by computing the

distance to the pulsar 1858+03 and the 80 MHz optical depth

expected in the direction of 3C396. Chapter VI ends with





7



an investigation of the large scale structure of ridge

recombination line emission. The last chapter concludes

the study by proposing future projects to study the galac-

tic ridge.
















CHAPTER II

INTERSTELLAR MEDIUM ASTROPHYSICS



Introduction


It is the purpose of this chapter to present a

foundation on which this project is based, particularly

the analysis of the spectra. The starting point is the

equation of radiative transfer. The coefficients of ab-

sorption and emission are parameters of the solution to

the transfer equation.

Since this study is based on radio spectral lines,

the line absorption coefficient will be examined in de-

tail. The line profile of a given element depends on the

thermodynamic state, temperature, abundance, electron

density, and motion of the gas creating the line.


The Transfer Equation and a General Solution of It


The transfer equation can be set up by considering

the radiation passing through a region as shown in Figure

II-1. The radiation entering this region has a frequency-

dependent specific intensity, Io(v), which can be both

thermal and nonthermal in origin. The line of sight enters


























I() I I io(v)
Q ^ _________--0-r---____ lo





OBSERVER
s=L s=0
-rr T'=O


Figure II-i.


Ray Path Geometry Through a Line-
emitting Region. The region is
shown shaded.









the region at a point where the path length, s, is zero.

The total path length through the region is L. At the

near boundary, where s = L, the specific intensity is

I(v). The change in specific intensity along a differ-

ential path length ds in a homogeneous medium is


dl(v) = KL(v) I(v,s) ds -KT(v) I(v,s) ds + jL(V ) ds + jT(v) ds


+ jN(v) ds. 11-1


The nonthermal emission coefficient is jN(v). In the

interstellar medium the nonthermal absorption processes

are not important. The thermal process, which will

generate or absorb radiation, is the free-free process;

it will be discussed later. The coefficients of ther-

mal continuum emission and absorption are jT(V ,s) andKT(v,s)

respectively. The applicable coefficients for spectral

lines are denoted by the subscript L.


Let j(v) = jT() + jL() and K(v) = KT(V) + L(v),then at any

point,


I(v) = IT() + IN() + IL() = Ic(v) + IL(). Equation II-1 then

becomes


dl(v) j(v) 3N
d ( + + V II -2










The optical depth, T, is defined by

L
t(v) K(v) ds = TC() + TL().



The thermal source function, S(v), is defined by


j(v) jL(v) + jT(v)
S( () = + II-3
-K(0)KL T


Since there is no nonthermal absorption coeffi-

cient and the thermal and nonthermal processes are in-

dependent, there is no nonthermal source function.

A general solution to the transfer equation can

be obtained by multiplying Equation II-2 by the inte-

grating factor exp(--) and integrating along the path

length to get


I(v) = (v ) e- + (l-e- ) + < (1-e-).


The brackets,< >, denote the average over the path

length. The intensity of the nonthermal emission gen-

erated within the line-emitting region is IN = jN>L

and L = T; so, the general solution is

I
I(v) = Ioe I (l-e ) + (-e-T). 11-4


Nonthermal emission decreases in intensity with increas-

ing frequency, in general.










In order to derive a solution for a specific

region, the source function-which depends on the thermo-

dynamic state of the region-must be found. It is im-

portant, therefore, to consider the conditions for thermo-

dynamic equilibrium. This is the subject of the next

section.


Thermodynamic Equilibrium


To be in thermodynamic equilibrium (TE), a gas

must have the following properties.


1. The temperature and pressure throughout the

region are constant and are equal to that of

its surroundings; hence, there will be no

flow of energy or mass through it.

2. The gas will radiate as a black body at that

temperature.

3. Detailed balancing must exist; that is, for

each process occurring in the gas, its inverse

must also occur by an equal amount (e.g.

ionization and recombination).

4. The velocities of the gas particles have a

Maxwellian distribution.

5. The population of any excited state of an

atom, ion, or molecule is determined by the

Boltzmann equation.










6. The relative population of any level of ion-

ization of a species to that of the neutral

species is given by the ionization (Saha)

equation.


The observed spectral line intensity depends on

how well these conditions are met. Radio recombination

line observations of a variety of regions indicate that

the interstellar medium is generally not in local thermo-

dynamic equilibrium (LTE); that is, the conditions of TE

are not met even on a small (local) scale (Hjellming and

Davies, 1970; Andrews, Hjellming, and Churchwell, 1971;

Gordon, 1971; Brown and Balick, 1973). If LTE does not

exist, the transfer equation parameters must be modified

to account for this. In the following work, a super-

script asterisk denotes an LTE quantity. The second con-

dition of LTE dictates that S*(v) = B(v,T), the Planck

function for a temperature T. For radio frequencies,

this simplifies to


B(2,T) kT 2
B(v,T) 2- v2 11-5
c


In the case of non-LTE, a correction parameter is defined

such that


S(v) = A(v) S*(v) = A(v) B(v,T).


II-6










In any steady-state situation (LTE or non-LTE),

the total per-unit volume emission and absorption must

balance. Only in LTE do emission and absorption balance

at each frequency so that, by Kirchhoff's Law, j(v) =

K(v) B(v,T).


The Thermal Continuum


Thermal emission and absorption are the result of

an encounter between a free electron and a positively-

charged ion after which the electron remains free (cf.

Mitchell and Zemansky, 1934, p. 154). From a classical

point of view, an electron moves in the field of an ion

(e.g. H ), is accelerated and absorbs a wave or is decel-

erated and emits a wave. As shown in Figure 11-2, the tra-

jectory of an electron with a given velocity is altered

by an amount determined by the impact parameter, p. For

radio frequencies, p is large enough that the entire

trajectory can be approximated by a straight line.

Oster (1961) has derived an expression for the

thermal absorption coefficient, which includes a quantum

mechanical correction applicable for temperatures in ex-

cess of 5.5 x 105 K. The derivation of K( v ) was ob-

tained by considering emission from a single electron with

specified velocity and impact parameter. Then jT( v ) was

found for an ensemble by integrating first over a suitable










range of p and then over a Maxwellian velocity distri-

bution. Then, KT(v ) = jT(v)/B(v,T) and, for hydrogen,



T(v) = 3.014 x 10-2 T-1.5 v-2 ln(4.955x10-2 T-2.5 -l1) N Z2 N. .
T o 0 e 1 1

11-7



The numerical constants are functions of fundamental con-

stants as shown in the Appendix, Equation A-1. The sum-

mation is over all ionic species present. Z. is the

effective charge number (ZH+ = 1) and Ni is the number
th
denisty for the i species.

Altenhoff et al. (1960) have derived an approx-

imate expression for KT(v),which is


K(x) = 8.235 x 10-2 T-1.35 -2.1 N Z2 N.. 11-8
T e e 1 i I


To bring this approximation into agreement with Oster's

more exact formulation, it is multiplied by a correction

factor a(Te,v). Using it, the optical depth then becomes


TT(V) = (v) ds = 8.235 x 10-2 a(T ,\) T-1.35 -2.1 ET

o1
11-9


ET is the continuum emission measure defined by









L
2
E (N Z2 N.) ds. II-10
T e 1 1
0

-6
The emission measure is expressed in units of pc cm

The factor a(T ,v) has been tabulated by Mezger

and Henderson (1967) for the ranges 100 < v < 105 MHz

and 3 x 103 < T < 15 x 103 K. For these ranges, a(T ,v)

varies between 0.7680 and 0.9971.

The emission measure is often given by

L
E = N2 ds. II-11
T e N


The effect of heavy elements on the size of ET

depends on the fractional ionization of hydrogen; x -

Ne/NH. The atomic species with an ionization potential
less than that of hydrogen (13.6 eV) may be totally
< -3
ionized. If x < 10 the contribution of these elements

to the electron pool will be an appreciable fraction of

the total unless their abundances are less than those

adopted by Spitzer (1968, p. 122). The most abundant

contributor to the electron pool, other than hydrogen,
-4
is carbon which has an abundance NC 4 x 10 NH.

The continuum emission spectrum from a region of

the interstellar medium can provide some information

about the conditions there. Consider Equation 11-3 for

the continuum; that is, IL = 0. Let I = I = 0 and,










assuming LTE, S(v) = B (v,T). Then, using Equation 11-5,


2kT 2 -T
I(v) v (1-e T). 11-12
2
c


When the frequency is high enough that the gas is opti-

cally thin, TT << 1, and


S2kT 2
I(v) = v T.


Since, by Equation 11-8, TT v 21, I(v) a v0) and the

spectrum is nearly flat. For low frequencies such that
2
the gas is optically thick, T, 1.5 and I(v) a The

turnover frequency, vT, is that for which TT = 1, al-

though sometimes the value of 1.5 is used. Setting TT

equal to one in Equation 11-8 and solving for vT gives



vT = 1.893 x 102 a(T ,v)-2 1 ET2.1 T-0.643 11-13


If in some region for which vT can be measured, either

ET or T is already known, the other parameter can be

found. One independent source of information is the in-

tegrated radio recombination line power. Radio

recombination lines will be considered in the next

Section.










Radio Recombination Lines


Radio recombination lines arise from transitions

between levels with large principal quantum numbers;

n 40. They result from the capture recombinationn) of

an electron by an ion into an upper level. The nomen-

clature for such a transition is Xn(An) where X designates

the recombining species; n is the principal quantum num-

ber for the transition which is specified by a for An = 1,

B for An = 2, etc. The predominant element from which

recombination lines are observed is hydrogen; however,

radio recombination lines of helium, carbon, and possibly

heavier elements have been observed (Gordon, 1974;

Chaisson, 1973; Pedlar and Hart, 1974).

The emission frequency, vmn, for the transition

between the mth and nth levels (m > n) is given by the

Rydberg formula (cf. Shore and Menzel, 1968, p. 13)


2 1 1 2 an
v = c R Z ( ) = 2c R Z2 An 11-14
mn x x 2 2 Rx x 3
n m n


R is the Rydberg constant for the species X and Z is
x x
its effective charge number. Radio recombination line

frequencies for hydrogen and helium have been tabulated

by Lilley and Palmer (1968).

The recombination line strength depends on the

number of atoms in the region which are excited to the










upper level of the transition. Since the gas may not be

in LTE, it is necessary to calculate the LTE populations

and the departure parameters,which indicate the degree to

which the gas is out of LTE. These matters will be dis-

cussed in the following subsections.


Thermodynamic Temperatures and Level Populations


Radio recombination line emission regions contain

excited neutral atoms, ions, and electrons. The number

densities (in units of cm-3) for hydrogen are NHo, NH*+

and N respectively. NH is the total hydrogen density.

In LTE there is an equilibrium situation where recombin-

ations are balanced by ionizations. What is desired is

a statistical relation between the kinetic temperature of

the gas, NH+, Ne and Nn, the number density of atoms in

the nth level of excitation.

For a gas in LTE, the population number density of

ions in the nth excited level and the i ionization state

is given by the Boltzmann equation (cf. Aller, 1963, p.

112):



Nn No gn exp(- Xin/kTex). -15



This equation is based on the condition that the

assembly has a Maxwellian velocity distribution. In










Equation 11-15, Noi is the LTE population of the ground

state; gn and go are the statistical weights of the ex-

cited and ground states respectively; Xin is the excitation

energy within the ith state of ionization; and Tex is the

excitation temperature. The statistical weights are the

number of sublevels in the nth level; g = (2J + 1). J

represents the total angular momentum--spin plus orbital--

of the level. Under the conditions of LTE, the excita-

tion temperature, T is equal to the kinetic temperature

of the gas.

The radio recombination lines,which have been

observed involve highly excited neutral atoms, for ex-

ample H I, He I, and C I. So,the desired population is

N expressed in terms of the electron density and the
no
respective number of the singly ionized species. For

the species X,


N (X) = 4.143 x 10-16 N.(X) N Te1 5 exp(- (X)/k T
n 1 e ex 2u7- T) exp(-Ein(X)/k Tex)"

11-16



The details of the derivation of this equation are given

in the Appendix; it results from placing the values of

the fundamental constants into Equation A-5. The par-

tition function, ul(T), is given by Equation A-2. Ein(X)

is the threshold energy which is (Equation A-6)




















RADIO FREQUENCY
e













e
X RAY


Figure 11-2.


The Free-Free Process. The elec-
tron, e, approaches the central
ion. The impact parameter, p,
is shown for two cases.










E1n(x)=-(XI-X ) = 157900 (Z 1/n2) k. 11-17


The kinetic energy of a particle with mass M is



EK = 1/2M v2 = 3/2 k TK.


If the gas has more than one constituent, each with a

different mass, the kinetic temperature of each constitu-

ent will be the same although, because of the different

masses, the RMS velocity of each constituent will be

different. The RMS velocity of the component with mass

M is


3k T
2 1/2 3 K 1/2 11-18



If the gas particle velocity distribution should

suddenly become non-Maxwellian, collisions involving a

deflection of 90 will thermalize them to an equilibrium

temperature unless the system is thermodynamically unstable

(Spitzer, 1968, p. 90). In a situation where there is a

nonequipartition of energy, the electrons will have the

largest kinetic temperature because of their low mass.

The energies of the electrons and ions will equalize first

because of the long-range interactions between them.

Following this, there will be elastic collisions between the

ions and neutral atoms. Collisions between electrons and atoms are not










as an effective means of equalizing the kinetic energies

as are collisions between ions and the electrons (Spitzer,

1968, p. 140).


Population and Depopulation Mechanisms of High n-Levels


The high n-levels are populated and depopulated by

several mechanisms which are discussed by Brockelhurst

(1970), Dupree and Goldberg (1970) and McDaniel (1964,

p. 588). The parameters, such as cross-sections, proba-

bilities, and coefficients are discussed in general by

these authors as well as by Spitzer (1968), Kaplan and

Pikelner (1970), and by the workers listed in Table II-1

who have calculated high n-level populations. The mecha-

nisms involved are as follows.


1. Electron recombination followed by a cascade.

2. Dielectric recombination (in atoms with more

than one electron).

3. Excitation and de-excitation induced by

electron collisions.

4. Collisional ionization followed by three-

body recombination leaving an excited atom

X': H + e X+ + e + e and X+ + e + e -

X' + e.

5. Stimulated and spontaneous emission.











6. Fine structure excitation; i.e. redistribution

of angular momentum with a level: nR nt'.


Excitation to high n-levels from the ground level by radi-

ative excitation is not important in H II regions (Dyson,

1969; Dupree and Goldberg, 1970).

To account for the effects of a departure from LTE

with respect to level populations, the departure coeffi-

cient, bn, is employed;



b = N /Nn 11-19
n n n


The departure coefficient is a function of the electron

temperature and density and the intensity of the radiation

field. When b is calculated for a type of region, the
n
radiation temperature is chosen on the basis of specific

environmental conditions of the region and bn is tabu-

lated or plotted as a function of n for specific values of

T and N For example, Dupree (1972a) has done this for
e e
departure coefficients in a cool region. The coefficients

were first calculated for an isolated region where Io(v)

= 0 and then the effects of an adjacent 104 K H II region

on the b values were demonstrated.
n
Statistical equilibrium requires that there are

as many transitions into a level as there are transitions

out of it. The equation of statistical equilibrium for










the nth level is of the form (Seaton, 1964; Dupree, 1972a,

b)


[ n-1 n-1 n+ p' n n-n
N A .+ Z C .+ C .+ ( B + z Bn jl(v)
nL jn n, j 1 nl m=n+1 n,m nn-
'=n 0 =n-P j=ntl m=n+l j= 1



= (N C +N C )+( N B
(Nn-p n-p,n n+p n+p,n m=n+ m m,n
p=l

1 m'
+ j N. B.) ) ( + [ N. A. + N N. (ar + d + )
io ,n j>n+l J ,n e 1 n n i,n
j=no0 j=n+l

II-20



The symbols used in Equation 11-20, which have not yet

been defined, are as follows:


A = Einstein spontaneous emission probability,
m,n
m > n;

B = Einstein absorption probability, m > n;
n,m
B = Einstein stimulated emission probability,
m,n
m > n;

Cj,k = collision probability for a transition
j ,k
between the jt and kth levels;

C n,i = collisional ionization probability from

level n;

C. = three-body recombination probability;
1,n










= rate of radiative recombination into level n;
n
an = rate of dielectronic recombination into level

n ( = 0 for hydrogen);

m' = the maximum level considered in cascades from

higher levels;

p' = the maximum p for bound collisional transitions,

p = |An ;

n = the lowest level of the atom considered for
o
the computation;

and N. = the ion number density.


Collisional Transition Probabilities and Cross-sections


In order to compute the departure coefficients,

means of simplifying Equation 11-20, limiting the number of

equations or limiting the range of the summations should

be considered. At some level n = m', the electron is close

enough to the continuum that the collisional ionization

probability, Cn,i, is much greater than the probability for

a spontaneous transition to lower levels, Am,n. The levels

beyond the m' cut-off level are in thermodynamic equilibrium

and, b = 1 for n > m'.
n
Sejnowski and Hjellming (1969) find that inclusion

of collisional transitions of p = A n > 1 will lower the

value of b by only a few percent from that considering

only transitions of p = 1. Consequently, many workers

limit their calculations to p' = 1.










Collisions involving a change of .- state within

a level, n,R n, + 1, occur frequently. In cold hydrogen
-3-3
gas with N > 10- cm fine structure collisions by pro-

tons as well as electrons,occur more frequently within

levels for which n > 100 than do radiative processes
2
(Pengelly and Seaton, 1964). Because of this, all n

k-states are equally populated. Also, at low temperatures,

fine structure excitation by electron collisions provide

the main cooling mechanism of the gas (Dalgarno and McCray,

1972). By this process, electron kinetic energy is con-

verted to low frequency radiation which leaves the gas,

thus removing energy from the thermal pool.

The total population of level n is divided into

2-states according to


N (2 + 1) N
n,R 2 n
n


(Dupree, 1969). The total population is N = n N
n 2. n,2 "
For hydrogen, the levels are degenerate and b = b

b n is independent of R.

The collisional transition rates, Cjk for
j,k
k = j + p, depend on cross-sections Qj,k by



C N= N . 11-21
j,k e Jk










The average is obtained by integrating over a Maxwellian

velocity distribution.

The bound-bound cross-sections have been computed

by means of one of three approximations or methods:

binary encounter method, impact parameter method, and a

method based on the correspondence principle. The binary

encounter method uses a classical approximation,which is

valid for electron-atom collisions if the time of inter-

action between the incident and bound electrons is short

compared to the orbital period of the bound electron and

the collision takes place in a region smaller than the size

of the atom. This time is sufficiently short when the

energy transferred between the two electrons is large.

With this condition, the energy transferred is assumed to

be equivalent to that transferred between two free elec-

trons. Because of this requirement, the binary encounter

method gives best results for ionizing collisions; the

results are poor for p = 1. In calculating cross-sections,

Brockelhurst (1970) uses the binary encounter method for

p > 6. The application of the theory involved is discussed

at length by Burgess and Percival (1968).

The impact parameter method for electron-atom

collisions uses the assumption that the incident electron

follows a straight-line trajectory past the target atom

with an impact parameter p (cf. Figure 11-2). Seaton










(1962) computes the collision cross-section for a bound

transition (n,R n', j') for both weak and strong cou-

pling. For weak coupling,



Qj,k Pkj(p) 2np dp,



where j = n,z, k = n', V', and po is the cut-off value of

p, which is approximately equal to the smaller of the two

atomic radii of the j and k states. Pkj(p) is the transi-

tion probability as a function of impact parameter.

For strong coupling, Seaton (1962) defines p,

such that Pkj ( 1) = 1/2. For p


mean value of 1/2 so that in this range, Qj,k = 1/2Tpi

For a transition of p = 1, most of Qj,k comes from P>pl

and



j,k = 1/2 p 2 + Pkj(p) 2 p dp.


Seaton (1962) calculates the cross-sections for

the weak- and strong-coupled cases and accepts the smaller

of the two values. Pkj(p) is calculated with the sym-

metrized, time-dependent perturbation theory. Brockelhurst

(1970) claims that the impact parameter method will not

work for p > 1. He assumes the case of strong coupling and










finds that most of Q comes from p < pi for which the per-

turbation theory is not valid.

The classical method only works for large p; the

semi-classical impact parameter method is valid only for

p = 1. The correspondence principle method can be used to

obtain cross-sections for intermediate transitions. All

variations of the correspondence principle state that for

large scale systems,where the uncertainty is much larger

than Planck's constant, h, the result of a quantum mechani-

cal approach is identical to that using a classical ap-

proach. Percival and Richards (1970a) show that the Bohr

correspondence principle can be applied to the energy

transfer of a collisional excitation for the case of a

weak interaction and if the quantum and classical first-

order perturbation theories are both valid. Also, the

collisional probability must be much larger than that of

a radiative transition, a condition which is satisfied for

n > 100. With the the Bohr principle for collisions, the

mean net energy transferred during the collision is equiva-

lent to the energy transferred from independent Fourier

components of order +p and -p of the classical atomic

orbital motion. Pjk(p) is computed from these Fourier

components.

For the case of a strong interaction, Percival and

Richards (1970b) show that the Heisenberg correspondence

principle relates the quantum mechanical matrix elements










with the Fourier components of classical atomic orbital

motion. The results for H II regions are published in a

separate paper (Percival and Richards, 1970c).

For collisional ionization, either the binary en-

counter method (Burgess and Percival, 1968) or the classi-

cal Thompson theory (Dyson, 1969; Seaton, 1964) can be

used to obtain the collision rates. From the Thompson

theory


-15 2 -1/2
C 3.46 x 10 15 n N T
n,i e


The inverse of collisional ionization is three-body re-

combination. Using the equation of detailed balancing


*
N C = N. N C.
n n,i 1 e i,n



Radiative and Dielectronic Recombination Coefficients


The radiative recombination coefficient for cap-

ture into level n is represented by ar. The partial re-

combination coefficient, a(k)' is that for captures into
th -
all levels above and including the k ; n > k. It is

given by a(k) = Za.- Usually k = 2. The total re-
(k) j=k J
combination coefficient a(1), includes radiative cap-

tures into the ground level. Seaton (1959) has derived

formulations for a(1) and a(2) using a symptotic expansion










of the Kramers-Gaunt factor. With a notation similar to

that of Spitzer (1968, p. 117),



(k) = 2.065 x 10-11 Z2 T-112 k (TK). 11-22


The function Ek(T) involves the asymptotic expansion.

Spitzer (1968, p. 117) has tabulated I1 and 2." They are

plotted as Figure 11-3.

Dielectronic recombination occurs in atoms,which

have two excitable electrons. This process is very im-

portant in establishing the high level populations of

carbon atoms and is, in fact, responsible for these levels

being over populated with respect to the LTE populations.

It takes place in two steps, the first of which is a

capture of an electron into a high level, n. The energy,

rather than being radiated, is transferred to a bound

electron,which rises to an excited level; n' << n. This

is an unstable condition and the second step is to dis-

pose of the energy of the lower excited level by means

of a downward transition to a stable level. The energy

difference is either radiated--leaving the atom with a

single, highly excited level--or it is transferred to the

highly excited electron which leaves the atom by auto-

ionization, a process called the Auger effect (McDaniel,

1964, p. 590; Shore and Menzel, 1968, p. 93). The radi-

ative process dominates (Dupree, 1969). The dielectronic





































10


Figure 11-3,


2 3 4
0o 10 T 'K o1


C Functions for Calculating the Radiative Recom-
bination Coefficient. C1 is for the total recom-
bination coefficient. These functions are tabu-
lated by Spitzer (1968, p. 117).











recombination mechanism is more effective at populating

highly excited states than is the radiative recombination

mechanism; i.e., ad > a at least for T 104 K and
m i e n n' e
-3
N > 1 cm-3 (Dupree, 1969).
e

Departure Parameters, b and 6


It will be shown later that the spectral line

strengths depend not only on the departure coefficient,

b but--since the transition involves two b 's--it also
n n
depends on the derivative of bn with respect to n. For

that reason, when bn is computed, a parameter related

to the derivative is also computed;


b b
6 m n 11-23
6mn b
m


where m > n.

Departure parameters have been published by sev-

eral workers. Information about this work is summarized

in Table II-1. The first column gives the reference to

the work. The parameters calculated by Dupree (1969)

and Hoang-Binh and Walmsley (1974) are for carbon. The

second and third columns give the range of n over which

the parameters are tabulated or plotted. Columns 4, 5

and 6 give the range parameters used in the calculations:

n m', and p' (cf. Equation 11-20). The seventh column

indicates the method by which the collision cross-sections








Table II-1
Collision Range of T [oK]
eR e
Reference n N n m' p' Model* and N [cm ]
in~~~~~~~~ ~ ~~~ 1 4i__ idA ___ u_________ ____________


Dyson (1969)


Sejnowski & Hjellming
(1969)


Brockelhurst
(1970)



Dupree (1972a)

Dupree (1969)
(Carbon)


80 300



20 240



40 300



50 250



20 300


2 400 4 IP(1)



2 240 20 IP(2)



2 >500 75 (3)



10 600 1 BE(4)



2 600 1 IP(5)


5 x 103 10 < Ne < 105

3 4
5 x 103< Te < 10
10 < Ne < 104

2.5 x 103 < Te < 2 x 104
10 < Ne < 105

20 < Te < 1000
0 < N < 103

Te =104 and 2 x 104
4
1 < N ~ 10
e


Hoang-Binh &
Walmsley (1974)
(Carbon) 80 250


? 500


BE(6)
1 CP


Te = 10 and 100
1 < Ne < 103


*The collisional approximations are BE = Binary Encounter, IP = Impact Parameter, CP = Correspondence
Principle. The references to the cross sections are as follows:
(1) Seaton (1962) (2) Seaton (1962) and Saraph (1964) (3) Brockelhurst (1970): IP for p = 1;
CP for 2 -< p < 6; and BE for p 6 including ionization (4) Flannery (1970) (5) Saraph (1964)
(6) Banks et al. (1973): BE for small p; CP for large p.










were calculated. The reference to the source of the cross-

sections is given in the parentheses following the designa-

tion. Since the departure parameters are functions of the

electron temperature and density, the last column of the

table gives the range of these quantities for which the

parameters were computed.

All, except Hoang-Binh and Walmsley (1974),do not

consider incident radiation in the calculations. Dupree

(1972) does, however, show the effect of an adjacent H II

region on a line-emitting region with a temperature of

20 K. The radiation field from the H II region increases

b slightly for n > 80.
n
The physical significance of departure from LTE

can be illustrated by showing the relation which n, T ,

and N have on b and 6 This is done with Figure 11-4
4 3
in which are drawn curves for T = 10 and 7.5 x 10 K
e
computed by Sejnowski and Hjellming (1969) and curves for

T = 100 K computed by Dupree (1972a). For each temper-
3 -3
ature, there are separate curves for Ne = 10 and 10 cm

As n decreases, collisions become less important

in determining level populations. For small n, radiative

processes dominate and the curves show an asymptotic be-

havior. The heavy portion of the 104 and 7.5 x 103 K

curves in Figure 11-4 is the radiative asymptote. De-

population by stimulated emission is much less frequent


















b
0.8

0.7


0.6

0.5
I
10

I 2
10

mn
105


-4
10


-5
10


Figure 11-4.


Departure Parameters for Hydrogen as
a Function of n, Te, and Ne. The curves
with a solid line are for Te = 10i4K and
with a dashed line are for Te = 7.5 x 103'K
(Sejnowski and Hjellming, 1969). The
dotted curve is for Te = 1000K from Du-
pree (1972a).


20 100 180 260











than by spontaneous emission (Gordon, 1974),except when an

intense radiation field is present.

As N decreases, collisions become less frequent
e
and the radiative asymptote extends to larger n. Then, for

hydrogen, there is a decrease in the rate of collisional

excitation of lower n-levels which causes a decrease in

level population with respect to the LTE values as n de-

creases. In the case of carbon, however, dielectronic re-

combination is efficient enough in populating upper levels

that the population will exceed the LTE values for n > 20
% 3 -3
and N < 10 cm- In any case, the populations approach

their LTE values as n - (Dupree, 1969).

It should be noted that for hydrogen, b and 6
n mn

are both positive in the radio regime. For any two levels

m and n, where m > n, b is closer to one than is b In

that sense, the upper level is "overpopulated" with respect

to the lower level. If the slope of the bn curve is large

enough, there is an enhancement of the line intensity over

that obtained from a region in LTE even though the level

population is less than its LTE value. This occurs be-

cause of stimulated emission induced by an exterior source

or the region's own thermal radiation field. An examin-

ation of the radio recombination line absorption coef-

ficien-t and non-LTE solutions of the transfer equation

will show how this happens.










Radio Recombination Line Absorption Coefficient

The rate, R(v) at which a bound-bound absorption

takes place between energy levels n and m (m > n) is equal

to the product of the number of absorbers, N the Einstein

absorption probability, Bnm, and the energy flux, I(v)

dQ/47. The degree of absorption is modulated by the spec-

tral line profile function, L(v). Then


dR
R(v) = Nn L(v) Bnm I( ) 4-


The energy per photon is h nm. The absorption coefficient,

KL (v), is defined such that the energy absorbed along an

increment of path length ds is


dR
dcA -L(V) I(v) d2 = -Nn L(v) Bm I(v) hvnm ds.


However, energy may be emitted by stimulated emission of

the amount


do
de = jl(v) dQ = NM tL(V) Bmn I(v) hymn 4 ds,


and by spontaneous emission by the amount


d s
d S = jS(v) dQ = Nm L(v) A hv ds.
S S m PL mn mn 4r










Smn(v) is the emission line profile, which is usually
identical to the absorption profile. Inserting deA, de ,
and deS in the transfer equation (Equation 11-2) gives



dI) Nm L(v) Amn [Nn L(v) Bn Nm L(v) B mn (


Let PL()) = >L(v) and define a total absorption coef-
ficient, KL(v), by


KL(v) (Nn Bnm B n) L(V) B


B N
-hv B (v) B ( mn Nm)
4 () Bnm B nm N


Using gn B = g Bmn (Mihalas, 1970, p. 83), the Boltzmann
n nm m mn
equation, and the departure coefficients, the absorption
coefficient becomes


hv b-24
KLU) L(v) Bnm bn N[ exp(-hv/k Tex)] 11-24



For LTE, b = b = 1 and the LTE absorption coefficient is
m n


K (V) = () B Nm [1 exp(-hv/k Tx)].
L ( f) = nm n ex


11-25










The Einstein probability can be written in terms

of the oscillator strength, fnm


2 2
B = 4 T e f I 1-26
nm hv Mc nm


In the above equation, e is the electron's charge; M is

its mass. Oscillator strengths for hydrogen radio re-

combination lines have been computed by Goldwire (1968)

and Menzel (1969).

Substituting Equation 11-26 into Equation 11-25

and assuming that hv << kTe, the absorption equation be-
ex
comes

2
h v T e e
ex


N can be obtained from the Saha-Boltzmann equation
n
(Equation A-5) and mn can be obtained from an approxi-
2 3
mation to the Rydberg equation; vmn = 2c RH Z An/n

Using these and the appropriate constants (RH = 1.097

x 105, Z = 1, u(T) = 1, and g = 2n2), the LTE absorp-

tion coefficient becomes



KL( -) = 1.071 x 107 (v) f N. N T-25 exp( /k T ) 11-27
where n nm 1 e e n ex


where cn /k = 157890/n2. The LTE optical depth is
n










TL(V) = 1.071 x 107 () fnm T-25 exp(En/k Te) EL. 11-28


EL is the line emission measure defined by

L

JEL N N ds
o

and N. is the number density of the recombining ion.

Following the notation of Dupree and Goldberg

(1970), mn is defined by

b
Bm E [1 m exp(-hv/k Te)] / [1 exp(-h-/k T )]. 11-29
mn b ex ex
n


Then KL(v) = b mn K L() and TL(v) = bn mn TL(V).

In most cases for radio recombination lines, hv/

k Tex is small enough that the exponentials involving them

are equal to 1 hv/k T Then,


k T bm he b k Tex b b n
5mn kI [1 ( ( )] = [1- ( b )].
n ex n m



Using mn, which was defined in Equation 11-23,

kT
Smn ( 1 ) (1 6 ) 11-30
mn )1-6 hv mn
mn


If Ab = b b is small, Ab db and
m n










= db d(ln b) An
mn bm dn m=n



which is a notation often used.

Gordon (1971) and Hjellming and Gordon (1971)

define a different B than that defined in Equation II-29;


kT kT
8 1 ex 6 ) = 1 ex d In b
Bmn hv 6mn hv dn A


This is given for reference only since the definition of

Bmn in Equation 11-29 will be used in the present work.

Line Profiles


There are three line profile functions,which are

relevant to radio recombination line analysis. They are

the Gaussian, Lorentzian, and the convolution of these

two, the Voigt function. The Gaussian function arises

from the Doppler shift resulting from thermal, turbulent,

expansion, and contraction motions of the line emitter

relative to the observer. A Lorentzian profile for a

radio recombination line results from pressure broadening.

Natural broadening, or radiation damping, also produces

a Lorentzian profile, but this is such a small effect that

it is impossible to detect in the observed line profiles.

If pressure broadening occurs, the effect would be blended

with the Doppler broadening and a Voigt profile would result.










Doppler broadening. Mainly, thermal broadening

will be considered. The thermal motion of particles in a

gas with a kinetic temperature TK has a Maxwellian velocity

distribution; so, the number of particles, out of a total

of N, which have a radial velocity between v and v + dv

is


d N(v) exp[-(v-v ) /v ] dv.


In this equation, v is the most probable velocity and

v is the velocity of the ensemble as a unit. v =

(2k TK/M)1/2 where M is the mass of the particles. The

intensity of the spectral line at a frequency Vl is pro-

portional to the number of emitters Doppler-shifted to

that frequency. The line profile is a Gaussian;


1(v) = exp[-(v-v ) /v2] dv. 11-31


The profile function is a probability distribution;


S$(v) dv = 1.


The power spectrum is usually displayed in terms

of antenna temperature as a function of velocity. Velo-

city is physically related to the emitter and the line

half-power width is then invariant to the line rest










frequency. The peak line antenna temperature, T'(c),

is that for which v = v The temperature profile is


T'(v) = T'(c) exp[-(v-v ) /v ]. 11-32
c 0


The integrated line power, which is the area under

the profile, is


P = I T'(v) dv = T'(c) vo F 11-33



Since v is not known, it is desired to substitute it with

a measurable quantity. When the line power drops to one

half of its peak intensity, v = v1 and


P P 2/2
T'(v) = 1/2 P- Pexp[-(v-v) /v2]
vo \ T 0 Vo TTc


The Doppler half-width is defined by AvD = 2(vl-vc); then

8k T
AvD = 2v 01n = (- D -M n 2) 11-34


TD is the Doppler temperature which is the effective

temperature of the gas if the line width were the result

of thermal motion only.

The integrated line power in units of K km/s is


P(K km/s) = (4 )1/2 T'(c)AVD.










However,P is usually given in units of "K kHz; so,


P(oK kHz) ( n 1 2)1 ( ) T'(c) Av
4 ln 2 c D


= 1.0645 ( ) T'(c) Av II1 -35


In Equation 11-35, v is the rest frequency measured in

kHz; the velocity of light, c, and the Doppler width are

both measured in km/s. Likewise, the integrated optical

depth, TL, is


TL = L( ) dv = 1.0645 T(vc )( ) Av 1-36



The Doppler temperature is given by



TD k n (AV )2 11-37


Doppler broadening includes the effects of thermal motion

by an amount AvT, turbulence of a scale smaller than the

antenna beam (Aturb), and expansion or contraction of the

emitting region (AVx). These velocity components combine

to give a total Doppler width of


Av = (Av2 + urb +Av /2 11-38
0 T turb










Pressure broadening and the Voigt profile. When

the electron density is high enough that the time be-

tween collisions is shorter than the life-time of the

excited atomic level, the energy distribution of the

radiative transition is altered. The wavetrain is

abruptly terminated by the passing electron. The line

profile emitted by the assembly of atoms is a Lorentz

function (Kuhn, 1962, p. 392). Peach (1972) has shown,

however, that the profile is not an exact Lorentz func-

tion, but it is close enough for an accuracy of about 10

percent. The main effect of the Lorentzian is to redis-

tribute the emitted energy to the wings of the line.

Griem (1967) has shown that proton-atom collisions are

not important.

Brockelhurst and Seaton (1972) have determined

that the pressure broadened half-power width for a m n

transition is


n 4.4 10 0 1 N
vP = 9.4 ( 0) ( ) e
e

Pressure broadening will occur in addition to Doppler

broadening. If the gas has an electron temperature T

and a Doppler temperature TD (Equation 11-37), the ratio

of pressure to Doppler broadening is, by Brockelhurst

and Seaton (1972),











0.12 7.4 Ne 2x104 1/2 04 0.1-
- 0.142 ( I ) 4) ) 11-39
AVD 10 D e



To obtain appreciable pressure broadening of radio re-

combination lines, TD > 2 x 104 and Te > 10 and Ne

> 104. So far, the only possible radio recombination line

sources known to have such parameters are dense knots in

emission nebulae such as M42, W51, and W3 (Hjellming and

Davies, 1970) or dense H II regions associated with type

I OH masers (Habing et al., 1974). No pressure broadening

has been seen, however (Gordon, 1974).

When Doppler and pressure broadening occur to-

gether, the resulting profile is the convolution of a

Lorentz with a Gaussian which is a Voigt profile, pV(v).

The convolution integral cannot be evaluated exactly. So,

the function is expressed in the form



(V) = A 1 H(a,w). II-40



where H(a,w) is the Voigt function

2
a _-y
H(a,.) a e- 2 dy
Sf (~-y)2+a2


and


v v a R
= y and a --
CY o U










The frequency dispersion, o, is given by a = vD 2(ln 2)1/2

= v ( -) (In 2)1/2. Posener (1959) has tabulated the
D c
Voigt function for values of astronomical interest.

Non-LTE Solutions of the Transfer Equation and Line
Enhancement

Two solutions of the transfer equation will be

developed here for use in the analysis of the spectra.

They will be derived starting with Equation 11-4. Since

the frequency of the spectral line is high, the Rayleigh-

Jeans approximation to the Planck function can be used to

convert specific intensity to brightness temperature.

From Equation 11-4, the brightness temperature is


T N -T
TB() = TL(v) + TC() = A(v) Te (l-eT) + (l-e-) Te
11-41



where T= TL(V) + TT(V).

Since, from Equation 11-6, A(v) = S(V)/B(v,T )

and jL(v) = b KL B(v,T ),
mT e


LT + b KL 1 + bm (L/KT)
A(v) = m II-42
T +n mn L 1 + bn mn (KL/T)


Optically thin H I region adjacent to a strong

source of radiation. Assuming that KL>> KT at the line









center (i.e. atv = vc),


b (L/T) bm
A(v) m = m
bn mn (KL/ K) bn mn


Since the region is optically thin, Equation 11-41 re-

duces to

b
TB(v) = Te ( m TT L) + TN + T (1-i T-L)'


When the frequency is off the line (i.e.v>>vc) only the

continuum is observed;

b
TB(vV)= TC( = bn Te T + TN + T (1-TT)



Subtracting this equation from the preceding one gives only

the line;

b
TL() = T () TC() = T T T
n mnne Lo

Since L = b Bmn T'



TL (v) = L(V) (bTe b Bm To). 11-43


This is the line brightness temperature from an optically










thin H I region for which KL >> KT. The second term in

Equation 11-43 will produce line enhancement if Bmn is

negative. To convert the brightness temperature to an-

tenna temperature, the right side of the equation must

be multiplied by the beam dilution factor, ,B' and the

main antenna beam efficiency, B. These are defined by

PB QS/ M and B M/A, where QS is the solid angle of
the line-emitting region; QM is the solid angle of the

antenna main beam; and qA is the total solid angle of

the antenna beam. The antenna temperatures are denoted

with a prime; so, Equation 11-43 becomes



TL( ) = PB B TK(v) (bm Te bn mn To). 11-44


Isolated region. This solution of the transfer

equation will be for the case of an isolated emitting

region for which the incident radiation is weak and can

be ignored. Then T = TN = 0 and Equation 11-41 reduces

to


TB(v) = A(v) Te [1-e ]T


Off the line, A(v) will be unity; so, the continuum tem-

perature is


TC(v) = T (1-e TT)










Taking A(v) from Equation 11-42 and converting TB(v)

to antenna temperature, T(v) = TB(v) TC() and


ST + b L -( T -T
TL( = B B Te b [l-e (1-e .
T + bn nm TL


11-45



Line enhancement. Equations 11-44 and -45 will

be used for the analysis of this study's observations.

The equations indicate how non-LTE conditions affect the

radio recombination line temperature. For LTE b = b

= n = 1; 6 = 0; and Equation 11-44 becomes TL())

= L (T -T ).
eL e
Figure 11-4 shows that as N decreases, collisions

become less frequent; so, the level populations and b

decreases. As the departure from LTE increases, 6mn be-

comes larger. When 6mn > (hv/kT), B is negative and

amplification of the line takes place. This process is

often referred to as the "partial maser" process. The

populations of the transition levels are not actually

inverted as is apparently true for, say, 1120 masers

(Sullivan, 1973).

The line enhancement can be seen from Equations II-

44 and -45 in which TL = bn mn TL is negative, Usually,

L << T because the path length involved with 'T
is much larger than that involved in











TL. Also, b is always positive. Therefore, if b

mn TL/T TI < 1, A(v) increases and the total absorption,
TL + TT' decreases.
The line enhancement will be lessened somewhat

because b < 1. However, for a decrease in electron
m
S3 -3
density from 10 to 10 cm 6 increases by more than a
mn
factor of ten while b decreases by about 14 percent.

The net result is a substantial increase in line power.

Equation 11-44 also indicates the possibility

that the recombination lines may be seen in absorption

against a strong radio source if the relative magnitudes

of b and 6 are such to make B positive. However even
n mn mn
for the case of a dense H II region (EL % 107 pc cm 6)

seen through a tenuous H II region, Brockelhurst and

Seaton (1972) have shown graphically that absorption

cannot be greater than 8 percent. So far, radio re-

combination lines have not been seen in absorption.



The Radiation Field


The radiation field intensity entering a recom-

bination line-emission region, and used in Equation 11-20

to determine the level populations,is computed from I (v)

= W(v) B(v,T). Usually the radiation source is a nearby

nebula whose blackbody temperature is 104 K. The dilu-

tion factor, W(v), is a product of three factors;










W(v) = WG WA(v) A(v). 11-46



WA(v) represents absorption of the radiation by the in-

tervening medium between the source and the line emission

region. If T is the intervening optical depth, WA(v)

= e If the nebula is adjacent to the line-emission

region, WA(v) = 1.

A(v) represents the departure of the radiation

source from a blackbody. It is defined in Equation 11-6.

The geometrical dilution factor, WG, can be

found as follows. Consider a radiation source which is

roughly spherical in shape with a diameter ds and located

at a distance rR from the line-emitter. The geometry

is shown in Figure II-5a. The line-emitting region sees

the radiation source as a disc of angular diameter
2
and filling a solid angle OR = R 2 /4. Then
2 2 R
Ti d2/4 d 2
S s s R 11-47
G = 2 2 4 -47
4 i rR 16 rR


If the two regions are contiguous, rR = 1/2 ds

and WG = 1/4. If, however, the radiation source is much

larger than is the line emitter, Figure II-5b applies and

WG = 1/2. Finally, if the source of radiation surrounds

the line emitter or if the line is emitted by a dense

H II region, WG = 1.





















.R .


a







bR






b


Figure 11-5.


Determination of the Geometrical Dilution
Factor, WG. Part a shows the line emit-
ting region and the continuum source to
be about the same size and rR > ds. Part
b shows the line emitting region adjacent
to a large continuum source.


LINE
EMISSION
REGION


CONTINUUM
SOURCE










A numerical example will illustrate the case

where rR > ds/2. Consider the radiation of temperature

TR entering a small line emitting region located 100 pc

from an H II region, such as NGC 7000 (the North American

Nebula), for which T = 104 K and r = 25 pc (Kaplan and
e s
Pikelner, 1970, p. 152). For this case A(v) = 1. Assume

that WA(v) = 1 also. Then, using Equation 11-47, it is

found that TR = 131 K. This indicates that, unless the

line-emitting region is directly adjacent to a large H II

region, the line-emitting region can be considered as

being isolated. In the case where the line emitter is

itself an H II region (T = 104 K), Dyson (1969) shows

that the effect of the free-free continuum on the departure

parameters is very small. Dupree (1972) and Hoang-Binh

and Walmsey (1974) demonstrate the effect of a large H II

region adjacent to the line-emitting region.



Ionization Rates


Knowing the radiative recombination coefficient,

a(T), the required rate of ionization per atom of species

X, (X, can be determined from the condition of detailed

balance. For hydrogen,


NH (H = N NHt a(T).


II-48










The radiative recombination coefficient may be obtained

from Equation 11-22. Habing (1970) has a formula for

computing CH which uses the assumption that a(T) T T0.65


= 1.27 x 10-15 2.1 T-0.3 TT(T)/ 21(v) dv.


11-49



In this formula, TT(VT) is the continuum optical depth

at a frequency vT and 21 (v) is the 21 cm line optical

depth discussed in the next section.

Ionization rates can be calculated theoretically

from a given source of ionization such as low energy cos-

mic rays (Dalgarno and McCray, 1972) and OB stars. These

rates can be compared to the ionization rate required by

a model for a line emitting region.

As an example the ionization rate within the

Stromgren sphere surrounding an OB star will be considered.

Let S be the total number of ultraviolet photons with
u
energy greater than hvL emitted from the central star.

vL is the frequency of the Lyman limit. Also let rS be

the Stromgren radius which is the radial distance from the

star at which the ionizing radiation flux is zero. Then


S = 4 r3 N NH+ a(T)
Su =3 = S e










(Spitzer, 1968, p. 116). From Equation 11-48, a(T)

= NH H/Ne NH+ and the ionization rate of an OB star is


3S
u TI-50
H H
4 rS NH



The ultraviolet flux, S from stars of spectral class 05

through B1 are tabulated by Spitzer (1968, p. 117).



The 21 cm Hydrogen Line


The well known 21 cm hydrogen line is emitted by a

magnetic dipole transition between the hyperfine levels

of the ground state. This is the result of the difference

of energy levels between the proton and electron spins

being parallel (upper state, F = 1) and antiparallel

(lower state, F = 0). The frequency of the transition

is 1420.405 MHz. The upper state is metastable; the

spontaneous emission probability is A10 = 2.85 x 10 15
-I
sec (Kaplan and Pikelner, 1970, p. 410). Usually the

metastable state is depopulated by collisional excitation

more frequently than it is by the 21 cm radiative transi-

tion. The F = 1 state is populated by atom-atom col-

lisions (Kaplan and Pikelner, 1970, p. 420).

Designating the lower and upper states by 0 and

1 respectively, the absorption coefficient is









KhN) (lv) [NO B N B]
10 (v) [N 0 01 1 10 ]


The parameters are analagous to those of the recombination

line. Since collisions play an important part in popu-

lating the upper state, it will be assumed for the moment

that the populations are LTE values as determined by the

Boltzmann equation. The statistical weights of the states

are gl = 3 and g0 = 1. Then, since g0 B01 g B10 and

hv10/kT << 1,

hv hv
10 Tf0 12 0 01
KlO(V) 40() k- s-No B



T is the excitation temperature which, for the 21 cm

transition, is called the spin temperature because the

transition results from a change of electron spin orien-

tation. The absorption coefficient is expressed in

terms of the spontaneous emission coefficient, A10;

2h v0
10 2 B 10


Then

3N h c
K 0 A (N) 11-51
10 k Ts A10 (10 ) II-51


The absorption coefficient is used to obtain the

neutral hydrogen column density along a line of sight.










For a cold gas, the total neutral hydrogen density, NHO,

is NO + N1. From the Boltzmann equation, the number den-

sity at some point s in the line of sight is



NH(s) = No(s) [1+3(1-hv10/k Ts)] = 4No(s),



since h 10/k = 0.07. Substituting in values for the

constants and integrating over the line profile and along

the line of sight gives


L L
L 1(v,s) ds dv T21(v) dv = 2.58 x 1015 H ds,

v o

II-52



where the path length, L, is in cm and v10 is in Hz.

If the line profile is Gaussian with a half-power

width vD and a peak optical depth 21, Equation 11-36 can

be used to obtain


a s-- ds = 4.126 x 101421 Av 11-53



It may be necessary to determine whether or not the

spin and gas kinetic temperatures are equal. Starting

with the equation of detailed balance,










N1 NH Co1 (TK) + N1 A10 + N1 B10 I(10TTR)


= NO NH C01(TK) + NO B 1 I( 10,TR).


C01(TK) and C o1 (TK) are the collisional excitation and de-

excitation probabilities, respectively, whose formulation

is given by Kaplan and Pikelner (1970, p. 420). Solving

the detailed balance equation for Ts in terms of TK and

the radiation temperature, TR, gives


557.3 T0.5 N + T
T -= K 11-54
557.3 T -0 5 N + 1
K H

T = TK if the hydrogen density is large enough that

557.3 T0.5 N >> TR For H I regions, TK > 64 K and
K H R K
N > 1 cm-3 is not unreasonable (Baker and Burton, 1975);
e
so 557.3 T0.5 N > 4.4 x 103 K which is much larger
K H
than one would expect TR to be. Therefore, unless NH
-3
< 1 cm Ts = TK.

Absorption spectra of the 21 cm line have been

obtained by three techniques. Hagen, Lilley, and McClain

(1955) obtained absorption spectra in the direction of

discrete sources by averaging the spectra from off-source

beams and subtracting this from the on-source beam spectrum.

The accuracy to which this method yields the true absorp-

tion spectrum depends on how uniform the hydrogen gas is

over the grid of beams.










Radhakrishnan et al. (1972a) and Hughes, Thompson,

and Colvin (1971) have observed absorption spectra in the

direction of radio sources (e.g. 3C391) with two-element

interferometers. The front-ends and synchronus detectors

are switched in such a way that the system is insensitive

to all but the absorption spectrum in the direction of the

source.

A third method has been used to obtain absorption

spectra in the direction of pulsars. Some of this work has

been done by Gordon et al. (1969), Manchester et al (1969),

Gordon and Gordon (1970), Guelin et al. (1969), and

Encrenaz and Guelin (1970). The pulsar is observed with

a gated, spectral line receiver which alternately stores

and integrates the spectrum with the pulsar on and with

it off, At the end of the integration period, the pulsar-

off spectrum is subtracted from the pulsar-on spectrum.

The difference of the two is the absorption spectrum. The

pulsar is a particularly useful beacon because its dis-

persion measure provides a value of the integrated electron

density along the line of sight. Therefore, two pieces of

information about the intervening interstellar medium are

known.



Pulsar Dispersion Measure


The arrival time of the pulsed radio emission from

pulsars is frequency dependent. The pulse travels at the










group velocity. The group index of refraction is a func-

tion of the electron density of the medium through which

the pulse travels. The difference of arrival time, t1 t2

of the respective frequencies v1 and v2 is, then, the in-

tegrated electron density along the path length L. This is

the dispersion measure, DM, which is calculated from the

equation (cf. Terzian, 1972)

L
OM N ds = 2.4 x 10-4 (t -t2) ( 2- 2) 1



II-55


From this equation, the unit of DM is pc cm-3
From this equation, the unit of DM is pc cm
















CHAPTER III

THE TELESCOPE SENSITIVITY AND
THE OBSERVING SCHEME



The observations, which will be discussed in the

next chapter, were made in such a way as to make the most

efficient use of the telescope. This chapter will discuss

the basis for the observational scheme. The important fac-

tor was the system sensitivity which is affected by the

microwave frontend of the receiver and the one-bit auto-

correlation spectrometer.

The general theory involved with the operation of

a one-bit autocorrelation spectrometer for astronomical use

was discussed in detail by Weinreb (1963). The operation

of the instrument used for the observations is discussed

in two reports. The IF filter system, which precedes the

correlator and determines the width of the spectral window,

is described by Mauzy (1968). The operation of the entire

system, with emphasis on the digital autocorrelator, is

described by Shalloway et al. (1968). A discussion of

system sensitivity for the different methods of observing

radio spectra is presented by Ball (1972).










System Sensitivity


The nominal system sensitivity is the RMS noise

temperature, AT, given by


yT
AT = ys III-1
/Af t.


The parameters are as follows:


y = a constant which depends on the system and the
method of its operation;

T = the system noise temperature;
sys
Af = the frequency resolution bandwith; and

t. = the integration time.


T is the combination of the noise temperatures
sys
received by the antenna as well as the power generated by

the antenna-transmission line system and the receiver.

The total bandwith, B, observed by the spectrom-

eter is divided up into N channels. Both B and N are

selectable. Each channel has the effect of measuring the

spectral power within a filter whose bandpass is of the

form (sin v)/v having a half-power width of


Af = 1.21 B III-2



(Shalloway et al., 1968). The spacing between channel

centers is B/N.










The factor y in Equation III-1 is greater than one

and is the degradation of the system sensitivity due to the

operation of the autocorrelator. It is a one-bit digital

machine which computes the autocorrelation function, p(nAt),

of the undetected IF power; nAt is the time delay at the

nth channel. The first step in obtaining the autocorrela-

tion function is to clip the signal, a random Gaussian volt-

age, v(t). The clipper generates a function, y(t); y(t) =

+Y if v(t) > 0 and y(t) = -Y if v(t) < 0. The function

y(t) is then sampled at a rate equal to 2B = I/At. The out-

put of the sampler is fed into an N-bit shift register and

N one-bit multipliers. The shift register is updated at

the time intervals of At. The output of each of the multi-

pliers updates a counter. At the end of the integration

time, each counter contains the sum which represents one

point of the autocorrelation function. The function is in

the form of an array which is transferred to the on-line

computer which computes the power spectrum by taking the

Fourier transform of it.

The effect of using one-bit quantization is to

increase AT above that which would result from a many-bit

quantization (lleinreb, 1963, p. 32). This is accounted

for in the constant y of Equation III-1. This constant

is actually a product of two factors; y = aB. The ef-

fect of one-bit quantization is that B = 1.39 (Weinreb,










1963, p. 49).1 The value of the other factor depends on

the weighting function used in computing the correlation

function. A uniform weighting function was used for which

a = 1.099 (Weinreb, 1963, p. 22). Therefore, v = 1.528

and Equation III-1 becomes


1.528 T
AT = s8 TSys 111-3
/Af t.




The Observing Scheme


There are several observing schemes which can be

used to observe spectral lines (cf. Ball, 1972). The ob-

serving scheme must provide a reference spectrum which is

used to remove the effects of the receiver passband from

the desired spectra.

The "Total Power" scheme can provide an efficient

means of obtaining the reference spectrum. Before observ-

ing a source on the program, a spectrum is obtained while

the antenna is pointed in a direction close to the program



1Burns and Yao (1969) have determined that the clip-
ping factor, B, is a function of the sampling rate. For
a sampling rate of 2B, 0 = T/2 = 1.571. If this is cor-
rect, y = 1.726 and the RMS noise of the data discussed in
Chapter IV should be multiplied by a factor of 1.13. How-
ever, the data reduction routines continue to use B = 1.39
(Cram, 1973, p. 133).










source and where the spectral line is not detected. This

spectrum is the receiver passband function. The nomen-

clature of the Total Power method is to designate the ref-

erence spectrum by "OFF" and a program spectrum by "ON."

If the receiver passband remains unchanged while the OFF

and ON spectra are observed, subtracting the OFF spectrum

from the ON spectrum removes the effect of the passband

from the desired spectrum. The efficiency of this observ-

ing scheme is realized if one OFF spectrum can be used with

more than one ON spectrum if there are several sources in

the observing program.

Let the observing time on an OFF source be tF and

that on an ON be tN. If an OFF source is followed by n

ON sources, each of which is observed for a time tN, the

total observing time, to, will be to = tF + n tN. The

times tF and tN and the ratio n must be chosen so that

the receiver passband remains unchanged during the time

to. The most efficient use of observing time for a given

number of program sources will be obtained if the ratio

a = tF/tN is chosen to minimize AT.

When an OFF spectrum with an RMS noise (AT)F is

subtracted from an ON spectrum with an RMS noise (AT)N the

RMS noise of the resulting spectrum will be


2 2 1/2
T = [(AT) + (AT) 2
N F










For spectral line observations, Tys is not expected to

change more than a few percent between the OFF and ON po-

sitions; then an average system temperature, Tave, can be

used to get


yT
ave
AT- ve
SAf


+


Referring to Equation III-1, the integration time is


tN tF
t. N F 111-4
1 tN tF


Since tF = a tN and to = (a + n)tN,


T
AT = ave (a + 1
/Af to

The value of a is determined when


+ n + n/a)


T is a minimum, for which


d(AT) .
da


Then (1-n/a2)(a+l+n+n/a)-1/2 = 0. The only reasonable

solution is a = / Therefore, the most efficient Total

Power observing pattern exists for


tF = /n tN


III-5















CHAPTER IV

THE RADIO RECOMBINATION LINE OBSERVATIONS



Introduction


The H157a radio recombination line spectra for

this study were obtained between 18 and 24 April, 1973

with the 43 meter radio telescope of the National Radio

Astronomy Observatory (NRAO) located in Green Bank,

West Virginia. The rest frequency of the line is 1.683200

GHz. The H157a line was chosen for two reasons. It is

conveniently located within the passband of the NRAO 10

cm cooled receiver which has good sensitivity and a rep-

utation of being very stable. Also, the frequency of this

line is such that the line strength is relatively strong.

The total power observing scheme was used to obtain the

spectra.

The sources used for the observations are the two

supernova remnants (SNR) 3C391 and 3C396 and the pulsar

1858+03. These were chosen on the basis of the objectives

discussed in Chapter I; they are located in the galactic



1NRAO is operated by Associated Universities, Inc.
under contact with the National Science Foundation.










ridge and independent data from the intervening inter-

stellar medium are available. A radio galaxy, 3C295 (Gold-

stein 1962), was also observed, but its galactic lati-

tude is 610 and no radio recombination line was observed

in its direction. The purpose of observing it was to dem-

onstrate that the recombination lines came from the ridge

gas and were not affected by instrumental effects.

Along with 3C391 and 3C396, a region adjacent to

each of them was also observed. These will be referred

to by the designation COMP. The COMP spectra were observed

for two reasons. The primary reason was to be able to

subtract out emission from the region immediately surround-

ing each of the SNR's and, thereby, make these ob-

servations compatible with those made with narrow beams.

The second reason for obtaining the COMP spectra was to

determine the angular scale over which the medium changes.

Table IV-1 lists the center position of each beam.

The right ascension and declination (1950.0 epoch) and

the galactic longitude and latitude are given for each of

the regions studied. The OFF beams are the reference

beams used for the Total Power observing scheme. Their

selection and use will be discussed in the following sec-

tion.

Figure IV-1 shows the continuumradiation environ-

ment of the regions observed. The continuum maps are adapted

from the 1414 MHz maps of Altenhoff et al. (1970). The













Table IV-1

Antenna Beam Positions


a(1950)

1 46m 47.s8

1 46m 01.s5

1 01m 35.s7

1 59m 59.s2

1 30m 0. S

1 58m 40.S0

1 12m O. O


6(1950)

S58 48."0

S 53' 14."1

S22' 30."0

' 16' 49."7

S30' 0."0

S27' 2."0

S00' 0.'0


beam for 3C391, 3C396, and their COMP beams

beam for the pulsar, 1858+03


Beam

3C391
COMP
3C391
3C396

COMP
3C396

OFFa

1858+03

OFFb


lII
11

31. 9

32. 1

39.02

38.9

35. 0

37.2

37. 4


bII
b1

0.0

-0. 2

-0.3

0.0

+5.00

-0.6

-4.02


athe OFF

bhe OFF
the OFF























Figure IV-1. The Continuum Radiation Environment of the Program Regions.
The size of the circles represents the antenna beam half-
power width, 18 arc minutes. The 1414 GHz continuum maps
are from Altenhoff et al. (1970). The counter is 1.74 K
of brightness temperature.

























































































































.1










size of the circle at each region is about equal to the half-

power width of the antenna beam, 18 arc minutes at the

H157a frequency.



Observing Scheme and Data Reduction


The receiver front-end was mounted at the prime

focus of the 43 meter paraboloid. A scaler horn was lo-

cated at the focal point of the reflector. The signals

of linear orthogonal polarizations were separated behind

the horn and fed into separate and identical receivers.

The box containing the front-end equipment was rotated so

that the E-plane of the horn was aligned normal to the

celestial equater. Each receiver was load-switched and

contained a two-stage parametric amplifier. The first

stage of the parametric amplifier and the switched load

were refrigerated.

The local oscillator frequency was generated by

a frequency synthesizer followed by a 6x frequency multi-

plier. The intermediate frequency (IF) sent to the con-

trol room from the front-end box was 150 MHz. The synthe-

sizer frequency was chosen to compensate for the Doppler

effects of all motions of the telescope with respect to

the recombination line emitter and so that the velocities

of the spectra would be with respect to the local standard

of rest (LSR). Since it was expected that the recombina-

tion line emission regions and galactic H 1 regions are










kinematically related, the 21 cm emission velocity lon-

gitude maps of Westerhout (1969) were consulted to deter-

mine the velocity offset needed to position the antici-

pated H157a and C157r spectral features in the 2.5 MHz

spectrometer passband. The velocity offset for the pul-

sar was +50 km/s. All other program regions involved an

off set of -10 km/s. The synthesizer frequencies were cal-

culated by the NRAO computer program DOPSET before any ob-

servations were made.

The 150 MHz IF from each receiver was monitored by

a continuum reciever. The remaining signal was fed to

the NRAO Model II autocorrelation spectrometer. The spec-

trometer creates 413 frequency channels which were divided

into three sections. One section of 192 channels was con-

nected to each of the front-end receivers. The remaining 29-

channel section of the spectrometer was connected to one

of the recievers to serve as a monitor. It was not used

for the acquisition of the data.

Each section of the spectrometer had a passband

of 2.5 MHz. This allowed a channel separation of 13.0

kHz. Each channel has a sin v/v profile with a half-pow-

er width of 15.8 kHz. At a frequency of 1.6832 Ghz, the

velocity resolution of each channel was 2.82 km/s.

The operation of the spectrometer system is de-

scribed by Shalloway et al. (1968) and the Total Power ob-

serving scheme is described in Chapter III. The autocorrelator










periodically transfers the autocorrelation function array

into the on-line computer which performs the Fourier trans-

form and loads the resulting spectrum array onto a seven-

track magnetic tape. This array, along with source coor-

dinates and control information, constitutes a record. The

control information contains a number designation of the

spectrum and whether it is a spectrum of a program (ON)

source or a reference (OFF) source, The identity of its

OFF spectrum is also specified.

Each source is observed for a length of time (tF

or tN as used in Chapter III) during which a sequence of

records is placed on the tape. This sequence of records

is treated as a unit during off-line analysis and is

called a "scan." At the beginning of the off-line pro-

cessing, the records of each scan are averaged together

and the spectrum of each scan is loaded on another mag-

netic tape. Then, for each ON spectrum, its OFF spectrum

is located and the antenna temperature spectrum is com-

puted from

T'(i) = (S(i) R(i)) T
R ( i ) sys

where S(i) and R(i) are the source (ON) and reference (OFF)

spectra, respectively, as a function of the channel, i.

The system temperature, Tys, is computed by the off-

line computer on the basis of a calibration noise source

whose power is periodically injected into each receiver.









The accuracy to which the system temperature is determined

depends on how well the calibration temperature is known.

During the observations the values determined during bench

testing of the receiver were entered into the DDP-116 pro-

gram. The calibration signal was injected into each re-

ceiver by a loop "antenna" placed in the waveguide at a

point directly following the coupler which separates the

two polarizations. During the bench testing the horn was

replaced by a termination. In spite of the careful match-

ing of the termination to the system, its characteristics

are somewhat different from those of the horn observing

the sky. Consequently the amount of calibration noise

power entering the system is not exactly the same in the

two situations (Turner,1973).

The calibration temperature discrepancy was cor-

rected by daily calibrations using the radio source Virgo

A whose continue spectrum is well known. Once a more ac-

curate calibration temperature was known, a correction fac-

tor was later applied in the off-line processing. The con-

tinum flux of Virgo A at 1.6832 GHz is 182 Jy which was

determined by plotting the fluxes published by Kellerman

et al.(1969) and interpolating to this frequency. The con-

tinum receiver connected to the IF was used to measure the

relative levels of Virgo A and the cold sky with and with-

out the calibration signal. Table IV-2 shows the results

of the daily calibration by which the system noise temperature,










T was also determined. The first calibration on 22
sys
April gave unexplainable bad results; so, the calibration

was repeated later. The spectra obtained in the meantime

showed no indication of a problem. The values obtained

during this bad calibration were not used to compute the

averages from which the correction factors were computed.

Before the telescope was pointed toward any source,

the receiver frontends were tuned so that the spectro-

meter passband was located in a portion of the reciever

passband (25 MHz wide at the 3 db points) which was rea-

sonably flat. The daily observing procedure began by peak-

ing up on Virgo A in order to check the pointing accuracy

of the telescope and obtain a correction factor for the

calibration temperature. No problems with the pointing

were found.

Before observing the first program region, a five-

minute OFF scan followed by a five-minute ON scan of M17

was made to check the operation of the spectrometer. The

OFF regions for this and the program regions were at least

four degrees above or below the galactic plane where it

was certain that the H157a line was not present. The H175a

spectrum of M17, an emission nebula, is very strong.

A five-minute scan gave adequate indication that the spec-

trometer was operating properly. The daily M17 spectra

were compared to look for degradation of the system. None

was noticed.













Table IV-2

Calibration and System Noise Temperatures


Date T T T T
Date ys B cal A sys B cal B

April
18 62.2 3.12 58.5 3.17

19 66.4 3.45 59.4 3.98

20 64.5 3.35 60.5 3.55

21 61.9 3.02 59.6 3.75

(22a 176.9 8.57 187.0 7.83)

22 60.4 3.21 58.0 3.21

23 84.5 4.40 84.4 3.59

Average 66.6 3.42 63.5 3.54

"Bench"values 3.23 3.63

Correction factor: 0.975 0.972


aThese are the results of the bad calibration and
are shown for reference only. They were not used to com-
pute the averages.









Most of the observing program for 3C391, its COMP,

3C396, and its COMP was performed with the Total Power

scheme using a sixty-minute cycle; a twenty-minute OFF

scan was followed by a ten-minute scan on each of the

four program regions in succession. The ratio of OFF-

time to ON-time was determined on the basis of Equation

111-5. To make the most efficient use of observing time,

it is desirable to make the scans as long as possible so

that the total time moving between regions is minimized.

The maximum length of the cycle was limited by the length of

time during which the receiver passband remained stable.

During the observations it was found that the quotient spec-

tra of the last few records of the sixty-minute cycle be-

gan to show a curvature of the baseline. This means that

the receiver passband had changed enough since the OFF scan

had been recorded that this OFF spectrum no longer repre-

sented the current passband well enough to produce a flat

continuum on which the spectral features are enhanced. The

curvature only affected the last few 3C396 COMP spectra;

so, the cycle was maintained at sixty minutes.

The observations of these four regions continued

until the RMS noise, expected after combining all scans of

each region, was about O.0020K. Most of the last two days

were used to observe the pulsar and 3C295. Each was ob-

served individually, usually with a fifty-minute cycle










consisting of a twenty-five minute OFF followed by a twen-

ty-five minute ON. The 3C295 spectra showed no spectral

features or abnormal baselines. Therefore, these data will

not be discussed further in this study.

After each day's observations had been completed,

the seven-track "telescope" tape was taken to Charlottes-

ville, Virginia where an IBM 360/55 computer was used to

average the records of each scan and load the resulting

spectra on a nine-track "user" tape for further analysis.

At the same time a summary of records and the quotient spec-

trum of each scan was printed. These tasks were performed

by the first phase of the TPOWER data reduction program

(Cram,1973). The printout was sent to the observatory the

next day. With these "quick-look" summaries, it was possi-

ble to determine which scans had problems which would make

them unusable. On the basis of these reports some scans

were deleted and reobserved on the last day of the program.

After all observations had been made, the final

spectra were produced using the second phase of TPOWER, a

system of routines controlled by a command language. It

was during this process that the correction factor, de-

termined from the daily noise tube calibration, was applied

to adjust the antenna temperature scale. The scans were

summed and weighted by the effective integrating time of

each scan. Then the spectra of the two receivers were av-

eraged together, thereby doubling the integrating time. The










resulting spectra were convolved with a five-channel "box-

car" function and plotted. The plotted spectra were stud-

ied in order to estimate the structure of the baseline.

Curvature of the baseline of each spectrum was removed with

a TPOWER routine which first computes a specified order

Chebyshev polynomial by means of a least squares fit to

a specified portion of the baseline. The routine then

subtracts the polynomial baseline from each spectrum and the

spectra are then plotted in their final form.

A final task was performed in order to facilitate

the analysis of the spectra. Gaussian functions were fitted

to the spectral features. The TPOWER routine begins with

initial estimates of the center channel and half-intensity

width of each feature and, from that, determines a best-

fit center, half-intensity width, and central intensity of

the features.



The Spectra


The basic spectra for the program regions are shown

in Figures IV-2, -3, and -4. Preceding the figures is

Table IV-3 which lists the total effective integration time

(ti) and the order of the Chebyshev polynomial base line

which was removed. The effective integration time was com-

puted with Equation 111-4. It includes the doubling ob-

tained by averaging the spectra of the two receivers.










The RMS noise, listed in the third column, is com-

puted from Equation III-3;

1.528 T
AT = s
P ti

T is the average system temperature of both receivers
sys
and is 65.0 oK. The effective bandwidth, B, is five times

the channel separation of 13.0 kHz because the spectra are

displayed after being convolved with a five channel box-

car function. Table IV-3 also provides a table of contents

which includes a list of the figure numbers for all spectra

including those showing the Gaussian fits to the spectral

features.

The lower portions of Figures IV-5 through 10 show

the results of fitting Gaussin functions through the spec-

tral features. The Gaussian functions of the individual

features are added together and the combined function is

plotted through the original spectrum. The upper portions

of these figures show the plots of the residuals which are

obtained by subtracting the combined Gaussian function from

the spectrum.

The two possible features at -118.8 and 58.2 km/s

on the COMP 3C396 spectrum are only about four times the

RMS. Therefore, it was decided not to present Gaussian

features fit to them.

The center velocity, half-power width, peak antenna

temperature and integrated line power of each feature in the












Table IV-3

Parameters of the Spectra


Basic
SOrder of T Spectra Gaussian Fitting
i RMS Baseline Baseline C Figuresb Compo- Figure
Source [hours] Noise [oK]a Removed [K] (Chapter IV) nents (Chapter IV)

3C391 7.12 .0024 2 10.7 IV 2 A 2 IV 5
3 6

3CO3 6.49 .0026 2 6.1 2 B 2 7
3C391 3 8

3C396 6.66 .0025 4 8.8 3 A 1 9

COMP
3C396 7.08 .0024 2 4.5 3 B

Pulsar
1858+03 7.10 .0024 2 3.7 4 2 10


tion.


After smoothing with a 5-channel boxcar and averaging both receivers.

The designation A refers to the upper portion of the figure, B to the lower por-









spectra are listed in Table IV-4 which follows the spectra.

Also listed for each parameter is its standard error (stan-

dard deviation of the mean) which, except for the standard

error of the integrated line power, is computed by the

Gaussian fitting routine. The standard deviation of the

line power, sp, is computed from the following equation

(Parratt 1961, p. 117) with Equation 11-35:

2 2
P 2 2 (P 2 1/2
sp = [( ) + ( )2s ]
P 5T T 9Av Av
2 2 2 2 1/2
S1.0645 [(A\)2 s2 + T s A1]

Here, sT is the standard error of the peak line temperature

and s A is the standard error of the line half-power width.

The discussion of the spectra will be a part of the

analysis of the emitting region. Chapter V will present

the analysis of the spectrum from the directions of 3C391

and its COMP beam. The model derived there will be applied

to the analysis of the medium in the direction of 3C396,

its COMP beam, and the pulsar 1858+03.





























Figure IV-2. H157a Spectra from the Direction
of 3C391 and Its COMP Beam. The
velocity resolution is 11.6 km/s.





















0.06 3C391


0.04


0.02


0.00

T'

K

0.06 COMP
3C391

0.04


-170 -130 -90 -50 -10


30 70 110 150
v km/s


____





























Figure IV-3. H157a Spectra from the Direction of
3C396 and Its COMP Beam. The veloc-
ity resolution is 11.6 km/s.




















































-170 -130 -90 -50 -10 30 70 110 150
v km/s


0.022


0.014


0.006


0.002

T'

K

0.022


0.014


0006


-0.002






















Figure IV-4. H157a Spectrum from the Direction of the Pulsar 1858+03.
The velocity resolution is 11.6 km/s.























1858+03


-120 -80 -40


0 40 80 120 160 200
v km/s


0.011

T'

0.007

oK

0.003



-0.001







Full Text

PAGE 1

RADIO SPECTRAL LINE STUDIES OF THE INTERSTELLAR MEDIUM IN THE GALACTIC PLANE By ANDREW WILKIN SEACORD, II A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1975

PAGE 2

\! To My PaAzntA With Gtcutitadu Louumz Vila.no Peck Szacotid and Wilkin Hmny Szacond N

PAGE 3

ACKNOWLEDGEMENTS I wish to express my appreciation to Dr. Stephen Gottesman who introduced me to the intriguing problem of the galactic ridge and who has provided many comments and suggestions during the course of my study of it. His encouragement and cooperation during the final stages of this project are greatly appreciated. The spectra observed as a part of this study were obtained with the 43 meter telescope at the National Radio Astronomy Observatory in Green Bank, West Virginia. I appreciate the opportunity to have used these facilities. I am grateful to the NRAO staff, particularly the operators and engi neers, whose support made the observations successful . I also wish to thank the other members of my doctoral committee whose comments and suggestions have been most helpful. Actually, many people within the Department of Physics and Astronomy deserve a word of thanks for their general support , for provi di ng travel funds, and for making computer time available to me. Finally, I owe many words of gratitude to my wife, Penny, who has been patient with me and often provided me 1 1 l

PAGE 4

with much-needed words of encouragement and to my daughter, Becky, who has provided many interesting moments during the course of my work here.

PAGE 5

TABLE OF CONTENTS Page ACKNOWLEDGEMENTS i 1 ' 1 ABSTRACT vii CHAPTERS I INTRODUCTION "• The Galactic Ridge ! Objective, Method and Scope of This Study 4 II INTERSTELLAR MEDIUM ASTROPHYSICS 8 Introduction 8 The Transfer Equation and a General Solution of It 8 Thermodynamic Equilibrium 12 The Thermal Continuum 14 Radio Recombination Lines 18 Thermodynamic Temperatures and Level Populations 19 Population and Depopulation Mechanisms of High n-Levels 23 Collision Transition Probalities and Cross-sections 26 Radiative and Dielectronic Recombination Coefficients 31 Departure Parameters b n and 6 mn 34 Radio Recombination Line Absorption Coefficient 39 Line Profiles ^ 3 Non-LTE Solution of the Transfer Equation and Line Enhancement 49 The Radiation Field 53 Ionization Rates j?6 The 21 cm Hydrogen Line 5 ° Pulsar Dispersion Measure 62

PAGE 6

Ill IV VI VII Page THE TELESCOPE SYSTEM SENSITIVITY AND THE OBSERVING SCHEME 64 System Sensitivity 65 The Observing Scheme 67 THE RADIO RECOMBINATION LINE OBSERVATIONS 70 Introduction 70 Observing Scheme and Data Reduction 75 The Spectra 83 MODELS FOR THE INTERSTELLAR MEDIUM IN THE DIRECTION 3C391 107 3C391 : A Summary of Previous Observations 107 The Models 1 19 The D-Method Models 119 Isolated Cold Region Models 147 The Hot Spot Models 156 H II Nebula Models 175 FURTHER CONSIDERATIONS AND APPLICATIONS OF THE MODELS 182 Discussion of the Models 182 The Pulsar 1858 + 03 200 The 3C396 and COMP Spectra 206 Distribution of the Recombination Line Sources Within the Ridge 210 SUGGESTIONS FOR FURTHER RESEARCH 216 APPENDIX 222 BIBLIOGRAPHY 2 26 BIOGRAPHICAL SKETCH 234 VI

PAGE 7

Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy RADIO SPECTRAL LINE STUDIES OF THE INTERSTELLAR MEDIUM IN THE GALACTIC PLANE By Andrew Wilkin Seacord, II August, 1975 Chairman: Stephen T. Gottesman Major Department: Astronomy This study develops a model for the source of radio recombination line emission from the galactic plane. The model is based mainly on recombination line spectra observed in the direction of the supernova remnant 3C391 . The H157 a recombination line was observed and used with the H92a spectrum observed by Cesarsky and Cesarsky (1973) to obtain a model consistent with both spectra. Other criteria— such as the 80 MHz optical depth, the contribution to the thermal continuum temperature, and ionization rates — were also considered. Four models were studied: a cold region illuminated by 3C391 ; an isolated cold region; a group of hot, dense H II regions; and a low density H II nebula. Of these, the best model is a small (^85 pc diameter) H II region with a temperature of 7500° K and an average elec-3 -3 tron density of 10 cm VI i

PAGE 8

The H157a spectrum in the direction of the pulsar 1858+03 was also observed. It was found that it is probable that one of the recombination line nebulae lies along the line of sight to the pulsar. In this case the pulsar would be at a distance of 2.6 kpc in order for its dispersion measure to agree with the measured value of 402 -3 pc cm The large scale distribution of the H II nebulae was also studied. The results of available radio recombination line spectra surveys from the galactic plane suggest that they exist within an annular region, which extends from 4 to about 8.7 kpc from the galactic center. VI 1 1

PAGE 9

CHAPTER I INTRODUCTION The Galactic Ridge As a radio telescope is scanned across the plane of the Milky Way at some galactic longitude, it is seen to be a source of radio emission. For radio frequencies of 100 MHz or greater, the intensity increases as the galactic equator is approached and peaks at the equator itself. Scans through the galactic plane made at several different frequencies reveal that the emission has two components, thermal and nonthermal. At frequencies below a few GHz, the emission is predominantly nonthermal with a spectral index between -2.7 and -3.0 (Westerhout, 1958; Matthews, Pedlar and Davies, 1973). At 85 MHz the radiation is almost completely nonthermal and the intensity In this work, the revised galactic latitude, b and longitude, l 11 , as adopted by the IAU (Blaauw et al . , I960) will be used throughout. The superscript II, however, will be dropped from the notation.

PAGE 10

distribution across the galactic plane has a total half-intensity width of about 5° at longitude greater than 12° from either side of the galactic center (Hill, Slee, and Mills, 1958). As the frequency increases, the overall intensity decreases and the width of the emission becomes smaller, approaching a thin ridge at centimeter wavelengths. At 1390 MHz the hal f i ntensi ty width is generally about 195. The thickness of the nonthermal emission region increases somewhat near the galactic center. This thin continuum emission region surrounding the galactic equator has been named the "galactic ridge." The galactic ridge has been recently mapped with an antenna resolution of 11 arc minutes at frequencies of 1414, 2695, and 5000 MHz by Altenhoff et aj_. (1970). Matthews, Pedlar, and Davies (1973) have used these maps to obtain the latitude distribution of the continuum intensities at three longitudes where no resolved sources are present. They plotted the distribution with the intensity scale at each of the frequencies adjusted so that, if the emission were entirely nonthermal, the three distribution curves would be congruent. The plots are congruent at latitudes greater than about 0?5, but inside this range, the intensity increases markedly with frequency due to thermal emission. These curves illustrate the two components of the galactic ridge continuum emission.

PAGE 11

It is known that hydrogen is concentrated within a thin region around the galactic equator (cf. Westerhout, 1958; Baker and Burton, 1 975) . The thermal emission comes from the free-free process between electrons and protons from ionized hydrogen. At frequencies below about 30 MHz, a scan cutting through the ridge shows an intensity profile with two peaks, one on each side of the equator, The decrease of intensity along the equator results from absorption of the strong nonthermal emission by the free-free process within the thermal emission region. The map by Shain, Komesaroff, and Higgins (1961) at 19.7 MHz shows the absorption lane prominently. The galactic ridge took on an interesting aspect when Gottesman and Gordon (1970) observed low intensity H157a radio recombination line spectra in two directions in which the Altenhoff et^ a]_. (1970) maps show no resolved sources. Gordon, Brown, and Gottesman (1972) found that at a longitude of 33°, the recombination line emission is limited to a latitude of less than 0?5 above the equator. In the region toward the galactic center, recombination line emission has been observed at a latitude of -0.62 (Lockman and Gordon, 1 973) . Previous to Gottesman and Gordon's observations, radio recombination lines had only been observed from known galactic H II regions. They have not been detected from the interstellar medium in the solar neighborhood .

PAGE 12

A number of workers have studied the galactic ridge in order to determine the nature of the radio recombination line sources and how they relate to the thermal continuum and 21 cm spectral line emission. The analysis of Cesarsky and Cesarsky (1971, 1973b) and Lockman and Gordon (1973) find that the electron temperature of the radio recombination line emission regions is close to 20 °K. 3 4 Other workers derive temperatures ranging from 10 to 10 °K (Gottesman and Gordon, 1 970 ; Gordon and Gottesman, 1 971 ; Matthews, Pedlar, and Davies, 1973; Jackson and Kerr, 1971; Chaisson, 1 974a; and Downes and Wi 1 son, 1 974) . Very recently two papers have appeared, whi ch conclude that these recombination lines originate from low density H II regions (Jackson and Kerr, 1975; Pankoni n, 1 975 ) . Objective, Method, and Scope of This Study The problem to which this study is directed is that of determining the type of region responsible for the recombination line emission in the galactic ridge. The first step will be to observe recombination line emission from regions in the ridge in which there are no H II regions visible on the Altenhoff et al_. (1970) maps and from which other data have previously been obtained. The recombination line spectrum will provide one piece of datum, the power contained in the spectral line. However, there are several

PAGE 13

parameters which define the type of region: electron temperature, electron, neutral hydrogen, ionized hydrogen number densities, size of region, and distribution of the region, or regions, within the galactic ridge. The latter may imply a relation between recombination line emission and other phenomena such as low frequency radio absorption and galactic structure. Since the observations made as a part of this study can only supply one bit of information, the identity of the recombination line source can be obtained only after other observations are utilized. Even so, there are still more parameters whose values are to be found than there are observed quantities. For that reason, the values of some parameters must be chosen at the onset of model computation. A type of model is hypothesized and theory or the conclusions from previous observation suggest what these values are. The second step will be to obtain, from one region in the ridge, a model solution which is consistent between two similar data. The ideal case for this is the region toward the supernova remnant 3C391 from which Cesarsky and Cesarsky (1973b) have observed an H92a recombination line spectrum. This region has undergone a considerable amount of scrutiny, particularly by those who have attempted to measure the 80 MHz optical depth (Caswell et al., 1971; Dulk and Slee, 1972; Goss, 1972).

PAGE 14

The third step in the process will be to assume that this model applies in the mean sense to the entire ridge and use the remaining data obtained for this study to check the model. The method is to calculate a parameter which depends on the model, to see if it agrees with what is observed. Finally, recombination line surveys will be combined in such a way as to examine the large scale structure of the recombination line ridge. The theoretical basis of this study will be developed in Chapter II. All equations used in the analysis will be derived and approximations and assumptions will be discussed. The third chapter will discuss theory behind the observing procedure. The method and conditions under which the observations were made and the display of the processed spectra will occupy Chapter IV. Chapters II, III, and IV provide a basis for the subsequent work. Four types of regions on which to base the models will be selected. Three of them result from previous observation of 3C391, a summary of which begins Chapter V. The Hot Spot model is based on unpublished observation of Altenhoff et aj. (1973). The remainder of Chapter V develops the four models. Chapter VI is involved with the selection of the most probable model and tests this by computing the distance to the pulsar 1858+03 and the 80 MHz optical depth expected in the direction of 3C396. Chapter VI ends with

PAGE 15

an investigation of the large scale structure of ridge recombination line emission. The last chapter concludes the study by proposing future projects to study the galactic ridge.

PAGE 16

CHAPTER II INTERSTELLAR MEDIUM ASTROPHYSICS Introduction It is the purpose of this chapter to present a foundation on which this project is based, particularly the analysis of the spectra. The starting point is the equation of radiative transfer. The coefficients of absorption and emission are parameters of the solution to the transfer equation. Since this study is based on radio spectral lines, the line absorption coefficient will be examined in detail. The line profile of a given element depends on the thermodynamic state, temperature, abundance, electron density, and motion of the gas creating the line. The Transfer Equation and a General Solution of It The transfer equation can be set up by considering the radiation passing through a region as shown in Figure II-l. The radiation entering this region has a frequencydependent specific intensity, I (v), which can be both thermal and nonthermal in origin. The line of sight enters

PAGE 17

ds I(v) I ( y ) OBSERVER s=L s=0 r=0 Figure II-l. Ray Path Geometry Through a Lineemitting Region. The region is shown shaded.

PAGE 18

10 the region at a point where the path length, s, is zero. The total path length through the region is L. At the near boundary, where s = L, the specific intensity is I(v). The change in specific intensity along a differential path length ds in a homogeneous medium is dl(v) = k l (v) I(v,s) ds -k t (v) I(v,s) ds + j,_(v) ds + j' T (v) ds + Jm(v) ds. II-l The nonthermal emission coefficient is j»>(v). In the interstellar medium the nonthermal absorption processes are not important. The thermal process, which will generate or absorb radiation, is the free-free process; it will be discussed later. The coefficients of thermal continuum emission and absorption are jj(v,s) andK T (v,s) respectively. The applicable coefficients for spectral lines are denoted by the subscript L. Let j(v) = j' T (v) + J L (v) and k (v) = k j {v) + K L (v),then at any point, I(v) = I T (v) + I N (v) + I L (v) = I c (v) + I L (v). Equation II-l then becomes dl(v) . . J(v) J N 1 1-2

PAGE 19

11 The optical depth, x, is defined by x(v) = I k(v) ds = t c (v) + r L (v, The thermal source function, S(v), is defined by S(v) j(v) J L (v) + J T (v) <(v) < L (v) + Kj(v) II-3 Since there is no nonthermal absorption coefficient and the thermal and nonthermal processes are independent, there is no nonthermal source function. A general solution to the transfer equation can be obtained by multiplying Equation 1 1 2 by the integrating factor exp(-x) and integrating along the path length to get I(v) = I Q (v) e" T + (l-e" T ) + ^y^ (1-e" 1 ). The brackets, < >, denote the average over the path length. The intensity of the nonthermal emission generated within the line-emitting region is 1^ = < J N >L and L = t ; so, the general solution is I.. Kv) I Q e" T + (l-e" T ) +-f (l-e" T ) II-4 Nonthermal emission decreases in intensity with increas ing frequency, in general .

PAGE 20

12 In order to derive a solution for a specific region, the source function — which depends on the thermodynamic state of the region — must be found. It is important, therefore, to consider the conditions for thermodynamic equilibrium. This is the subject of the next section. Thermodynamic Equilibrium To be in thermodynamic equilibrium (TE), a gas must have the following properties. 1. The temperature and pressure throughout the region are constant and are equal to that of its surroundings; hence, there will be no flow of energy or mass through it. 2. The gas will radiate as a black body at that temperature . 3. Detailed balancing must exist; that is, for each process occurring in the gas, its inverse must also occur by an equal amount (e.g. ionization and recombination). 4. The velocities of the gas particles have a Maxwellian distribution. 5. The population of any excited state of an atom, ion, or molecule is determined by the Bol tzmann equati on .

PAGE 21

13 6. The relative population of any level of ionization of a species to that of the neutral species is given by the ionization (Saha) equation. The observed spectral line intensity depends on how well these conditions are met. Radio recombination line observations of a variety of regions indicate that the interstellar medium is generally not in local thermodynamic equilibrium (LTE); that is, the conditions of TE are not met even on a small (local) scale (Hjellming and Davies, 1970; Andrews, Hjellming, and Churchwell, 1971; Gordon, 1971; Brown and Balick, 1973). If LTE does not exist, the transfer equation parameters must be modified to account for this. In the following work, a superscript asterisk denotes an LTE quantity. The second con' dition of LTE dictates that S*(v) = B(v,T), the Planck function for a temperature T. For radio frequencies, this simplifies to v,T 2kT 2 c 1 1-5 In the case of non-LTE, a correction parameter is defined such that S(v) = A(v) S*(v) = A(v) B(v,T). II-6

PAGE 22

14 In any steady-state situation (LTE or non-LTE), the total per-unit volume emission and absorption must balance. Only in LTE do emission and absorption balance at each frequency so that, by Kirchhof f ' s Law, j(v) = k(v) B(v,T). The Thermal Continuum Thermal emission and absorption are the result of an encounter between a free electron and a positivelycharged ion after which the electron remains free (cf. Mitchell and Zemansky, 1934, p. 154). From a classical point of view, an electron moves in the field of an ion (e.g. H ), is accelerated and absorbs a wave or is decelerated and emits a wave. As shown in Figure II-2, the trajectory of an electron with a given velocity is altered by an amount determined by the impact parameter, p. For radio frequencies, p is large enough that the entire trajectory can be approximated by a straight line. Oster (1961) has derived an expression for the thermal absorption coefficient, which includes a quantum mechanical correction applicable for temperatures in excess of 5.5 x 10 °K. The derivation of <( v ) was obtained by considering emission from a single electron with specified velocity and impact parameter. Then j,( v ) was found for an ensemble by integrating first over a suitable

PAGE 23

15 range of p and then over a Maxwellian velocity distri bution. Then, < T (v) = jy(v)/B(v,T) and, for hydrogen K (v) = 3.014 x 10" 2 r K5 v~ 2 ln(4.955x!0" 2 T" 2 " 5 v" 1 ) N g I Z 2 N. 1 1-7 The numerical constants are functions of fundamental constants as shown in the Appendix, Equation A-l. The summation is over all ionic species present. Z. is the effective charge number (Z R+ 1) and N. is the number den i sty for the i species. Altenhoff et al (1960) have derived an approximate expression for Kj(v), which is Ky (v ) .235 x 10" 2 T" 1 ' 35 v" 2 ' 1 N $ Z 2 N . . II e e i i i To bring this approximation into agreement with Oster's more exact formulation, it is multiplied by a correction factor a(T ,v). Using it, the optical depth then becomes t t (v) = I (v) ds = 8.235 x 10" 2 a(T e ,v) T" 1 35 v" 2J E ] II-9 E T is the continuum emission measure defined by

PAGE 24

16 (N E Z. N.) ds. e . i "i 11-10 The emission measure is expressed in units of pc cm The factor a(T ,v) has been tabulated by Mezger 5 and Henderson (1967) for the ranges 100 ± v < 10 MHz and 3 x 10 3 < T < 15 x 10 3 °K. For these ranges, a(T ,v) — g — 3 g varies between 0.7680 and 0.9971. The emission measure is often given by X T N 2 ds e 11-11 The effect of heavy elements on the size of E T depends on the fractional ionization of hydrogen; x = N / N ,. . The atomic species with an ionization potential e H r less than that of hydrogen (13.6 eV) may be totally ionized. If x ^ 10 , the contribution of these elements to the electron pool will be an appreciable fraction of the total unless their abundances are less than those adopted by Spitzer (1968, p. 122). The most abundant contributor to the electron pool, other than hydrogen, -4 is carbon which has an abundance N ~ 4 x 10 N ,, . The continuum emission spectrum from a region of the interstellar medium can provide some information about the conditions there. Consider Equation 1 1 3 for the continuum; that is, I. = 0. Let I = I.. = and, L ON

PAGE 25

17 ssuming LTE, S(v) = B (v,T). Then, using Equation II-5, if \ 2kT 2 n Q " T T I(v) = — — v (1-e c 11-12 When the frequency is high enough that the gas is optical ly thin, t t << 1 , and ., v 2kT 2 I(v) = — gv t t c Since, by Equation 1 1 -8 , t t a v" ' , I(v) * v" ' and the spectrum is nearly flat. For low frequencies such that the gas is optically thick, t t X 1 • 5 and ! ( v ) « v . The turnover frequency, \>j, is that for which t t = 1, although sometimes the value of 1.5 is used. Setting t t equal to one in Equation 1 1 -8 and solving for v T gives v T . 1.893 x 10 2 a(T .v)2 ' 1 El 2 '' T" ' 643 11-13 If in some region for which v T can be measured, either E T or T is already known, the other parameter can be found. One independent source of information is the in tegrated radio recombination line power. Radio recombination lines will be considered in the next section.

PAGE 26

Radio Recombination Lines Radio recombination lines arise from transitions between levels with large principal quantum numbers; n ? 40. They result from the capture (recombination) of an electron by an ion into an upper level. The nomenclature for such a transition is Xn(An) where X designates the recombining species; n is the principal quantum number for the transition which is specified by a for An = 1 , B for An = 2, etc. The predominant element from which recombination lines are observed is hydrogen; however, radio recombination lines of helium, carbon, and possibly heavier elements have been observed (Gordon, 1974; Chaisson, 1973; Pedlar and Hart, 1974). The emission frequency, v , for the transition between the m and n levels (m > n) is given by the Rydberg formula (cf. Shore and Menzel , 1968, p. 13) v = c R Z (-0-— o) mn xx v 2 2' n m 2c R Z 2 ^ x x n 3 11-14 R is the Rydberg constant for the species X and Z is x x its effective charge number. Radio recombination line frequencies for hydrogen and helium have been tabulated by Lil ley and Palmer (1968) . The recombination line strength depends on the number of atoms in the region which are excited to the

PAGE 27

19 upper level of the transition. Since the gas may not be in LTE, it is necessary to calculate the LTE populations and the departure parameters, which indicate the degree to which the gas is out of LTE. These matters will be discussed in the following subsections. Thermodynamic Temperatures and Level Populations Radio recombination line emission regions contain excited neutral atoms, ions, and electrons. The number _ 3 densities (in units of cm ) for hydrogen are N„ , N H+ , and N respectively. N„ is the total hydrogen density. In LTE there is an equilibrium situation where recombinations are balanced by ionizations. What is desired is a statistical relation between the kinetic temperature of the qas, N..+ , N , and N , the number density of atoms in hr e n the n level of excitation. For a gas in LTE, the population number density of ions in the n excited level and the i ionization state is given by the Boltzmann equation (cf. Aller, 1963, p. 112): N . = N . — exp(x • /kT ) . m oi g Y v A l n ex 11-15 This equation is based on the condition that the assembly has a Maxwellian velocity distribution. In

PAGE 28

20 Equation 11-15, N . is the LTE population of the ground state; g and g are the statistical weights of the exa n 3 o cited and ground states respectively; x-„ "is the excitation energy within the i state of ionization; and T gx is the excitation temperature. The statistical weights are the number of sublevels in the n level; g = ( 2 J + 1). J represents the total angular momentum — spin plus orbital — of the level. Under the conditions of LTE, the excitation temperature, T , is equal to the kinetic temperature r ex of the gas. The radio recombination lines, which have been observed involve highly excited neutral atoms, for example H I, He I, and C I. So, the desired population is N expressed in terms of the electron density and the no v respective number of the singly ionized species. For the spec i es X , N>) 4.143 x lO" 16 N*(X) N e T^ 5 ^ exp(-e in (X)/k T ex ) 11-16 The details of the derivation of this equation are given in the Appendix; it results from placing the values of the fundamental constants into Equation A-5. The partition function, u,(T), is given by Equation A-2. e in ( x ) is the threshold energy which is (Equation A-6)

PAGE 29

21 RADIO FREQUENCY <— • X RAY Figure 11-2. The Free-Free Process. The elec tron, e, approaches the central ion. The impact parameter, p , is shown for two cases.

PAGE 30

22 e ln (x)=-( Xl -X 1n ) 15 7900 (Z^/n 2 ) k. 11-17 The kinetic energy of a particle with mass M is e K = 1/2M v" = 3/2 k T K . If the gas has more than one constituent, each with a different mass, the kinetic temperature of each constitu ent will be the same although, because of the different masses, the RMS velocity of each constituent will be different. The RMS velocity of the component with mass M i s 2 1/2 , 3k T K J/2 TT , = ( jjj — ) . H-1 If the gas particle velocity distribution should suddenly become non-Maxwel 1 ian , collisions involving a deflection of 90° will thermalize them to an equilibrium temperature unless the system is thermodynamical ly unstable (Spitzer, 1968, p. 90). In a situation where there is a nonequi partition of energy, the electrons will have the largest kinetic temperature because of their low mass. The energies of the electrons and ions will equalize first because of the long-range interactions between them. Following this, there will be elastic collisions between the ions and neutral atoms. Collisions between electrons and atoms are not

PAGE 31

23 as an effective means of equalizing the kinetic energies as are collisions between ions and the electrons (Spitzer, 1968, p. 140). Population and Depopulation Mechanisms of High n-Levels The high n-levels are populated and depopulated by several mechanisms which are discussed by Brockel hurst (1970), Dupree and Goldberg (1970) and M^Daniel (1964, p. 588). The parameters, such as cross-sections, probabilities, and coefficients are discussed in general by these authors as well as by Spitzer (1968), Kaplan and Pikelner (1970), and by the workers listed in Table II-l who have calculated high n-level populations. The mechanisms involved are as follows. 1. Electron recombination followed by a cascade 2. Dielectric recombination (in atoms with more than one electron). 3. Excitation and de-excitation induced by el ectron collisions. 4. Collisional ionization followed by threebody recombination leaving an excited atom X': H + e > X + + e + e and X + e + e + X' + e. 5. Stimulated and spontaneous emission.

PAGE 32

24 6. Fine structure excitation; i.e. redistribution of angular momentum with a level: n £ -* n & ' . Excitation to high n-levels from the ground level by radiative excitation is not important in H II regions (Dyson, 1969; Dupree and Goldberg, 1970). To account for the effects of a departure from LTE with respect to level populations, the departure coefficient, b , is employed; b = N /N n n n 11-19 The departure coefficient is a function of the electron temperature and densi ty and the intensity of the radiation field. When b is calculated for a type of region, the radiation temperature is chosen on the basis of specific environmental conditions of the region and b n is tabulated or plotted as a function of n for specific values of T and N . For example, Dupree (1972a) has done this for e e departure coefficients in a cool region. The coefficients were first calculated for an isolated region where I Q (v) 4 = and then the effects of an adjacent 10 °K H II region on the b values were demonstrated, n Statistical equilibrium requires that there are as many transitions into a level as there are transitions out of it. The equation of statistical equilibrium for

PAGE 33

25 the n th level is of the form (Seaton, 1964; Dupree, 1972a, b) n-1 n-1 n+p' t "fo ? Vj + T . C n,j + ^ +1 C n,1 + £-., L n,m r , n,n,, j=n ' J J=n-p ° j=n+l m=n+l jI I(v = y (m C + N C ) + ( y N B *[ n-p n-p,n 'n+p n+pV v m f^ +1 m m,n p=l + L NB n ) I(v)+ £ N A + N e N i (a^a n + C. jn ) j= n J J' j= n +l 11-20 The symbols used in Equation 11-20, which have not yet been defined, are as follows: A = Einstein spontaneous emission probability m , n n ,m m , n m > n ; = Einstein absorption probability, m > n; = Einstein stimulated emission probability, m > n ; C. , = collision probability for a transition J > k between the j and k levels; C . = collisional ionization probablity from n , i 1 evel n ; C. = three-body recombination probability;

PAGE 34

and 26 a, r = rate of radiative recombination into level n; n a = rate of di electronic recombination into level a n n ( = for hydrogen ) ; m ' = the maximum level considered in cascades from hi gher 1 evel s ; p' = the maximum p for bound collisional transitions, p = | An | ; n = the lowest level of the atom considered for o the computation; N. = the ion number density Collisional Transition Probabilities and Cross-sections In order to compute the departure coefficients, means of simplifying Equation 11-20, limiting the number of equations or limiting the range of the summations should be considered. At some level n = m ' , the electron is close enough to the continuum that the collisional ionization probability, C ., is much greater than the probability for a spontaneous transition to lower levels, A m , n The levels beyond the m ' cut-off level are in thermodynamic equilibrium and, b = 1 for n >m'. n Sejnowski and Hjellming (1969) find that inclusion of collisional transitions of p = | A n | > 1 will lower the value of b by only a few percent from that considering only transitions of p = 1 . Consequently, many workers limit their calculations to p ' = 1 .

PAGE 35

27 Collisions involving a change of Istate within a level, n,£ + n , I + 1, occur frequently. In cold hydrogen qas with N > 10 cm , fine structure collisions by progtons as well as el ectrons, occur more frequently within levels for which n > 100 than do radiative processes 2 (Pengelly and Seaton, 1964). Because of this, all n £-states are equally populated. Also, at low temperatures, fine structure excitation by electron collisions provide the main cooling mechanism of the gas (Dalgarno and McCray, 1972). By this process, electron kinetic energy is converted to low frequency radiation which leaves the gas, thus removing energy from the thermal pool. The total population of level n is divided into Jl-states according to n,£ (2& + 1 ) (Dupree, 1969). The total population is N = E N . For hydrogen, the levels are degenerate and b n,A b ; b is independent of I. n r The collisional transition rates, C. , for = j + p, depend on cross-sections Q. , by C j,k N e «j.k (v)> 11-21

PAGE 36

28 The average is obtained by integrating over a Maxwellian velocity distribution. The bound-bound cross -secti ons have been computed by means of one of three approximations or methods: binary encounter method, impact parameter method, and a method based on the correspondence principle. The binary encounter method uses a classical approximation, which is valid for electron-atom collisions if the time of interaction between the incident and bound electrons is short compared to the orbital period of the bound electron and the collision takes place in a region smaller than the size of the atom. This time is sufficiently short when the energy transferred between the two electrons is large. With this condition, the energy transferred is assumed to be equivalent to that transferred between two free electrons. Because of this requirement, the binary encounter method gives best results for ionizing collisions; the results are poor for p = 1. In calculating cross-sections, Brockelhurst (1970) uses the binary encounter method for p > 6. The application of the theory involved is discussed at length by Burgess and Percival (1968). The impact parameter method for electron-atom collisions uses the assumption that the incident electron follows a straight-line trajectory past the target atom with an impact parameter p (cf. Figure 1 1 -2 ) . Seaton

PAGE 37

29 (1962) computes the collision cross-section for a bound transition (n,£ •+ n', £') for both weak and strong coupling. For weak coupling, 'j.k P k -(p) 2ttp dp, where j = n,£, k = n', I ' , and p is the cut-off value of p>which is approximately equal to the smaller of the two atomic radii of the j and k states. P^Cp) is the transiK J tion probability as a function of impact parameter. For strong coupling, Seaton (1962) defines p, such that P..(p-.) = 1/2. For pp, , and / Q j,k = 1/2 ^ p l + J P. .(, ) 2, r dp P 1 " J Seaton (1962) calculates the cross-sections for the weakand strong-coupled cases and accepts the smaller of the two values. P u -(p) is calculated with the symmetrized, time-dependent perturbation theory. Brockelhurst (1970) claims that the impact parameter method will not work for p > 1. He assumes the case of strong coupling and

PAGE 38

30 finds that most of Q comes from p < p-i for which the perturbation theory is not valid. The classical method only works for large p; the semi -cl assical impact parameter method is valid only for p a 1. The correspondence principle method can be used to obtain cross-sections for intermediate transitions. All variations of the correspondence principle state that for large scale systems, where the uncertainty is much larger than Planck's constant, h, the result of a quantum mechanical approach is identical to that using a classical approach. Percival and Richards (1970a) show that the Bohr correspondence principle can be applied to the energy transfer of a collisional excitation for the case of a weak interaction and if the quantum and classical firstorder perturbation theories are both valid. Also, the collisional probability must be much larger than that of a radiative transition, a condition which is satisfied for n ~ 100. With the the Bohr principle for collisions, the mean net energy transferred during the collision is equivalent to the energy transferred from independent Fourier components of order +p and -p of the classical atomic orbital motion. P . . ( p) is computed from these Fourier components . For the case of a strong interaction, Percival and Richards (1970b) show that the Heisenberg correspondence principle relates the quantum mechnical matrix elements

PAGE 39

31 with the Fourier components of classical atomic orbital motion. The results for H II regions are published in a separate paper (Percival and Richards, 1970c). For collisional ionization, either the binary encounter method (Burgess and Percival, 1968) or the classi cal Thompson theory (Dyson, 1969; Seaton, 1964) can be used to obtain the collision rates. From the Thompson theory C . = 3.46 x 10" 15 n 2 N T~ 1/2 n , i e The inverse of collisional ionization is three-body recombination. Using the equation of detailed balancing N C . = N. N C. n n n , i l e i , n Radiative and Dielectronic Recombination Coefficients The radiative recombination coefficient for capture into level n is represented by a . The partial recombination coefficient, a,.v, "> s that for captures into all levels above and including the k ; n > k. It is given by a,.\ = Z°°a . . Usually k = 2. The total reIKj j=k J combination coefficient , a,,\, includes radiative captures into the ground level. Seaton (1959) has derived formulations for a,,x and a,^ using a symptotic expansion

PAGE 40

32 of the Kramers-Gaunt factor. With a notation similar to that of Spitzer (1968, p. 117) , a (k) = 2.065 x 10" 11 Z 2 T" 1/2 C k (T R 11-22 The function £ k (T) involves the asymptotic expansion. Spitzer (1968, p. 117) has tabulated £, and C 2 Tne y are pi otted as Figure 1 1 -3 . Dielectronic recombination occurs in atoms, which have two excitable electrons. This process is very important in establishing the high level populations of carbon atoms and is, in fact, responsible for these levels being over populated with respect to the LTE populations. It takes place in two steps, the first of which is a capture of an electron into a high level, n. The energy, rather than being radiated, is transferred to a bound electron, which rises to an excited level; n' << n. This is an unstable condition and the second step is to dispose of the energy of the lower excited level by means of a downward transition to a stable level. The energy difference is either radiated — leaving the atom with a single, highly excited level — or it is transferred to the highly excited electron which leaves the atom by autoionization, a process called the Auger effect (McDaniel, 1964, p. 590; Shore and Menzel , 1968, p. 93). The radiative process dominates (Dupree, 1969). The dielectronic

PAGE 41

33 E O 3 O O J2 o d) io 0) s+-> rrq; oi ro s> +-> l/> •iOJ O •a -c •>W dlrcn-i— i/i •r,— j= o. i— co rs <£> C_) +-> -M CO i— C C i— rt5 0) OJ O -i-r<_) l_> SS_ T-rQJ O 4Mtsl M_ (+_ 4_ +J > +-> O O -Q O -rTC +->-)-> -o 3 fB HI O U_ C C 4-> •r-r(O U£U3i —

PAGE 42

34 recombination mechanism is more effective at populating highly excited states than is the radiative recombination d r 4 mechanism; i.e., a_ > a , at least for T ^ 10 °K and n n e N > 1 cm" 3 (Dupree, 1969). e v r Departure Parameters, b and 6 — t 2 — n mn It will be shown later that the spectral line strengths depend not only on the departure coefficient, b , but — since the transition involves two b 's — it also n n depends on the derivative of b with respect to n. For r n that reason, when b is computed, a parameter related to the derivative is also computed; 11-23 where m > n. Departure parameters have been published by several workers. Information about this work is summarized in Table II-l. The first column gives the reference to the work. The parameters calculated by Dupree (1969) and Hoang-Binh and Wal msley ( 1 974 ) are for carbon. The second and third columns give the range of n over which the parameters are tabulated or plotted. Columns 4, 5 and 6 give the range parameters used in the calculations: n , m', and p' (cf. Equation 11-20). The seventh column indicates the method by which the collision cross-sections

PAGE 43

35 I v Iv I v l v |v I v I v co CD | V r— i— O 00 I V I V lv Iv CD i—. I v X o LO I lv iv ca c_> D_ i — CO *f CD «d" O CD 3

PAGE 44

36 were calculated. The reference to the source of the crosssections is given in the parentheses following the designation. Since the departure parameters are functions of the electron temperature and density, the last column of the table gives the range of these quantities for which the parameters were computed. All, except Hoang-Binh and Walmsley (1974), do not consider incident radiation in the calculations. Dupree (1972) does, however, show the effect of an adjacent H II region on a line-emitting region with a temperature of 20 °K. The radiation field from the H II region increases b si i ghtly for n > 80. n ? j The physical significance of departure from LTE can be illustrated by showing the relation which n, T , and N have on b and 6 . This is done with Figure 1 1 4 e n mn 4 3 in which are drawn curves for T = 10 and 7.5 x 10 °K e computed by Sejnowski and Hjellming (1969) and curves for T = 100 °K computed by Dupree (1972a). For each temper3-3 ature, there are separate curves for N = 10 and 10 cm r e As n decreases, collisions become less important in determining level populations. For small n, radiative processes dominate and the curves show an asymptotic be4 3 havior. The heavy portion of the 10 and 7.5 x 10 °K curves in Figure 1 1 4 is the radiative asymptote. Depopulation by stimulated emission is much less frequent

PAGE 45

37 260 Figure II-4. Departure Parameters for Hydrogen as a Function of n, T e , and N e . The curves with a solid line are for T e = 10 4o K and with a dashed line are for T e = 7.5 x 1 Jo K (Sejnowski and Hjellming, 1969). The dotted curve is for T e = 100°K from Dupree (1972a).

PAGE 46

38 than by spontaneous emission (Gordon, 1974), except when an intense radiation field is present. As N decreases, collisions become less frequent e and the radiative asymptote extends to larger n. Then, for hydrogen, there is a decrease in the rate of collisional excitation of lower n-levels which causes a decrease in level population with respect to the LTE values as n decreases. In the case of carbon, however, dielectronic recombination is efficient enough in populating upper levels that the population will exceed the LTE values for n > 20 and N < 10 cm" . In any case, the populations approach their LTE values as n + °° (Dupree, 1969). It should be noted that for hydrogen, b and 6 mn are both positive in the radio regime. For any two levels m and n, where m > n, b is closer to one than is b . In that sense, the upper level is "overpopulated" with respect to the lower level. If the slope of the b n curve is large enough, there is an enhancement of the line intensity over that obtained from a region in LTE even though the level population is less than its LTE value. This occurs because of stimulated emission induced by an exterior source or the region's own thermal radiation field. An examination of the radio recombination line absorption coefficient and non-LTE solutions of the transfer equation will show how this happens.

PAGE 47

39 Radio Recombination Line Absorption Coefficient The rate, R(v) at which a bound-bound absorption takes place between enrgy levels n and m (m > n) is equal to the product of the number of absorbers, N , the Einstein absorption probability, B , and the energy flux, I(v) r nm dn/4iT. The degree of absorption is modulated by the spectral line profile function, tj> . ( v ) . Then dQ. R(v) = N n ^(v) B nm I(v) ^ The energy per photon is hv . The absorption coefficient, i k, (v), is defined such that the energy absorbed along an increment of path length ds is f HO de A =-. L (v) I(v) dfl -N n * L (v) B nm I(v) hv nm ^ ds. However, energy may be emitted by stimulated emission of the amount dp, de, jj(v) du = N M * L (v) B mn I(v) hv mn ^ds and by spontaneous emission by the amount ds s = j s (v) d^ N m * L (v) A mn hv mn ^ ds

PAGE 48

40 \h (v) is the emission line profile, which is usually r mn identical to the absorption profile. Inserting de A , de j and de ^ in the transfer equation (Equation 1 1 2 ) gives Let <$>. ( v) = . (v) 5j L n nm m mn L 4tt B N hv , / \ R n mn m -rcf> (v ) B U 4tt nm V B nm N n Usinq q B = q B (Mihalas, 1970, p. 83), the Boltzmann y y n nm 3 m mn equation, and the departure coefficients, the absorption coefficient becomes M=%~ * L < v) B rm b n N n [1 -b ex P(^ /kT ex^ 11-24 For LTE, b = b = 1 and the LTE absorption coefficient is hv k l (v) = f
PAGE 49

41 The Einstein probability can be written in terms of the oscillator strength, f nm nm a 2 2 4 tt e hv Mc nm 11-26 In the above equation, e i s the el ec tron ' s charge; M is its mass. Oscillator strengths for hydrogen radio recombination lines have been computed by Goldwire (1968) and Menzel (1969) . Substituting Equation 11-26 into Equation 11-25 and assuming that hv << kT , the absorption equation be comes * h v it e * ex m N can be obtained from the Saha-Bol tzmann equation n (Equation A-5) and v can be obtained from an approxi2 3 mat ion to the Rydberg equation; v mn = 2c R H Z An/n . Using these and the appropriate constants (R., = 1.097 x 10 5 , Z = 1, u(T) = 1, and g p = 2n 2 ), the LTE absorption coefficient becomes k*(v) = 1-071 x 10 7 *(v) ff m N. N e r 2 5 exp(e n /k T e /k T ) 11-27 "X where e /k = 157890/n . The LTE optical depth is

PAGE 50

42 t*(v) = 1.071 x 10 7 db and m n

PAGE 51

43 mn d_b _ d(1n b) b„, dn A n , m = n which is a notation often used. Gordon (1971) and Hjellming and Gordon (1971) define a different 3 than that defined in Equation 11-29; k T mn (1 ex ) 1 k T Ht\T ex d 1 n b dn An This is given for reference only since the definition of B in Equation 11-29 will be used in the present work. mn Li ne Prof i 1 es There are three line profile f uncti ons, whi ch are relevant to radio recombination line analysis. They are the Gaussian, Lorentzian, and the convolution of these two, the Voigt function. The Gaussian function arises from the Doppler shift resulting from thermal, turbulent, expansion, and contraction motions of the line emitter relative to the observer. A Lorentzian profile for a radio recombination line results from pressure broadening. Natural broadening, or radiation damping, also produces a Lorentzian profile, but this is such a small effect that it is impossible to detect in the observed line profiles. If pressure broadening occurs, the effect would be blended with the Doppler broadening and a Voigt profile would result

PAGE 52

44 Doppler broadening . Mainly, thermal broadening will be considered. The thermal motion of particles in a gas with a kinetic temperature T., has a Maxwellian velocity distribution; so, the number of particles, out of a total of N, which have a radial velocity between v and v + dv i s d N(v 0/ 2 2 exp[-(v-v c ) /v J dv In this equation, v is the most probable velocity and v is the velocity of the ensemble as a unit. v = c o 1 / ? (2k T„/M) where M is the mass of the particles. The K intensity of the spectral line at a frequency v^ is pro portional to the number of emitters Doppl er-shi f ted to that frequency. The line profile is a Gaussian; 1 2 2 (v) = exp[-(v-v c ) /v ] dv o ,r^~ 11-31 The profile function is a probability distribution
PAGE 53

45 frequency. The peak line antenna temperature, T'(c), is that for which v = v . The temperature profile is T'(v) = T'(c) exp[-(v-v c ) 2 /v^]. 11-32 The integrated line power, which is the area under the profile, is P = I T 1 (v) dv = T' (c) v Q JH 11-33 Since v is not known, it is desired to substitute it with o a measurable quantity. When the line power drops to one half of its peak intensity, v = v, and T'( Vl ) = 1/2 ^-Vv -^ p exp[-( V v c ) 2 /v2]. O V IT V TT The Doppler half-width is defined by Av D = 2(v,-v ); then 8k T, AV D = 2v o T^TT {r-jf D In 2) 1/2 11-34 T n is the Doppler temperature which is the effective temperature of the gas if the line width were the result of thermal motion only. The integrated line power in units of °K km/s is P(°K km/s) <4-rir2» 1/2 t, ' c »«d

PAGE 54

46 However, P is usually given in units of °K kHz; so, (°K kHz) = (^-^) 1/2 (*£) T'(c) Av, 4 In 2 1 .0645 [-£) T' (c) Av Q . 11-35 In Equation 11-35, v is the rest frequency measured in kHz; the velocity of light, c, and the Doppler width are both measured in km/s. Likewise, the integrated optical depth , t , , is V t l (v) dv = 1.0645 t(v c )(^) Av d _ n _ 36 The Doppler temperature is given by M , n2 D " 8k In 2 ^ A V 11-37 Doppler broadening includes the effects of thermal motion by an amount Av T , turbulence of a scale smaller than the antenna beam (Av, , ), and expansion or contraction of the turb emitting region (av ). These velocity components combine to give a total Doppler width of 2 2 Av n = (Av T + Av a 2x1/2 i + Av turb x 11-31

PAGE 55

47 Pressure broadening and the Voigt profile . When the electron density is high enough that the time between collisions is shorter than the life-time of the excited atomic level, the energy distribution of the radiative transition is altered. The wavetrain is abruptly terminated by the passing electron. The line profile emitted by the assembly of atoms is a Lorentz function (Kuhn, 1962, p. 392). Peach (1972) has shown, however, that the profile is not an exact Lorentz function, but it is close enough for an accuracy of about 10 percent. The main effect of the Lorentzian is to redistribute the emitted energy to the wings of the line. Griem (1967) has shown that proton-atom collisions are not important. Brockelhurst and Seaton (1972) have determined that the pressure broadened half-power width for a m •* n transition is Av p = 9 4 W { ^r] Pressure broadening will occur in addition to Doppler broadening. If the gas has an electron temperature T and a Doppler temperature T n (Equation 11-37), the ratio of pressure to Doppler broadening is, by Brockelhurst and Seaton (1972),

PAGE 56

48 Av Av~ 0.142 — ) 100 ; 7.4 io 4 2xloV /2 JO 4 , " 1 11-39 To obtain appreciable pressure broadening of radio recombination lines, T D > 2 x 10 , and T g > 10 , and N g > 10 . So far, the only possible radio recombination line sources known to have such parameters are dense knots in emission nebulae such as M42 , W51 , and W3 (Kjellming and Davies, 1970) or dense H II regions associated with type I OH masers (Habing et al . , 1974). No pressure broadening has been seen, however (Gordon, 1974). When Doppler and pressure broadening occur together, the resulting profile is the convolution of a Lorentz with a Gaussian which is a Voigt profile, 4> ., ( v ) . The convolution integral cannot be evaluated exactly. So, the function is expressed in the form w(v) 1 \T ' Av D v/T H(a,oj) . 11-40 where H(a,w) is the Voigt function 2 H(a,co) = •y (or-y) 2 + a 2 dy and y = and a Av^ a

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49 The frequency dispersion, o, is given by o = v D 2 ( 1 n 2 = v D (— ) (In 2)~ 1/2 . Posener (1959) has tabulated the Voigt function for values of astronomical interest. 1/2 Non-LTE Solutions of the Transfer Equation and Line Enhancement Two solutions of the transfer equation will be developed here for use in the analysis of the spectra. They will be derived starting with Equation 1 1 -4. Since the frequency of the spectral line is high, the RayleighJeans approximation to the Planck function can be used to convert specific intensity to brightness temperature. From Equation 1 1 4 , the brightness temperature is L, T B (v) = T L (v) + T c (v) A(v) T e (l-e" L ) + f (1-e L ) T Q e 11-41 where t = t ( v) + t t (v ) . and j l (v) b m Since, from Equation 1 1 6 , A(v) * L S(v)/B(v,T e ) v,T ), e A(v K T + b K. ! m L 1 + b m (K L /K T ) : T + b 3 k. l + b 3 (< /< ) ' T n mn L n mn L T 11-42 Optically thin H I region adjacent to a strong source of radiation . Assuming that «,>> kj at the line

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50 center (i.e. a t v = v ) , a(v; b m < J V b B (k. / k t ) b 3 n inn v L' J n mn Since the region is optically thin, Equation 11-41 reduces to T n (vl b n e mn T e (t t ^ l ) + T n + T q (1-t t -t l ). When the frequency is off the line (i.e.v>>v c ) only the continuum is observed; T B< v >T c (v) -rTT . T T + V T o< lJt T> n mn Subtracting this equation from the preceding one gives only the line, T L (v) -T B (v) -T C (v) = b-f-T e T L -T o x L . n mn Since i L =b n B mn t*, T l (v)=t l (v) (b m T e -b n B mn T o ) 11-43 This is the line brightness temperature from an optically

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51 thin H I region for which < L >> < T The second term in Equation 11-43 will produce line enhancement if 3 mn is negative. To convert the brightness temperature to antenna temperature, the right side of the equation must be multiplied by the beam dilution factor, p g , and the main antenna beam efficiency, n g . These are defined by P B e si s /Q M and n B e Q M /^ A , where 9^ is the solid angle of the line-emitting region; q m is the solid angle of the antenna main beam; and fi ft is the total solid angle of the antenna beam. The antenna temperatures are denoted with a prime; so, Equation 11-43 becomes T L (v) Tin T l/ ( v ) ( b T r| B K v ; v m e b 3 T ) mn o 11-44 Isolated region. This solution of the transfer equation will be for the case of an isolated emitting region for which the incident radiation is weak and can be ignored. Then T = T M = and Equation 11-41 reduces J O IN to -(t,+t ) T D (v) A(v) T [1-e T ]. Off the line, A( v ) will be unity; so, the continuum temperature is T c (v) = T e (l-e T ).

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52 Taking A(v) from Equation 11-42 and converting Tg(v) to antenna temperature, T L (v) = T B (v) T" c (v) and T L (v) = P B n B T e J m L__ t t + b 3 T, T n nm L -(t.+t ) -t t [1-e L T ] (1-e '; 11-45 Line enhancement . Equations 11-44 and -45 will be used for the analysis of this study's observations. The equations indicate how non-LTE conditions affect the radio recombination line temperature. For LTE b n = b m 0; and Equation 11-44 becomes T. (v = t, (T -T L e o Fiqure 1 1 4 shows that as N decreases, collisions ° e become less frequent; so, the level populations and b^ decreases. As the departure from LTE increases, 6 mn becomes larger. When 6 > (hv/kT), 6 mn is negative and amplification of the line takes place. This process is often referred to as the "partial maser" process. The populations of the transition levels are not actually inverted as is apparently true for, say, H^O masers (Sullivan, 1973). The line enhancement can be seen from Equations II 44 and -45 in which t l = b n 3 mn t l is negative. Usually, •k t. << t t because the path length involved with t T is much larger than that involved in

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53 t. . Also, b is always positive. Therefore, if b L m in 3 t, /t t < 1, A(v) increases and the total absorption, mn L T I t, + t-t, decreases . The line enhancement will be lessened somewhat because b < 1. However, for a decrease in electron m 3 -3 density from 10 to 10 cm ',5 increases by more than a J mn factor of ten while b decreases by about 14 percent. The net result is a substantial increase in line power. Equation 11-44 also indicates the possibility that the recombination lines may be seen in absorption against a strong radio source if the relative magnitudes of b and 6 are such to make 3 positive. However even n mn mn for the case of a dense H II region (E. ^ 10 pc cm ) seen through a tenuous H II region, Brockelhurst and Seaton (1972) have shown graphically that absorption cannot be greater than 8 percent. So far, radio recombination lines have not been seen in absorption. The Radiation Field The radiation field intensity entering a recombination line-emission region, and used in Equation 11-20 to determine the level popul ations, i s computed from I Q (v) = W(v) B(v,T). Usually the radiation source is a nearby 4 nebula whose blackbody temperature is 10 °K. The dilution factor, W(v), is a product of three factors:,

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54 W(v) = W Q W A (v) A(v 11-46 W.(v) represents absorption of the radiation by the intervening medium between the source and the line emission region. If t is the intervening optical depth, w /\( v ) = e" T . If the nebula is adjacent to the line-emission region, W. (v ) = 1 . A(v) represents the departure of the radiation source from a blackbody. It is defined in Equation II-6. The geometrical dilution factor, Wg, can be found as follows. Consider a radiation source which is roughly spherical in shape with a diameter d g and located at a distance r R from the line-emitter. The geometry is shown in Figure 1 1 -5a . The line-emitting region sees the radiation source as a disc of angular diameter 2 and filling a solid angle Q it e R /4. Then tt d^/4 4 7T rl 16 r IK. 4tt ' 11-47 If the two regions are contiguous, r R 1/2 d g and W~ = 1/4. If, however, the radiation source is much b larger than is the line emitter, Figure 1 1 5 b applies and W„ = 1/2. Finally, if the source of radiation surrounds G the line emitter or if the line is emitted by a dense H II region, W g = 1 .

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55 LINE EMISSION REGION CONTINUUM SOURCE Figure 1 1 5 . Determination of the Geometrical Dilution Factor, Wq. Part a shows the line emitting region and the continuum source to be about the same size and r R > d s . Part b shows the line emitting region adjacent to a large continuum source.

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56 A numerical example will illustrate the case where r D > d /2. Consider the radiation of temperature K S T R entering a small line emitting region located 100 pc from an H II region, such as NGC 7000 (the North American Nebula), for which T = 10 °K and r = 25 pc (Kaplan and Pikelner, 1970, p. 152). For this case A(v) = 1. Assume that W.(v) = 1 also. Then, using Equation 11-47, it is found that T D = 131 °K. This indicates that, unless the line-emitting region is directly adjacent to a large H II region, the line-emitting region can be considered as being isolated. In the case where the line emitter is itself an H II region (T = 10 4 °K), Dyson (1969) shows that the effect of the free-free continuum on the departure parameters is very small. Dupree (1972) and Hoang-Binh and Walmsey (1974) demonstrate the effect of a large H II region adjacent to the 1 i neemi tti ng region. Ionization Rates Knowing the radiative recombination coefficient, a(T), the required rate of ionization per atom of species X, £v» can be determined from the condition of detailed A balance. For hydrogen, N H c H N e N H+ a(T) 11-48

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57 The radiative recombination coefficient may be obtained from Equation 11-22. Habing (1970) has a formula for computing c H which uses the assumption that a(T) a T C H = 1.27 x TO" 15 v^ 1 T"°3 t t (v t )/ J T 21 (V) dv 11-49 In this formula, t t (v t ) is the continuum optical depth at a frequency v y and t 2 ,(v) is the 21 cm line optical depth discussed in the next section. Ionization rates can be calculated theoretically from a given source of ionization such as low energy cosmic rays (Dalgarno and McCray, 1972) and OB stars. These rates can be compared to the ionization rate required by a model for a line emitting region. As an example the ionization rate within the Stromgren sphere surrounding an OB star will be considered Let S be the total number of ultraviolet photons with u energy greater than hv, emitted from the central star, v, is the frequency of the Lyman limit. Also let r<. be the Stromgren radius which is the radial distance from the star at which the ionizing radiation flux is zero. Then S u * J* r S N e V a(T)

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58 (Spitzer, 1968, p. 116). From Equation 11-48, a(T) = N.. Cm/M N..+ and the ionization rate of an OB star is H H' e H T 3S H 4 tt r 3 N 4 * r S H 11-50 The ultraviolet flux, S , from stars of spectral class 05 through Bl are tabulated by Spitzer (1968, p. 117). The 21 cm Hydrogen Line The well known 21 cm hydrogen line is emitted by a magnetic dipole transition between the hyperfine levels of the ground state. This is the result of the difference of energy levels between the proton and electron spins being parallel (upper state, F = 1) and antiparallel (lower state, F = 0). The frequency of the transition is 1420.405 MHz. The upper state is metastable; the -1 5 spontaneous emission probability is A,„ = 2.85 x 10 sec" 1 (Kaplan and Pikelner, 1970, p. 410). Usually the metastable state is depopulated by collisional excitation more frequently than it is by the 21 cm radiative transition. The F = 1 state is populated by atom-atom collisions (Kaplan and Pikelner, 1970, p. 420). Designating the lower and upper states by and 1 respectively, the absorption coefficient is

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59 hv 10 K io ( ^ = TT^ (v) [N o B oi " N i B io ] The parameters are analagous to those of the recombination line. Since collisions play an important part in populating the upper state, it will be assumed for the moment that the populations are LTE values as determined by the Boltzmann equation. The statistical weights of the states are g, = 3 and g Q = 1 . Then, since g Q B Q1 g ] B 1Q and hv-, Q /kT << 1 , 10 !v) hVl0 *t»\ hVl ° N B (v) FT" N o B oi s 4 TT T is the excitation temperature which, for the 21 cm s transition, is called the spin temperature because the transition results from a change of electron spin orientation. The absorption coefficient is expressed in terms of the spontaneous emission coefficient, A,q5 3 2h v 10 '10 10 Then 3N h c' o 10 (v) = o" t A ln <(.(v tt k T v iri '10 s 10 11-51 The absorption coefficient is used to obtain the neutral hydrogen column density along a line of sight.

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60 For a cold gas, the total neutral hydrogen density, N HO is N n + N, . From the Boltzmann equation, the number den sity at some point s in the line of sight is N H (s) = N Q (s) [1+3(1 -hv 1Q /k T s )] 4N Q (s since hv, n /k = 0.07. Substituting in values for the constants and integrating over the line profile and along the line of sight gives k 1q (v,s) ds dv =/ x 21 (v) dv = 2.58 x 10" ID I -^ ds : 11-52 where the path length, L, is in cm and v , Q is in Hz. If the 1 i ne prof i 1 e is Gaussian with a half-power w idthv and a peak optical depth t 9 , , Equation 11-36 can be used to obtain r L N u (s) 21 ,14 £ ds = 4.126 x 10'" t 21 Av D 11-53 It may be necesary to determine whether or not the spin and gas kinetic temperatures are equal. Starting with the equation of detailed balance,

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61 N l N H C 10 ( V + N l A 10 + N 1 B 10 ^lO'V l N H C Q1 (T K ) + N Q B Q1 I(v 10f T R ) Cm ( T„ ) and C, n (T„) are the collisional excitation and de(J I K I U In excitation probabilities, respectively, whose formulation is given by Kaplan and Pikelner (1970, p. 420). Solving the detailed balance equation for T in terms of T.. and the radiation temperature, T R , gives 557.3 T°5 N H + T R 557.3 T~ 0,5 N H + 1 11-54 T = T., if the hydrogen density is large enough that S K. 557.3 T°' 5 N H >> T R . For H I regions, T K > 64 °K and O N > 1 cm is not unreasonable (Baker and Burton, 1975); e so 557.3 T?' 5 N u > 4.4 x 10 3 °K which is much larger K n than one would expect T R to be. Therefore, unless N H << 1 cm" 3 , T T„. s l\ Absorption spectra of the 21 cm line have been obtained by three techniques. Hagen, Lilley, and McClain (1955) obtained absorption spectra in the direction of discrete sources by averaging the spectra from off-source beams and subtracting this from the on-source beam spectrum The accuracy to which this method yields the true absorption spectrum depends on how uniform the hydrogen gas is over the grid of beams.

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62 Radhakri shnan ej^ aj_. (1972a) and Hughes, Thompson, and Colvin (1971) have observed absorption spectra in the direction of radio sources (e.g. 3C391 ) with two-element interferometers. The front-ends and synchronus detectors are switched in such a way that the system is insensitive to all but the absorption spectrum in the direction of the source . A third method has been used to obtain absorption spectra in the direction of pulsars. Some of this work has been done by Gordon e_t aJL (1969), Manchester et a_l_. (1969) : Gordon and Gordon (1970), Guelin ejt a]_. (1969), and Encrenaz and Guelin (1970). The pulsar is observed with a gated, spectral line receiver which alternately stores and integrates the spectrum with the pulsar on and with it off. At the end of the integration period, the pulsaroff spectrum is subtracted from the pulsar-on spectrum. The difference of the two is the absorption spectrum. The pulsar is a particularly useful beacon because its dispersion measure provides a value of the integrated electron density along the line of sight. Therefore, two pieces of information about the intervening interstellar medium are known . Pulsar Dispersion Measure The arrival time of the pulsed radio emission from pulsars is frequency dependent. The pulse travels at the

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63 group velocity. The group index of refraction is a function of the electron density of the medium through which the pulse travels. The difference of arrival time, t, t ? of the respective frequencies v , and v „ is, then, the integrated electron density along the path length L. This is the dispersion measure, DM, which is calculated from the equation (cf. Terzian, 1972) DM e f N e ds = 2.4 x 10 -4 ( t -, 1 2 ) (v~ 2 -v~ 2 ) -1 11-55 From this equation, the unit of DM is pc cm -3

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CHAPTER III THE TELESCOPE SENSITIVITY AND THE OBSERVING SCHEME The observations, which will be discussed in the next chapter, were made in such a way as to make the most efficient use of the telescope. This chapter will discuss the basis for the observational scheme. The important factor was the system sensitivity which is affected by the microwave frontend of the receiver and the one-bit autocorrelation spectrometer. The general theory involved with the operation of a one-bit autocorrelation spectrometer for astronomical use was discussed in detail by Weinreb (1963). The operation of the instrument used for the observations is discussed in two reports. The IF filter system, which precedes the correlator and determines the width of the spectral window, is described by Mauzy (1968). The operation of the entire system, with emphasis on the digital autocorrel ator , is described by Shalloway et al . (1968). A discussion of system sensitivity for the different methods of observing radio spectra is presented by Ball (1972). 6 4

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65 System Sensitivity The nominal system sensitivity is the RMS noise temperature, AT, given by Y T AT = sys /Af t. III-l The parameters are as follows: Y = a constant which depends on the system and the method of its operation; T = the system noise temperature; sys Af = the frequency resolution bandwith; and t . = the integration time. 1 T is the combination of the noise temperatures sys received by the antenna as well as the power generated by the antenna-transmission line system and the receiver. The total bandwith, B, observed by the spectrometer is divided up into N channels. Both B and N are selectable. Each channel has the effect of measuring the spectral power within a filter whose bandpass is of the form (sin v)/v having a half-power width of Af = 1.21 J 1 1 1 2 (Shalloway e_t aj_. , 1968). The spacing between channel centers is B / N .

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66 The factor y in Equation III-l is greater than one and is the degradation of the system sensitivity due to the operation of the autocorrelator. It is a one-bit digital machine which computes the autocorrelation function, p(nAt), of the undetected IF power; nAt is the time delay at the n channel. The first step in obtaining the autocorrelation function is to clip the signal, a random Gaussian voltage, v(t). The clipper generates a function, y ( t ) ; y(t) = +Y if v(t) > and y(t) = -Y if v(t) < 0. The function y(t) is then sampled at a rate equal to 2B = 1/At. The output of the sampler is fed into an N-bit shift register and N one-bit multipliers. The shift register is updated at the time intervals of At. The output of each of the multipliers updates a counter. At the end of the integration time, each counter contains the sum which represents one point of the autocorrelation function. The function is in the form of an array which is transferred to the on-line computer which computes the power spectrum by taking the Fourier transform of it. The effect of using one-bit quantization is to increase AT above that which would result from a many-bit quantization (Weinreb, 1963, p. 32). This is accounted for in the constant y of Equation III-l. This constant is actually a product of two factors; y = a$. The effect of one-bit quantization is that B = 1.39 (Weinreb,

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67 1963, p. 49). The value of the other factor depends on the weighting function used in computing the correlation function. A uniform weighting function was used for which a = 1.099 (Weinreb, 1963, p. 22). Therefore, y = 1.528 and Equation III-l becomes 1 .528 T AT sys /Af t. 1 1 1 3 The Observing Scheme There are several observing schemes which can be used to observe spectral lines (cf. Ball, 1972). The observing scheme must provide a reference spectrum which is used to remove the effects of the receiver passband from the des i red spectra . The "Total Power" scheme can provide an efficient means of obtaining the reference spectrum. Before observing a source on the program, a spectrum is obtained while the antenna is pointed in a direction close to the program Burns and Yao (1969) have determined that the clipping factor, 3, is a function of the sampling rate. For a sampling rate of 2B, 3 = tt / 2 = 1.571. If this is correct, y = 1-726 and the RMS noise of the data discussed in Chapter IV should be multiplied by a factor of 1.13. However, the data reduction routines continue to use 3 = 1-39 (Cram, 1973, p. 133).

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68 source and where the spectral line is not detected. This spectrum is the receiver passband function. The nomenclature of the Total Power method is to designate the reference spectrum by "OFF" and a program spectrum by "ON." If the receiver passband remains unchanged while the OFF and ON spectra are observed, subtracting the OFF spectrum from the ON spectrum removes the effect of the passband from the desired spectrum. The efficiency of this observing scheme is realized if one OFF spectrum can be used with more than one ON spectrum if there are several sources in the observing program. Let the observing time on an OFF source be tr and that on an ON be t M . If an OFF source is followed by n IN ON sources, each of which is observed for a time t^, the total observing time, t , will be t Q = t p + n t [N| . The times t F and t N and the ratio n must be chosen so that the receiver passband remains unchanged during the time t . The most efficient use of observing time for a given o number of program sources will be obtained if the ratio a = tr/t., is chosen to minimize AT. r IN When an OFF spectrum with an RMS noise (AT) is subtracted from an ON spectrum with an RMS noise (AT) N the RMS noise of the resulting spectrum will be T = [(AT)jj + (AT)*] 1/2

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69 For spectral line observations, T is not expected to b y o change more than a few percent between the OFF and ON positions; then an average system temperature, T , can be used to get /Af v r N V/ Referring to Equation III-l, the integration time is t . i t N t F 1 1 1-4 Since t F = a t N and t = (a + n ) t N , AT ave 1/2 /Af t a + 1 + n + n/a The value of a is determined when T is a minimum, for which dun = o da ? -1/2 Then (1 -n/a ) (a + 1+n+n/a) ' = 0. The only reasonable solution is a = /7T . Therefore, the most efficient Total Power observing pattern exists for t F = /n t N 1 1 1 5

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CHAPTER IV THE RADIO RECOMBINATION LINE OBSERVATIONS Introd u ction The H157a radio recombination line spectra for this study were obtained between 18 and 24 April, 1973 with the 43 meter radio telescope of the National Radio Astronomy Observatory (NRAO) located in Green Bank, West Virginia. The rest frequency of the line is 1.683200 GHz. The H157a line was chosen for two reasons. It is conveniently located within the passband of the NRAO 18 cm cooled receiver which has good sensitivity and a reputation of being very stable. Also, the frequency of this line is such that the line strength is relatively strong. The total power observing scheme was used to obtain the spectra . The sources used for the observations are the two supernova remnants (SNR) 3C391 and 3C396 and the pulsar 1858+03. These were chosen on the basis of the objectives discussed in Chapter I; they are located in the galactic NRAO is operated by Associated Universities, Inc under contact with the National Science Foundation. 70

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71 ridge and independent data from the intervening interstellar medi urn are avai 1 abl e . A radio galaxy, 3C295 (Goldstein 1962), was also observed, but its galactic latitude is 61° and no radio recombination line was observed in its direction. The purpose of observing it was to demonstrate that the recombination lines came from the ridge gas and were not affected by instrumental effects. Along with 3C391 and 3C396, a region adjacent to each of them was also observed. These will be referred to by the designation COMP. The COMP spectra were observed for two reasons. The primary reason was to be able to subtract out emission from the region immediately surrounding each of the SNR's and, thereby, make these observations compatible with those made with narrow beams. The second reason for obtaining the COMP spectra was to determine the angular scale over which the medium changes. Table I V 1 lists the center position of each beam. The right ascension and declination (1950.0 epoch) and the galactic longitude and latitude are given for each of the regions studied. The OFF beams are the reference beams used for the Total Power observing scheme. Their selection and use will be discussed in the following secti on . Figure I V 1 shows the continuumradiation environment of the regions observed. The cont i nuum maps are adapted from the 1414 MHz maps of Altenhoff et aj_(1970). The

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0| -I 72 O O O i— ro I— Qra i — _

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O 4Q-O rO ^O) o _c i— .c +-> «* r— C Ul C rol 3 O d) -i+-> r,— E 4->| ro •i-iS4Q. -a U ro 4E rj

PAGE 82

74

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75 size of the circle at each region is about equal to the halfpower width of the antenna beam, 18 arc minutes at the H157a frequency. Observing Scheme and Data Reduction The receiver front-end was mounted at the prime focus of the 43 meter paraboloid. A scaler horn was located at the focal point of the reflector. The signals of linear orthogonal polarizations were separated behind the horn and fed into separate and identical receivers. The box containing the front-end equ i pment was rotated so that the E-plane of the horn was aligned normal to the celestial equater. Each receiver was load-switched and contained a two-stage parametric amplifier. The first stage of the parametric amplifier and the switched load were refrigerated. The local oscillator frequency was generated by a frequency synthesizer followed by a 6x frequency multiplier. The intermediate frequency (IF) sent to the control room from the front-end box was 150 MHz. The synthesizer frequency was chosen to compensate for the Doppler effects of all motions of the telescope with respect to the recombination line emitter and so that the velocities of the spectra would be with respect to the local standard of rest (LSR). Since it was expected that the recombination line emission regions and galactic H 1 regions are

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76 ki nemati cal ly related, the 21 cm emission velocity longitude maps of Westerhout (1969) were consulted to determine the velocity offset needed to positon the anticipated H157a and C157a spectral features in the 2.5 MHz spectrometer passband. The velocity offset for the pulsar was +50 km/s. All other program regions involved an off set of -10 km/s. The synthesizer frequencies were calculated by the NRAO computer program DOPSET before any observations were made. The 150 MHz IF from each receiver was monitored by a continuum reciever. The remaining signal was fed to the NRAO Model II autocorrelation spectrometer. The spectrometer creates 413 frequency channels which were divided into three sections. One section of 192 channels was connected to each of the front-end recei vers . The remaining 29channel section of the spectrometer was connected to one of the recievers to serve as a monitor. It was not used for the acquisition of the data. Each section of the spectrometer had a passband of 2.5 MHz. This allowed a channel separation of 13.0 kHz. Each channel has a sin v/v profile with a half-power width of 15.8 kHz. At a frequency of 1.6832 Ghz, the velocity resolution of each channel was 2.82 km/s. The operation of the spectrometer system is described by Shalloway e_t aj_. (1968) and the Total Power observing scheme is described in Chapter III. The autocorrelator

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77 periodically transfers the autocorrelation function array into the on-line computer which performs the Fourier transform and loads the resulting spectrum array onto a seventrack magnetic tape. This array, along with source coordinates and control information, constitutes a record. The control information contains a number designation of the spectrum and whether it is a spectrum of a program (ON) source or a reference (OFF) source, The identity of its OFF spectrum is also specified. Each source is observed for a length of time ( t F or t N as used in Chapter III) during which a sequence of records is placed on the tape. This sequence of records is treated as a unit during off-line analysis and is called a "scan." At the beginning of the off-line processing, the records of each scan are averaged together and the spectrum of each scan is loaded on another magnetic tape. Then, for each ON spectrum, its OFF spectrum is located and the antenna temperature spectrum is computed from T . m _ /S(i) R(ih T T L (i) ( ^ ) T sys , where S(i) and R(i) are the source (ON) and reference (OFF) spectra, respectively, as a function of the channel, i. The system temperature, T , is computed by the offline computer on the basis of a calibration noise source whose power is periodically injected into each receiver.

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The accuracy to which the system temperature is determined depends on how well the calibration temperature is known. During the observations the values determined during bench testing of the receiver were entered into the DDP-116 program. The calibration signal was injected into each receiver by a loop "antenna" placed in the waveguide at a point directly following the coupler which seperates the two polarizations. During the bench testing the horn was replaced by a termination. In spite of the careful matching of the termination to the system, its characteristics are somewhat different from those of the horn observing the sky. Consequently the amount of calibration noise power entering the system is not exactly the same in the two situations (Turner, 1973). The calibration temperature discrepancy was corrected by daily calibrations using the radio source Virgo A whose continum spectrum is well known. Once a more accurate calibration temperature was known, a correction factor was later applied in the off-line processing. The continum flux of Virgo A at 1.6832 GHz is 182 Jy which was determined by plotting the fluxes published by Kellerman e_t aj_. ( 1 969 ) and interpolating to this frequency. The continum receiver connected to the IF was used to measure the relative lev Is of Virgo A and the cold sky with and without the calibration signal. Table IV-2 shows the results of the daily calibration by which the system noise temperature,

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79 T » was also determined. The first calibration on 22 sys April gave unexpl ai nabl e bad results; so, the calibration was repeated later. The spectra obtained in the meantime showed no indication of a problem. The values obtained during this bad calibration were not used to compute the averages from which the correction factors were computed. Before the telescope was pointed toward any source, the receiver frontends were tuned so that the spectrometer passband was located in a portion of the reciever passband (25 MHz wide at the 3 db points) which was reasonably flat. The daily observing procedure began by peaking up on Virgo A in order to check the pointing accuracy of the telescope and obtain a correction factor for the calibration temperature. No problems with the pointing were found . Before observing the first program region, a fiveminute OFF scan followed by a five-minute ON scan of Ml 7 was made to check the operation of the spectrometer. The OFF regions for this and the program regions were at least four degrees above or below the galactic plane where it was certain that the H157a line was not present. The H175a spectrum of M17, an emission nebula, is very strong. A five-minute scan gave adequate indication that the spectrometer was operating properly. The daily M17 spectra were compared to look for degradation of the system. None was noticed.

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80 Table I V2 Calibration and System Noise Temperatures Date T D sys B Apri 1 18 62.2 19 66.4 20 64.5 21 61 .9 (22 a 176.9 22 60.4 2 3 84.5 Average 66.6 "Bench" val ues Correction factor: cal A 3.12 3.45 3. 35 3.02 8.57 3.21 4.40 3.42 3.23 0.975 T cal B 58

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Most of the observing program for 3C391 , its COMP, 3C396, and its COMP was performed with the Total Power scheme using a sixty-minute cycle; a twenty-minute OFF scan was followed by a ten-minute scan on each of the four program regions in succession. The ratio of OFFtime to ON-time was determined on the basis of Equation 1 1 1-5. To make the most efficient use of observing time, it is desirable to make the scans as long as possible so that the total time moving between regions is minimized. The maximum length of the cycle was limited by the length of time during which the receiver passband remained stable. During the observations it was found that the quotient spectra of the last few records of the sixty-minute cycle began to show a curvature of the baseline. This means that the receiver passband had changed enough since the OFF scan had been recorded that this OFF spectrum no longer represented the current passband well enough to produce a flat continuum on which the spectral features are enhanced. The curvature only affected the last few 3C396 COMP spectra; so, the cycle was maintained at sixty minutes. The observations of these four regions continued until the RMS noise, expected after combining all scans of each region, was about 0.002°K. Most of the last two days were used to observe the pulsar and 3C295. Each was observed individually, usually with a fifty-minute cycle

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82 consisting of a twenty-five minute OFF followed by a twenty-five minute ON. The 3C295 spectra showed no spectral features or abnormal baselines. Therefore, these data will not be discussed further in this study. After each day's observations had been completed, the seven-track "telescope" tape was taken to Charlottesville, Virginia where an IBM 360/55 computer was used to average the records of each scan and load the resulting spectra on a nine-track "user" tape for further analysis. At the same time a summary of records and the quotient spectrum of each scan was printed. These tasks were performed by the first phase of the TPOWER data reduction program (Cram, 1973). The printout was sent to the observatory the next day. With these "quick-look" summaries, it was possible to determine which scans had problems which would make them unusable. On the basis of these reports some scans were deleted and reobserved on the last day of the program. After all observations had been made, the final spectra were produced using the second phase of TPOWER, a system of routines controlled by a command language. It was during this process that the correction factor, determined from the daily noise tube calibration, was applied to adjust the antenna temperature scale. The scans were summed and weighted by the effective integrating time of each scan. Then the spectra of the two receivers were averaged together, thereby doubling the integrating time. The

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83 resulting spectra were convolved with a five-channel "boxcar" function and plotted. The plotted spectra were studied in order to estimate the structure of the baseline. Curvature of the baseline of each spectrum was removed with a TPOWER routine which first computes a specified order Chebyshev polynomial by means of a least squares fit to a specified portion of the baseline. The routine then subtracts the polynomial baseline from each spectrum and the spectra are then plotted in their final form. A final task was performed in order to facilitate the analysis of the spectra. Gaussian functions were fitted to the spectral features. The TPOWER routine begins with initial estimates of the center channel and halfintensity width of each feature and, from that, determines a bestfit center, halfintensity width, and central intensity of the features . The Spectra The basic spectra for the program regions are shown in Figures I V 2 , -3, and -4. Preceding the figures is Table I V3 which lists the total effective integration time (t.) and the order of the Chebyshev polynomial base line which was removed. The effective integration time was computed with Equation III-4. It includes the doubling obtained by averaging the spectra of the two receivers.

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84 The RMS noise, listed in the third column, is computed from Equation 1 1 1 3 ; 1 .528 T AT = All 3t i T is the average system temperature of both receivers sys and is 65.0 °K. The effective bandwidth, 8, is five times the channel separation of 13.0 kHz because the spectra are displayed after being convolved with a five channel boxcar function. Table I V 3 also provides a table of contents which includes a list of the figure numbers for all spectra including those showing the Gaussian fits to the spectral features . The lower portions of Figures IV5 through 10 show the results of fitting Gauss in functions through the spectral features. The Gaussian functions of the individual features are added together and the combined function is plotted through the original spectrum. The upper portions of these figures show the plots of the residuals which are obtained by subtracting the combined Gaussian function from the spectrum. The two possible features at -118.8 and 58.2 km/s on the COMP 3C396 spectrum are only about four times the RMS. Therefore, it was decided not to present Gaussian features fit to them. The center velocity, half-power width, peak antenna temperature and integrated line power of each feature in the

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85 --— S— s—

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86 spectra are listed in Table I V4 which follows the spectra Also listed for each parameter is its standard error (standard deviation of the mean) which, except for the standard error of the integrated line power, is computed by the Gaussian fitting routine. The standard deviation of the line power, s p , is computed from the following equation (Parratt 1961 , p. 117) with Equation 11-35: s p = [(gy) s T + 3P 9Av *l ] ,/2 Av J = 1.0645 [(Av) 2 s 2 + T 2 s 2 v ] 1/2 . Here, s T is the standard error of the peak line temperature and s, is the standard error of the line half-power width. Av The discussion of the spectra will be a part of the analysis of the emitting region. Chapter V will present the analysis of the spectrum from the directions of 3C391 and its COMP beam. The model derived there will be applied to the analysis of the medium in the direction of 3C396, its COMP beam, and the pulsar 1858+03.

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Figure IV-2. H157a Spectra from the Direction of 3C391 and Its COMP Beam. The velocity resolution is 11.6 km/s

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88 0.06 0.0 4 0.02 0.00 0.0 6 0.0 4 0.02 0.00

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Figure IV-3. H157a Spectra from the Direction of 3C396 and Its COMP Beam. The velocity resolution is 11.6 km/s.

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90 0.022 0.014 0.006 -0.002 0.022 0.014 O006 • -0.002

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o •rlO O) c: sz o 4-> -i+-> E =5 O .— to E S3 s>. +-> 4-> U -rO) o CL O U~l i— QJ 3 >

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92

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Figure IV-5. Gaussian Profiles Fitted to the 3C391 Spectrum with a Single Gaussian Through the Hi ghVel oci ty Feature. The lower curve is the fitted spectrum. The upper curve is thatof the residuals.

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94 0.06 0.04 0.02 0.00 0.06 0.04 0.02 • 0.00

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Figure IV-6. Gaussian Profiles Fitted to the 3C391 Spectrum with Two Gaussians Through the HighVel oci ty Feature. The lower curve is the fitted spectrum. The upper curve is that of the residuals.

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0.06 0.04 0.0 2 0.00 0.06 96 I < 1 1 1 1 1 I H 0.04 0.02 0.00 -170 -130 -90 -50 -10 30 70 110 150 v km/s

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Figure IV-7. Gaussian Profiles Fitted to the 3C391 COMP Spectrum with a Single Gaussian Through the Hi ghVel oci ty Feature. The lower curve is the fitted spectrum. The lower curve is that of the residuals.

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98 0.06 0-04 0.02 0.00 T' 0.0 6 0.04 0.02 0.00

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Figure IV-8. Gaussian Profiles Fitted to the 3C391 COMP Spectrum with Two Gaussians Through the High-Velocity Feature. The lower curve is the fitted spectrum. The upper curve is that of the residuals.

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100 0.06 0.04 0.02 0.00 T' o K 0.06 0.04 0.02 0.00 COMP 3C39I — r v^ — \pv — V v""w \ 170 -130 -90 -50 -10 30 70 110 150 v km/s

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Figure I V-9 . Gaussian Profiles Fitted to the 3C396 Spectrum. The lower curve is the fitted spectrum. The upper curve is that of the residuals.

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102 0.022 0.0 14 0.006 -0.002 •

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Figure IV-10. Two Gaussian Profiles Fitted to the Pulsar 1858+03 Spectrum. The lower curve is the fitted spectrum. The upper curve is that of the residuals

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104 0-0 1 I 0.007 0.003 -0.00 I T' "K 0.0 1 I 0.007 0.003 -0.001 1 1 i

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105 <3 E <] X I— O r— CO r— CO i — i— o CM CM CM CO CO LO LT> ud en «tfr— . — CM oi en co uo oo CO i — i — CO ID Ol OO CM CO o oi en cm oo a~> CM CO LO "3co o

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106 ? N <3 nz U"> CO <£> i— O r— C\J I CO o

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CHAPTER V MODELS FOR THE INTERSTELLAR MEDIUM IN THE DIRECTION OF 3C391 3C391 : A Summary of Previous Observations The radio source 3C391 has the nonthermal spectrum of a supernova remnant (SNR). Caswell et aj_. (1971) give its spectral index between 750 MHz and 5 GHz to be -0.57. However, there is a turnover of the spectrum and the spectral index between 80 and 408 MHz is about 0.13. Low frequency optical depths derived from radio recombination lines and also the continuum spectrum itself strongly indicate that the low frequency decrease in the flux is due to absorption by the interstellar gas rather than due to a process within the SNR itself (Bridle and Kesteven, 1971; Gordon, 1972; Dulk and Slee, 1972). Assuming that the absorption is due to the interstellar gas, the optical depth at 80 MHz may be determined by assuming that the flux from the SNR follows a power law in the frequency range between 80 MHz and 5 GHz. The spectral index is constant over this range and was found to be -0.50 using the fluxes measured at 408 MHz 5 GHz. The observed flux at 80 MHz, S(80) was compared to that expected 107

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from the power law, S (80). For 3C391 , t(80) In |°^1 = 0.84 (±0.3) The antenna beamwidth used was (axfi) = 3.7 x 4.8 arc minutes which is the size of the angle subtended by 3C391 . Caswell et aj_. (1971) compute this optical depth to be 1.3 assuming a spectral index of -0.57 between 80 MHz and 5 GHz. Goss (1972) makes use of the H109a recombination line power and the 5 GHz brightness temperature computed by Jackson and Kerr (1971) to compute t(80) for the direction 1 = 31?2, b = 0°. This position is 0?7 away from 3C391 , but the optical depth was assumed to be valid for the direction of 3C391 . For an electron temperature range of 10? to 10 4 , Goss finds that x(80) is close to one. From the observations tabulated by Caswell et al . (1971) it is evident that the size of 3C391 does not change appreciably with frequency. Their Mills Cross observations at 408 MHz give its size as (axfi) = 3.2 x 3.4 arc minutes. Their drift scan map shows the source to be "an intense compact feature within a weaker more extended structure." Their angular resolution is (ax5) = 2.06 x 3.47 arc minutes. Bridle and Kesteven (1971) detect the same

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109 structure with a 10.63 GHz map. Goss (1972) determined the size at 31.4 GHz to be 3.2 arc minutes in diameter; his resolution was 2.8 arc minutes. A map made by Chaisson (1974a) at 15.5 GHz with a resolution of 2.1 arc minutes shows an arc of structure which is generally surrounded by the lowest intensity isophote of the 10.63 GHz map of Bridle and Kesteven (1971) which they claim to be of low r e 1 i a b i 1 i ty . The distance to 3C391 can be estimated by two methods, from an empirical relation between radio surface brightness and from 21 cm absorption profiles. The 21 cm absorption profile obtained by Radhakri shnan et aj_. (1972) (shown in Figure V-l) has an absorption peak with a radial velocity of 107 km/s with respect to the local standard of rest (LSR). This is only slightly higher than the maximum velocity of 105 km/s predicted by the Schmidt rotation model for the galaxy (Schmidt, 1965) and observed by Burton (1970) at a galactic longitude of 31?9. Assuming pure circular galactic rotation of the gas, the maximum velocity corresponds to emission from the tangent point where the line of sight at 31?9 intersects the 5.3 kpc (radius) orbit. Also, it is assumed that the sun is 10 kpc from the galactic center. The distance from the sun to this point is 8.5 kpc. Since the 21 cm line is seen in absorption against the SNR at this velocity, it can be concluded that the SNR

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no lies no nearer than 8.5 kpc. Because of the distance ambiguity of the radial velocity-distance relation (cf. Mihalas and Routly, 1968, p. 126), it cannot be determined with certainty whether the other absorption features lie in front of the tangent point or behind it. Therefore, all that can be said is that the distance to 3C391 is greater than 8.5 kpc and may be as close as 13.4 kpc. Ilovaisky and Lequeux (1972) derive a distance-luminosity function for galactic supernova remnants. They use 3C391 as a calibrator for which the distance was assumed to be 11 ± 2 kpc. There is considerable disagreement as to the nature of the interstellar medium in the direction of 3C391 . The intent of this study is to provide another data point which can be used with previous studies to generate a consistent model. Of particular interest are the recombination line observations of Cesarsky and Cesarsky (1973a, b) who observed wide (^30 km/s) H92a, H157a, and H166a lines in the direction of 3C391 . Using their H92a line and x(80) measured by Dulk and Slee (1972), they employ a unique technique to derive an electron temperature of 20°K with the gas in non-LTE. Chaisson (1974a) takes issue with this solution and presents an argument in support of the low frequency absorption resulting mainly from an H II region imbedded in the two-phase medium and the remaining absorption from the partially ionized cool component of the medium.

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Ill The wide recombination lines are indicative of thermal broadening of a hot emitter. For that reason Chaisson suggests that the recombination lines are emitted by the H II regions from which most of the absorption results. Downes and Wilson (1974) also conclude that the line emission can come from H II regions rather than from H I regions. The structure seen in Chaisson's map could possibly be evidence for there being one or more H II regions in the intervening medium. However, it is not certain that the arc of sources which surrounds the east, south, and west of 3C391 exist in the medium. Since they are included within an isophote of Bridle and Kesteven's (1971) map, the sources form a structure within 3C391 itself. Also, both Chaisson (1974a) and Bridle and Kesteven (1971) find an increase i n the continuum flux at high frequencies above 10 GHz over that expected from the spectral index at lower frequencies. This excess flux, they suggest, could be due to at least one H II region imbedded in the intervening medium. Goss (1972) does not, however, see any evidence for an excess at 31.4 GHz and suggests that the low frequency absorption and radio recombination lines are both due to a diffuse medium with a temperature ranging from 10 2 to 10 4 °K. Figure V-l shows the 21 cm absorption and emission spectra obtained by Radhakri shnan e_t al_. (1972) placed

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U 1IB] SCU QJ +-> O) ro t*— Q. U +->|x: OJ C Z3 <3" 1/1 QJ 0)|+1 Q ro O OJ i — CTi CO) S4 CO £ ro +J T3 TD OUCfOr->,Or— OO SZ (_> 3 -C O QJ . — to >rO -^ CO C\J -iTD 2 QJ 4U -o x: ro =3 I— -C +-> -o . CC CO ro rS~ £ x: +-> o I— o s_ C O O T3 • •r+->•!C 1/1 I C QJ O O +J o SX2 re) r— QJ 4XI rC QJ QJ -i— XT 4a. owo U S-C Q +-> ,r-

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113

PAGE 122

114 above this study's H157a spectrum. Figure V-2 shows the recombination line spectra which the Cesarskys (1973a, b) observe placed above that of this study's. The Cesarskys' H147a and H166a spectra show a possible feature at a velocity of -45 km/s which they suggest may be real and may be the C157a line. The velocities in both figures are with respect to the local standard of rest. On this study's spectrum, the horizontal bar labeled with a C indicates the range of velocities in which the C157a line would be if observed, assuming that it is colocated with one or more of the hydrogen line emission regions. The vertical arrows labeled with CO indicate where the CO features, seen by the Dickels (1974), are located. Downes and Wilson (1974) looked for but did not detect H134a and C134a recombination lines. Their signalto-noise ratio is such that the H134a peak temperature is less than 0.02 °K and that of the C134a is less than 0.03 °K. Since T|_ a 1/n, these limits are close to the expected H134a peak temperature and probably above the expected C134a peak temperature. The work which has been discussed is summarized in Table V-l. The conclusion which one might draw from this discussion is that there are three possible models of the interstellar medium — at least in the direction of 3C391 — which must be considered: a group of cold, partially ionized H I regions, one or more H II nebulae, or an

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Figure V-2. Comparison of Radio Recombination Line Spectra from the Direction of 3C391. The upper two spectra, H166a and H157a, are from Cesarsky and Cesarsky (1973a). The H92a spectrum is from Cesarsky and Cesarsky (1973b). The bottom spectrum is the H157a spectrum of the present study. All velocities are with respect to the local standard of rest .

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116 0.04 0.05 150 -100 -50 50 100 v

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117 C !-

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118 O -!->

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119 extended diffuse H II region. The remainder of this chapter will discuss this study's observations in terms of parameters consistent with these models. The Models The D-Method Models This is the method used by the Cesarskys (1973b) in obtaining a model from their H92a recombination line spectrum. They began by assuming a transfer equation solution suitable for an H I region observed in front of a strong continuum source. From Equation 11-43, which assumes that i^ and tj<<1 , the line temperature is T, (v) = x*(v) (b T -b 6 T ) . L v ' L v ' v m e n mn R ; The radiation temperature is the sum of several terms: T R = T T +T N +T SNR +2 7 T$nr 1S the radiation temperature of 3C391 as experienced by the line-emitting region and Tj and T R are the respective temperatures of the thermal and nonthermal ambient radiation fields excl udi ng that of 3C391 . The 2.7 °K term represents the universal thermal background radiation The line power emitted along a unit pathlength, As, is

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120 AP(v) = k l (b m T e -b n 3 nm T R )As V-l The line absorption coefficient (Equation 11-34) is k* = f K *(v)dv = fl.071xl0 7 (v)Ajl f nm Tg 2 5 exp(e n /kT)Ni N e dv, Line Line where Nis the number density of the recombining ion (hydrogen). The thermal continuum absorption coefficient at 80 MHz along As is found by combining Equations 1 1 7 with the defintion of optical depth; At(80) = 4.710T" 1 5 1 n (0. 6194 Tj5 )As. V-2 Dividing Equations 11-28 and V-l by t(80) and Ar(80) respectively provides two functions, D and D Q , defined as fol lows : n D P, o " t t (80 so , a a (b T b S T D ) exp(e n /kT D = 2.274xl0 4 n R Mf 1-n.e n mn R_ . i^_ZL_ o B n mn ) E. ln(0.6194 T^5 ) V-3 and

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so , At(80) 121 a a (b T b 3 T n ) exp(c /kT) As. D = 2.274xl0 4 n R ^f iiSLS n mn R' ^ n L . *" L , V -4 ln(0.6194 Tg J ) AS T The antenna main beam efficiency, n B > is the ratio of main beam solid angle to total antenna solid angle (Kraus, 1966, p. 73). Neglecting the contribution of carbon to the electron pool, the Cesarskys use • j N j H H + + N He + 1 .1 N H+ ' so, E T = 1.1 E L if the absorption and line emission media are colocated and their respective path lengths, L T and L,, are equal. If C + is present, the abundance is N C =N C + 4xl0" 4 N H (Spitzer, 1968, p. 122). Then ZZ? Nj« 1.1N H is a good approximation only if the fractional ionization, x = N /N , is high. N u is the total hydrogen density; e H H 'H+ N H N H0 H N, This technique computes D for models specified by choices of T R , N e , and T e . For each model, curves of D(T e ) for specific values of N and T R will be computed and compared to the observed quantity, D Q , in such a way that

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122 N and T can be specified. The estimate of T n will remain e e K R the same for all models. There are two problems, however. First, there is one "known," D Q , and three "unknowns," T R , N , and T . This is complicated by the fact that b n and 8 are functions of these variables and, particularly mn r at low temperatures (T < 1 00 °K) , this functional relationship is not well known. There is controversy regarding the bound-bound collision cross-sections at these temperatures which correspond to kinetic energies near the reaction threshold (Percival and Seaton, 1972; Dupree, 1972b). N e is not completely eliminated in Equation V-4 since b n and 3 m n are nonanalytic functions of it. The second problem involves the rel ation between D Q and D. D is a ratio of integrated quantities; D is a ratio of differentials. The Cesarskys (1975b) show that, for the H92oc line, two extreme conditions provide limits for D. D(H92a) will be largest for an LTE solution for which T e = 20 °K. Since for this case b n = B mn = l , N e does not have to be specified and does not enter into the solution. The conditions for which D will be a minimum are a non-LTE situation where T e =0 °K and N e <10" 3 cm -3 . At these electron densities collisions need not be considered and the departure parameters are better known. These conditions dictate that 6
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123 The Cesarskys' spectral line gives D Q =6.8 °K kHz from which they conclude that the most probable temperature of the emitter is 20 °K with a possible range of temperature of 0L=4 pc cm" 6 . There is another matter which should be considered. In a two-phase medium, the line emission from each phase must be considered. In an earlier paper, the Cesarskys (1971) showed that for a two-phase medium like those proposed by Field, Goldsmith, and Habing (1969) and Hjellming, Gordon, and Gordon (1969), the line emission will come predominantly from the cold regions. The fraction of the line power from the hot region increases as T g decreases because T L <* T" 2 5 N 2 . However, emission from the hot component will generate a weak and wide (^80 km/s) line because of the temperature dependence and the fact that this gas is widely distributed so that there is line broadening resulting from differential galactic rotation along the line of sight. Consequently the hot region spectral feature will be difficult to identify because it will be immersed in the baseline noise or be hidden by one or more features from the cool regions. For these reasons, the Cesarskys (1975b) consider emission only from a cool region .

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124 It should be noted that the antenna beam used to obtain this study's spectra is 18 arc minutes in diameter which is considerably larger than Cesarskys' beam (2.3 arc minutes) and 3C391 (3.5 arc minutes). It will be assumed that t(80) is constant within the 18 arc minute beam and, therefore, the D-method can be used for this study. The comparison of the high-velocity features of the 3C391 and its COMP spectra (Figure IV-2) shows a variation of line power by a factor of 2.9 over a distance of 17 arc minutes. The 42 km/s feature varies oppositely by a factor of 1.2 and is not seen in the H92a beam which is 2.3 arc minutes wide. Since a high-velocity feature appears on both the 3C391 and its COMP beams, it seems to originate from an extended region. Using this feature only, a solution consistent with the H92a observations may be possible using the D-method. However, considering that the solution is expected to yield a low temperature, it is relevant to consider the 21 cm hi ghveloci ty absorption features. There are two prominent features, one at about 80 km/s and another at 107 km/s. This raises the question as to whether the H157ct high-velocity feature is single, centered on 99.82 km/s and having a half-power width of 35.7 km/s or is double with components at 80.66 km/s (Av = 18.4 km/s) and 102.8 km/s (Av = 29.2 km/s). It is not certain whether or

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125 not this feature is double on the H92a spectrum. There may be a component at about 80 km/s, but its peak is no more than two times the RMS noise. The Cesarskys 1 (1973a) H166a and H157a spectra show some structure other than the 103 km/s feature which has an intensity roughly three times the RMS. Taking each spectrum by itself, it is not clear that there should be a feature around 80 km/s. Taken together, the spectra indicate a strong possibility of there being a feature there. To see what effect the choice of situation has on the solution, both possibilities will be considered. Assuming that the feature is single and using t(80) from Dulk and Slee (1972) , D o (H157a) = o.ll UO.If = 1M4 (±2 ' 89) ° K kHz Taking only the 103 km/s feature if a double feature exists, D o (H157a) = HI (±o:3) } = 9 37 (±2 ' 37 > ° K kHz ' Assuming that the uncertainty quoted for t(80) is one standard deviation, the standard deviation of D Q is computed by

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126 'D P 2 s 2 + t(80) 2 s 2 T P 1/2 (Parratt, 1961 , p. 117) . For the H157a line, Equation V-4 becomes D = 3.83X10 5 (b m T e b n 3 mn T R ) exp(6.4059/T e ) ^ E ± T In (0.6194 T^ 5 ) E T V-5 T and T n are brightness temperatures. D and D are in e K J r Q terms of antenna temperature; therefore, the constant multiplier in Equation V-5 contains the factor hb = °8 > the beam efficiency of the 43-meter NRAO telescope at the H157a frequency (Gottesman and Gordon, 1971). T R may be computed from the transfer equation solved for the situation shown in Figure V-3. If the gas between 3C391 and the line emitter has a kinetic temperature T e and continuum optical depth tj, then T R = T e (1-eT) + W T SNR + T N + 2.7 V-6 The dilution factor, W, is given by l d2 U A(v) -i|4 ex P (-T T ) R R V-7 (cf. Equation 11-46)

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127 LINE EMITTING REGION OBSERVER Figure V-3. Geometry for Determining T R . This is a schematic drawing and is not to scale.

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128 Ilovaisky and Lequeux (1972) determine the diameter of 3C391 to be 11 pc. For this discussion, it can be assumed that Tq NR = 10 4 °k and the gas between 3C391 and the line emitter has the characteristics of the intercloud medium of the Fi eld, Gol dsmi th , and Habing (1969) two-phase model; then, T e 7500 °K and N e = 0.012 cm" 3 . Also it will be assumed that N e = N H+ , E T = 1.1R R = 1 . 6x1 0" 4 R R . Since 3C391 is between 8.5 and 13.4 kpc from the sun (Caswell et a]_. , 1971), the range on R R is 0
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129 At the line emitter 3C391 subtends a solid angle of 2 2 n+ which is equal to ird /4 R R . Then WT 2 s c n, c S ' SNR 2kv 2 Q S 4^ The following table shows that T R ^ T N + 2 7 except when the line emitter is adjacent to the SNR. The optical depth is computed from Equation II-8. Since the optical depth is so small, T D is not attenuated and is equal to the observed *R [pc] Table V-2 Calculation of Radiation Temperature L T Lpc cm J 5.5

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130 o

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131 Figure V-4 shows the plot of Equation V-5 with all of the parameters which have been drived inthe discussion. Two extreme cases are plotted. The solid curve is the LTE function for which b> n = b m mn 1 . The dashed curve i s -4 -3 the col 1 i si onless non-LTE case where N e <10 cm . The nonLTE departure parameters were computed by Dupree (1972). Their values are listed in Table V-3. Table V-3 Non-LTE Departure Parameters with N e < 1 [from Dupree (1972)] T e [o K ] b 3 n mn 10 20 100 1000 0.0583 0. 1150 0.2720 0. 5350 -0.0438 -0. 1442 -1.0414 -8.7328 The values of D , 9.37±2.37 and 11.44±2.89, are indicated on the plot by labeled horizontal lines with error bars. Considering both values of D , it is seen that the solution is an LTE gas with a temperature of 1 4 . 3±0 . 3 °K. The solution disagrees markedly in one respect from that of the Cesarskys 1 (1973b) which is a non-LTE solution with a most-probable temperature of 20 C K. To obtain this solution for this study's spectrum, D Q would have to be 25.3 °K kHz, a value well outside of the error quoted. The value of D Q (H157a) which the Cesarskys predict is about 25 °K kHz.

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132 If t(80) = 1.3, as derived by Caswell et al_. (1971), the H157a solution would be closer to T R , but the H92a solution would yield a temperature of less than 10 °K. This is evident from Cesarskys' (1973b) Figure 3. Recent unpublished observations suggest that t(80) may be close to 0.4 (Chaisson, 1974b). If this is the correct value, D (H157a) will be about 24.2 °K kHz which will give a nonLTE solution with a temperature of about 60 °K if P is 7.87 °K kHz and an indeterminate solution if P is 9.61 °K kHz. However, with x(80) = 0.4, D Q (H92a) will be 14.2 °K kHz giving a non-LTE solution with a temperature of about 1000 °K. Even though the surrounding hot medium will generate a wide, shallow line, it is necessary to determine what effect this component will have on the solution. For this purpose, Equation V-5 can be modified to n AP(c) + AP(h) v q U At (80, c) + At (80, h) ' where, letting x equal to c for the cold component and h for the hot component, AP(x) 1.84X10 6 r 2 5 exp(6.4049/T x ).(b m T -b S^, T r )aE Lx and At(80,x) 4.710 T" 1,5 ln(0.6194 T 1 ' 5 ) AE,. X XIX

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133 Because of the different number densities in, and vastly different path lengths through, the two regions, it cannot be assumed that their emission measures will be equal. Therefore, they will not cancel and at least two more variables are added to the solution. To solve equation V-9, four combinations of emission measures will be chosen. For each combination Ej = 1.1 E , for both components. The cold region non-LTE departure parameters are those computed by Dupree (1972) and listed in Table V-3. For the surrounding hot region, T e will be _3 7500 °K and N is assumed to be near 0.01 cm for the purpose of computing the departure parameters. The values of T and N correspond to those of the stable, hot phase of the two-phase model of Field, Goldsmith, and Habing (1969). _ 3 Also, N = 0.01 cm is analogous to the col 1 i s ionless case for which Dupree (1972) computed the "N e = 0" parameters. Sejnowski and Hjellming (1969) have computed departure parameters for T = 7500 °K, but in order to obtain them for _3 N = 0.01 cm , they must be estimated by extrapolating the radiative asymptote through n = 157 (cf. Figure II-5). With this extrapolation, the departure parameters for the _3 non-LTE hot medium are b = 0.75 and 6 = 10 , which makes m mn b 3 = -68. 95. n mn For each of the models examined, D(H157a, Te(c)) will be computed for each of the possibilities:

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134 (a) cold region in LTE, hot region in non-LTE (b) cold region in LTE, hot region in LTE (c) cold region in non-LTE, hot region in non-LTE (d) cold region in non-LTE, hot region in LTE. In each of these cases, the continuum emission measure is assumed to be related to the electron density by E T (c) = N e (c) 2 p L L and E T (h) N e (h) 2 (1 p L ) L, where L is the total line-of-sight path length through both media and p. is the line-of-sight filling factor of the cold regions. L will be assumed to be equal to the distance to 3C391 although this may not be a good assumption since the electron density may be appreciably different in the ridge gas than it is in the solar neighborhood. From Figure I V 2 it is seen that a portion of the line of sight does not lie within the ridge. It is not clear what the ridge boundary is like and how much N Q differs in and out of the ridge. Pulsar e 3 data seem to indicate that, in the solar neighborhood, N = _3 0.03 cm (Guelin, 1974). The error in choosing L = 8.5 kpc should have little effect on the solution if it is shown that the hot region does not contribute much to the recombination line power. Model 1 is essentially the same as that previously examined; it is based on the Cesarskys' (1973b) model for a cold region emitter: E> (c) = 4 pc cm" and T e = 20 °K. Model 2 is based on Model 1 but with E.(c) reduced by a factor of 10 to see what the effect of doing so would be. The electron density remains the same and the path

PAGE 143

135 length through the cold region becomes 85 pc. If the region is spherical, it would subtend an angle of 34 arc minutes. This is a reasonable size because the line strength observed by the COMP beam is about one-third of that observed by the 3C391 beam. This suggests that the size of the emitting region may be about 34 arc minutes in diameter. Model 3 is based entirely on the results of Field, Goldsmith, and Habing (1969). A filling factor of P[ _ = 0.006 is used for this analysis which falls into the range they suggest. The cold region emission measure is based on the information given in their Figures 1 and 2. Model 4 was inspired by the results of Chaisson's (1972) recombination line observations in the direction of W3A, and H II region located at 1 = 134° and b = 0°. His spectrum indicates the presence of hydrogen, helium, carbon, and a blend of heavy element recombination lines. The residuals he obtains by removing Gaussian features from his spectrum suggest the possibility of an additional weak, narrow component of the prominent hydrogen feature. He concludes that the weak, narrow feature originates from a cold region where the heavy elements (Mg, Si, S, and Fe) are located. The prominent feature, then, originates from the H II region. Using the H I and carbon lines and assuming the abundances given by Spitzer (1968, p. 122), he obtains T e = 100 °K, N R = 10 cm" 3 , N R+ = 1.8 (±1.4)xl0~ 4 cm" 3 and -3 -3 N = 7 ( ± 5 ) x 1 cm for the cold region. This gives most

PAGE 144

136 -5 probable values of N„N Ui = 1.26x10 cm and N I Z K e H+ e l 'e"H 5 -6 e j j J 1.1N N Ui = 1.39x10 cm . Chaisson does not suggest a e H + value for p. , but a value of 0.01 will be used here because of the argument used for Model 2 . Figures V-5, 6, 7, and 8 show the D-curves plotted for each model and combination of thermodynamic condition. The parameters used for these are summarized in Table V-4. D was computed with a WATFIVE computer program. The LTE cold curves, labeled a and b in each figure, are asymptotic at T = T R = 13.4 °K. As the plots show, neither D Q = 9 . 37 (±2 . 37 ) °K kHz nor D Q 11.44 (±2.89) °K kHz is large enough to obtain a non-LTE solution for the cold region. The result, therefore, remains the same as that for Figure V-4. The hot region has no effect on the solution and the error in assuming that L = 8.5 kpc is of no consequence. The RMS electron densities shown in Table V-3 are computed from ^ 2 E T (x)/s x 1/2 where s x is either s c = p.L or s n = C ~ P L ) L Table V-3 also shows the fraction, in percent, of the line power which originates from each of the two phases. It is apparent that the hot medium has little effect on the line power except when E (c) << E(h) .

PAGE 145

137 >— CO I cnj a> £ o

PAGE 146

*+O l/l O 4, o sa» •.^ ^ +-> +-> O rO O) CZ S-C U_ QLU +-> C UJ I— O O I— C _l r— I— o o o u u a -a ca aj o o I— I— to E t—

PAGE 147

139 o CO

PAGE 148

4O CD O 4-^ C dJ 0) o s.c hO O (— OH-DTJ (D tt) CD I— -o c/1 O n3 CU -a t(DT5 Ul •• P 0) l» W 4-> s: -c 3 o a. o i — -a i — o_ — o i— O 2 O Q O (— 4O O I— I— I— I— O O _1 _l £Z C

PAGE 149

141

PAGE 150

CD (J <_>
PAGE 151

143

PAGE 152

UJ -P C UJ I— o o t— i CUJ " OH"D X) C _J r— i— O O " »u u -o -a i — i— UJ UJ o o (— t— I— I— o o •O'r m s

PAGE 153

145 i i iw

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146 Two matters must be considered. First, there is the question of the validity of the cold region departure < -4 parameters. They were chosen for the case where N ^ 10 Because of the proximity of the neighboring curves, there is some tolerance; that is, N e can be somewhat larger without affecting the departure from LTE. The amount of tolerance depends on n and T . At n = 157 and T g = 10 °K, the electron -3 -3 density of the medium should not exceed 10 cm . If T = _3 1000 °K, the electron density could be as large as 0.1 cm The error from using col 1 i s i onl ess values for a medium where _3 N is actually about 0.1 cm — as was done for Models 1, 2, and 3 — is probably not large. Percival and Seaton (1972) claim that there are large uncertainties in the bound-bound collision cross-sections. For that reason the Cesarskys (1973b) estimate that the col 1 i s ionl es s condition may be valid at electron densities which are 60 times that expected from Dupree's (1972) curves. If the D-curve for N g = 0.1 _3 cm were used, it would fall between curves a and c because collisions will be more frequent and bring the level populations closer to their LTE values. Consequently, the D -lines will not intersect the non-LTE curves and the solution is not affected. Finally, there is the question as to the use of the D-method for the H157a spectrum since the antenna beam is much larger than 3C391 . As mentioned before, Figure 1 of

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147 Cesarsky and Cesarsky (1973b) predicts that D obtained with an antenna beam 2.3 arc minutes in diameter should be 25 °K kHz. The fact that the 18 arc minute beam yields a much smaller D may indicate clumping or small scale structure within the emitting region seen by the antenna beam. Because of beam dilution the antenna power is less than it would be if the line emitting region filled the beam. Since the D-method gives uncertain results, another approach will be taken . Isolated Cold Region Models Since the background radiation is quite small, a solution to the transfer equation can be used which is applicable to an isolated region. The solution will be left in terms of line power in order to avoid the use of a parameter with a large uncertainty such as x(80). A solution which is consistent between the H92a and the H157a spectra will be found by adjusting emission measures until both observed line powers are duplicated. By doing this, both T and E, can be determined if there is a consistent solution From Equation 11-45, the peak line antenna tempera t u r e is T L (v c ) = P B n B T e A(vJ (1 e" (T L +T T)) = (i eT T) V-10

PAGE 156

148 where , A(v, VVl V T T 1+b t. /t t 1+b B n T,/T T n m n L T Both optical depths are peak values (i.e., at v v c ) and * * t, = b 3 T. . The optical depths t, and t t are computed L n mn L r L ' from Equations 11-28 and 1 1 8 respectively. Let « L> « M , and n, be the solid angles of the line emitting region, antenna main beam, and total antenna beam respectively. Then p b = ^J% and n B = VV For these models 1t is assumed that the line emitting region fills the H157a and H92a beams. So, in both cases p B = 1. It is also assumed that E T = 1.1 E.. The integrated line power, P, in °k kHz, is computed by the use of Equation V-10 and Equation 11-35; P 1.0645 T L (v c )~ Av D In computing the Hl57a line power, a single highvelocity Gaussian feature with full half-power width of 35.73 km/s (Av D 200.6 kHz) is used. The expected H157a and H92a line powers are computed for a range of emission measures. Both LTE and col 1 i s i onl es s non-LTE functions are computed. The departure parameters used are those computed by Dupree (1972), the same as those used for the D-method and tabulated in Table V-3.

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149 A consistent solution between this study's spectrum and the H92a spectrum is found graphically. Figure V-9 shows non-LTE curves and Figure V-10 shows the LTE curves. Each curve is the integrated line power, P, plotted as a function of electron temperature and is labeled by the value of the emission measure used in computing it. In both figures, the H157acurves are placed above those for H92a and the horizontal lines represent the respective observed line powers: P Q (H92a) = 5.7 (±2.4) °K kHz and P (H157a) = 9.61 (+0.26, -1.27) °K kHz. The error with P Q (157a) is determined by considering the possibility of the feature being double. The error bar for both P -lines is represented in the figures by shaded strips. The intersection of the P -lines with each of the o respective curves in Figures V-9 and 10 is plotted in Figure V-ll. Each of the four curves in Figure V-ll is, then, the locus of points which relate the temperature and emission measure required to produce the observed line power. Error strips are not shown. An intersection of the respective LTE or non-LTE loci defines a consistent solution. The LTE curves do not intersect; therefore, the emitting medium is not in LTE. The non-LTE loci do, however, intersect. The angle between them is small, making the solution difficult to identify. The most probable values are T e = 53 (±6) °K and E L = 9.9 (±0.6) pc cm" 6 . This solution does not involve the errors in either locus. The

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Figure V-9. Non-LTE Integrated Line Power as a Function of T e for an Isolated Cold Region. The upper curves are for the H157a line; the lower curves are for the H 9 2 a line. Each curve is labeled with the emission measure (in pc cm"6) used to compute it. The observed line power is represented by the line labeled with P and the shading represents the error bars.

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151 100 ' ' ' I HI57a * (— i — i — i i i i | i 1 — * — • — i > * 50 H92a \20 50 1000

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Figure V-10. LTE Integrated Line Power as a Function of T e for an Isolated Cold Region. The upper curves are for the H157a line; the lower curves are for the H92a line. Each curve is labeled with the emission measure (in pc cm -6) used to compute it. The observed line power is represented by the line labeled with P and the shading represents the error bars .

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153 1000

PAGE 162

154 1000 100 10

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155 error strip for each locus would encompass both lines and render an uncertainty to the solution equal to the range of the loci. Hence the solution presented is only a best estimate. This solution temperature does not seem to agree very well with the most probable temperature derived by the Cesarskys (1973b) but is closer to the 21 cm spin temperature of 60 °K derived by Radha kr i shnan and Goss (1972). But considering the uncertainties, the agreement is all right. This result, along with the alignment of the high-velocity recombination line features with the 84 and 105 km/s 21 cm absorption features, tends to indicate that their regions of origin are spatially related. The beam dilution was taken to be unity for both lines because the variation of the H157a line power between its two beams seems to indicate that the emitting region fills both of them. This suggests a H157a emission region size of about 85 pc . This, however, is quite a bit larger than the average size of the 21 cm absorption regions (5 pc) derived by Baker and Burton (1975). If 85 pc is equal to the total path length through the recombination line emission region, the emission measure of 9.9 pc cm derived _3 here results in an RMS electron density of 0.34 cm . It is possible that the recombination lines may come from a _3 cluster of 5 pc clouds. Since N u >0.34 cm , Equation 11-54 H shows that the spin and electron kinetic temperatures are equal .

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156 It is difficult to explain the width of the recombination line features if the emitter has a temperature of 53 °K. Since 53 °K is the gas kinetic temperature, the thermal line broadening should be 1.6 km/s (Equation 11-34). This is obviously much smaller than the observed line width which is between 30 and 36 km/s. Turbulence within the cold emission region will cause some broadening, but the average total half-intensity width of 21 cm absorption features seen in the galactic plane is about 3 km/s; this includes all broadening effects (Radhakrishnan and Goss, 1972). The widths of the 80 and 107 km/s features observed in the direction of 3C391 are 10 and 3 km/s respectively. The 80 km/s feature may be a blend of two or more features (Radhakrishnan and Goss, 1972) . The discrepancy between the 21 cm absorption and radio recombination line widths suggests considering other model s . The Hot Spot Model s The observational results of Matthews, Pedlar, and Davies (1973), Gottesman and Gordon (1970), and Chaisson (1974a) suggest that the recombination line emission regions must be hotter than 1000 °K. One possibility is a distribution of very small, hot H II regions. Altenhoff et al . (1973) have observed 10 discrete continuum sources within 36 arc minutes of the position (1= 24°, b = 0°) at which

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157 Gottesman and Gordon (1970) and Gordon and Gottesman (1971) observed recombination lines. This is a region in which Altenhoff et al . (1970) indicate there are no resolved continuum sources. The 10 discrete sources were observed at a wavelength of 11 cm and with a spatial resolution of 4.4 arc minutes. According to Guelin (1974), three or more of these sources produce 25 km/s-wide recombination lines. These are about as wide as those observed in the direction of 3C391. Weak and broad (Av = 80 km/s) lines were observed between these sources. Assuming that these "Hot Spot" sources are widely distributed throughout the galactic ridge, expected line powers will be calculated; the calculations will be based on as much evidence as is available. The expected line powers of H157a and H92a emission will be compared to those observed . Guelin (1974) provides no more information on these sources than has been mentioned except that it was implied that they are separated by a minimum of 4.5 arc minutes. With this spacing, 13 of them can be distributed within the 18 arc minute H157a antenna beam. The maps of the region surrounding 3C391 made by Caswell et al . (1971), Dulk and Slee (1972), Bridle and Kesteven (1971), and Chaisson (1974a) show no collection of discrete sources within 18 arc minutes of 3C391. Since Guelin does not mention the intensity of the sources, one cannot be certain of their

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158 absence since, in all cases, the maps were presented with the background removed. The structure seen by Chaisson (1974a) with a 2.1 arc minute beam at 15.5 GHz is enclosed by the weakest isophotes of all other maps and is probably, therefore, structure within the SNR rather than of the intervening medium. Braes and Miley (1972) have produced an aperture synthesis map of a l°xl° region in which Cygnus X-3 is located. This is in the Cygnus X region, a distribution of H II regions appearing roughly circular in shape with a diameter of about six degrees. There are about 100 known thermal continuum sources in this region (Wendker, 1970). Radio recombination lines have been observed in several places between these continuum sources (Gottesman and Gordon, 1970; Seacord and Gottesman, unpublished observations). The map of Braes and Miley (1972) does show some structure. There are a few extended sources, one of which coincides with an OB association (Cygnus 0B2 or VI Cygni) and the others coincide with known H II regions. There are about 15 discrete sources having a flux density considerably less than that of Cyg X-3, 0.021 Jy. However, no more than five of these sources exist in any 18x18 arc minute area over the entire l°xl° map and some of these sources are closer together than 4.5 arc minutes. There are two cases for which discrete H II regions have been studied. Habing et aj_. (1973) have observed H II

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159 regions which are related to Type I OH maser sources. One of these, ON 1 , had previously been studied by Winnberg et al_. (1973). ON1 is located at 1 = 69?5 and b = -1?0 o _ c and has an emission measure of about 2.7x10 pc cm , 5 -3 an electron density of about 10 cm , and a diameter of 0.03 pc or less. Winnberg ejt aj_. scanned a field of 24x24 arc minutes around ON 1 with a resolution of 1.3 arc minutes and found no other sources. If the sources observed by Altenhoff et aj_. (1973) are like that associated with ON1 , it is apparent that they do not occur frequently out of the galactic ridge. This agrees with the finding that radio recombination lines are not observed in the general interstellar medium outside the ridge. Another case on which to base hot spot models is the hot, dense condensations, or "knots," within emission nebulae. An example of this kind of H II region is in the core of M42. For the knots there, Hjellming and Davies (1970) find that the average temperature, electron density, 4 4-3 7 and emission measure are 10 °K, 2x10 cm , and 10 pc cm" respectively. These averages are made over an effective path length of 0.02 pc which is assumed to be the size of the sources . A series ofmodels for a distribution of 0.02 pc H II regions will be considered. They are assumed to be distributed across and along an 18 arc minute beam such that 13 of them are observed in

PAGE 168

160 it and there is one of them in the 2.3 arc minute H9 2a beam. The distribution has a centroid located at a distance of 8.5 kpc from the sun. The angle subtended by one of these sources is about 0.48 arc seconds. Therefore, the beam dilution factor, p B , is 1.65x10" for the H157a beam and 6.80xl0" 6 for the H92a beam. The integrated line power, P, is computed with Equation V-10. LTE and non-LTE cases are computed for which N = 1 , 10, 10 2 , and 10 cm" . The temperatures and range of emission measures are based on the results of Hejllming and Davies (1970) and Habing et al_. (1974); T g = 7.5xl0 3 and 10 4 °K with 10 4 < E L < 5x10 pc cm" . The non-LTE departure parameters were computed by Sejnowski and Hjellming (1969) and are tabulated in Table V-5. The slope of the b n curve, 6 mn = Cv b n )/b m' is alS ° shown Figures V-12 and 13 show P(H157a) and P(H92a) plotted as a function of emission measure. Figure V-12 is for T e 7500 °K and Figure V-13 is for T g = 10000 °K. In both figures the H157a curves are plotted above those of H92a and the vertical line labeled with P Q represents the observed line power: P Q (H157a) = 9.61 (+0.26, -1.27) °K kHz and P (H92 a ) = 5.7 (±2.4) °K kHz. The shaded strip represents the error bars. The non-LTE curves are labeled with the electron density for which the departure parameters were computed. The H92a LTE curves are drawn dashed. For both temperatures, the H157a LTE line powers are less than

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161 Table V-5 Departure Parameter Used in the Hot Spot Models [Computed by Sejnowski and Hjellming (1969)]

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Figure V-12. Integrated Line Power as a Function of Emission Measure for a 7500°K Hot Spot. The upper curves are for the H 1 5 7 a line. The lower curves are for the H92a line. Each non-LTE curve is labeled with the electron density for which it is calculated. The observed line power is represented by the line labeled with P and the shading represents the error bars.

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163 pc cm 10 o ioo P K kHz

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Figure V-13. Integrated Line Power as a Function of Emission Measure for a 10 000°K Hot Spot. The upper curves are for the H157a line. The lower curves ar for the H 9 2 a line. Each non-LTE curve is labeled with the electron density for which it is calculated. The observed line power is represented by the line labeled with P and the shading represents the error bars.

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165 pc cm 0.01 10 100 P K kHz

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166 0.01 °K kHz and, therefore, are not shown in the figures. Figure V-12 indicates a consistent solution for T e 7500 °K because the P Q (H157a) and P Q (H92a) lines -3 ^ intersect the respective N g = 100 cm curve at E L ^ 2.6x10 pc cm" . The intersection is well within the error limits The situation for T = 10000 °K is not as simple. The solution occurs between the curves for N e = 10 and 100 cm . Simple interpolation is impossible because the dependence of the departure parameters on the electron density inverts the E.-P relationship of the HI 57 a and H92a curves _3 between N = 10 and 100 cm . Consequently, in order to obtain a consistent solution, the functions were replotted as P being a function of N , using E L as a parameter. This is shown in Figure V-14. The locus of the points of intersection between the P lines and the constant E. curves is shown in Figure V-15. The H157a and H92a loci intersect 7 -3 at the values E L = 4x10 and = 81 cm . The intersection occurs within the range dictated by Figure V-13. Table V-6 compares the derived models with the prototypes on which they were based. The agreement between the emission measures is not good and there is no agreement between the electron densities. , the electron density averaged over the path length L, is at least two orders of

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Figure V-14. Integrated Li ne non-LTE Power as a Function of Electron Density for a 10 000°K Hot Spot The upper curves are for the H157a line. The lower curves are for the H92a line. Each curve is labeled with the emission measure for which it was calculated. The observed line power is represented by the line labeled with a P and the shading represents the error bars.

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168 K kHz

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169 pc cm Fi gure V-l 5 . Hot Spot Model Solution for T ? = 10 000°K The solution is given by the intersection of the two curves; E[_ = 4 x 10' pc cm" 6 and < N e > ' ' =81 cm" 3 1

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170 Table V-6 Comparison of Models with Their Prototypes Emission Nebula Knots This Study /'^ T T 10000 °K 4x10 pc cm 81 cm" 3 4.47xl0 4 cm" 3 0.16 4.8 M42 (Hjellming and Davies, 1970) 10000 °k 1 . 26x1 pc cm 2.5xl0 4 cm" 3 3 -3 6x10 cm M.2 >1 1 H II Region Associated with Type I OH Masers This Study < N >/ 1//2 M e ' e 7500 °K

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171 magnitude too small for either type of region. can be compared with the RMS electron density computed by dividing the emission measure by the path length, 0.02 pc. .2 1/2 Table V-6 also shows the ratio / and the 1.683 GHz continuum optical depth. The ratio would indicate the amount of clumping along the line of sight if and 2 1/2 were obtained by independent measurements. Winnberg et al (1973) made no determination of . Hjellming and Davies (1970) solved Equation V-10 for several a-, B-, y-, 6 , and e-lines to obtain values of < T e > > < ^ e > ' a n ^ *" ^ I > ' So, and the RMS value of it should be independent except for the fact that they both were obtained from radio recombination line observations. This is true in the case of this study also. An independent determination of would be through the measurement of a pulsar's dispersion measure (cf. Equation 11-55). Since it is probable that there are more electrons along the line of sight than are responsible for the radio recombination line emission, the does not represent an average over the line of sight and, 2 1/2 therefore, / does not measure clumping. However, it is apparent that the ratio should represent the same quantity in both the analysis in this study and in that of Hjellming and Davies (1970) and would, therefore, indicate agreement or disagreement with the type of source being considered. There is clearly a lack of agreement, not only with this ratio, but with all parameters der i ved .

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172 The number of sources in the H157a beam was chosen to be the maximum number allowed with a minimum intersource spacing of 4.5 arc minutes. To determine what effect a smaller number of sources in the 18 arc minute beam would have on the solution, Figure V-16 shows the line poweremission measure curves for 13, 10, and 5 sources in the beam. The number of sources enters into the calculation by the beam dilution factor, p g . Each curve in Figure V-16 is labeled by the number of sources. The upper set of curves is for T = 7500 °K; the lower curve is for T e = 10000 °K. -3 In both sets, the dashed curves are for N e = 100 cm and -3 the solid curves are for N e = 1 cm Reducing the number of sources has the effect of reducing the electron density and increasing the emission measure. So, the size of the emitting region increases greatly. The recombination line emission seems to favor an electron density more like that of the more tenuous gas surrounding the dense knots of an emission nebula. This is in agreement with the conclusion of Brockelhurst and Seaton (1972). Consequently, the results of this analysis do not seem to support the possibility that radio recombination lines observed from the galactic ridge are emitted by a collection of small (^0.02 pc), dense, hot, H II regions like ON 1 or the condensations in emission nebulae.

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Figure V 1 6 . The Effects of the Number of Sources on the Hot Spot Model Curves. The non-LTE, H157a integrated line power is plotted as a function of emission measure for T e = 7500°K (upper set of curves) and T e = 10000°K (lower set of curves). The dashed curves in both sets are for N e = 100 cm 3; the solid curves are for N e = 10 cm-3. Each curve is identified by the number of sources in the antenna beam. The observed line power is identified by P .

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174 10 F 10 I0 7 " 10 pc cm 10 8 ^ 10 1 1 1 1 1 1 III 1 — 1 — 1 1 lll'l 1 1 — 1 III" T e = 75 00°k

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175 H II Nebula Models Chaisson (1974a) suggests that the H92a line observed by the Cesarskys (1973b) is emitted by a hot (7500 °K), _3 low density (N = 22 cm ) H II region embedded in a twophase medium rather than by the cold regions which the Cesarskys' results suggest. There are several arguments in favor of this which also apply to this study's spectra. The line widths (^30 km/s) are like those observed from emission nebulae which are much larger than those of the corresponding 21 cm absorption features (£10 km/s). Also, the 80 MHz optical depth measured by Dulk and Slee (1972) can, in part, be caused by a partially ionized cool region with the remaining absorption caused by an H II nebula. In fact, recent unpublished observations indicate that t(80) may be 0.4 (Chaisson, 1974b) which is the value it would be if the recombination lines were emitted only by the nebula (Chaisson, 1974a). He detects an excess continuum flux at 15.5 GHz which could result f rom t he nebula. However, it should be noted that Goss (1972) points out that his 31.4 GHz observations do not indicate an excess flux. Finally, the Cesarskys (1973b) derive an ionization rate of Cu^" 10 ~ sec from their cold-region model. However, Hjellming, Gordon, and 1 5 1 Gordon (1969) derive Cu^lO sec from pulsar dispersion measurements and Hughes, Thompson, and Colvin (1971) obtain a range of 1.9x10 < u < 4.0x10 sec from 21 cm absorpti on spectra .

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176 These arguments suggest considering an H II nebula model. The model curves were computed with Equation V-10 for LTE and non-LTE cases. The temperature, T , was restricted to 7500 °K and the non-LTE models were computed -3 for electron densities of 10 and 100 cm using the departure parameters of Sejnowski and Hjellming (1969) given in Table V-5. Because of the variation of the line intensities between the 3C391 and its C0MP beam, it was assumed that the nebula subtends an angle of about 34 arc minutes, giving it a diameter of 85 pc at a distance of 8.5 kpc. Therefore, since it completely fills the H157a and H92a beams, the beam dilution factor is unity for both. Figure V-17 shows the integrated line power plotted as a function of emission measure. The H157a curves are drawn solid; those for the H92a line are drawn dashed. Each set consists of three curves, the LTE curve and the non-LTE curves labeled by the electron density used to compute it. The observed integrated line power for each line is indicated by the horizontal lines labeled with P (line). As for J o the previous models, P Q (H157a) = 9.61 (+0.26, -1.27) °K kHz and P Q (H92a)=5.7 (±2.4) °K kHz. A constant solution is obtained when the lines of P„(H157a) and P (H92a) intersect their respective LTE or oo non-LTE curve at a common value of Ei. Because of the large errors involved, it is difficult to ascertain exactly what

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177 100 'K kHz ICT I0 4 pc cm Figure V 1 7 H II Nebula Model Solution. The integrated line power is plotted as a function of the line emission measure. The H157a curves are drawn solid; those for the H92a line are drawn dashed. The non-LTE curves are labeled with the electron density used in their computation. The observed line powers are identified by P (line) and the error bars are indicated by the shaded strip.

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178 values apply to the solution. Perusal of the areas of intersection in Figure V-17 is easier with a table. Thus, Table V-7 indicates the value of emission measure where the lower error limit, P (line), and upper error limit intersect each of the curves. From the figure and table, it appears likely that the solution is close to E^ = 3.8x10 pc cm with N e = 10 cm -3 . Since P Q (H157a) = 9.6 °K kHz is closer to the upper error limit, the solution E, = 3.3x10 pc cm with _3 N e = 100 cm is not unlikely. The table and figure indicate that an LTE solution is unlikely since a common intersection is so close to both error limits. There is astrophysical evidence which also suggests thi s . The likely range of parameter values agrees well with those Chaisson (1974a) obtains by an independent method: = 22 cm" and E T ^4x10 pc cm" , giving E L = Ey/1.1 % o _ c 3.6x10 pc cm . This solution also compares favorably with the parameters applicable to visible H II nebulae. For example, the North American nebula complex (W80) has a _ o diameter of about 50 pc and % 10 cm (Kaplan and Pikelner, 1970, p. 153) which gives E T ^5xlO pc cm Finally, it should be noted that, in a very recent paper, Jackson and Kerr (1975) conclude that radio recombination lines originate from small, lowdensity H II nebulae like this study's H II nebula model except that the electron temperature they derive is close to 4400 °K. Their results

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180 will be discussed in more detail in Chapter VI where the ridge as an entity will be discussed. Also, as this was being written, Pankonin (1975) reported on his H248arecombi nation line observation of 3C391. His line indicates nothing which has not been observed before and he favors a low density H II region as the source of the line. Table V-8 summarizes the models derived in this chapter. The discussions made during the derivation of these models suggest that the most likely models are the Isolated Cold Region and the H II Nebula models. In the following chapter, additional arguments will be considered from which a most probable model, including the distribution of the emission sources, will be considered.

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181 «3o o n i — i — CM 01 O *Jr-. wa ci — i CM CM O O o o o o IT) O r-~ O E

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CHAPTER VI FURTHER CONSIDERATIONS AND APPLICATIONS OF THE MODELS Discussion of the Models This discussion pertaining to the models suggests that the most plausible condition is that the recombination line-emitting region is either an isolated cold (T e ~53 °K) region or a low density emission nebula similar to W80. In order to decide on a final model, several more parameters will be calculated and compared to the results of other work. Three of these parameters are the 1683 and 80 MHz optical depths and the continum antenna temperature at 1683 MHz, which are expected to be observed from each of the model s . The D-Method models i n vol ve a two-component medium which will be considered to be a cold region surrounded by hot gas. The continuumantenna temperature, T-j, is given by T T = n B U e (h)[l-exp(-T f (h))] + ^ T e (c)exp(-T f (h))[l exp(-i(c))] + T (h)exp(-T(c)-T f (h))[l-exp(x b (h)]} 182

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183 In this equation, ft is the solid angle subtended by the cold region. For these models it is greater than the antenna main beam solid angle, ft..; so, ft/^ M = 1All optical depths are for the thermal continuum. That of the hot gas in front of the cold region is x . ( h ) and that of the hot gas behind the cold region is x.(h). It will be assumed that the hot gas is distributed equally in front of and behind the cold region; so, i f (h) = T b (h). For a single component recombination line emitter with a continuum optical depth x T and observed through an optically thin gas (t^3x10~ ), Equation VI-1 reduces to n B ^ T e [l-ex P (-T T )], where ft, is the solid angle subtended by the line-emitting region. For all except the Hot Spot models, ft,/ftw = 1For the Hot Spot models Q,/ft M = 1.65x10" . The continuum optical depths at 1683 and 80 MHz and the 1683 MHz continuum antenna temperature for each of the models are listed in Table VI-1. If the isolated cold region were surrounded by a hot gas, the situation would be similar to that of the two-component model. From the table it is evident that this hot region would contribute a neglibible amount to the antenna temperature. If the Hot Spots are surrounded by a low density HII nebula, the observed contribution to the continuum temperature would be about 4 °K.

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184 Gordon and Gottesman (1971) observed H1973 recombination lines in Cygnus X and at two positions in the ridge, at longitudes of 23?9 and 25?1. The ratio of H157a to H 1 970 integrated line powers are 3.21, 3.95, and 4.56 for longitudes 23?9 and 25?1 and Cygnus X respectively. The H 1 9 7 3 line powers expected in the direction of 3C391 are calculated for each model using the parameters of Table V-7 and Equation V-10 except for the D-Method models where the integrated line power is given by P(H1 973 ) = 1 .0645Av d [0.8t£(H1976) (T -T r )] The power ratio, P(H1 57a)/P ( HI 976) , for each model is listed in the fifth column of Table VI1. The velocity half-power width of the line used to calculate line power was taken to be same as that observed for the H157a line, 35.7 km/s or 201.8 kHz. A comparison of the power ratios in Table VI-1 with those found by Gordon and Gottesman is made with the assumption that a radical change from region to region within the ridge is not to be expected. Acutally, their ratios did not change much between the ridge regions and that in Cygn us X . Another parameter which will be used to evaluate the models is the hydrogen ionization rate, c u = H N,. + J a H e H a ? (T)/N„ (Equation 11-48). The partial recombination coefficient, a ? (T), is computed from Equation 11-22. Then

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185 a 2 (r 2.065X10" 11 T" 1/2 c 2 (T; and ?p(T) is obtained from Figure 1 1 3 . Because radiative recombination directly into the ground state generates a photon which immediately ionizes a neighboring hydrogen atom, the ionization rate from outside sources is related to recombination into all levels expect the ground level. Therefore, the partial recombination coefficient is used rather than the total coefficient (Dalgarno and McCray 1972). In calculations of ^ H for the Hot Spot and HII Nebula models, total ionization in the regions is assumed. Then N H = N H + . Also, N e = l.lNj,+ to account for electrons contributed by helium. Then /N L , reduces to . e n n e For the D-Method and Isolated Cold Region models, N e H H + is assumed to be equal to N . The total hydrogen density is nearly equal to the neutral hydrogen density, N u0 H which is related to the 21 cm line integrated optical depth by means of Equation 11-52; r , -u r ^u°( s ) J x 21 (v)dv = 2.58x10 ID / -~ ds. line o s y\ 2 -|(v) dv is obtained from Table 2 of Radha kri s hnan et al. (1972b). There are three hi ghvel oc i ty features in the absorption spectrum which are located at 80, 101, and 107 km/s. In calculating N H for the cool models, is assumed that

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186

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187 N ^ N o and, since the effective beam width of the 21 cm H H measurements is no greater than the 18 arc minute beam used in this study, all three features are assumed to be represented by a blend in the H157a spectrum. N H is, therefore, computed using all of them. Then < j > l H >=3.876xl0 14 — j-— h 2] {' 3.876xl0 14 T — [1 .0045 £t 01 (i)Av n J L L ' uu ^ .21 v ' /uv Di l 18.97— j-^. The path length, L, is in pc. is listed in the sixth column of Table VI-1. The fractional ionization, xeN 6 /N h , is listed in the seventh column. A perusal of this table indicates problems with most of the models. For all cases except the H II Nebula, the recombination line emission region contributes a negligible fraction to the observed continuum temperature. In these cases the thermal portion of the total continuum is the 2.7 °K universal background radiation. The 80 MHz continuum optical depth computed on the basis of model parameters shows considerable disagreement with measured values. The value oft (80) for D-l is equal to that obtained by Caswell et al_. (1970). That for the Isolated Cold Region model is a bit lower than the value of 0.83 derived by Dulk

PAGE 196

and Slee (1972). t(80) for the H II Nebula model is equal to that which Chaisson (1974a) suggests. Of the eight models considered here, only three produce H157a/H197B ratios which are reasonably close to those obtained by Gordon and Gottesman (1971), model D-l and the Isolated Cold Region and H II Nebula models. The extremely high ratio for model D-4 results from P ( H 1 9 7 3) a t*-E l 1 .07xl0" 3 pc cm" 6 . If the recombination lines arise from a cold region, one would expect to find a hydrogen density between 10 and 3 1 5 1 100 cm and an ionization rate near 10 sec . These values are in agreement with the twocomponent model of Field, Goldsmith, and Habing (1969) for which the ionization rate is 4x10 sec as expected from 2 Mev cosmic ray protons. The ionization fraction of their model is -4 about 5.5x10 . None of the D-models agree very well with their model. For a temperature near 14.3 °K, their model 3 would require that N H ^ 100 cm . Lock man and Gordon (1973' obtain a low temperature ( T< 1 00 ° K ) solution from H159a recombination lines observed in the galactic plane in regions near the center. They did not detect the H2003 line. So, P (HI 59a) /P (H200B ) was greater than would be expected from a gas in LTE. This result, the results of Cesarsky and Cesarsky (1973b), Table VI-1, and the discussion immediately following the derivation of the D-Method models suggest that a 14.3 °K, LTE solution is not valid.

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189 The Isol ated Cold Region model fares only a little better with respect to N M , x, and £„. The Field, Goldsmith, and Habing (1969) model suggests that N H ~30 cm and x ,-3 1.3x10 if T = 53 °K and the carbon abundance is a f a c e tor of ten less than what is usually assumed (Spitzer, 1968, p. 122). The depletion of carbon (due to accretion on negatively-charged grains) is supported by preliminary analysis of ultraviolet observations wi th satellites (Dupree, 1974) and, perhaps, by the fact that the C157a line was not observed in this study. Nevertheless the number densities and fractional ionization of the model derived here are too small. The agreement is not improved by introducing a timedependent source of ionization as suggested by Gerola et al . (1974) . The hydrogen densities derived for the Hot Spotmodels are much too small since the prototypes require that N > 3 3 10 cm . It is difficult to imagine how such an electron density can be obtained in the Hot Spot models where N,, ^ _ 3 100 cm . Evaluation of these models by means of the ionization rates is not possible since the source of energy is not known. Of all models studied, the H II nebula produces the most promi s i ng set of parameters. The value of N,, derived is reasonable when compared to that of W80 (Kaplan and Pikelner 1970, p. 153). If the region is ionized by a hot star, total ionization is to be expected. The ionization

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190 rate due to a star with a flux S at frequencies greater u 3 than that of the Lyman limit is (Equation 11-50) 3S ' H 4irr 3 N u s H S is tabulated by Spitzer (1968, p. 116). The Stromgren radius, r , for this model is 42.5 pc. Then, for an 05 1 3 1 star, Cm = 3.61x10 sec . If r is decreased to 25 pc, H s p 1 2 1 the size of W8 0, c H = 1.77x10 sec . In either case, a cooler star provides a smaller ionization rate. In order to obtain the variation of line strength between the 3C391 and C0MP beams, the size of the nebula cannot be very much less than 85 pc. So, in order to exist with an ionization 1 2 1 rate of 3.27x10 sec or less, the nebula must be a small cluster of nebulas together occupying a region 85 pc in diameter. The arguments favoring and rejecting the models studied here are summarized in Table VI2. It is apparent that the best model for the source of radio recombination lines from the galactic ridge, at least in the direction of 3C391 , is a low-density H II region. Other aspects of this model — particularly the relation of H I absorption spectra to it — will now be discussed. A recent paper by Jackson and Kerr (1975) supports this conclusion except that for their model they derive a temperature of 440 (+600) °K. Also, they claim that

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191 OO

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iI — 4r— 192 00 QJ CL o jd -a

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193 ac EJ3 -C CO

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194 consideration of non-LTE will not affect their results significantly. Their argument for assuming LTE was discussed in an earlier paper (Jackson and Kerr, 1971) where they show that the ratio of the line temperature, T. , expected from a non-LTE gas to the line temperature, Tj* , expected from an LTE gas is roughly equal to b . They assume that b =1 for the HllOa line (v = 4874 MHz). However, consulting Figure II-4, it is seen that b -, -, -, < 0.7. The best way of determining how much the kinetic temperature is above 4400 °K, is to observe higher order lines with rest frequencies close to that of the HllOa line. The line strength depends on 3 3 v mn as well as b . The gas in these nebulae may not be far out m of LTE because the solution shown in Figure V-7 indicates only a small difference between the non-LTE solutions for _ o o _ c N = 10 and 100 cm if E , ~ 3 x 1 pc cm . In this case there e L r _ 3 is no appreciable difference between the 100 cm non-LTE and the LTE solutions. Hjellming and Davies (1970) determine that the recombination line emission regions in nebulae such as M42 and Ml 7 are in non-LTE even though they are denser and hotter than is the model derived by Jackson and Kerr (1975). It has been pointed out, however, that the recombination lines arise from the more tenuous gas which surrounds the dense knots. It is difficult to explain why such knots would be out of LTE since, with their temperature and electron density, collisional excitation must dominate for

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195 large n. So, the recombination lines must come from the low density, non-LTE gas surrounding them. It is interesing to note that the model developed in this study is not very much different than the model developed by Matthews, Pedlar, and Davies (1973). The average temperature and emission measure which they compute from H166a lines are 6.2xl0 3 °K and 2.7xl0 3 pc cm" 6 respectively as compared to 7.5x10 °K and 3.8x10 pc cm derived by this study. Like Jackson and Kerr, they assume a condition of LTE. Another difference between their model and that of this study is the size of the region. The positions which they observed were spaced 5° to 10° apart and they could not observe the change in the spectrum over a distance as small as that observed here. From their observations they conclude that the emission regions are extended 3° to 5° in the galactic plane which is equivalent to a size in order to 0.5 to 10 kpc. This study suggests that the structure is much smaller, being in the order of 85 pc or smaller. This is in agreement with the conclusions of Jackson and Kerr (1975). It has been mentioned that high spatial resolution maps of the region surrounding 3C391 (cf. Chaisson 1974a) do not indicate the presence of discrete sources like the ten such sources reportedly observed by Altenhoff et al . (1973) near the direction 1 = 23?92, b = -0?32. But, the possibility of their existence in the direction of 3C391

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196 cannot be discounted completely. If they do exist, it is not completely clear how they fit into the model proposed by this study. Without knowing more about the size, distribution, continuum flux density, and recombination line spectra of these sources, it can only be assumed that they form a small-scale structure within the H II nebulae. Perhaps they are dense condensations associated with star formation. If that is so, it is not clear why they are not seen in Chaisson's (1974a) map. The structure seen in his map is included within the isophotes of the map of Bridle and Kesteven (1971) and it appears to be structure within 3C391 and not of the intervening medium. There must be an explanation for the alignment of the 21 cm absorption features, which are associated with cold gas, with the H II nebulae. This can be explained by cool cloud-like regions which lie close to the nebulae. Being close to each other, they will both have the same central velocities. This arrangement is similar to the structure of the Orion nebula described by Balick, Gammon, and Hjellming (1974). The distribution of the H II nebulae must explain the observed central velocities and relative intensities of the recombination line features from the region around 3C391 including the COMP beam. If circular galactic rotation can be assumed, the Schmidt rotation model (Schmidt, 1965) can position the emission regions along the line of sight. In

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197 the region inside the sun's orbit, there is a twofold ambiguity between velocity and radial distance. FiguresVI-1 and 2 show the positions of the emission regions. Figure VI-1 is for the case where the high-velocity feature is single; Figure V I 2 is for the case where it is double. In both figures, an outer boundary with a radius of 8 kpc is drawn in to encompass the low-velocity feature. The outer boundary of the ridge has been generally thought to be a circular disc or an annulus with an outer radius of 7 kpc (Matthews, Pedlar, and Davies, 1973). If the ridge is a disc-shaped distribution of H II nebulae with a nebular density which drops off rapidly beyond 7 kpc, the boundary may not be well defined. At longitudes progressively near the tangent to the boundary, the lines of sight will pass through fewer nebulae and recombination line emission will become patchy. So, even though recombination lines have not been detected in the direction of the tangent to the 8 kpc circle (1 = 53°)(Gordon and Cato, 1972), it is reasonable to expect that some nebulae lie this far from the galactic nucleus. The Hughes, Thompson, and Colvin (1971) 21 cm absorption spectrum shows an absorption feature at 42 km/s. There is a weak feature at this velocity in the Radhakri shnan e t a! , (1972b) spectrum. From this it can be assumed that either 3C391 is beyond 14 kpc or the H II nebula and its neighboring cold cloud responsible for the 42 km/s features are located at a distance of about 3.1 kpc from the sun.

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198 >>

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199 O l/l cn-o

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200 If these H II nebulae are to explain the spectra of other galactic ridge surveys, it must be hypothesized that the ridge phenomena are caused by a distribution of H II nebulae. It is confined to a diskor ring-shaped region extending about 70 pc above and below the plane (Gordon, Brown, and Gottesman, 1972) and limited to a gal actocentri c radius of about 8 kpc. In order to test this hypothesis, the spectra from the directions of the pulsar, 3C396, and its COMP beam will be examined. Following this, the results of this study will again be compared with other surveys of the plane. The Pulsar 1858+03 Determining the distance to a pulsar is not a trivial task. The distance to the stronger pulsars can sometimes be determined from the 21 cm absorption spectrum which is obtained by subtracting the pulse-off spectrum from the pulse-on spectrum (Gordon e_t a_l_. 1969). An estimate of the distance can be calculated from the dispersion measure, DMe / N ds = L, / e e if the mean electron density can be determined independently. The determination of is particularly difficult in the galactic plane. A recent survey of interstellar

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201 medium research (Guelin, 1974) indicates that the average electron density within about 2 kpc of the sun is 0.03 _3 cm . is much larger in the ridge. The reason for choosing the pulsar 1858 + 03 as a direction in which to observe radio recombination lines is that it appears to lie in the ridge. Its dispersion measure 3 is 402 (+2) pc cm (Manchester and Taylor, 1972) which is, by far, the largest dispersion measure known as of late 1971 . The H157a spectrum from the direction of 1858+03 is shown in Figure IV-5. The centervel oci ty , peak temperature, half-power width, and the integrated line power of each feature are brought from Table I V3 and shown in Table VI-3. In addition to these parameters, the Doppler temperature of each feature, computed from Equation 11-37, is shown in the last column of this table. The Doppler temperature is the kinetic temperature of the gas if the line broadening were entirely thermal. Table VI-3 Spectral Parameters for 1858+03 v c km/s

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202 Figure V I 3 shows the possible positions along the line of sight of the other two features as determined by the Schmidt rotation model. The near positions are indicated by dots and labeled A and B. The far positions are indicated by x's and labeled A 1 and B' respectively. A boundary with an 8 kpc radius is shown. The technique to determine the distance to the pulsar will be to compute the expected dispersion measure at points along the line of sight. The distance at which the _ 3 computed dispersion measure equals 402 pc cm will be the distance to the pulsar. The line of sight passes through four different types of region, each with its own electron density and path length. They are the region outside the ridge, cold H I regions, the diffuse hot gas, and the H II nebula. The arrangement with one H II nebula is shown schematically in Figure VI-4. The parameters of the regions used to computed the distance, R, to the Pulsar 1858+03 are listed in Table VI-4. Up to the ridge boundary, the accumulated disper_3 sion measure is 73 pc cm . If a nebula is encountered, the dispersion measure will rise sharply. If the path length through the nebula is 32.4 pc, the dispersion measure on _ 3 the far side of it will be 402 pc cm . The diagram in Figure VI-4 purposely shows the line of sight not passing through the center of the nebula to illustrate that the path length through it may be less than its diameter. If the line of

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203 C C tM•r-

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204 QJ i— -C ro C O o +-> < c 3 00 ro c so -a cn-t-> CD o a: c CD CO .c Tro i• cnoo ro O •r+ O CO LT> O CO

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205 Table VI -4 Parameters of the Regions Shown in Figure IV-4 Parameter Value Reference Path Length, Sun to Ridge Boundary Outside Ridge e 3 2.6 kpc 0.03 cm Figure VI-4 Guelin (1974) Diameter of Cold H I Region 5 pc Baker and Burton (1975) Ne in H I Regions 0.055 cm" Field, Goldsmith, and Habing (1969) Average Distance Between H I Region 333 pc Radhakrishnan and Goss (1972) in Surrounding Hot Region (b=0) 0.17 cm" Baker and Burton (1975) Path Length through H II Nebula adjustable Figure VI-4 in H II Nebula 10 cm" This work

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206 sight misses the nebula entirely, the distance to the pulsar must be 15.9 kpc, which is beyond A'. This distance is based on a path length through a two-phase medium for which the dispersion measure increases 30.17 pc cm for each kpc of path length. The distance of 15.9 kpc is also considerably larger than that derived for other pulsars, particularly those for which their lines of sight encounter H II regions (Terzian, 1972; Prentice and ter Haar, 1969). Although it is possible that the line of sight encounters neither nebula, it seems more likely that it passes through part of one since, otherwise, the pulsar would lie beyond the solar orbit. Hence, a distance of 2.6 kpc to 1858+03 appears most reasonable. The 3C396 and COMP Spect r a These spectra will be used as a final check on the model which has been developed. The spectra are shown in Figure IV-3. The results of Gaussian profile fitting for the 3C396 spec trum are shown in Figure I V9 and the parameters of the fit are given in Table IV-4. The reliability of what may be features at velocities of 58 and -119 km/s is relatively poor since the peak temperatures are only about four times the RMS noise. Nevertheless, by comparing the 3C396 with its COMP spectrum in Figure IV-3, it is seen that the peak temperature of the 54 km/s feature decreases by more than a factor of two in the distance between the two antenna beam centers which, is 30 arc minutes.

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207 The absorption spectrum in the direction of 3C396 observed by Hughes, Thompson, and Colvin (1971) consists of a wide envelope extending from -10 to 93 km/s and appears to be a blend of many features. There is no structure at 53 or 58 km/s. The 21 cm emission spectrum is flat and extends over the same range as does the absorption spectrum. A means of checking the model with the 3C396 spectrum is to compute the optical depth at 80 MHz and compare it with that measured by Dulk and Slee (1972). The line emission measure is computed from Equati ons V-l and 11-41 graphically by using Figure V-17. Using the graph sets T = 7500 °K. If a non-LTE gas with N = 10cm" 3 is cone J e o _ r _ o sidered, E. = 2.30x10 pc cm" . A gas with N > 100 cm" o _ c is in LTE; then, E. = 2.15x10 pc cm . It will be assumed tha t the recombi nation line emission region is similar to that in the direction of 3C391 (ie. non-LTE with N e 3 10 cm ). Therefore it will be assumed that E T = 1.1E. = 3 -3 2.53x10 pc cm in the direction of 3C396. By the use of Equation V-2, t(80) 0.237. Dulk and Slee obtain t(80) = 0.49 (+0.3). The values differ by a factor of two, but are in agreement within the errors quoted. The di ssagreement is about the same as it is between Dulk and Slee's value for the line of sight to 3C391 and that derived from the H II Nebula model as shown in Table VI-1. In this case, however, the agreement with Chaisson (1974a, b) is good. It is possible that Dulk and Slee's values for t(80) may be consistently large by a factor of two.

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208 Distribution of the Recombination Line Sources Within the Ridge The major objective of this study has been to determine what kind of region is responsible for radio recombination line emission observed from the galactic plane. The best model is a distribution of small, low-density H II nebulae. The last property of the model to be considered in this study is the distribution of these nebulae within the ridge. The recombination line surveys which have been mentioned can be used to obtain a rough picture of this distribution. The accuracy of this picture is severely limited by the resolution of the spectra — which prevents the separation of blended line features — and non-circular orbital motion (e.g. streaming) of the interstellar gas. Even if circular motion is assumed, placement of the nebulae along their lines of sight is confused by the two-fold ambiguity of the radial velocity -distance relation. In spite of these limitations, a "first order" approximation to the distribution can be drawn from the radio recombination line observations in the ridge from longitudes between 5° and 130°. The surveys not included in this summary are those from regions closer than 5° from the galactic center made by Lockman and Gordon (1973) and Matthews, Davies, and Pedlar (1973). They are excluded because there is no radial velocity from which a position can be obtained. All Doppler shifts are due to streaming of the gas. Both

PAGE 217

209 surveys show considerable variation of the line profile over a distance of a few tenths of a degree. Table VI-5 lists the observations made in the range 5° < 1 < 1 30 ° . If at some longitude no line was observed, "none" is entered in the last comumn. For several longitudes, it is apparent that a wide feature is a blend of unresolved features. For these cases, the central velocity of the end features has been estimated by using the turnover points of the blend. At the bottom of the table is the list of references— each with the spectral line observedindicated in the third column. The possible locations, based on the Schmidt circular rotation model, are shown in Figure VI-5. The near and far positions are indicated by a dot and an x respectively. The blended features are shown as bars to indicate the range in which the features are located. Since Cygnus X has been observed to be a region of extensive radio recombination line emission (e.g., Gottesman and Gordon, 1970; Seacord and Gottesman, unpublished observations) its angular extent is shown. It lies along the local spiral arm or spur. In Figure VI-5 the ridge boundaries, defined by the extent of the nebula positions, are drawn. It is seen that the radio recombination line ridge is an annulus with an inner radius of 4 kpc and an outer radius of 8.7 kpc. The distribution shown in the figure is quite similar to that of the giant H II region as shown by Mezger (1970). He

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210 Table V I 5 Radio Recombination Line Sources in the Ridge 1

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211 Tabl e VI5 conti nued Reference Velocity Features [km/s] 44.2 47.5 52.2 70.0 90 no 130 GC, P MPD GC GC GC GC GC 40 to 87 None None -4 None None None Key to References and Spectral Line MPD Matthews, Pedlar, and Davies (1973); H166a GC Gordon and Cato (1972); H157a JK Jackson and Kerr (1975); HllOa GG Gottesman and Gordon (1970); H157a S This study ; HI 57a GBG Gordon, Brown, and Gottesman (1972); H157a CC Cesarsky and Cesarsky (1973a); H166a and H157a Cesarsky and Cesarsky (1973b); H92a Pankonin (1975); H248a

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u c a) s.

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213

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214 defines the "giant" H II regions as those which satisfy the criterion S(5 GHz)-D 2 > 400; S(5 GHz) is the 6 GHz n — continuum flux and D is the nearer of the two kinematic n distances. Relevant to the distribution of the H II nebulae is their relation to H I gas. Comparing the 21 cm absorption and emission spectra and the H157a spectra of 3C391 (Figure V 1 ) and those of 3C396 indicated that the 21 cm spectra show more features and are extended over a greater velocity range than is so for the recombination line spectra. However, it should be noted that there is an H I emission peak close to each recombination line peak. A comparison of the radial velocities of H I and H II in the galactic pi ane within 60° of either side of the galactic center shows that it is generally true that H II features coincide with H I emission peaks (Mezgere_taJ_., 1970). One or more H I regions may be physically related to an H II nebula in a way that has been suggested for the Orion Nebula (Balick, Gammon, and Hjellming, 1974). When relating radial velocities to distance, it must be noted that Burton (1971) has shown that the Schmidt model does not adequately explain the observed H I velocitylongitude maps. Gas streaming causes the rate of change of velocity along the line of sight, dv/ds, to change across a spiral arm. The velocity -distance relation is, therefore, very complicated. Consequently, Figure V I 5 does not represent

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215 the actual nebular distribution well enough to relate the nebulae to spiral arm structure. The only conclusion that can be drawn from the figure is that, except for Cygnus X, these nebulae are restricted to an annulus with an outer radius of about 8.7 kpc and an inner radius of 4 kpc. It is possible that the wide spectrum from the longitude of 44?2 may be a blend of features from a string of H II regions along the inner edge of the Sagittarius spiral arm. If this is correct, this string of nebulae and the Cygnus X complex do indicate a possible relationship between the radio recombination line ridge and galactic spiral structure .

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CHAPTER VII SUGGESTIONS FOR FURTHER RESEARCH The results of this study do not conclusively determine a model for the radio recombination emission regions However a distribution of low density (Ne = 10 cm ) H II nebulae seems to fit the data of this and previous studies most consistently. There are a number of aspects of the problem which need further study. These will be considered in this final chapter. Assuming the conclusion that recombination line emission arises from low density nebulae is correct, the aspects of the general model which remain in question are as foil ows : 1 . Size of the nebulae 2 . Radial and z-direction distribution of the nebul ae 3. Thermodynamic conditions in the nebulae; LTE vs. non-LTE 4. Energy source within the nebulae 5. Relation between nebulae and low frequency absorption 216

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217 6. Relation of nebulae to H I gas, both hot and cold; and galactic structure in general 7. General relation with discrete sources seen by Altenhoff et al_. , 1973 The size of the nebulae can best be found by high resolution radio recombination line observations. The Altenhoff e_t aj_. (1970) continuum maps show considerable structure in the plane. To avoid confusion with recombination line emission from large H II complexes like W51, W49, or M42, observations should be confined to the regions like those observed by Matthews, Pedlar, and Davies (1973), Gottesman and Gordon (1970), Jackson and Kerr (1975) and Gordon and Cato (1972) which are between the continuum sources . An approach taken in making the observations would be to use a large single aperture telescope to observe a high frequency line; say the H 92a at 8.3 GHz. The observations could be made in a rectangular pattern of contiguous beams. The extent of the grid should be thirty minutes of arc in longitude if possible and about thirty minutes above and below the plane. Several of these grids should be observed along the plane. One of the grids should encompass the region observed by Gordon and Gottesman (1970) and Altenhoff e_t aj_. (1973). One grid should be placed at the outer boundary of the ridge and at least one placed at an intermediate longitude. The outer boundary

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218 of the ridge is not expected to be discrete; rather, as the longitude increases the number of nebulae decreases until the line of sight density drops to zero. This longitude, which is expected to be between 40 and 45°, can be estimated by observing individual regions between the continuum sources in this longitude range. At the ridge boundary there will be a minimum number of nebulae along the line of sight; thus, there is a better chance that one nebula can be identified individually. This technique should provide a better estimate of size and distribution — radially and normal to the plane — of the nebulae. It would also provide a comparison between the ridge in general and the region in which Altenhoff et al . (1973) have observed recombination lines from discrete sources and the hot H I gas. Unfortunately the recombination line power decreases as the line frequency becomes large. For that reason each beam position would require at least 7 hours of integration time in order to get a AT of 0.002°K if the 43-meter NRAO telescope is used, making the entire program a large undertaking. It would have to be broken up into units in order to fit in with telescope scheduling. More information on the thermodynamic condition of the gas in the nebulae can be obtained by observing several higher order ( A n > 1 ) lines all with frequencies within a small range. A set of a-, 3-, yand higher order lines should be observed from several regions between continuum

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219 sources along the plane to ascertain if the thermodynamic state varies within the ridge. From each line the temperature of the gas is determined using the assumption that LTE applies. As An increases, the excited states of the atom are more apt to be col 1 i si onal ly excited and the measured excitation temperature approaches the electron kinetic temperature of the gas. The amount by which the measured temperature is less than the kinetic temperature is a measure of the departure from LTE. This technique was applied to denser nebulae by Hjellming and Davies (1970) from which the line powers are considerably larger than they are from the nebulae under consideration in this model. Consequently, the highest order of line observed will be smaller and the kinetic temperature may have to be estimated by extrapolation. The results of this experiment should be compared to the work of Matthews, Pedlar, and Davies (1973) and Jackson and Kerr (1971, 1975) who find the radio recombination line emission regions to be close to LTE. The results should also be compared to those of Brown and Balick (1973) who determine that this region should be considerably out of LTE. Once the size, kinetic temperature and thermodynamic condition within the nebulae are better understood, it is hoped that a source of energy can be confirmed. A possibility which has been suggested in Chapters V and VI is that the energy source is one or more OB stars. Perhaps

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220 if the distribution of these nebulae within the ridge disc can be determined, it will provide some confirmation of this. However, the distribution in the galaxy of OB stars at distances greater than 2 kpc is not known. Obtaining a better understanding of the relation between the radio recombination nebulae and cold H I regions may come from observing with a matrix of beams at a high frequency line with the matrix centered on a source for which Radhakri shnan et_ aj_. (1972b) or Hughes, Thompson, and Colvin (1971) have published a 21 cm absorption spectrum. The velocity resolution of the observation must be as good as the absorption spectrum. Confirming or rejecting a relation between low frequency absorption and radio recombination line emission in the ridge will require collaborative observations at low frequencies. Presently the 80 MHz optical depths of Dulk and Slee (1972) are open to question (cf. Chapter VI). As mentioned in Chapter VI, the distributions of these nebulae can not be reliably determined using circular galactocentric orbits. A density wave model like that developed by Burton (1971) or Burton and Shane (1970) would produce a more correct distribution, but the nebulae must be positionally related to 21 cm emission regions. Such a relationship will be difficult to develop. If the energy source of the nebulae can be determined — for example if it is an OB association — the type of source may suggest a distri buti on .

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221 The purpose of this study, as stated in Chapter I, has been to determine how the intersteller medium in the ridge differs from that in other regions of the galaxy such that it is a well-defined source of radio recombination lines. Although considerable uncertainty still exists there is a trend— which this study s upports— toward the acceptance of a distribution of distinct low density H II regions. As this chapter shows, there is much work that can be done to reduce the uncertainty. The location and proposed model of the ridge suggest that it is an extreme population I region. The relation between galactic structure and star formation has been shown; cf. Shu (1973) and Biermann (1973). It follows that the ridge may well prove to be an important component of spiral galaxies.

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APPENDIX The following are details of Equation 1 1 7 and the derivation of Equation 11-16. The Thermal Continuum Absorption Coefficient The general form of the thermal continuum absorption coefficient (Equation 1 1 7 ) in terms of fundamental constants is Kj(v) = 32-rr 2 e 6 4u 2 v 2 3/2tt M 3 c V|
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223 and summing over all levels, co oo * r — c — Q N N* = > N*. = > N*. — exp(x . AT ) = -21 u .(j). n=o n=o The partition function is oo u-(T) = > g exp(x . /k T ). i v ' / y n i n ex A-2 Therefore , N . g -4 1 = — ?W exp(-v. /k T ) ^ u i ^ T > in ex A-3 N. can be found from the ionization, or Saha, equati on ; 1.5 N.N /2ttM k T„\ '*•' 2u.(T) \ / N is the neutral atom number density and u(T) is the o o partition function for the neutral atom. The ionization energy, x-» ^ s tne energy required to remove the last electron to form the i state of ionization. Setting i = in Equation A-3 and dividing the Saha equation by it gives •1.5 N. N e /N Q N. N e (2tt M k T K ) . * * " n n no o

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224 N is the quantity desired; so, the LTE number density of no ^ n-level excited atoms is 16 * N r 1.5 ^n 'i N e T 2UTTFT in ex' A-5 All number densities are in units of cm e . = ( y . y. ) is the threshold energy which is in A i A i n aJ the total energy of the electron in its orbit and equal to the sum of the kinetic and potential energies, e„ + V. For hydrogenic ions, electron shielding of the nucleus can be ignored. If a is the Bohr radius of the n level, a n a a ~ , r n r n 7 2 7 \F d r = f -h dr " VJ J r n 2 2 The orbital kinetic energy is e K = 1/2M aw ; to is the orbital angular frequency. Then H , /M ^ Mr Z e' 2 a and e . Z e /2 a. Since Z e' a n M a M a Z e a M n 2 2 _ n 2 (h/2iT) 2 n Ml~7

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Finally 225 •M Z 2 e 4 i n 2 n 2 (h/2n) 2 2 ? Also, e. /k = 157890 Z /n from substitution of the values of the fundamental constants.

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BIBLIOGRAPHY Aller, L. H. 1963, As trophys j c s: The Atmospheres of the Sun and Stars . 2nd edition. New York: The Ronald Press Co. Altenhoff, W., Mezger, P. G., Wendker, H., and Westerhout, G. 1960, Veroffentlichungen der Universitats Sternwarte zu Bonn, No. 59, 48. Altenhoff, W. J., Downes, D., Goad, L., Maxwell, A., and Rinehart, R. 1970, A. and Ap. Supplement , 1_, 319. Altenhoff, W., Churchwell, E. B., Mebold, U., and Walmsley, M. 1973, personal communication, cited by Guelin (1973). Andrews, M. H., Hjellming, R. M., and Churchwell, E. 1971, Ap. J . , 1_67, 245. Baker, P. L. and Burton, W. B. 1975, Ap. J . , 198 , 281. Baker, P. L. and Diadiuk, V. 1973, Ap. Letters , 13, 199. Balick, B., Gammon, R. H., and Hjellming, R. M. 1974, P.A.S.P . , 86_, 616. Ball, J. A. 1972, "Measurements with Radio Frequency Spectrometers," in Methods of Experimental Physics , Volume II, M. I.. Meeks, ed. New York! Academi c Press (Preprint 1972) . Banks, D., Percival, I. C, and Richards, D. 1973, Ap . Letters , 14., 161 . Biermann, P. 1973, A. and Ap . , 2_2, 407. Blaauw, A., Gum, C. S., Pawsey, J. L., Westerhout, G. 1960, M . N . R . A . S . , 121 , 123. Braes, L. L. E. and Miley, G. K. 1972, Nature, 237, 506. 226

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227 Bridle, A. H. and Kesteven, M. J. L. 1971, A. J . , 76, 958. Bridle, A. H. and Venugopal , V. R. 1969, Nature , 224 , 545. Brockelhurst, M. 1970, M.N.R.A.S . , 148 , 417. Brown, R. L. and Balick, B. 1973, Ap. J . , 185 , 843. Burgess, A. and Percival, I. C. 1968, Advances in Atomic and Molecular Physics , 4, 109. Burns, W. R. and Yao, S. S. 1969, Radio Sci . , 4, 431. Burton, W. B. 1970, A. and Ap. Supplement , 2, 261. Burton, W. B. 1971, A. and Ap . , 1_0, 76. Burton, W. B. and Shane, W. W. 1970, "Neutral Hydrogen in the Sagittarius and Scutum Spiral Arms," p . 39 7 in The Spiral Structure of Our Galaxy , IAU Symposium 38, W. Becker and G. Contopoulos eds, Dordrecht: D. Reidel. Caswell, J. L., Dulk, G. A., Goss, W. M., Radhakri shnan , V. and Green, A. J. 1971, A. and Ap . , T_2, 271. Cesarsky, C. J. and Cesarsky, D. A. 1971, Ap. J ., 169 , 293. Cesarsky, D. A. and Cesarsky, C. J. 1973a, Ap . J . (Letters) , 183 , LI 43. Cesarsky, D. A. and Cesarsky, C. J. 1973b, Ap. J . 184 , 83. Chaisson, E. J. 1972, Nature Phys . Sci . , 239 , 83. Chaisson, E. J. 1974a, Ap. J . , 189 , 69. Chaisson, E. J. 1974b, personal communication. Cram, T. R. 1973, TPOWER and SPOHER User's Manual , NRAO Computer Div. Internal Report No. 10. Dalgarno, A. and McCray, R. A. 1972, Ann. Rev. A. and Ap 10, 37 5.

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228 Dickel, H. and Dickel, J. 1974, Personal communication Downes, D. and Wilson, T. L. 1974, A. and Ap . , 34_, 133 Dulk, G. A. and Slee, 25, 429. 1972, Austral . J. Phys. , Dupree, A. K. 1969, Ap. J ., 158, 491. Dupree, A. K. 1972a, Ap. J ., 173 , 293. Dupree, A. K. 1972b, Ap. Letters , 1_2, 125. Dupree, A. K. 1974, Personal communication. Dupree, A. K. and Goldberg, L. 1970, Ann. Rev. A. and Ap 8, 231 . Dyson, J. E. 1969, Ap. J . , 155 , 47. Encrenaz, P. and Guelin, M. 1970, Nature, 227 , 476. Field, G. B., Goldsmith, D. W., and Habing, J. J. 1969, Ap. J. (Letters) , 155 , L149. Flannery, M. R. 1970, J. Phys. B: Atom. Molec. Phys ., 3. 1610. Gerola, H., Kafatos, M., and McCray, R. 1974, Ap . J . , 189 , 55. Goldstein, S. J. 1962, A. J ., 6_7_, 171. Goldwire, H. C. Jr. 1968, Ap. J. Supplement , 1_7, 445. Gordon, M. A. 1971, Ap. J . , 167 , 21. Gordon, M. A. 1972, Ap. J . , 174 , 361. Gordon, M. A. 1974, "The Radio Characteristics of H II Regions and the Diffuse Thermal Background," Chapt. 3 in Galactic and Extragala c tic Radio Astronomy , G . L . Verschuur and K. ~l~. Ke I I erman eds . , New York: Spr i nger-Verl ag . Gordon, M. A., Brown, R. L. and Gottesman, S. T. 1972, Ap. J . , 1_78, 119. Gordon, M. A. and Cato, T. 1972, Ap. J . , 1 76 , 587.

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229 Gordon, C. P., Gordon, K. J., and Shalloway, A. M. 1969, Nature , 222 , 129. Gordon, K. J. and Gordon, C. P. 1970, Ap. Letters , 5_, 153. "" ~ Gordon, M. A. and Gottesman, S. T. 1971, Ap. J . , 168 , 361 . Goss, W. M. 1972, A. and Ap. , 1_8, 484. Gottesman, S. T. and Gordon, M. A. 1970, Ap. J. (Letters) 162, L9 3. Guelin, M. 1974, Galactic Radio Astronomy , IAU Symposium No. 60, p. 51, Dordrecht: Re i del. Guelin, M., Guibert, J. Huchtmeir, W., and Wei i a chew, L. 1969, Nature, 221 , 249. Habing, H. J. 1970, personal communication cited by Hughes, Thompson, and Colvin (1971). Habing, H. J., Goss, W. M., Matthews, H. E.,and Winnberg, A. 1974, A. and Ap . , 3_5, 1. Hagen, J. P., Lilley, A. E., McClain, E. F. 1955, Ap. J . , 122 , 361. Hill, E. R., Slee, 0. B., and Mills, B. Y. 1958, Austral . J. Phys . , 11, 5 30. Hjellming, R. M. 1972, "Interstellar Cloud Properties Revealed by Pulsars," p. 69 in The Physics of Pulsars , A. M. Lenchek ed . , New York: Gordon and Breach . Hjellming, R. M. and Davies, R. D. 1970, A. and Ap . , 5, 53. Hjellming, R. M. and Gordon, M. A. 1971, Ap. J . , 164 , 47. Hjellming, R. M., Gordon, C. P., and Gordon, K. J. 1969, A. and Ap . , 2, 202. Hoang-Binh, D. and Walmsley, C. M. 1974, A. and Ap . , 35_, 49. "" " Hodge, P. W. 1974, P.A.S.P., 86, 845

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230 Holden, D. J. and Caswell, J. L. 1969, M . N . R . A . S . , 143 , 407. Hughes, M. P., Thompson, A. R. , and Colvin, R. S. 1971, Ap . J . Suppl ement , 23 , 323. Ilovaisky, S. A. and Lequeux, J. 1972, A. and Ap . , ]_8, 169. Jackson, P. D. and Kerr, F. J. 1971, Ap. J . , 168 , 29. Jackson, P. D. and Kerr, F. J. 1975, Ap . J . , 196, 723. Kaplan, S. A. and Pikelner, S. B. 1970, The Interstellar Med i urn , Cambridge, Mass.: Harvard University Press . Kellermann, K. I., Paul i ny-Toth , I. I. K. , and Williams, P. J. S. 1969, Ap. J . , 1_5_7, 1 . Komesaroff, M. M. 1961, Austral . J. Phys . , U_, 515. Kraus, J. D. 1966, Radio Astronomy , New York: McGrawHill. Kuhn, H. G. 1962, Atomic Spectra , New York: Academic Press . Large, M. I., Mathewson, D. S., and Haslam, C. G. T. 1961, M . N . R . A . S . , 123 , 123. Lilley, A. E. and Palmer, P. 1968, Ap. J. Supplement , 16. 143. Lin, C. C, Yuan, C, Shu, F. H. 1969, Ap. J . , 155 , 721. Lockman, F. J. and Gordon, M. A. 1973, Ap. J . , 182 , 25. Manchester, R. N., Murray, J. D., and Radhakri shman , V. 1969, Ap. Letters , 4, 229. Manchester, R. N. and Taylor, J. H. 1972, Ap. Letters , 1_0, 67. Mathewson, D. S., Healy, J. R. , and Rome, J. M. 1962a, Austral . J. Phys . , 1_5_, 354. Mathewson, D. S., Healy, J. R., and Rome, J. M. 1962b, Austral . J. Phys . , 15_, 369.

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231 Matthews, H. E., Davies, R. D., and Pedlar, A. 1973, M.N.R.A.S . , 165 , 173. Matthews, H. E., Pedlar, A., and Davies, R. D. 1973, M.N.R.A.S ., 165 , 149. Mauzy, R. 1968, Au tocrrel a ti on Receiver, Model II: IF Filter System , NRAO Electronics Division Internal Report No. 77. McDaniel, E. W. 1964, Colli si on Phenomena in Ionized Gases . New Yorkl John Wiley and Sons, Inc. Menzel, D. H. 1969, Ap . J. Supplement , 1_8, 221. Mezger, P. G. 1970, "The Distribution of H II Regions," in The Spiral Structure of Our Galaxy , I AU Symposium 33, Dordrecht: D. Reidel. Mezger, P. G. and Henderson, A. P. 1967, Ap. J . , 147 , 471. Mezger, P. G., Wilson, T. L., Gardner, F. F., and Milne, D. K. 1970, A. and Ap . , 4, 96. Mihalas, D. 1970, Stel 1 ar Atmospheres . San Francisco: W . H. Freeman and Co. Mihalas, D. and Routly, P. M. 1968, Galactic Structure . San Francisco: W. H. Freeman and Company. Mitchell, A. C. G. and Zemansky, M. W. 1934, Resonance Radiation and Excital Atoms . New York: The Macmi 1 1 an Company . Oster, L. 1961, Rev. Mod. Phys ., 3_3, 525. Pankonin, V. 1975, A. and Ap. , 38, 445. Parratt, L. G. 1961, Probability and Experimental Errors in Science . New York: John Wiley and Sons. Peach, G. 1972, Ap. Letters , 1_0, 29. Pedlar, A. and Hart, L. 1974. M.N.R.A.S . , 168, 577. Pengelly, R. M. and Seaton, M.J. 1964, M.N.R.A.S ., 127 , 165. Percival, I. C. and Richards, D. 1970a, J. Phys. B: Atom Mol ec. Phys . ,3,315.

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232 Percival, I. C. and Richards, D. 1970b, J. Phys . B : Atom. Molec. Phys ., 3, 1035. Percival, I. C. and Richards, D. 1970c, Ap. Letters , 4, 235. Percival, I. C. and Seaton, M. J. 1972, Ap . Letters , U_, 31 . Posener, D. W. 1959, Austral . J. Phys . , 1_2, 134. Prentice, A. J. R. and ter Haar, D. 1969, M.N.R.A.S . , 146 , 423. Radhakrishnan, V., Brooks, J. W . , Goss, W. M., Murray, J. D., and Schwarz, U. 0. 1972a, Ap . J. Supple ment , 24_, 1 . Radhakrishnan, V. and Goss, W. M. 1972, Ap. J. Supplement , 24, 161 . Radhakrishnan, V., Goss, W. M.. Murray, J. D., and Brooks, J. W. 19 72b, Ap . J. Supplement , 24, 49. Saraph, H. E. 1964, Proc. Phys. Soc , 8_3, 763. Schmidt, M. 1965, "Rotation Parameters and Distribution of Mass in the Galaxy," Chapt 22 in Galactic Structure , A. Blaauw and M. Schmidt eds., Chicago: Univ. of Chicago Press. Seaton, M. J. 1959, M.N.R.A.S ., 119 , 81. Seaton, M. J. 1962, Pr oc. Phys. Soc , 7_9, 1105. Seaton, M. J. 1964, M.N.R.A.S ., 127 , 177. Sejnowski, T. J. and Hjellming, R. M. 1969. Ap. J., 156 , 915. Shain, C. A., Komesaroff, M. M., and Higgins, C. G. 1961, Austral . J. Phys . , 1_4, 508. Shalloway, A. M., Mauzy, R., Greenhalgh, J., and Weinreb, S. 1968, Autocorrelation Receiver Model II : Operational Description , NRA0 Electronics Division Internal Report No. 75. Shore, B. W. and Menzel , D. H. 1968, Principles of Atomic Spectra. New York: John Wiley and Sons, Inc.

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233 Shu, F. H. 1973, Amer. Scientist , 61_, 524. Sorochenko, R. L. and Berulis, J. J. 1969, Ap. Letters , 4, 173. Sorochenko, R. L. and Berulis, J. J. 1971, Soviet Astron A. J . , 14, 683. Spitzer, L. Jr. 1968, Diffuse Matter in Space . New York: John Wiley and Sons ( I ntersci ence ) . Sullivan, W. T. Ill, 1973, Ap . J . Suppl . Series , 2_5_, 393. Terzian, Y. 1972, "Dispersion Measures and Distances to Pulsars," p. 85 in Physics of Pulsars , A. M. Lenchek ed . , New York: Gordon and Breach. Turner, B. E. 1973, Personal communication. Weinreb, S. 1963, A Digital Spectral Analysis Technique and its Application to Radio Astronomy , M . I . T . Research Laboratory of Electronics, Technical Report 41 2. Wendker, H. J. 1970, A. and Ap . , 4, 378. Westerhout, G. 1958, B_JUJi. , 1_4, 215. Westerhout, G. 1969, Maryland-Green Bank Galactic 21 cm Line Survey , 2nd Ed, College Park, Md . : Univ. of M a ry 1 a n d . Whiteoak, J. B. and Gardner, F. F. 1974, A. and Ap ., 3_7, 389. Wilson, T. L. 1972, A. and Ap . , ]_9_, 354. Winnberg, A., Habing, H. J., and Goss, W. M. 1973, Nature Phys. Sci. , 243_, 78.

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BIOGRAPHICAL SKETCH Andrew Wilkin Seacord, II was born on 19 September, 1941 in Milwaukee, Wisconsin. His family moved to LaGrange, Illinois in 1942, to Evansville, Indiana in 1946, and to Western Springs, Illinois in 1952. He graduated from Lyons Township High School in June, 1959. He attended the University of Illinois from September, 1959 to February, 1964, and graduated with a Bachelor of Science in Electrical Engineering. His graduate work began at the Pennsylvania State University in March, 1964, where he worked in the Ionosphere Research Laboratory and the Radio Astronomy Observatory. The thesis work for his Master of Science degree in Electrical Engineering was done under Dr. John P. Hagen. The degree was conferred in September, 1966. Shortly before his thesis was completed, he began working for the National Aeronautics and Space Administration, at the Goddard Space Flight Center. There he served as an engineer with solar flares patrol instrumentation for the Apollo program. On 9 March, 1968, he married Penelope Lee Connors. They presently have two children, Rebecca Louise and William Wilkin. 234

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235 In June, 1970, he began work toward the doctorate at the University of Florida. During the first two years, he worked with Dr. T. D. Carr on the 26.3 MHz array telescope. However, for several years, he had been interested in galactic structure and the interstellar medium. So, when Dr. S. T. Gottesman joined the faculty, Mr. Seacord began studying under him. The Doctor of Philosophy is expected to be conferred in August, 1975.

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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. % l>yfyo>v«4~ p/tU-M ! / Stephen T. Gottesman, Chairman Assistant Professor of Astronomy I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Thomas D. Carr Professor of Physics and Astronomy I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. 2& Alexanden^G.^Smi th Professor of Physics and Astronomy

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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. »C\ rr-x Va-c^ ,i..^l'.M. WV Michael T. Parkinson Associate Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Thomas L. bai ley Professor of Physics arft Electrical Engineering This dissertation was submitted to the Graduate Faculty of the Department of Physics and Astronomy in the College of Arts and Sciences and to the Graduate Council, and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. August, 1975 Dean, Graduate School

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