; 72-15,181 I i BUERGER, Edith Guderley, 1943- l ABUNDANCES AND IONIZATION STRUCTURE IN LOW EXCITATION PLANETARY NEBULAE.
j The Ohio State University, Ph.D., 1971 Astrophysics
j
University Microfilms, A XEROX Company, Ann Arbor, Michigan
THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED ABUNDANCES AND IONIZATION STRUCTURE
IN
LOW EXCITATION PLANETARY NEBULAE
DISSERTATION
Presented in Partial Fullfillinent of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University
By
Edith Guderley Buerger, B.A., M.Sc.
* * rr * *
The Ohio State University 1971
Approved by
Department *6f Astronomy PLEASE NOTE:
Some pages have indistinct print Filmed as received.
University Microfilms, A Xerox Education Company ACKNOWLEDGEMENTS
I would like to extend my thanks to my adviser, Dr. Stanley
J. Czyzak, for suggesting this problem, for his interest in my pro gress and for his continued support.
I appreciate the continued interest which Dr. Lawrence H.
Aller has shown, as well as the time he spent discussing the prob lem with me.
I am grateful to Dr. J. P. Harrington for explaining his computer program and letting me use it.
I thank the Instruction and Research Computer Center of The
Ohio State University for providing the necessary computing time to carry out the calculations.
The patience, understanding and encouragement given me by my husband will always be remembered.
This work was done while the author was a research assistant supported by the National Science Foundation.
ii VITA.
October 6 , 1943 Horn - Braunschweig, Germany
1965 B.A., Earlham College Richmond, Indiana
1965-1967 NDEA Fellow at Ohio State University
1967 M.Sc., The Ohio State University, Columbus, Ohio
1967-1968 Research Assistant in the Department of Astronomy at Ohio State University
1968-1969 NDEA Fellow at Ohio State University
1969-1971 Research Assistant in the Department of Astronomy at Ohio State University
PUBLICATIONS
"Periodic Changes in the Spectrum Variable UR 234." Publica tions of the Astronomical Society of the Pacific, Vol. 79, pp. 589-592, December, 1967
"Spectrophotometric Studies of Gaseous Nebulae. XVIII The Bright Intermediate-Excitation Planetary NGC 6210." (Co author with L.H. Aller and S.J. Czyzak) The Astrophysical Journal, Vol. 162, pp. 7S3-787, December, 1970
iii TABLE OF CONTENTS
Page ACKNOWLEDGEMENTS...... ii
VITA ...... iii
LIST OF T A B L E S ...... vi
LIST OF ILLUSTRATIONS...... viii
INTRODUCTION ...... 1
Chapter 1. PHYSICAL PROCESSES ...... 0
General Considerations The Ionization Equilibrium The Energy Balance The Radiation Field Line Intensities
2. MATHEMATICAL CONSIDERATIONS ...... 15
Absorption Coefficients Recombination Coefficients Recombinations Free-Free Transitions Collisional Losses The Transfer Problem Quadrature Points
3. RESULTS OF VARYING NEBULAR PARAMETERS ...... 27
Physical Size Effect of Sulfur, Chlorine and Argon on Thermal Structure Collisional Cooling Effects Energy Distribution of Central Star Effect of Density Changes Importance of Diffuse Field
iv TABLE OF CONTENTS (cont.)
Chapter Page A, A FEW PARTICULAR NEBU L A E ...... 6 8
IC 418 IC 3568 IC 4593 NGC 7662
5. CONCLUDING REMARKS...... HI
LIST OF REFERENCES...... 117
APPENDIX...... 122
v LIST OF TABLES
Table Page
1. Defining Parameters for Models 1-5: Effect of Changing Distance...... 41
2. Relative Line Intensities for Models 1-5: Effects of Changing Distance ...... 44
3. Ionic Abundances Integrated Over the Nebular Volume of Models 1 and 3: Effect of Changing Distance...... 46
4. Defining Parameters for Models 6-11: Effect of S, Cl, and Ar on Thermal S t r u c t u r e ...... 49
5. Relative Line Intensities for Models 8-10: Effect of Adding S, Cl, and A r ...... 52
6 . Defining Parameters for Models 12-14: Effect of the Ultraviolet Flux Distribution ...... 54
7. Relative Line Intensities for Models 12-14: Effect of Ultraviolet F l u x ...... 57
8 . Defining Parameters for Models 15 and 16: Effect of Changing Density ...... 58
9. Relative Line Intensities for Models 15 and 16: Effect of Changing Density ...... 59
10. Ionic Abundances Integrated over the Nebular Volume of Models 15 and 16: Effect of Density Changes...... 62
11. Defining Parameters for Models 17 and 18: Effect of Diffuse Field ...... 63
12. Relative Intensities for Models 17 and 18: Effect of Diffuse Field ...... 65
13. Relative Line Intensities in IC 4 1 8 ...... 80
14. Defining Parameters for Specific Nebulae ...... 81
vi LIST OF TABLES (cont.)
Table Page
15. IC 418: Ionic Abundances Integrated Over Nebular Volume for Homogeneous M o d e l ...... 83
16. IC 418: Ionic Abundances Integrated Over Nebular Volume for Inhomogeneous Model ...... 83
17. Relative Line Intensities in IC 3568 94
18. IC 3568: Ionic Abundances Integrated Over Nebular Volume . 95
19. Relative Line Intensities in IC 4593 ...... 103
20. IC 4593: Ionic Abundances Integrated Over Nebular Volume . 104
21. Relative Line Intensities in NGC 7662 ...... 109
22. Chemical Composition of Planetary Nebulae ...... 115
23. Atomic Parameters for Statistical Equilibrium Equations . . 125
24. Parameters for the Absorption Coefficients Represented by a Power L a w ...... 128
25. Parameters for the Absorption Coefficients Represented by the Quantum Defect Method ...... 129 2 26. Atomic Parameters for Fine Structure Transitions of P ions 131
27. Atomic Parameters for Fine Structure Transitions of P ions 133
28. Atomic Parameters for Forbidden Transitions Between Ground T e r m s ...... 134
29. Atomic Parameters for Permitted Transitions ...... 136
vii LIST OF ILLUSTRATIONS
Figure Page
1. Electron Temperature for Models 1 and 3: Effect of Changing Distance ...... 42
2. Ionization Distribution for Selected Ions of Models 1 and 3: Effect of Changing Distance...... 43
3. Electron Temperature for Models 4 and 5: Effect of Changing Distance ...... 47
4. Ionization Distribution for Selected Ions of Models 4 and 5a; Effect of Changing Distance ...... 48
5. Electron Temperature for Models 6 and 7: Effect of S, Cl, and Ar on Thermal Structure...... 50
6 . Electron Temperature for Models 8-10; Effect of S, Cl, and Ar on Thermal Structure...... 51
7. Electron Temperature for Model 11: Coolant Series .... 53
8 . Electron Temperature for Models 12-14: Effect of Ultraviolet Flux Distribution ...... 55
9. Flux Distribution of Central S t a r ...... 56
10. Electron Temperature for Models 15 and 16: Effect of Changing D e n s i t y ...... 60
11. Ionization Distribution for Selected Ions of Models 15 and 16: Effect of Changing Density...... 61
12. Electron Temperature for Model 17: Effect of Diffuse F i e l d ...... 64
13. Electron Temperature for Model 18: Effect of Diffuse F i e l d ...... 67
14. IC 418: Electron Temperature Distribution for Homogeneous M o d e l ...... 84
15. IC 418: Ionization Distribution for Homogeneous Model . . 85
viii LIST OF ILLUSTRATIONS (cont.)
Figure Page
16. IC 418: Electron Temperature Distribution for Inhomogeneous M o d e l ......
17. IC 418: Ionization Distribution of Sulfur for Inhomogeneous Model ......
18. IC 3568: Electron Temperature Distribution ...... 96
19. IC 3568: Ionization Distribution ...... 97
20. IC 4593: Electron Temperature Distribution ...... 105
21. IC 4593: Ionization Distribution ...... 106
22. Energy Level Diagrams for Ions Giving Rise to Forbidden L i n e s ...... 123
ix Introduction
Determination of the abundance of the elements has been one of
the fundamental problems of astrophysics. For planetary nebulae, the
classical discussion of abundance determination is by Bowen and Wyse
(1939) who obtained abundances for NGC 7027 which do not deviate
significantly from the solar abundances. Considering the assumptions
which had to be made due to the lack of cross-section data, transi
tion probabilities, etc., their results are indeed noteworthy. Since
the initial efforts of Bowen and Wyse, many other investigations have
been carried out which have utilized recent calculations of atomic
parameters.
With the advent of accurate transition probabilities and col
lision strengths, the recent efforts at abundance determinations
have considerably refined the earlier attempts to determine the ionic
abundances. Thus, Aller and Czyzak (1968) have solved the equations
of statistical equilibrium for a given temperature and density to
obtain equations for the relative number in each state of ionization
for ions represented by forbidden lines. Piembert and Costero (1969)
have improved on this method by removing the restriction that only
one temperature is present in the nebula.
Nevertheless, after obtaining the ionic abundances the above methods still rely on an assumption concerning the distribution of ions in unseen states. The usual assumption is that the fraction of an element in a particular state of ionization depends only on the ion ization potential of the previous state of ionization. This assumption neglects, for example, the different absorption coefficients for dif ferent ions as well as the density fluctuations which exist in many nebulae. Consequently, uncertain corrections often have to be applied for ions in "unseen" states.
The development of theoretical models and their application to particular nebulae (Harrington 1969, Flower 1969b) enables detailed abundance studies to be carried out. Using theoretical models to cal culate the run of ionization and electron temperature with radius eliminates the greatest source of uncertainty in abundance determin ations — namely, that of estimating the fraction of an element pre sent in an unobservable state of ionization. With the addition of inhomogeneities into the model, a more realistic determination of the distribution of the ions in the ionization states is possible; thus the abundances m$y be determined more accurately.
In this study theoretical models are used to determine the chemical composition of a few fairly low excitation planetaries.
Theoretical models are sensitive to chemical abundances which may be adjusted to match the model to the nebula. Thus, by a successful comparison of the calculated emission line intensities with the ob served nebular lines, one may obtain the abundances of the elements.
This method is a large step forward in abundance analysis. In Chapter
A, theoretical models are presented which successfully predict the line spectra of a few particular fairly uniform nebulae. The calcula tions are made with the computer program developed by Harrington (1968) and later (1969) applied to the high excitation nebula NGC 7662. This program has been revised to include the elements sulfur, chlorine and argon, for which accurate atomic parameters have now become available
(Krueger and Czyzak 1970). The effect of sulfur, chlorine and argon on the thermal structure of the nebulae was investigated. In addition for particular cases, some of the more uncertain nebular parameters
■are investigated, to determine their effect on the structure (i.e., the ionization distribution and electron density and temperature) of the nebula and consequently on the abundances.
The nebulae are idealized as spherical gas shells surrounding a hot central star. The gas consists of H, He, C, N, 0, and Ne in addi tion to the elements mentioned above, and is assumed to be in a steady state. The radiation field of the central star is described by model tr if atmosphere calculations of Bohm and Deinzer (1965, 1966) and Bohm
(1969). The equations of ionization equilibrium and energy balance are solved simultaneously at each point in the nebula and for each density considered. Thus, one obtains the ionization structure, the electron temperature, and the electron density throughout the nebula.
The line intensities, integrated over the nebular volume, of some of the emission lines of H, He, [N II], [0 II], [0 III], [Ne III],
[Ne IV], [Ne V], [S II], [S III], [Cl III], [Cl IV], [Ar III], and
[Ar IV] are then obtained. Comparing these emission lines with the observations then gives information on the abundances of the elements. f
4
The effect of inhomogeneities present in many planetary nebulae
can also be included. They have been described as two densities exist
ing simultaneously in a particular shell surrounding the star. This
method is definitely useful for comparison with observations inte
grated over the nebular volume (e.g., photoelectric spectrum scanning).
Consideration of these fairly simple models can be of value in deter
mining how well the observations can be predicted without the necessity
of more complicated geometrical models,
A considerable amount of work has been done recently on theore
tical models for planetary nebulae and a few attempts have been made
which compare a theoretical model with actual observations. Purely
theoretical studies have been carried out by Goodson (1967), Harring
ton (1968), and Flower (1969a). Actual comparison of theoretical
models with observations have been made by Harrington (1969), Flower
(1969b), and Kirkpatrick (1970). Harrington and Kirkpatrick each
studied high excitation nebulae, whereas Flower concerned himself
with both a high and a low excitation nebula. Each of the above made
theoretical models for NGC 7662; Harrington was the most successful
in predicting its observed spectrum.
Kirkpatrick concerned himself primarily with presenting more
accurate models of the observed density distribution (axially symme
tric models rather than spherically symmetric ones), but, as a result
of the complicated geometry, was forced to neglect the diffuse radia
tion field. This assumption limits him to making models of optically
thin nebulae. Flower assumes spherically symmetric nebulae. Rather than solv ing the ionization equilibrium equation and the thermal balance equa tion simultaneously, he calculates an ionization distribution for a constant electron temperature, then solves the thermal balance equa tion. He uses an "on the spot" approximation for the diffuse field.
We have chosen to study the nebulae IC 418 (excitation class 3),
IC 4593 (excitation class 4) and IC 3568 (excitation class 5) in some detail. Each of these nebulae has a fairly regular structure, although density fluctuations have recently been detected in IC 418 (Kron and
Walker, 1970) and are most likely also present in IC 4593 and IC 3568.
IC 418 has been observed more extensively than the other two nebulae — consequently more emphasis will be placed on it. For each of the nebulae the temperature for the central star which best reproduces the observations has been determined. Using distances and densities ob tained from the literature, models for each of the nebulae have been constructed which give the ionization structure and the run of elec tron temperature with distance from the central star. These models include the elements sulfur, chlorine and argon; sulfur is shown to have a sizeable effect on the thermal structure of low excitation nebu lae. This is an effect not previously considered. The resultant line intensities, integrated over the nebular volume, are calculated and com pared with the observations. Thus, the abundances of hydrogen, helium, carbon, nitrogen, oxygen, neon, sulfur, chlorine and argon are deter mined for each of the nebulae. Chapter 1
PHYSICAL PROCESSES
General Considerations
The physical processes occurring in a planetary nebula may be divided into two parts: (a) the processes acting to establish equili brium in a small volume element and (b) the transfer of radiation through the nebula.
The radiation field gives rise to ionizations which heat up the nebular gas by supplying it with energetic electrons. All ionizations are from the ground state, as a result of the low density of the gas as well as the large dilution factor of the stellar radiation. These two effects imply that the time between either radiative or collisional ex citations is much greater than the lifetime of any level which is not highly metastable. Electron-electron collisions distribute the energy of an electron ejected by photo-ionization among the other electrons of the gas; this results in a Maxwellian velocity distribution. The three major ways of losing energy are recombinations with ions, free-free transitions and collisions with ions having low lying energy levels.
The electron-ion collisions give rise to ions in metastable levels which cool the gas by emitting energy in the radiation of the "forbidden lines" as the atom returns to the ground level. This is the major cool ant of the gas. The electron temperature of the gas is found by balancing the electron gains against the losses. The equations appli
cable to these considerations will be discussed in the following sec
tions as will also the equations describing the transfer of radiation.
The Ionization Equilibrium
Let . represent the number of atoms per cubic centimeter of the til i kind which are ionized (j-1) times. One can then write the equa
tion for equilibrium between two states in the form
N N. .... a , . (T ) = N, , : J a. . (v) dv, (1) e i, j+1 ij e ij JVj.^ hv V ij * where N is the electron density. (X. . (T ) is the total recombination e 1 j e coefficient upon all levels of the ion as a function of the electron
temperature, is the mean intensity of the radiation, a ^ (v) is the
continuous absorption coefficient from the ground state and is the
ionization potential. th The total number of atoms of the i kind is
KR Ai " U Nlj ’ (2) where N„ denotes.the number of hydrogen atoms per cubic centimeter, A. H 1 fcH is the number of atoms of the i kind relative to hydrogen, and where
the sum extends over the k states of ionization being considered.
These two equations, considered for each element, are sufficient to de
termine N . , explicitly as a function of N and T . ij e e For hydrogen and neutral helium, equation (1) has an additional term arising from photoionizations caused by the Lyman-alpha line radi ation of singly ionized helium. The determination of the intensity of this line radiation is rather difficult. Thus, since it is uncertain whether He II Ly-a radiation is absorbed by H and He, it has been as sumed that 50 per cent of the He II Ly-a radiation is absorbed in the region where it is formed (i.e., "on the spot")by H or He. The ioni zation equilibrium equations for hydrogen and helium are then coupled, but may still be solved explicitly. As a result of these uncertainties this radiation is neglected for all other ions.
Since the number of electrons per cubic centimeter is just the sum of the electrons contributed by each atom, the equation of ioniza tion equilibrium is just
Ne = L l i l (i- » Nij V • (3>
Nearly all electrons come from the ionization of hydrogen and helium.
The Energy Balance
All of the energy of the electron gas comes from photoioniza tions. Define <1*^^ as the amount of energy contributed to the gas by th ionization of the ij ion. Then V. . 0 . . = . 4ir j a. . (v) ( 1 ----- 1) dv . (4) rj V:ij V xj V
Since we have considered photoionizations of H and He by He II Ly-a, we must include this process for gaining energy. The rate of energy gain for this process,P_, represents the product of the number of Ij Ly -a photons produced and the energy of the ejected electron.
The energy losses for the electron gas may be written as (Menzel and Aller 1941) where P(T^) is the loss by recombinations, is the loss by free- free transitions and C . .. is the loss by radiative de-excitations. I jk Balancing energy gains against losses in a small volume element thus gives
[?/ i j *13 + PL ] - Ne fi/ii ® (Ie> +
Pff(V + £ Cijk)] ' (6)
The left hand side and the first two terms in the right hand side of equation 6 are dominated by hydrogen and helium. Equations (3) and
(6 ) may then be solved simultaneously for the electron density and tem perature by two-dimensional Newton-Raphson iteration in the volume ele ment under consideration, for a mean intensity of radiation, . The convergence criteria used were
AN /N < 10" 5 and AT /T < 10" 5 ; (7) e e e e they were usually satisfied in several iterations. The truncation errors introduced by this approximation are negligibly small compared with the errors resulting from other approximations used in the solution of this problem.
An actual planetary would have an additional mechanism for los ing energy which has been neglected here — namely, work done by expan sion of the gas. Sofia and Hunger (1968) have shown that at least for some nebulae this might be important. 10 The Radiation Field
The radiation field in a planetary nebula consists of two parts: the radiation coming directly from the central star, and the dif fuse field produced by the gas itself, Thus the mean intensity,
Jy, which appears in equations (1 ) and (4) may be written as:
Jv ■ Jv + JvD (8)
The stellar mean intensity passing through a shell of gas at a distance
R from the star may be written as
Fv E* 2 - V R> ’
The diffuse component of the radiation is given by
4tt Jv°(R) = jv (r) exp[-Tv (R, r) ] dr , (10) where j^(r) is the emission coefficient at point r and Tv(R,r) is the *■* optical depth along the path joining R and r.
For the stellar fluxes, model atmosphere calculations have been used when available; otherwise the blackbody approximation has been used.
The most important source of the diffuse radiation emitted by the gas is the Lyman continuum which results from recombinations directly to the ground state of H I and He I or He II. Furthermore, the Balmer 11
continuum of He II (the radiation emitted by recombinations to the n«s2
level) is absorbed by hydrogen. The emission coefficient, j , for re
combinations is just the recombination rate times the energy of the
photon.
As mentioned previously, the line radiation due to the Lyman
series of He II also contributes to the diffuse field and has been in
cluded. It is assumed to be absorbed "on the spot" rather than includ
ing it in the transfer problem.
The two photon emission of He I and He II has also been included
— for He II it arises from the 2s-ls transition and for He I it arises 12 3 2 from the (ls2s S-ls ) and (ls2s S-ls ) transitions. The frequency dis
tribution of the two photon emission for He II is given by Spitzer and
Greenstein (1951); for He I it is given by Dalgarno (1966).
The mathematical solution to the transfer problem will be dis
cussed in Chapter 2.
Line Intensities
In order to compare the theoretical models of the planetary neb
ulae with observations, calculations of intensities of several of the
forbidden lines often seen in planetaries have been made. Some of the
ions for which such calculations have been made are [N II], [ 0 II],
[0 III], [Ne III], [Ne IV], [Ne V], [S III], [Cl III], [Cl IV], [Ar III], and [Ar IV], Calculations are also carried out for emission due to Hf3,
He I U 4 7 1 and \5876, and He II U 6 8 6 and X5412. 3 The electron density dependence of lines arising in the p con
figuration makes these ions especially valuable for study. The line 12
ratios of 11 ^ and V 52QQ of ^ bave lonS been recognized to
be indicative of density fluctuations. Since the recent work of
Krueger, Aller and Czyzak (1970) and Saraph and Seaton (1970) it is now 3 possible to study line ratios in the 3p series. The line ratios
X4068 \6716 „ r„ X5517 r„, , M 7 1 1 X.6716 + X.6730’ X6730 of [SL II], TT X55377T 7 of L[C1 I11] J and TTV/n X4740 and
1?Y4 237 of are density indicators. Thus, by studying
line intensities as observed in the nebula, one can obtain estimates of
the densities in the regions in which the lines have been formed.
Unfortunately some of the line ratios quoted here have only
scanty data available from the observational standpoint; many fall be yond the wavelength region in which IIa-0 plates are sensitive (i.e.,
X6716, \6730, X5517, X5537, *.7171, X.7237) . [S II] X4068 is often blended with a mercury line and the line X4-711 of [Ar IV] is usually blended with X4713 of He I. But, progress is being made as astronomers are beginning to observe more seriously in the longer wavelength region
(e.g., Aller and Walker 1970; Czyzak, Walker and Aller 1969),
To be able to calculate the line intensities accurately for a given electron temperature and density requires solution of the appro priate equations of statistical equilibrium. Thus, accurate transition probabilities and collision strengths are needed. Transition probabil ities for all the ions in question have been available for quite some
time, but collision strengths for S, Cl and Ar have only recently 3 become available. In particular, for 3p configurations, only colli- 4 2 2 sion strengths between the terms S, P and D had been available. 13
However, the necessary fine structure collision strengths are now avail
able (Krueger and Czyzak 1970, Blaha 1968),
The atoms may enter metastable levels by collisions; forbidden
line radiation is emitted in transitions to lower levels. For posi
tive ions one can integrate over the Maxwellian velocity distribution
of the electrons. Then one obtains the rate of collisional excitation
from a level i to a level j
ft 5 ‘ q. . = 8.62 • 10 t " ° exp (-1.1605 -- 1) , (11) ij
where is the statistical weight of the lower level, is the sep- -4 aration of the energy levels in electron volts, t=10 T^, and is
the collision strength for the excitation by electron impact to level
j from level i. The corresponding rate of de-excitation from level j
to level i is
n . . c q.. = 8.62 * 10 - 2J- t " ° . (12) Ji ao.
The equations for statistical equilibrium of a five level atom may be written in the following form:
5 5 N £ N.q. . + 2 N.A. . = N. (N 2 q..+E A..) e i=l 1 i>i 1 1J J e i=l J1 i
with j=2,3,4,5, along with the condition that the populations of the
levels sum to the abundance of the ion.
The atomic parameters used for the solution of these equations
are given in the Appendix in Table 23. The transition probabilities
for N II, 0 II, 0 III, Ne III, Ne IV, and Ne V are from Garstang
(1968). The collision strengths for these elements have been taken 14 from Saraph, Seaton and Shemming (1969). The transition probabilities for the 3p^ ions were taken from Czyzak and Krueger (1963) and the col lision strengths are from Krueger and Czyzak (1970). Values of the energy level separations are given in electron volts; they are from
Moore (1949) .
The equations of statistical equilibrium were solved by matrix inversion to obtain the populations of the various levels. The line intensities were calculated at each radial point and then integrated over the nebular volume.
Diagrams showing the transitions giving rise to the calculated forbidden lines are given in the Appendix in Figure 22. Chapter 2
Mathematical Considerations
This chapter deals with some of the parameters which have been
used in the previous equations but which have not yet been discussed
in detail. Some of the parameters which will be considered are the ab
sorption and l-ecombination coefficients necessary for the solution of
equation (1). To solve the equation of energy balance one needs to
consider recombination rates, free-free transitions and collisional
cooling mechanisms.
The integration of equations (1) and (4) is done by numerical
quadrature. The determination of the quadrature points for these equa
tions is discussed in this chapter. A discussion of the method for
solving for the radiation field at every point in the nebula is given.
Absorption Coefficients
The absorption coefficients for hydrogen and helium are well
known and have been used in the form given by Hummer and Seaton (1963,
1964). For most of the heavier elements, with a 2p^ ground configura
tion, the absorption coefficients were taken from Henry (1970). He has
used an equation of the form -S -(S+l) a,,(v) “ C [a(-^-) + (1-a) (-v- ) ] , (14) J i j i j where is the ionization potential of the ion in question. The 16
results of Henry are in satisfactory agreement with the few available
absorption coefficients obtained experimentally. In general, Henry's
absorption coefficients were slightly higher than those of Seaton
(1958) at low energies and lower than those calculated by Seaton at
high energies. The values of C, a, S and are listed in Table 24
which is given in the Appendix.
The total absorption coefficient for each ion must include the
different possible thresholds (i.e., for each ion. This has been
included in the calculations and may be seen in the tables where sev
eral entries are listed for a particular ion, 2 The absorption coefficients for highly ionized atoms with 2s
and 2s ground states were taken from two sources. For C III and C IV,
parameters for C, a and S were given by Silk and Brown (1971). For
N IV, N V, 0 V, and 0 VI, the quantum defect method, as given by
Burgess and Seaton (1960a), was used by Harrington (1967); his results were verified and have been adopted here.
The results of Silk and Brown were also used for the absorption
coefficient of Ar I (3p^ ground state configuration); they were found
to be in good agreement with experimentally obtained absorption coef
ficients (Marrl967, Huffman 1964).
For the sulfur, chlorine, and argon ions with 3p^ or 3s^ ground
state configurations, no results were available. Thus, the quantum de
fect method of Burgess and Seaton was used for some new calculations of
the absorption coefficients. 17
The quantum defect method makes use of interpolated or extra polated quantum defects to determine the asymptotic forms of atomic wave functions from known energy levels of the initial atomic system.
The quantum defect takes into account the change in energy levels due to electrons surrounding the nucleus, relative to what one would expect from the hydrogenic approximation. Thus, for example, for "s" terms the quantum defect should be larger than for "p" terms, since there is less shielding for the former.
The quantum defect method may be used both in calculation of atomic transition probabilities and of photoionization cross-sections; here we are concerned with the latter.
One can write the equation for the cross-section of a N-electron atom as
2 1 - ■ " I "2 <>»* + k > — s <15>
(Burgess and Seaton 1960a) where
S = E'lJWa) R ^(i,k2) dT|2 (16) and where a is the Bohr radius, a the fine structure constant, I „ the o nf 2 threshold ionization energy, k the electron kinetic energy, and oj the statistical weight of the initial energy level of the atom (or ion). 2 ^ (a) refers to the wave-function of the initial atom, while (i,k ) represents the final wave-function of the system. R is the vector sum of the radius vectors of the N electrons. The summation in (16) is carried over all initial and final states.
It is shown that 18
— = £ . C,|/Tp (r) r G t (r) dr |2 (17) “ * = j? ± 1 njC kj£
where P is the radial function for the n / electron. If more than one ne value of the final state is allowed, equation (17) includes a summation
over them. The C^,, are algebraic factors, obtained from integrations
over spin and angular co-ordinates; they can be found in the literature
(Brtes 1946). Let
g W j e ' j O = InJZ f t Since 2 A ira a> „ o a- (njE) = — r-S (I - + O S C | f t v M V 3 nf =^+ ^ i r Gkjj, (r) dr|2 , (u) then 2 2 Aircxa I - + k ° / rug 3V (n*} 3 ( j 2 > X %L=R±X V nJ? |g(vJ?;€ 'Jl,)|2 . (20) This is the equation as given by Burgess and Seaton. The expression given for g is g(v^; e'j?1) = - 5 g ^ * cosir [v + w'(e') + & ' (V,f) x(vX;€'J2')] (21) with g Mv^e'r) = ^.(v) + ^ e,v aM , + , . v 2 ttt2' . (V) (23) 1 + e v y x and 19 |(V,^) = 1 + - ^ 3 • (24) v de In the above equations V is the effective quantum number,n'(e') is the quantum defect of the n ’/' series, u is the quan tum defect of the n/ series, and e' is the energy of the ejected elec tron divided by the square of the charge on the remaining ion. Burgess and Seaton have carried out extensive calculations, given in tabular form, for many of the parameters needed in equations (22) and (23) . However, one must still determine // and graphi cally for each ion; for this one needs many energy levels for the ion. Unfortunately, these data are often not complete; thus, well-known methods (Edlen 1964) were used to extrapolate for the missing levels. After some manipulation, the above equations may be rewritten in a more useable form for configurations A where -f= 0 or 1 (i.e., sq or pq configurations)." _g aij(y) = aw _i 10"18 ty '**'*“1 cos2 + a IQ"18 fy cos2 hr (d + T n j r i T aj , m + .i+d]) • (25) where y is the ratio of the photon energy to the ionization energy. For-/s=0, the first term of the expression vanishes. The parameters I have obtained for each ion are given in Table 25 which may be found in the Appendix. Since the experimental data on absorption coefficients are rather limited, one is heavily dependent on the theoretical results. However, theoretical methods have proved reliable in previous determinations of absorption coefficients, so they may be relied upon. 20 Recombination Coefficients The recombination coefficients needed for the solution of equa tion (1) have been used in the approximate form given by Seaton (1959) for hydrogenic ions. This form relies on using asymptotic expansions for the Gaunt factors. The total number of recombinations per unit volume per unit time into all levels may be expressed as NeN ^ a „ ( \ ) where OfjjCx) = 5.197 • 10"4 Z \ ° ' 3 (0.4288 + 1/2 In*. +0.469X'*333) (26) 2 and where = 15.789Z /t, Z is the residual charge, and t is the scaled 4 electron temperature (10 t = T ). The maximum errors expected from this approximate form have been compared with the results of more ac curate calculations by Burgess (1958) and were found to be about 2.5 per cent for (t/Z ) < 2. Burgess and Seaton (1960c) have calculated the recombination coefficients for He I and have shown that the above expression is at most a few per cent small. Except for the work of Burgess and Seaton (1960b) on oxygen, there are little additional data available on recombination coefficients. Thus, the above general equa tion was used for all ions. For the more highly ionized atoms, the higher energy levels which receive most of the recombinations tend to be hydrogenic, so the approximation is satisfactory (Harrington 1967). Recombinations The energy loss rate by recombinations, which enters the energy balance equation, has also been taken from Seaton (1959). The total 21 kinetic energy loss per unit volume per unit time of electrons recom bining to all levels may be expressed as NeN.^p(X) where P(X) = 2.85 10“ 2 5 Z2t°‘5[-.0713 + 1/2 lnX- ’33) . (27) X, 2 and t are all defined as above. This expression has been used for all ions. The errors are comparable to those involved in equation (26). Free-Free Transitions The loss rate for free-free transitions has been used in the form given by Allen (1963) p ff(t) = 1.435 • 10" 2 5 t° ' 5 (j-1) 2 Ne Ntj (28) Free-free transitions do not play an important role in the cooling of planetary nebulae. Collisional Losses The inelastic collisions of electrons with ions are the most effective way in which the electron gas loses energy. The energy is completely lost to the electron gas if the atom returns to the ground state by a radiative process. The rates for collisional excitation and de-excitation have al ready been given by equations (11) and (12) respectively. Knowing these rates, the equations of statistical equilibrium can be solved for the transitions of interest. In gaseous nebulae transitions of interest may be grouped into three categories: (1 ) the case where collisional de-excitations may be neglected, because ^ q ^ is so small compared to the Einstein trans- sition probability, 4.^; (2 ) the case of a two-level atom with 22 collisional de-excitation; (3) the case of a three-level atom with col lisional de-excitation. For each of the above cases, the equations of statistical equilibrium may be solved to obtain the populations of the levels and consequently the energy loss rate, which may be expressed in the form E. . = N N. .C. (N ,T ) (29) ij e ij ljk e e For example, for the first case, the energy loss rate is the energy of the transition times the rate of upward transitions. Thus, Cijk = 1.602 • 10-12 elk ,lk , (30) th where k denotes the various energy levels of the ij ion which are under consideration. For a two-level atom the energy loss rate is E = 1.602 • 10- 1 2 e12 N2 , (31) which can also be written in a form as equation (29) upon solving the statistical equilibrium equations for N^. There are several categories of energy levels which are impor tant for collisional cooling of gaseous nebulae. One of these consists 2 3 of the energy levels formed by the terms of the atoms of np , np or np^ ground configurations. Secondly, one has the fine structure levels 2 4 5 present in the ground term in the np, np , np or np ground state 2 3 3 configurations -- namely the P or P levels. (The ground term of np has no fine structure.) For h'igh excitation nebulae one also has cooling by allowed transitions of the more highly ionized atoms with 2 2s or 2s ground configurations (Hummer and Seaton 1964). 23 2 4 For the ions having p or p ground configurations (e.g., N II, 0 III, Ne III, Ne V, S I, S III, Cl^ll Cl IV, Ar III, Ar V) both the fine structure levels and the terms are important coolants. For these ions, certain approximations are made which depend on the fact that the radi ative transition probabilities of the fine structure levels are much smaller than those of the transitions between terms. However, the col lision strengths are of the same order of magnitude for the terms and the fine structure levels. If the electron density is large enough so that collisional de excitation of the terms is important (i.e., if N q..»A..), then the v e ji/ collisional de-excitation of the fine structure levels will dominate the radiative de-excitations so much that the radiative de-excitations of the fine structure levels can be neglected. Then the terms are treated as a three-level atom. For lower densities, one can neglect the collisional de-excitation of the terms and treat the fine structure levels as a three-level atom. Table 26 gives the atomic parameters for 5 the fine structure transitions of ions with np or np ground state con- 3 figurations. Table 27 gives the parameters for the ions with P ground states. Table 28 gives the atomic parameters for the ions with low lying terms. (These tables may all be found in the Appendix.) When the fine structure levels within a terra are treated as one level, the equations given by Osaki (1962) have been used to obtain mean values for A , . and e ... ji ij In Tables 26, 27, and 28 the same references were used to obtain the atomic parameters as in Table 23. The atomic parameters given in these tables are good to about 10-15 per cent. 24 Table 29 gives the parameters for the permitted transitions. They have been taken from Bely, Tully and Regemorter (1963), Varsavsky (1961), Osterbrock (1963), or Hummer and Seaton (1964). Some of these parameters are only estimates, but no better results are currently available. The Transfer Problem The integral for the diffuse component of the stellar radiation field (equation 1 0 ) cannot be solved without specifying the solution everywhere; thus an iterative procedure is used. The choice of the first approximation depends upon the optical thickness of the nebula. For an optically thick nebula, the radiation is assumed to be absorbed in the same region in which it is produced (on the spot); thus the dif fuse intensity is set equal to the source function, j^(R)/k^(R). All models were assumed to be optically thick beyond 54.4 ev. (i.e., they absorb the He II Lyman continuum). For frequencies at which the nebula is optically thin, the diffuse field is neglected. The construction of models thus proceeds as follows. The equa tions of equilibrium are solved for the stellar radiation alone, start ing at the inner boundary of the nebula and proceeding to the outer edge, using the Runge-Kutta method to evaluate the optical depth inte gration. The equations are then resolved to include the diffuse field in the "on the spot" form. The values of jv (R) and t v (R) from this first approximation are then used to evaluate J ^ for each R and a new model is then constructed with this diffuse field. This process is continued to convergence. 25 Since the nebula is considered to be spherically symmetric, equation (1 0 ) becomes a double integral which was evaluated numerically using a Lobato quadrature over angle. Quadrature Points The integrals of equations (1) and (4) were broken into 40 in tervals chosen to fall at the ionization potentials of the ions con sidered. In order to keep the number of intervals of integration to a manageable size, the ionization limits of the following ions were con sidered coincident: H I and 0 1, 0 II and S III, Ne II and Ar III, N III and 0 IV, S IV and N IV. This introduces no appreciable error since the differences in the ionization potentials of the relevant ions are small. The quadrature points for the stellar radiation were Gaussian in the first 39 intervals; the last interval (from 158 electron volts to °o) was spanned by a three point Laguerre quadrature. A total of 79 points was required. For the recombination radiation the roots and weights were cal culated using a Gaussian quadrature over a finite interval with a weighting function of exp(-l.1605x), chosen to approximate the fre quency dependence of this radiation, x is the variable of integration. For this a total of 34 points was used. The quadrature points for He I and He II two photon emission were obtained using weighting functions which approximated the fre quency dependence of the two photon emission. In each interval the various types of radiation were treated separately, since they differ in their frequency dependences. The re sultant quadratures seem to be quite accurate. The effect of increas ing the quadrature from 50 stellar quadrature points and 28 recombina tion quadrature points, used when H, He, C, N, 0, and Ne were the only elements under consideration, to the present number, resulted in a change of less that .3 per cent in the temperature for all radial points. Chapter 3 Results of Varying Nebular Parameters In the process of fitting the models to the observations, the dependence of the models on some of the parameters was investigated. The results of these investigations are presented here. The parameters which specify a nebula are the following: the physical size (i.e., the angular dimension as well as■the distance to the nebula), the abundances of the elements, the temperature (or energy distribution) and the radius of the central star, and the hydrogen den sity of the nebula. The dependence of the final models on each of these parameters is given here. Physical Size Distance determinations to many planetary nebulae are dependent upon basic theoretical assumptions concerning planetary nebulae. For excellent discussions on distance determinations, see Minkowski (1964) or Aller and Liller (1968). For some planetary nebulae distances have been determined by more direct means. For example, the distance to NGC 246 is determined by a spectroscopic parallax method, Aller and Liller (1968), the distances of some nebulae are obtained from their membership in the Megellanic Clouds (Webster 1969), and the distance of the nebula in M15 is found from the distance to the cluster. For planetaries no such methods are feasible, so one must rely on theore tical assumptions 27 28 The nebulae which will be considered in the next chapter have been studied by various investigators who make different assumptions about the nebulae. The resultant distances may vary by as much as a factor of two or three. Thus, since the distances are so poorly es tablished, X investigated the effect of varying the nebular distance. The angular size (which is usually well known) in conjunction with the distance gives one the true size of the nebula. Thus, the size of the nebula varies linearly with the distance. Two types of nebulae were considered in this study: radiation bounded and density bounded. For a radiation bounded nebula (e.g., IC 418) the stellar radius is related in the following way to the neb ular radius r» =<(iW 3/'2 t32> since W * L * 1/3 Thus, upon changing the nebular radius, the stellar radius must be changed as in equation (32) to ionize the nebular matter as far as is observed. In a density bounded nebula (as IC 3568, which will be discussed later), the nebula does not depend on the size of the star; that is, one runs out of nebular material before running out of radiation. For this case, the formalism of Harman and Seaton (1966) was used, where the nebular and stellar radii are both linearly related to the distance. So when considering the effect of a distance variation upon the calculated model, one must also adjust the stellar radius. For each of 29 the series of models presented here, the radial points were spaced similarly, in order to have the integration proceed in the- same way. Since the final comparison of a model with observation lie's in match ing the line intensities, the resultant line intensities will be pre sented for each case. Let us first consider Models 1-3 which show the effects of varying the distance for a radiation bounded nebula. The parameters which specify the nebulae are given in Table 1. The electron tempera ture distribution for Models 1 and 3 is given in Figure 1 -- the rise of the electron temperature to the edge of an optically thick nebula (which will be explained later) was already noted by Aller, Baker, and Menzel (1939). The ionization distribution of a few elements is given in Figure 2. The resultant line intensities are given in Table 2. Table 3 gives the ionic abundances, integrated over the nebular volume, of a few ions. Model 1 is the upper entry; Model 3 the lower entry. A glance at Table 2 shows that differences do, in fact, arise by changing the distance. The electron temperature distribution for the two extreme cases is nearly the same; however, the ionization dis tribution has changed somewhat, giving rise to the different line in tensities. This can be seen from Figure 2 and in Table 3. One would expect certain variations in the ionization distribu tion. Since the stellar radiation may be expressed as * Fv R* 2 - V R) Jv m = 0 (W ) e • (34) one immediately sees that the total stellar flux seen at point R is different for the two cases. From the considerations for a radiation 30 bounded nebula, R.v will increase faster than R. Thus, the dilution factor differs for the two cases. The optical depth also differs in the two cases; this difference varies with the frequency. In other words, the stellar radiation field, as well as the diffuse field, undergo complicated changes upon changing the size of the nebula. Models 4 and 5a, whose defining parameters are also given in Table 1, show the effect of changing the distance on a density bounded nebula. The electron temperature distribution, shown in Figure 3, ap pears to have undergone drastic changes, but basically, it is mainly a shift in the location of the transition region. Figure 4, giving the ionization distribution of several ions, also shows this. Since the radius of the star has been varied in direct proportion to the ra dius of the nebula, the dilution factor appearing in equation (34) re mains the same for the two models. However, the stellar field for the larger nebula has been sharply reduced by the optical depth factor. Thus, since there is less ionizing radiation for the large nebula, the transition region from He III to He II moves closer to the star. In other words, if the size of the nebula is increased by doubling the radius and at the same time the size of the central star is doubled, with its flux per unit area remaining the same, then the amount of en ergy flowing into the nebula will be quadrupled, but the volume of the nebula will have been increased eight times. Thus, the fraction of the nebular volume that is ionized will be decreased. The resultant line intensities are given in Table 2. One of the most obvious differences in the line intensities be tween Models 4 and 5a lies in the helium lines. The He II lines have become much weaker and the He I lines have become enhanced. This is easily explained, for the He III zone is now considerably smaller. To show the effect of the stellar radius on the helium lines, an additional model has been constructed with the stellar radius varying as 3/2 (R , ) . The electron temperature for this model (curve b)is also neb given in Figure 3, Now the transition region again lies in the same region as for Model 4; consequently the He II lines have again attained their former strength. The electron temperature of the entire model is somewhat higher than for Model 4, following the same trend set by Models 1 and 3. Since the electron temperature is higher, many of the high excitation lines formed in the inner region have now become somewhat stronger. The question arises whether the changes in line intensity men tioned above are significant, when compared to observations. We will discuss the models in turn. In Models 1-3 only the weaker lines have undergone large changes (e.g., lines of [S II], [Cl III], and [Ar IV]), In these models these lines are very weak; the observational data on such weak lines are not sufficiently accurate to make these distinc tions. Even if observations were this accurate, density fluctuations could mask this effect. The 0 III lines show a difference of about 25 per cent for the two extreme distance cases. This is larger than the observational errors of 15 per cent claimed by Czyzak and Aller (1971). Models 4 and 5a show variations of comparable size as Models 1-3, except for the helium lines, where the difference is considerably larger. Under these circumstances, a change in the stellar radius, as 32 shown by Model 5b, would bring the results into somewhat better agree ment, although discrepancies still are present (e.g., Ne V). Clearly, uncertainties in the distance to a nebula may affect the resultant model, although the effect is more important in some nebulae than in others. For the particular nebulae which will be con sidered, we are dealing with objects more like those of Models 1-3, in which the uncertainties in the distance appear to have relatively minor significance. Effect of S, Cl, and Ar on Thermal Structure Since one of the goals of this study was to include sulfur, chlorine and argon in models for planetary nebulae which had heretofore only included the elements H, He, C, N, 0, and Ne, it seemed advisable to determine how this addition would affect the thermal structure of the nebula. Clearly, the size of the effect depends on the abundances chosen for S, Cl, and Ar (the higher the abundance of the elements, the greater the cooling effect). The size of the effect will also be shown to depend on the excitation class of the nebula. In the high excitation category two models were constructed -- Model 7 includes S, Cl and Ar; Model 6 neglects these elements. The defining parameters for these two models appear in Table 4; the varia tion of electron temperature with radial distance for these two models is plotted in Figure 5. Note the fairly high abundances of sulfur, chlorine and argon, obtained from Aller and Liller (1968), used for these two models. As can be seen, the electron temperature distribu tion has been affected, but only slightly (about 3 per cent in the 33 inner regions and increasing to a maximum of about 1 0 per cent in the outer regions), This effect is most noticeable on the strong [0 III] and [Ne III] lines, which change by about 25-30 per cent; such a change is not unexpected, since these lines are formed predominantly in the region where the change in electron temperature is the greatest. But, it should be noted that the addition of sulfur, chlorine and argon in even these rather large amounts results in a very small change on the thermal structure of this high excitation nebula. We now consider the effect of sulfur, chlorine and argon on the structure of a low excitation nebula with a central star of about 35000°K. The defining parameters for Models 9-10, which may be com pared, may also be found in Table 4. The electron temperatures are plotted in Figure 6 . The line intensities are given in Table 5 ■— clearly, the addition of moderate amounts of sulfur can result in large changes to the thermal structure and thus to the line intensities. Note that although the sulfur abundance has been increased by a factor of 10 from Model 10 to Model 9, the increase in the intensities of the sul fur lines is not comparable since the temperature is lowered. For the models containing sulfur, chlorine and argon, it is seen that the stellar radiation does not penetrate as far into the nebula; this is because there are now more absorbers present and the optical depth be comes larger. A qualitative explanation for the difference between high exci tation and low excitation nebulae may be found by examining the line intensities in these two types of nebulae. For a low excitation 34 nebula the [s II] and [S III] lines are much stronger than in high ex citation nebulae, where sulfur is present mostly in the higher ioniza tion states. Thus the lower ionization states of sulfur are more important coolants than the higher states. For a high excitation nebula, the lines of [Ar IV] are the chief coolants among these ele ments. The chlorine lines are all fairly weak in both types of nebu lae; thus one would expect a smaller cooling effect. Collisional Cooling Effects Since the primary method of cooling a nebula is by collisions, the cooling effect of different elements was evaluated. That is, a model was constructed for a low excitation nebula and the coolants were added one at a time. The resultant electron temperature plotted against distance from the central star is given in Figure 7, These models do not correspond to reality, since if an element is present, it will cool the nebula in some way. The purpose of the models shown is merely to illustrate in which region of a low excitation nebula each element cools most effectively. The defining parameters for this sequence of models is given in Table 4; the model we are discussing is Model 11. As can be seen, if there were no cooling by collisions, the electron temperature would be rather high (curve a). Adding oxygen (curve b) has a very large effect on the electron temperature; oxygen cools even more in the inner regions (in the form of [0 III]) than it does in the outer regions. Adding nitrogen, shown by curve c, lowered the electron temperature mainly in the outer regions of the nebula -- 35 (i.e., cooling in the form of the strong [N II] lines, 6548 and 6584). Adding neon (curve d) lowered the temperature slightly throughout the nebula, mainly in the form of [Ne III], The effects of adding carbon, chlorine or argon to the above elements were so minimal, that the elec tron temperature nearly coincided with the previous curve. The effect of adding a moderate amount of sulfur (shown by curve e) was again sizeable, as might be expected from the previous discussion. It should be noted that sulfur plays an important part in the cooling of a Iciw excitation nebula even when oxygen and nitrogen are present in higher amounts than in the previously presented low excita tion models. The importance of accurate transition probabilities and collision strengths for sulfur cannot be overestimated. Without the recent calculation of these parameters it would not have been possi ble to determine the size of the effect played by sulfur in the cooling of these low excitation objects. Energy Distribution of Central Star The ionization distribution and the electron temperature of a nebula are strongly dependent upon the ultraviolet flux distribution i of the central star. Figure 8 shows the electron temperature distri bution of the three models, Models 12-14, which were constructed making different assumptions concerning the flux distribution of the central star. In order to ionize the same amount of hydrogen in each case, the stellar radius had to be adjusted. The defining parameters for the three models shown appear in Table 6 . 36 For Model 13 a 38000° blackbody energy distribution was used for the central star. Model 14 shows the effect of using a stellar atmos phere with an effective temperature of 38000°K which does not include the opacities of carbon, nitrogen, oxygen and neon (Buerger, 1971), whereas Model 12 uses a central star in which these opacity sources have been included (Bohm and Deinzer 1965, 1966). The stellar energy distributions are plotted in Figure 9. The line intensities for these models are given in Table 7. The effects of the different energy distributions on the final models are sizeable. Not only is the electron temperature affected strongly, but also the distribution of the ions in their ionization states. Thus, the importance of the assumptions made concerning the ultraviolet flux distribution of the central star cannot be overem phasized. Effect of Density Changes The electron density of a planetary nebula can be obtained from the total emission in lip or from relative intensities of some forbid- , den lines, as has been discussed earlier. The electron densities ob tained in this way often differ considerably from one ion to the next and, at times, even for different line ratios of the same ion (e.g., [S II] of IC 418). The latter effect must be due to difficulties in observation, but the different electron densities obtained for dif ferent ions have been interpreted in the following way: ions are pre sent in filaments of different densities. 37 Probably the density varies from point to point in the nebula; however, the exact density distribution is an uncertain parameter. In order to assess the effects of variations in density, two models were constructed in which the density was changed, but most other para- 4 meters were left unchanged. Model 15 has a density of 1 x 10 hydro gen atoms/cc; in Model 16 the density has been doubled. The stellar radius was changed in order to maintain a nebula of the same angular extent. The defining parameters for these two models are given in Table 8 . The line intensities appear in Table 9; the electron temper ature distributions have been plotted in Figure 10. The effect of the change in density on the electron temperature distribution is seen to be fairly small. However, the resultant line intensities have undergone some fairly noticeable changes -- e.g., the [0 II] line intensities have decreased and the [0 III] lines have increased by doubling the density; similarly, the [S II] lines have decreased while the [S III] lines have increased. In Figure 11 the ionization distribution of a few relevant ions have been plotted and in Table 10 the relative abundances of the ions integrated over the nebular volume are given. Thus, one sees that the ionization distri bution has changed for the two cases. At first glance, it would appear that since the electron tem perature for the more dense model is higher in the outer regions of the nebula than for the low density model, for example, the [0 II] lines would be stronger for the denser model. This, however, is not the case, as is seen from Table 9. The forbidden lines do, in fact, increase with density but not as rapidly as Hj3. Thus when the 38 forbidden lines are normalized to Hp, the forbidden lines appear small er than for the previous case. In addition to the change in electron temperature, Figure 11 and Table 10 show that the ions have redistrib uted themselves in the ionization states. The upper entry in Table 10 refers to the low density model, while the lower entry refers to the high density model. The changes occurring here are of a similar type as those occurring by changing the size of the nebula; both the dilu tion factor and the optical depth factor in the equation for the mean intensity of the stellar radiation (equation 34) are affected. Attempts were made to adjust the abundances of the elements, in order to bring about better agreement of the line intensities of Model 16 with those of Model 15. Most of these were quite unsuccessful, Consequently, the density chosen for a particular model will affect the other parameters needed to satisfy the observational results. Importance of the Diffuse Field Although the diffuse field is not a defining parameter in a planetary nebula, the effects of different assumptions about it on the final structure of a planetary nebula deserve some discussion. Let us first consider a high excitation nebula, which is opti cally thin to radiation below 54.4 ev. Table 11 gives the defining parameters for this model (17a-c). In Figure 12 one sees the differ ence in the electron temperature distribution obtained by neglecting the diffuse field (curve a), using the "on the spot" approximation (curve b) and iterating on the diffuse field (curve c). 39 The addition of the diffuse field lowers the electron tempera ture, since one is adding lower energy electrons to the electron gas. The position of the transition region is seen to change noticeably from curve a to b or c; if one neglects the diffuse radiation field, there is less energy available, so one runs out of ionizing radiation more quickly. The resultant line intensities are given in Table 12. The "on the spot" approximation gives results which are about as accurate as the observations, except perhaps for the forbidden [0 III] lines, where the line intensities differ by about 20 to 25 per cent. The "no dif fuse field" model shows considerably larger differences, mainly since the transition region has been moved. These differences are defin itely larger than the errors in observation would permit. The same study has been made of a low excitation, optically thick nebula (Model 18) for which the defining parameters are also given in Table 11. The electron temperature for the three cases is plotted in Figure 13, For the low excitation nebula, neglecting the diffuse field (curve a) is a much more serious matter than in the pre vious optically thin model. The "on the spot" approximation is quite good, except in the innermost regions. The line intensities for these models also appear in Table 12. These models show that neglecting the diffuse field results in models which have a considerably different thermal structure if the other parameters (e.g., stellar radius) are not adjusted. The models made by Kirkpatrick (1970) for optically thin nebulae have 40 neglected the diffuse field; for optically thin nebulae, adjusting the stellar radius may remedy the problem. However, for optically thick nebulae no such easy solution exists. Here the "on-the-spot" approxi mation is considerably better, although a few iterations on the diffuse field still improve the results, especially for the inner region of the nebula. As can be seen, many parameters strongly influence the structure of a planetary nebula. The distances must be determined, reliable ob servations of the line intensities must be available, the energy dis tribution of the central star is needed, and the gas density must be known. If all of the above are fairly well determined, one may make models which will give reliable abundances of the elements. In the next chapter the models of a few specific nebulae will be discussed. Table 1 DEFINING PARAMETERS FOR MODELS 1-5: EFFECT OF CHANGING DISTANCE Model Number __ 1 .. 2 3 4 6a ,5b . O w H 6 * 3.8 • 104 3.8 • 104 3.8 ■ 104 1. ■ io5 1.- • 105 1. ■ 10 V Re 8.768 5.061 2.000 .236 .472 .6675 R 1 (psc)a .024 .0167 .009 .029 1 .058 .058 R 2 (psc) .075 .052 .028 .0775 .150 .150 Nu (atonts/cc) n 1.0 • 104 1.0 • 104 1.0 • 104 2.0 • 103 2.0 * 103 2.0 - 103 He/H .16 .16 .16 .16 .16 .16 C/H .0002 .0002 .0002 .0002 .0002 .0002 N/H .00003 .00003 .00003 .0004 .0004 .0004 0/H .00025 .00025 .00025 .00045 .00045 .00045 Ne/H .000063 .000063 .000063 .000075 .000075 .000075 S/H .000004 .000004 .000004 .00004 .00004 .00004 Cl/H .0000003 .0000003 .0000003 .0000003 .0000003 .0000003 Ar/H .000002 .000002 .000002 .000002 .000002 .000002 aR 1 and R 2 are respectively the inner and outer radii of the nebula 8 - ______l______I______I______I______!______L .3 A .5 .6 .7 .8 .9 FRACTIONAL RADIUS Fig.l - Electron Temperature for Models ] and 3: Effect of Changing Distance, FRACTIONAL ABUNDANCE .0 .2 .8 .4 0 .6 Fig.2- Ionization Distributionfor SelectedIons Effectof ofModels Changing1 and Distance.3: He 2+ 4 .5 RCINL RADIUS FRACTIONAL 6 .7 oe Number Model He 8 9 •c* w Table 2 RELATIVE LINE INTENSITIES FOR MODELS 1-5: EFFECTS OF CHANGING DISTANCE Model Number Emission Lines 1 2 3 4 5a 5b H I 4861 100. 100. 100. 100. 100. 100. He I 4471 2.95 2.95 2.98 4.6 6.3 4.6 5876 8.52 8.52 8.62 13.0 17.5 13.1 He II 4686 — — — 59.9 26.8 59.4 [N II] 5755 .16 .19 .05 6548 34.5 34.9 35.7 2.8 3.3 .9 6584 100. 101.0 103.4 8.2 0.5 2.8. [0 II] 3726 121.6 121.8 122.8 4.1 5.1 2.1 3729 53.8 53.9 54.3 2.6 2.9 1.2 [0 III] 4363 .61 .55 .45 15.9 15.2 13.9 4959 47.0 43.0 36.6 426. 504. 402. 5007 137.6 126.7 107.2 1249.. 1476. 1177. [Ne III] 3343 .33 .24 .26 3869 7.79 6.84 5.26 77.6- 83.8 69.8 3967 2.32 2.04 1.57 23.2 25.0 20.8 [Ne IV] 4714 _ .63 .31 .74 4716 -- — .18 .09 .21 4724 -- — — .69 .35 .81 4726 - -- .65 .32 .75 [Ne V] 3346 6.9 4.1 13.2 3426 — -> 18.6 11.2 35.5 & Table 2 (cont.) RELATIVE LINE INTENSITIES FOR MODELS 1-5: EFFECTS OF CHANGING DISTANCE Model Number 1 2 3 4 5a 5b [S II] 4068 .82 1.05 1.57 .49 .64 .13 4076 .27 .34 .51 .16 .21 .04 6716 1.52 1.97 2.98 .97 1.2 .25 6730 2.4 3.11 4.72 1.46 1.8 .39 [S III] 6312 1.01 .98 .01 1.8 1.9 .8 [Cl III] 5517 .70 .61 .46 .98 .97 .59 5537 1.06 .93 .71 1.04 1.02 .66 [Cl IV] 5325 ... .018 .013 .015 [Ar III] 5791 .015 .015 .014 .030 .026 .016 [Ar IV] 4711 .27 .015 .014 10.2. 8.0 9.2 4740 .25 .19 .11 7.9 6.2 7.3 Lnp* 46 Table 3 IONIC ABUNDANCES INTEGRATED OVER THE NEBULAR VOLUME OF MODELS 1 and 3: EFFECT OF CHANGING DISTANCE IONIZATION STATES Element I II III IV Hydrogen .0245 .9527 .0330 .9474 Helium .6895 .2997 .6935 .3000 Carbon .2621 .7114 .0202 -- .4115 .5763 .0086 Nitrogen .0143 .6979 .2305 .0245 .0183 .7392 .2050 .0095 Oxygen .0145 .7139 .2397 __ .0188 .7623 .1918 -- Neon .0213 .7550 .1958 _ _ .0313 .8085 .1373 — Sulfur MOT .1130 .7913 .0811 - .1999 .7521 .0398 Chlorine .0041 .4188 .5610 .0120 .0140 .5901 .3889 .0420 Argon .0101 .7051 .1761 .0730 .0123 .7524 .1721 .0317 Models 20 o (O O X 0) H A .5 .6 .7 .8 .9 1.0 FRACTIONAL RADIUS Fig.3 - Electron Temperature for Models 4 and 5: Effect of Changing Distance. •-j cc 0.2 0.8 .CTIOuAL ABUNDANCE0.6 0.4 1.0 r .3 i Fig.4 - Ionization Distribution for Selected Ions of Models 4 and 5a; Effect of Effect 5a;for Ionsofand Models4IonizationSelected- Distribution Fig.4 Changing Distance.Changing .4 T FRACTIONAL RADIUS .5 j 6 ,r .6 T 0 " " ni '■■■. I 0 " T .8 0 r ~ .9 Model ---- 5a 4 ------CD Table 4 DEFINING PARAMETERS FOR MODELS 6-11: EFFECT OF S, Cl, & Ar ON THERMAL STRUCTURE Model Number 6 7 8 9 10 11 4 T, (°K) 1.0 • 105 1.0 • 105 3.5 • 104 3.5 ■ 104 3.5 • 10 3.8 • 104 “A , .236 .236 5.072 5.072 5.072 7.0 R 1 (psc) .029 .029 .025 .025 .025 .0154 R 2 (psc) .0775 .0775 .075 .075 .075 .056 o o r“l N„n (atoms/cc) 2.0 • 103 2.0 • 103 1.0 • 104 1.0 • 104 1.0 • 104 He/H .16 .16 .16 .16 .16 .16 C/H .0002 .0002 .00025 .00025 .00025 .0002 N/H .00025 .00025 .000055 .000055 .000055 .00025 O/H .00045 .00045 .00025 .00025 .00025 .0004 Ne/H .000055 .000055 .000063 .000063 .000063 .00007 S/H — .0001 .00004 .000004 .00004 Cl/H -- .000003 — .0000003 ,0000003 .0000003 Ar/H .000008 .000002 .000002 .000002 •t* 01 X X a> h- j .025 .035.045 .055 .065 .075 R (PSC) Fig.6 - Electron Temperature for Models 8 - 10: Effect of S, Clj and Ar on Thermal Structure. »-* 4 52 Table 5 RELATIVE LINE INTENSITIES FOR MODELS 8-10: EFFECT OF ADDING S, Cl, and Ar Model Number Emission Lines 8 9 10 H I 4861 100. 100, 100. He 1 4471 3.4 3.4 3.4 5876 9.9 10.0 9.8 [N II] 6548 40.9 19.1 39.1 6584 118.5 55.4 113.3 [0 II] 3726 66.5 19.4 61.0 3729 29.1 8.5 26.7 [0 III] 4363 .41 .06 .30 4959 39.0 14.9 33.8 5007 114.3 43.6 96.7 [Ne III] 3868 3.5 1.1 2.9 3967 1.00 .31 .86 [S II] 4068 _ _ 2.81 .73 4076 .92 .24 6716 — 8.0 1.5 6730 — 12.7 2.4 [Cl III] 5517 .17 .38 5537 — .28 .59 [Ar IV] 4711 _ _ .03 .07 4740 -- .03 .06 53 31 r 30 Curves 1 0 o 9 ro I O X 8 7 •* 6 5 .035.015 055 R (PSC) Fig. 7 - Electron Temperature for Model 11: Coolant Series a)No Forbidden Lines, b) 0 , c)0+N, d)0+N+Ne+C, e)0+N+Ne+C+S+C1+Ar 54 Table 6 DEFINING PARAMETERS FOR MODELS 12-14: EFFECT OF THE ULTRAVIOLET FLUX DISTRIBUTION Model Number 12 13 14 I* <°w 3.8 • 104 3.8 • 104 3.8 • 104 5.657 2.5 3.78 R 1 (pac) .0154 .0154 .0154 R 2 (pac) .055 .055 .055 Njj (atoms/cc) 1. • 104 1. * 104 1. • 104 He/H .16 .16 .16 C/H .0002 .0002 .0002 N/H .00004 .00004 .00004 O/H .00025 .00025 .00025 Ne/H .000031 .000031 .000031 S/H ,00001 .00001 .00001 Cl/H .00000015 .00000015 .00000015 Ar/H .000005 .000005 .000005 Ul Lrt .06 (PSC) R R 13 — 13 Models .02 .03 .04 .05 - Electron Temperature for Models 12 - 14: Electron Models Temperature - for - 14: 12 Effect Ultraviolet of Distribution* Flux *8 Fig 8 II 1 0 <0 01 X I- I 10 LOG -11.0 -9.0 -5.0 O L - -7.0 3.0 Fig.9 -ofCentralFluxStar.Fig.9 Distribution . 04 . 00 1.0 0.0 0.6 0.4 0.2 Hell Bohm & Deinzer & Bohm Black Body Black Buerger R Q E G CS 10 X FREQUEMGY C/S ± Hell -16 1.2 _ l > 1.4 1.6 Ul a\ 57 Table 7 RELATIVE LINE INTENSITIES FOR MODELS 12-14: EFFECT OF ULTRAVIOLET FLUX Model Number Emission Lines 12 13 14 H I 4861 100. 100. 100. He I 4471 2.9 4.1 6.2 5876 8.4 11.9 17.4 [N II] 6548 47.1 33.0 21.7 6584 136.3 95.6 62.9 [0 II] 4363 .73 .81 4.56 4959 49.6 62.1 202.9 5007 145.3 181.9 594.4 [Ne III] 3868 4.1 3.5 22.9 3967 1.2 1.0 6.8 [S II] 4068 2.5 2.9 2.5 4076 .82 .95 .83 6716 4.6 5.7 4.3 6730 7.3 9,0 6.9 [Cl III] 5517 .33 .27 .39 5537 .50 .41 .61 [Ar IV] 4711 .67 .47 6.58 4740 .62 .44 6.12 Table 8 DEFINING PARAMETERS FOR MODELS 15 and 16: EFFECT OF CHANGING DENSITY Model Number 15 16 T* (°K) 3.8 • 10^ 3.8 • U f VRg 8.0 16.0 R 1 (psc) .025 .025 R 2 (psc) .075 .075 (atoms/cc) 1. ■ 104 2. ■ 104 He/H .16 .16 C/H .0002 .0002 N/H .00004 ,00004 O/H .00025 .00025 Ne/H .000031 .000031 S/H .00001 .00001 Cl/H .0000003 .0000003 Ar/H .000002 .000002 59 Table 9 RELATIVE LINE INTENSITIES FOR MODELS 15 AND 16: EFFECT OF CHANGING DENSITY Model Number Emission Lines 15 16 H I 4861 100. 100. He I 4471 2.9 3.0 5876 8.4 8.5 [N II] 6548 48.0 45.8 6584 139.1 132.8 [0 II] 3726 129.7 99.1 3729 57.4 39.5 [0 III] 4363 .61 .79 4959 46.8 54.7 5007 137.1 160.3 [Ne III] 3868 3.8 4.9 3967 1.1 1.5 [S II] 4068 2.2 1.5 4076 .71 .51 6716 3.95 1.51 6730 6.3 2.9 [S III] 6312 2.7 3.3 [Cl III] 5517 .72 .72 5537 1.1 1.5 [Ar IV] 4711 .25 .37 4740 .23 .43 1 2 Models /\ o 10 10 I O X Q) h* 8 ± ± ± _L j .025 .035.045 .055 .005 .075 R(PSC) Fig.10 - Electron Temperature for Models 15 and 16: Effect of Changing Density. 1.0 2+ UJ 0.8 o Model < Q 0.6 Cl < —< 9 x 0.4 H O < iC u_ 0.2 0 .025 .035 .055 .065 R(PSC) Fig.11 - Ionization Distribution for Selected Ions of Models 15 and 16* Effect of Changing Density. 62 Table 10 IONIC ABUNDANCES INTEGRATED OVER THE NEBULAR VOLUME OF MODELS 15-16: EFFECT OF DENSITY CHANGES IONIZATION STATES Element I II III IV Hydrogen .0230 .9523 .0526 .9261 Helium .6910 .2942 .6997 .2871 Carbon .2692 .7074 .0180 .2007 .7615 .0294 Nitrogen .0139 .6995 .2277 .0216 .0465 .6656 .2227 .03661 Oxygen .0141 .7166 .2334 .0465 .6774 .2494 -- Neon .0208 .7590 .1893 ■»e* .0510 .7077 .2171 n e * Sulfur M e * .1144 .7976 .0744 - .0985 .7740 .1079 Chlorine .0044 .4304 .5512 .0103 .0015 .3238 .6496 .0196 Argon .0100 .7062 .1771 .0665 .0384 .6764 .1544 .1019 I 63 Table 11 DEFINING PARAMETERS FOR MODELS 17 and 18: EFFECT OF DIFFUSE FIELD Model Number 17a-c 18a-c T*(°K) 1. * ID5 3.5 * 104 V * 0 .236 5.072 R 1 (pac) .029 .025 R 2 (pac) .0775 .0675 Nt, (atoms/cc) n 2. • 103 1. ■ io 4 He/H .16 .16 C/H .0002 .0002 N/H .00025 .00025 O/H .00042 .00042 Ne/H .000075 .000075 S/H .00004 .00004 Cl/H .0000003 ,0000003 Ar/H .000002 .000002 Curves .03 .04 .05 .06 .07 .08 R (PSC) Fig.12 - Electron Temperature for Model 17: Effect of Diffuse Field. Table 12 RELATIVE INTENSITIES FCR MODELS 17 and 18: EFFECT OF DIFFUSE FIELD Model Number Emission Lines 17a 17b 17c 18a 18b 18c H I 4861 100. 100. 100. 100. 100. 100. He I 4471 5.6 4.5 4.7 3.4 3.6 3.5 5876 15.4 12.2 13.0 9.9 10.6 10.3 He II 4686 40.1 62.9 60.1 MM MM 5412 3.8 6.0 5.7 MM -M «*M [N II] 5755 .14 .15 .10 1.9 .53 .5 6548 2.0 2.1 1.8 90. 49. 46.. 6584 6.0: 6.0 5.2 263. 141. 134. [0 II] 3726 5.8 5.5 4.05 38.4 12.4 11.7 3729 3.3 3.0 2.3 16.8 5.3 5.1 [0 III] 4363 23.7 22.2 15.4 .24 .03 .05 4959 576.6 499.7 404.3 39.9 13.9 17.1 5007 1689. 1463. 1184. 117.0 40.8 50.1 [Ne III] 3343 .43 .48 .34 Mm • • 3869 106. 100. 79.3 2.35 .61 .85 3967 31. 29. 23.2 .69 .18 .25 [Ne IV] 4714 .61 .77 .66 MM _ _ MM 4716 .18 .22 .19 -M — M M 4724 .67 .85 .73 M. — M- 4726 .63 .79 .68 -M — -M Table 12 (cont.) RELATIVE INTENSITIES FOR MODELS 17 and 18: EFFECT OF DIFFUSE FIELD Model Number Emission Lines 17a 17b 17c 18a 18b 18c [Ne. V] 3346 6.8 . 8.0 7.2 M a* 3426 18.1 21.3 19.1 —— [S II] 4068 .69 .69 .49 3.4 1.5 1.5 4076 .22 .22 .16 1.1 .5 .5 6716 1.2 1.2 .96 8.8 5.4 5.3 6730 1.8 1.8 1.4 14.1 8.6 8.6 [S III] 6312 2.3 2.5 1.81 2.3 .6 .7 [Cl III] 5517 1.2 1,2 1.0 .25 .09 .1 5537 1.2 1.2 1.0 .39 .15 .16 [Cl IV] 5325 .02 .02 .02 -- m m — [Ar III] 5191 .04 .04 .03 — — — [Ar IV] 4711 11,0 12.1 10.4 .05 .02 .02 4740 8.6 9.5 8.1 .05 .02 .02 o\ o\ 91- Curves o X o . .025 .035 .045 .055 .065 R (PSC) Fig.13 - Electron Temperature for Model 18; Effect of Diffuse Field. Chapter 4 A Few Specific Nebulae IC 418 IC 418 is one of the few nebulae which exhibits a fairly regu lar form. It appears to be quite symmetrical and the lower resolution instruments which were available to the earlier observers gave slitless images which appeared homogeneous in the images of [0 II], [0 III], [N II], and [Ne III]. The A? image, shown by Aller (1956), seems to indicate that inhomogeneities might, in fact, be present on a small scale. The recent high resolution observations of.Kron and Walker (1970) verify that inhomogeneitiea are indeed present. Slitless image observations by Wilson and Aller (1951), to gether with the expansion velocities (Wilson 1953), show that [0 III] as well as [Ne III] show a concentration toward the central star, as does the image of Hf3, although the latter extends farther from the central star than do the former. However, the images of [0 II] and [N II] have the appearance of hollow shells, surrounding the [0 III] and the [Ne III] region. The recent work of Weedman (1968a) is in good agreement with these earlier results; he finds that the [0 III] 2 emission decreases as exp(-r/4.6") . Thus, in this low excitation 68 69 nebula, the [0 III] and the [Ne III] lines cool the inner regions of the nebula whereas the [0 II] and [N II] lines are the chief coolants of the outer region. The angular size of IC 418 has been fairly well established using slitless image observations. Wilson (1950) reported an angular diameter obtained from the HP, He I and [0 III] images of about 10.5". The images of [S II] and [0 I] extended to 12". Measurements of the isophotic contours given by Aller (1956) resulted in a size of about 14" for the [0 III] image and 15" for the HP image. Andrillat and Houziaux (1968) report an angular size of 14" by 11", whereas the radio observations made by Colomb with an interferometer and reported by Menon (1968) gave 2Vl as the angular separation of the central star and the inner edge of the nebula and a total angular diameter of 1 2 " of arc. The angular diameter that has been adopted is 13" of arc. The inner radius used has been taken from the deconvolution of Kron and Walker; this lies about 3.5" from the central star. Aller (1956) was one of the first astronomers who stated that IC 418 might be a Stromgren sphere, although there was no definitive proof. Since that time several criteria have been established which help determine whether a nebula is optically thick or thin. Seaton (1960) used the presence of fairly strong [0 i] lines as a criterion of an optically thick nebula, O'Dell (1963b) has observed a weak \6300 [0 I] line in IC 418, so this criterion is not necessarily sat isfied. Seaton (1966) classified IC 418 as an optically thick nebula, since it shows no He II lines. The lack of coincidence of [0 III] 70 emission with HP emission is taken by Weedman (1968a) to imply that IC 418 is optically thick. In an optically thin nebula, [0 III] emission is an excellent tracer of ionized gas, for it then coin cides with H0 emission. Terzian (1968) and Thompson (1968) have both observed IC 418 at radio frequencies and find it is optically thick at all observed frequencies. Consequently, there is strong evidence that IC 418 is, indeed, optically thick as Aller origin ally suggested. The high surface brightness together with the small dia meter suggests that this nebula is probably in the early stages of ionization with neutral material surrounding the ionized region. The question of the optical thickness of the nebula is of signifi cance since distance determinations are aften made using assump tions concerning the optical depths of the nebulae. Distance determinations for optically thick nebulae assume that all the radiation from the star is transformed into a nebular emission spectrum. Assuming that the stars of all optically thick nebulae have the same absolute magnitude, the photographic magnitude of the nebula, being a measure of the bolometric magni tude of the star, is a distance indicator. For optically thin nebulae, the distance obtained using this method is overestimated. For optically thin nebulae, the basic assumption is that all the mass of the nebula is visible and that all planetary neb ulae have the same mas, M . Since the distance may be shown to * o depend on the ionized mass raised to the two-fifths power (Minkowski 1964), the 71 distance to an optically thick nebula would be overestimated. For an optically thick nebula, the ionized mass is then less than the total mass M . o Distance determinations using the photometric method are based on mean nebular parameters. It is assumed that the gas is completely ionized, has a given electron temperature, filling factor and a con stant helium to hydrogen ratio. Since a constant nebular mass is as sumed, this method is reliable only for optically thin nebulae. Distance determinations for planetary nebulae, assuming that all nebulae are optically thick, have been made by Minkowski (1961), Vorontsov-Velyaminov (1950) and Berman (1937) — their results for IC 418 are 1760, 2200, and 1800 parsecs, respectively. Distance deter minations, under the assumption that all nebulae are optically thin, were made by Shklovsky (1956) and O'Dell (1963a) — they obtain 1300 and 1880 parsecs respectively. Camm (1939) used radial velocities, interpreted by means of galactic rotation and obtained a distance of 2040 parsecs to IC 418 whereas Zanstra (1931) used an empirical rela tion between the photographic brightness and the temperature of the central star. This gave him a distance of 440 parsecs. Seaton (1966) has made a careful study of the distances to nebulae by evaluating the optical depth c f each nebula both from spectrophotometric data as well as from the appearance; he obtains distances to various nebulae. For IC 418 he obtains a distance of 760 parsecs. Cahn and Kaler (1971) obtain a distance of 1710 parsecs, using the photometric method. Liller, Welther and Liller (1966) have studied the expansion velocities of a few selected nebulae. Unfortunately, IC 418 was not included in their sample. Expansion velocities, together with the gas velocities, obtained from line widths, give an accurate distance deter mination. Thus, they are able to obtain information on the validity of the various distance scales. Despite a fairly small sample, they conclude that the distance scales of Berman and Minkowski appear to be the most consistent, whereas the scales of Seaton and O ’Dell, for ex ample may be overestimated by factors of 2-3. Thus, the distance which I have adopted for IC 418 is 1800 parsecs. The distance, together with the angular diameter, gives the ab solute dimensions for the nebula; thus the inner radius is .0154 and the outer radius is .056 parsecs from the central star. Observations of IC 418 have been reported by several sources. Wyse (1942) and O'Dell (1963b) report observations ranging from X.3727 to HO!; Liller and Aller (1963) and Aller and Kaler (1964) made careful observations in the blue region from X3727 to \5007. Aller and Walker (1970) observed the nebula from X4640 and X5900. Andrillat and Houziaux (1968) observed the nebula from about 5800$ to 8000$; however, many of the lines they observed appear to be overexposed and could not be included. Unpublished photoelectric results for the visible region are also available from observations by Czyzak and Aller. Fluxes for a few particular lines or the intensities of line ratios have also been determined. Osterbrock, Capriotti and Bautz (1963) measured the the fluxes of the [N II] lines ralative to HCt. White (1952) also 73 determined the ratios of the [N II] lines to HO!, Seaton and Osterbrock I(X 37^9) (1957) have obtained the ratio of ~j~(\3726)'* density depend ent line ratio. Observations of the [0 III] lines have been made by Liller and Aller (1954) and Capriotti and Daub (1960). Weedman (1968b) made careful determinations of the M l o n ' ratio of [S II] and Aller et \o/ j U al. (1970) observed the [Cl III] ratio For some lines the observations show quite a large amount of scatter; one might expect some scatter from difficulties in data reduc tions. Additional scatter may be introduced when observations have been made at different parts of the nebula. For this study, we need the line intensities integrated over the nebular volume. Thus, accurate photoelectric intensities are preferred,since photographic results generally refer to a small particular part of the nebula. The study by Aller and Kaler, although photographic, has been made for two re gions of the nebula. They also give integrated values for most lines. To a large extent, their results have been adopted here. For the wavelength region beyond 5000X, the line intensities obtained by Aller and Walker (1970) were largely used; their observations did not extend as far as the [S II] lines at X6716 and \6730. Thus the photo electric results of Czyzak and Aller (1971) were used to obtain the relative intensity of the doublet, and the results of Weedman (1968b) were incorporated to obtain the intensity of each component. The adopted line intensities, corrected for reddening, are given in Table 74 13. They were corrected using the reddening correction given by O'Dell (1963b) of c=0, .42 which was applied to the reddening law as given by Seaton (1960). Central star temperatures of planetary nebulae present a diffi cult problem. The spectral features of the central star of IC 418 sug gested to Aller (1956) that it was an Of star with a temperature of ii around 33,200 K. Bohm (1968) quotes a blackbody temperature of 43,000°K, as determined by Harman and Seaton (1966). Other estimates of blackbody temperatures for the central star have been given. Some of these temperature determinations are: 25,000°K (Berman 1937), 36,000°K (Zanstra 1931), and 53,000°K (O'Dell 1963a). Some initial models were constructed employing blackbody tem peratures of 43,000°K and 53,000°K. However, these models resulted in a concentration of ions to higher states of ionization than can be justified on the basis of the observations. A more reasonable ioniza tion distribution was obtained using a blackbody temperature of 35,000°K for the central star. o Based on these preliminary calculations, the 38,000 K model II atmosphere of Bohm and Deinzer (reported by Goodson, 1967) was adopted to provide a more realistic distribution of the ultraviolet flux of the central star. The stellar radius was adjusted so that the final • model is radiation bounded. Electron density determinations of planetary nebulae are valu able to obtain the hydrogen density for a particular nebula. These have been obtained from various lines. The recent work of Krueger, 75 Aller and Czyzak (1970) gives the following densities using an electron temperature of 11,300°K: j(\6730) ^ gives an electron density of 1.12 X 104 /cc, T (\67l6)+°i(X.6730) of 11 ^ Sives 2.45X104 /cc, I(\3726) 0f [0 II] giveS 4 *17 X 1q4/cc^ x(xM'17) 0f [C1 111 ] slves 4 4 1.48 X 10 /cc and the H3 flux gives an electron density of 1.66 X 10 , As has been mentioned previously, these varing densities may be inter preted as actual fluctuations in the density within the nebula. How ever, the difference given by the [S II] lines is partly due to obser vational difficulties in obtaining accurate estimates of the intensity of \4068. Other difficulties result from the length of the wavelength interval involved. Photographic plates of different spectral and in tensity sensitivity must be used in the two regions. This may intro duce observational error. Further, the differential reddening over such a long wavelength interval is large and uncertain. Thus, inten sity ratios of lines which fall near each other are preferred. Further more, the results of O'Dell were employed for [S II] and they appear to be very inconsistent with the observations of all other sources. The electron density derived from the [0 II] lines appears to be unreliable since the theoretical transition probabilities are inconsistent with the observations (Seaton and Osterbrock 1957). The most reliable of the above ratios is the [S II] ^ ^ 6 7 3 0 ^ ratio as obtained by Weedman A (1968b). Thus, the hydrogen density adopted for IC 418 is 1 X 10 atoms per cubic centimeter. 76 The abundances which have been used are given in Table 14. They have been chosen to suitably match the observed line intensities. The electron temperature distribution obtained has been given in Figure 14. It is slightly lower than that generally quoted for this nebula. The high electron temperature of about 18,000°K quoted in earlier work was due to poorly known collision strengths as well as a poor measurement of the intensity of \4363. Kaler (1970) gives an elec tron temperature of about 10,000°K for the region in which the [0 III] lines are formed. LeMarne and Shaver (1969) obtain an electron tem perature of 12,500°k by combining radio data and the isophotic con tours of Wilson and Aller. As can be seen from Figure 15, [0 III] is present predominantly in the inner region of the nebula; referring to Figure 14, one sees that the temperature in that region is lower. However, it must be realized that this temperature is based on an ob- I 3 63) servationally determined line ratio of 9j + ~i (50'07)~ ^ HI]- For low excitation nebulae, the \4363 line may be in considerable error, since it is so weak. A method of estimating the electron temperature for low excita tion objects which relies on the intensity ratio of the [N II] lines was given by Seaton (1960). It has been updated by Rubin (1969) using the latest collision strength data and is given by 1^6584) 6.16 exp (25000/Te) I(\5755) = 1 + 0.282 X -2 - 1/2 where X = 10 N T and T is the electron temperature, e e e 77 For the observed ratio of the line intensities, and using an 4 electron density of 1 X 10 /cc, one obtains a value for the electron temperature of 10,500°K in the region in which the [N IX] lines are formed. Our results are in good agreement with this estimate. One must emphasize that the \5755 line is also very weak and lies in a region of the spectrum which has not been observed as much as might be desired. So one again is hindered, by lack of observations, in apply ing this method to all low excitation nebulae. The general distribution of the electron temperature with in creasing distance from the central star may be explained as follows: initially we have a fairly flat electron temperature which gradually starts to rise due to the selective absorption of the low frequency radiation at smaller radii. The sharp drop at .038 parsecs occurs since no He ionizing radiation is left. The subsequent rise in the electron temperature is again caused by the selective absorption of the low frequency radiation. It is enhanced at the outer edge, where the diffuse field is received from only 2-jr steradians, and thus acts less to lower the electron temperature. The final drop in the electron temperature shows that all ionizing radiation has been used up. The ionization distribution shown in Figure 15 agrees favor ably with the general trend of the contours given by Aller (1956). [0 III] and [Ne III] are concentrated toward the central star and there is a sharp rise of [0 II], [N II], and [SIII] toward the outer edge. The ionic abundances integrated over the nebular volume may be found in Table 15. 78 The line intensities obtained for this homogeneous model are pre sented in Table 13. The agreement with observations is quite good, especially if one recalls the observational accuracy. This model pre dicts the observations in a much better way than do the models by Flower (1968, 1969b). Since inhomogeneities have been found in this nebula, a model was made which included them. The isophotic contours of Kron and Walker (1970) were analyzed to determine where the inhomogeneities are located. They are found in the inner region and extend to .043 parsecs, The condition placed upon the addition of the inhomogeneities is that the total HP emission remain the same for the homogeneous and inhomo- geneous cases. This condition implies 2 2 2 D 1 V ! + D 2 V 2 = N and V1 + v2 “ 1 where and are the densities for the inhomogeneous model, and are the volumes occupied by the densities and N is the hydrogen density for the homogeneous model. The densities were chosen to be 3 4 5 X 10 and 3 X 10 and the corresponding volumes were.9143 and .0857. The electron temperature distribution (Figure 16) is seen to be af fected only slightly for the two densities present in the same shell, but the ionization equilibrium has been changed. An example of the amount of the changes is shown in Figure 17 for sulfur. When consid ering the changes which have occured to the ionization distribution, one must consider the fact that at a particular point, R, the radia tion field is the same for each density. The changes to the ionization distribution which are seen are thus the density effects on the ioniza tion equilibrium. The high density material is found to have ions in lower ionization states, since more recombinations occur for higher densities. Despite the slightly increased amount of [S II] , [0 II] and [N II] in the high density phase, the line intensities for these ions are found to be much weaker in the high density phase than in the low density phase. This effect has already been explained in a pre vious section. Combining the line intensities obtained from each phase one obtains the values listed in Table 13. The ionic abundances integrated over the nebular volume appear in Table 16. The comparison of the calculated line intensities of both the homogeneous and inhomogeneous models with the observations show no dis crepancies which are greater than the observational errors. The in homogeneous model is somewhat better for the [N II] lines, but only slightly. Thus it has been shown that models can be constructed which successfully predict the observations. 8 0 Table 13 RE1ATIVE LINE INTENSITIES IN IC 418 Observations Homogeneous Inhomogeneous Corrected for Emission Lines Case Case Reddening H I 4861 100. 100. 100. He I 4471 2.9 2.9 3.5 He II 4686 .001 ,001 — [N II] 5755 2,2 2.5 3.1 6548 46.9 48.3 51. 6584 136.0 140.0 160. [0 II] 3726 122.6 119.3 135. 3729 54.2 52.4 50. 7318 3.2 3.7 7319 9.5 11.2 97 7329 5.2 6.1 7330 5.1 5.9 [0 III] 4363 .77 .77 1.03 4959 51.0 45.4 44.6 5007 149.5 133.1 135. [Ne III] 3869 4.3 3.6 4.0 3967 1.3 1.1 1.3 [S II] 4068 2.5 3.3 2.3 4076 .8 1.1 .9 6716 4.7 5.1 5.8 6730 7.4 8.3 9.7 [S III] 6312 2.6 2.8 2.8 [Cl III] 5517 .33 .27 .27 5537 .50 .46 .61 [Ar III] 5191 .05 .05 .05 7135 9.0 8.9 7751 .41 .31 [Ar IV] 4711 .69 .45 weak or 0 4740 .64 .46 weak or 0 7170 .01 .01 7237 .01 .01 7267 .01 .01 Table 14 DEFINING PARAMETERS FOR SPECIFIC NEBULAE IC 418 IC 3566, IC 4593 _NGC_2662_ T* <°K) 3.8 • 10 4.625 * 10 4.0 • 10H 1.0 - 10 1.0 .236 V R0 5.657 4.0 Distance (psc) 1800. 2540. 1430. 1000. R 1 (psc) .0154 .015 .005 .029 R 2 (psc) .056 .110 .035 .0775 5.0 • 10* , ,3 1.4 10: NH (atoms/cc) 104 8.0 • 102 5.0 * 10“ 3.0 • 103 2.0 ■ 10 3.0 - K T x' 8.2 10“ .08 Mneb/M0 .178 .47 .013 He/H .16 .16 .16 .16 C/H .0002 .0002 .0002 .0002 N/H .00004 .000055 .00001 .00040 O/H .00025 .00027 .00025 .00045 Table 14 (cont.) DEFINING PARAMETERS FOR SPECIFIC NEBULAE IC 418 IC 3568 IC 4593 NGC 7662 Ne/H .000031 .000063 .000063 .000075 S/H .00001 .00001 .000004 .00004 Cl/H .00000015 .0000001 .0000003 .0000003 Ar/H .000005 .000001 .000001 .000002 83 Table 15 1C 418: IONIC ABUNDANCES INTEGRATED OVER NEBULAR VOLUME FOR HOMOGENEOUS MODEL IONIZATION STATES Element I ii III IV V Hydrogen .1578 • 10"1 .9735 Helium ,6806 .3085 .5499 • 10-5 Carbon .2623 .2913 .6880 .1799 • io - 1 .3426 ■ io - 9 Nitrogen .8646 • io “2 .7250 .2298 .2151 ■ io"1 .1501 . 10-1C Oxygen .8882 ■ io “2 .7423 .2346 .9220 • IO"6 .2003 • 10"1’ Neon .1511 • 10-1 ,7848 .1876 .1133 • 10“7 .4245 ■ io - 11 Sulfur .4594 • IO-4 .1174 .8034 .7224 • io"1 .1331 • io - 2 1 Chlorine .5948 • io"2 .4628 .5188 .1029 • io - 1 .1118 H o Argon .5358 • io - 2 .7380 .1793 .6437 • 10"1 .2075 •io-8 Table 16 IC 418: IONIC ABUNDANCES INTEGRATED OVER NEBULAR VOLUME FOR INHOMOGENEOUS MODEL IONIZATION STATES Element I II III IV V Hydrogen .3214 ■ io"1 .9591 Helium .6754 .3199 .1354 • 10"* Carbon .2552 ■ IO”* .2877 .6779 .3167 • IO"1 .1339 •10-8 Nitrogen .2040 • IO-1 .6928 .2342 .3799 • IO"1 .5735 . 10-1C t Oxygen .2061 O .7048 .2610 .2324 . io -5 .9929 • io"1* Neon .2894 • 10“l .7364 .2234 .2875 • IO-7 .2128 ■ to -17 1 Sul fur .4598 H o .1269 .7615 .1027 .3396 • io - 2 Chlorine .5746 . io - 2 .4394 .5341 .1875 • io"1 .3869 • 10"4 1 H -1 O Argon .1315 I .7034 .1681 .9738 • io - 1 .6198 • 10"8 0 O I— .02 .03 .04 .05 R (PSC) Fig,14 - IC 418; Electron Temperature Distribution for Homogeneous Model. _ ’ * * •...... ' r > C > v,«t* ci* • u ' H,He C N •i 0 • He - .i.i S - Cl Ar ■ RCPSC) Fig.15 - IC 418; Ionization Distribution for Homogeneous Model. Nh = Nh = I X 10 .04 .05 . R R (PSC) .03 NH = 3 X 10 X 3 = NH \ Nh= Nh= 5 X 10' .02 Fig.16 - IC 618: Electron Temperature Distribution for Inhomogeneous Model. Inhomogeneous for Distribution Temperature 618: Electron IC - Fig.16 8 ll 12 r 12 10 o o 01 X FRACTIONAL ABUNDANCE 1.0 .4 .8 .6 .2 Fig.17 - IC Ionization 418: Distribution of Sulfurfor Inhomogeneous Model. i .3+ .020 3+ 2 + 00 .040 .030 2 + / 1 t (PSC) Ft o — low Densities high n — one I ------ .050 »4 oa IC 3568 88 This nebula is a very symmetrical and uniform planetary, often referred to as the "theoreticians planetary". It consists of a bright inner region surrounded by a somewhat fainter shell. The intensity dis tributions across the Hp and [0 III] images are given by Aller (1956); since the stellar image was over-exposed, no reliable determination of the emission could be made near the central star. Weedman (1968a) has determined that the distribution of the emission does not show a re solved inner boundary for the nebular material. The angular diameter of about 18" was determined from the HP and [0 III] images. Wilson (1950) was not able to include this nebula in his studies, since it is too far north for observations with the 100" telescope. Thus, the study of the isophotic contours by Minkowski (Aller 1956) contains the only available data. We have adopted their results. Seaton (1966) classified IC 3568 as an optically thick nebula, since it shows no He II lines. The study of Weedman (1968a) shows that the distribution of [0 III] and hydrogen appears to be identical. (This confirms the early work of Minkowski.) Thus, Weedman classifies this nebula as optically thin, having most of its gas in ionized form. Khroraov (1965) has found weak [0 1] lines at \6300 and \6363 in this nebula. These lines are sufficiently weak that they probably imply the nebula is not optically thick. Thus, the above criteria appear to in dicate that IC 3568 is optically thin. 89 Distance determinations for this nebula range from 670 parsecs (Zanstra 1931) to 2560 parsecs (O'Dell 1963a). Berman (1937) obtained a distance of 2340 parsecs and Minkowski (1961) placed the nebula at 2540 parsecs. Cahn and Kaler (1971) obtain a distance of 2650 parsecs. The distance I have adopted is 2540 parsecs. The absolute radius for the nebula is thus .11 parsecs. Observations of IC 3568 cover the wavelength region from 3350$ to 5900$ with additional observations of the region near H2 . Lee et al. (1969) report observations from \3350 to X.5007. These observa tions are superior to those reported by Aller (1951). Observations by Aller and Walker (1970) of the region from 5200$ to 5900$ show very few lines. The region around Hcc was observed by White (1952) who found no trace of the [N II] lines. Capriotti and Daub (1960) have observed the [0 III] lines \4959 and \5007. The line intensities adopted ap pear in Table 17. The study by Lee et al. (1969) indicated that the agreement be tween the observed and the theoretical Balmer decrement was very good; thus the nebula does not seem to be reddened, and no reddening correc tion was applied. Estimates of the temperature of the central star of IC 3568 show some scatter. The central star shows absorption lines comparable to those found in 05 stars; thus, Aller (1968) gives it a temperature of 50,000°K. Earlier estimates of the blackbody temperature were given at 30,000°K (Berman 1937), 32,000°K (Harman and Seaton 1966), 90 35,000°K (O'Dell 1963a) and 38,000°K (Zanstra 1931). Attempts at con structing models were made using blackbody temperatures ranging from 30,000°K to 40,000°K. These models gave electron temperatures which were so low that the observations could not be reproduced. A black body temperature of about 56,000°K was found to give a more reasonable set of line intensities. Subsequent to these preliminary calculations, the 46,250°K 11 model atmosphere of Bohra and Deinzer (1966) was used to describe the energy distribution of the central star. The stellar radius was adopt ed so that a region existed in which the [0 II] lines might be formed. Since IC 3568 has not been observed as extensively as has IC 418, fewer line ratios are available on which one can base electron density determinations. If one recalls the discussion on IC 418, most of the line ratios quoted are beyond \5007; IC 3568 has not been observed in the wavelength region to use sulfur as a density criterion. The region in which the chlorine lines are found has been observed, but IC 3568 showed no detectable lines of chlorine. Thus, one is dependent upon the electron density determinations obtained from the [0 II] lines and from the total H|3 flux. The HP flux, as reported by Lee et al. (1969), gives an electron density of 800/cc, whereas the observed [0 II] line ratio gives an electron density of 5000/cc. These densities are ob tained under the assumption of a constant electron temperature through out the nebula of about 12,400°K. The [0 II] lines are formed in the outer shell of the nebula. We have thus adopted a two shell model for this nebula — the inner region consists of material at a density of 800/cc, and the density adopted for the outer region is 5000 hydrogen atoms per cubic centimeter. The abundances which are in the best agreement with the obser vations are given in Table 14. The electron temperature for the region in which the [0 III] lines are formed has been determined by Lee et al. (1969) as being about 12,400°K. Kaler (1970) gives an electron temperature of 10,200°K for this nebula, using the observational results of Lee et al. as well as those of Capriotti and Daub (1960). The electron tempera- . ture obtained here lies between these results; it is given in Figure 18 as a function of the radial position in the nebula. The initial sharp drop of the electron temperature is due to the depletion of He II ionizing radiation; this can be seen from Figure 19, which gives a plot of the ionization distribution. The electron temperature dis tribution is seen to be fairly constant throughout the nebula. The ions have distributed themselves entirely in one degree of ionization; i.e., no stratification seems to be apparent. [0 III] has the same distribution as H, just as the observations of Aller (1956) and of Weedman (1968a) have shown. A very small amount of [0 II] has started to appear at the outside edge of the nebula, which accounts for the presence of the [0 II] lines which have been observed. The ionic abundances integrated over the nebular volume are given in Table 92 It is interesting to compare the results of Table 18 with those obtained by Lee et al (1969) for the ionic abundances. The agreement for the ionic abundances is really quite good, especially considering the assumptions involved in the method used by Lee et al. The largest discrepancy is a factor of 4 which arises for the abundance of [Ne III] and a factor of 2.5 for [Ar IV]. In the absence of theoretical models the total abundance of an element can only be determined on the basis of some assumptions about the distribution of ions in unseen ioniza tion states. This presents a severe restriction. The method used by Lee et al. also presupposes a constant electron temperature throughout the nebula. This Is a good assumption for IC 3568 but for many other nebulae it is rather questionable. Thus, models yield much more unam biguous abundance determinations than were possible heretofore. The line intensities obtained for this nebula are given in Table 17. The agreement between the model and the observations is seen to be very good except for the He II L46S6 line. The observa tions of Lee et al. (1969) report an intensity of 1.2 whereas our com puted model gives an intensity of .001. The only way to obtain a He II line of the strength observed would be to make the He III region of the nebula considerably larger. This can be achieved by using a hotter star in conjunction with a larger stellar radius. The result of this type of change is a model which does not predict the observations nearly as well as the current model. The conclusion that has been drawn (Czyzak and Aller 1971) is that the observed He II line is 93 actually due to scattered starlight, rather than being formed in the nebula. The nitrogen lines predicted by the model are sufficiently weak that they would probably not have been observable by White (1952), since they are so near H2 . Thus, the parameters chosen predict line intensities which are in good agreement with observations. Table 17 94 RELATIVE LINE INTENSITIES IN IC 3568 Emission Lines Model Observed H I 4861 100. 100. He I 4471 7.8 6.8 5876 21.8 21. He II 4686 .001 1.2 [N II] 5755 .09 weak or 0 6548 1.7 weak or 0 6584 4.9 weak or 0 [0 II] 3726 8.7 7.9 3729 4.4 4. 7318 .18 7319 .54 7329 .29 7330 .29 [0 III] 4363 8.9 13.8 4959 341.8 359. 5007 1001. 1069. [Ne III] 3343 .15 weak or 0 3869 72.9 75.9 3967 21.7 18. [S II] 4068 .26 .44 4076 .09 .10 6716 .67 6730 .89 [S III] 6312 1.3 [Cl III] 5517 .54 weak or 0 5537 .63 weak or 0 [Cl IV] 5325 .001 weak or 0 8045 .10 [Ar III] 5191 .03 .3 7135 3.97 7751 .15 [Ar IV] 4711 2.4 2.2 4740 1.9 2.1 7170 .03 7237 .03 7267 .03 Table 18 IC 3568; IONIC ABUNDANCES INTEGRATED OVER NEBULAR VOLUME IONIZATION STATES Element I II III IV V VI Hydrogen .7703 • 10-3 .9990 Helium .1201 • 10 "2 .9983 .1680 • IO-3 Carbon .1897 • io“5 .8106 • 10~2 .4833 .5074 .2189 • io-6 Nitrogen .3786 • 10-5 .8099 ■ 10 "2 .4504 .5403 .1025 - IO"7 Oxygen .6952 • 10'5 .1475 • 10_l .9848 .1380 • 10"3 .3443 • IO'9 r-t 1 O Neon .4802 • 10 .4097 • 10~l .9586 .1790 • IO"5 .3313 • rH Sulfur .6598 • IO-5 .5268 • 10‘2 .1547 .5901 .2480 .1273 • 10-7 Chlorine .8474 • 10'3 .5132 • IO'1 .4810 .3701 .9439 • IO”1 .1581 • 10"7 Argon .9729 • io-6 .5924 • IO-2 .1961 .7971 .3886 • io-6 .1102 • 10"11 vO Ln 12 NH = 8 X IO3 Nh= 5X IO3 J I I i 1 I I I I I I I I I 1 I I t I l .02 .03 .04 .05 .06 .07 .08 .09 .10 .11 R (PSC) Fig.13 - IC 3568; Electron Temperature Distribution, . I - • f / V #*■4x -J V v / .-"I . .■* N' \ \ ^ / / / . / ctiX 7' -• r > il # • / K?U / • • K *5. \ / ^ V *♦ K \ ,\ ■ A Y < f f/i. _L 1------d.______i .015 .035 .055 .07 5 .095 ,||5 RCPSO Fig.19 - IC 3568; Ionization Distribution. IC 4593 98 This nebula consists of a disk of bright matter which fades out somewhat toward the edges. Fainter material appears to exist in a shell surrounding the bright inner region. Isophotic contours of this nebula show a bright central core which appears double in the light of [0 III]. Wilson (1950) attempted to determine the angular size of IC 4593 from isophotic contours, but found the images too diffuse. However, other investigators have reported angular diameters for this nebula. Curtis (1918) gives an angular diameter of 11" X 15", O'Dell (1963a) gives 12.8", and Liller and Aller (1954) obtain an angular diameter of 10". The angular diameter which I have adopted is 10". Seaton (1966) has classified IC 4593 as a nebula which may be optically thin, although it does not have any He II lines. The nebula has been observed over a large spectral range by Vorontsov Velyaminov et al. (1965); they have observed a line of [0 I] at \6300; Khromov (1965) has also observed a weak [0 I] line. This might indicate that the nebula is optically thick, but is not conclusive. However, the ob servations of Vorontsov Velyaminov et al. are not in good agreement with any other observations of this nebula. Thus, until further observa tions have been made of this spectral region, their results will not be used. Weedman (1968a) did not include this nebula in his study. The result of Seaton concerning the optical thickness has been adopted to make a distance determination possible. Distance determinations have been given by various sources, ranging from 870 parsecs (Vorontsov Velyaminov 1950) to 2810 parsecs (O’Dell 1963a). The distance given by Shklovsky (1956) is 1200 parsecs and that given by Zanstra (1931) is 1050 parsecs. Cahn and Kaler (1971) obtain a distance of 2960 parsecs. The distance adopted here has been taken from the compilation by Aller (1961); this distance of 1430 parsecs is in good agreement with that ob tained by Shklovsky, using the criterion for optically thin nebu lae. The adopted distance gives a nebula which extends .035 parsecs from the central star. 1C 4593 has been observed extensively in the blue region of the spectrum by Aller (1951), Liller and Aller (1954), Seaton and Osterbrock (1957), Capriotti and Daub (1960) as well as by Vorontsov Velyaminov et al. (1965). A considerable amount of unpublished data by Czyzak and Aller (1971) are also available; these consist of several photographic plates as well as some photo electric data for both the blue and the visible region of the spec trum. White (1952) has observed the nebula in a search for the [N II] lines surrounding HCt which he could not find. Unfortunately, Aller and Walker (1970) did not include this nebula in their study. Consequently, except for the study by Vorontsov Velyaminov et al. (1965), only photoelectric results are available for the longer wavelength region of the spectrum. Thus, reliable intensities for the weak lines are not available. The situation for chlorine is particularly unfortunate, since no chlorine line intensities have been obtained at all. Thus, we can only give an upper 100 limit to the chlorine abundance. The adopted line intensities, cor rected for reddening using a value of c=0.27 obtained from Kaler (1970), are given in Table 19. Blackbody estimates for the temperature of the central star have been given by several investigators. Zanstra (1931) gives 22,000°K, Berman (1937) gives 25,000°K, and O'Dell (1963a) gives the central star a temperature of 27,000°K. Aller (1968) has classified the star as 06 and quotes a temperature of 41,000OK- 2,000°. Using a black body energy distribution the most satisfactory fit to the observations was obtained by using 49,000°K central star. Subsequent calculations were made using non-gray model atmos pheres — it was found that the best fit to the observations could be obtained using a model of 40,000°K, obtained by interpolating in the model atmospheres of Bohm and Deinzer. The stellar radius was adjusted to give a region in which [0 II] became sufficiently abundant to form the observed [0 II] lines. Electron density estimates for this nebula have been reported by several sources; surprisingly small fluctuations exist in the re sults. Seaton and Osterbrock (1957) based their estimate on the [0 II] line ratio and obtained a value for the electron density of 2700/cc; Kaler (1970) reported an electron density of 2650/cc, also from the [0 II] lines. Liller and Aller (1954) reported an electron density of 2900/cc from Ip and 2100/cc from the Balmer continuum using a distance of 2000 psc. There are as yet no observations of chlorine or sulfur lines, to determine whether these ions are formed in filaments of dif ferent densities. 101 Since there are insufficient data currently present to make an inhomogeneous or a two-shell model, a uniform model was constructed with a hydrogen density of 3000 atoms per cubic centimeter. The abun dances giving the best fit to the observations are given in Table 14, The electron temperature of the region in which the [0 III] lines are formed has been given as 7700°K by Liller and Aller (1954). Seaton and Osterbrock (1957) quote an electron temperature of 12000°K. In this nebula the \4363 line of [0 III] used in the determination of the electron temperature is quite weak, as it was in IC 418, Conse quently electron temperature estimates based on this line ratio may vary considerably. Thus a theoretical model which reproduces the observations well is a much better indicator of the electron temper ature than are the older methods based on line ratios alone. The electron temperature obtained here is plotted in Figure 20, It should be pointed out that the model which has been constructed is almost radiation bounded but not quite. One thus would expect to see only weak [0 I] lines, unless they arise from high density condensations. As yet, no such condensations have been observed in this nebula. The ionization distribution is given in Figure 21 and the ionic abundances integrated over the nebular volume are given in Table 20, The line intensities obtained for this model are given in Table 19. The He I \5876 line appears to be strong in the model, but the observed value is based upon one photoelectric observation and may thus not be reliable. The calculated [N II] lines are sufficiently weak that they would probably not be observable, since they lie in 102 the wings of Ha. The [S II] \40G8 and X4076 lines have not been found in this nebula, nor were any lines found in the photoelectric observa tions at \6716 or \6730. The model is thus in satisfactory agreement with the few avail able observations. More observations of this nebula should be made at wavelengths longer than 5000&. Table 19 103 RELATIVE LINE INTENSITIES IN IC 4593 Emission Lines Model Observed H I 4861 100. 100. He I 4471 7.7 7.1 5876 21.8 13. [N 11] 5755 .15 weak or 0 6548 3.9 weak or 0 6584 11.2 weak or 0 [0 II] 3726 58.7 45,6 3729 34.3 32.2 7318 .74 7319 2.3 7329 1.2 7330 1.2 [0 III] 4363 3.5 2.99 4959 184.1 166. 5007 539.4 544. TNe III] 3343 .04 3869 31.8 27.6 3967 9.5 12. [S II] 4068 .65 weak or 0 4076 .21 weak or 0 6716 2.7 weak or 0 6730 3.1 weak or 0 [S III] 6312 .89 weak or 0 [Cl III] 5517 1.1 weak or 0 5537 1.1 weak or 0 [Ar III] 5191 .03 weak or 0 7135 5.4 3.65 7751 .26 [Ar IV] 4711 .57 weak or 0 4740 .44 weak or 0 7170 .01 7237 .01 7267 .01 Table 20 IC 4593: IONIC ABUNDANCES INTEGRATED OVER NEBULAR VOLUME IONIZATION STATES Element I II III IV V VI Hydrogen .1852 • 10"2 .9979 Helium .2972 • io"1 .9690 .6125 • 10"5 00 1 Carbon .4377 * 10 “4 .1417 .8219 .3489 .5506 • 1-1o Nitrogen .2261 • 10’3 .2053 .7423 .4984 - 10_I .9492 • IO"10 Oxygen .2971 • io -3 .2264 .7719 .4409 • 10'5 .2193 • io -12 H r-* 1 Neon .2683 • 10“2 .4014 .5947 .9602 ■ 10"7 .5256 - r—1O Sulfur .1376 • 10“3 .9304 - 10_1 .6478 .2531 .4512 • 10‘2 .4926 ■ 10"10 Chlorine .1026 • 10"1 .3975 .5517 .3880 *10-1 .2265 ■ 10-3 .1306 • 10‘10 Argon .1105 • io"3 .2532 .5320 .2114 .2776 • 10-7 .1959 ■ 10‘14 o 4> __I_ i______i______i______i______i__ ___i__ .005 .010 • .015 .020 .025 .030 .035 R(PSC) Fig.20 - IC 4593: Electron Temperature Distribution. H,He — c ... N " — o- — Ne --- S ------Cl ------Ah ------ . 0/ S' VAr \ / ,015 .020 R(PSC) Fig.21 - IC 4593: Ionization Distribution. o NGC 7662 107 NGC 7662 is a high excitation nebula with a fairly regular structure which makes it suitable for a study such as we are under taking here.The discussion on this nebula will be rather brief since it has been discussed in great detail by Harrington (1969). All de fining parameters needed to make a theoretical model have been taken from his discussion. They are listed in Table 14. The nebula is only mentioned here to report the abundances obtained for sulfur, chlorine and argon which had not been obtained by Harrington. The atomic parameters have been updated from those used by Har rington and the latest data for the absorption coefficients were used. These changes resulted in very small changes to the thermal structure of the model. The addition of sulfur, chlorine and argon cooled the nebula slightly. Consequently a few of the abundances had to be ad justed to obtain better, agreement with the observations. The line intensities adopted by Harrington have been used here, except for lines for which more recent observations by Aller and Walker (1970) are available. For the lines of sulfur, chlorine and argon, which fall in the wavelength region of 5400X to 5900X, the re sults of Aller and Walker (1970) have also been adopted. For the sul fur lines at \6716, \6730, and \6312 the results of Minkowski and Aller (1956) have been adopted. The intensities of the [S II] and [Ar IV] lines in the blue region of the spectrum have been taken from Aller, Kaler and Bowen (1966). The adopted line intensities are given in Table 21. The nitrogen and neon abundances were adjusted from those used by Harrington (1969); the oxygen abundance was found to be in good agreement with his determination. The abundance estimates of Aller and Czyzak (1968) for sulfur and argon are in very good agreement with those obtained using theoretical models, especially if one considers the difficulties involved in obtaining abundances without the aid of theoretical models. For sulfur they are within 50 percent of our value and for argon their estimate is within a factor of 3 of our results. Such good agreement is hardly expected for elements which are present in such small amounts. For chlorine a much lower abundance is ob tained here than that found by Aller and Czyzak. However, until the recent observations of Aller and Walker (1970) no reliable observa tions of the chlorine lines were available. Thus, earlier abundance determinations are rather doubtful. Table 21 109 RELATIVE LIKE INTEKSITIES IH NGC 7662 Emission Lines______Model______Observations H I 4861 100. 100. He I 4471 4.8 2.9 5876 13.3 8.7 He II 4686 58.6 63. 5412 5.5 7.9 Eh ii] 5755 .15 .19 6548 2.8 2.8 6584 8.1 6.1 [0 II] 3726 4.1 6.7 3729 2.3 3.8 7318 .09 7319 .26 7329 .14 7330 .14 [0 III] 4363 14.9 16.9 4959 410.5 434. 5007 1203. 1270. [Ne III] 3343 .31 0.46 3869 74.1 80. 3967 22.1 23. [Ne IV] 4714 .63 .71 4716 .18 .08 4724 .68 .74 4726 .63 .42 [Nc V] 3346 6.8 5.5 3426 18.2 12. [S II] 4068 .46 1.04 4076 .15 6716 .94 6730 1.4 1.5 [S lit] 6312 1.9 1.8 [Cl III] 5517 1.0 .64 5537 1.0 .59 Table 21 (cont.) 110 RELATIVE LINE INTENSITIES IN NGC 7662 Emission Lines______Model______Observations [Cl IV] 5325 .02 .09 8045 .65 [Ar III] 5191 .03 .18 7135 2.7 7751 .10 [Ar IV] 4711 9.9 9. 4740 7.7 8.6 7170 .20 7237 .15 7267 .17 Chapter 5 Concluding Remarks A careful study of a few specific nebulae has been carried out. Theoretical models of the thermal structure of three rather low exci tation nebulae have been constructed which reproduce the observations successfully. The elements sulfur, chlorine and argon have been in cluded in the construction of the models and their abundances have been determined. The effect which these elements have on the thermal structure of nebulae was assessed. For high excitation objects (as, for example, NGC 7662) it was found that the addition of sulfur, chlorine and argon had only a very small effect on the thermal structure of the nebula. For a lew excitation nebula it was found that sulfur is an important coolant, mainly in the form of the [S II] lines. This effect has not been determined prior to this study, possibly because investigators generally have studied high excitation nebulae, rather than the low ex citation objects to which we have directed our attention here. Thus, it appears that sulfur should be included in future studies of plane tary nebulae. It was also shown that the assumptions made concerning the energy distribution of the central star can have sizeable effects on the 111 112 thermal structure of the nebula and consequently on the abundance determinations. Thus, it is very important that better determinations of the ultraviolet flux of central stars be obtained. Reliable esti mates of the effective temperatures of the central stars are also needed, in order to make reliable and accurate models for the nebulae. It should be emphasized that the abundances quoted in Table 22 which have been obtained for the nebulae are only as accurate as the nebular parameters. Only the abundances of elements represented by forbidden lines are being discussed here, as the abundances of hydrogen, helium and carbon were held fixed throughout the calculations. A short discussion concerning the errors one might expect for each of the nebu lae discussed will follow. This discussion will presuppose that the observed line intensities are fairly reliable. A discussion of the ef fect of errors in the flux distribution of the central star on the model nebula will not be included, since this effect is so difficult to assess. It has been pointed out that differences in the opacity sources included in the model atmosphere for the central star will have significant effects on the structure of the nebula. The approach taken here was to find a flux distribution which resulted in a nebular model which fit the observations with some success and then to alter the abundances until the best fit was obtained. As has been pointed out in Chapter 3, the variation of the nebu lar parameters affects the thermal structure of the nebula and conse quently the abundances of the elements. However, it should also be stressed that changing the abundance of one element (e.g., sulfur) may 113 change the structure of the nebula to such an extent, that the model no longer fits the observations at all. Thus it must be realized that a quantitative discussion of errors in abundances resulting from un certainties in nebular parameters is rather complex. However, an at tempt will be made to indicate the reliability of the nebular para meters used and, where possible, to estimate the effect this might have on the abundances. For IC 418, the distance appears to be fairly well determined; thus one would expect no significant errors in the abundances as a result of an unreliable distance determination. The electron density of this particular nebula seems to be somewhat more uncertain, as was shown by the currently available observational data on line ratios. Although the electron density determinations indicated some spread, the inhomogeneous model covered the range of densities. Nevertheless, errors in the abundances of neon, sulfur, chlorine and argon of about a factor of two may be expected for this type of uncertainty. Oxygen and nitrogen are somewhat more accurate, since a change in their abundance by a factor of two could conceivably alter the thermal structure of the entire nebula. For IC 3568 the distance again seems well determined; thus no significant changes in the abundance determination as a result of distance errors would be expected. With the information currently available on densities, there is little reason to expect any differ ences in the electron density. However, higher resolution studies might show inhomogeneities which have not yet been seen. Including 114 such an effect in the model might cause changes in the abundances com parable to those discussed above, XC 4593 is somewhat more problematic and, consequently, the abun dances obtained for this nebula are perhaps the least reliable. The stellar flux distribution necessary to obtain a realistic nebular model had to be obtained by interpolating in available atmospheres — this casts some uncertainty on the validity of the source of the radiation. The nebula appears to be almost optically thick, although not quite. Thus, the distance used for the nebula may be to small by as much as a factor of two. This change would have an effect on the abundance de termination which might easily be off by a factor of two or three. This study has shown that the importance of further reliable observational data cannot be overemphasized. The abundances of the elements can be determined well only if the observational data are re liable and complete. Thus observations at wavelengths longward of X5007 are needed to determine the chlorine abundance in planetaries. Currently, progress is being made in obtaining more and better data in the visible region of the spectrum. The use of the image tube should make studies of the visible part of the spectrum much more feasible than ever before. Additional reliable observational data in the visible region are thus of great importance to make accurate abun dance determinations for planetary nebulae. The abundances which have been determined for the different nebulae are given in Table 22, along with the abundances for plane taries given by Aller and Liller (1968). The abundance of helium Table 22 115 CHEMICAL COMPOSITION OF PLANETARY NEBULAE Lor N IC 418 IC 4593 IC 3568 NGC -7662 Planetarles Hydrogen 12.0 12.0 12.0 12.0 12.00 Helium 11.20 11.20 11.20 11.20 11.25 Carbon 8.3 8.3 8.3 8.3 8.7 Nitrogen 7.6 7.0 7.74 8.66 8.5 Oxygen 8,4 8.4 8.43 8.65 9.0 Neon 7.49 7.80 7.80 7.88 8.2: Sulfur 7.00 &.60 7.00 7.60 8.0: Chlorine 5.18 <3.5 <5.0 5.5 6.5: Argon 6.70 6.0 6.0 6.3 6.9: and carbon used for the construction of the models is that found in early type stars. In general, the abundances of the heavier ele ments obtained in this study are lower than those given by Aller and Liller. This may indicate that the method of obtaining abundances from forbidden lines used prior to methods employing theoretical models systematically overestimated the abundances of the elements. Until more models for particular nebulae are constructed, for nebulae of all degrees of excitation, one cannot make any definite statements concerning the significance of the abundances obtained here. Thus, it appears to be worthwhile to continue this type of in vestigation for nebulae featuring a fairly regular structure. More observations in the visible region of the spectrum are needed, as well as reliable information concerning the energy distribution of the central star. Studies concerning the exact density distribution present in a nebula are valuable, but unless accurate calculations of the radiation field are also included, they do not appear to be of primary importance. List of References 117 LIST OF REFERENCES Allen, C.W. 1963, As.trophysical Quantities (2d ed.; London: Athlone Press), p.100 Aller, L.H. 1951, Ap. J .. 113, 125 . 1956, Gaseous Nebulae (London; Chapman and Hall) ______. 1961, Landolt-BBrnstein: Numerical Data and Functional Relations in Science and Technology ed. H.H. Voigt (Springer Verlag) 1, 566 ______. 1968, Planetary Nebulae. I.A.U. Symposium 34. p.339 Aller, L.H., Baker, J.G., and Menzel, D.H. 1939, Ap.J., 90, 601 Aller, L.H., and Czyzak, S.J. 1968, Planetary Nebulae, I.A.U. Sym posium 34, p.209 Aller, L.H., Czyzak, S.J., Walker, M.F. and Krueger, T.K. 1970, Proc. Natl. Acad, of Sci.. 6 6 . 1 Aller, L.H., and Kaler, J.B. 1964. Ap.J.. 140, 936 Aller, L.H., Kaler, J.B., and Bowen, I.S. 1966, Ap.J.. 144, 291 Aller, L.H., and Liller, W. 1968, in Nebulae and Interstellar Matter. ed. B.M. Middlehurst and L.H. Aller (Chicago: University of Chicago Press), p.483 Aller, L.H. and Walker, M.F. 1970, Ap.J.. 161. 917 Andrillat, Y. and Houziaux, L. 1968, Planetary Nebulae. I.A.U.Symp. 34. 6 8 . Bates, D.R. 1946, M.N.R.A.3.. 106, 432 Bely,0., Tully,J., and Regemorter, H.V. 1963, Ann. Phys., J3,303 Be man, L, 1937, Lick Obs. Bull.. 18, 57 Blaha, M. 1968, Annls. Astrophvs.. 31. 311 BBhm, K.H. 1968, Planetary Nebulae. I.A.U. Symposium 34. p.297 . 1969, Astronomy and Astrophysics, 1, 180 BBhm, K.H., and Deinzer, W. 1965, Zs.f. Ap ., 61, 1 ______. 1966, ibid., 63, 177 118 119 Bowen, I.S., and Wyse, A.B. 1939, Lick Obs. Bull.. 19. 1 Buerger, P.F. 1971, Private Communication Burgess, A. 1958, M.N.R.A.S., 118, 477 Burgess, A., and Seaton, 11.J. 1960a, M.N.R.A.S.. 120. 121 ______. 1960b, ibid., 121, 76 ______. 1960c, ibid.. 121, 471 Cahn, J.H. and Kaler, J.B. 1971, Ap.J. Supp., 22, 319 Camm, G. 1939, M.N.R.A.S.. 99. 71 Capriotti, E.R., and Daub, C.T. 1960, Ap.J.. 132. 677 Curtis, H.D. 1918, Lick Obs. Publ.. 13, 57 Czyzak, S.J., and Aller, L.H. 1971, Private Communication Czyzak, S.J., and Krueger, T.K. 1963, M.N.R.A.S.. 126. 177 Czyzak, S.J., Walker, M.F., and Aller, L.H. 1969, Mem.Soc.Roy.Sci. Liege. 17, 271 Dalgarno, A. 1966, M.N.R.A.S.. 131, 311 Edlen, B. 1964, Handbuch der Physik, 27. ed.S. FlUgge (Springer Verlag) Flower, D.R. 1968, Ap.J. Letters. 2j 205 ______. 1969a, M.N.R.A.S.. 146. 171 ______. 1969b, ibid., 243 Garstang, R.H. 1968, Planetary Nebulae. I.A.U. Symposium 34_. p. 143 Goodson, W.L. 1967, Ss.f.Ap., 6 6 , 118 Harman, R.J., and Seaton, M.J. 1966, M.N.R.A.S., 132, 15 Harrington, J.P. 1967, Ph.D. Dissertation at The Ohio State University ______. 1968, Ap.J., 152, 943 ______. 1969, ibid., 156, 903 Henry, R.J.W. 1970, Ap.J., 161, 1153 120 Huffman, R.E. 1964, Autoionization Spectra of Gases Observed In the Vacuum Ultraviolet. Physical Sciences Research Papers. 6 6 . 1 Hummer, D.G., and Seaton, M.J., M.N.R.A.S., 125, 439 ______. 1964, ibid., 12_7, 217 Kaler, J.B. 1970, Ap.J.. 160, 887 Khromov, G.S. 1965, Soviet A.J., _9, 431 Kirkpatrick, R.C. 1970,' Ap.J., 162, 33 Kron, G., and Walker, M.F. 1970, Private Communication Krueger, T.K., Aller, L.H., and Czyzak, S.J. 1970, Ap.J.,160, 921 Krueger, T.K., and Czyzak, S.J. 1970, Proc. Roy.Soc.Lond.A..318.531 Lee, P.,Aller, L.H., Czyzak, S.J., and Duvall,R. 1969, Ap.J.,155,853 LeMarne,A.E., and Shaver,P.A. 1969, Froc. of Ast. Soc. Austr., _1, 216 Liller, W., and Aller, L.H. 1954, Ap.J., 120. 48 . 1963, Proc. of Natl. Acad, of Sci.Am., 49, 675 Liller, M.H., Welther, B.L., and Liller W. 1966, Ap.J., 144. 280 Marr, G.V. 1967, Photoionization Processes in Gases, (Academic Press, New York) Menon, T.K. 1968, Planetary Nebulae, I.A.U. Symposium 34, p. 123 Menzel, D.H. and Aller, L.H. 1941, Ap.J.. 94, 30 Minkowski, R. 1961, Landolt-BBrnstein: Numerical Data and Functional Relations in Science and Technology ed.H.H. Voigt (Springer Verlag) _1, 566 ______. 1964, P.A.S.P., 76, 197 Minkowski, R., and Aller, L.H. 1956, Ap.J., 124, 93 O'Dell, C.R. 1963a, Ap.J., 138, 67 ______. 1963b, ibid., 1018 Osaki, T. 1962, Pub. Astr. Soc. Japan. 14, 111 Osterbrock, D.E. 1963, Planetary and Space Science. llf 621 Osterbrock, D.E., Capriotti, E.R. and Bautz, L.P. 1963, Ap.J.,138.62 Rubin, R.H. 1969, Ap.J.. 155, 841 Saraph, H.E., and Seaton, M.J. 1970, M.N.R.A.S.. 148. 367 Saraph, H.E., Seaton, M.J., and Shemming,J. 1969, Phil.Trans.Roy.Soc. Lond.A.. 264. 77 Seaton, M.J. 1958, Rev.Mod.Phys., 30, 979 ______. 1959, M.N.R.A.S., 119. 81 . 1960, Rep.Prog. Phys., 23. 313 ______. 1966, M.N.R.A.S.. 132, 113 Seaton, M.J., and Osterbrock, D.E, 1957, A p .J.. 125, 6 6 Shklovsky, l.S. 1956, Astr. Zh., 33, 322 Silk, J., and Brown, R.L. 1971, Ap.J., 163, 495 Sofia, S., and Hunter, J.H. 1968, Ap.J., 152, 405 Spitzer, L. and Greenstein, J.L. 1951, Ap.J., 114, 407 Stromgren, B. 1939, Ap.J., 89, 526 Terzian, Y. 1968, Planetary Nebulae. I.A.U. Symposium 34, p.87 Thompson, A.R. 1968, Planetary Nebulae, I.A.U. Symposium 34, p.112 Varsavsky, C.M. 1961, Ap.J. Supplement. J6 , 75 Vorontsov-Velyaminov, B.A., Kostyakova, E.B., Dokuchaeva, O.D. and Arkhipova, V.Pl 1965, Soviet A.J., _9, 364 Webster, E.L. 1969, M.N.R.A.S.. 143. 79 Weedman, D.W. 1968a, Ap.J.. 153. 49 ______. 1968b, P.A.S.P.. 80, 314 White, M.L. 1952, Ap.J.. 115, 71 Wilson, O.C. 1950, Ap.J.. Ill, 279 ______. 1953, ibid.. U_7, 264 Wilson, O.C. and Aller, L.H. 1951, Ap.J., 114, 421 Wyse, A.B. 1942, Ap.J., 95. 356 Zanstra, H. 1931, Zs.f. Astroph., _2, 331 APPENDIX 123 D ' h \ i Ux rn V Vi r xA rv ?3 N 4 5 Vi V <>» vS CJ D (v s $ I 2 r i.r u i O © S Vi 2 VV "" '/x ( n * NV N > 2 . 'n \ • Vi U 4r - '* % D VS 0 3 P ---- 1■r 2 I -1 P 3;/s O Fig.22 - Energy Level Diagrams for Ions Giving Rise to Forbidden Lines. 124 *P V r \ ■“7iln V s 4P • >U r ■*&*n »/*. u ’D N rn Z «? l ■Vx "s . 5P ■ 34 ?p ® S Vi. r N 8 NS & i? * * ln i 'd - Vu U VS O J. t 3p Fig.22 (cont.) Table 23 125 ATOMIC PARAMETERS FOR STATISTICAL EQUILIBRIUM EQUATIONS Ion i j Gij (electron volts) "u J k 2 6 1 0.0061 0,401 2.1 10 -12 1 3 .0163 0.279 1.3 10 7 1 4 1.8988 0.348 4.2 ■ 1 0 ' 1 5 4.0524 0.033 0.0 2 3 0.0102 1.128 7.5 • 10 ' N II 2 4 1.8927 1.045 1.0 -2 2 5 4.0464 0,114 3.4 • 10 -3 1.742 3.0 ■ 10 3 4 1.8825 -4 3 5 4.0362 0.190 1.6 • 10 4 5 2.1537 0.376 1.0 -5 1 2 3.3236 0.858 4.2 10 1 3 3.3262 0.572 1.8 10 -2 1 4 5.0170 . 0.285 6.0 10 1 5 5.0172 0.143 2.0 0 II 2 3 0.0026 0.894 1.3 10 2 4 1.6933 0.743 1.0 -2 2 5 1.6935 0.302 6.1 10 •2 3 4 1.6907 0.418 6.1 10 ■1 0.279 1.0 10 3 5 1.6909 •11 4 5 0.0002 0.263 6.0 10 ■5 1 2 0.0141 0.376 2,6 10 •11 1 3 0.0380 0.213 3.5 10 -6 1 4 2.5131 0.266 1.9 10 1 5 5.3538 0.037 0.0 0.0240 0.948 9.8 -5 0 III 2 3 IP-3 2 4 2.4991 0.797 7.1 10 -1 2 5 5.3395 0.112 2.3 10 -2 3 4 2.4751 1.329 2.1 10-4 3 5 5.3157 0.186 7.1 10 4 5 2.8407 0.310 1.6 3 1 2 0.0806 0.527 6.0 10 8 1 3 0.1149 0.131 2.0 10 1 1 4 3.2037 0.700 10 1.7 3 1 5 6.9113 0.091 5.1 10 3 Ne III 2 3 0.0343 0.185 1.2 10 -2 2 4 3.1231 0.420 5.2 10 2 5 6.8308 0.055 2.2 3 4 3.0888 0.140 1.2 10' 3 5 6.7964 0.018 0.0 4 5 3.7077 0.188 2.8 Table 23 (cont.) 126 ATOMIC PARAMETERS FOR STATISTICAL EQUILIBRIUM EQUATIONS eij Ion 1 J (electron volts) ... Ajl i 2 5.0769 0.624 5.9 ■ 10"' i 3 5.0800 0.416 5.6 • 10" i 4 7.7060 0.142 5.3 ■ 10" i 5 7.7073 0.285 1.3 2 3 0.0031 0.817 • 10 tie TTTiv 1.4 2 4 2.6292 0.246 1.1 • 10“ 2 5 2.6304 0.619 4.0 • 10" 3 4 2.6261 0.234 3.9 • 10" 3 5 2.6273 0.342 4.4 • 10" 4 5 0.0012 0.446 2.3 • 10 1 2 0.0513 0.244 1.3 • 10; 1 3 0.1379 0.210 5.2 • 10 1 4 3.7558 0.153 1.9 ■ 10 1 5 7,9221 0.024 0.0 2 3 0.0865 0.578 4.6 • 10" uAtie \Jv 2 4 3.7045 0.459 1.0 2 5 7.8708 0.073 4.2 3 4 3.6179 0.764 3.8 • 10" 3 5 7.7842 0.121 6,8 ■ 10" 4 5 4.1663 0.185 2.6 1 2 1.8416 1.229 1.8 • 10" 1 3 1.8455 1.843 4.7 • 10" 1 4 3.0410 0.494 1.3 ■ 10“ 1 5 3.0470 0.087 3.4 * 10" C* TT 2 3 .0039 2,558 3.3 ■ 10" D IX 2 4 1.1994 1.7270 1.9 ■ 10" 2 5 1.2053 2.194 1.8 • 10“ 3 4 1.1954 1.541 8.7 • 10" 3 5 1.2054 4.323 2.1 • 10" 4 5 .0060 0.796 1.0 • 10" 1 2 .0369 1.215 4.7 • 10" 1 3 .1032 0.569 4.7 ■ 10I 1 4 1.4037 0.548 9.1 • 10" 1 5 3.3682 0,114 0.0 O TT T 2 3 .0664 2.797 2.4 • 10" o 1XX 2 4 1.3668 1.643 9.1 ■ 10" 2 5 3.3314 0.342 .9 3 4 1.3004 2.738 6.4 • 10" 3 5 3.2650 0.569 1.6 • 10" 4 5 1.9645 1.278 2.5 Table 23 (cont.) 127 ATCMIC PARAMETERS FOR STATISTICAL EQUILIBRIUM EQUATIONS Ion i j CiJ (electron volts) JV V 1 2 2.2386 1.255 7.1 10l4 1 3 2.2469 1.882 10.1 io-J 4 3.6977 1 0.427 3.7 10 l 1 5 3.7985 1.255 9.6 lole 3.2 111/'I Till T T 2 3 .0083 3.189 io J 2 4 1.4581 1.240 3.5 io" 2 5 1.4699 1.905 3.9 io”J 3 4 1.4498 1.381 1.1 10"* 3 5 1.4616 3.329 3.6 10~6 4 5 .0118 1.341 7.6 10 1 2 .0609 0.475 2.1 I 3 .1663 0.400 2.8 10““ 1 1 4 1.7070 0.224 2.2 10 I 5 4.0362 0.037 0.0 2 3 .1054 1.501 01 TITIV 8.2 10~l 2 4 1.6461 0.673 7.9 10"2 2 5 3.9753 o.no 2.6 3 4 1.5407 1.122 1.9 10~2 3 5 3.S700 0.184 3.8 10 2 4 5 2.3292 1.256 3.2 1 2 .1379 2.239 3.1 • 10 ~_l 1 3 .1947 0.531 2.7 • 10 6 1 4 1.7372 2.634 .3 1 5 4.1251 0.378 4.3 • l0l3 2 3 0.0568 1.176 5.1 ArA** TTTill • 10 2 2 4 1.5993 1.581 8.3 • 10"£ 2 5 3.9872 0.227 4.0 £ 3 4 1.5426 0.527 2.9 • 10"S 3 5 3.9304 0.076 0.0 4 5 2.3879 0.823 3.1 1 2 2.6151 0.570 2.8 io:2 2.6312 L 3 0.854 2.2 1 0 1 1 4 4.3219 0.212 9.7 10"1 1 5 4.3443 0.423 2.6 .0160 Ar IV 2 3 1.345 2.3 1 0 1 2 4 1.7067 1.008 6.8 l0-\ 2 5 1.7292 1.239 9.1 1.2 10- i 3 4 1.6907 0.865 1 0 1 3 5 1.7132 2.498 6.7 10 £ 4 5 .0224 0.601 5.2 10 "5 Table 24 128 PARAMETERS FOR THE ABSORPTION COEFFICIENTS REPRESENTED BY A POWER LAW Transition CS (LS-L'S*) (electron volts) C I (Jp)-c II (2P) 11.264 12.19 3.317 2.0 C II (2P)-C III ^S) 24.376 4.6 1.950 3.0 C III (XS)-C IV (2s) 47.871 1.84 3.0 2.6 C IV (2s)-c V (V) 64.476 .71 2.7 2.2 N I (AS)- N II (3P) 14.54 11.42 4.287 2.0 N II (3P) - N III (2P) 29.605 6.67 2.86 3.0 N III (2P) - N IV (1S) 47.426 2.06 1.626 3.0 0 I (3P) - 0 II <*S) 13.614 2.94 2.661 1.0 0 I CP) - 0 II (p) 15.204 3.85 4.378 1.5 0 I CP) - 0 II ( P) 18.630 2.26 4.311 1.5 0 II(As) - 0 iii(3p) 35.146 7,32 3.837 2.5 0 III(3P) -0 IV (2P) 54.934 3.65 2,014 3.0 0 IV (2P) - 0 V (1s) 77.394 1.27 0.831 3.0 Ne I (1S) - Ne II (2P) 21.559 5.35 3.769 1.0 Ne II (jPJ-Ne III (3P) 41.07 4.16 2.717 1.5 Ne II CP)-Ne III CD) 44.274 2.71 2.148 1.5 Ne II ( P)-Ne III ( S) 47.981 0.52 2.126 1.5 Ne III (3P)-Ne IV (*S) 64.0 1.8 2.277 2.0 Ne III CP)-Ne IV (~D) 68.806 2.5 2.346 2.5 Ne III ( P) -Ne IV ( P) 71.434 1.48 2.225 2.5 Ne IV (AS) -Ne V (3P) 97.16 3.11 1.963 3.0 Ne V (3P) - Ne VI <2P) 126.4 1.4 1.471 3.0 Ar I (*S) - Ar II (2P) 15.76 32.7 4.2 1.6 Table 25 PARAMETERS FOR THE ABSORPTION COEFFICIENT PRESENTED BY THE QUANTUM DEFECT METHOD TRANSITION 9, - 1 s . , r ^ij(ev) V - 1 o-t-1 S I(3P)-S i i 4 s ) 10.35 .169 1.95 2.88 2.24 3.14 2.14 .362 .069 1.15 S iqP)-S II(-D) 11.82 .211 1.95 2.88 2.82 3.14 2.14 .362 .069 1.15 S ICP)-S II( P) 13.4 .127 1.95 2.33 1.69 3.14 2.14 .362 .069 1.15 S IK^SJ-S III(3P) 23.4 .387 2.22 2.81 10.89 2.31 2.31 .362 .044 1.53 S III(3P)-S IV(2P) 35.10 .231 2.32 2.86 4.29 2.28 2.67 .362 .035 1.87 S IV(2P)-S V^S) 47.3 .085 2.33 2.84 1.12 1.67 2.72 .362 .048 2,15 S V ( 1S)-S VI(2S) 72.5 ------1.36 1.60 2.75 .310 .000 2.17 S VI(2S)-S Vll(1s) 88.0 ------.53 1.60 2.71 .310 .00 2.36 Cl 1 (2 ?)-cl II(3P) 13.01 .242 1.67 2.95 .012 3.60 2.39 .362 .077 1.02 Cl If_P)-Cl II(.D) 14.45 .135 1.67 2.95 .010 3.60 2.39 .362 .077. 1.02 Cl I( P) -Cl II( S) 16.47 .027 1.67 2.95 .001 3.60 2.39 .362 .077 1.02 ci i i (^p )-ci m d s ) 23.80 .156 2.20 2.84 4.44 2.32 2.41 .362 .045 1.51 Cl II(,P)-C1 IIICD) 26.04 .195 2.20 2.84 5.55 2.32 2.41 .362 .045 1.51 Cl IIC,P)-C1 iii( P) 27.51 .117 2.20 2.84 3.33 2.32 2.41 .362 .045 1.51 Cl III(AS)-C1 IV(3?) 39.9 .259 2.30 2.87 5.71 2.25 2.45 .362 .031 1.75 Cl IV(3P)-C1 V(2p) 53.50 .139 2.33 2.87 2.07 2.32 2.78 .362 .035 2.02 129 % V 1 . Cl V ( P)-C1 VI( S) 67.8 .062 2.33 2.98 .72 1.67 2.68 .362 .056 2.24 Table 25 (cont.) PARAMETERS FOR THE ABSORPTION COEFFICIENT PRESENTED BY THE QUANTUM DEFECT METHOD TRANSITION ytj(ev) r V,<*/ Cl VI^SJ-Cl VII(2S) 96.70 -- —— 1.00 1.60 2.80 .310 .00 2.25 Cl VII(2S)-C1 Villas) 114.27 -- —— .41 1.60 2.73 .310 .00 2.42 Ar II(2P)-Ar III(?P) 27.62- .424 2.16 2.80 8.51 2.40 2.27 .362 .053 1.40 Ar II(,P)-Ar 111(h)) 29.35 .235 2.16 2.80 4.73 2.40 2.27 .362 .053 1.40 Ar II( P)-Ar III( S) 31.74 .047 2.16 2.80 .95 2.40 2.27 .362 .053 1.40 Ar III(~V) -Ar IV^S) 40.90 .111 2.30 2.89 2.55 2.25 2.27 .362 .034 1.73 Ar IIKgPj-Ar IV(h>) 43.53 .235 2.30 2.89 3.19 2.25 2.27 .362 .034 1.73 Ar III( P)-Ar IV( P) 45.23 .047 2.30 2.89 1.91 2.25 2.27 .362 .034 1.73 U 3 Ar IV( S)- Ar V( P) 59.79 .188 2.33 2.89 3.27 2.28 2.57 .362 .034 1.91 Ar V(3P)-Ar VI(2P) 75.00 .099 2.33 3.00 1.29 2.34 2.88 .362 .046 2.13 Ar VI(2P)-Ar VII^S) 91.30 .047 2.33 3.07 .52 1.65 2.80 .362 .062 2.32 Ar VII(LS)-Ar VIII(2S) 124.00 ------.77 1.60 2.83 .310 .00 2.32 Ar VIII(2S)-Ar IX^S) 143.0 ------.37 1.60 2.75 .310 .00 2.46 N IV^S)- N V(2S) 77.45 ------1.67 2.23 1.82 .310 .00 1.68 N V(2S)-N VI(lS) 97.86 ------.55 2.22 1.86 .310 .00 1.86 0 v^sj-o vi(2s) 113.87 ------1.05 2.23 1.83 .310 .00 1.73 0 vi(2s)-o VH(lS) 138.08 ------.39 2.22 1.88 .310 .00 1.88 130 Ne VI(2P)-Ne VII(ls) 157.91 .023 2.3 1.78 .16 2.24 1.99 .362 .031 1.76 Table 26 131 ATOMIC PARAMETERS FOR EINE STRUCTURE TRANSITIONS OF P IONS ION TRANSITION (electron volts) *4 AJi 2 2 C II Pl/2" P3/2 .0079 1.432 2 4 2.4 •10"6 2 2 N III Pl/2‘ P3/2 .0216 1.097 2 4 4.8 • 10"5 2 2 0 IV Pl/2" P3/2 .0679 .810 2 4 5.2* 10“4 Ne II 2p3/2" 2pi/2 .0970 .244 4 2 8.6 • 10"3 2 2 -3 S IV Pl/2 ” P3/2 .1178 1.670 2 4 7.7 • 10 2 2 Cl V Pl/2" P3/2 .1850 1.052 2 4 Ar II 2P - 2P 3/2 1/2 .1776 .635 4 2 — 2 2 Ar VI Pl/2 P3/2 .274 .798 2 4 Table 27 132 ATOMIC PARAMETERS FOR,FINE STRUCTURE TRANSITIONS OF P IONS ION TRANSITION U) A ^ ‘j j 1 (electron volts) n .0061 .401 1 3 2.1 •ID'6 3P» N II P„ - % .0162 .279 1 5 1.3 • !0"lS 3p: .0102 1.128 3 5 7.5 • io~6 - .0141 .376 1 3 2.6 • 10"5 3po \ 0 III - .038 .213 1 5 3.5 • K f 1] 3po 3p2 - .024 .948 3 5 9.8 • io“5 \ 3p, - .0806 .527 5 3 6.0 • 10'3 3p, Ne III - .1149 .131 5 1 2.0 • io"8 3po - .0343 .185 3 1 1.2 • 10-3 3p0 - .244 1 • 10’3 3po \ .0513 3 1.3 Ne V - .1379 .122 1 5 5.2 • io'9 3po 3p2 - .0865 .578 3 5 4.6 • io"3 3p> 3p2 1 3P - .0369 1.215 1 3 4.7 !-» O 0 3p, S III - 3P .1032 .569 1 5 4.7 • io-8 3p0 V t I - .0664 2.797 3 5 2.4 © 3 3 LO - .0864 2.172 5 3 7.5 • H T J 3p2 P 1 Cl II - .1235 .443 5 1 4.8 • io"7 S 3po - 3P .0371 .933 3 I 1.4 • io"3 3p, *0 - .0609 .475 1 3 2.1 • IO"3 3pi -7 Cl IV Ui U .1663 .400 1 5 2.8 • 10 o o o 3p2 - .1054 1.50 3 5 8.2 . 10"3 \ 3p2 Tabic 27 (cont.) 133 ATOMIC PARAMETERS FOR FINE STRUCTURE TRANSITIONS OF P IONS ION TRANSITION u)t (electron volts) •AiJ “’j AJi 3 P - 3 p 2 1 .1379 2.239 5 3 3.1 • IO-2 3 3 Ar III P 2 - P 0 .1947 .531 5 1 2.7 ■ 10"6 3 3 -3 1.176 1 5.1 • 10 P 1 - P 0 .0567 3 3 3 8.0 • 10“3 P0 " P1 .0945 .257 1 3 Ar V 3Pr0 - 3P 2 .2520 .320 1 5 1.3 ’ 10"° 3 3 P 1 - P 2 .1571 1.040 3 5 2.7 - 10~2 t Table 28 134 ATOMIC PARAMETERS FOR FORBIDDEN TRANSITIONS BETOEEN GROUND TERMS ION TRANSITION Jl ii) (electron volts) ij “i Aji 3P - 1.885 3.136 9 5 4.0 •10'3 N II 3P - Ls 4.046 .342 9 1 3.4 ■10“2 *D - 2.154 .376 5 1 1 . 1 4s - 2d 3.326 1.43 4 10 9.7 •io -5 4 2 0 II S - P 5.017 .428 4 6 4.8 • io ‘ 2 1 2d - 2P 1.692 1.74 10 6 1.7 • O 3P - XD 2.481 2.391 9 5 2.8 •10-2 0 III 3P - l s 5.339 .335 9 1 2.3 1io"1 *D - xs 2.841 .310 5 1 1.6 3P - 1D 3.185 1.27 9 5 .22 Ne III 3P - XS 6.831 .164 9 1 2.2 1D - 1S 3.708 .188 5 1 2.8 ' 4S - 2D 5.080 1.04 4 10 2.6 • 10-3 Ne IV 4S - 2P 7.707 .431 4 6 1.1 2D - 2P 2.628 1.46 10 6 7.3 • io”1 3P - LD 3.641 1.38 9 5 5.2 • 10_1 Ne V 3P - *S 7.871 .218 9 1 4.2 1D - LS 4.166 .185 5 1 2.6 Table 28 (cont.) 135 ATOMIC PARAMETERS FOR FORBIDDEN TRANSITIONS BETWEEN GROUND TERMS ION TRANSITION SI U) u) (electron volts) ij 1 j Aji 4 2 -3 S - D 1.843 3.072 4 10 1.0 • 10 S II 4s - 2P 3.046 1.481 4 6 .3 2d - 2P 1.445 9.789 10 6 .4 3P - ld 1.318 4.928 9 5 .09 S III 3P - xs 2.675 1.025 9 1 .87 - Ls 1.965 1.278 5 1 2.3 3P - D 1.425 3.855 9 5 .12 Cl II 3P - h 3.374 .456 9 1 1.4 lD - 2.070 1.149 5 1 2.3 6S - 2d 2.240 3.137, 4 10 3.4 • 10‘3 Cl III As - 2P 3.706 1.882 4 6 .77 2d - 2P 1.464 7.885* 10 6 . 66 3P - XD 1.571 2.019 9 5 .28 Cl IV 3P - 1s 3.971 .331 9 1 2.7 2d - S 2.329 1.256 5 1 3.2 3P - XD 1.709 4.742 9 5 .4 Ar III 3P - S 3.989 .680 9 1 4.1 XD - S 2.388 5 1 3.1 *s - 2D 2.760 1.424 4 10 1.2 ■ 10‘2 Ar IV As - 2P 4.341 .635 4 6 2.0 2d - 2P 1.719 5.609 10 6 1.3 3P •• 1.817 1.083 9 5 .73 Ar V . 3P - 1s 14.966 .142 9 1 6.8 - 1s 13.061 1.115 5 1 3.8 Table 29 ATOMIC PARAMETERS FOR PERMITTED TRANSITIONS ION TRANSITION £*J (electron volts) - a tto-L C II 2P- AP 5,33 0.2 C II h - h 9.28 1.64 C III Xs - 3P 6.49 0.3 C IV 2s - 2p 8.00 5.5 N III 2P - 4P 7.09 0.2 N IV 8.34 0.3 2_ 2 N V S - P 10.00 3.6 0 IV 2P - AP 8.82 0.2 0 V xs - 3p 10.22 0.3 0 VI 2s - 2p 11.99 2.6 S IV 2p -*d 8.96 1.2 2 2 S V P - D 11.69 1.9 2 2 S VI S - P 10.32 0.3 Cl V 2P - 2d 14.00 1.5 Cl V 10.75 1.2 Cl VI 1s - 3P 12.3 0.3 2 2 Cl VII S - P 15.25 9.1 2 2 1.34 Ar II 2P “ 13.5 Ar II 2P - IsAP 16.4 0.3 2 '4 Ar VI ,P - 2P 12.49 1.2 Ar VI 2P - 2d 16.4 1.18 Ar VII W p 14.0 0.3 Ar VIII 2s - 2p 17.36 8.5