This dissertation has been microfilmed exactly as received 68-2996
HARRINGTON, James Patrick, 1939- MODEL PLANETARY NEBULAE: ELECTRON TEMPERATURES AND IONIZATION STRATIFICATION.
The Ohio State University, Ph.D., 1967 Astronomy
University Microfilms, Inc., Ann Arbor, Michigan MODEL PLANETARY NEBULAE: ELECTRON TEMPERATURES
AND IONIZATION STRATIFICATION
DISSERTATION
Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University
By
James Patrick Harrington, S.B. , M.Sc.
******
The Ohio State University 1967
Approved "by ACKNOWLEDGMENTS
It is a pleasant duty to extend my gratitude to Dr. Eugene
Capriotti, both for suggesting this problem, and in acknowledgment of his advice, encouragement, enthusiasm, etc.
Thanks are also due to Dr. Stanley Czyzak for discussions con cerning collision cross sections, and for communicating to me the results of certain calculations in advance of publication.
The numerical calculations in this dissertation were made on the IBM 709^ of the Ohio State University Computing Center, and acknowledgment is due for the allotment of the necessary computing time.
This work was carried out while the author held a National
Aeronautics and Space Administration Traineeship. VITA
December 21, 1939 Born - Salem, Ohio
196l...... S.B., The University of Chicago, Chicago, Illinois
1964 ...... M.Sc., The Ohio State University, Columbus, Ohio
I96I4—1967 .... National Aeronautics and Space Administration Trainee
PUBLICATIONS
"Variations in the Maxima of Long-Period Variables." The Astronomical Journal, Vol. 70, pp. 569-575» October, 1965
"Theoretical H-Beta Line Profiles and Related Parameters for Rotating . B Stars." (Co-author with George W. Collins, II) The Astrophysical Journal, Vol. lH6, pp. 152-176, October, 1966
iii TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS ...... ii
VITA ...... iii
PUBLICATIONS ...... iii
LIST OF T A B L E S ...... vi
LIST OF ILLUSTRATIONS...... vii
INTRODUCTION ...... 1
PART I. FORMULATION OF THE PROBLEM
Chapter
I . THE PHYSICAL PROCESSES WHICH ESTABLISH EQUILIBRIUM ...... U
General Considerations The Ionization Equilibrium Absorption Coefficients Recombination Coefficients The Energy Balance The Solution for Electron Density and Temperature
II. THE TRANSFER P R O B L E M ...... 28
The Stellar Radiation The Diffuse Radiation The Quadrature Formulae for Frequency Integration The Integral Equation of Transfer and the Iteration Procedure
III. THERMAL STABILITY ...... ^3
PART II. THE MODELS CONSTRUCTED
IV. DEFINING PARAMETERS OF THE MODELS...... U8
iv TABLE OF CONTENTS (CONTINUED)
Chapter Page
V. RESULTS AND DISCUSSION...... 52
Electron Temperature and Ionization Structure Line Intensities and Radio Emission Thermal Stability
LIST OF REFERENCES...... 7 6
v LIST OF TABLES
Table Page
1. Parameters for the Absorption Coefficients Represented by a Power L a w ...... * ...... 12
2. Parameters for the Absorption Coefficients Represented by the Quantum Defect Me t h o d...... 13 p 3. Atomic Parameters for Fine Structure Transitions of P I o n s ...... 20
U. Atomic Parameters for Fine Structure Transitions of ^P Ions ...... 21
5. Atomic Parameters for Forbidden Transitions Between Ground T e r m s ...... * ...... 22
6. Atomic Parameters for Permitted Transitions...... 23
7. Defining Parameters for Six M o d e l s...... 50
8. Abundances of the Elements...... 51
9. Optical Depths of the Mo d e l s...... 66
10. Logarithms of the Fractions of the Elements in Various Stages of Ionization, Integrated over the Volume of the Six Nebular Models ...... 67
11. H0 and Radio Emission of the Six Models ...... 72
12. Integrated Intensity of 21 Emission Lines Relative to H6 = 1 0 0 ...... 73
vi LIST OF ILLUSTRATIONS
Figure Page
1. The Geometry of the Transfer Problem...... 1+1
2. The Electron Temperature of Model No. 1 ...... 53
3. The Electron Temperature of Model No. 2 ...... 5*+
1+. The Electron Temperature of Model No. 3 ...... 55
5. The Electron Temperature of Model No, k ...... 56
6. The Electron Temperature of Model No. 5 ...... 57
7. The Electron Temperature of Model No. 6 ...... 58
8. The Ionization Structure of the Inner Region of Model No. 1 ...... 59
9. The Ionization Structure of the TransitionZone from Ionized to Neutral Material in ModelNo. 1 ...... 60
10. The Ionization Structure of Model No. 2 ...... 6l
11. The Ionization Structure of Model No. 3 ...... 62
12. The Ionization Structure of Model No. 1 + ...... 63
13. The Ionization Structure of Model No. 5 ...... 6U
ll+. The Ionization Structure of Model No. 6 65
15. The Thermal Stability of Model No. U ...... 75
vii INTRODUCTION
This dissertation describes the construction of a number of models of planetary nebulae, These nebulae are idealized as homo geneous masses of gas bounded by tvo concentric spheres and illumi nated by central stars whose flux is represented by the recent model atmosphere calculations of BBhm and Deinzer (1 9 6 5, 1966), The gas is considered to be composed of the six most abundant elements: H, He,
C, N, C, and Ne, The gas is further considered to be motionless and in a steady state. The ionization structure, electron temperature, and electron density are obtained, as well as various auxiliary quanti ties such as the total emission in some of the lines of H I, He I,
He II, [N II), [0 II], [0 III], [Ne III], [Ne IV], and [Ne V], The thermal stability of the gas is also investigated.
Two features of real nebulae have been neglected here which
could have significant effects upon the ionization structure. The
first of these is the lack of homogeneity observed in many of those
objects for which large scale photographs are available. Condensa
tions may not be present in all planetary nebulae, however; and in any
case, it is necessary to be able to solve the homogeneous case before
attempting to include the condensations. But it should be pointed out
that these results cannot be applied to nebulae with pronounced density
fluctuations,
It is also true that the assumption of a static, uniform density distribution is unrealistic, for such a configuration is dynamically unstable and will expand rapidly into the surrounding vacuum. A theoretical determination of the density distribution of the nebula at various stages in its evolution can be obtained only by solving the complex gas dynamics problem — which may involve the pressure of trapped Lyman-alpha radiation or even magnetic fields. On the other hand, observational material does not seem good enough to de rive empirically the run of density with radius. For these reasons this work has been limited to constant density models.
The recent work on the dynamics of planetaries by Mathews
£1 9 6 6) circumvented the problem of ionization structure by assuming the gas to be composed of totally ionized hydrogen with a trace of
0++ as coolant. Such an assumption makes it impossible to take into account the large drop in temperature, illustrated by the models pre- sented here, at the transition zone between the He and He'1' regions.
This temperature change would doubtless have important dynamical ef fects. These calculations may be considered as a step toward further work on the dynamics problem, for models such as these (but with density varying from point to point) may be used to form an evolving
sequence. PART I
FORMULATION OF THE PROBLEM CHAPTER I
THE PHYSICAL PROCESSES WHICH ESTABLISH EQUILIBRIUM
General Considerations
The treatment of the ionization structure of a gaseous nebula may be conveniently divided into two parts. In this chapter the phys
ical processes which establish equilibrium in a small volume element
of gas subjected to a given radiation field will be discussed. The
subsequent chapter will be devoted to the transfer problem whose solu tion determines the radiation field of the nebula.
The electrons which are ejected when atoms are ionized by
ultraviolet radiation provide the source of energy for the nebular
gas. Two conditions underlie most of the phenomena peculiar to gaseous
nebulae: the extreme dilution of the ionizing radiation and the low
density of the gas. This means that the time between either radiative
or collisional excitations is much greater than the lifetime of any
level which is not highly metastable. The one possible exception is
the 2p level of hydrogen, where absorption of the intense Lyman-alpha
radiation might give some competition to the downward radiative
transitions. Calculations, however» show that neither the 2p nor the
metastable 2s level could be significantly populated relative to the
ground state without giving rise to an anomalous Balmer decrement;
this is not observed (Capriotti, 1964a, 1964b), 5
An. extremely metastable level which deserves some considera tion is the 2^S level of helium. This level will be discussed below where we consider the sources of diffuse radiation; it is enough to note here that while some photoionizations may occur from this level, they will not appreciably affect the ionization or energy balance.
Thus, to a very good approximation, we may assume all photoionizations of the various ions to occur from their respective ground states.
Once an electron is ejected by photoionization, electron- electron collisions (which are very frequent compared to other pro cesses; the electron gas always has very nearly a Maxwellian distri bution — see Bohm and Aller, 19^7) distribute its energy among the other electrons of the gas. At the same time there are three proces ses by which the gas is losing energy: electrons recombine with ions, they radiate energy in free-free transitions, and they lose energy in
collisions with atoms having low-lying energy levels.
To find the electron temperature, we balance the energy gains
against the losses. This follows from our assumption of a static nebula in equilibrium. Any actual planetary nebula would be expanding
and, in the first place, this would introduce an additional term in the
energy equation to account for the work done by the expansion of the
gas. Second, the thermal energy of the gas is so large compared to the
rate of energy losses and gains that the time constant for relaxation
to equilibrium would be measured in years. Thus, if dynamical changes
occur rapidly enough, it would not necessarily be true that the gas
would be in equilibrium. Only a dynamic model could include these
effects. Although their importance would be sensitive to the particular model and its initial conditions, calculations "by Sofia
(1967) indicate they can he important for at least some nebulae.
The Ionization Equilibrium
Let Nj. j denote the number per cubic centimeter of atoms of the i^k kind in the state of ionization — is the number of neutral atoms. Further, let us define
= /°° itIL J(v) ai.(v) &v , (1) hv where J(v) is the mean intensity of radiation, a^j(v ) ‘blie absorption coefficient from the ground state, and the ionization potential.
Balancing ionizations against recombinations, we have
where a^(t) is the recombination coefficient as a function of the electron temperature. If we define
0ij(Ne» ^ = Re a ^ t ) » ^ then we can write equation (2) as
N, 1+n = 6 N, . (U) itJ+l ij id
The total number of atoms of the i^ kind is
k NH Ai “ I Nij » (5) j “1 where Ng denotes the number of hydrogen atoms per cubic centimeter, and Aj is the number of atoms of the ith kind relative to hydrogen; the sum extends over the k states of ionization under consideration. By combining equations (U) with (5)» can be written as
(6) k t I xin n=l where
for j i 2, and Xil = 1.
Equation (6) will serve for C, N, 0, and Ne, but for hydrogen and helium we wish to include photoionizations by the Lyman-alpha radiation of ionized helium. We assume that the photons of this resonance line will be scattered without diffusing far until they are absorbed by neutral hydrogen or helium. It would be difficult to estimate the intensity of this radiation; instead, we will assume each He+ Ly-a photon to be absorbed on the spot by hydrogen or helium.
The number of Ly-a photons produced per cubic centimeter per second is given by
Ne % e ++ (1 - X) “B (t) » where ag(t) is the effective recombination coefficient to the n=2 level of He+ and X is the fraction of recombinations that terminate on the 2s level. The line photons will be absorbed by hydrogen and helium in proportion to the respective numbers of neutral atoms,
NHo a n d NHeo, and in proportion to the absorption coefficients. Let y denote the ratio of these absorption coefficients:
_ aHe° (v) = 10.6 y “ Ta^o(v) T r, 7T J+0.8 e.v
The equations for ionization equilibrium then are 8 r v Ne NH+ °H° * % ° *H° + Ne NHe++ "X ^ “b |nh o + YNHeo (7)
(8) Ne NHe+ aHe° ” NHe° *He° + Ne NHe++ (l“X) aB Nh° + YNHe°
Ne NHe++ °He+ = NHe+ *He+ (9)
NflO + (%►) = Ng (10)
% e ° + ^ %e + ^ + ( % e ++) = AHe NH (1 1 )
Here, Age = NHe^NH» rela^iVe abundance of helium.
Upon elimination of all the unknowns except Ngo from equations
(7)-(ll), we are left with a quadratic equation. Let us define S(t) =
(l-Xjog/o^o. We can replace a^Qo with ago — this approximation is
good to about 5 per cent at the temperatures of interest. Then, after
some algebra, the solution of (7)-*(ll) may be written:
(12) NgO « 1/2 ' ^ ' hc] "« where
AHe f 1 V S~| He 1 + * 9He+ 9H° 1 * 9He+ He 1 + 8He+ b = 1 L ~ 1 + “J (13) X + 0H° . i ||eHeo . (l + eHo)
Age
1 + 1 + 0H e+ {1 ~ 0 H e + S} c = ilk) (1 + 0Ho) |jL + 0Ho - ” ^®He° + 1 + 0He+)J and
Nh+ = % - Nh o (15)
NHe° - [«[1 + 0Heo (1 + ©He+)] + Y(1 + 0He+)
-1 • ~ NH° t1 + 0H ° O 1 % o J *He
KHe+ “ '"He * "He0 ’'1 + ®H=+ >'1 <” >
MHe++ “ ^ H e “ IfHe°^®He+ ^ 1 + ®He+ ^ *
All the 0's in the above equations are defined as in equation (3).
The complexity of (12)-(18) would be greatly reduced if we
assumed that the He+ Ly-a photons were absorbed only by hydrogen.
This is a good approximation in all respects except*one: it leads
to values of NHeoin the inner regions which are too large; four times
too large in a typical case.
Of course, in the outer regions of the nebula, where there is
almost no He++, we can use equation (6) for H and He as well as for
the other elements.
The number of electrons per cubic centimeter is just the stun
of the number of electrons contributed by each atom; thus, the equa
tion of ionization equilibrium is
6 k g(Ne , t) = Ne - I I (j-1) NjJN t) = 0 (19) i=l J=2 J 10
Absorption Coefficients
For hydrogen and ionized helium the absorption coefficients were taken from the fundamental- work of Menzel and Pekeris (1935).
They can he written in the form employed by Hummer and Seaton (19^3) as
aHo(y) = 6.318 * 10”18 f(y) ; y = e/13.595 (20)
aHe+(y) = 1.5TT • 10~18 f(y) ; y = e/5U.U03 (21)
where e is the frequency in electron volts. The function f(y) is given by
f . U tan**1( /y-1) *) exp A k ------j = r f — j f(y) = ---- 1------i f(l) = 1. (22) y1* {1 - exp [-2ir//y-l]}
Later we will also need the absorption coefficient from the n=2 level of He+ to evaluate the diffuse radiation due to the Balmer continuum of ionized helium. It is given by
[aHe+ (y)]n=2 “ 3,bj2 • lO”18 f(w) ; w = e/13.595 (23) where
f 8 tan~1(V^l) eXP L " v^ZT j (w+2) (w+U) f(w) = -j— 58------J= -“ — ' ■ (2b) w {1 - exp [—U tt/ /w-l]} 15w
and
f(l) - 1. 11
For neutral helium two formulae were used. The first is that employed hy Hummer and Seaton (196U) and is good for e < 5^.^ e.v.
It is
aHe°(y) = T.35 * 1 0 - 18 exp {-0 .7 3 (y-1 .8 0 8)} , (25) where y = e/13.595
This formula gives values which are too small at higher frequencies and so I have used a power law form which is continuous with (-25); the exponent was obtained hy the quantum defect method. This second formula is ajjgoCy) = 1.152 • 10“1T y-2.58 . y = e/2^.58i (26) for e > 5^.U e.v. It fits the calculations of Huang (19^8) fairly well.
The absorption coefficients for most of the heavier ions have been taken from Seaton (1 9 5 8). His results may be written as follows:
aiJ (y) = aQ • IO"18 (ay"s + (l-a)y~^s+1^) (2 7 ) where y = (v/Vfj ) and is the ionization potential of the ion in question. The values of aQ, a, and s are listed in Table 1.
For the more highly ionized atoms with 2s^ and 2s ground states, no such results were available. For these ions I used the quantum defect method as presented by Burgess and Seaton (1960a).
The energy levels needed were taken from Moore (19^9). The formula given by Burgess and Seaton may be simplified to read
aij(y) = aQ • lCfl 8 y"s cos2 I"v(q + .9»3l(y-l)yj (2 8) 12
Table 1
PARAMETERS FOR THE ABSORPTION COEFFICIENTS REPRESENTED BY A POWER LAW
Ion a a s o
C II 1*.3 1.67 2 N I 8.9 3.1 1 N II 6.8 2 .0 6 2 N III 2.0 0.9 2 0 I 2.533 U.O 1 0 II 8.1 2 .U5 2 0 III 3.U8T 1.30 2 0 IV 1.11 1 .8 2 3 Ne I 5.0 U.3 1 Ne II 5.0 3.8 2 Ne III 1.698 2.07 2 Ne IV 2.95 1.20 2 Ne V l.hh 1.0 2.31
where y = 11:16 values for the constants aQ , s, q f and r are listed in Table 2.
This covers all the states of ionization considered: C II -
C IV, N I - N V, 0 I - 0 VI, and Ne I - Ne V. Carbon was assumed to be at least once ionized since the ionization potential of Cl is less than that of hydrogen.
The accuracy of the absorption cross sections for the ions
of C, N, 0, and Ne may not be high since there is little experimental
data and since the theory is complex and severe approximations must be
made to obtain results.
Recombination Coefficients
The recombination rate coefficients for hydrogen-like atoms
have been accurately computed by Burgess (1958). Seaton (1959) gives 13
Table 2
PARAMETERS FOR THE ABSORPTION COEFFICIENTS REPRESENTED BY THE QUANTUM DEFECT METHOD
Ion ao s q. r
C III 2.1*1*8 2.2U0 1.789 1.600 C IV 0.8381* 2.220 1.8U6 1.81*0 N IV 1.673 2.23k 1.821 1.676 N V O.5H89 2.218 1.86U 1.861* 0 V l.ol*5 2.230 1.825 1.728 0 VI 0.3868 2.218 1.878 1.883
the approximate formula
scaled electron temperature: 10^t = Te. Comparison of this expres sion with the results of Burgess shows that it should be accurate to a few percent for all (t/Z^) < 2. Burgess and Seaton (1960c) have also calculated the recombina tion coefficients for He° and have shown that equation (29) gives results which are only 3 per cent too small at t=l and 8 per cent too small at t=2. There is a complete lack ofrecombination coefficients for other ions, with the exception of some work done on 0++ by Burgess and Seaton (1960b). Thus, equation (29) was used exclusively. This is probably a fair approximation, especially for the more highly ionized atoms, for the high energy levels tend to be hydrogenic, and these levels receive more of the recombinations the higher the charge or the lower the temperature. Ik The Energy Balance Energy Gains The electron gas gains energy hy the photoionization of atoms. Let us define = J(v) aij(v) h (v “ dV (30) Then the rate of energy gain due to the ionization of the atoms ij will he . In the case of the photoionization of H° and He° by He+ Ly-a photons, we do not have the intensity; rather, the rate of energy gain, P(Ly-a), is the number of Ly-a photons produced multi plied by the energy of the ejected electron: P(Ly-a) = NeNHe++ (^--X) aB " VH^ £ N^o + yN^ o^j + h( u12 - vHe) Hh o *KYNHeo J (31) Here, is "t*1® frequency of the Ly-a photons (v12 = ^0.8 e.v.), is the ionization potential of H o and vHe that of He ° . Energy Losses Recombinations The electron gas loses energy by recombinations. The loss rate for hydrogen-like atoms due to this process is given by Seaton (1959) as 8(X) » 2.85 • IQ”25 Z2 f 5 {-.0713 + 1/2 lnX + .6Hox"*333} (32) 15 where X is defined as in equation (29). The accuracy of (32) should be comparable to that of (2 9). Free-Free Transitions The loss rate for free-free transitions as given by Allen (1963) is Bff(t) = 1.U35 * 1CT25 t*5 (j-l)2 Ne N (33) where we have set the Gaunt factor to unity. This process is not very important. Collisional Losses The most effective means of removing energy from the electron gas is provided by inelastic collisions between the electrons and ions. The energy expended by an electron impact is permanently lost to the gas if the atom returns to the ground state by a radiative rather than collisional process. For positive ions (as opposed to neutral atoms) we may inte grate over the Maxwellian velocity distribution of the electrons to obtain the rate of collisional excitation from level i to level j per atom per electron, with the result U ij(t) = 8,629 * 10”8 t_*5 (nij/u)i) exp(“1.1^05eij/t) (3b) where is the statistical weight of the lower level, is the separation of the levels in electron volts and is a dimensionless constant called the collision strength (Seaton, 1958). The corres ponding rate for downward collisional transitions is 16 Dji(t) = 8 .6 2 9 • 10"8 t” 5 (nij/Wj) (35) It will "be seen that for transitions of interest in gaseous nebulae we will need to consider three cases: l) the case where Ne is so small compared to the Einstein transition probability, Ajj, that we can neglect collisional de—excitations; 2) the case of a two-level atom with collisional de-excitations; 3) the case of a three-level atom with collisional de-excitations. In the first case the rate of energy loss, E^j, is just the energy of the transition times the rate of upward transitions: EiJ - He Mij | °k (36) where ck ^ = 1 .6 0 2 • 10“12 elk ulk (37) til and k denotes the various energy levels of the ij ion which are under consideration. In the second case, we consider two levels with populations and N^. The equations of statistical equilibrium are Ne % u12 = N2 The rate of energy loss due to radiative transitions is EiJ = 1 .6 0 2 • 10-12 e12 N2 A21 (U0) 17 Using the solution of (38)—(39) for N2 we can thus write EiJ - Ne NiJ C* (1,1) where t) = 1 .6 0 2 • 10-12 ------(1*2) 1 ♦ Ne(U12+D21)/A21 In the third case, the equations of equilibrium are N1 (Ne U12 + Ba U13> = N2 (He D21 + A21> + H3 (lfa D31 + A31> (1,3) K2 (Ne U23 + Ne D21 + A21^ = N1 ^We U12^ + W3 ^Ne D32 + A 32^ + N2 + N3 = N±j (U5) and the loss rate is Ei(. = 1 .6 0 2 • 10“12 [N3 (e3lA31 + e32A 32^ + N2 ^e21A2 1 ^ Upon solving the system (i^3)— (^+5) for Ng and we may express E^j as in equation (Ul) if we define C* as c*(w_, t) = 1.602 • 1 0 - [ --- — — + -— — ~I (^7 ) LC2 +C3 C5 + C6 J where the C's are defined as follows: C1 = ^e31A31 + e32A32^ ^U13^Ne ^D21 + U23^ + A21^ + NeU12U23* C2 = (l>32 — U^2 ) + A 32} 'fNg (D2i + U^2 + Agi) c3 = {Be (D21 + U23 + U12> + A21} (Ka (I)31 + U12 + “lS* + A31} 18 CH “ e21 ^ 1 ^ U12 + U13^ ^Ne D32 + A32^ + U12 ^Ne °31 + A3 1 ^ C5 * *Ne ^D21 " D31^ + A21 *“ A31* *He ^D32 " U12^ + A32* C6 85 *Ne ^U23 D21 + D32^ + A21 + A32* *Ne ^U12 + U13 + °31^ +A31* Now the energy levels which are important for collisional cooling of gaseous nebulae may be grouped into three categories. The first of these is comprised of the levels formed by the terms of those atoms with 2p2 , 2p3, or 2p^ ground configurations. These energy levels lie a few electron volts above the ground term. Transitions between such terms are forbidden to electric dipole radiation since they have the same electron configuration. The second category is comprised of the fine structure levels within the ground term of those ions having 2P or ground states. The separation of these levels is of the order of a tenth to a hun dredth of an electron volt, and thus transitions between them ~ also forbidden — give rise to infrared radiation. The importance of these levels was first emphasized by Burbidge, Gould, and Pottasch (1963). In the third category we have permitted transitions of the 2 more highly ionized atoms with 2s or 2s ground configurations, such as the 2S ’- 2P transition of C IV, These levels lie 8 to 12 e.v. above the ground state, but, as pointed out by Hummer and Seaton (196U), they can be quite important in the inner regions of high excitation planetaries. 19 Our treatment of N II, 0 III, Ne III, and Ne V, in which both the fine structure levels and the terms are important, depends upon the fact that the radiative transition probabilities of the fine structure levels are much smaller than those of the transitions be tween terms (transition probabilities vary as the cube of frequency). Thus when the density of the model is high enough for collisional de-excitation of the terms to be of importance, the collisional de excitation of the fine structure levels will so dominate the radiative de-excitations that we can neglect the infrared cooling entirely. We then treat the S, P, and D terms as a three-level atom. On the other hand, at lower densities we can neglect the col lisional de-excitation of the terms and view the 3p ground term as a three-level atom. Of course, the cross-over density will vary from ion to ion. In treating the losses due to the ions with permitted transi tions, we can always neglect downward collisions. Table 3 gives the atomic parameters for the fine structure transitions of ions with 2P ground states; Table h gives the para meters for the ions with ground states. In Table 3, the values for both the collision cross sections and the transition probabilities were taken from Osterbrock (1965). The values of the energy level separations in this and the following tables have been taken from Moore (19U9). The values of the collision cross sections listed in Table were taken from Osterbrock (1965); the transition probabilities are from Garstang (l95l). 20 Table 3 ATOMIC PARAMETERS FOR FINE STRUCTURE TRANSITIONS OF 2P IONS Ion Transition i -► j flij wi “J Aji £ij . 1 0-6 C II 2p -+ 2p 1.1*1* 2 1* 2 .It .00793 1 /2 3 /2 N III 2P -»■ 2P 0 .9 2 2 k U.8 • 10"5 .02163 1/2 3/2 0 IV 2p h. 2P 0.92 2 1* 5.2 • 10" ^ .0U792 1 /2 3 /2 - 3 Ne II 2p -v 2P 0.32 1+ 2 8 .6 * 10 .09695 3 /2 1 /2 In the cases where the terms are regarded as one level, I have followed Osaki,, (1962) in defining the mean values of A^j and 4----- r y (2J +1) A(L S J -*■ L S J ) (W) ™ (2L n+ l ) (2S_+l)(.— n Jt Jt n n n n m m m n m and A I (2Jn ♦ 1) A(LnSnJn . Jm ) (LnSnJn * J j (1*9) J J n m Table 5 lists the atomic parameters for the ions with low-lying terms. The collision cross sections are from Saraph, Seaton, and Shemming (1 9 6 6) and from Czyzak, Krueger, Saraph, and Shemming (1967). The transition probabilities are from Garstang (1951» I960) and from Seaton and Osterbrock (1957). 21 Table U ATOMIC PARAMETERS FOR FINE STRUCTURE TRANSITIONS OF 3P IONS Transition Ion i - J EiJ niJ “i WJ AJi 3po - 3pi 0.00609 0.1*3 1 3 2.1 * 10“6 N II 3p0 - 3p2 0 .0 1 6 2 8 O.1 9 1 5 1.3 • 10“12 3 3 P -*■ P 1 2 0 .0 1 0 1 9 1 .01* 3 5 7.1* * 10-6 3 3 Po - P1 0 .01U1 0.31 1 3 2.6 • 10“5 0 III 3Pro - 3P 2 0 .0 3 8 0 0.25 1 5 3.5 • 10-11 3p 3p rl *2 0.021*0 0.96 3 5 9.8 • 10“5 3p -► 3p 2 *1 0 .0 8 0 5 8 0.1*2 5 3 6.1 * 10“3 3 3 Ne III 0.111*93 0.10 5 1 2.0 • 10“8 \ + 3po O.0 3 U3 U 0.13 3 1 1.13 » 10“3 3po - 3pi 0.0513 O.ll* 1 3 1.3 • 10“3 Ne V 3po - 3p2 0.1379 0.21 1 5 5.2 • 10-9 3pi - 3p2 O.O865 0.61* 3 5 l*.6 • 10“3 22 Table 5 ATOMIC PARAMETERS FOR FORBIDDEN TRANSITIONS BETWEEN GROUND TERMS Transition Ion “i i + J eiJ nij “j AJi 3P -*■ "h) 1.8851 3.13 9 5 U.03 • 10"3 N II 3P XS U.OU63 0.31* 9 1 3.1*2 • 10“2 ■h) 1S 2.1537 0.38 5 1 1.08 **s 2d 3.3255 1.1*3 1* 10 9.72 • 10”5 1* 2 0 II S -v *p 5.0170 0.1*28 1* 6 1*.80 . 10“2 2d -> 2p 1.6923 1.70 10 6 1.71 * 10-1 3P -► 1D 2.U81 2.39 9 5 2 .8 1 • 10“2 0 III 3P ■* 1S 5.339 0.31* 9 1 2.31 • 10”*1 ■4) ^ 2.81*1 0.30 5 1 1 .6 0 3.1851 1 .2 6 9 5 2 .6 0 • 10”1 Ne III 3p -> 1s 6.8309 O.16U 9 1 2.205 ■S 1s 3.7076 0 .1 8 8 5 1 2 .8 0 U« 2 5 -*■ D 5.0795 l.Ol* 1* 10 2.59 • IO- 3 1*_ 2 Ne IV S -► P 7.7071 0.1*27 1* 6 1 .0 6 2d ■> 2p 2 .6 2 8 1 1.1*2 10 6 7 .2 7 • 10"1 3.61*12 1.38 9 5 5.20 • 10-1 Ne V 3P -> ^ 7.8707 0 .2 1 8 * 9 1 U.21 l*.l663 0.185 5 1 2 .6 0 23 Finally, Table 6 gives the parameters for the permitted transi tions. The collision cross sections for C IV, N V, and 0 VI are taken from Hummer and Seaton (196U) ; the others are from Osterbrock (1963). Table 6 ATOMIC PARAMETERS FOR PERMITTED TRANSITIONS Transition Ion «ij i - i EiJ C II 2P -* Up 5.3298 0.20 C II 2P 2D 9.28U5 1 .6U C III 1S ** 3p 6.U923 0.30 C IV 2s -*■ 2P 8.003U 5.5 N III 2 P - Up 7.0873 0.20 N IV 1S 3p 8.3U03 0.30 N V 2s **■ 2P 9.9971 3.6 0 IV 2P ^P 8 .817U 0.20 0 V IS •* 3p 10.2173 1.5 0 VI 2g ■+ 2p 11.9923 2.6 The accuracy of the cross sections listed in Table 5 should be of the order of *20 per cent; those are the most important transitions under most nebular conditions. The cross sections listed in the other tables are in many cases only estimates, and they will be of inferior accuracy. With regard to the neutral atoms, He° and Ne° have no low ly ing levels, but 0° and N° do. Seaton (1958) has given values for the 2k collision cross sections of these atoms, but they can only be regarded as estimates. Unless the nebula is optically thick there will be no significant amounts of neutral material; even in this case Hjellming (1966) found the effect of 0 I and N I cooling to be negligible. I have therefore neglected these atoms. In the extreme case of a low density nebula excited by a very high temperature star, the region of transition from H+ to H° becomes quite extensive, and in this region H° coexists with high temperatures. To cover this contingency, collisional cooling by H° has been included. The collision cross section has been taken from Hummer (1963). Re taining only the terms for excitation to the n=2 and n=3 levels, we can express E^j as in equation (Ul) if we define C as C(t) * 8.U9U • lCT19 exp (-.12 lnl) {exp (-.75*) + 0.1972 exp (-.8 8 9)> 1 (50) where \ » 15.789/t. The Equation of Energy Balance Let us define f(Ne , t) as the rate of energy gains minus the rate of energy losses per cubic centimeter. Then from the preceding sections we have I wiJ *ij + PCI*-*) The equation for energy balance is simply f(Ne, t) - 0 . (52) 25 The Solution for Electron Density and Temperature Equations (19) and (52) were solved simultaneously for Ne and t by two-dimensional Newton-Raphson iteration. This method can be found in most books on numerical analysis; see, for example, Hildebrand (1956), Supposing we have approximations and tthen the (i+l)^h approximations can be written Ne (i"**1) = + A(Ne) , (53) t^i+1^ = t ^ + A(t) (5*0 where A(N, e > - (55) (t) =FfL is-3N e - g3£-1/3N e J D , (56) and 3Ne 3t 3Ne 3t From the definitions of the functions f and g we have: H - H - (58) 3 |g- - i-n <59) 3Ne i j 3He It - 8P'f3t“ ) + l} [jiJ - se (B + 6ff+ c* + I ck>J "It1 - Ne ti , i3 [ # ♦ f r M £ ] (6 0, 26 jg_ . iEitol + ^ Q la . „e{6 + c. + i ok)J 3 % j aiC 3C* I »iJ Furthermore, if we define Q^j(t) as we can write L aw I “ Qik) Xik 3 N ij _ „ k=l w - (63) I *ik k=l and _iigjr . HlJ k-i__ I • --- Xik (61() B* J i for the elements for which we use equation (6) for Here, L represents the highest stage of ionization of the 1th element. The derivatives of equations (l2)-(l8) will, of course, he rather cumber some as will those of C* as defined by (U7); there is no need to reproduce them here. The point is that the expressions are analytic and the derivatives are straightforward; they can be evaluated for a given Ne and t and used in (55)-(5T) to obtain the next Ne and t. At the inner edge of the nebula the first guess for Ne and t were 27 taken to be Ke » Njj + Njje and t = 1.0* After that* the values at the last radial distance were used as a first guess for the next step. An iteration process of this type has second order convergence. In this case only a few iterations were necessary to satisfy the con ditions []A(Ne)/Ne]< 10"6 ijd(t)/t]< 10”6 . CHAPTER II THE TRANSFER PROBLEM The Stellar Radiation The material in the nebula will be subjected to radiation from two sources: that which reaches it directly from the central star and diffuse radiation emitted by the nebular gas. Thus the mean intensity j(v) appearing in the 4 and ¥ integrals can be written as J(v) = J* + Jv (65) where J* is the stellar component and the diffuse. O Let irFv be the physical flux emitted by each cm of the surface of a star of radius R*. Then* neglecting absorption* the flux passing through a spherical surface at a distance R from the star (where R is large enough so that the angle subtended by the star is negligible) is given by ttFv (R*/R)2 . (66) In passing through the nebular gas some of the light will be absorbed. Absorption only by H°, He°, and He+ is considered since the abundances of the other absorbers are so low. For even though the opticaldepths in the continue of some of the heavier ions like 02+ may exceed unity, such absorption occurs at frequencies greater than 5U.U e.v. and hence has negligible effect in competition with He+ . It is not so clear, however, that we can neglect the absorption due to C2+ and N2+ longward 28 of 5b(U e.v., tut calculations with the completed models show that there would be no significant effects. Let R1 denote the inner radius of the nebula and define R xgo (R) = aHo(v0) J NHo(r) dr , (67) R1 where vQ is the frequency of the absorption edge, with corresponding definitions for Tgeo and Tjje+. Then the optical depth of the material between the star and an element of gas at R is t v (R) = Tiff th °(r ) + Tiff ‘rHe° a__ +(v) + -BS — THe+(R) (68) 1.577 • 10”18 The mean stellar intensity ar R can then be written J*(R) = (PVA ) (R*/R)2 e~Tv(R ) (69) In most previous work, the ultraviolet flux of the central star, Fv, was assumed to be that of a black body ~ mainly for want of any better information* Recently, BShm and Deinzer (1965, 1966) have calculated model atmospheres for stars with temperatures and surface gravities in the range expected for the central stars of planetaries. The frequency distribution of the emergent flux differs considerably from Planck's law, especially in the region beyond 51*.1* e.v., which is critical in high excitation nebulae. I have therefore used the fluxes from BShm and Deinzer's models and employed Planck's law only for the sake of comparison. 30 The Diffuse Radiation The primary source of the diffuse ionizing radiation emitted by the gas is the Lyman continuum which results from electronic recom binations directly to the ground state of H I, He I, or He II. In addition, the radiation emitted by recombinations to the n=2 level of He II — the Balmer continuum — has nearly the same spectral distri bution as the H I Lyman continuum and thus is readily absorbed by hydrogen. The rate of those recombinations to the n***1 level which emit radiation in the frequency range e to e + de (where e is measured in electron volts) is 0 r-l.l605(e - e H Fn (e) de = 8 1 .8 6N i ^ e 2 t~3/2 expj^------ (e) de (TO) I. where is the number of ions per cubic centimeter, 10 • t the electron temperature, en the energy difference between the continuum and the n^** level, u»n and the statistical weights of the n*'*1 level and the ion, respectively, and an(e) the continuous absorption coef ficient from the n^*1 level. The emission coefficient, j£, is just the energy of the photon times the rate of recombinations and is thus W j t B 1.3115 • 10"10 Nj Ne e3 t*3/2 exp |j— ■ K K (c) (ti) 31 For the K I and He II layman continua, (uq/u^) = 2; for the He I Lyman continuum (u>a/ui^) = 1/2; and for the He II Balmer continuum = 8. For temperatures below 20,000°, almost all recombinations originate within a few electron volts of the series limit, so recom binations to levels other than those considered above will contribute very little ionizing radiation. The only line radiation that can contribute to the diffuse field is that of the Lyman series of He+ and the (is np ^-P -► Is2 lg ) series of He°, Since the absorption coefficients in these lines are several orders of magnitude higher than the continuous absorption coefficients of H° and He°, these photons will be scattered in the lines and degraded before they can be absorbed by H° or He°. Thus we need only consider the He+ Lyman-a and He° (is 2p Is2 lg ) radiation. There is one remaining source of ionizing radiation that we need to consider. This is the two-photon continuum which arises from the He+ 2s Is transition and the (2s Is) and (2s -*• Is) transitions of He°, A recombination to any level other than the ground state leads either to the production of a line photon as dis cussed above, or to the population of a metastable level which gives rise to two-photon emission. In the limit of low densities, the per centage of recombinations which terminate in either process is just an atomic parameter determined by the solution of the cascade equa tions. For higher densities, there are collisional transitions between the 2s and 2p states which become important. 32 Let us first consider He*. Of those recombinations terminating on the n«2 level, a fraction X will populate the 2s state and the remaining (l-X) will give rise to He+ Ly-a» It does not appear that the collision cross section for 2s to 2p transitions has been calcu lated for this ion. However, for H°, the collisional process does not dominate until we have densities of the order of > 1.7 • 10^ (Seaton, 1955). But in H°, proton impacts produce 90 per cent of these transitions, while these collisions would be unimportant in He+ due to this ion's repulsive Coulomb field. Furthermore, the two- photon transition rate will be higher in He+ than in H°, since transition probabilities increase with the separation of the levels. Thus, if the 2s - 2p electron collision cross section of He+ is com parable to that of H°, we can neglect collisional effects for Ne < 10^, which covers all models where there is an appreciable amount of ITHe + + . The frequency distribution of the 2s -► Is emission of H° is given by Spitzer and Greenstein (l95l) in terms of the function 'J'(y); we will assumethe frequency distribution for He+ 2s Is is the same. Then if we define T(He+ ; e) = 0.013 7(e/Uo.8), we have U0.8 J T(He+ ; e) de = 2 (72) o and we can write the emission coefficient for two-photon transitions as Uirje(He+ ; 2s - Is) - {1.602 .- 10”12 e Ne NHe++) • (X(He+ ) «B (He+ ) T(He+ ; e)> (73) 33 With respect to the He* Ly-a radiation, even nebulae which are optically thin in the H° Lyman continuum will be optically thick in the He* Lyman continuum and the He* Ly-a radiation will be trapped. Now the transfer problem for H° Ly-a is very difficult and consider able simplifications must be made to obtain results; the problem for He* ly-a is further complicated by two circumstances: the radiation can be absorbed by H° and He°, and it can be converted to 0 III and N III radiation by the Bowen fluorescent mechanism. It is observed, however, that the images of nebulae in the 0 III fluorescent lines are never significantly larger than the images in He II XU686 (see, for example, Aller, 1956); this implies that the He* Ly-a radiation does not escape from the region where it is being produced. Seaton (i960) has shown that only about 30 per cent of the He* Ly-a is converted to 0 III radiation in most nebulae. Even after the He* Ly-a radiation is degraded to 0 III radiation, or further, to N III radiation, there is still an energetic quantum remaining which can ionize either H° or He°. Due to the insolubility of the transfer problem and in view of the preceding discussion, the assumption is made for the purpose of these calculations that all the He* Ly-a photons are absorbed on the spot by H° or He°. For comparison, a few models have also been con structed assuming no absorption of He+ Ly-a by H° ar He°. The He* Ly-a radiation is important to the energy balance of the inner regions of high excitation nebulae, and it is unfortunate that such an approximation must be made. By assuming on-the-spot absorption of this radiation, it has its greatest effect in the 3U interior of the He region, whereas, in fact, we would expect most of the ahBorption to occur near the edge of the He region or Just beyond, where the abundances of H°, He°, and He4* are higher. This would tend to reduce the temperatures of the innermost regions and increase the sharpness of the temperature drop at the transition zone observed in many of the models calculated here. The values of aB(He+ ) and X(He+ ) were taken from Hummer and Seaton (196U). We can fit the expressions we need for equations (7 3 ) and (31) by the following interpolation formulae: X(He+ ) «B (He+) - 10"13 t**'5 (U.355 - .63U9 In t) (7 U) [1 - X(He+ )] ctB(He+ ) = 10“13 t“ ‘ 5 (10.93 - 2.953 In t) (75) Further, we find that the expression needed in equations (13) and (lU) S(t) * [l - X(He+ )] Og(He+ )/ (H°), can be approximated over the range 1 < t < 2 to within a few percent by S(t) = constant ** 2.55 • (76) In the case of He°, about three-fourths of the recombinations are upon triplet levels and populate the 23S state. Of the remaining one-fourth, some will populate the 2^-S level and others will give rise to the 2^P -*■ l-^S line. This line, which is analogous to Ly-a, does not build to great intensities, however; this is because the separation of the 2^S and 2^P levels is great enough that radiative transitions to the 2^S occur after a relatively small number of scatterings (Aller, 1956). Thus, in effect, all recombinations upon the singlets terminate upon 21S, and we can neglect the 2^~P - 1*S line. 35 The transition rate from 2^S to 1^-S toy the two-photon process is exceedingly slow: A = 2.2 • 10"^ sec“^ (Mathis, 1957). Since the rate of collisions from 2^S to 2^-S is about 5.7 * 10“9 Ne sec-1 (Osterbrock, 196U), most atoms in the 2^S state will reach the ground state via the 2^S state if the electron density is greater than about U • 10^ cm”3. Indeed, observations indicate that the 2^S level is depopulated at densities lower than this, but the mechanism by which this occurs is not understood (see, for example, Robbins, 1966, and Capriotti, 1967). We may ask if helium atoms spend enough time in the 2^S state for an appreciable number of photoionizations to occur from this level. An upper limit on the number of atoms in the 2^S state relative to the ground state can be obtained by neglecting depopulation by photoioni zation or "unknown" mechanisms; the number in the 2^S is then N(2 3S , . ». % e * <3A> «H.ott) (77) 2.2 • 10"5 + 5.7 • 10“9 Ne Using the values of Ne , NHeo, NRe+, and t from the various nebular models constructed here, it was found that while the ratio N(2^S)/Ngeo was usually small, it could attain values of 50 per cent or greater. But even in this case, considering that the ratio of H to He is about 6:1, neglect of photoionizations from this levelwould have an insig nificant effect on the energy balance of the gas. We will assume, then, that all recombinations which are not directly to the ground state lead to two-photon emission. We further take the unknown frequency distribution of the (2^S -► l^S) emission 36 to be the same as that of the (2^8 *► r^s) emission which has been calculated by Dalgarao (1966) in terms of the function A(y). Then, if we normalize this function by defining T(He°i e) = 1.055 • 10" 3 A(e/20.6), so that 20.6 J T(He°; e) de = 2 (78) o we can write the two-photon emission coefficient of He° as Uirje(He°;2s Is) * 1,602 • 10~12 e N N + oB(He°) T(He°;e) (79) 6 It© The effective recombination coefficient og(He°) to the n=2 level of helium has been calculated by Burgess and Seaton (1960c); from these results it can be seen that an error of only about 6 per cent is made if we replace dg(He°) with a ^ H 0). Thus we can use the expression of Hummer and Seaton (1963) in equation (79). ag(H°) ■ 1.628 • 10”13 t” *5 (1 - .7196 In t + .58U t*333) (80) This completes our discussion of the sources of the diffuse radiation field. The Quadrature Formulae for Frequency Integration In order to evaluate the integrals over frequency in equations (l) and (30), the range from 13.595 e.v. to « was divided into twenty intervals which were chosen to fall at the ionization potentials of the various ions (H° and 0° and also (P+ and N^+ were considered coin cident). Within each interval each type of radiation — stellar, 37 recombination, He+ two-photon and He° two-photon — were treated separately, due to their differing frequency dependence. For the stellar radiation the quadrature points within the first 19 intervals were Gaussian, and the last interval (138 e.v. to «•) is spanned by a three-point Laguerre quadrature; a total of 90 points was employed. For the recombination radiation the frequencies and weights were chosen as the ones appropriate for Gaussian quadrature over a finite range with a weighting function exp (-1.1 6 0 5 x); this absorbs the exponential term in equation (7 1 ) for t = 1 .0 and reduces it at other temperatures. A total of 28 points was employed. Finally, for the two-photon emission, the frequencies and weights were appropriate for a weighting function (y - y2 )*^ — where y = e/2 0 .6 or e A 0 .8 — which approximates the frequency depen- * dence of the two-photon emission. Twelve points were used for the He+ radiation and four for the He° radiation. The quadratures should be quite accurate. In some earlier calculations an inferior scheme was employed, and the difference be tween the temperatures obtained by the two methods was less than .9 per cent for all values of the radius. The Integral Equation of Transfer and the Iteration Procedure The mean intensity of the stellar radiation at a point R has already been given by equation (69). The mean intensity of the dif fuse radiation is - r .. 38 UrrJ^Cft) = f «)v(r) exp{-rv(^,r)} d£ (8l) volume where JVC£) is the emission coefficient at a point r and xv(R,r) is the optical depth along the path joining 5 and r. Since we cannot evaluate this integral without specifying the solution everywhere, we must use an iterative procedure. Our choice of a first approximation will depend upon the optical thickness of the entire nebula. If the model is optically thin, we may set J®(R) = 0; if it is optically thick we may assume that radiation is absorbed in the same region in which it is produced (on-the-spot absorption) and set the diffuse intensity equal to the source function, i.e., JB (R) = j (?0/k (5). While some of the models v v \> constructed here were optically thin at frequencies less than e.v., all were optically thick for e > 5^.** e.v. Thus, in practice, two approximations were employed: l) on-the-spot absorption of all diffuse radiation; and, 2) on-the-spot absorption of the He Lyman continuum and neglect of all other diffuse radiation. Starting at the inner boundary of the model we solve the equa tions of equilibrium for the stellar radiation alone, evaluate jy/kv and re-solve the equations including the diffuse radiation which is considered absorbed on the spot. Then we may re-evaluate and repeat the process until the new j is near enough the previous value.1 ■'■It should be noted here that in some cases where the helium is almost completely twice ionized at the inner boundary, we cannot use approximation (l). This is because diffuse quanta capable of ioniz ing neutral hydrogen are being produced at a rate higher than the recombination rate of hydrogen; thus successive evaluations of Jv/^v result in driving kv -*• 0 and J-p 00. All we need do in this case is to neglect some of the diffuse radiation. 39 The integrals of the type in equation (6 7) are, of course, only formal since Njjo(r) is a function of T^o(r), Tne°(r )* a**4 THe+^r^* The gration is carried out by the Runge-Kutta method. With this procedure we work our way to the outer boundary, storing the values of JV(R), tho(R), THeo(R), and Tge+(R)• The values of the emission coefficient and optical depth thus obtained enable us to evaluate the integral for J^(R). We then con struct another model using these values of the diffuse radiation and repeat the process to convergence. For the actual evaluation of the mean intensity we have J®(R) = 1/2 J1 I^Tr, u) dp = 1/2 I lJ(R, ]i±) w± , (8 2) -1 i*l where I?(R, m ) is the intensity traveling in a direction which makes an angle G with the radius vector, and p = cos 6. The integral was replaced with a Lobato quadrature over angle with and wi the ab scissas and weights, respectively; seven angles were employed. t We may write the intensity in a particular direction as n x(R,R2,p.) IV (R, pA) = / J^rtR.x,^)] exp{-tv(R,x,pi)> dx , (83) 0 where Figure 1 will define the quantities r, x, and R2. xv(R*x,p^) is the optical depth between R and a point at a distance x along a path defined by p^. The evaluation of the integral in equation-(83) required some care, inasmuch as the integrand is not always well behaved. The fol lowing procedure was adopted. The emission coefficients and optical depths along a radius are known at specific values, r^, of the radius, values which were determined by the step size of the Runge-Kutta inte gration. The spherical shells of radii rn cut the line of sight defined by at points x^. We can express the intensity as the sum of the integrals, Ik, from x^ to Xj^^: xw+n Ik = / Jv(x) exp{-rv(x)} dx (8M xk To find the optical depth, Tv(xk ), we make the approximation (85) where the optical depth between rQ ^ and r^ along the radius vector, Tv (rn ) - Tv^rn-1^» is knovm‘ Next, we assume that t v (x ) = a + bx and jv(x) = c exp(dx) over the range rn to *"n+ ^, where a, b, c, and d are constants. The integration of (8U) then gives the expressions: Ik (86) This formula gives reasonable values for Ik even when jv(x) is chang ing rapidly. In some cases the denominator can go to zero, but then so does the numerator; removal of this singularity is trivial. In the case of the models constructed for this thesis, the first approximation, followed by three re-integrations generally re- suited in a model whose temperature distribution as a function of Ul R 2 n+ Fig. 1.— The Geometry of the Transfer Problem radius differed everyvhere by less than 1 per cent from the preceding model. Such accuracy is much greater than the basic Uncertainties in atomic parameters* etc., vhich underlie these models. CHAPTER III THERMAL STABILITY The condensations observed in many planetary nebulae have been the subject of considerable speculation. Zanstra (1955) proposed that they are the result of "thermal instability"; this occurs if a denser region is able to cool so much more rapidly than the surrounding mate rial that there is a pressure drop entailing further contraction. The model Zanstra chose to illustrate this process was of such a simplified nature, however, that it was questionable whether these effects would occur in real nebulae. Daub (1963) found that the inclusion of a mix ture of elements as coolants removed the instability. Field (1965) developed a general theory of thermal instability and showed that the criterion employed by Zanstra and Daub was not generally valid. Re cently, Sofia (1966) reconsidered the problem using Field's stability criterion and including the effects of infrared cooling which had been neglected in earlier work. His results were also negative. Owing to the method employed here in solving the equations of ionization equilibrium and energy equilibrium, it proved easy to check the thermal stability of the gas at each point. This provides the most realistic test of the Zanstra mechanism yet made since all previous work represented the radiation field by Planck's law; this is certainly not the case, since stellar radiation reaching a point deeper in the nebula has undergone selective absorption by the intervening material. kk The condition for instability given by Field is ££"] < 0 (8 7) [3T j p where L represents the rate of energy losses minus gains per gram and the derivative is taken at constant pressure. In terms of the func tion f(Nfi, t) defined by equation (51)* f ® -pL. The inequality (8 7) can thus be written n&l >0 (88) U » U p Since ve wish to hold the pressure constant and to satisfy the ionization equation (19) while varying the temperature, we must also regard the number density of hydrogen atoms as a variable. Thus «(*.. t, V - |i dt + |£_ dNe + ! § - « „ . (89) e H Inspection of the function f shows that 3f _ f (90) 3 % Nh which vanishes at equilibrium where f=0. Thus we have df ■ ££ dt + ££- dN . (9 1) at 3N e e Now we wish to vary N with t in such a way that the gas pres- © 3 ure is constant. If the plasma were a perfect gas so that we could write p = const. Ne t, then we would have that dNe = -(Ne/t) dt at constant pressure. Although the perfect gas law has been assumed in all previous work, it is not strictly applicable. This can be seen if we consider what happens when we compress a given volume: we will force some electrons into recombining with the ions and the total number of particles will change. Although this effect will be impor tant only where a significant amount of the hydrogen is neutral, we can easily write the correct criterion. The pressure is p(Ne , Nh , t) « (Ne + NionS) 10U k t t (92) so that dp = 10U k Qb dNe + (He + Nions) dt ^ “ j (93) where ^ions “ I • i The variation of N„ with N and t is obtained by solving equation (19) Ja e for NH and taking partial derivatives to obtain dH„ f i t at + a n ~ 1 (9**) H Ne Lst eJ Setting dp * 0 in (93) and employing (9*0 we find that we can write dNe * -R' (Ne/t) dt (95) where R* is given by *’ ■ + i8]E + %“ tT (96) By using equation (95) in equation (9l)» we can express the criterion for instability, (87), in terms of known derivatives of the functions k6 f and g. Thus, the gas will he thermally unstable if the following inequality is satisfied: [ l - K*[r] I s ; ] * 0 • (97) PART II THE MODELS CONSTRUCTED CHAPTER IV DEFINING PARAMETERS OF THE MODELS In order to construct a particular model, it is necessary to specify the following quantities: i) The frequency distribution of the emergent flux of the central star, Fv . ii) The radius of the central star, R*. iii) The inner and outer radii of the nebula, R1 and R2. iv) The density of the nebular gas, N^. v) The abundance of the other elements relative to hy drogen, Ai# Having constructed a model, the resultant electron temperature, elec tron density, the number of atoms in the various stages of ionization determine the intensities of the emission lines in the nebular spec trum, Ideally, we would like a large grid of models with various densities and optical thicknesses, having central stars of different effective temperatures and gravities, all varying independently. Then we could solve the problem in reverse: from the observed intensities of the lines, deduce the physical conditions of a particular nebula. Aside from considerations of computing time, this is at present much too ambitious a program, both because of the limited number of stellar atmospheres available and because the neglect of potentially important effects such as condensations and expansion cooling could introduce U8 *9 considerable inaccuracies. What will be done here is to construct a sequence of models based upon parameters which presently appear typi cal of observed nebulae. These models may be compared with observations for major inconsistencies. They will also indicate what processes are important for further work. The models have been based upon the relationship between nebu lar radius, stellar temperature, and stellar luminosity as deduced by Seaton (1966). They represent an evolutionary track along which all the nebulae have the same mass: ^ e b s We are itirther restricted to stellar temperatures equal to those of the model atmos pheres calculated by Bohm and Deinzer (1965, 1966). The inner radius has been taken to be one-third the outer radius. This enables us to find the defining parameters (i) through (iv). Originally, it was intended to use the six Bohm and Deinzer model atmospheres with stel lar temperatures ranging from 38,000° to 106,000°, but the model constructed with = U6,250° proved to be so optically thick that it was 98.3 per cent neutral gas. Hence, the 38,000° atmosphere was not used. Later, the pre-print of a thesis by Goodson (1966) became available which listed the emergent fluxes of the Bohm and Deinzer models in tabular form, and included a previously unpublished model with an effective temperature of 7^*000°. This model was included in the sequence, whose defining parameters are listed in Table 7. It should be noted that the values of the surface gravity for which the model atmospheres were calculated, when combined with the values of the stellar radius from Table 7, give values of the stellar I mass which vary from 0 .2 6 M0 to 1.5 Mfi; thus the sequence is not 50 completely self-consistent. However, the surface gravities are not too bad, usually being within 50 per cent of the desired value for a 1,2 M# star (the value of 0 ,2 6 MQ for model #1 is exceptionally bad, but in this case it makes little difference as the radiation beyond 5**.l* e*v., which is sensitive to the surface gravity, is practically non-existent), Table 7 DEFINING PARAMETERS FOR SIX MODELS T* NH (atoms/cc) eff R*/R0 R1(psc.) KS(pso.) 1 1*6,250° 0.759 5.12 . 10k O.Oll* 0.0U2 2 63,100° 1.380 9.36 • 103 0.025 0.075 3 7U,000° 1.202 3.79 * 103 0.033 0.099 1* 91,200° 0.813 1.22 • 103 0.0U9 O.1U7 5 1 0 6,000° 0.231* 2 .2 9 • 102 0.085 0.255 6 100,000° 0.115 8.81* . 101 0.120 0.360 The abundances of the heavy elements relative to hydrogen were taken from Aller (1961). The relative abundance of helium to hydrogen was taken to be 16 per cent, which is also the mean of the values quoted by Osterbrock (1961*). The abundances are listed in Table 8. In addition to the six models, others were constructed for the purpose of illustrating the effect of certain process or of changes in parameters. Thus a version of model #3 was made in which the cool ing due to fine structure transitions was neglected. In one version of model #U, the flux from the central star was represented by black body radiation; in another, the absorption of the He+ Ly-a line radiation was neglected. Finally, a version of model #5 was con structed which employed the chemical composition used by Daub (1963) and another was made with an abnormally high abundance of helium. Table 8 ABUNDANCES OF THE ELEMENTS Element Aller (1961) Daub (1963) He/H 0 .1 6 0.15 C/H 0 .000U0 0.00020 N/H 0.00011 0.000i+l 0/H 0.00089 0.000U9 Ne/H 0.00050 0.00010 CHAPTER V RESULTS AND DISCUSSIONS Electron Temperature and Ionization Structure _ The electron temperatures obtained for the various models are plotted in Figures 2 through 7 as a function of radius. In each figure the solid line represents one of the "standard” models of the sequence defined by the parameters in Table 7, with the Aller (l96l) composition and a model atmosphere representation for the central star. The other lines represent the variant models discussed in the last chapter. In addition, the first approximation for the "standard" model is indicated. The ionization structures of the six "standard" models are illustrated in Figures 8 through lU. Each curve represents the frac tion of the atoms of a given element which are in the specified state of ionization. In cases where a curve approaches very close to unity it is discontinued; this should cause no confusion since there is a redundancy in having all the other stages of ionization plotted. We will discuss each model in turn. For the purposes of this discussion it will be useful to list the values of tr o {R2), THeo(R2), and +(R2) (which are the optical thicknesses of the nebulae at the ne absorption edges of the ion in question, due to absorption by that ion only: see equation (6 7)) for the six models. These values are found in Table 9. The fractions of elements in various stages of ionization 52 11,000 10,000 9,000 8,000 e 7,000 on-the-spot approximation 6,000 5,000' Model No. I 4,000 .0140 .0142 .0144 .0146 .0148 .0150 .0152 .0154 .0156 .0158 .0160 R(psc.) — Fig. 2.— The Electron Temperature of Model Eo. 1 10,000 9,000 on-the-spot approximation 7,000 .025 .030 .035 .040 .045 .050 .055 .060 .065 J070 .075 R ( p S C . ) ** Fig. 3.— -The Electron Temperature of Model No. 2 14,000' Model No. 3 no fine structure / cooling first approximation (no diffuse field) .03.04 .05 .06 .07 .08 .09 .10 R(psc) Fig. The Electron Temperature of Model No. 3 VI VI 20,000 18,000 16,000 Blockbody 14,000 12,000 No Ho Lyman o< first approximation (no diffuot fiold ) 10,000 8,000 Model No. 4 6,000 .05 .06 .07 ! 5 § .13 .14 R (psc ) —*■ Fig. 5.— The Electron Temperature of Model No. ^ 20,000 18,000* 16,000 Daub abundances 14,000* helium abundance twice normal - { Ah#«0.32) 12,000* first approximation 10,000* — ' (no diffuse field) 8 ,000* Model No. 5 6,000 .08 .10 .12 .14 .16 .2 0 .22 .24 .26 R(psc) —» Fig, 6,— The Electron Temperature of Model No. 5 10,000* T. 9,000* 8 £ 00* 7 £00* 1.0 2+ ^.2+y nt+ Model No. I (inner region) .0140 .0*42 .0144 .0148 .0194 006 R ( p«C ) —m. Fig. 8.— /The Ionization Structure of the Inner Region of Model No. 1 •oVI 1 XZ N 1 W \\ Model No Transition Zont bntwon loniztd and Nnutrol Rtgiont '• \ •• IV- VA .01968 .0(980 R(psc) Fig. 9.— The Ionization Structure of the Transition o\ Zone from Ionized to Neutral Material in Model No. 1 o ,3+ 2+ Ha .2+ Na Model No. 2 .030.025 .035 .040 .045 .0 5 0 .055 .0 6 0 .065 .070 .073 R ( psc ) — a* Fig, 10.— The Ionization Structure of Model No. 2 ,4 + 2+ Ha 0 4 0 090 R (p$C ) __ _ •090 ■100 2a«°n Str^ of „ e of Model p\ to *1 ‘ON ispoji jo sjnq.on.ijs aoxjnzTHoi stjj;— »jjx ‘S tj ir st «*--{ 9*d)a ir er »/• t+ .V- .3+ *.<>• Model No. 5 .8+ .065 .065 .105 145 .165 185 .205 .245 R(psc) Fig, 13.— The Ionization Structure of Model No. 5 o\ *r n R i w m s n v 8+ N*1.2+ Ht Model No. 6 M A+ 20 .22 .24 .30 3 4 ' 3 6 R(psc)—» Fig. lit.— The Ionization Structure of Model No, 6 ON vn 66 Table 9 OPTICAL DEPTHS OF THE MODELS Model th o (R2) xjjeo(R2) THe+(R2) #1 212* 30.0* 66.9* i2 8.23 1.08 364 #3 1.03 0.167 191 #4 0.363 0.040 55.9 #5 0.510 0.042 17.8 #6 1.70 0.227 15.8 *The values for model #1 refer only to the region between R1 and R = 0.01584 psc. have also been integrated over the nebular volume. The logarithms of these integrated values for the six models are given in Table 10. The antilogs of each row should sum to unity; however, there may be errors of a few percent in the entries due to the errors involved in the nu merical integration of the rapidly changing ionic abundances. Model #1. As can be seen from Tables 9 and 10, this model is so optically thick that 98.3 per cent of the material is neutral. Al though R2 was to be 0.042 parsecs, the calculation extended only to R = 0.01584, where the temperature had dropped below 3,000°. There is too little radiation beyond 54.4 e.v. to have any effect at the inner edge. The temperature rises steadily as the selective absorp tion preferentially removes the lower frequency radiation, until at Tjjo(0.015575) * 17.2 the increasing amount of absorption by neutral 67 Table 10 LOGARITHMS OF THE FRACTIONS OF THE ELEMENTS IN VARIOUS STAGES OF IONIZATION, INTEGRATED OVER THE VOLUME OF THE SIX NEBULAR MODELS Model Element I II III IV V VI H 1.9931* 2 .1 7 8 1 He 1.9932 2 .1 8 2 0 ¥.2791 No. 1 C 1.9938 2 .1 0 1 1 ^ .1 8 8 2 • * N 1.9930 1.2039 2 .0 8 9 6 ¥.3399 • • • • 0 1.9935 3.1926 2 .15H1 ¥ .1 1 1 6 • • • • Ne 1.9930 3.5253 2.1035 • • • • • • H ¥. 1664 1.999U He ¥ .9 6 9 3 1.9993 ¥.2100 CM £5 O C • 3.1*75** 1 .U881 1.8383 ¥.3623 N ¥.001*3 3.3010 1.3966 1.871*3 7.6509 • • 0 ¥.9U01 3.8717 1.9965 ¥ .1 1 9 6 ¥ .2 6 1 7 ♦ • Ne 5*. 1199 2 .2 6 6 2 1.9919 5.171*1 • • » • H ¥.1*367 1.9999 He ¥.381*2 1.9979 ¥.6321 C ¥.1*1*11 1 .0 6 1 0 1 ,91*66 No. 3 ¥.2312 N S’. 1865 ¥. 21*31* 2.91*28 1.9600 ¥.2771 • • 0 7 .6 5 2 0 ¥.3107 1.9970 ¥ .5 8 0 3 6.81*11 • • Ne 7.9388 ¥ .7 1 8 3 1.9971 ¥.9757 ¥ .8 1 2 1 • • H ¥.3011 1.9999 He ¥.131*9 1-902U 1.2975 C No. k ¥ .8 8 8 7 2.7089 I.9U62 2.8176 N 9.3993 5.63**7 2.5776 1.9535 2 ,8 0 1 0 5.1*573 0 1.171*5 ¥.9781 1.9008 1.1731* 2.708U ¥.1*1*27 Ne 7.3669 3.3U78 1.9271 1.1761 ¥.8635 9.9265 H ¥.9628 1.9996 He ¥ .6 6 2 6 1 .8 9 0 8 1 .3 U06 ¥.8212 1 .9 2 6 8 2.5980 No. 5 C 1 .0 5 8 5 N ¥ * 8689 ¥.5113 2.9U68 1.91*00 2 . 5998 ¥.1722 0 ¥.2313 ¥.U38U 1.8905 1.2808 2.1*205 ¥.2011* Ne ¥.3369 ¥.7763 1.9331 1.1229 ¥. 1*388 ¥. 5210 H ¥.7372 1.9976 He 3 . 5987 1.9927 2.0915 No. 6 C 2.2178 1.61*78 1.7312 ¥ . 6 1U8 N 1.1797 2.0002 1.5825 1.7833 ¥.1*358 • • 0 5.9277 2.3392 1.9856 ¥.0195 £.501*8 • • Ne ¥.0277 2.6310 1.9796 ¥.1*599 7.7571 • • material overcomes this trend and the temperature drops sharply. The discontinuity in the slope of the descending branch at R * 0,01582 is not a real effect, but rather is due to the artificial assumption that the carbon is at least once ionized. The electrons from C+ contribute to the cooling but their photoionization is not included in the heat ing integral ; the discontinuity occurs when the material becomes so neutral that C+ becomes the major source of electrons. It is seen that for this model the on-the-spot approximation for the diffuse radiation is quite good. Model ff2 , This model is also optically thick, but not so much so that the material is neutral at the outer edge. As in the last model, the temperature rises as we go outwards due to selective absorption. This rise is accentuated at the outer edge where the diffuse radiation is received from only 2ir steradians; the diffuse radiation depresses the temperature because the electrons ejected upon absorption of it have, on the average, less energy them those ejected by stellar radiation. This rise in temperature at the outer boundary is found in the follow- , ing models also. The higher temperature at the inner edge is due to the absorption of radiation beyond the ionization limit of He+ ; this radiation is extinguished almost immediately. The on-the-spot approximation is somewhat worse for this model — the error incurred can exceed 600°. If we were to extend the inner boundary of our model towards the central star, the discrepancy would become progressively larger, since the diffuse radiation produced in the highly transparent inner regions escapes and is absorbed out fur ther where the amount of neutral material is greater. 69 Model #3. The optical thickness of this model is small enough that the second alternative of neglecting most of the diffuse radiation was employed as the first approximation. The He++ region is still rather insignificant. It can he seen that at this density the neglect of the cooling due to the collisional excitation of the fine structure levels gives temperatures which are ahout 500° too high. At lower densities the effect would he greater; at higher densities, less. Model #U. In this model the He++ region has become important. Com parison of Figures 5 and 12 show that the transition from the high temperature inner zone to the outer zone — which is similar to the previous models -- corresponds to the transition from He++ to He+ . A number of factors combine to increase the temperature in the He++ region. First, it can be seen from Figure 5 that the ionization of H° and He° by He+ Ly-a is quite important. Secondly, the presence in the inner region of stellar radiation with frequencies beyond 5U.U e.v, results in the production of rather energetic electrons through the photoionization of He+ . Finally, it should be noted from Figure Ox O4. 12 that the transition of oxygen from 0J to 0 is coincident with ++ + the transition of He to He ; this is simply due to the depletion of ^ OX the high frequency radiation, which is absorbed by He . Since 0 is the most important cooling ion, this transition has a great effect on the electron temperature. When a black body spectrum of the same effective temperature is substituted for the model atmosphere fluxes, it can be seen that 70 the increase in the high frequency radiation results in a model in which the helium is twice ionized throughout nearly the whole nebula. Model #5. This model is similar to model #k. Since the density is lower, the transition from He++ to He+ is not as sharp as in the pre vious case. The great sensitivity of these models to the assumed chemical composition can be seen in Figure 6. The use of the Daub composition raises the temperature everywhere by more than 2,000°. This is pri marily because of the reduction in the amount of oxygen relative to the Aller composition: oxygen is the most important coolant. The effect of increasing the helium abundance is what one would expect; the initial temperature is greater because the heating effect of the high frequency radiation and the He+ Ly-a is concentrated in a smaller volume. Model #6. Because of the decreasing stellar luminosity toward the end of the sequence defined by Table 7, the optical depth at the Lyman edge, t h o ( R 2 ) , is again greater than unity for this last model, in spite of the decreasing density of the material. Thus the on-the-spot approximation was used. Line Intensities and Radio Emission Calculations were made for the intensities of particular recom bination lines of H I, He I, and He II and for some forbidden lines of [N II], [0 II], [0 III], [Ne III], [Ne IV], and [Ne V]. For the emis sion due to H6, He I X5876 and XUU7 1 , and He II XH686 and X5^12, the 71 results of Pengally (196U) vere fitted to polynomials in temperature. For the calculation of the forbidden lines, the general equations for a five-level atom with collisional de-excitation were solved by matrix inversion to obtain the population of the various levels. These line intensities were computed at each radial point and then integrated over the nebular volume. For HB, the total emission was computed; the other lines are expressed relative to HB = 100. The total radio emission was also calculated for a frequency of 2650 Me (at which most planetaries will be optically thin) using the expression of Oster (1 9 6 1)). His result may be written as follows for free-free encounters of electrons with H+ , He+ , and He++; UttJ(2650 Me) * 3.709 * 10”39 Ne t“ *5 {[(%+) + ( % e+)] (1+0.1525 In t) + U(NHfi++)(0.9295+0.1525 In t )} ergs/cm3 cps (98) In Table 11 we list the total emission of the nebular models in HB, and also the ratio of the radio frequency emission to HB. Table 12 gives the relative intensities of the other emission lines. Thermal Stability Of all the models constructed in this study, none were found to have any regions in which the gas was thermally unstable. The first term in the criterion for instability as given by inequality (97), (3f/3t), was found to be always negative. The second term, -R'(Ne/t)(3f/3Ne), could be positive or negative. However, when the second term was positive, it never attained more than 20 per cent of the magnitude of the first (negative) term. 72 Table 11 HB AND RADIO EMISSION OF THE SIX MODELS HB Emission Model (ergs/ sec) Uirj(2 6 5 0 Me)/HB #1 5.77 • 103U 3.136 • lO”1*4 #2 7.37 • 1035 3 .1 6 3 * 10~lU #3 2.80 • 1035 3.156 ♦ lO"1*4 #k 8.8k * 103*1 3.22k * lO”1*4 #5 1.5k • lO3*4 3 ,7 1 5 * io"1*4 #6 7.52 • 1033 3.099 * lO"1^ Although no evidence for thermal instability was found, it may be of some interest to discuss the conditions under which the positive term attained its maximum value relative to the negative term. This maximum invariably occurred at that radius where the transition from o. o+ O-3'1' to 0 took place; this point coincides with the transition from X X X. He to He . This is, of course, the point where one would expect instabilities to occur, since 03+ cools very ineffectively, while the 02+ ion is a most efficient coolant (see, for example, Seaton (i960) and Osaki (1962)). The two terms of the instability criterion are platted as a function of radius in Figure 15 for the "standard11 model #k. This . behavior is typical of all the models which contain any He++. The value of R' is never less than 0.95 except in regions where the hy drogen becomes neutral; then we can have R* < 0.20 and it is essential to include this correction term. Table 12 INTEGRATED INTENSITY OF 21 EMISSION LINES RELATIVE TO H6 = 100 Number of Model Emission Line No. 1 No. 2 No. 3 No. 1+ No. 5 No. 6 Ai+l+7l 7.1+5 7.33 7.31 6 .2 0 5.91+ 7.30 He I A5876 23.0 2 2 .6 22.5 1 9 .0 1 8 .1 22.7 Al+686 • • • • 0.71 2 8 .2 35.0 2.38 He II A5^12 « « • • 0.05 2 .11+ 2.63 0.17 A5755 0 .1+1+ • • • ♦ • • • • 0 .0 2 [N II] A65I+8 6.21+ 0.27 0 .0 2 0 .0 1 0 .0 6 1 .2 1 A658U 1 8 .1 0.79 0.07 0 .0 2 0.17 3.50 A3726 9.61+ 2 .5U 0 .8 1 0 .6 2 1.97 7.71 [0 II] A3729 3 .6 2 1 .1 0 0 .1+1+ 0.1+9 2.1+3 1 0 .6 AU363 5.29 5.78 fc.95 9.11 11.9 3.1+1 [0 III] Al+959 1+03 1+86 1+69 519 576 382 A5007 1181 11+32 1371+ 1521 1687 . 1119 A331+3 0 .1 2 0 .1 6 0.13 0 .8 0 0.98 0 .0 8 [Ne III] A3869 168 212 203 311 371+ 159 A 39 67 1+9.0 6 2 .1 59.3 9 1 .0 109 1+6 .6 AI+71 I+ • • • • • • 0.85 p. 1+5 • t Al+716 • • • • • • 0.25 t|.13 • • [Ne IV] Al+72l+ • • • ♦ • • 0.91+ 0.50 • • A1+726 ♦ • • • • • 0 .8 8 0.1+7 ; • • A33U6 • • • * • • » 0.36 1.13 • • [Ne V] A3 U26 • • • • • • 0.95 3 .0 0 • • It should he emphasized that since the models constructed here are by no means exhaustive, the negative result of this search for instabilities is not conclusive. For example, one would want to exam ine the effect of much higher stellar temperatures (although a model constructed with the 180,000° model atmosphere of B8hm and Deinzer re sulted only in a positive term which was 26 per cent of the negative term). The variations of the models do indicate the following con- * elusions, however: the use of the Daub composition and the replacement of the model atmosphere fluxes with black body fluxes both make the gas more stable against condensations, as does the neglect of the on-the- spot absorption of He+ Ly-a. An increase in the abundance of helium, on the other hand, decreases the stability; this effect may be due to the increase in He+ Ly-a which accompanies an increase of AHe. It is difficult to say what effect a more exact treatment of the He+ Ly-a radiation would have on the thermal stability of the gas. +1 - least stable point Model No. 4 .050 .060 .070 080 .080 .100 .110 .120 .130 .140 .150 R (psc) Fig. 15.— The Thermal Stability of Model No, LIST OF REFERENCES LIST OF REFERENCES Allen, C. W. 1963» Astrophysical Quantities (2d ed.; London: Athlone Press;, p. 100 Aller, L. H. 1956, Gaseous Nebulae (London: Chapman and Hall) . 1961, The Abundance of the Elements (New York: Interscience), p. 192 Eohm, D. and Aller, L. H. 19^7, Ap. J . , 105,-131. B5hm, K. H. and Deinzer, W. 1965, Zs. f. Ap., 6l, 1 ______. 1966, ibid., 63, 177 Burbidge, G. R., Gould, R. J., and Pottasch, S. R. 1963, Ap. J., 138, 9^5 Burgess, A. 1958, M.N.R.A.S., 118, M*7 Burgess, A. and Seaton, M. J. 1960a, M.N_.R_.A.S_., 120, 121 ______. 1960b, ibid., 121, 76 ______. 1960c, ibid., 121, U71 Capriotti, E. R. 196Ua, Ap. 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