Radiative Heating and Co oling
in Circumstellar Envelop es
von
Dipl Phys Peter Woitke
aus Berlin
Vom Fachbereich Physik
der Technischen Universitat Berlin
zur Erlangung des akademischen Grades
Doktor der Naturwissenschaften Dr rer nat
genehmigte Dissertation
Berlin D
Promotionsausschu
Vorsitzender Prof Dr D Zimmermann
Berichter Prof Dr E Sedlmayr
Priv Doz Dr J P Kaufmann
Tag der mundlichen Prufung
Wei man wo man innehalten mu entsteht geistige Festigkeit
Gibt es geistige Festigkeit dann entsteht innere Ruhe
Hat man innere Ruhe dann entsteht Gelassenheit
Hat man Gelassenheit dann entsteht besonnenes Nachdenken
Gibt es besonnenes Nachdenken so kommt das Gelingen
Konfuzius Buch der Riten
Zusammenfassung
o e bis zu etwa m bilden auf den ersten orp erpartikel von einer Gr Kleine Festk
orende Komp onente der Materie in unserem Kos Blick eine unbedeutende eher st
atzliche mos Genauer b etrachtet kommt diesen Staubteilchen jedo ch eine grunds
Bedeutung zu Aufgrund ihrer gro en Wirkungsquerschnitte fur die Wechselwirkung
mit Licht pr agen sie in ganz entscheidender Weise das Erscheinungsbild des heutigen
Universums Sie b eein ussen wesentlich die dynamischen thermischen und chemi
schen Eigenschaften des Gases in der Interstellaren Materie und sind ohne Zweifel
mitverantwortlich fur den kosmischen Kreislauf und die chemische Evolution der
Materie Man kann ohne Ub ertreibung sagen da es ohne die Existenz der Staub
teilchen weder die Erde no ch den Menschen ja vielleicht nicht einmal die Sonne
geb en wurde
Die Bildung dieser Staubteilchen aus der Gasphase erfordert relativ sp ezi sche ther
mo dynamische Bedingungen Neb en hohen Dichten sind insb esondere niedrige ab er
nicht zu niedrige Temperaturen unterhalb der Sublimationstemp eratur des b etrach
teten Festk orp ermaterials erforderlich Diese Voraussetzung ist absolut zwingend
Fragt man nach der Existenz solcher Bedingungen in astrophysikalischen Ob jekten
so liegen diese vor allem in den zirkumstellaren Hullen von kuhlen Riesensternen vor
demzufolge gelten die massiven Winde dieser Ob jekte als Hauptpro duktionsst atten
des Staub es im Universum Bei Riesensternen mit E ektivtemp eraturen unterhalb
von etwa K ist der Mechanismus der Staubbildung und des Massenverlustes
nicht zuletzt durch die Arb eiten der Berliner Arb eitsgrupp e von Prof Dr Sedlmayr
hinreichend verstanden Bei genugend gro en radialen Abst anden vom Stern
erreicht das Gas Temperaturen die niedrig genug sind um den Phasenub ergang vom
oglichen Die entstehenden Staubteilchen nehmen orp er zu erm Molekul zum Festk
durch Absorptions und Streuprozesse den Impuls des Strahlungsfeldes teilweise
auf und geb en diesen durch St o e an das Gas weiter Dieser Impulseintrag treibt
den stellaren Wind
Neb en diesen Sternen gibt es eine Reihe von weiteren staubbildenden Ob jekten
Insb esondere existiert eine zahlenm a ig eher unbedeutende Klasse von R Coronae
Borealis RCB Sternen die sich nicht recht in das obige Bild einordnen lassen Bei
diesen Ob jekten kommt es in unvorhersagbaren zeitlichen Abst anden immer wieder
zur Bildung von riesigen Staubwolken die den gesamten Stern vorub ergehend ver
onnen so da dieser fur das blo e Auge fur Monate o der Jahre vom Himmel decken k
zu verschwinden scheint Beobachtungen legen nahe da der Staubbildungsproze
are Beobachtungen liegen fur Wolf Rayet Sterne Ahnliche wenn auch nicht derart sp ektakul
und Novae Explosionen vor i
ii ZUSAMMENFASSUNG
b ei diesen Sternen in einer Entfernung von nur einigen wenigen Sternradien statt n
den mu obwohl die RCB Sterne E ektivtemp eraturen von etwa K b esitzen
aftiger als die Sonne sind Die vorliegende Arb eit die also hei er und viel leuchtkr
nimmt diese Beobachtungsergebnisse ernst
G angige Metho den zur Temperaturb estimmung ergeb en in so geringen radialen Ent
fernungen vom Stern sehr hohe Temperaturen so da die Staubbildungstheorien
a t sich die Staubbil durch die RCB Sterne auf eine harte Prob e gestellt werden L
dung in der N ahe dieser Sterne mit den ublichen Theorien erkl aren Setzt man die
Gultigkeit der Theorien voraus so mussen entweder die Beobachtungen falsch sein
o der es mussen in der N ahe dieser Sterne zumindest zeitweilig viel niedrigere
Temperaturen als erwartet herrschen
Kann es in der N ahe von hei en Sternen zu thermo dynamischen Bedingungen kom
men die Staubbildungsprozesse zulassen Angeregt durch diese Fragestellung unter
sucht die vorliegende Arb eit den thermischen Zustand dunner Gase unter dem Ein
u von stellaren Strahlungsfeldern Es handelt sich hierb ei zun achst um allgemeine
nicht RCB spezi sche grundlegende Studien Eine Metho de zur zeitabh angigen
Temperaturb estimmung von Gasen in zirkumstellaren Hullen wird entwickelt die
von vornherein so konzipiert ist da sie als elementarer Bestandteil von komplexeren
Mo dellrechnungen in zukunftigen Arb eiten verwendet werden kann
Das thermo dynamische Konzept dieser Metho de b eruht auf einer non LTE Beschrei
bung des Gases in der jedo ch eine Geschichtsabh angigkeit der Konzentrationen der
Molekule und der Besetzungsdichten vernachl assigt wird Stattdessen wird ein ki
netisches Gleichgewicht steady state vorausgesetzt Es wird gezeigt da diese
a t ohnliche thermo dynamische Beschreibung des Gases zul Annahme eine gew
Die folgenden radiative Prozesse werden in dieser Arb eit b erucksichtigt Linienub er
g ange von Atomen und einfach geladenen Ionen Vibrations und Rotationsub er
g ange von p olaren diatomischen bzw linearen Molekulen Quadrup ol Ub erg ange
von H gebunden frei Ub erg ange von Atomen aus dem Grundzustand und im
Falle von Wassersto aus angeregten elektronischen Niveaus ferner Photo disso
ange Diese Prozesse ergeb en in der Summe die ziationsprozesse und frei frei Ub erg
radiativen Heiz und Kuhlraten d h die W armemengen die das Gas durch Ab
sorptionsprozesse pro Zeit aufnimmt bzw durch Emissionsprozesse verliert Die ra
diativen Heiz und Kuhlraten bilden somit die Grundlage zur thermo dynamischen
Mo dellierung des Gases
Drei Anwendungen der entwickelten Metho de werden vorgestellt
Zun achst werden die stabilen Gleichgewichtszust ande des Gases in den zirkumstella
ren Hullen von RCB Sternen b estimmt Diese Zust ande zeichnen sich dadurch aus
da sich die radiativen Heiz und Kuhlraten ausgleichen Strahlungsgleichgewicht
Es wird jedo ch festgestellt da das Strahlungsgleichgewicht eine zwar notwendige
ab er nicht hinreichende Bedingung zur Berechnung des thermischen Zustand des
Gases darstellt Unter gegeb enen Druck und Strahlungsfeldb edingungen k onnen
aumliche Ko existenz von hei en atomaren osungen existieren d h eine r mehrere L
ZUSAMMENFASSUNG iii
oglich thermische Phasen neb en kalten molekularen Phasen erscheint prinzipiell m
Bifurkationen
Der Relaxationsproze des Gases zum Strahlungsgleichgewicht wird in den zirkum
stellaren Hullen von C Sternen untersucht Hierb ei wird insb esondere das Verhal
ten des Gases hinter Sto wellen diskutiert die durch eine Pulsation des zentra
len Sterns verursacht werden Es ergibt sich da nach der Passage einer solchen
Sto welle nur ein hinreichend dichtes Gas in der Lage ist den Strahlungsgleichge
wichtszustand nach einiger Zeit wieder zu erreichen Bei Teilchendichten cm
alt sich das Gas zunehmend adiabatisch so da schlie lich die Bedingung des verh
Strahlungsgleichgewichtes ihre b estimmende Bedeutung fur die Temperaturstruktur
dieser Sternhullen verliert
angige thermische Verhalten des Gases in den zirkum Schlie lich wird das zeitabh
stellaren Hullen von pulsierenden RCB Sternen genauer untersucht Es wird eine p e
ahe des Sterns fortlaufend durch rio dische Situation studiert in der das Gas in der N
Sto wellen erhitzt und komprimiert wird und in der Zwischenzeit reexpandiert In
einem b estimmten Dichtebereich kann dab ei das Gas durch einen Stufen Proze
b estehend aus radiativer Kuhlung gefolgt von adiabatischer Expansion Tempera
turen erreichen die weit unterhalb der Strahlungsgleichgewichtstemperatur liegen
anden von etwa R treten hierb ei zeitweilig Tempera Schon b ei radialen Abst
angig von der Sto wellengeschwindigkeit Diese turen unterhalb von K auf abh
Arb eit stellt daher die Hyp othese auf da die Kondensation von Ru teilchen in der
aren ahe der RCB Sterne durch Sto wellen verursacht wird wodurch die sp ektakul N
onnten ost werden k Verdunklungsereignisse dieser Ob jekte ausl
alt somit grundlegende Erkenntnisse ub er das thermo Die vorliegende Arb eit enth
dynamische Verhalten der Gase in zirkumstellaren Hullen Neue alternative Wege
zur Staubbildung werden aufgezeigt iv
Abstract
This thesis investigates the thermal state of diluted gases b eing exp osed to stellar
radiation elds On the basis of a steady state non LTE description the radiative
heating and co oling rates of the gas are determined considering the typical densities
present in circumstellar envelopes
The following radiative pro cesses are examined line transitions of neutral and singly
ionized atoms vibrational and rotational transitions of p olar diatomic and linear
molecules resp ectively quadrup ole transitions of H b ound free transitions from
the electronic ground states and in case of hydrogen from excited electronic levels
photo disso ciation and free free transitions
A thermo dynamic description of the gas is developed which allows for a time
dep endent determination of the temp erature structure in the circumstellar envelopes
of co ol and warm stars and can b e included into more complex e g hydrodynamic
mo del calculations Three applications of this description are presented
First the stable radiative equilibrium states of the gas are calculated for the circum
stellar envelopes of R Coronae Borealis RCB stars It is found that the condition
of radiative equilibrium is not su cient in order to determine the temp erature of the
gas More than one temp erature solution may exist for xed conditions of pressure
and radiation eld Thus a spatial co existence of hot atomic and co ol molecular
phases is principally conceivable thermal bifurcations
Second the relaxation pro cess towards radiative equilibrium is studied in the cir
cumstellar envelopes of C stars The character of the thermal relaxation b ehind
propagating sho ck waves which are caused by a pulsation of the central star is
discussed It is found that the gas must b e su ciently dense in order to b e capable
to reestablish radiative equilibrium after the passage of such sho cks For densities
cm the b ehavior of the gas b ecomes more and more adiabatic so that
nally the condition of radiative equilibrium lo oses its signi cance concerning the
determination of the temp erature structure
Third the time dep endent b ehavior of the gas in the circumstellar envelopes of
pulsating RCB stars is investigated more detailed A mo del for sho ck levitated
atmospheres is developed where the gas is p erio dically heated and compressed by
sho ck waves and re expands b etween the sho cks Within a distinct density interval
the gas is found to undergo a two step co oling pro cess consisting of radiative co ol
ing at high temp eratures followed by adiabatic expansion at low temp eratures In
this case a considerable sup erco oling of the gas o ccurs temp orarily pro ducing tem
p eratures b elow K far b elow the values exp ected from radiative equilibrium
at radial distances as small as R despite of the high e ective temp eratures of
these stars Thus this thesis states the hypothesis that the onset of dust formation
close to RCB stars is caused by sho ck waves which might trigger the sp ectacular
decline events v
Contents
Zusammenfassung i
Abstract v
List of Symbols x
List of Figures xiv
List of Tables xv
Introduction
The Imp ortance of Temperature Determination for
Mo dels of Dust Formation
Critical Comments on the Usual Metho d of Temperature Determination
The Puzzle of Dust Formation around RCB Stars
Requirements for Di erent Approaches of Temperature Determination
Aim and Structure of this Work
The Thermo dynamic Concept
First Law of Thermo dynamics and Equation of State
LTE and Non LTE
Steady State
Radiative Heating and Co oling
Bound Bound Transitions
Escap e Probability Metho d for an N Level System without
Continuum
Numerical Iteration Scheme
Discussion of the Applicability of Sob olev Theory
An Exemplary Two Level Atom
Lines of Atoms and Ions
Rotational Transitions of Linear Polar Molecules vi
Rotational Heating and Co oling by CO
Fast Approximate Metho d
Vibrational Transitions of Diatomic Polar Molecules
Vibrational Heating and Co oling by CO
Fast Approximate Metho d
Quadrup ole Transitions of H
Bound Free Transitions
The Rate Equations for an N Level System with Continuum
Fast Approximate Metho d
The H Atom
Other Neutral Atoms
Photo disso ciation and Radiative Asso ciation
The H Heating Co oling Rate
Free Free Transitions
Overview of the Considered Radiative Pro cesses
Further Heating and Co oling Pro cesses
The Calculation of the Equation of State
Calculation of the Particle Concentrations
Calculation of the Internal Energy
Thermal Bifurcations in the Circumstellar Envelopes of RCB Stars
The Mo del
De nition of the Radiative Equilibrium Gas Temperature
Element Abundances
Approximation of the Radiation Field
Results
Degree of Ionization
Chemistry
Radiative Heating and Co oling Rates
Radiative Equilibrium Temperature Solutions
Discussion
Radiative Co oling Time Scales in the Circumstellar Envelopes of
C Stars
The Mo del vii
De nition of the Radiative Co oling Time Scale
Element Abundances
Approximation of the Radiation Field
Lo cal Velocity Gradient
Results
Comp osition of the Gas
Internal Energy
The Radiative Co oling Time Scale and the Role of the Various
Heating and Co oling Pro cesses
Dep endence on the Radiation Field
Dep endence on the Velocity Gradient
Comparison to Analytical Heating Co oling Functions
Bowen s Heating Co oling Function
LTE Heating Co oling Function
Results of the Comparison
The Transition from Isothermal to Adiabatic Sho cks
Discussion
Sho ck Induced Condensation around RCB Stars
The Mo del A Fixed Periodically Sho cked Fluid Element in a Con
stant Radiation Field
Sho ck Transitions
Re Expansion Phases
Thermo dynamics
The Mo deling Pro cedure
Overview of Introduced Parameters
Examined Range of Parameters
Results
Cyclic Variations in the Periodically Sho cked Fluid Elements
Dep endence on Density
Dep endence on Sho ck Velocity
Preconditions for Carb on Nucleation
Dep endence on Radial Distance
Discussion
Advantages of the Mo del
Criticism
Interpretations of Observations with Regard to the Mo del viii
Conclusions
A Current Status of RCB Research
A General Observations
A Observations During the Decline Events
A Mo dels
A Historical Mo dels
A Mo del Calculations
A Empirical Mo dels
References
Danksagung
Leb enslauf ix
List of Symbols
symbol description unit page
A Einstein co e cient for sp ontaneous emission s
ul
B frequency integrated Planck function erg s cm
B rotational constant Hz
B Einstein co e cient for absorption erg cm s
lu
B Einstein co e cient for stimulated emission erg cm s
ul
Planck function erg s cm Hz B
C rate co e cient for collisional excitation s
lu
rate co e cient for collisional de excitation s C
ul
D disso ciation p otential erg
mol
E activation energy of a chemical reaction erg
a
total disso ciation p otential energy erg E
diss
E total electronic excitation energy erg
el
E total ionic p otential energy erg
ion
total rotational excitation energy erg E
rot
E total translational energy erg
trans
E total vibrational excitation energy erg
vib
E energy di erence b etween upp er and lower state erg
ul
G free enthalpy of formation at standard pressure erg
f
p
inc
incident continuous intensity from direction erg s cm Hz str I
J mean sp ectral intensity erg s cm Hz
frequency integrated mean intensity erg s cm J
J rotational quantum number
cont
continuous mean intensity at line center erg s cm Hz J
ul
frequency
J line averaged continuous mean intensity erg s cm Hz
ul
J nucleation rate i e the number of seed particles cm s
forming p er volume and p er second
solid angle
temp erature indep endent rate co e cient for col cm s
ul
lisional de excitation
e
mean escap e probability P
e
e
P simpli ed mean escap e probability
Q total net radiative heating function erg s cm
rad
b
total net radiative heating rate p er mass erg s g Q
rad
b
Q total net heating rate p er mass due to presence erg s g
dust
of dust grains
bf
Q net radiative heating function due to b ound free erg s cm
rad
transitions
Q net radiative heating function due to free free erg s cm
rad
transitions x
symbol description unit page
Q rotational net heating function erg s cm
rot
Q vibrational net heating function erg s cm
vib
chem
net radiative heating function of a photo erg s cm Q
rad
chemical reaction
R stellar radius cm
total rate co e cient for transition i j s R
ij
S sup ersaturation ratio of the gas with resp ect to
graphite
L
line source function erg s cm Hz S
ul
S T Saha function of level i cm
i g
T temp erature K
vibrational transition moment cgs TM
T rotational excitation temp erature K
rot
T vibrational excitation temp erature K
rot
black b o dy temp erature K T
bb
T e ective stellar temp erature K
e
T unique kinetic temp erature of the gas K
g
T radiation temp erature K
rad
RE
T radiative equilibrium temp erature K
g
sp eci c volume cm g V
W dilution factor
Z partition function of an singly ionized atom
II
Z rotational partition function
rot
Z vibrational partition function
vib
atomic mass unit m g amu
C
atm standard atmospheric pressure dyn cm
photo recombination co e cient to level i cm s
i
departure co e cient from LTE b n n b
i i i
i
T rate co e cient for collisional ionization from level cm s
i g
i
sp eed of light cm s c
e internal energy of the gas erg g
element abundance by number
El
gas emission co e cient erg s cm Hz str
j dimensionless mean intensity at
ul ul
number of rotational degrees of freedom f
rot
g g statistical weights of lower and upp er level
l u
rate co e cient for collisional de excitation cm s
ul
Planck s constant erg s h
h enthalpy p er mass unit h e p erg g
h erg s h
abs
hh i mean absorb ed photon energy erg
em
mean emitted photon energy erg hh i
k Boltzmann constant erg K
k rate co e cient of a forward chemical reaction dep ends
f
rate co e cient of a reverse chemical reaction dep ends k
r
gas absorption co e cient cm
Planck mean absorption co e cient cm
B
intensity mean absorption co e cient cm
J
wavelength cm
mass of neutral atom of element El g m
El xi
symbol description unit page
m electron mass g
e
m reduced mass for collisions b etween the consid g
red i
ered sp ecies and collision partner i
cos
dip ole moment of a molecule cgs
D
n particle density in LTE chemical equilibrium cm
p opulation of level i cm n
i
n level p opulation of the lower level cm
l
n level p opulation of the upp er level cm
u
El
n total neutral atom particle density cm
at
El
n singly ionized atom particle density cm
II
n total density of H nuclei in all ionic atomic and cm
H
P
m molecular forms
El El H
El
total helium particle density in atomic or ionized cm n
He
form
critical density n value for thermal cm n
H cr
p opulation
n critical density n value for optical depths cm
thick H
e ects
n electron density cm
e
n total particle density of molecule mol cm
mol
total particle density of one sp ecies cm n
sp
frequency Hz
i
threshold frequency for photoionization Hz
thr
line center frequency Hz
ul
eigenfrequency of the harmonic oscillator Hz
mol
j th vibrational eigenfrequency of a molecule Hz
j
gas pressure dyn cm p
p standard pressure dyn cm
p vapor pressure of neutral atoms over the bulk dyn cm
sat
material
pro le function of the considered transition Hz
ul
r radial distance to the center of the star cm
mass density of the gas g cm
El
s stoichiometric co e cient of molecule mol for ele
mol
ment El
Stefan Boltzmann constant erg cm K
f
photo disso ciation cross section cm
total cross section for rotational de excitation cm
bf
b ound free absorption cross section from level i cm
i
S
Sob olev optical depth
ul
S
e mean Sob olev optical depth
ul
radiative heating co oling time scale s
co ol
angle b etween the considered ray and the radial
direction
characteristic temp erature of vibrational K
transitions
v vibrational quantum number
hydrodynamic gas velocity cm s v
v terminal wind velocity km s
1
dv
lo cal mean velocity gradient s
dl xii
List of Figures
Possible radiative equilibrium temp eratures over dilution factor in a
Planck type radiation eld
Sketch of an RCrB decline event
Temperature structure in hydrodynamic mo dels using LTE cooling
Temperature structure in hydrodynamic mo dels using co oling
The p o ols and uxes of energy in the gas
The co oling rate p er mass of an exemplary two level atom
Temperature dep endence of the line co oling rate
Dep endency of the line co oling rate on the radiation eld
Rotational co oling rate and excitation temp erature of CO
Vibrational co oling rate and excitation temp erature of CO
The quadrup ole co oling rate of H
The b ound free plus b ound b ound co oling rate of hydrogen in the
case without continuous radiation eld
The b ound free plus b ound b ound co oling rate of hydrogen in the
case with continuous radiation eld
Details of the hydrogen co oling rate
The b ound free and free free co oling rates of H without continuous
radiation eld
The b ound free and free free co oling rates of H with continuous
radiation eld
Overview of the considered heating and co oling pro cesses
Element abundances of R Coronae Borealis
Heating co oling rates as function of the gas temp erature
Thermal bifurcations in RCB envelopes for p and dyn cm
Thermal bifurcations in RCB envelopes for p and dyn cm
Thermal bifurcations in RCB envelopes for p and dyn cm
The comp osition the internal energy and the net heating function of
the gas as function of temp erature and density xiii
Radiative co oling time scales for C star envelopes for the case J
Most e cient co oling pro cess referring to Fig
Radiative co oling time scales for C star envelopes in the case J
B K
Most e cient co oling pro cess referring to Fig
Radiative co oling time scales calculated from the analytical heat
ing co oling function prop osed by Bowen
Radiative co oling time scale calculated from LTE
Ballistic tra jectories of xed uid elements in the envelope of a pul
sating star
Schematic description of the thermo dynamic pro cesses o ccurring in a
xed uid element of the CSE of a pulsating star
Time variations of the thermo dynamic state variables in a xed p e
rio dically sho cked circumstellar uid element
Details of the p erio dic time variations
Cyclic variations of density and temp erature in xed uid elements
Minimum gas temp eratures and the p ossibility of carb on nucleation
to o ccur at a radial distance of r R
xiv
List of Tables
Observational and theoretical constraints on the nucleation distance
in RCB envelopes
Atomic line heating and co oling considered sp ecies and transitions
Vibrational and rotational heating and co oling considered sp ecies
and molecular data
Bound free heating and co oling considered sp ecies and atomic data
Overview of further heating and co oling pro cesses
Molecular data for the determination of the internal energy
Abundant molecules in the circumstellar envelopes of RCB stars
Imp ortant heating co oling pro cesses for RCB abundances
Results of the sho ck induced condensation mo del as function of radial
distance and sho ck velocity xv xvi
Chapter
Introduction
The Imp ortance of Temperature Determination for
Mo dels of Dust Formation
Compared to the usual organizational forms of matter in space like stars and the
interstellar medium ISM there are remarkable and exceptional thermo dynamic
conditions in extended circumstellar envelopes CSEs Here the densities are lower
by orders of magnitude than in the interior and the atmospheres of stars but due
to mass loss still higher by orders of magnitude than in the ISM If the central
star is su ciently co ol that its radiation eld do es not ionize the surrounding CSE
temp eratures can exist which are on one hand low enough to ensure the stability
of complex chemical structures but on the other hand high enough to bridge the
energy barriers during their formation
Therefore the circumstellar envelopes of co ol stars are a cosmic lab oratory where
large amounts of complex chemical and physical pro cesses can o ccur These pro
cesses are of fundamental imp ortance for the evolution of matter esp ecially for
the transition from molecules to dust grains i e the primary formation of solids in
space The high densities combined with low but not to o low temp eratures provide
almost ideal preconditions for condensation pro cesses
Besides the CSEs of co ol stars only a few classes of astrophysical ob jects are known
which show similar thermo dynamic conditions These are the rare explosive phe
nomena like novae and sup ernovae sho ck waves in the most dense parts of the ISM
probably connected with star formation or comet impacts on planets Therefore
the CSEs of co ol stars are supp osed to b e the main pro duction sites of small solid
particles dust grains in space These particles are carried into the ISM by stellar
winds and nally can b e observed everywhere in the universe This ubiquitous ex
istence of dust particles is of greatest imp ortance for the app earance of the present
universe for the circulation of matter for the formation of stars and planets and
last but not least also for the existence of life including mankind
The chemical reactions which successively lead to increasing complexity in gases and
nally to the formation of seed and dust particles show such a strong temp erature
dep endence that even slight temp erature deviations can change the formation rates
by orders of magnitude Therefore e ective nucleation is generally restricted to a
small temp erature window of a few hundred degrees b elow the sublimation temp er
CHAPTER INTRODUCTION
ature of the solid material which is to b e considered Thus it is immediately clear
that the results of theoretical mo del calculations of dust formation critically dep end
on the prop er determination of the temp erature in the medium Questionable or
insu cient metho ds for temp erature determination can easily induce severe errors
in the results of such calculations
For the mo deling and the understanding of dust formation from the gas
phase the most precise information ab out the thermo dynamic state of
the gas is absolutely required
How profound is our knowledge of the true thermo dynamic state of the gases es
p ecially their temp erature in astrophysical ob jects In exceptional cases a direct
determination of the temp erature strati cation from observations of the ob jects
might b e p ossible However theoretical metho ds are usually required which may
still su er from large intrinsic uncertainties see next section Astonishingly most
mo del calculations concerning dust formation in co ol stellar envelopes use rather
simple and not very reliable metho ds for temp erature determination At least the
exp enditure for the theoretical temp erature determination often seems to b e inad
equate compared to the detailed treatment of the chemistry and dust formation
pro cesses in such investigations
These questions get even more imp ortant when the gas elements are sub ject to
dynamic pro cesses which directly a ect the internal energy of the gas For instance
in sho ck waves in the heating by magneto acoustic waves or during fast expansions
accompanied by adiabatic co oling Which temp erature do es one use in such cases
How reliable are those results In this way new p otential sites e g close to hot
stars for dust formation not previously considered might b e discovered
Critical Comments on the Usual Metho d of Tempera
ture Determination
The temp erature structure in extended CSEs is usually calculated by means of
the solution of a radiative transfer Assuming that solely radiative pro cesses are
imp ortant for the heating and co oling of the gas the gas will relax to radiative
equilibrium RE where the total amount of absorb ed radiative energy is lo cally
balanced by the total amount of emitted radiative energy everywhere in the envelope
Z Z
RE J d d
The meaning of the physical quantities is explained in the List of Symbols on page x
In case of lo cal thermo dynamic equilibrium LTE the emissivity can b e eliminated
by means of Kirchho s law B T
Z Z
RE
LTE J d B T d
This statement refers to b oth the classical and kinetic treatment of the problem
THE USUAL METHOD OF TEMPERATURE DETERMINATION
In order to arrive at a short notation appropriate means of the absorption co e cient
RE
and can b e de ned such that Eq simpli es to J B T and
J B J B
the temp erature of the gas in radiative equilibrium can b e expressed by
J
RE
J T
B
As far as the assumptions of RE and LTE are appropriate Eq together with
a solution of the radiative transfer determines the prop er temp erature structure in
the considered astrophysical ob ject The uncertainty of this metho d lies within the
calculation of the gas absorption co e cients This calculation however is one
of the key problems in astrophysics and a very di cult task esp ecially in those
cases where the gas is su ciently co ol for molecule formation Considering the
H O molecule for example which is one of the most abundant sp ecies in a co ol
gas with a solar elemental comp osition over a billion of line transitions are known
ranging from the near IR to the micro wave sp ectral region Furthermore electronic
and b ound free transitions in the UV may b e imp ortant In principle all these
transition must b e taken into account for a prop er calculation of Additional
questions concerning the individual line pro les and the e ects caused by Doppler
shifts due to hydrodynamic velocities and self shielding come into play
Therefore due to the lack of knowledge of the exact frequency dep endency of
the simplifying assumption is often made In this case we have J B which
J B
gives the black b o dy temp erature
grey T J
bb
Alternatively Eq can b e derived directly from Eq by assuming
const is henceforth called the quasi grey assumption In the following
J B
the intrinsic uncertainty of this approximation for the resulting temp erature struc
ture is explored First Eq obviously has always exactly one temp erature solu
tion for a given J whereas Eq may have two or more stable solutions b ecause
the fraction can b e temp erature dep endent itself Thereby the quasi grey
J B
assumption ignores the p ossibility of thermal bifurcations which we will encounter
later in this work
The maximum e ect caused by true frequency dep endent absorption can b e esti
mated by considering the extreme case of a function
This case is not as arti cial as it might rst app ear b ecause there is often one sp ecial
radiative pro cess which dominates the heating and co oling of the gas and which has
a certain characteristic wavelength We consider the e ects in a diluted Planck eld
of type
J WB T
rad
CHAPTER INTRODUCTION
which is a useful approximation of the radiation eld if the gas element mainly
receives light from a distant black b o dy source In a spherically symmetric optically
thin CSE with T T we nd the dilution factor to b e
rad e
s
R
A
W r
r
Allain has shown that even for an optically thick CSE Eq still provides
a reasonable t to the results of frequency dep endent radiative transfer calculations
Winters In general the parameter T is smaller than T and W is larger
rad e
RE
T or T B compared to Eq For the assumed we nd WB
rad
h h
RE
T ln exp
k W k T
rad
which can b e compared to the black b o dy temp erature given by Eq
T T W
rad bb
Figure depicts the results for some arbitrarily chosen central wavelengths
c First if the radiation eld is not diluted W the RE temp erature always
equals T In this case typical for the deep atmosphere and the interior of the
rad
star the frequency dep endency of is meaningless for the resulting temp erature
which makes the temp erature determination very reliable
Farther out in the envelope however where W a wide spread of p ossible
temp erature solutions exist dep ending on the central wavelength note the logarith
mic scaling of the temp erature axis The p ossible solutions lie b etween the UV
limit h Max fk T k T g T T and the IR limit h Minfk T k T g
rad UV rad rad
T WT The black b o dy temp erature T cf Eq is just one solution in
IR rad bb
b etween concerning a sp ecial type of frequency dep endency of
The theoretically determined temp erature structure is therefore very sensible for
the frequency dep endency of For example at r R the result is K if the
interaction b etween matter and radiation eld mainly takes place at m but
is K if m
RE
r R m T K
If we compare these values to the small temp erature window where e cient dust
condensation may take place it is obvious that an uncertainty concerning the fre
quency dep endency of as considered ab ove can easily change the temp eratures to
values well ab ove or well b elow p ossible condensation resp ectively Let s generously
assume T K for the temp erature where e cient nucleation from
cond
the gas phase may o ccur The corresp onding radius intervals r in the optically
cond
thin limit are then given by
m r R
cond
grey r R
T K
cond
cond
m r R
cond
THE USUAL METHOD OF TEMPERATURE DETERMINATION
Figure Radiative equilibrium temp eratures over dilution factor W in a Planck type
radiation eld with T K cf Eq according to a function type gas absorp
rad
tion co e cient The radius axis b elongs to the optical thin limit pure radial dilution
according to Eq The shaded region indicates the range of p ossible RE temp eratures
b etween the UV and the IR limit where the central wavelength see lab els on solid curves
is small and large resp ectively The dashed line shows the black b o dy temp erature
CHAPTER INTRODUCTION
These estimates clearly indicate that the assumption has a decisive in uence
J B
on the calculated temp erature structure with severe consequences for the mo deling
of dust formation in the circumstellar envelopes of co ol stars
A lot of simplifying or even unphysical assumptions have b een made in this section
so that the calculated numbers are meaningless However what is imp ortant is the
clear trends in the results Uncertainties concerning the frequency dep endency of
can easily change the results of the theoretically determined temp erature structure
by K The results are even less reliable if the assumptions of RE and LTE
b ecome questionable which is another topic of this work More detailed studies on
the imp ortant heating and co oling pro cesses are required to tackle the problem of
theoretical temp erature determination in CSEs Which sp ectral regions are imp or
tant and which are the corresp onding rates Remembering the strong temp erature
dep endency of nucleation from the gas phase such investigations can lead to a new
and distinct theoretical view on dust formation in CSEs
The Puzzle of Dust Formation around RCB Stars
An example to the ab ove conclusion can p ossibly b e found in the CSEs of R Coronae
Borealis stars
More than two centuries ago the German astronomer Eduard Pigott discovered that
the th brightest star in the northern constellation Coronae Borealis previously
known to b e ab out th magnitude had suddenly disapp eared from the sky The star
remained invisible for several months and nally recovered slowly In the following
years Pigott observed similar disapp earances at irregular intervals He published
his article on this remarkable star exactly years ago Pigott which es
tablished a new class of ob jects the class of irregular variable stars R Coronae
Borealis b ecame its rst member Since then the unpredictable R Coronae Borealis
RCB type decline events with decreases in visual brightness of up to magnitudes
within a few weeks and the eye catching shap e of the light curve always attracted
much interest and fascination in the astrophysical community The uniqueness and
distinctiveness of this extreme type of variability in contrast to the broad variety of
stellar parameters among the RCB stars immediately suggests that there must b e
one unique physical mechanism which triggers all the events
Over the years much observational e orts have b een undertaken and more and
more complete observational data have b een collected covering the RCB decline
events photometry sp ectroscopy and p olarimetry These observations form a com
prehensive data set for this extreme type of stellar variability A summary of the
observations is outlined in App endix A
In spite of the completeness and the quality of observational data our understanding
of the physical pro cesses causing the RCB decline events is still rather p o or Since
Loreta s and O Keefe s basic suggestion that the decline events are
caused by the sudden o ccurrence of dust somewhere in the line of sight towards the
THE PUZZLE OF DUST FORMATION AROUND RCB STARS
chromosphere ?
R CrB star stellar pulsation ~ 7000 K 4 ~10 L shock waves Rcond = ?
zone of possible nucleation
t 0 dust growth & cloud formation t1 & cloud acceleration
t 2 radial dilution
decline event
visual brightness
t0 t1 t2 time
Figure Sketch of the physical pro cesses the geometry and the time evolution of an
RCB decline event A visualization like this is always a mixture of observational facts and
interpretation In this case the stellar parameters the pulsation of the star the o ccur
rence of sho ck waves and dust clouds and the shap e of the light curve are supp orted by
observations The geometry of the scenario and the nucleation zone refer to the hypothesis
of this work
CHAPTER INTRODUCTION
observer the progress during the last six decades concerning a physical explanation
of the phenomenon has b een slow The most favorable picture to day is that clouds
of carb on dust o ccasionally form from the gas phase near to the star which are then
radially accelerated by radiation pressure in random directions away from the star
If the dust cloud forms in the line of sight it successively eclipses the star and blo cks
the stellar light In the late phases of a decline the dust cloud moves outward and
disp erses due to radial dilution The star slowly returns to normal light This overall
picture sketched in Fig and the fact that it is some form of carb on dust which
actually condenses is now generally accepted
All further details however e g the distance of the dust clouds to the star in
the early phases of a decline the physics and chemistry of the decline phase the
survival of the dust clouds close to the star the dynamic b ehavior of dust clouds in
the circumstellar envelope etc are still controversial Esp ecially so are the physical
reasons for the o ccasional onset of dust formation and for the formation of dust
clouds rather than spherical dust shells Thus the whole phenomenon is still waiting
for a convincing explanation
One key towards a b etter understanding of the RCB decline events is given by the
radial distance to the stellar photosphere where fresh carb on dust condenses from
the gas phase By means of a reliable determination of this quantity many of the
prop osed mo dels and scenarios could b e ruled out immediately The distance to
R CrB is ab out p c and in order to detect a dust cloud with a diameter of one
stellar radius an angular resolution of arc seconds would b e required
Therefore according to the present state of observational techniques this highly
controversial quantity cannot b e observed directly However there exist some indi
rect observational clues on the nucleation distances indicating that dust formation
o ccurs rather close to the photosphere
Temporal evolution of emission lines During a typical decline event a rich
chromospheric emission line sp ectrum app ears similar to a solar eclipse A sp ecial
temp oral evolution of three distinguishable classes of emission lines can b e observed
see App endix A It is natural to suggest that the expanding dust cloud subse
quently covers the regions resp onsible for the line emissions In this case the dust
cloud must form b elow these regions Additionally the emission lines are apparently
less p olarized than the continuum see App endix A which supp orts this scenario
Corresp onding observational estimates claim a nucleation distance of R
Clayton et al
Dust acceleration time scale In many cases strongly blue shifted km s
absorption lines have b een detected just in the b eginning of a decline see Ap
p endix A Since such blue shifts are only seen in the context of decline events
radiation pressure on dust seems to b e resp onsible for the acceleration of the gas
Accelerations to velocities of a few hundred km s within a few weeks however are
only p ossible rather close to the star where the radiation ux is su ciently intense
Whitney et al yielding nucleation distances of R Go eres
THE PUZZLE OF DUST FORMATION AROUND RCB STARS
Decline time scale The initial decline phase typically lasts a few weeks If the
changes of the brightness and the sp ectra observed during this phase are caused by
an optically thick dust cloud radially expanding and subsequently obscuring the
stellar disk the dust cloud must b e lo cated close to the star Clayton et al
Only in this case the tangential pro jections of the measured radial velocities can b e
as large as one stellar radius p er week From this argument Feast estimates
the initial radial distance of the dust cloud to b e R
Dust dilution time scale The recovery phase in late decline is supp osed to b e
a consequence of the dust cloud moving away from the star at a constant velocity
while radially diluting By simultaneous measurement of the expansion velocity via
absorption line blue shifts and the gradient of light increase an absolute distance of
the cloud can b e derived compatible with nucleation distances of R Go eres
IR ux constancy The fact that the IR uxes only show minor changes during
the declines can b e used to estimate the angular coverage of a single dust cloud
Forrest et al The semi cone angle together with the condition that such a
cloud must have at least the size of the stellar disk in order to o ccult the star leads
to a minimum distance of the dust clouds at the b eginning of the declines which
provides an estimate for the nucleation distance of R Go eres et al
Pulsation phase correlation There is some observational evidence for certain
individual ob jects that the decline events always b egin at xed pulsation phases
e g Pugach Lawson et al For such a correlation a physical connection
b etween the photosphere of the star and the condensation zone with a constant time
delay is required The closer the condensation zone to the star the more plausible
this type of connection app ears The character of this argument is only qualitative
Although at least one weak link can b e found in every chain of the ab ove arguments
the clear common tendency of the observational ndings is that dust condensation
in RCB envelopes in fact o ccurs fairly close to the photosphere of the star
The small nucleation distances derived from observations at rst sight
seem to contradict the basics of dust formation theory
According to all known theories classical nucleation theory chemical pathway cal
culations or the mo deling of chemical reaction networks the formation of a solid
b o dy demands temp eratures well b elow the sublimation temp erature of the consid
ered solid material This yields lower temp eratures than K under the density
conditions present in CSEs for all high temp erature condensates including graphite
and SiC Considering the typical chemical conditions present in the envelopes of
RCB stars temp eratures b elow K are inevitably required for carb on nucleation
Go eres Sedlmayr
In standard mo dels for CSEs cf last section such temp erature conditions are
only present outside ab out R for T K Furthermore if we ask for the
e
CHAPTER INTRODUCTION
Table Observational and theoretical constraints on the nucleation distance in RCB
envelopes Distances are given in units of stellar radii
observations theory
temp oral evolution and p o
larimetry of chromospheric
su ciently low gas tem
emission lines
p erature for nucleation
dust acceleration time scale
decline time scale
su ciently low dust tem
dust dilution time scale
p erature for dust stability
IR ux constancy
pulsation phase correlation close
minimum distance required to assure the stability of small carb on dust particles in
an optically thin stellar radiation eld the result is ab out R for T K
e
Fadeyev In the case of hot RCB stars with e ective temp eratures up to
K one derives distances as large as R and so the problem of nearby dust
formation app ears to b e even more serious
To summarize Table two contradictory p oints of view can b e distinguished On
one hand the observational astronomers argue for dust formation near to the star on
the basis of several supp orting indep endent scienti c ndings On the other hand
theoretical mo dels for dust formation predict large nucleation distances as a con
sequence of thermo dynamic constraints This obvious con ict b etween theory and
observation traces through the whole literature and constitutes the central problem
of the present understanding of the RCB type decline events Once one accepts that
dust formation o ccurs close to the star as indicated by observations there are only
two ways out of this dilemma
i There are fundamental errors in the current dust formation theory Carb on dust
can b e formed from the gas phase already at temp eratures of K as
present at R according to the standard mo dels of CSEs
or
ii There is a mistake in the previous applications of standard dust formation the
ory to RCB envelopes concerning the temp erature determination of the gas The
conventional theories on dust formation are applicable but have to b e discussed in
the context of more careful metho ds for the temp erature determination taking into
account the dynamic conditions in the CSEs of RCB stars
All RCB stars measured thus far seem to b e pulsating variables Lawson Kilkenny
The pulsation p erio ds are of the order of days and the radial velocity
variations at the photosphere range from ab out km s to km s cf Ap
p endix A The stellar pulsation creates sho ck waves which further steep en up in
REQUIREMENTS FOR DIFFERENT APPROACHES
the atmosphere and propagate into the CSE e g Bowen Fleischer et al
Consequently a xed uid element in the envelope is hit by sho ck waves time and
time again The sho cks dissipate mechanical energy and furthermore initiate a more
or less p erio dical compression and re expansion of the gas b oth of which may cause
strong deviations from RE in the gas phase If however one of the three usual
assumptions for temp erature determination RE LTE grey gas opacities is aban
doned the radial range of nucleation distances prescrib ed by observations could
easily op en up
Requirements for Di erent Approaches of Tempera
ture Determination
The most promising scienti c metho d to gain insight into the complex pro cesses
of astrophysical ob jects is the mo deling by full time dep endent computer simula
tions The basis of these mo dels is a hydro and thermo dynamic description of the
gas where the physical interactions are formulated in terms of ordinary di erential
equations which are integrated in time Besides hydro and thermo dynamics these
mo dels may include radiative transfer chemistry and dust formation according to
the chosen degree of approximation Thereby all the necessary physical and chemi
cal pro cesses can b e investigated simultaneously so that just the complex interplay
among these pro cesses can b ecome the main topic of examination Therefore com
puter mo dels can help to build up a higher level of completeness in science
However the results of this rather new metho d of scienti c computing cannot b e
b etter than our physical understanding of the basic pro cesses involved and our ability
to abstract and to simplify In this context the b est description of a physical
pro cess under investigation is not necessarily the most accurate and detailed one
but a description which correctly describ es the most imp ortant features of the
pro cess while using the least amount of resources Such a description must b e
su ciently simple in order to b e included in more complex mo del calculations as
part of the investigations of astrophysical ob jects
Concerning the temp erature determination in the CSEs of pulsating stars such as
Miras and long p erio d variables on the AGB and RCB stars Cepheids and RV Tauri
stars near to the Instability Strip the presence of sho ck waves caused by the stellar
pulsation calls for a time dep endent treatment of the thermo dynamics As argued
ab ove strong deviations from RE may o ccur and the gas temp erature structure
cannot b e obtained by means of radiative transfer calculations alone
According to the present state of approximation in such mo dels Bowen
Fleischer et al Feuchtinger et al the gas temp erature structure is cal
culated as follows Bowen rst carries out a frequency integrated radiative
transfer using the basic assumptions of RE LTE and grey gas opacities as de
scrib ed in Sect which determines the instantaneous RE temp erature structure
Secondly he calculates the current gas temp erature by assuming a lo cal relaxation
CHAPTER INTRODUCTION
log Radius cm
Figure Adopted from Bowen
Figure Temperature structure adop
ted from Feuchtinger et al
toward RE at a nite rate Fleischer et al consider the isothermal limiting
case assuming that the gas instantaneously relaxes to RE everywhere in the enve
lop e Applying radiation hydrodynamics Feuchtinger et al use an approach
which is similar to Bowen s but more consistent In order to treat the thermo dy
namics in these mo dels the total net heating rate due to radiative gains and losses
Q is an imp ortant ingredient Q vanishes in RE and otherwise determines the
rad rad
time scale for relaxation toward RE
Concerning the calculation of Q in the mo dels cited ab ove crucial assumptions
rad
have b een made LTE or in contrast Q yielding simple analytical ex
rad
pressions for Q But dep ending on which of these assumptions is applied the
rad
gas temp erature structure turns out to b e quite di erent In the LTE case the
calculated radiative heating co oling rates are very e cient resulting in gas tem
p eratures usually very close to the RE temp erature structure except for some thin
gas temp erature p eaks at the lo cations of the sho ck fronts cf Fig In con
trast at small densities in the Q case rather broad regions of enhanced
rad
gas temp eratures b ehind the sho cks are pro duced almost entirely decoupled from
the RE temp erature structure cf Fig These results are typical examples for
sho ck waves of predominantly isothermal or predominantly adiabatic charac
ter resp ectively The resulting gas temp erature structure a ects all other results
of these mo del calculations e g the mass loss rate and even the mo del stability
Woo d An imp ortant feedback b etween the gas temp erature structure and
the dynamics of these envelopes is given by the condensation of seed particles from
the gas phase nucleation which is very sensitive to the gas temp erature It triggers
the further evolution of the dust comp onent and hence the acceleration of the gas
due to radiation pressure on dust grains Fleischer et al
Considering the determination of Q in other astrophysical environments exten
rad
sive mo del calculations have b een made for stationary plane parallel sho cks e g in
AIM AND STRUCTURE OF THIS WORK
the interstellar medium where all physical prop erties are purely determined by the
distance from the sho ck front but are not explicitly time dep endent Hollenbach
McKee Fox Woo d Hollenbach McKee Gillet Lafon
Neufeld Hollenbach This situation allows for a very accurate physical de
scription including non LTE ionization non equilibrium chemistry and radiative
transfer However this scheme cannot b e easily applied to the time dep endent
mo dels for pulsating stars for essentially two reasons First the sho cks in the en
velopes of pulsating stars are not stationary e g a xed uid element will start to
re expand after it has b een compressed by an propagating sho ck as opp osed to the
stationary situation and second the detailed description given in the pap ers cited
ab ove is much to o elab orate to b e included within time dep endent hydrodynamic
mo dels at least at the present state of computer sp eed
Thus there is a great need for a realistic calculation of Q On the one hand it
rad
must b e physically based on the relevant heating and co oling pro cesses taking into
account imp ortant features such as the non LTE p opulation of excited states or
radiative trapping On the other hand it must b e su ciently simple to b e included
in time dep endent hydrodynamic mo dels
Aim and Structure of this Work
The basic aim of this work is to gain more theoretical insights on the temp erature
structure of circumstellar envelopes esp ecially those of pulsating stars where sho ck
waves propagate through the envelopes This work fo cuses on the problem of the
oretical temp erature determination in a given radiation eld radiation transfer
calculations are explicitly not considered and are not p erformed
For this purp ose the radiative heating and co oling of the gas in circumstellar en
velopes is investigated from the very b eginning examining densities from to
cm Preceding studies in the literature are usually not applicable in this den
sity range typical for CSEs However there is detailed knowledge available at b oth
extremes of this density interval For large densities extensive calculations of gas
absorption co e cients in stellar atmospheres exist which in case of LTE determine
the net radiative heating rate At low densities the imp ortant radiative heating
and co oling pro cesses are known from studies of interstellar clouds and interstellar
sho ck waves This work derives advantages from b oth and intends to close the gap
b etween these density limits
Due to the time dep endent conditions present in CSEs of pulsating stars the de
termination of the temp erature strati cation must involve time dep endent hydro
dynamic mo del calculations The p ossibility of a fast and prop er inclusion of the
calculated heating and co oling rates into hydrodynamic mo dels is an essential con
straint for these investigations It is the aim of this work to lay the foundations for
a more reliable treatment of the time dep endent thermo dynamics in such hydrody
namic mo del calculations
CHAPTER INTRODUCTION
Although dust formation is rarely discussed in this work explicitly the work is guided
by the certainty that the formation of solids in CSEs requires large densities and
very sp ecial temp erature conditions The question which always stands b ehind the
investigations and is the basic motivation for this work is
Where in the envelope such thermo dynamic conditions may o ccur
The work is organized as follows
Chapter describ es the basic concept for the treatment of the time dep endent
thermo dynamics The level of approximation for this work is xed and the internal
energy of the gas is de ned according to the basic assumption of steady state non
LTE
Chapter contains the calculations of the various heating and co oling rates con
sidering arbitrary radiation elds Computational metho ds are developed which
include the imp ortant e ects of non LTE and of optical thickness in sp ectral lines
Due to the wide temp erature range to b e considered a variety of di erent radiative
pro cesses is investigated rotational and ro vibrational transitions of p olar molecules
and of H line transitions of neutral atoms and ions b ound free transitions free
free transitions and photo chemical reactions Sp ecial attention is paid to which kind
of atomic and molecular data has to b e known for a reliable determination of the
corresp onding rates A short list of radiative pro cesses which have not b een consid
ered so far but might b e of further interest for the heating and co oling of the gas
completes the theoretical part of this work
Chapter outlines some common features for the following applications The
technical details of the calculation of the various particle concentrations and the
internal energy of the gas are explained The particle concentrations and the state
of the gas are determined from the element abundances the mass density of the
i
gas its temp erature T and the continuous radiation eld J
g
In the following chapters three applications of the theoretical metho ds are presented
Chapter examines the top ology of the radiative equilibrium temp erature solu
tions in the CSEs of RCB stars It is shown that the condition of radiative equilib
rium may not b e unique but can have two or more stable temp erature solutions
These thermal bifurcations in principle allow for a spatial co existence of hot and
co ol phases in the circumstellar envelope
Chapter investigates the relaxation towards radiative equilibrium esp ecially in
resp onse to propagating sho ck waves For this purp ose radiative co oling time scales
for a carb on enriched gas typical for C stars are calculated as function of gas den
sity and temp erature The imp ortance of the di erent heating co oling pro cesses
is discussed and the most e cient pro cess in the various density and temp erature
AIM AND STRUCTURE OF THIS WORK
regions is determined The results of the co oling time scales are compared to those
derived from formerly applied analytical heating co oling functions in previous re
search The character of the thermal relaxation of the gas after the passage of sho ck
waves is discussed providing new fuel to the controversy ab out whether the sho cks
in CSEs b ehave predominantly isothermally or adiabatically
Chapter again considers RCB stars A physical mechanism is presented which
may b e essential for the o ccasional onset of dust formation in the circumstellar
envelopes of pulsating RCB stars A mo del for xed uid elements which are
p erio dically hit by strong sho ck waves pro duced by the stellar pulsation is developed
and the thermal energy balance the chemistry and the nucleation in such uid
elements are investigated According to this mo del the preconditions for e ective
carb on nucleation may b e temp orarily present quite near to the photosphere of a
pulsating RCB star despite their high e ective temp eratures Thus this work might
bridge the gap b etween observations and theory concerning RCB stars as outlined
in Sect
Chapter summarizes the results and presents the conclusions of this work
App endix A gives an overview of the current status of observational knowledge on
the fascinating class of RCB stars and summarizes previous mo dels This App endix
provides an imp ortant background for the investigations in Chapter and
Chapter
The Thermo dynamic Concept
This chapter intends to state the basic assumptions of this work and to clarify the
meaning of some terms frequently used
The central physical quantity of this work is the temp erature of the gas The velocity
distribution of the gas at rest is assumed to b e given by a unique Maxwellian distri
bution characterized by a single kinetic temp erature which is henceforth called the
gas temp erature and denoted by T Di erences in the kinetic temp eratures of
g
di erent kinds of particles e g electrons and atoms are neglected The pro cesses re
sp onsible for the relaxation towards the Maxwellian distribution are elastic collisions
which distribute the total translational energy present among the gas particles The
corresp onding relaxation time scale is assumed to b e considerably shorter than any
other time scale inherent in the physical system under investigation With regard
to this relaxation the most critical pro cess is the equalization of the translational
energies b etween light and heavy particles b ecause of the ine cient energy trans
fer rates of such collisions For conditions in stellar atmospheres Mihalas Weibel
Mihalas see p and p for a more comprehensive discussion arrive at
the conclusion that the existence of a unique Maxwellian can safely b e assumed
First Law of Thermo dynamics and Equation of State
The aim of this work is to develop a time dep endent metho d for temp erature de
termination based on the rst law of thermo dynamics
dE Q W
Equation states that the change of internal energy dE is given by the amount
of heat transfered to the gas Q counted p ositive for gains plus the work done to
the gas W counted p ositive when the surroundings deliver work to the gas
How are the gas temp erature and the internal energy related to each other The an
swer of this questions seems to b e trivial given by the well known caloric equation of
state but in fact deserves some further discussion for diluted gases under astrophys
ical conditions Besides the translational degrees of freedom a real gas consisting
of neutral atoms electrons ions and molecules has additional p ossibilities to store
energy which are henceforth called the p o ols of energy The p opulation of excited
CHAPTER THE THERMODYNAMIC CONCEPT
electronic vibrational and rotational states represent such p o ols Furthermore en
ergy is stored in p otential form according to the binding forces b etween electrons
and atoms ionization p otential and b etween the constituting atoms of molecules
disso ciation p otential Consequently the internal energy of the gas is de ned as
E E E E E E e
trans ion diss el vib rot
The details of the evaluation of the various energy terms are stated in Chapter
Eq In case of Lo cal Thermo dynamic Equilibrium LTE the relationship
b etween the gas temp erature T and the internal energy e is well de ned The
g
degree of ionization the concentration of the molecules and the p opulation of the
excited levels can b e determined by means of Saha equations the law of mass action
and Boltzmann distributions resp ectively All energy terms in Eq can b e
calculated straightforwardly yielding e e T or more generally e as function
g
of any suitable set of two lo cal state variables
LTE and Non LTE
Considering the diluted gases in CSEs however LTE is not valid Radiative pro
cesses alter the state of the gas in various ways Figure sketches this situation
The gas is represented by the big grey b ox containing the internal p o ols of energy
The gas interacts with the radiation eld via the exchange of photons and with the
dust comp onent e g via inelastic collisions At the same time internal pro cesses
grey arrows redistribute the total internal energy among the various degrees of
freedom of the gas Work can b e done to the gas W p dV as indicated by the
arrow on the l h s in the gure and the gas may exchange heat with its surroundings
not shown Examples for the such pro cesses are heat conduction viscous pro cesses
and sho ck dissipation
The radiative pro cesses generally drive the gas away from LTE whereas the in
ternal pro cesses pro cesses not involving photons or dust grains that is drive the
gas toward LTE In general the energy transp ort b etween the radiation eld and
the translational energy p o ol is indirect so to sp eak b ecause a transmitting state
is involved Usually a two step pro cess is required for example a collisional ion
ization followed by radiative recombination or the absorption of a photon followed
by collisional de excitation These transmitting states e g the excited electronic
states of the atoms are considerably a ected by the radiation eld and hence the
state of the gas cannot b e calculated by thermo dynamic considerations
Thus a non LTE treatment for the diluted gases in CSEs is required Balancing
the gains and losses by all collisional radiative and chemical pro cesses the change
Except for the case that the radiation eld exactly equals a Planckian of the gas temp erature
Direct links b etween the radiation led and the translational energy p o ol do also exist but
are usually less imp ortant e g free free emission bremsstrahlung
LTE AND NON LTE
electronic ionization excitation radiation
−p dV vibrational translation excitation
dissociation rotational dust
excitation
Figure The p o ols and uxes of energy Black full arrows indicate energy
uxes via photons From top to b ottom photoionization radiative recombi
nation b ound b ound free free vibrational and rotational emission absorption
photo disso ciation radiative asso ciation Black dashed arrows show energy ex
change rates b etween dust and gas via collisions and surface chemical reactions
Full grey arrows represent internal energy uxes via collisional de excitation
or particle creation destruction resp ectively Dotted grey arrows show additional
examples of internal energy uxes not explicitly considered in this work such as
pumping by uorescence ro vibrational pumping or the excitation of vibrational
states via chemical reactions
CHAPTER THE THERMODYNAMIC CONCEPT
j
of chemical sp ecies i in quantum state j can b e expressed of the particle density n
i
by e g Mihalas Weibel Mihalas p
j
X X
d n
l j j j l
i
l
n R n R
i
k k i i k
dt
fk l g fij g fk l g fij g
Di usion pro cesses are neglected in Eq and d dt denotes the Lagrangian
derivative with resp ect to time considering a comoving moving frame The rate
j l
co e cient R Hz denotes the total rate of all collisional radiative and chemical
i k
pro cesses which destroy a particle of sp ecies i in state j and create a particle of
sp ecies k in state l For simple collisional absorption or emission pro cesses we have
i k For chemical pro cesses i k the quantum indices j and l usually refer to the
ground state The rate co e cients may contain further particle densities collision
partners or chemical reactants or the radiation eld J dep ending on which type
of pro cess is considered The rate co e cients will b e quanti ed in Chapter
Steady State
If the gas is exp osed to static outer conditions radiation eld prop erties of the
dust comp onent volume etc the gas will relax towards a steady state i e the
concentrations of all sp ecies in all quantum states b ecome time indep endent
j
d n
i
dt
In case of a steady state the l h s of Eq vanishes The particle densities can b e
calculated by solving the coupled algebraic equations of type Eq for all sp ecies
and states under investigation The results are time indep endent and can formally
b e expressed by
j j
n n T J
g
i i
Consequently the caloric equation of state of the gas writes as e e T J
g
Therefore in addition to the two lo cal state variables su cient in LTE the radiation
eld o ccurs as additional external parameter for the determination of the particle
densities and the equation of state The resulting steady state of course di ers
from LTE in general In the limiting case of large densities however where the
internal e g collisional pro cesses dominate LTE is valid and the dep endency on
J disapp ears
Other terms used in the literature are kinetic equilibrium or statistical equilibrium
In Chapter it will b e stated that another external parameter enters into the determination
dv
of the steady state which is the mean lo cal velocity gradient This parameter is involved in
dl
the calculation of the rate co e cients for b ound b ound transitions concerning optically thick lines
However the in uence of this parameter is small
The in uence of the dust comp onent on the state of the gas is neglected in Eq
STEADY STATE
The condition of a static environment can b e relaxed to some extend concerning
those time dep endent situations where the changes of the outer conditions o ccur
slowly In this case the gas rapidly accommo dates to the varying environment This
work assumes that this accommo dation in fact o ccurs instantaneously
The internal relaxation of the gas toward steady state is assumed to o ccur
on a short time scale compared to the changes of the outer conditions
In this case a caloric equation of state exists as stated ab ove although the gas is
not in LTE A thermo dynamic description of the gas is appropriate Of course the
Considering the excited electronic states of atoms and ions the relaxation time scale is given by
the radiative lifetimes of the levels at small densities and is even shorter at large densities Allowed
electronic transitions have radiative lifetimes of typically s but actually the slowest rate within
the level system decides up on the relaxation time toward a complete steady state which can b e
as large as s for the meta stable levels included in this work The radiative lifetimes of the
vibrational and rotational levels of p olar diatomic molecules are found to b e typically s
These time scales are to b e compared to typical hydrodynamic time scales which for the CSEs
of pulsating stars are approximately given by one pulsation p erio d which is ab out s for Miras
and s for RCB stars Hence the relaxation of the electronic vibrational and rotational states
can b e assumed to b e fast
Considering the relaxation towards ionization equilibrium balance b etween ionization and recom
bination rates the corresp onding time scales are very di erent If photoionization dominates
e g considering the case J B K the relaxation time scales are found to b e s
for all atoms under investigation indep endent of density and temp erature If collisional ionization
dominates the relaxation time scale strongly dep ends on the density In the case J B K
the relaxation lasts s at a density of cm s at cm and s at cm
concerning low temp eratures If the temp erature is high enough to cause considerable collisional
ionization the relaxation time scale b ecomes much shorter Hence the relaxation of the degree
of ionization is fast in those cases where is it imp ortant for the calculation of the radiative heat
ing co oling rates but can exceed the hydrodynamic time scales otherwise
The chemical pro cesses usually introduce the largest time scales to the gas disregarding dust
formation pro cesses The chemical relaxation time scales can indeed exceed hydrodynamic time
scales in CSEs e g Beck et al However the chemical pro cesses themselves do not provide
rst order radiative heating and co oling rates the most imp ortant heating and co oling pro cesses
usually involve the degrees of freedom discussed ab ove which can b e assumed to b e p opulated
according to a steady state
Thus one has to conclude that the relaxation of the gas in CSEs towards its steady steady may
not always b e complete A full time dep endent non LTE approach would b e required for a more
accurate mo deling of the gas In this case the rst law of thermo dynamics has to b e applied to the
translational energy alone e E A caloric equation of state do es not exist in the usual sense
trans
or b ecomes obsolete One has to determine the internal energy distribution pro cesses in this case
e g how much translational energy is consumed or lib erated during a chemical reaction A time
dep endent treatment of ionization and chemistry within hydrodynamic mo dels seems to b e out of
question at least at the current state of computer sp eed This would introduce a large number of
new sti di erential equations to the usual set of hydrodynamic equations to b e solved which
requires much more computational e orts However the p ossibility to include the results of this
work into such calculations is regarded as essential concerning future investigations Therefore
the assumption of a steady state is made nevertheless It represents an appropriate compromise
b etween accuracy and exp ense it accounts for the most imp ortant non LTE e ects but keeps
things simple enough for a thermo dynamic description compatible to hydrodynamic calculations
For a further discussion of this topic see Mihalas Weibel Mihalas p
CHAPTER THE THERMODYNAMIC CONCEPT
particle densities can only b e determined if the radiation eld is known that is
there are as many new external parameters as are required to sp ecify J However
the circumstellar envelopes are optically thin by de nition maybe except for some
strong sp ectral lines The radiation eld is mainly provided by external sources
eventually mo di ed by dust in the CSE which is also considered as external
Therefore it seems appropriate to prescrib e the radiation eld in CSEs e g by a
radially diluted photospheric radiation eld or to use the results of radiative transfer
calculations concerning the dust comp onent The gas itself is not or at least not
much resp onsible for the radiation eld Therefore the non LTE radiative transfer
problem decouples in CSEs in contrast to the situation in stellar atmospheres and
the thermo dynamic b ehavior of the gas can b e studied in the prop osed way
The rst law of thermo dynamics Eq is further sp ecialized in the following
According to the de nition of the internal energy cf Eq the internal energy
transfer rates do not cause any heating or co oling since they only transfer energy
b
from one internal p o ol to another cf Fig The net heating rate Q Q dt is
given by the sum of energy uxes from the remaining external p o ols the radiation
b
eld and the dust that is to the gas Let Q denote the total net heating rate
rad
p er unit mass and time due to radiative pro cesses which is given the amount of
b
net absorb ed photon energy and Q the total net heating caused by the presence
dust
of dust grains Disregarding other heating co oling mechanisms as heat conduction
convection and viscous pro cesses which are usually negligible at the low densities
in CSEs the rst law of thermo dynamics writes
de dV
b b
p Q Q
rad dust
dt dt
p is the gas pressure and V the sp eci c volume The gas temp erature can b e
regarded as an implicit result of the solution of Eq inferred from the caloric
equation of state
The main task of the following Chapter will b e to quantify all the imp ortant
internal and external rates as far as p ossible By means of these rates the steady
b
state of the gas is determined The net radiative heating rate Q is a kind of useful
rad
byproduct of these calculations
As a consequence of the steady state assumption the internal energy and the net
heating function are entirely determined by means of lo cal physical quantities which
are readily available in hydrodynamic mo del calculations It is hence p ossible to
b
tabulate Q and e as function of two state variables say and T and a suitable
rad g
parametric sp eci cation of J Thus it is guaranteed that the prop osed time
dep endent metho d of temp erature determination can b e included in hydrodynamic
mo del calculations with regard to future investigations
Chapter
Radiative Heating and Co oling
The determination of the total radiative heating co oling rate of the gas in CSEs
requires a quantitative analysis of all radiative pro cesses o ccurring in the considered
uid element a di cult and principally in nite task What is feasible however
is the investigation of the most imp ortant heating and co oling pro cesses mainly
relying on the exp erience of preceding studies
From stellar atmosphere calculations it is generally known that b ound free tran
sition usually are the primary cause for the shap e and the magnitude of the ab
old and hence for the heating and co oling of the sorption co e cient e g Uns
gas The additional consideration of line transitions line blanketed mo dels is
only a second order e ect in this context In predominantly neutral stellar at
mospheres the b ound free transitions of H are imp ortant Below ab out K
molecules enter into comp etition and so on dominate the absorption co e cient by
their electronic bands vibrational and rotational sp ectra esp ecially those molecules
with p ermanent dip ole moment J rgensen
Concerning interstellar conditions Hollenbach McKee p ointed out
that forbidden ne structure lines meta stable transitions and some low lying p er
mitted line transitions of the various neutral and singly ionized metal atoms of
ten provide the dominant co oling mechanism for a sho cked non molecular gas If
present p olar molecules contribute by their large amount of allowed vibrational and
rotational line transitions Bound free transitions mainly of hydrogen Ly and
K esp ecially for large densities H are imp ortant at temp eratures T
Reviewing these exp eriences it is imp ortant to tackle at least all the heating and
co oling pro cesses mentioned ab ove in order to unify the picture of imp ortant heat
ing co oling rates for CSEs A general theoretical description must b e developed
which is applicable to b oth stellar atmospheres as well as interstellar density con
ditions
In the following the net heating rate of one particular radiative pro cess and its reverse
pro cess is always discussed simultaneously which is di erent from other approaches
concerning interstellar matter e g Spitzer where the heating and co oling
rates are usually discussed strictly apart
This statement refers to a static atmosphere where self shielding diminishes the in uence of
sp ectral lines the e ects of lines in a moving medium may b e larger by orders of magnitude
CHAPTER RADIATIVE HEATING AND COOLING
Bound Bound Transitions
The most basic form of interaction b etween matter and radiation eld is given by
the absorption and emission of line photons In this section a general theoretical
metho d for the calculation of the heating co oling rate of an arbitrary N level
system of b ound states is presented The metho d is applicable to line transitions of
atoms and ions to vibrational and rotational transitions of molecules and also to
quadrup ole transitions of H It has the following features
a The calculation of the level p opulation is p erformed under the assumption of
steady state non LTE It is absolutely necessary to consider non LTE under
the density conditions present in circumstellar envelopes esp ecially for allowed
atomic transitions
b Compared to interstellar conditions the high densities encountered in circum
stellar envelopes may cause large optical depths in the lines which signi cantly
change the heating co oling rates due to radiative trapping These e ects are
tackled by applying an escap e probability metho d
c Since propagating sho ck waves may b e present in the circumstellar envelopes
of pulsating stars large velocity gradients o ccur In contrast to steady plane
parallel sho cks e g in the ISM the explicit time dep endence and the geome
try of the ow requires a di erent metho d to calculate the escap e probabilities
as outlined by Hollenbach McKee This work uses the Sob olev theory
in case of spherical symmetry
d Line absorption is completely taken into account The intense continuous
radiation eld in circumstellar envelopes changes the co oling rates signi cantly
and can in fact lead to net line heating in contrast to interstellar conditions
Escap e Probability Metho d for an N Level System without Con
tinuum
An atomic or molecular N level system is considered The quantity to b e determined
is the total rate of energy which is transferred to from the radiation eld via line
emission absorption This total energy transfer rate is calculated in two steps
First the level p opulations n are calculated by means of the statistical equations
i
steady state non LTE and secondly the energy transfer rate is determined The
statistical equations are given by
X X
n R n R
i ij j j i
j i j i
and can b e solved together with the equation for the conservation of the total particle
P
n The rate co e cients are de ned by density of the considered sp ecies n
i sp i
BOUND BOUND TRANSITIONS
R A B J C
ul ul ul ul ul
R B J C
l u l u ul l u
where u and l lab el an upp er and lower level resp ectively The rate co e cients
for stimulated emission B and absorption B can b e calculated from those for
ul l u
sp ontaneous emission A by applying the Einstein relations Similarly the rate
ul
co e cients for collisional excitation C can b e calculated from those for collisional
l u
excitation C by applying a detailed balance relation
ul
c
A B
ul ul
h
ul
g
u
B B
ul l u
g
l
g
u
C C exp E k T
l u ul ul g
g
l
The frequency integrated mean intensity
Z Z
I d d J
ul ul
inc
is not solely given by the incident continuous intensities I which are regarded
as known but is mo di ed by line emission and absorption in the considered res
onance region itself which b ecomes imp ortant for optically thick lines An exact
solution of this problem can only b e achieved by frequency dep endent non LTE ra
diative transfer calculations in the moving medium which go es far b eyond the scop e
of this work Fortunately there are approximate escap e probability techniques avail
able which can account for the most imp ortant resonance e ects This work uses the
Sob olev approximation in the case of spherical symmetry e g Puls Hummer
for a detailed description see Woitke
e cont e L
J P J P S
ul
ul ul ul
ul
Z
cont inc
J I d
ul ul
h g n
u l
ul
L
S
ul
c g n
l u
S
Z
exp
ul
e
d P
ul
S
ul
dv
g c A
k
u ul
S
n n
l u
ul
g dl
l
ul
dv
v v
k
dl r r
e L cont
P is the mean escap e probability and S the line source function J is the
ul ul
ul
continuous mean intensity at line center frequency caused by incident radiation
ignoring radiation transfer e ects in the considered resonance region itself
A discussion of the applicability of Sob olev theory to the sho cked envelopes of pulsating stars
is given on p
CHAPTER RADIATIVE HEATING AND COOLING
cont
J in principle results from the calculation of a continuous radiative transfer
ul
without the considered line For simplicity the incident intensities are assumed to
S
b e isotropic in Eq is the so called Sob olev optical depth of the line and
ul
dv
k
the velocity gradient on a considered ray The following approximation is used
dl
in order to avoid the elab orate and time consuming integration over the solid angle
D E
dv
in Eq An appropriate mean velocity gradient is de ned and the escap e
dl
probabilities are calculated according to
v v dv
x
dl r r
v v
x max
r r
g c A dv
u ul
S
e
n n
l u
ul
g dl
l
ul
S
e
exp
ul
e
e
P
ul
S
e
ul
e e
The error of this pro cedure vanishes for the two imp ortant cases P and P
e
and reaches a maximum value of around P Di erent ow geometries and
cases with vanishing velocity gradients can b e tackled by using di erent expressions
E D
dv
in Eq as summarized in Neufeld Kaufman for
dl
For the numerical solution of the statistical equations it is very advantageous
L
to eliminate the unknown line source functions S It is straight forward to show
ul
that
cont
J
e
ul
e
n A n B n B J n A P
u ul u ul l l u ul u ul
ul
L
S
ul
g
u
e
e
j n j n A P
ul u ul l ul
ul
g
l
cont
where j c h J is a dimensionless quantity which characterizes the
ul
ul
ul
lo cal continuous radiation eld By means of Eq all ab ove equations can b e
combined into the following set of e ective rate co e cients
e
e e
R A P j C
ul ul ul ul
ul
g E
u ul
e
e e
R A P j C exp
l u ul ul ul
ul
g k T
l g
e
The level p opulations can now b e calculated by solving Eq with R instead
ij
of R where the line source functions do not app ear anymore
ij
Strictly sp eaking the in uence of incident continuous radiation increases or decreases compared
to Eq if the considered uid element mainly receives the light from a particular direction
where the velocity gradient is smaller or larger than the mean velocity gradient resp ectively the
e ect of the incident intensities is prop ortional to the escap e probability in that particular direction
but the isotropic re emission is prop ortional to the mean escap e probability
BOUND BOUND TRANSITIONS
Finally after having determined the level p opulations the net heating rate can
readily b e calculated either from
X X
g E
u ul
Q E C n n exp
coll ul ul u l
g k T
l g
l u l
or by multiplying the radiative rate in Eq by E and summing up the
ul
contributions from all transitions
cont
X X
J
e
ul
e
P Q E n A
rad ul u ul
ul
L
S
ul
l u l
X X
g
u
e
e
E A P n j n j
ul ul l ul u ul
ul
g
l
l u l
Both expressions are equivalent and must yield the same result Q Q since
coll rad
the gains and losses from the translational and the radiative p o ol the only consid
ered source terms balance each other cf Fig This equality demonstrates the
physical meaning of the basic assumption of steady state A fast relaxation of the
degree of excitation of the considered sp ecies is assumed such that Q Q
coll rad
is assured Equation shows the mo di cations from the usual expression
P
E n A caused by optical thickness and incident continuous radiation note
ul u ul
however that the level p opulations are also a ected
Numerical Iteration Scheme
The solution of the statistical equations still requires an iteration since the escap e
probabilities dep end on the level p opulations In most cases a direct iteration
converges rapidly but there are also cases in which this pro cedure fails The fol
lowing scheme which may b e called a decelerated iteration converges for all
considered mo del atoms under all considered density temp erature and radiation
eld conditions where X it means quantity X at iteration step it
S e
e
e
Put P Q
rad
ul ul
e e
Calculate R it and R it according to Eqs and
ul l u
Determine n it from the statistical equations
j
Calculate Q it and Q it according to Eqs and
coll rad
De ne j Q it Q it j
rad rad
S
e
Calculate according to Eq
ul
S S S S
e e e e
Put it it it exp maxf it g
ul ul ul ul
e
e
Calculate P it from Eq
ul
Go back to step unless
Take Q as nal result if the p opulation is close to LTE otherwise
rad
rely on Q to avoid the errors pro duced by the subtraction of large
coll
almost equally large numbers
CHAPTER RADIATIVE HEATING AND COOLING
The total line heating co oling rate dep ends on the following physical parameters
The particle density of the considered sp ecies n the particle densities of the col
sp
lision partners the gas temp erature T the continuous background radiation eld
g
E D
dv
cont
J and the lo cal mean velocity gradient
dl
ul
The following atomic or molecular data are required the statistical weights g and
i
energies E of the considered levels the Einstein co e cients for sp ontaneous emission
i
A and the rate co e cients for collisional de excitation C T where the particle
ul ul g
densities of the collision partners enter into the calculation
The presented metho d is an universal and rapidly converging to ol for the calculation
of the total line heating co oling rate of an arbitrary N level system It is applicable
to arbitrary conditions of density temp erature and radiation eld and can b e applied
to a variety of ow geometries as far as the involved escap e probability concept
makes sense
Discussion of the Applicability of Sob olev Theory
The application of Sob olev theory to the sho cked envelopes of pulsating stars requires
some critical remarks
Sob olev theory is applicable only in case of large velocity gradients where
the sizes of the resonance regions where an emitted line photon can still b e
re absorb ed are small compared to typical scale heights of the envelope In
case of thermal broadening this condition can b e written as
D E
dv
d ln T d ln n d ln
sp
dl
max
v dr dr dr
th
Regarding the results of time dep endent mo dels for the envelopes of long
p erio dic variable stars Bowen Fleischer this condition seems to
b e just even ful lled The thermal velocities are a few km s the mean
velocity gradient cf Eq is typically to km s R and the scale
height the r h s of Eq is typically R Problems can o ccur very close
to the star where the scale height can b e much smaller and close to sho ck
fronts in the p ost sho ck regions where the temp erature gradients can b e fairly
large
Due to the strict division b etween considered line and continuum line overlaps
are an intrinsic problem of Sob olev theory The Sob olev theory requires that
the emitted line photons of one particular transition cannot b e re absorb ed by
any other line transition anywhere else in the envelope
v
c
The maximum relative shift of the lines due to hydrodynamic velocities the
l h s of Eq is ab out for AGB stars which is usually much smaller
BOUND BOUND TRANSITIONS
than the relative spacing of the considered sp ectral lines the r h s of Eq
where is the frequency di erence of two considered lines This condition
b ecomes more serious for the vibrational bands of diatomic p olar molecules
where the r h s of Eq is given by hB h cf Sect
But even in this case condition remains valid Problems can o ccur in
electronic bands of molecules where the spacing of the individual lines is even
more narrow or in very narrow spaced atomic multiplets
The problem of non monotonic velocity gradients in the saw tooth like veloc
ity elds of CSEs of pulsating stars coupled with the question of the di erence
b etween lo cal and global escap e is ignored Thereby the radiative heating
of e g one p ost sho ck layer by line emissions from the p ost sho ck region of
another sho ck wave is neglected in this work
Close to the lo cation of a velocity discontinuity caused by a sho ck front most
of the disturbing absorb er are missing in the directions across the discontinuity
Therefore larger escap e probabilities can o ccur in this case and the radiative
co oling of otherwise optically thick lines increases which may b e imp ortant
just for the hot emitting p ost sho ck regions directly b ehind sho ck waves
The advantages of the presented escap e probability metho d however clearly out
weigh these short comings As long as no b etter and comparable simple metho ds are
available the Sob olev theory is just the appropriate compromise b etween simplicity
and accuracy of the physical description Using this theory the results of the line
heating co oling rates are entirely determined by lo cal physical prop erties which are
available in hydrodynamic mo dels still including the most imp ortant line transfer
e ects The only real alternative would b e to ignore optical depth e ects of the lines
completely which would induce much larger errors
An Exemplary Two Level Atom
In order to demonstrate the basic features of the line heating co oling functions an
exemplary two level atom is examined The following typical atomic parameters
and physical conditions are considered
g g E k K A Hz C n Hz cm
H
dv km s
n n n n
H H He sp
dl R
The resulting line heating co oling rate p er mass Q as function of the total
rad
hydrogen particle density n is depicted in Fig for the case of negligible con
H
cont
tinuous radiation eld J The gure demonstrates the fundamental density
dep endence of the two level co oling rate Three di erent cases can b e distinguished
dep ending on the relation b etween the density of the gas n and two critical den
H
sities n and n which are de ned b elow
cr thick
CHAPTER RADIATIVE HEATING AND COOLING
I II III
Figure The co oling rate full lines left axis and the excitation temp erature
de ned by k T E ln g n g n in units of T dashed lines right axis
exc u u g
ul l l
of an exemplary two level atom in the case of negligible continuous radiation eld
cont
J
Figure The temp erature dep endence Figure The dep endency of the line
of the line co oling rate p er mass for n co oling rate p er mass on the radiation eld
H
cont
Full circles indi for n cm and J cm and T K
H g
cate p oints already depicted in Fig
BOUND BOUND TRANSITIONS
I n n The line is optically thick and the co oling rate is limited by
H thick
radiative trapping where only a fraction of the emitted line photons can escap e
the lo cal surroundings LTE is valid In the limiting case n the
H
e S
escap e probability scales as P Thereby Q b ecomes density
rad
indep endent
I I n n n The atom is thermally p opulated LTE and the line is
cr H thick
optically thin The co oling rate is simply given by the thermal rate of emitted
photons leading to Q
rad
I I I n n The atom is p opulated sub thermally non LTE and the co oling
H cr
rate is limited by the rate of energy transferred from the gas via collisions
which yields Q In the limiting case n each exciting collision
rad H
is followed by sp ontaneous emission
n denotes the usual critical density for thermal p opulation and n corresp onds
cr thick
S
to The critical densities are de ned by
A
n
cr
C n
H
dv n g
H
n
thick
A g c dl n
sp
In some cases the two critical densities will overlap n n If this happ ens
cr thick
as e g in case of the large Einstein co e cients of allowed transitions the co oling
rate directly changes from the Q to the Q const b ehavior
rad rad
Figure shows the freezing out of the concerned degree of freedom A minimum
temp erature of ab out T E k is required for e cient collisional excitation and
g
hence for e cient line co oling A further increase of the temp erature do es not
increase the co oling rate much However since at the same time other co oling lines
enter into comp etition and b ecome much more e cient the considered sp ectral
line gets less imp ortant in comparison Hence a sp ectral line requires very sp ecial
density and temp erature conditions in order to b e an e cient co olant The depicted
temp erature dep endence is unique for all densities The Q n curve is simply
rad H
shifted up and downwards in Fig according to that temp erature dep endence
Line transitions can cause net co oling of the gas but can in fact also cause net
heating in the case of intense continuous radiation elds as shown in Fig This
result is straightforward but fundamentally di erent from the exp erience with inter
stellar matter where the continuous radiation eld may b e neglected and where line
transitions generally cause radiative co oling Equation expresses the linear
dep endency shown in Fig
All the discussed dep endencies of the two level line heating co oling function are
quite general and approximately apply also to the other heating co oling mechanisms
outlined in this work
However the considered line can still b e interesting for radiative heating as far as E k T
l g
concerning a multi level atom
CHAPTER RADIATIVE HEATING AND COOLING
Lines of Atoms and Ions
For a general discussion of the imp ortance of line heating and co oling in circumstellar
envelopes the selection of sp ecies and lines is crucial The selection dep ends on the
considered elemental abundances and on the considered density and temp erature
conditions As argued ab ove one should esp ecially include a variety of lines with
di erent E values di erent sp ectral regions Furthermore the availability of
ul
atomic data esp ecially the collision rates can b e problematic The selection of lines
in this work is mainly based on the exp erience of Hollenbach McKee Since
their work relates to interstellar conditions mainly the few low lying levels of the
more abundant atoms and ions are taken into account
In case of larger densities where the p opulation is generally closer to LTE more
lines enter into comp etition and the chosen selection may b e insu cient Esp ecially
for high gas temp eratures even the p opulation of very high lying levels may b ecome
imp ortant which immediately causes troubles due to the rapidly increasing number
of transitions to b e considered However since sp ectral lines have generally proven
to b e imp ortant solely at small densities a troublesome expansion of the line list
would b e rather fruitless b ecause the b ound free heating and co oling rates will
dominate at larger densities anyway
Therefore only a few further lines which satisfy the conditions
large elemental abundance
neutral or singly ionized
low excitation energy E
l
large A di erent E values and
ul ul
collisional data available
have b een additionally included esp ecially from Mendoza and the references
therein The completeness of the mo del atoms is another necessary precondition for
non LTE investigations If for example a transition with the principal quantum
number jump lo oks interesting all the transitions up to levels should b e
taken into account Table summarizes the selection of sp ecies and line transitions
in this work comprising sp ecies and lines The list includes most of the existing
low lying energy levels of the considered sp ecies
The rates for collisional de excitation are assumed to b e given by the collision rates
with free electrons which are usually dominant unless the degree of ionization is
lower than and with neutral H atoms
e H
C T n T n T
ul g e g H g
ul ul
x
ul
x x
T T T
g g ref
ul ul
The collisional rates are often represented as Eq so that for one collision rate
usually two parameters and are to b e collected for electrons and H atoms
ul ul
There are often di erent ts for di erent temp erature regimes T ref
BOUND BOUND TRANSITIONS
Table Atomic line heating and co oling considered sp ecies and transitions
Levels m Ref
ul
H n n n
He S S S P P
He n n n n n
P P P C
P D S
C P P
P P
S D P N
N P D S
P P P O
P D S
S D D O
P P P Si
P D S
P P Si
P P
P P P S
P D S
S S D D P P
D D D Fe
D F F
Fe D D D
D F F D
Levels are listed in the order of energy rst level ground level Levels without
lower index are multiplets which are treated as single levels
Order of transitions for two level atoms for three level
atoms for four level atoms
for ve level atoms
Hollenbach McKee and references therein
H
Mendoza and references therein cm s is assumed
ul
Luttermoser Johnson
Einstein co e cients from Mihalas collisional de excitation rates from
Mihalas Stone
CHAPTER RADIATIVE HEATING AND COOLING
In conclusion the selection of lines has b een p erformed in view of the imp ortance for
the gas not from the observational p oint of view At rst sight an astronomer
would probably suggest to consider those lines which can b e seen These lines
however refer to an optical depth i e to a particular shell of the CSE
usually close to the photosphere of the star where the density is large In contrast
the lines listed in Table may not even o ccur in the stellar sp ectra
The calculation of the various line heating co oling functions straightforwardly pro
ceeds according to the metho ds outlined in Sect Each row in Table
P
n n The re is thereby considered as closed multi level system with
i sp
row
sults roughly are a sup erp osition of several two level type functions as depicted
in Fig The b ehavior of lines with larger excitation energy however is usually
more complex since the p opulation of the lower level changes and the upp er level
can b e pump ed by another transition etc In a real physical situation the concentra
tions of the carriers of the lines n n additionally dep end on the temp erature
sp H
the density and the radiation eld The same o ccurs for the electron concentration
which is of crucial imp ortance for the collision rates
Rotational Transitions of Linear Polar Molecules
As so on as molecules b ecome abundant in the gas phase they usually dominate
the radiative energy exchange Esp ecially the ro vibrational transitions of abun
dant p olar molecules have proven to b e imp ortant under interstellar conditions
e g Neufeld Kaufman in the atmospheres of co ol stars and even in the
outer atmosphere of the sun e g Ayres The general problem of the treat
ment of molecules in radiative transfer arises from the large number of line transitions
to b e considered For non LTE investigations a huge amount of molecular data has
to b e collected individual radiative lifetimes collision rates etc This pro cedure is
only feasible for a very few well known molecules and subsets of transitions Fortu
nately there are some approximate analytical expressions available for certain types
of molecules e g diatomic molecules Since we are not interested in any sp ec
troscopic details but in the total e ect of molecules for the radiative heating and
co oling of the gas these analytical approximations are just appropriate
Concerning the rotational transitions of linear p olar molecules the basic mo del of a
rigid rotator provides the statistical weights g and energies E of the levels The
J J
Einstein co e cients for the allowed dip ole transitions with selection rule J J
for sp ontaneous emission can b e derived from the rotational constant B and the
dip ole moment Chin Weaver The rates for collisional de excitation C
D ul
are adopted from Hollenbach McKee
E J J hB
J
g J
J
J
J J
A
J J
D
h c J
BOUND BOUND TRANSITIONS
X
g hB E
l l
C n v exp
ul i th i
k T k T
g g
i
q
k T m v
g red i th i
where JB is the frequency of the transition v the most probable ther
J J th i
mal velocity and is the total collisional cross section which is usually estimated
to b e cm
The molecular data required for the calculation of the rotational heating co oling rate
are the total collisional cross section the rotational constant B and the dip ole
moment which can b e taken from various molecular data tables e g Landolt
D
ornstein Hellwege Table summarizes these molecular data of the con B
sidered molecules in this work
Table Vibrational and rotational heating and co oling considered
sp ecies and molecular data
Sp ecies K Hz B MHz D cm
D
CO
OH
CH
C H
HCN
CN
C N
SiC
SiN
SiO
SiS
CS
The vibrational heating co oling function of this molecule cannot b e
treated according to Sect since it is not diatomic
Estimated The corresp onding net vibrational heating function how
ever is not signi cantly a ected by the choice of this parameter
cf Sect
Since only applications for carb on enriched cases are made H O is not
considered in this work In the case water it is almost absent
C O
from the gas phase
The calculation of the rotational heating co oling function can b e p erformed simi
larly to the last paragraph Instead of solving the statistical equations for all
considered rotational level p opulations n however which would also b e p ossible
J
but to o elab orate for our purp ose I use the following approximate metho d prop osed
CHAPTER RADIATIVE HEATING AND COOLING
by Kruger et al The rotational states are assumed to b e p opulated according
to a Boltzmann distribution with a yet unknown rotational excitation temp erature
T
rot
k T
rot
Z
rot
hB
g E
J J
mol
n n exp
J
Z k T
rot rot
By means of Eqs and and by replacing the sums
over the rotational states in Eq by integrals it can b e shown that the total
rate of collisional energy transfer simpli es to
X
mol
n v Q n k T T
i th i coll rot g
i
The rotational temp erature is found by iteration until the b oth results for Q
coll
and Q from Eqs and are equal The evaluation of the radiative net
rad
heating rate according to Eq which prop erly includes the optical depth e ects
in the individual lines is thereby carried out over the rst J k T hB
max rot
P P
J
max
u J l J yielding ab out typically rotational states by
l u
J
of the total thermal emission rate in the optically thin limit
Rotational Heating and Co oling by CO
For example the rotational heating co oling function of CO is brie y discussed which
is of sp ecial imp ortance due to its large abundance The molecular data of CO are
outlined in Table and the following physical conditions are considered
dv
km s R n n n n
H H He CO
dl
cont
The density dependence of the Figure depicts the results for the case J
rotational co oling function is generally similar to a two level type co oling function
with the critical densities n cm and n cm for CO cf Eqs
cr thick
and b elow Due to the increasing p opulation of the higher rotational levels
and the smaller radiative life times of these levels however the rotational co oling
function scales as Q T which is di erent from a two level type co oling function
rot
g
Fast Approximate Metho d
For certain applications even the rather simple metho d describ ed ab ove for the
calculation of the rotational heating co oling functions may b e to o time exp ensive
e g for the mo del calculations in Chapter In such cases the following t to the
upp er results can b e used if the continuous radiation eld in the micro wave sp ectral
BOUND BOUND TRANSITIONS
Figure Rotational co oling rate and excitation temp erature of CO in case
cont
Arrows indicate the trend for increasing gas temp erature J
cont
WB T W k T c Rayleigh Jeans approximation region ts like J
rad rad
n n
cr H
Q Q
rot rot LTE
n n
H thick
B
D
n k T WT T Q
mol g rad g rot LTE
c h
B k T
g
D
n
cr
h c v
th
k T dv n
g H
n
thick
B dl n
mol
D
P
n v is the mean thermal velocity with resp ect to the con v n where
i th i th
i
H
centrations of the collision partners Considering typical astrophysical relevant
molecules the critical densities for thermal p opulation of the rotational states n
cr
range b etween ab out cm e g SiS and cm e g HCN
Equation is a very useful t formula with acceptable accuracy error
at the critical densities elsewhere Q is the rotational heating co oling
rot LTE
e
e
function in case of LTE T T and vanishing optical depths which can P
rot g
ul
b e analytically derived from Eq Equation expresses the dep endencies
of the rotational heating co oling function up on the temp erature and the radiation
CHAPTER RADIATIVE HEATING AND COOLING
eld As the rotational frequencies are lo cated in the micro wave sp ectral region
h k hB k K for CO radiative heating via rotational pumping solely
o ccurs in case WT T which seems unlikely to o ccur in circumstellar envelopes
rad g
cf the IR limit in Fig The opp osite case is much more probable the ro
tational transitions will almost always cause net radiative co oling According to
the comparatively weak temp erature dep endence rotational heating co oling is es
p ecially signi cant at low gas temp eratures The relevance of a considered molecule
mol
scales as n B which is imp ortant for the choice of the molecules to b e taken
D
into account
Vibrational Transitions of Diatomic Polar Molecules
The allowed vibrational transitions of p olar molecules also provide an e ective heat
ing co oling mechanism for the gas The vibrational sp ectra of p olyatomic molecules
are already rather complex so that no closed analytical expressions for the mean ra
diative life times and the collision rates are known Therefore this work restricts to
the vibrational transitions of diatomic p olar molecules with selection rules v v
J J for sp ontaneous emission Fortunately these molecules are usually
the most abundant p olar molecules in the gas phase e g CO The vibrational
heating co oling by p olyatomic molecules probably is a second order e ect The
corresp onding wavelengths typically range from m to m
For the level energies the most simple mo del of a harmonic oscillator and a rigid
rotator is applied which is su cient for the purp ose of this work
E h v J J hB
v J
g g
v J J
v v
J
J J
v v
TM A v P branch
J J
h c J
v v
J
J J
v v
A v TM R branch
J J
h c J
X
k T atm
g
n exp B A T C
i i i
g
exp T
g
i
h k
m
red i
A
i
amu
m
red i
B A
i i
amu
v v T
g
v v C exp C
vv
T
g
An exception is the H O molecule in case of an oxygen rich elemental comp osition of the gas
Note that overtone transitions are not considered here cf discussion in Sect
BOUND BOUND TRANSITIONS
v is the vibrational quantum number the eigenfrequency of the harmonic oscillator
and TM its transition moment which is related to the mean radiative life time of
the rst excited vibrational state via A A The analytical
J J J J
representation of the Einstein co e cients is adopted from Nuth Donn The
analytical representation of the rate of collisional de excitation of the rst vibrational
state C is taken from Millikan White The Landau Teller co e cients
A and B are to b e determined by exp eriments or can b e estimated for simple
i i
systems diatomic molecule plus atom or diatomic molecule as collision partner
according to Eqs and The corresp onding collisional cross sections
for vibrational de excitation are much less than the geometric cross sections of the
molecules and show a strong temp erature dep endence The collisional de excitation
rates for higher quantum numbers v v according to Eq are estimated by
surprisal analysis Elitzur
Equations form a useful set of approximate analytical expressions for
the required molecular data in terms of a few basic quantities which are the eigen
frequency the rotational constant B and the transition moment TM or the mean
life time of the rst excited vibrational state resp ectively The rst two data
can easily b e obtained from various molecular data tables whereas the latter is
available only for a few well known molecules from lab oratory exp eriments or ab
inito quantum mechanical calculations Typical values for range from ab out
to Hz The obvious advantage of the analytical expressions ab ove is their broad
applicability to diatomic p olar molecules The disadvantage is the mo dest accuracy
Of course more accurate Einstein co e cients and collisional data can b e used for
particular molecules if available
As in the last section the ro vibrational states are assumed to b e p opulated accord
ing to Boltzmann distributions
Z
vib
exp h k T
vib
mol
n g vh J J hB
J
n exp
v J
Z Z k T k T
vib rot vib rot
The rotational temp erature is considered as known from the calculation of the ro
tational heating co oling function and the vibrational excitation temp erature T is
vib
again found by iteration until the results for Q and Q derived from Eq
rad coll
and Eq are equal Equation is thereby evaluated solely for the vi
brational states and restricted to the rst v k T h vibrational lev
max vib
P P P
v
v
max
els u v l v yielding ab out of the total collisional
l u
v
v
P P P
J
v
max
max
rate Equation is evaluated according to u fv J g
l u
v J
l fv J g which yields of the total thermal emission rate in the
optically thin limit
CHAPTER RADIATIVE HEATING AND COOLING
Figure Vibrational co oling rate and excitation temp erature of CO in case
cont
J
Vibrational Heating and Co oling by CO
In order to illustrate the outlined pro cedure the vibrational heating co oling function
of the CO molecule is calculated The molecular data for CO are given in Table
and the considered velocity gradient and gas abundances are given in the last sec
tion In case of CO more accurate collisional data are available The rate
co e cients for collisions with H atoms are taken from Glassgold see Neufeld
Hollenbach and for collisions with H molecules from Rosenberg et al
see Hollenbach McKee Landau Teller co e cients have b een explicitly mea
sured for the CO He collisions Millikan White
cont
Figure depicts the results for the case J The vibrational co oling rate
essentially is a two level type co oling function and consequently shows all the fea
tures discussed in Sect The higher vibrational levels v are usually
not very signi cant According to the large Einstein co e cients of the vibrational
transitions the maximum p ossible emission rate in the optically thin LTE case
is never realized b ecause the emission is limited either by insu cient collisional
pumping or by radiative trapping which is the typical b ehavior of allowed tran
sitions Consequently the vibrational co oling function directly changes from the
Q to the Q const case at ab out n cm for CO The basic slop e
rad rad
cr
of the temp erature dep endence is the same as depicted in Fig although for
BOUND BOUND TRANSITIONS
the higher vibrational levels cause some mo di cations The temp eratures T
g
main di erence to an ordinary two level type co oling function arises from the weak
sensibility of the vibrational heating co oling to optical thickness since the emit
ted photons are spread among the ne structure of the P and R branch of the
vibrational band
Fast Approximate Metho d
Similar to the rotational heating co oling function in the previous section a fast t
formula is designed which can b e applied in time critical mo del calculations It is
assumed that the background radiation eld is constant over the vibrational band
cont
and equals J
n n
cr H
Q Q
vib vib LTE
n n
H thick
cont
J
h n
mol
Q
vib LTE
exp T B T
g g
n
H
n
cr
C
k T h n dv
g H
n
thick
dl hB hc n
mol
p
n n n
cr thick
cr
and b etter than The accuracy of formula is ab out at n n
H
cr
elsewhere The dep endency of the vibrational heating co oling function on the ra
diation eld is expressed by Eq once more indicating that radiative heating
cont
B and radiative co oling otherwise Q is the energy o ccurs in case J
vib LTE
e
e
exchange rate in case of LTE T T and negligible optical depths P
vib g
ul
Although this maximum p ossible rate is usually not realized see ab ove it scales
the results as formulated in Eq As far as n n is valid the vibrational
cr thicks
heating co oling rate is almost entirely indep endent from the mean life time This
allows for the determination of the vibrational heating co oling rates also of those
diatomic p olar molecules for which the values are not exactly known Consider
ing typical values for and C for diatomic p olar molecules and gas temp eratures
K the critical densities for thermal excitation of the vibrational states
n are of the order cm which due to radiative trapping are usually
cr
signi cantly reduced n n by up to orders of magnitude Considering the
cr
cr
Consequently strong non LTE e ects concerning the p opulation of the vibrational states of
p olar molecules can b e exp ected in circumstellar envelopes in contrast to the p opulation of the
rotational states Chemical reactions might b e a ected by these e ects since many reactions are
extremely temp erature dep endent and the vibrational energies of the reactants may b e involved
This situation may have severe consequences for the chemistry and also for the nucleation of dust
grains in these envelopes A rst approach to handle reactants of di erent temp eratures has b een
presented by Cherchne et al
CHAPTER RADIATIVE HEATING AND COOLING
mostly realized case Q the imp ortance of a molecular sp ecies under exami
vib
nation concerning its contribution to the total heating co oling of the gas scales as
n h C
mol
Quadrup ole Transitions of H
Unp olar molecules may usually b e neglected considering the total energy exchange
b etween the gas and the radiation eld b ecause these molecules have extremely
small radiative transition probabilities The H molecule however may b e su
ciently abundant in order to comp ensate for this Its ro vibrational quadrup ole
transitions are known to b e signi cant in warm interstellar clouds and are lo cated
roughly b etween m and m
The radiative heating co oling function of H is calculated analogously to Sect
Since no analytical expressions are available an extensive list of individual transi
tion probabilities must b e used which means a much larger exp ense compared to
Sect The level energies are derived from the sp ectroscopic constants
E v J hc w v w x v B J J D J J
e e e v e
D cm w x B v
e e e v
as given by Hub er Herzb erg and the Einstein co e cients for sp ontaneous
emission of the forbidden ro vibrational quadrup ole transitions v v J
fJ J J g are taken from Turner et al where all transitions with v
and J are taken into account comprising pure rotational and ro
vibrational transitions
The collisional vibrational de excitation rates for H atoms and H molecules
are adopted from Lepp Shull and references therein Those for He atoms
are estimated according to Eq with A and B The
He He
collisional rates for the higher vibrational states are again estimated according to
Eq The collisional de excitation of the rotational states is not considered
in detail here instead the rotational temp erature of H is assumed to equal
the gas temp erature According to the calculations of Lepp Shull this
approximation is reliable unless the gas density is lower than cm
If the optical depths in the lines are neglected as assumed in the work of Lepp Shull
their results can b e repro duced within a maximum factor of generally
much b etter for all considered gas temp eratures and for densities larger than
cm proving that the presented metho d including the introduction of exci
tation temp eratures works prop erly Figure shows the results once more for the
D E
dv
cont
case J n n n and km s R According
H H He
dl
to the assumption T T the rotational co oling rate is prop ortional to for all
rot g
cm where even the quadrup ole transitions b ecome densities unless n
H
optically thick The vibrational co oling rate is more imp ortant for high temp era
tures T K and high densities where it exceeds the rotational co oling rate g
BOUND BOUND TRANSITIONS
Figure The total quadrup ole co oling rate thick full lines the vibrational
co oling rate thin full lines the rotational co oling rate dotted lines left axis
and the vibrational excitation temp erature dashed lines right axis of H in case
cont
J
by ab out one order of magnitude The critical density for thermal p opulation of the
vibrational states of the H molecule strongly dep ends on the gas temp erature and
ranges from to cm
The total contribution of H heating and co oling roughly stays prop ortional to the
gas density over the whole considered density range of circumstellar envelopes with
an accuracy of ab out one order of magnitude This b ehavior is a natural conse
quence from the low transition probabilities of the quadrup ole transitions and is
typical for forbidden lines In comparison to other heating co oling rates which
decrease as Q for small densities the H quadrup ole heating co oling is esp e
rad
cially signi cant at low density e g interstellar conditions
CHAPTER RADIATIVE HEATING AND COOLING
Bound Free Transitions
Bound free transitions photoionisation and radiative recombination generally pro
vide imp ortant heating and co oling rates as so on as considerable fractional ionization
is present in the gas which o ccurs in the following two cases
i A strong UV radiation eld is present which causes b oth photoionisation
and net radiative heating of the gas This case is generally realized in the
overwhelming part of the ISM except for the dense and shielded molecular
clouds where the interstellar UV radiation eld interacts with the gas
ii The gas is dense and hot so that collisional ionization causes considerable
fractional ionization High temp eratures K for hydrogen are usu
ally required for e ective collisional ionization which followed by radiative
recombination preferably causes net co oling of the gas the details however
dep end on the relation b etween the gas temp erature and the present UV ra
diation eld see b elow The comp etitive pro cesses of collisional ionization
and three b o dy recombination are furthermore resp onsible for keeping the ion
ization equilibrium close to LTE in stellar atmospheres In return the large
b ound free opacities in the case of LTE control the radiative transfer and the
radiative heating and co oling of the gas as e g in the atmospheres of hot stars
Considering the physical conditions in CSEs large fractional ionization esp ecially
o ccurs around warm and hot stars where the photospheric UV radiation eld is
already su ciently intense to cause considerable b ound free radiative heating can
b e exp ected
The conditions in the predominantly neutral CSEs of co ol e g AGB stars do
generally not favor large b ound free heating co oling rates There are however
imp ortant exceptions from this rule First if chromospheric activity is present
radiative heating by b ound free transitions of the gas can b e e ective Second if the
interstellar UV radiation eld can p enetrate into the considered layer of the CSE it
will cause considerable fractional ionization and radiative heating Third concerning
the hot p ost sho ck gas layers in the CSEs of pulsating stars collisional ionization
followed by radiative recombination can b e an imp ortant co oling pro cesses
The Rate Equations for an N Level System with Continuum
A level system consisting of N b ound electronic states plus one additional level
denoted by II for the rst ionized state of the considered sp ecies is exam
ined Besides the b ound b ound pro cesses discussed b efore the pro cesses of pho
toionisation radiative recombination collisional excitation and three b o dy recom
bination are taken into account Analogously to Sect the level p opulations
n n n n are derived from the statistical equations assuming that the
N II
net pro duction rates of all considered states vanish steady state non LTE This
BOUND FREE TRANSITIONS
assumption is more restrictive in this section b ecause the time scale for relaxation
P P
n R can b e much larger com R towards ionization equilibrium n
i i II II i II
i i
pared to the time scales for relaxation of the excited b ound states under certain
circumstances e g in the case of low fractional ionization and might exceed other
e g hydrodynamic time scales Furthermore the level system is assumed to b e
closed in the sense that other pro cesses which might provide additional source
terms for the particle densities of the neutral and singly ionized atoms e g chemical
reactions charge exchange reactions are neglected
The rate co e cients for the b ound b ound transitions are given by Eqs and
whereas those for the b ound free transitions are formulated according to
Mihalas
Z
J
bf
d n T R
e i g i II
i
h
i
thr
Z
B C
J n
e
bf
h
B C
exp d n T R
e i g II i
i
A
k T
g
S T c h
i g
i
thr
m k T Z
i e g II
exp S T
i g
g h k T
i g
bf
i lab els a b ound state is the b ound free absorption cross section and T
i g
i
the rate co e cient for collisional ionization we only consider collisions with electrons
i
here is the energy di erence b etween the i th level and the continuum
i
thr
h the corresp onding threshold frequency S T the Saha function and Z the
i i g II
partition function of the ionized state
The total radiative heating co oling function of such a multi level system com
prises contributions from b ound b ound transitions which are calculated according
to Eq and from b ound free transitions
Z
N
X
n n h
II e
bf bf
h
n J Q J exp d
i
rad i
k T
g
S T c
i g
i
i
thr
It is imp ortant to note that the evaluation of the radiative heating co oling rate de
p ends on the de nition of the internal energy In this work the total absorb ed emitted
photon energy is calculated and the ionization energies o ccur as p otentials in the
i
internal energy cf Eq Concerning other publications the radiative heating
and co oling rates o ccasionally refer to the p o ol of translational energy alone In
this case no internal ionization p otentials are considered but an additional factor
h h app ears in Eq describing the gain or loss of pure translational
i
energy
Besides the data for the level energies the statistical weights g and the partition
i i
bf
function Z only and T are required for the calculation of the b ound free
II i g i
CHAPTER RADIATIVE HEATING AND COOLING
radiative heating co oling functions The so called photo recombination co e cients
are principally not needed since they can b e deduced from the Einstein Milne
relations for b ound free transitions which are already included in Eqs
For the actual solution of the outlined system of equations all integrals are evaluated
numerically The solution of the statistical equations for the level p opulations
including n requires an inner iteration of the escap e probabilities of the b ound
II
b ound transitions where the same pro cedure as outlined in Sect is applied
The system of equations is well de ned for given total particle density n n
sp II
P
n given temp erature T given radiation eld J and given electron density n
i g e
Another outer iteration is necessary to achieve the physical condition of charge
P
conservation n n comprising all ions under consideration which in return
e II
yields the electron density
According to the outlined equations the degree of ionization of the gas and the
b ound free heating co oling rates are calculated simultaneously Optical depths ef
fects are not included concerning the b ound free transitions in contrast to the
b ound b ound transitions discussed b efore where it was p ossible to apply Sob olev
theory This problem could only b e handled by means of non lo cal non LTE
radiative transfer calculations Since the basic approach of this work is to determine
the radiative heating and co oling of single gas elements we ignore these e ects
cont cont
assume the gas to b e optically thin in the continuum and put J J where J
is the continuous background radiation eld
Fast Approximate Metho d
A useful and quite illuminating form for the general b ound free rates and heat
ing co oling functions can b e derived by introducing the photo recombination co ef
cients as
Z
bf b
h
i
T exp d a T
i i
i
k T
S T c
i
i
thr
The second part of Eq provides a common t formula where the parameters
a and b are o ccasionally stated in the literature As far as stimulated b ound free
i i
emission can b e neglected which usually is a very accurate approximation in the
UV the recombination rates can b e re written as
n
e
R n T T
II i e i g i g
S T
i g
The photo recombination co e cients however are very useful for quick approximate calcu
lations see b elow
Such optical depth e ects are exp ected to reduce the b ound free heating co oling rates and
drive the gas towards LTE ionization already at comparatively smaller gas densities
Note however that the corresp onding wavelengths of recombinations to highly excited states
can b e lo cated in the optical or even IR sp ectral region
BOUND FREE TRANSITIONS
h
If the continuous radiation eld ts like J WB T W h c exp
rad
k T
rad
Wien approximation also the ionization rates can partly b e expressed in terms of
the photo recombination co e cient
R WS T T n T
i II i rad i rad e i g
By determining the derivative k T from Eq it can b e shown that
i
the net b ound free heating rate then reduces to
N
X
abs em bf
n WS T T hh i n n T hh i Q
i i rad i rad II e i g
i i rad
i
abs
hh i b k T
i i rad
i
em
b k T hh i
i i g
i
abs em
where hh i is the mean absorb ed and emitted photon energy resp ectively
Equation is exact as far as the upp er conditions are valid and the derivative
dZ dT can b e neglected The big technical advantage of Eqs and
II
bf
is that no integrals o ccur and that instead of a function only two
i
parameters a and b have to b e known for each considered b ound free transition
i i
Equation demonstrates that even if the number of b ound free absorb ed pho
tons equals the number of free b ound emitted photons as in the case of negligible
collisional ionization the net rate of transferred energy do es usually not vanish in
contrast to all b ound b ound type transitions discussed in the previous section The
reason lies within the integration over the absorb ed emitted photon sp ectrum since
the mean absorb ed photon energy usually di ers from the mean emitted photon
energy In the case of thermo dynamic equilibrium however where J B T and
g
n n n S the net b ound free radiative heating rate according to Eq or
i II e i
according to Eq is indeed zero as demanded by detailed balance
The most simple case o ccurs if solely the ground state of the neutral atom is con
sidered and if the collisional ionization rates are neglected From the condition of
steady state n R n R it follows that in this case the net heating co oling rate
II II II
simpli es to
bf
n n T b k T T Q
II e g rad g
rad
In this case radiative heating o ccurs for T T and radiative co oling otherwise
rad g
indep endent of the value of the dilution factor W which corresp onds to the UV
limit depicted in Fig
CHAPTER RADIATIVE HEATING AND COOLING
The H Atom
According to its overwhelming abundance hydrogen is always imp ortant for b oth
the total degree of ionization and the radiative heating and co oling of the gas
However the high lying energy levels of hydrogen make it almost inaccessible for
collisional excitation and collisional ionization at lower temp eratures so that the
imp ortance of hydrogen is mainly restricted to high temp eratures
For demonstration a pure hydrogen plasma is examined in the following consisting
of the rst three b ound levels and the ionized state The following data of hydrogen
and physical conditions are considered
n
G
II
bf
eV n g n Z
n n II
n
n
dv
n n n n n n n km s R
H II e II
dl
The b ound free absorption cross sections are taken from Mihalas with the
Y n
abbreviation X Y X in cgs units G are the b ound free Gaunt factors
II
which are of the order of unity The collisional ionization rate co e cients T are
n g
taken from Luttermoser Johnson and the references therein The hydrogen
b ound free transitions II Lyman continuum II Balmer continuum and
II Paschen continuum are calculated by means of the exact equations given in
Sect The treatment of the hydrogen b ound b ound transitions Ly
H and H has b een describ ed in Sect
Figures and show the resulting total b ound free plus b ound b ound co oling
rates of hydrogen Two gures are presented here since the degree of ionization and
the heating co oling rates strongly dep end on the continuous back ground radiation
eld which is chosen to b e zero in the rst and to equal a Planckian of K in
the second gure Note the scaling of the y axis which is di erent from the other
plots b efore
Compared to the magnitude of the heating co oling rates discussed so far hydrogen
heating and co oling is found to b e unimp ortant for T K However for higher
g
gas temp eratures hydrogen co oling so on b ecomes e cient and nally hydrogen
provides the dominant co oling rate of the gas at temp eratures ab ove K
This temp erature dep endency is a consequence of the high lying energy levels of
hydrogen which can b e collisionally excited or ionized solely in the case of high gas
temp eratures
The total hydrogen co oling rate is found to scale roughly as Q which is an
indicator for strong non LTE e ects in the level p opulations
Luttermoser et al have shown that a three level mo del for hydrogen is su cient in co ol
stellar environments for describing accurately b oth the emergent H sp ectrum and the contribution
of hydrogen to the electron density
LTE without optical depth e ects implies Q for sp ectral lines as already discussed in
Sect Concerning the b ound free heating co oling functions we have n n S n in LTE
II e i i
which according to Eq also implies Q as far as hydrogen is mostly neutral
BOUND FREE TRANSITIONS
Figure The total b ound free plus b ound b ound hydrogen co oling rate
full lines left axis and the degree of ionization dashed lines right axis in the
case without continuous radiation eld
Figure Same as Fig but with an underlying continuous radiation eld
CHAPTER RADIATIVE HEATING AND COOLING
The degree of ionization in Fig shows a step like b ehavior This is an e ect
caused by the varying optical depths of the hydrogen lines With increasing gas
density Ly and for larger densities also H b ecome optically thick Consequently
the e ective radiative b ound b ound rates according to Eqs and b e
come negligible compared with the collisional rates forcing the upp er level of the
considered transition to achieve thermal p opulation with resp ect to the lower level
Therefore the collisional ionization rate from that upp er level is increased by or
ders of magnitude leading to successively enhanced electron concentrations from
the right to the left in Fig In Fig this b ehavior is smeared out since the
rates for photoionisation enter into comp etition
The hydrogen net co oling rates in the case J B K depicted in Fig
are found to b e larger than in the case J since photoionisation pro duces
considerably higher electron concentrations providing more collision partners
Further details concerning the relative contributions of the di erent transitions and
the level p opulations are presented in Fig considering the case T K and
g
J The b factors for departures from LTE are calculated as
n S T n b n n
i i g i i
e i
n
H
q
n
e
n S S S
H
Hydrogen b ound free co oling is found to dominate in hot dense media whereas
emission in hydrogen lines dominates the co oling of a hot thin gas which is a
straightforward consequence of the increasing optical depths in the hydrogen lines
with increasing gas density The transition b etween these two cases o ccurs at a
particular density which dep ends on the considered gas temp erature and velocity
gradient In Fig this transition density is ab out n cm The bumps
H
on the total co oling rates depicted in Fig corresp ond to these transitions
The Lyman continuum always provides the most imp ortant b ound free co oling
rate The relative contributions of the other continua with resp ect to the Lyman
continuum scale as n which can b e analytically derived from Eq
Ly is usually the most imp ortant hydrogen co oling line as has already b een p ointed
out by Hollenbach McKee and Neufeld Hollenbach although Ly
e
e
is optically thick for all considered densities P n for the chosen velocity
H
gradient However for large densities H b ecomes more e cient than Ly b ecause
the H transition do es not involve the ground level and hence is not so much
a ected by optical thickness Therefore there is in fact a small density interval
where H is the most e cient co oling pro cess already more imp ortant than Ly
and still more imp ortant than b ound free transitions around n cm in
H
This is di erent from all co oling rates discussed so far since we have not yet considered a
change of the concentrations of the collision partners If the concentration of the collision partners
are constant an increase of the background continuous radiation eld always implies reduced net
co oling rates and nally causes net radiative heating
BOUND FREE TRANSITIONS
Figure Details for the case T K and J Upp er panel rela
g
tive contributions of the di erent b ound b ound and b ound free transitions
Lower panel b factors for the hydrogen levels
Fig Ly is also always optically thick similar to Ly and hence always much
less imp ortant than H
The lower panel of Fig shows the gradual change from almost LTE ionization
and LTE population b at large densities to pronounced non LTE conditions at
n
small densities caused by the decreasing relevance of the collisional pro cesses The
b factor indicates that the degree of ionization of hydrogen is always sub thermal
II
provided that T T for a Planck eld J B T Consequently the ground
rad g rad
state is p opulated hyper thermally which is an imp ortant result for the CSEs of co ol
stars since it keeps the gas predominantly neutral also at fairly high temp eratures
and low densities where hydrogen would b e strongly ionized according to LTE
The p opulations of the excited hydrogen levels are completely decoupled at small
densities and are thermally coupled to the ground state for large densities caused by
the strongly decreasing escap e probabilities of the lines photons LTE p opulation
is established in direction of increasing gas densities successively from the lower
to the higher excited levels nally also for the ionized state For complete LTE
ionization however extremely large densities are required e g n cm
H
for T K where the three b o dy recombination rates b ecome relevant g
CHAPTER RADIATIVE HEATING AND COOLING
The situation in the case J B K is quite di erent Here the net b ound free
co oling rates dominate over net b ound b ound co oling rates unless the gas density is
smaller than cm for all considered temp eratures Negative and p ositive net
contributions from the di erent transitions may o ccur at the same time although
the sum of all contributions always results in a net co oling
To summarize hydrogen is mainly an imp ortant high temp erature co olant approx
imately contributing as Q For large densities the Lyman continuum is most
e ective whereas at small densities Ly dominates
Other Neutral Atoms
Concerning other atoms than hydrogen solely the electronic ground states of the
neutral atoms are considered in this work for practical reasons Furthermore since
all b ound free transitions from the ground states are lo cated in the UV the ap
proximate metho d outlined on p can b e used for the calculation of the photo
recombination rates and the b ound free heating co oling functions The approxi
mate photo recombination rates derived from Eq are found to show reason
able agreement with those calculated from Eq when applying the photoioni
sation cross sections of the various metal atoms from Schmutzler Therefore
the application of the approximate metho d is rather accurate and very practical
since it avoids the time consuming numerical frequency integrations
The rates of collisional ionization are determined from the analytical expression
given by Allen
q
T o T eV exp
g g
k T
g
where o is the number of optical electrons of the neutral atom Table summarizes
the data used for the determination of the the b ound free heating co oling rates
At the end of this section the imp ortant features of the developed metho ds for the
b ound free transitions are once more summarized The metho ds are used for the
determination of the electron concentration the concentrations of the various ions
and the calculation of the b ound free heating and co oling rates
Ionization equilibrium steady state non LTE is assumed to determine the
particle densities of the considered atoms ions and electrons The rates of
photoionisation and recombination collisional ionization and three b o dy
recombination are taken into account for each atom
A couple of simplifying assumptions are used for other atoms than hydrogen
Hydrogen is treated more accurately including of the rst two excited levels
Bound free optical depths e ects are ignored
PHOTODISSOCIATION AND RADIATIVE ASSOCIATION
Table Bound free heating and co oling considered sp ecies and atomic data
Sp ecies eV g Z a b o
II
He
C
N
O
S
Mg
Si
Fe
Na
Allen
Beck and references therein
For simplicity the partition function Z is approximated by the
II
statistical weight of the ground state of the ionized atom
The number of optical electrons is the number of electrons in
the last o ccupied quantum state
In conclusion the outlined metho ds are approximate but simple and applicable to
the wide range of density conditions present in circumstellar envelopes The gradual
change from almost LTE ionization at large densities to non LTE ionization at small
densities can b e repro duced
Photo disso ciation and Radiative Asso ciation
Radiative gains and losses of the gas can also b e caused by chemical reactions
According to the de nition of the internal energy in this work cf Fig solely
those reactions which are accompanied by an absorption or emission of a photon
photo disso ciation or radiative asso ciation contribute to the radiative heating or
co oling resp ectively
The disso ciation p otentials of the molecules of astrophysical interest typically range
from to eV exceptions CO eV and N eV which already gives a
rst impression of the concerned wavelength region of the radiative pro cesses un
der investigation Compared to typical molecular ionization p otentials of more than
ab out eV the disso ciation energies are substantially smaller Thus as far as hard
Pure gas phase reactions which do not involve photons do not contribute to the radiative
heating co oling of the gas even if they are exothermic or endothermic Such reactions only
convert disso ciation p otential energies into translational ro vibrational and electronic excitation
energies and might b e considered as additional source terms for these p o ols but do not directly
a ect the total internal energy of the gas
CHAPTER RADIATIVE HEATING AND COOLING
UV radiation is absent photo disso ciation is exp ected to b e more e cient for the
heating of the gas than photoionisation in the molecular domain of circumstellar en
velopes even if the corresp onding photo cross sections are smaller In the following
a photo chemical reaction of prototype
k
f
AB h A B
k
r
is considered where A and B lab el an atom ion molecule or electron and AB the
corresp onding comp osite sp ecies k and k are the rate co e cients of the forward
f r
and reverse reaction resp ectively Irresp ective of the fact that photo disso ciation is
mostly initiated by absorption in electronic bands e g the Lyman and Werner
bands of H which are in principle narrow spaced b ound b ound transitions the
f
photo disso ciation cross sections are assumed to b e given in a quasi continuous
way
Z
J
f
k d
f
h
n n k it is found From the detailed balance consideration n k
r f
A B AB
J B
Z
E B T n
a
AB
f
d A T exp k T
r
n n h k T
A B
T
The second part of Eq is the usual Arrhenius law for the backward reaction
is the particle with E b eing the activation energy Coming back to the rst part n
a
X
density of sp ecies X in chemical equilibrium i e the rst fraction in Eq can
b e determined by means of the law of mass action from the corresp onding free
enthalpy of formation at standard pressure p
n k T G T
f
AB
exp
n n p k T
B A
T
G T G T G T G T
f f f f
AB A B
The contribution of the photo chemical reaction to the total net heating func
tion of the gas is given by
Z
n
AB
chem f
A
n J n n Q B T d
AB A B g
rad
n n
B A
T
g
n are the actual particle densities which in this context have to b e determined from
X
the steady state solution of a chemical reaction network Esp ecially interesting are
those reactions which have the largest net photo rates this do es not necessarily
imply that the involved molecule has b e to abundant
The general problem of the treatment of these pro cesses is the large amount of
di erent sp ecies to b e considered and the p o or availability of appropriate molecular
PHOTODISSOCIATION AND RADIATIVE ASSOCIATION
f
data Therefore the so far outlined equations do not lo ok very promising
since they can solely b e applied to a very few well known molecules e g H CO
This problem can b e avoided by the following consideration analogously to the
approximate metho d for the b ound free transitions discussed in the last section We
assume the continuous radiation eld to t like J WB T and consider the
rad
case maxfk T k T g D h applying Wien s law as b efore By di erentiating
g rad
AB
Eq with resp ect to k T the following expression can b e derived
n n
A B
chem abs em
Q n W k T hh i n n k T hh i
AB r rad A B r g
rad
n
AB
T
rad
abs
hh i G T E k T
f rad a rad
em
hh i G T E k T
f g a g
Compared to Eq this expression for the net heating rate of a considered
photo reaction can easily b e applied to the results of chemical reaction networks
since only the Gibbs energies G and the Arrhenius co e cients A and E
f a
have to b e known The considerations nd the mean absorb ed and emitted photon
energies to b e of order G E k T D E i e the molecule disso ciation
f a a
AB
energy plus the activation energy of the radiative asso ciation reaction The net
heating rate vanishes in the case of thermo dynamic equilibrium as exp ected
For a comprehensive discussion of the imp ortance of photo disso ciation and radiative
asso ciation for the thermal balance of the gas in CSEs the steady state results of
chemical reaction network calculations are required providing the various concentra
tions of the sp ecies under examination Such investigations go b eyond the scop e of
this work and must b e left to future investigations However an imp ortant example
is considered in the following section
The H Heating Co oling Rate
The negative hydrogen ion H shows exceptionally large photo rates in circumstellar
envelopes Beck Its b ound free and free free transitions are furthermore
well known to b e the most signi cant contributor to the gas opacity in the stellar
atmospheres of warm stars as the sun Therefore it is imp ortant to consider the
radiative heating and co oling by H in more details
The concentration of H in circumstellar envelopes is mostly controlled by the fol
lowing two reactions Beck et al
k
f
H h H e
k
r
k
f
H H H e
k
r
f
Considering the frequency integration in Eq the soft end of almost entirely deter
mines the rates and the heating co oling function di cult to measure in lab oratory exp eriments
CHAPTER RADIATIVE HEATING AND COOLING
Thus the concentration of H in steady state kinetic equilibrium is always
prop ortional to the electron concentration
k n k n
r H r H
n n
e
H
k n k
f H f
Reaction H b ound free is the pro cess to b e considered for the radiative
heating and co oling of the gas The ab ove outlined metho ds are straightforwardly
applied although the b ound free transitions of H are classi ed as photoionisation
and radiative recombination rather than as photo disso ciation and radiative asso ci
ation The reaction rate co e cients k and k are calculated by means of the
f r
exact Eqs and where A lab els the H atom B the electron and AB
the negative ion H The b ound free absorption cross section of H is interpolated
from tables given by Wishart Stimulated recombinations are treated in LTE
for simplicity The rate co e cients of the second reaction are taken as
s Schmetekopf et al k cm
f
n n
H
H
k k
r f
n n
H e
Considering a gas of solar elemental abundances the required particle densities
n n and n are determined by means of the metho ds outlined in Chapter
e H H
density is calculated afterwards from Eq Accordingly collisional The n
H
ionization and photoionisation of metal atoms with low ionization p otentials Na
Mg Fe are imp ortant low temperature electron donators and provide electron
concentrations of at least for T K leading to considerable H particle
g
densities
The radiative heating and co oling by H comprises b ound free and free free contri
butions The b ound free heating co oling rate is calculated according to Eq
and the free free heating co oling rate is determined by
bf
Q H Q H Q H
rad
rad rad
Z
Q H n n J B T d
H e g
rad
where the free free cross section is tted on the dip ole length calculations of
Stilley Callaway
T h
g
exp cm
k T
g
This is an approximate pro cedure since it neglects the feedback on the former particle densities
In contrast the consideration of a pure hydrogen plasma leads to a systematic underestimation
of the H concentration and heating co oling rates The resulting electron concentrations are
smaller in this case esp ecially around T K just where the heating and co oling of H turns
g
out to b e most signi cant
Kirchho s law B T is applicable to free free transitions also in non LTE since they
g
solely refer to the thermal motion of the gas
PHOTODISSOCIATION AND RADIATIVE ASSOCIATION
Figure The total b ound free free free co oling rate full lines the
free free co oling rate short dashed lines left axis and the concentration long
dashed lines right axis of H in the case without continuous radiation eld
Figure Same as Fig but with underlying continuous radiation eld
CHAPTER RADIATIVE HEATING AND COOLING
Figures and depict the results for the two cases J and J B K
In b oth cases the radiative co oling rate of H scales as Q n n which implies at
H e
least Q Therefore the radiative heating and co oling of H is only imp ortant
for a dense medium e g in stellar atmospheres The step like b ehavior of the H
concentration and total co oling rates for T K and T K in Fig
g g
corresp ond to the step like degree of ionization of hydrogen cf Figs and
whereas for lower gas temp eratures the electron concentration is controlled by the
metals with low ionization p otentials For even lower gas temp eratures not shown
in the gures hydrogen is mostly lo cked in H and the concentration and the
heating co oling rates of H rapidly vanish
The calculated H concentrations for the case J B K cf Fig are
considerably smaller due to the large photoionisation rates k However since the
f
the co oling rates radiative co oling rates of H are related to n n and not to n
H e
H
remain similar The di erences b etween Figs and are mainly caused by the
di erent electron concentrations Free free heating co oling of H is always found to
b e less imp ortant than b ound free heating co oling For even more intense radiation
elds not shown the b ound free transitions e ectively destroy the negative ion
so that radiative heating by H is rarely found to b e signi cant only if an active
chemistry at large densities quickly restores the H ions
In conclusion H is mainly an imp ortant co olant for large densities and medium
temp eratures where the pro duct of electron and atomic hydrogen density is large
Free Free Transitions
If the gas is almost fully ionized and the density is large n cm free
H
free emission Bremsstrahlung b ecomes an e ective co oling pro cess We use the
ordinary expression for free free emission for a partially singly ionized gas given in
Allen and include for consistency the reverse pro cess of free free absorption
by means of the relation B T
g
n h
e
q
T exp
g
k T
g
T
g
Z
J
d Q T
g
rad
B T
g
Free free transitions always concern the whole electromagnetic sp ectrum Conse
quently free free transitions principally provide one of lasting p ossibilities for ra
diative heating if the incident radiation eld mainly consists of IR photons where
other radiative heating pro cesses b ecome imp ossible However the gas must b e
considerably ionized Q n for such heating
rad e
OVERVIEW OF THE CONSIDERED RADIATIVE PROCESSES
Overview of the Considered Radiative Pro cesses
Figure summarizes the radiative heating co oling functions considered in this
work To what extend this selection is complete must b e left to the reader s dis
cretion The shown sp ectral p ositions of the various radiative pro cesses already
provide a rst impression of their net e ect and their imp ortance for the heating
and co oling of the gas In general radiative heating o ccurs in the case J B T
g
and co oling in the opp osite case For the depicted case of a diluted Planck type
radiation eld the radiative pro cesses at short wavelengths b ound free transitions
sp ectral lines are resp onsible for radiative heating and those at long wavelengths
vibrational and rotational transitions sp ectral lines for co oling Considering the
thermal relaxation of the gas towards radiative equilibrium the gas temp erature
will tune in such a way that these gains and losses balance each other Note that
the formation of molecules in the gas intensi es the interaction b etween matter and
radiation eld at long wavelengths thus reinforcing radiative co oling and conse
quently leading to lower radiative equilibrium temp eratures this e ect can in fact
cause thermal bifurcations in the gas as discussed in Chapter
As part of this summary the imp ortant features of the developed metho ds are once
more placed together
All considered radiative pro cesses are treated in non LTE The non LTE de
scription of the molecules is restricted to individual vibrational and rotational
excitation temp eratures The heating co oling rates are calculated in steady
state
All corresp onding reverse pro cesses are taken into account relying on detailed
balance considerations Consequently each considered pair of forward and
reverse pro cess can app ear as b oth net radiative heating and co oling Which
case actually o ccurs dep ends up on a sp eci c relation b etween the gas temp er
ature and the radiation eld In the case J B T as in thermo dynamic
g
equilibrium all discussed net heating co oling rates vanish Q
rad
J Es The heating co oling rates are formulated for arbitrary radiation elds
p ecially simple expressions are derived for diluted Planck elds of type J
WB T
rad
are included for all b ound b ound type transitions and Optical depths e ects
neglected otherwise
These relations are not exact but usually correct also in non LTE since the corresp onding
B T S source functions considering e g a two level atom generally satisfy J
g
The t parameters W and T can b e di erent for di erent sp ectral regions e g UV and IR
rad
as far as the considered heating co oling pro cesses merely refer to such region
CHAPTER RADIATIVE HEATING AND COOLING
Balmer continuum m
Paschen continuum m
H b ound free m
H ro vib quadrup ole ca m
free free transitions all wavelengths
Figure Overview of the considered heating and co oling pro cesses The full
line shows an assumed continuous mean intensity J according to Eq with
W r R for pure radial dilution and T K The dashed line
rad
is the Planck function for T K The arrows indicate the energy exchange
g
b etween matter and radiation eld favoring radiative heating at short and ra
diative co oling at long wavelengths resp ectively The lower panel indicates the
wavelength regions of the considered heating co oling pro cesses
FURTHER HEATING AND COOLING PROCESSES
Further Heating and Co oling Pro cesses
The theoretical part of this work ceases with some remarks on those heating and
co oling pro cesses which have not b een taken into account
Of course the prop er inclusion of all radiative pro cesses is principally desirable The
construction of such an utmost complete set however is a long lasting pro cess
and cannot b e carried out by one single work The selection of heating co oling
functions of this work may b e extended in two ways First to include a larger
number of pro cesses more sp ecies more lines etc of the already considered types
of pro cesses and second to take into account further types of pro cesses
A pure quantitative extension will probably not lead to substantial changes com
pared to the forthcoming results of this work since the most promising candidates of
each considered type of pro cess are already included What may b e crucial however
are the additional types of pro cesses not considered so far which might prove to
b e imp ortant under certain circumstances As the author started to study a few of
them more or less serious sp eci c obstacles o ccurred which prevent a simple quan
titative discussion for the time b eing Together with some valuations and remarks
these obstacles shall b e xed in this section
Table lists some interesting candidates of di erent types of pro cesses the ex
p ected sp ectral region of absorb ed emitted photons the exp ected e ect for the gas
the faced obstacles for the determination of the corresp onding heating co oling rates
and some comments Extensive explanations are not included as Table is mainly
given for reasons of completeness and to op en the discussion Some additional com
ments however are necessary concerning the energy gains and losses caused by the
presence of dust grains
Due to the large time scales involved in the dust formation pro cess the dust com
p onent here esp ecially the total dust surface should b e treated time dep endently
e g Fleischer et al and cannot b e determined by any steady state consider
ations Therefore the heating and co oling pro cesses caused by the presence of dust
are not explicitly included in this work
Dust grains provide a similar external p o ol as the radiation eld see Fig
Consequently energy transfer rates directly o ccur by collisions The corresp ond
ing rates can easily b e added to the total net radiative heating rate of the gas if
the appropriate informations ab out the dust comp onent total surface dust tem
p erature drift velocities etc are available The rates for thermal accommo da
tion energy transfer via inelastic gas dust collisions are given for example in
Burke Hollenbach and those for drift heating energy transfer via gas colli
sions with moving dust grains caused by radiation pressure in Goldreich Scoville
or in Kruger et al
More di cult to determine are the heating co oling rates caused by surface chemical
reactions which principally exchange all types of gas internal energies esp ecially
disso ciation and ionization p otentials with the dust comp onent Very detailed
knowledge ab out these reactions is required
CHAPTER RADIATIVE HEATING AND COOLING
Table Overview of further heating and co oling pro cesses not explicitly considered in
this work
sp ectral
region
pro cess imp ortant for what obstacles comments
estimated
number of trans
photo rad heating in concurring
several eV
disso ciation molecular domain reaction channels and
rates
rad heating in
probably strong
number of trans rad molecular domain
electronic non LTE e ects
several eV lifetimes coll rates maybe imp ortant
trans of short rad lifetimes
line overlaps counterbalance for rot
molecules typically s
and vib co oling
and small coll rates
unp olar p olyatomic
vib trans selection rules rad
molecules may b ecome
of lifetimes analytical
m heating and co oling
p olar during
p olyatomic expressions coll
vibration
molecules rates
A co e s ab out one
rad lifetimes
vib
magnitude smaller
analytical
m heating and co oling
overtone
than for ground tone
expressions
trans
trans
selection rules rad
rot trans
lifetimes analytical
GHz co oling
of non linear
expressions
molecules
heating and co oling
bf trans
indirect
complete mo del
several eV
from excited
de excitation of
atoms
states
b ound levels
bf and
trans of
concurring
several eV heating and co oling
negative
reaction channels and
ions
rates
molecules
indirect excitation of
reaction rates
electronic and vib
reaction heats energy
di cult co oling
gas phase
levels of reaction
distribution among
approximately
dep ends
chemical
pro ducts probably
the various degrees of
pro ceeds on chemical
reactions
followed by emission
freedom on the
time scale
i e co oling
reactants
bf b ound free free free rad radiative rot rotational vib vibrational A
co e s Einstein co e cients for sp ontaneous emission inverse of the radiative lifetimes coll
rates collisional de excitation rate co e cients trans transitions corresp onding
photo cross sections
p olyatomic molecules except for H O are generally less abundant than diatomic
molecules e g CO in CSEs
see the simple approximate metho ds prop osed in this work
FURTHER HEATING AND COOLING PROCESSES
Table continued from page
sp ectral
region
pro cess imp ortant for what obstacles comments
esti
mated
absolute cross sections
applies to p olar and
for Raman scattering
unp olar molecules and
rad heating of
Raman UV and Stokes and
to arbitrary
molecular gases by
scattering
optical Anti stokes of
wavelengths however
inelastic scattering
individual molecules
cross sections are
e g of H
small
dust
according to
thermal
heating and co oling temp erature di erence
accommo da
b etween gas and dust
tion
e cient mechanism
dust drift
heating
see Kruger et al
heating
gain or loss of
reaction rates
disso ciation p otential
reaction heats energy
energies heating and
distribution among
dust surface
di cult
co oling desorption of
the various degrees of
reactions
excited reactants
freedom of the
heating
reaction pro ducts
Chapter
The Calculation of the Equation of State
The calculation of the equation of state provides the basic link b etween the micro
physics and the thermo dynamic description of the gas Having once determined
the microphysical quantities the particle densities as function of a suitable set
of thermo dynamic state variables e g temp erature and density all macroscopic
prop erties of the gas can b e determined by means of statistical metho ds Thus the
mo deling of the gas can b e p erformed on a higher thermo dynamic level without
going back into the details of microphysics
This chapter describ es the assumptions and the numerical techniques used to de
termine the particle densities and the internal energy of the gas as function of T
g
and As p ointed out in Chapter this work do es not rely on LTE but con
siders a steady state Consequently two additional external parameters enter into
the usual thermo dynamic description the radiation eld J and the mean velocity
E D
dv
Since the techniques are the same for all following applications they gradient
dl
are summarized in this separate chapter
Calculation of the Particle Concentrations
The basis for the calculation of the particle concentrations are the element abun
dances In this work a mixture of the elements H He C N O Na Mg Si S and Fe
is considered Since di erent types of stars with di erent abundances are considered
in the forthcoming applications C stars RCB stars the assumed abundances are
stated separately cf Sect and Sect The following basic assumptions
are made in order to calculate the various particle densities of the neutral atoms
ions electrons and molecules
Neutral and singly ionized atoms are taken into consideration The ratios b e
tween the particle densities of ions and neutral atoms are calculated by means
of the statistical equations Eq taking into account the rates of pho
toionization recombination collisional ionization and b o dy recombination
steady state non LTE as describ ed in Chapter
For simplicity the ratios b etween the particle densities of molecules and neu
tral atoms are calculated according to chemical equilibrium Negative ions
This is of course a simplifying assumption An improvement of the mo del may b e achieved by
calculating the steady state solution kinetic equilibrium of a complete and reliable chemical
CHAPTER THE CALCULATION OF THE EQUATION OF STATE
are treated like molecules except for H cf Sect The chemistry
comprises sp ecies Gail Sedlmayr where some larger pure carb on
molecules have additionally b een included using the thermo chemical data
from Go eres Sedlmayr
The particle densities are nally found by means of nested Newton Raphson and
iteration techniques until the conservation of charge and elements is assured The
D E
dv
following scheme is applied where the quantities T J and are given
g
dl
Estimate the electron density n and all neutral atom densities in the
e
El
electronic ground state n
Calculate the b ound free and free b ound rates R and R which de
i II II i
p end on T J and n
g e
Perform an inner iteration for each atomic sp ecies in order to solve the
coupled equations for the level p opulations and the escap e probabilities
compare Sect i e
e e
a Calculate the b ound b ound rates R and R which dep end on
ul l u
e
e
T J the escap e probabilities P and the densities of the collision
g
ul
partners
El
b Determine the level p opulations and the ion particle densities n
II
from the statistical equations
e
e
c Calculate the escap e probabilities P which dep end on the level
ul
E D
dv
p opulations and the lo cal velocity gradient
dl
d Go back to step a unless the pro cedure has converged
Calculate the particle densities of the molecules n by assuming chemi
mol
El
cal equilibrium according to the total neutral atom densities n and the
at
gas temp erature T
g
Calculate the current errors of charge and element conservation i e
P P
e El
n s n n
mol e
mol El
mol II
B C
P
H H H
H
B C
P
n s n n
mol
B C
mol
mol at II
m
El El
H Fe
El
B C
F n n n
e
B C
B C
A
P
Fe Fe Fe
Fe
P
n s n n
mol
mol
mol II at
m
El El
El
Perform one Newton Raphson iteration step i e solve D F n F for
the corrections n and put n n n where the comp onents of the
vector n are shown as the argument of F in the upp er equation
Go back to step unless all further corrections b ecome small
max f n n g
j j
j e H Fe
reaction rate network which however go es b eyond the scop e of this work Concerning the RCB
element abundances most of the imp ortant reaction channels probably involve the abundant pure
carb on molecules which are all radicals and whose reaction rates are only p o orly known
CALCULATION OF THE INTERNAL ENERGY
The successful convergence of this iteration scheme critically dep ends on the quality
of step i e the rst estimate of the electron and neutral atom densities The inner
iteration step d is necessary if b ound free transitions from excited levels
are included In this case the degree of ionization of a considered atom and hence
the electron density may dep end on the escap e probabilities as demonstrated for
hydrogen in Sect If no such b ound free transitions are considered as for all
other elements than hydrogen in this work the system of equations decouples and
the p opulations of the multi level atoms without continuum can b e determined after
having solved the ab ove iteration scheme Esp ecially all excited states of ions can
b e calculated afterwards since only the rst ionization stage is taken into account
The describ ed metho d yields all particles densities including the considered level
p opulations as function of the mass density the gas temp erature T the contin
g
D E
dv
uous background radiation eld J and the velocity gradient
dl
Calculation of the Internal Energy
For dynamic considerations the prop er determination of the internal energy is as
imp ortant as the determination of the radiative heating and co oling rates Hav
ing once determined the particle densities as outlined ab ove the evaluation of the
internal energy is comparable simple and not very time consuming
According to the de nition of the internal energy in this work cf Chapter the
internal energy comprises of translational ionization and disso ciation p otential and
electronic vibrational and rotational excitation energies The di erent terms are
calculated as follows
n k T E
g trans
X X
El El El El El
E n n
ion
II II III II III
El El
X
E n D
diss mol
mol
mol
X
n E E
ij ij el
i j
mol
X X
g h
j
j
E n
vib mol
mol
h
j
j
mol
exp
mol
k T
vib
mol
X
f
rot
mol
n k T E
mol rot
rot
mol
Warning One should not take the radiative heating and co oling rates out of this work and
consider e f k T at the same time The de nitions of the internal energy and the radiative
heating and co oling rates refer to each other For example the internal energy in the molecular
domain turns out to b e negative in this work
CHAPTER THE CALCULATION OF THE EQUATION OF STATE
n is the total gas particle density atoms ions electrons molecules
II III
is the ionization p otential of the rst second ionization stage the latter only given
for reasons of completeness D is the total disso ciation p otential of a molecule
mol
measured from the vibrational ground state i e the energy required to totally
disso ciate the molecule into its constituting atoms at K By de nition neutral
atoms have zero p otential energies
n is the particle density of sp ecies i in the j th excited electronic state and E
ij ij
mol
the corresp onding energy di erence to its electronic ground state is the j th
j
eigenfrequency of a molecule and g the corresp onding degeneracy Equation
j
assumes indep endent harmonic oscillators Equation is the classical limit for
large rotational temp eratures which is su cient in this context f is the number
rot
of rotational degrees of freedom for linear molecules otherwise
As far as p ossible the vibrational and rotational excitation temp eratures T and
rot
T are calculated by means of the metho ds outlined in the Sects and
vib
For those molecules which are not considered therein the excitation temp eratures
are assumed to equal the gas temp erature
For the calculation of the disso ciation p otential vibrational and rotational excitation
energies only the abundant molecules must b e taken into account Table lists
the selected molecules and summarizes the necessary molecular data The selection
comprises the most abundant molecules in b oth cases C star and RCB star ele
El
ment abundances Additional data for the ionization p otentials and the excited
II
electronic levels E can b e found in the Tables and
i j
In summary the outlined metho ds for the calculation of the particle densities and
the internal energy provide the caloric and thermal equations of state in the form
D E
dv
e e T J E E E E E E
g trans ion diss el vib rot
dl
D E
X X
El El
dv
p p T J n n n n k T
g e mol g
at II
dl
El mol
A similar expression can b e written for the total radiative net heating rate of the
gas which is calculated according to Chapter as function of the various particle
densities which dep end on density and temp erature the radiation eld and the
velocity gradient
E D
dv
Q Q T J
rad rad g
dl
Equations to de ne the thermo dynamic system which is ex
amined in the following parts of this work
D E
dv
Together with the two external parameters J and any suitable set of two
dl
indep endent state variables is su cient to determine the thermo dynamic state and
hence all gas prop erties Equations to are formulated in terms of T
g
but other useful choices can b e e g p p T or p h dep ending on the problem g
CALCULATION OF THE INTERNAL ENERGY
Table Molecular data for the determination of the internal energy
molecule D eV cm and degeneracy f
mol rot
mol
H
CO
CH
C H
C H
CH
C
C
C
C
C
C
N
CN
C N
C N
HCN
SiC
Si C
SiC
SiO
SiS
CS
The total disso ciation p otential energy can b e determined from the JANAF tables
P
El
Chase et al according to D H mol s H El at K
f f
El
mol mol
For the larger carb on molecules C and C the disso ciation p otentials are taken
from ab initio quantum mechanical calculations scaled binding energies from
Raghavachari Binkley
Values for diatomic molecules are taken from Hub er Herzb erg according to
x Values for p olyatomic molecules from Chase et al Values
e e e
for C C C and C from Raghavachari Binkley The reader may verify
the relation f N N for linear and non linear molecules resp ectively vib
CHAPTER THE CALCULATION OF THE EQUATION OF STATE
In practice such dep endencies are co ded by numerical inversion One computer
routine carries out the determination of thermo dynamic state of the gas as stated
ab ove yielding the values of all state variables as function of T If e g a
g
formulation in p h is needed another computer routine nds the corresp onding
values for and T which yield p h by Newton Raphson iteration g
Chapter
Thermal Bifurcations in the
Circumstellar Envelopes of RCB Stars
As a rst application of the thermo dynamic description developed in this work the
top ology of the radiative equilibrium solutions is investigated
Radiative equilibrium RE is de ned as the equality of radiative gains and losses
Supp osing that other heating and co oling pro cesses are negligible as heat con
duction and heating by magneto acoustic waves or cosmic rays RE is the main
criterion for the thermal stability of gases under astrophysical conditions Since
any long term physically realized solution must b e thermally stable the condition
of RE provides the basic equation for the determination of the gas temp erature in
the case of static conditions
However as shown in this chapter the condition of RE may not b e unique but
can have two or more stable temp erature solutions These multiple solutions are
commonly called thermal bifurcations Thermal bifurcations are well known to
o ccur in the outer solar atmosphere Ayres Muchmore Ulmschneider
Muchmore in late type stars Kneer and in the interstellar medium
e g Biermann et al
This chapter investigates the circumstellar envelopes of RCB stars Here the ques
tion of whether or not low temperature solutions already exist at small radial dis
tances to the star is of sp ecial scienti c interest The o ccurrence of such solutions
might b e related to the formation of dust in these envelopes which causes the sp ec
tacular RCB type decline events cf Sect and App endix A
The main intention of this chapter is to demonstrate that thermal bifurcations in
principle can lead to di erent co existing phases of the gas in pressure equilibrium
The temp eratures of these phases can easily di er by several thousands of degrees
The phenomenon of thermal bifurcations is exp ected to o ccur frequently in all par
tially molecular gases
A secondary criterion for thermal stability is that the gas must o er resistance against com
dp
pression along a RE tra jectory Otherwise the gas is unstable against collapsing
d
RE
CHAPTER THERMAL BIFURCATIONS IN RCB STARS
The Mo del
De nition of the Radiative Equilibrium Gas Temperature
In the following the gas temp eratures where radiative heating and co oling balance
RE
each other are determined The RE gas temp eratures T are calculated for given
g
values of the gas pressure p the radiation eld J and the mean lo cal velocity
E D
dv
according to gradient
dl
D E
RE
dv
Q p T J
rad
g
dl
Equation is an implicit de nition of the RE gas temp erature which may of
course b e non unique For stability one has to require that the derivative of Q
rad
with resp ect to temp erature is smaller than zero Otherwise a small enhancement
in temp erature T will increase net heating that is the gas element will absorb
g
even more radiation and will heat up further
Q
stable RE
rad
unstable RE
T
g
RE
T T
g
g
The bifurcation p oints where solutions app ear or disapp ear satisfy Eq and
have zero rst derivatives They only exist for certain values of the other parameters
e g for sp ecial radiation elds
Element Abundances
The element abundances of the prototype star R Coronae Borealis are considered
adopting the values from Cottrell Lambert RCBs are chemically p ecu
liar stars showing strong hydrogen de ciency and considerable carb on enrichment
cf App endix A Mg and Ne are assumed to have the solar abundances given by
Allen Figure summarizes the choice of the element abundances in this
work for all mo dels with regard to RCB stars
Na H S Si Al Fe Mg Ne N O C He
−6 −5 −4 −3 −2 −1 0 log ε
Figure Assumed element abundances of R Coronae Borealis
Other RCB stars show considerable individual deviations from these abundances esp ecially
for H He and C N O Lambert Rao
RESULTS
Approximation of the Radiation Field
The radiation eld is an imp ortant ingredient for the mo del entering into b oth the
determination of the particle densities and the calculation of the radiative heating
and co oling rates In this chapter a two parameter approximation of the radiation
eld is used The radiation eld is tted by a radially diluted black b o dy eld of
the e ective temp erature T of the central star
e
q
r R B T J r
e
Absorption b etween the outer edge of the photosphere and the lo cation of inter
est are neglected i e the CSE is assumed to b e optically thin T is set to b e
e
K which is a representative value for this class of stars cf App endix A The
approximation reasonably ts the stellar sp ectrum in the optical and IR region with
a maximum deviation of a factor but leads to somewhat to o high intensities
for nm Asplund et al which is a consequence of the large UV optical
depths in the stellar atmosphere Furthermore the stellar photosphere is assumed to
b e the dominant source for radiation at all wavelengths chromospheric emissions
continuous UV emissions from sho cked gas layers in the circumstellar envelope and
also IR emissions from extended circumstellar dust shells cf App endix A are ig
nored Such e ects would enhance the mean intensities in the UV and IR sp ectral
regions resp ectively as compared to Eq
Results
Before studying the structure of the RE solutions some of the microphysical results
of the RCB applications are stated rst In the following the typical features for the
ionization and the chemistry of the gas are summarized and the role of the various
heating co oling pro cesses is discussed Regarding the abundances cf Fig the
results can di er a lot from those of a hydrogen rich gas with nearly solar abundances
as encountered in the interstellar medium or for example in the atmospheres of
AGB stars
Degree of Ionization
Fractional ionization usually turns out to b e large irresp ective of the gas temp er
ature This is a consequence of the large rates of photoionization according to the
assumed radiation eld with its strong UV intensities The most abundant ele
ment helium however is mostly neutral unless the gas temp erature is larger than
K where the rates of collisional ionization come into play Consequently
the degree of ionization equals almost for T K and is approximately given
g
by the C He ratio at lower gas temp eratures At very low temp eratures T K
g
carb on is mainly present in the form of molecules and the electrons are provided by
CHAPTER THERMAL BIFURCATIONS IN RCB STARS
other elements mainly Si and Mg According to the somewhat p o or t of the UV
part of the stellar sp ectrum these results are still preliminary
The character of the results changes for very large densities n cm
He
or g cm or p dyn cm For such densities as present in the at
mospheric layers of the star the state of the gas is close to LTE due to the large
e ciency of the various collisional pro cesses esp ecially the three b o dy recombina
tion rates Consequently the degree of ionization smo othly reaches the results of
Saha ionization with increasing densities
Chemistry
Molecules b ecome abundant in the gas phase approximately b elow a dividing line in
the gas temp erature density plane reaching from T K at n cm
g He
to T K at n cm With decreasing temp erature the rst
g He
molecules to o ccur are CO and N
Table Abundant molecules in the circumstellar envelopes of RCB stars
Elements most abundant molecules abundant molecules
pure C C C C C C
C mono cyclic ring
C N N CN C N C N NCN C N
O CO NO O CO C O
Si S SiC Si C SiC SiO SiS CS SiN Si Si Si N SO SN S
H C H HCN H C H CH OH HN HS SiH CH C H
Mg atomic MgO MgN MgS MgH
Fe atomic FeO
resulting from equilibrium chemistry based on the element abundances given in
Fig for the range n cm and T K
He g
molecules with n n somewhere in the n T plane
C He g
mol
molecules with max fn n g where El includes all elements the
mol El
El
molecule is comp osed of
Table reviews the more abundant molecules for the hydrogen de cient and
carb on rich element comp osition considered here The chemistry is divided into the
following subgroups The most abundant group contains the pure carb on molecules
with small chains which are all radicals and mono cyclic rings With decreasing
gas temp erature the concentrations of the more complex carb on molecules increase
See Go eres Sedlmayr for more detailed information concerning the carb on
chemistry Oxygen is mostly blo cked by the formation of CO and consequently all
other molecules containing oxygen are not abundant Esp ecially H O is practically
A substantial improvement of the mo del may b e achieved by using a detailed mo del sp ectrum
for RCB stars in future investigations
RESULTS
absent from the gas phase The next group are comp ounds formed out of nitrogen
and carb on The most imp ortant nitrogen molecule however is N Furthermore
there are several abundant silicon sulphur and hydrogen b earing molecules all
formed out of these elements and the abundant and unblocked elements C and N
except for SiO Iron and magnesium b earing molecules are unimp ortant
Radiative Heating and Co oling Rates
The question of imp ortant contributors to the heating and co oling of such sp ecial
gas has to b e investigated carefully No preceding studies are available for this
case Table summarizes the results of this work concerning the role of the
various radiative pro cesses for a typical choice of the parameters T r R and
e
D E
dv
The absolute value of the total radiative heating co oling rate as given in
dl
the rst line of each panel in Table increases with increasing gas temp erature
by many orders of magnitude and mo derately decreases with decreasing density
The imp ortance of the individual heating and co oling pro cesses strongly dep ends
on temp erature and density Usually one sp ecial radiative pro cess dominates in
a certain temp erature density regime All basic radiative pro cesses may cause
heating or co oling and change the sign at di erent temp eratures which dep end on
the relation b etween J and the source function at the characteristic wavelength
of the pro cess cf Fig Together with the strongly varying concentration of
b oth the carriers of the heating co oling rates and the collision partners esp ecially
the electron density a very complex picture app ears which shows the following
features
Free free heating co oling is imp ortant for large densities n cm
He
Bound free transitions mainly of He and C provide the most imp ortant
heating co oling pro cess at large densities n cm where all the
He
b ound b ound type transitions are optically thick
The heating co oling rates of line transitions cover the whole temp erature den
sity plane and are generally imp ortant They dominate the heating and co ol
ing of the gas for not to o large densities n cm and not to o low
He
temp eratures The most imp ortant contributors are He C N S and
Fe b ecause of the high fractional ionization in the mo del cf Sect
As so on as p olar molecules b ecome abundant in the gas phase their large num
b er of allowed transitions vibrational and rotational dominates the radiative
heating and co oling of the gas This happ ens b elow the dividing line describ ed
in Sect CO plays the overwhelming role concerning the heating and co ol
ing of the gas by molecules since it is the most abundant p olar molecule by
approximately two orders of magnitude Further imp ortant molecules are CS
and SiS For larger densities n cm the vibrational transitions
He
are imp ortant whereas for smaller densities the pure rotational transitions are
more signi cant
CHAPTER THERMAL BIFURCATIONS IN RCB STARS
Table Imp ortant heating co oling pro cesses for RCB abundances as function of tem
p erature T and density n Parameters are chosen as T K r R and
g He
e
D E
dv
km s R
dl
cm cm cm cm cm
K
He bf He bf HeII HeI CII HeII CII HeII
C bf C bf He bf HeI HeI
K
He bf C bf He bf C bf CII HeI CII HeI CII OII
O bf O bf SiII SiII NII
K
C bf CII C bf CII SiII CII NII CII NII
O bf SiII NII SII OII
K
C bf C bf CII NII NII CII NII OII
O bf O bf SiII SII FeII
K
C bf CO vib CII C bf CII NII SII NII
Si bf C bf O bf SII FeII
K
Si bf Mg bf CS vib CII C bf CO rot CII CO rot SII
Fe bf CO vib SiII SiII SII NII
CO vib
K
Mg bf CO vib CO rot FeII CO rot FeII
SiS vib
Fe bf Na bf SiS vib SiII HCN rot He bf
CO rot
K
Mg bf SiS vib SiS vib SiII FeII CO rot CO rot FeII
Fe bf Na bf CO vib SiII FeII SiII SiII
Each panel of the table has two entries
The rst line is the resulting total net radiative heating rate p er mass of the gas
Y
Q erg s g where X Y means X
rad
A list of the three most e cient heating co oling pro cesses is stated b elow in order
of decreasing absolute net rates co oling heating I lines of neutral
atom II lines of ionized atom free free bf b ound free vib vibrational
rot rotational transitions
R R is assumed in this context
RESULTS
Figure Heating co oling rates as function of the gas temp erature for p
E D
dv
dyn cm n cm T K r R and
He
e
dl
km s R The thick full line shows the total net heating rate The other
dashed and dotted lines depict the free free rate Q the total b ound free rate
Q all atoms ions the total line heating co oling rate Q all atoms and
Lines
bf
ions the total vibrational rate Q all molecules and the total rotational rate
vib
Q all molecules The circles denote stable radiative equilibrium temp erature
rot
solutions
Radiative Equilibrium Temperature Solutions
The solutions of the radiative equilibrium problem are related to the changes of sign
of the total net radiative heating function Q as a function of the gas temp erature
rad
I will brie y explore the reasons for these changes of sign in the following
The heating co oling rates as functions of the gas temp erature are shown in Fig
for a sample choice of the parameters The sums of the rates of all kinds of pro cesses
free free b ound free lines vibrational and rotational transitions are depicted For
su ciently high temp eratures all radiative pro cesses cause net co oling Consider
ing the direction to lower gas temp eratures the di erent pro cesses subsequently
change the sign at di erent temp eratures For the parameters chosen in Fig
one nds b ound free K sp ectral lines K free free K vibra
CHAPTER THERMAL BIFURCATIONS IN RCB STARS
tional K and rotational K Finally for su ciently low temp eratures all
radiative pro cesses cause net heating
Thus there always exists at least one stable solution for the radiative equilibrium
problem Considering the direction to low temp eratures the rst solution to o ccur
is henceforth called the high temp erature solution For the high temp eratures in
Fig the line transitions provide the dominant heating co oling pro cess Conse
quently the high temp erature solution K is usually close to the temp era
ture where the total line heating co oling rate Q changes its sign
Lines
The high temp erature solution refers to a predominantly molecule free
partially ionized gas The temp erature is xed by the change of sign of
Q for small and Q for large densities resp ectively
Lines bf
The change of sign of Q is caused by the temp erature dep endent comp eting
Lines
pro cesses of line absorption followed by collisional de excitation and collisional ex
citation followed by line emission The change of sign of Q is caused by the
bf
comp eting pro cesses of photoionization followed by collisional three b o dy recom
bination and collisional ionization followed by radiative recombination If molecule
formation was not p ossible in the gas phase the high temp erature solution would
b e the only solution and the radiative equilibrium problem would b e unique
However once the gas has reached a su ciently low temp erature molecules b ecome
abundant Their large number of allowed vibrational and rotational transitions
lo cated in the IR and microwave sp ectral region enters into comp etition with the
other atomic transitions which substantially increases the e ciency of the interaction
b etween the gas and the radiation eld at long wavelengths Thereby the app earance
of molecules causes reinforced co oling for the present b ecause of the comparable
faintness of the central star at these wavelengths as sketched in Fig Much lower
temp eratures are required to cause a change of sign of the molecular heating co oling
functions
The additional temp erature solutions are caused by the presence of
molecules Two types of stable solutions are found The medium
temp erature solutions result from an equilibrium b etween atomic heating
and molecular co oling The low temperature solutions are caused by a
change of sign of the dominant molecular heating co oling function
For example in Fig one nds a second stable solution at K where
the radiative heating by lines and b ound free transitions is balanced by vibrational
co oling At ab out the third stable solution K the vibrational heating co oling
function changes its sign Additional unstable solutions exist at K and
K
Strictly sp eaking even Q and Q may change the sign more than once b ecause of the
bf Lines
sup erp osition of the numerous transitions
RESULTS
p dyn cm
p dyn cm
Figure Thermal bifurcations in RCB envelopes for p dyn cm upp er
panel n cm and dyn cm lower panel n
He He
RE
cm The radiative equilibrium temp erature solutions T are
g
shown versus dilution factor W in a Planck type radiation eld with T K
e
dv
km s R Full and dotted black lines indicate stable and and for
i h
dl
unstable solutions resp ectively The radius axis b elongs to the optical thin limit
pure radial dilution according to Eq The UV and IR limit and the black
b o dy temp erature T are the same as shown and explained in Fig bb
CHAPTER THERMAL BIFURCATIONS IN RCB STARS
p dyn cm
p dyn cm
Figure Same as Fig but for p dyn cm upp er panel n
He
cm and dyn cm lower panel n cm
He
RESULTS
p dyn cm
p dyn cm
Figure Same as Fig but for p dyn cm upp er panel n
He
cm and dyn cm lower panel n cm
He
CHAPTER THERMAL BIFURCATIONS IN RCB STARS
The general top ology of the radiative equilibrium solutions is depicted in the Figs
to The temp erature solutions are shown as function of the dilution factor W
This factor is related to a distinct radial distance in the case of pure geometric
dilution according to Eq but its meaning is more general W characterizes
the departure from an equilibrium and hence is an appropriate variable to study
the structure of the bifurcations W together with RE implies complete ther
mo dynamic equilibrium TE according to the concept of this work In the case
where W the radiation eld is a non diluted Planck eld J B T and the
rad
RE
only solution of RE is given by T T Every collisional or photo pro cess
rad
g
is directly balanced by its corresp onding reverse pro cess which characterizes TE
Since all reverse pro cesses are included by means of detailed balance considerations
cf Chapter this work accurately describ es this b ehavior
All calculated RE temp erature solutions are lo cated b etween the IR limit T
IR
WT and the UV limit T T nicely con rming the simple results of Sect
e UV e
where LTE and a type gas absorption co e cient have b een considered
Thermal bifurcations are found to o ccur under the following conditions
A high temp erature stable solution must b e p ossible i e a radiative equilib
rium state of the gas mainly consisting of atoms and ions
R is required to make p ossible a molecule rich low r W
temp erature solution as motivated by the IR limit
p dyn cm n cm is required to limit the in uence of the
He
b ound free heating co oling rates compared to the molecular heating co oling
rates For to o large densities Q dominates the heating and co oling even for
bf
low temp eratures cf Table and consequently molecule formation do es
not pro duce additional solutions
Under these conditions the gas is always found to b e at least bi stable Up to
simultaneous temp erature solutions may exist dep ending on the pressure and the
dilution factor The stable temp erature solutions e g K K K
and K for p dyn cm and W can di er by several thousands
of degrees usually yielding one high temp erature atomic solution and one or more
low temperature molecular solutions
Another result of the mo del is that the RE gas temp eratures are density dependent
which can b e seen by comparison of the Figs to The general tendency is
that a thin gas tends to b e co oler than a dense gas considering the same branch
of solution This is caused by the increasing imp ortance of sp ectral lines and rota
tional transitions compared to b ound free and vibrational transitions for decreasing
density resp ectively The former transitions have longer characteristic wavelengths
compared to the latter yielding lower RE temp eratures according to Fig
A violation of this criterion o ccurs at small pressures p dyn cm and large dilutions
W in Fig In this case the high temp erature solution drops b elow K where it
enters into the molecular regime and disapp ears Only one low temperature solution remains in
this case
DISCUSSION
Discussion
The circumstellar envelopes of RCB stars show a multi stable character Co ol gas
phases mainly consisting of molecules can principally co exist b esides hot phases
mainly consisting of atoms and ions Both phases are in radiative equilibrium and
in pressure balance with each other
The multi stable character of the gas causes a kind of co oling trap Once the
gas has reached a su ciently low temp erature molecules are formed which cause
reinforced radiative co oling The gas then co ols down to much lower temp eratures
until the heating and co oling by molecules alone pro duces another solution of the
radiative equilibrium problem and stabilizes the low temp erature
Thermal bifurcations are found to o ccur in a large range of examined parameters
concerning b oth the radial distance to the star and the gas pressure These ndings
indicate that the o ccurrence of thermal bifurcations is not restricted to the CSEs of
RCB stars but is a common phenomenon in partially molecular gases However the
thermal bifurcations are exp ected to o ccur mainly in the CSEs of warm stars with
T K where the atomic high temp erature solution still exists cf criterion
e
of the item list on the previous page
Concerning the CSEs of co ol stars as C and M stars on the AGB the radiative
equilibrium gas temp eratures are exp ected to b e much lower than the black bo dy
temp eratures The gas in these circumstellar envelopes is molecule rich Conse
quently the solutions of the radiative equilibrium problem should b e similar to
the low temperature solutions discussed ab ove However these envelopes are dust
enshrouded and hence optically thick The approximation of the radiation eld used
in this chapter is not appropriate for this case and the results can b e di erent
Nevertheless the consequences of the multi stable character of the gas reach far
as F Kneer wrote in view of this instability I conclude that RE stellar
atmospheres with T K may not exist in principle I would not go that
e
far but consider for example a gas element which slowly moves outwards in a CSE
with a temp erature structure similar to that depicted in Fig The motion
of the element shall b e slow so that RE remains valid The gas element mainly
consists of atoms and ions as long as the high temp erature solution is realized
The gas temp erature slowly decreases with increasing radial distance down to ab out
K until suddenly at ab out R in Fig a certain amount of molecules has
b een formed just su cient to destabilize the radiative equilibrium The gas then
quickly co ols down towards the second low temperature solution at K The
nal chemical comp osition and the amount of dust formed in the gas element will
crucially dep end on the relation b etween the chemical and the co oling time scale
during this transition In the end the chemistry freezes out and dust formation
b ecomes imp ossible again If this scenario proves to b e true it would change our
general theoretical view of the chemistry and the dust formation pro cesses in stellar
How such a su ciently low temp erature can b e reached is left op en for the present
CHAPTER THERMAL BIFURCATIONS IN RCB STARS
envelopes quite dramatically Other topics related to the multi stable character of
the gas could b e inhomogeneities cloud formation or a hysteresis like b ehavior of
the gas in the CSEs of pulsating stars
The results of this chapter refer to the assumption of static RE Chapter will
calculate radiative co oling time scales which give an impression on the applicabil
ity of RE under dynamic conditions For example the low temperature solutions
can easily b e destabilized by adiabatic heating co oling rates which diminishes the
meaning of the low temperature results of this chapter to some extent
In contrast to the general nding that thermal bifurcations should o ccur principally
a reliable determination of the gas temp erature is di cult In the static case it really
dep ends on the details of i the chemistry ii the heating co oling functions and iii
the radiation eld Each radiative pro cess which is additionally taken into account
RE
may change the results for T substantially This is fundamentally di erent from
g
the results of Chapter as will b e discussed therein
Chapter
Radiative Co oling Time Scales in the
Circumstellar Envelopes of C Stars
The second application of the thermo dynamic metho ds developed in this work in
vestigates the relaxation towards radiative equilibrium A gas element in non RE
is considered The element b eing hotter or co oler than in RE will consequently
radiate away excess internal energy radiative co oling or gain radiative energy by
net absorption radiative heating resp ectively The key quantity which describ es
the e ciency of this relaxation is the time scale for radiative co oling or heating
which is de ned b elow
The character of the thermal b ehavior of the gas under dynamical conditions can
b e discussed by comparing this time scale henceforth called the radiative co oling
time scale with the other hydrodynamic or chemical time scales involved in the
considered pro cess If the radiative co oling time scale is shorter than the others
the gas quasi instantaneously relaxes towards RE and consequently the condition
of RE can b e used to determine the gas temp erature If it is comparable or larger
than the others the temp erature of the gas dep ends on the history of the pro cess
and must b e calculated time dep endently
In the following the applicability of RE for the determination of the gas
temp erature under dynamic conditions is investigated
Concerning the chemistry and the dust formation in the CSEs of pulsating stars the
character of the thermal relaxation of the gas in resp onse to propagating sho ck waves
is of sp ecial imp ortance The pulsation in the interior of the star pro duces waves
which steep en up to sho ck waves in the atmosphere and propagate into the CSE
e g Bowen Fleischer et al Thus the gas elements in the envelopes of
pulsating stars are hit by sho ck waves time and time again The sho cks dissipate
mechanical energy and heat up the gas to considerably high temp eratures The gas
must b e able to radiate away this excess internal energy b efore the next sho ck hits
the element Otherwise it will never b ecome su ciently co ol to allow for complex
chemical and dust formation pro cesses
Following this consideration one would exp ect the stellar pulsation to hinder dust
formation In fact from observations just the opp osite conclusions can b e drawn
Many of the dust forming ob jects are known to b e pulsating stars Moreover a
CHAPTER RADIATIVE COOLING TIME SCALES IN C STARS
strong correlation b etween the o ccurrence of an IR excess indicating dust formation
and a light variability indicating stellar pulsation can b e observed for late type
stars Jura i e stellar pulsation favors dust formation Therefore an e cient
relaxation of the sho ck heated gas in circumstellar envelopes seems to b e con rmed
by observations
This chapter considers the CSEs of pulsating C stars It picks up the controver
sial question of whether the sho cks in these CSEs b ehave predominantly isother
mally or adiabatically more informations ab out this controversy can b e found in
Sect A clari cation of this question is an imp ortant step towards the principal
understanding of dust formation in the CSEs of pulsating stars
The Mo del
De nition of the Radiative Co oling Time Scale
An arbitrary physical quantity y shall b e considered The time evolution of y is
assumed to b e given by the rst order ordinary di erential equation
dy
f y
dt
The equilibrium values of the physical quantity y are implicitly de ned by f y
First order Taylor expansion of Eq in time yields y t t y t t f y
If y is to relax toward equilibrium i e y t t y the rst order estimate of the
time required for the relaxation is
y y
t
f y
The radiative co oling time scale is de ned analogously considering pure radiative
de
b b
heating co oling according to Q with Q Q
rad rad rad
dt
D E D E
dv dv
RE
E D
e T J e T J
g
g
dl dl
dv
D E
T J
co ol g
dl
dv
b
Q T J
rad g
dl
RE
T is one RE temp erature solution as de ned in Chapter Apart from the prob
g
RE
lem of RE multi stability T and thereby are completely determined by the
co ol
g
thermo dynamic quantities and T the continuous background radiation eld J
g
D E
dv
and the lo cal velocity gradient In the following the aim is to calculate for
co ol
dl
the entire density and temp erature range encountered in the sho cked envelopes of
pulsating C stars
THE MODEL
Element Abundances
The element comp osition of C stars is assumed to b e solar except for carb on The
solar abundances are adopted from Lambert Rao and references therein
The abundance of He is assumed to b e and the abundance of Mg is taken
He H
from Allen Carb on is assumed to b e overabundant with resp ect to oxygen
by which according to Frantsman Eglitis is a representative
C O
value for C stars
Approximation of the Radiation Field
For the applications with regard to C stars the mean back ground intensities are
assumed to b e given by an non diluted Planck eld W that is
J B T
rad
This is done for three reasons First the CSEs of C stars are supp osed to b e dust
enshrouded and hence not optically thin Eq represents the limiting case of an
optically thick CSE Second the assumption considerably simpli es the evaluation
of the co oling time scale as de ned ab ove According to Eq there is always
RE
exactly one trivial RE temp erature solution given by T T Thereby is
rad co ol
g
well de ned according to Eq Third the calculations are p erformed in order
to determine and not to nd the sp eci c temp erature solutions of RE The
co ol
latter of course requires more detailed knowledge ab out J The parameter T is
rad
assumed to vary b etween and K for C star envelopes considering K as
a representative value for the e ective temp eratures of these stars Even for the
extreme cases J and J B K the results for the co oling time scales are
remarkably similar cf Sect Therefore the choice of the radiation eld is
not crucial for the determination of Chromospheric emissions and continuous
co ol
emissions from sho cked gas layers in the CSE are again ignored
Lo cal Velocity Gradient
Regarding the typical saw tooth like velocity structures in mo del calculations for
the sho cked envelopes of co ol pulsating stars e g Fig of Winters et al the
E D
dv
as de ned by Eq has more or less a certain mean velocity gradient
dl
characteristic value of the order of v R varying by ab out one order of magnitude
throughout the whole considered circumstellar shell except for the very thin sho ck
D E
dv
fronts Therefore this parameter is xed and set to km s R The
dl
in uence of this parameter is small cf Sect
CHAPTER RADIATIVE COOLING TIME SCALES IN C STARS
Results
Before discussing the results for the radiative co oling time scales some of the micro
physical results shall b e stated rst the comp osition of the gas degree of ionization
and chemistry the internal energy and the role of the di erent radiative pro cesses
Comp osition of the Gas
The comp osition of the gas is roughly depicted in Fig The upp er panel of
the gure shows contour lines of the concentrations of H and e in the temp era
ture density plane indicating whether the gas is predominantly molecular atomic
or ionized The two extreme cases J and J B K are considered on the
left and right hand side of Fig resp ectively
The resulting electron density is very imp ortant for the calculation of the radiative
heating co oling rates It has a decisive in uence on the b ound free rates and as
collision partner also on the b ound b ound collision rates The degree of ionization
of the gas is found to strongly dep end on the radiation eld
In the case T fractional ionization is solely caused by collisional ionization
rad
which is mainly balanced by radiative recombination Since the rates of the b oth
pro cesses linearly dep end on n cf Eq and the density dependence
e
cancels out and the contour lines are horizontal lines on the left hand side of Fig
The deviations from straight lines at high temp eratures are caused by collisional
excitation from excited states of hydrogen For large densities a two step collisional
pro cess H e H e and H e H e turns out to b e more e cient
than a direct collisional excitation H e H e cf Sect Fractional
ionization is found to b e negligible for T K in the case J
g
For T K right hand side photoionization of metal atoms with low ion
rad
ization p otentials Si Mg Fe Na additionally pro duces free electrons and is more
imp ortant than collisional ionization for T K Since the photoionization rates
g
are density independent but the radiative recombination rates do dep end on the
density the contour lines are approximately vertical lines for T K on the
g
right hand side of Fig A degree of ionization as large as to is re
tained for low temp eratures dep ending on the density Thus fractional ionization
is found to b e much larger than in LTE at low temp eratures for this radiation eld
The three b o dy recombination rates are found to b e negligible compared to the
radiative recombination rates in the entire temp erature density plane under in
vestigation Consequently LTE Saha ionization is never achieved For exam
ple the gas remains predominantly neutral for the radiation elds under examina
tion unless the temp erature is as large as K Only for very large densities
n cm the calculated fractional ionization of the gas approaches LTE
He
Such degree of ionization is exp ected to cause considerable e ects with regard to grain charge
and grain drift for example
RESULTS
Figure The comp osition the internal energy and the net heating function of the gas
as function of temp erature and density The upp er diagrams are contour plots of the H
concentration log n n dotted lines and the electron concentration log n n
H H e H
full lines The middle diagrams show the total internal energy of the gas e on a
linear scale ranging from ab out to erg g The zero line is additionally
shown as a dashed line and the sign of e is indicated The lower diagrams show the
total net heating function of the gas log jQ j erg g s On the right hand side
rad
Q is p ositive b elow and negative ab ove the dashed T K line The left column
g
rad
considers the case J whereas the right column considers the case J B K
E D
dv
km s R All calculation are made for
dl
CHAPTER RADIATIVE COOLING TIME SCALES IN C STARS
The chemical comp osition of the gas is calculated by assuming chemical equilibrium
with resp ect to the neutral atom densities in this work cf Chapter Consequently
the concentrations of the molecules are found to b e very similar compared to the re
sults of previous works using chemical equilibrium The results of the application of
chemical equilibrium to C stars envelopes have thoroughly b een describ ed elsewhere
e g Gail Sedlmayr Some mo di cations are caused by the somewhat di er
ent neutral atom densities due to the upp er results concerning the ionization which
yield somewhat di erent molecule concentrations e g less silicon b earing molecules
if Si is strongly ionized as in the case T K However these mo di cations
rad
are small The H concentration for example is depicted by the dotted contour
lines in the upp er panel of Fig
Internal Energy
The determination of the internal energy of the gas is an imp ortant ingredient for
the calculation of the co oling time scale Moreover it is essential for any time
dep endent treatment of thermo dynamics for example in hydrodynamic mo dels It
provides the basic link from the energy content which is mo di ed by radiative
heating and co oling to the temp erature of the gas
The middle diagrams of Fig depict contour lines of e in the temp erature density
plane The internal energy di ers a lot from that of an ideal gas e f k T
Three di erent regimes can b e distinguished which refer to the predominantly molec
ular atomic and ionized state of the gas The regimes are divided by considerable
energy barriers in b etween In o der to overcome such a barrier to p erform a phase
transition a considerable amount of energy is to b e added or to b e removed from the
gas while the temp erature only changes gradually Within one phase the internal
energy approximately dep ends linearly on the temp erature close to the b ehavior of
an ideal gas The internal energy always increases monotonically with temp erature
All comp onents of the internal energy cf Eq except for the electronic excita
tion energy are found to signi cantly contribute to the total internal energy of the
gas at least in a particular region of temp erature and density E is imp ortant
ion
for high temp eratures where it reaches ab out E at K E dominates
trans diss
the internal energy at low temp eratures ab out E at K the internal
trans
energy is negative in the molecular regime E is ab out E as so on as H
rot trans
is more abundant than H The contribution of E dep ends on T indicating that
vib rad
the p opulation of the vibrational states of the molecules is strongly a ected by the
radiation eld Its maximum contribution is found to b e E for T
trans rad
and E for T K E is found to b e negligible E
trans rad el trans
The internal energy of the gas is not completely determined by temp erature and
density The dep endence on the radiation eld is signi cant The ionic p otential
energy E and the vibrational excitation energy of molecules E are mainly re
ion vib
sp onsible for these dep endences The dep endence of e on the velocity gradient is
principally also present but negligible
RESULTS
The Radiative Co oling Time Scale and the Role of the Various
Heating and Co oling Pro cesses
The radiative co oling time scales as function of and T are depicted in Fig for
g
T and in Fig for T K The dashed arrows will b e discussed later
rad rad
in Sect and are not of interest for the present For completeness the net
b
radiative heating function Q is additionally shown in the lower panel of Fig
rad
Typical values for are found to range from s for a hot and dense gas
co ol
to years for a warm and thin gas The co oling time scale strongly dep ends
on b oth the gas temp erature and the gas density The temp erature dep endence is
found to comprise orders of magnitude at large and orders of magnitude at small
densities considering gas temp eratures of K The density dependence
is also strong orders of magnitude at high and orders of magnitude at lower
temp eratures considering densities of cm
These dep endences result from a sup erp osition of the di erent heating and co oling
functions which are a ected by the varying particle concentrations by minimum
gas temp eratures required for the e cient excitation of the upp er states by non
LTE e ects and by radiative trapping In general a dense gas heats and co ols
more e ciently than a thin gas However a simple approach like Q fails
rad
to provide a reasonable t to the results There are even cases where a dense gas
heats and co ols less e ciently than a thin gas This o ccurs for T K and
g
n cm In this region all imp ortant radiative heating and co oling rates
H
are of b ound b ound transition type and the total rate is strongly reduced by the
large optical depths in the lines The heating co oling e ciency is no more a question
of the strength but of the number of lines taken into account Therefore the results
are uncertain in this region For su ciently low gas temp eratures T K
g
low continuous radiation elds and high densities the heating and co oling of the
gas is p ossibly controlled mostly by the presence of dust grains e g via thermal
accommo dation since dust formation can take place e ciently in this region
The most e ective heating co oling pro cesses are stated in the Figs and The
following picture app ears
For high temp eratures T K hydrogen co oling dominates
g
For small densities co oling by Ly and H is e cient whereas for large densi
ties b ound free co oling of hydrogen turns out to b e more imp ortant due to the
large optical depth in the hydrogen lines Free free co oling is also imp ortant
for high temp eratures and large densities
For intermediate temp eratures there is a zone of considerable smaller heat
ing co oling rates i e larger co oling time scales In this zone the temp erature
is already to o low in order to excite the H atoms but still to o high for con
siderable molecule concentrations The remaining radiative pro cesses are lines
of neutral and singly ionized metal atoms CI OI SiI FeI FeII and also SI
CI I and OI I At very large densities n cm b ound free
H
transitions of H dominate the radiative heating and co oling
CHAPTER RADIATIVE COOLING TIME SCALES IN C STARS
Figure Contour lines of the radiative co oling time scales full lines The
digits on the curves lab el log days The dashed arrow indicates the critical
co ol
isobar co oling track with a maximum radiative co oling time scale of one year
D E
dv
on the track Parameters T and km s R
rad dl
H I H−bf
− C I Fe II H −bf C I O I O I Si I Fe I
CO−vib
C2H−rot
HCN− SiO−vib rot H2
SiS−vib CO−rot rot
Figure Most e cient co oling pro cess referring to Fig schematically
rot rotational vib vibrational I lines of neutral atom II lines of ionized
atom bf b ound free
RESULTS
Figure Contour lines of log days as in Fig but for T K
co ol rad
The critical co oling track ends at T T where radiative equilibrium is re
g
rad
established Note that the co oling time scale remains p ositive and steady al
though the net radiative heating function Q changes its sign at T K
g rad
CII H I H−bf
− C I H −bf O I
Fe II CO−vib CO−rot C2H−rot
H2−vib H2−vib HCN− rot H2 SiS−vib CS− rot rot CO−rot
Fe II
Figure Most e cient heating co oling pro cess referring to Fig
CHAPTER RADIATIVE COOLING TIME SCALES IN C STARS
For low temp eratures T K as so on as CO is abundant
g
p olar molecules dominate the radiative heating and co oling Vibrational and
rotational transitions of CO SiS HCN C H CS H and also of SiO and
CN are imp ortant For small densities rotational heating co oling dominates
whereas for large densities the vibrational heating co oling is more imp ortant
as exp ected from the larger critical densities for the thermal p opulation of the
vibrational states cf Sect
Dep endence on the Radiation Field
The co oling time scale as function of J is exp ected to vary b etween the values shown
in the Figs and supp osed that J B K for C star envelopes
The radiative co oling time scale is found to b e only marginally a ected by the choice
of the radiation eld The maximum deviation b etween the extremes T and
rad
T K is found to b e dex which o ccurs in the region controlled by H
rad
at large densities and warm temp eratures However the usual deviation is much
smaller The standard deviation is found to b e dex This is a surprising result
For example at T K in Fig the radiative pro cesses change from co oling
g
to heating One would exp ect the pro cesses resp onsible for heating to have a di erent
e ciency than those resp onsible for co oling But this turns out to b e wrong The
e ciency of radiative heating and co oling is an inherent feature of the gas mainly
controlled by temp erature and density
Dep endence on the Velocity Gradient
The calculations have b een rep eated with the one tenth and ten times the usually
assumed value of km s R Signi cant di erences are only found for T
g
K and n cm where the evaluation of Q has to b e taken with
H rad
D E
dv
care anyway as stated in Sect Q never varies more than linearly with
rad
dl
For smaller densities or higher temp eratures the dep endence is much smaller
Comparison to Analytical Heating Co oling Functions
In the following the results of this work are compared to previous analytical ap
proaches to determine the radiative heating and co oling rates in circumstellar en
velopes As p ointed out in Sect these approaches lead to considerable di er
ences in the hydrodynamic mo del calculations concerning for example the resulting
temp erature structures
RESULTS
Bowen s Heating Co oling Function
The following analytic expression of the net radiative heating rate has b een prop osed
by Bowen
k
b
T T Q
rad g rad
C
Bowen strictly assumes Q throughout the circumstellar envelope The heat
rad
ing co oling pro cesses which b ehave in such a way are limited by the collisional energy
transfer as in the limiting case of small densities cf Sect Consequently
one might call Eq the strict non LTE heating co oling rate A temp erature
indep endent co oling time scale is furthermore assumed The parameter C re ecting
the radiative co oling time scale is chosen to b e g s cm For higher temp era
tures co oling by emission in Ly is additionally taken into account This H co oling
rate is calculated as describ ed by Bowen although several assumptions are
involved here which with regard to this work seem to b e questionable as for ex
ample the assumption of a constant density independent escap e probability for
Ly
LTE Heating Co oling Function
R R
b
b b b
Starting from the exact expression Q J d d where is the
rad
b
true absorption cross section p er mass and is the sp ectral emissivity p er mass of
the gas the following expression for the net radiative heating rate can b e obtained
b b
by means of the assumption of LTE B T
g
b
b b
Q T T T T
rad J g B g
rad g
b b
T is the intensity mean and T the Planck mean absorption cross sec
J g B g
tion p er mass of the gas is the Stefan Boltzmann constant This analytical form
of the net radiative heating rate has b een used by Feuchtinger et al assuming
b
a constant grey gas absorption cross section As far as is density independent
Q results Using Eq means to assume that all radiative pro cesses refer to
rad
collisionally p opulated levels of the considered atoms and molecules which in gen
eral requires very large densities cf Chapter Furthermore if one uses s which
have b een calculated by opacity sampling metho ds with resp ect to sp ectral lines for
example the numerous lines of molecules the included lines are assumed to b e opti
cally thin Due to the lack of Planck mean opacities Rosseland mean opacities are
used in the following for b oth opacities in Eq The Rosseland mean opacities
b
T are interpolated from tables provided by Scholz Scholz Tsuji
R g
Results of the Comparison
The resulting radiative co oling time scales according to Bowen s and according to
the LTE heating co oling function are depicted in the Figs and resp ectively
In fact these two assumptions contradict each other
CHAPTER RADIATIVE COOLING TIME SCALES IN C STARS
Figure log days calculated from the analytical heat
co ol
ing co oling function prop osed by Bowen T K is con
rad
sidered
Figure log days as in the upp er gure but calculated from
co ol
the LTE heating cooling function No critical co oling track exists here
since the radiative co oling time scale is always much shorter than one year
RESULTS
In this context e k T with amu is assumed for b oth approaches
g
under discussion These results can b e compared to Fig
Neither the co oling time scales derived from Bowen s nor from the LTE rate show
much agreement with the results obtained in this work Bowen s heating co oling
function yields a strong density dep endence vertical contour lines
co ol
whereas the LTE heating co oling function yields more or less density independent
co oling time scales roughly horizontal contour lines Compared to the heat
ing co oling rates calculated in this work Bowen s rate usually gives much smaller
values up to a factor of in the low density low temperature regime whereas
the LTE rate usually gives much larger values up to a factor of in the low
density high temp erature regime The b est that can b e said is that the co oling
time scales calculated in this work usually lie b etween the values derived from the
two analytical formulae
Some rough agreements are found nevertheless For high temp eratures T
g
K Bowen s rate gives ab out the same slop e and order of magnitude com
pared to the results of this work As Bowen s rate to some extent treats hydrogen
co oling more detailed and since hydrogen co oling is dominant at high temp eratures
according to this work this agreement was to b e exp ected The LTE rate pro duces
a similar temp erature dep endence as found in this work very e ective heating
and co oling for high temp eratures an intermediate minimum for the predominantly
atomic phase at warm temp eratures and a re increase of the heating co oling e
ciency in the molecular regime at low temp eratures Best agreement with the LTE
co oling time scale is found on the left hand side of the diagrams at large densities
yielding similar co oling time scales within ab out orders of magnitude However
detailed agreement is not achieved not even for these large densities This disagree
ment might b e caused by missing radiative pro cesses in this work However more
probably it is b ecause
i Rosseland means of have b een used instead of Planck means
ii according to this work the gas is still far from b eing in LTE at n
H
cm esp ecially with regard to the degree of ionization cf Fig
and
iii the LTE heating co oling function neglects optical depths in the lines
In summary b oth analytical heating co oling functions yield p o or agreement with
the results of this work Bowen s rate seems to underestimate and the LTE rate
seems to overestimate the heating co oling e ciency by orders of magnitude This
stresses the necessity to use more detailed mo del calculations for the radiative heat
ing and co oling The prop osed analytical functions are insu cient to describ e the
radiative heating and co oling in the circumstellar envelopes of co ol stars
CHAPTER RADIATIVE COOLING TIME SCALES IN C STARS
The Transition from Isothermal to Adiabatic Sho cks
The calculated radiative co oling time scales of this work allow for a quantitative
discussion of the character of the thermal relaxation b ehind propagating sho ck waves
in the circumstellar envelopes of pulsating stars
A gas element b eing hit by a strong sho ck wave of velocity v is almost instanta
s
neously heated up to high temp eratures K v km s for an ideal gas
s
consisting of H He After the passage of the sho ck the gas radiates away the
excess internal energy dissipated by the sho ck i e it relaxes to RE in principle
However considering the more or less p erio dically sho cked envelopes of pulsating
stars only a limited time for this relaxation is available b efore the next sho ck
wave hits the element This time is given by ab out one stellar pulsational p erio d
P Furthermore the propagating sho ck waves initiate a considerable compression
and re expansion of the gas accompanied by considerable adiabatic heating co oling
The time scale of these pro cesses is also given by P Thus the relation b etween the
radiative co oling time scale and the stellar pulsational p erio d P determines the
co ol
character of the thermal b ehavior of the gas in the CSEs of pulsating stars
P isothermal sho cks
co ol
P adiabatic sho cks
co ol
If is much smaller than P the gas quickly relaxes to RE b ehind the sho cks
co ol
The adiabatic heating co oling rates are meaningless RE is established in the over
whelming parts of the circumstellar envelope except for some thin temp erature
p eaks at the lo cation of the sho ck fronts cf Fig Apart from these p eaks the
temp erature structure of the CSE can b e calculated by assuming RE
If however exceeds P a chromospheric situation results The gas cannot
co ol
radiate away the energy dissipated by one sho ck within one pulsational p erio d Con
sequently the gas subsequently heats up due to the sho cks cf Fig roughly
co oling adiabatically in the meantime The temp erature structure must b e calcu
lated time dep endently
In analogy to the situation in stationary sho cks e g Neufeld Hollenbach an
isobar co oling track in the temp erature density plane is considered in the following
The total co oling time along such a track is roughly given by the maximum radiative
co oling time scale on the track The isobar co oling track with yr on the
co ol co ol
track henceforth called the critical co oling track is depicted in Fig Fig
and also in Fig as dashed grey arrow considering one year as a typical p erio d of
pulsating C stars The deviations from a straight line are caused by changes of the
mean particle mass due to phase transitions cf Fig The critical co oling track
is an estimate for the dividing line b etween the sho cks of predominantly isothermal
and predominantly adiabatic character Gas elements which are sho cked to the left
of the critical co oling tracks can reestablish RE b efore the next sho ck arrives
those which are sho cked to the right of the critical co oling track will b e hit by the
next sho ck b efore RE can b e achieved
DISCUSSION
According to the results of this work a transition of the character of the sho ck waves
is to b e exp ected to o ccur around p ost sho ck densities of cm changing
from predominantly isothermal to predominantly adiabatic with decreasing density
Around T K the co oling gas element sp ends most of its total co oling time
g
The co oling time scale in this temp erature region is found to vary by orders
of magnitude for the entire range of considered densities which is orders of
magnitude Therefore a sharp transition is not exp ected to o ccur rather a gradual
change over a broad range of densities For example if P was demanded
co ol
for isothermal sho cks densities as large as cm would b e required A
nal answer to these questions can only b e obtained by means of time dep endent
hydrodynamic mo del calculations
According to Bowen s rate the transition from isothermal to adiabatic sho cks al
ready o ccurs at cm The LTE rate predicts the sho cks to b e close to
the isothermal limiting case for all densities This explains the di erences b etween
the mo del calculations of Bowen and Feuchtinger et al concerning the
resulting temp erature structures
Discussion
The results of this chapter strongly suggest to include time dep endent thermo dy
namics in the mo del calculations for co ol stellar envelopes esp ecially in the case
of pulsating stars The basis for the thermo dynamic description is a realistic cal
culation of the relevant heating and co oling rates Simple analytical expressions
previously used are not su cient in this context
A time scale discussion can b e p erformed in order to clarify whether or not the con
dition of radiative equilibrium RE can b e used to determine the gas temp erature
By comparing the radiative co oling time scale as depicted in the Figs and
co ol
with the other time scales controlling the physical pro cess under consideration
it can b e decided whether the temp eratures may b e calculated from radiative trans
fer calculations assuming RE or whether for instance a simple adiabatic co oling
law is more appropriate
The general tendency of the results obtained in this work is that the condition
of RE can only partly b e used in order to determine the temp erature of the gas
For the large densities close to the star the radiative co oling time scales are found
to b e of the order of days so that RE is probably established and a temp erature
determination on the basis of RE is justi ed However in general time dep endent
e ects as adiabatic co oling can throughout b e imp ortant The lower the density to
b e considered the more questionable the determination of the temp erature on the
basis of RE b ecomes For instance at n cm in the warm atomic phase
H
the radiative co oling time scale approaches the order of one year already close to the
expansion time scale in stationary wind mo dels for C stars e g Kruger et al
Concerning the sho cked envelopes of pulsating stars the thermo dynamics should b e
CHAPTER RADIATIVE COOLING TIME SCALES IN C STARS
treated time dep endently and if only for the existence of the sho ck waves Addi
tionally apart from the lo cations of the sho ck fronts strong deviations from RE are
exp ected to o ccur roughly at densities n cm connected with the gradual
H
transition of the character of the sho cks changing from approximately isothermal
to approximately adiabatic
Time dep endent thermo dynamic e ects can cause substantial changes in the tem
p erature structures of co ol stellar envelopes How far the chemical and the dust
formation pro cesses are a ected is to b e investigated The prop er inclusion of the
heating and co oling rates calculated in this work into the time dep endent hydro
dynamic mo del calculations can b e achieved by tabulating Q and the internal
rad
energy e as functions of T and further parameters characterizing the lo cal con
g
tinuous radiation eld and the lo cal mean velocity gradient According to the strong
temp erature dep endence of the chemical and the nucleation pro cesses pronounced
e ects are conceivable Another topic which might b e related to the results of this
work is the formation of chromospheres
The following chapter will demonstrate what severe consequences a time dep endent thermo
dynamic mo deling may have with regard to dust formation
Chapter
Sho ck Induced Condensation around
RCB Stars
The third and last application in this work studies a distinct time dep endent ther
mo dynamic pro cess The most complex level included in the work is achieved
time dep endent non LTE in the steady state approximation and non RE
The circumstellar envelopes of pulsating RCB stars are considered A thermo dy
namic description for xed uid elements which are p erio dically hit by sho ck waves
is developed As the sho cks compress the gas it re expands in the meantime which
causes considerable adiabatic co oling The internal energy balance the temp erature
of the gas and the p ossibility for e ective carb on nucleation to o ccur in such uid
elements are investigated Sp ecial attention is paid to the minimum radial distance
required for such nucleation The calculations provide a hypothesis for the physical
cause of the sp ectacular RCB decline events which are supp osed to b e caused by
dust formation close to these relatively hot stars The astronomical background
and the scienti c meaning of these studies are further describ ed in Sect and in
App endix A
The Mo del A Fixed Periodically Sho cked Fluid Ele
ment in a Constant Radiation Field
A chosen uid element in the circumstellar envelope of a pulsating star is time
and time again hit by propagating sho ck waves caused by the stellar pulsation
The sho cks accelerate heat chemically alter and compress the gas Bowen
Fleischer et al Feuchtinger et al Between the sho cks the uid element
follows a roughly ballistic tra jectory while co oling chemically relaxing and re
expanding Gillet Lafon Bowen
Thus the CSEs of pulsating stars are astrophysical sites where a complex inter
play of di erent physical pro cesses takes place hydrodynamics thermo dynamics
chemistry and dust formations A complete mo deling of these pro cesses o ccurring in
circumstellar sho ck waves is a very challenging work which go es far b eyond the scop e
of this thesis Instead the mo del calculations presented in this chapter study the
thermo dynamic consequences of the hydrodynamic situation of p erio dically sho cked
gas A simple gas b ox description suitable for the thermo dynamic investigations is
CHAPTER SHOCK INDUCED CONDENSATION IN RCB STARS
developed for this situation which according to observations is apparently common
among the dust forming ob jects Jura
The circumstellar envelope is assumed to oscillate in a p erio dic manner Further
more mass loss is neglected Mass loss leads to an additional outward movement
accompanied by additional expansion which causes additional adiabatic co oling
Therefore the neglection of mass loss systematically underestimates the e ect dis
cussed in this chapter which is the temp oral sup erco oling of the gas b elow its RE
temp erature during the phases of re expansion According to these assumptions
the uid element exactly returns to its starting p oint cf Fig and all hydrody
namic and thermo dynamic quantities vary p erio dically in time with the pulsation
p erio d of the star We will concentrate on this most simple case which is considered
as b eing typical for the envelopes of pulsating stars
vs.t
∆r(t) r
∆r
t
Figure Lagrangian tra jectories schematically upp er panel and distance
b etween two neighboring uid elements lower panel in the sho ck levitated cir
cumstellar envelope of a pulsating star without mass loss v is the sho ck velocity
s
in the lab oratory frame On average the acceleration by sho cks is balanced by
gravitational deceleration The re expansion of the gas b etween the sho cks is
caused by the phase shift of the ballistic tra jectories according to the gravitation
of the star
The p erio dic situation as sketched in Fig can b e divided into two phases the
sho ck transition and the re expansion of the gas Both pro cesses are examined in
the following in order nd an appropriate prescription of the p erio dic b oundary
conditions which the gas elements are exp osed to
Fleischer et al have p ointed out that a multi perio dic or even chaotic oscillation of the
envelope is p ossible even if the stellar pulsation is p erfectly p erio dic
THE MODEL
Sho ck Transitions
The sho ck transitions are treated by applying the Rankine Hugoniot relations e g
Landau Lifschitz The sho ck front is considered as in nitesimally thin and the
actual transition pro cess as instantaneous The jump conditions for a plane parallel
p erp endicular sho ck v to the front are given by the equations of the conservation
of mass momentum and energy in a comoving frame of the sho ck front Negligible
magnetic elds and vanishing contributions of the radiative ux are assumed
v v
p v p v
h h v v
Equations relate the hydrodynamic and thermo dynamic prop erties of the up
stream ow index pre sho ck to those of the downstream ow index p ost
sho ck h e p is the enthalpy and e the internal energy p er mass of the gas
Together with the equation of state cf Chapter the p ost sho ck quantities can
b e calculated from the pre sho ck thermo dynamic state and the sho ck velocity v
Strictly sp eaking the so de ned p ost sho ck state refers to a de nite time after the
passage of the sho ck wave when the gas has just relaxed to its steady state so that
the equation of state is applicable again This time is assumed to b e small compared
to pulsation p erio d P and the radiative co oling time scale
Due to the non trivial equation of state involved the actual solution of the system
of equations requires an iteration The following simple iteration scheme is
applied which is found to converge reliably
Start with a compression ratio of four v v
Put h h v v and p p v v v
Calculate the p ost sho ck density according to the equation
E D
dv
of state in the form p h J
dl
De ne j v v j
Perform one iteration step by v v v
Go back to step unless
D E
dv
The radiation eld J the velocity gradient and the element abundances are
i
dl
additional parameters for the calculation of the equation of state cf Chapter
These parameters are set to xed values during the calculations and are assumed to
b e equal on b oth sides of the front cf Sect
The calculated compression ratios for strong sho cks v pre sho ck sound
sp eed are found to b e larger than the maximum value of for an ideal gas Typical
values range from to dep ending on the sho ck velocity This e ect is caused
by the disso ciation and ionization p otential energy terms in the equation of state
According to the theoretical description outlined the gas is completely disso ciated
CHAPTER SHOCK INDUCED CONDENSATION IN RCB STARS
and partially ionized by strong sho cks Since the dissipated sho ck energy is partly
consumed in order to break the chemical b onds and to ionize the atoms the p ost
sho ck gas temp eratures are found to b e lower compared to an ideal gas Typical
values are K to K for sho ck velocities km s to km s The p ost
sho ck temp eratures are higher compared to a hydrogen rich gas b ecause of the more
massive hydrogen de cient gas
Re Expansion Phases
Between the sho cks the change of the internal energy of the gas is calculated via the
rst law of thermo dynamics cf Sect In order to nd an appropriate descrip
tion of the re expansion pro cess a suitable state variable is chosen whose explicit
time dep endence can b e prescrib ed Of course this is an approximate pro cedure A
consistent physical description would include time dep endent hydrodynamics and
cannot by limited to single uid element considerations The following approach is
adapted to the exp erience of hydrodynamic mo del calculations
t MOD P
p t p p p
P
p is the pre sho ck and p the p ost sho ck gas pressure cf Eq MOD the
mo dulo function and the adiabatic index of the gas which is assumed to b e
in this context The main idea of this approach is to assume that the gas pressure
monotonically decreases b etween the sho cks with a p ower law in time The approach
is motivated as follows
i In the limiting adiabatic case of negligible radiative heating co oling where
pV const the volume varies like a saw tooth function in time
t MOD P
V t V V V
min min
P
V is the pre sho ck volume and V the minimum volume of a xed uid element
min
during one p erio dic cycle V equals the p ost sho ck volume V in the adiabatic
min
case Equation provides a go o d t to the volume variations found in time
dep endent hydrodynamic mo del calculations e g Bowen see his Fig It is
readily obtained if the gas actually b ehaves purely ballistic as sketched in Fig
In this case the Lagrangian tra jectories r t are second order p olynomials and the
distance b etween two neighboring uid elements r varies like a saw tooth func
tions in time Supp osed that the amplitude of radial motion is small compared to
the absolute radial distance the enclosed volume V r r is prop ortional to r
and therefore also a saw tooth function In general taking into account radia
tive heating and co oling the calculated volume variation do es not di er much from
Eq as demonstrated in Fig The main di erence is that in this case V
min
is smaller than V which will b ecome more clear in the next paragraph
THE MODEL
ii According to Eq the gas pressure varies on a time scale of P Hence fast
radiative co oling with P automatically pro ceeds isobaricly Consequently
co ol
the uid element compresses by fast co oling which esp ecially o ccurs shortly after
the passage of a sho ck wave where the gas is hot and co ols very e ciently This
matches well with the results of stationary sho ck mo dels e g Hollenbach McKee
and Neufeld Hollenbach where the initial sho ck compression of
ab out a factor is followed by a subsequent p ost sho ck compression which amounts
up to a factor The reason for this b ehavior is that the ow is subsonic b ehind
the front so that pressure balance can establish The cited calculations show that
p const is valid within accuracy in the whole p ost sho ck region in the case of
a stationary ow Therefore again considering the p erio dic sho cks V in general
min
do es not coincide with V but is substantially smaller due to subsequent p ost sho ck
compression by radiative co oling The p ost sho ck co oling usually pro ceeds so fast
that Eq is still a go o d approximation of the resulting overall volume variation
iii According to the assumption that the pressure variation is monotonically de
creasing b etween the sho cks the amplitude of pressure variation is given by the jump
conditions Eq There is no need to introduce additional free parameters in
order to describ e the amplitude of the cyclic variations caused by the sho cks Esp e
cially V is a result of the calculations If the volume was chosen to b e prescrib ed
min
one free parameter that is V would b e additionally required
min
The results are not much a ected by the assumed slop e of the pressure time
dep endence Additional calculations have b een carried out with di erent values
for and also calculations where the volume was chosen to b e prescrib ed using
Eq with V as additional free parameter The results are very similar
min
only the existence of a p erio dical p erturbation of the uid element and its amplitude
characterized by the sho ck velocity are apparently imp ortant
Thermo dynamics
The time evolution of the internal energy e of the considered uid element during
the re expansion phases is straightforwardly calculated via the rst law of thermo
dynamics
de dV
b
p Q
rad
dt dt
Since the gas pressure is chosen to b e prescrib ed it is more convenient to consider
the sp eci c enthalpy h e p and to solve instead
dp dh
b
V Q
rad
dt dt
Equation is solved by implicit numerical integration with adaptive step size
control The key for this calculation is the determination of the state of the gas and
Consequently the tra jectories are in fact not purely ballistic In the hot p ost sho ck regions
the pressure gradient provides a non negligible hydrodynamic force
CHAPTER SHOCK INDUCED CONDENSATION IN RCB STARS
D E
dv
the net radiative heating co oling rate as function of p h J and which yields V
dl
and Q at every instant of time so that Eq b ecomes an ordinary di erential
rad
equation The gas temp erature T as function of time is an implicit result of these
g
calculations
The Mo deling Pro cedure
A schematic description of the thermo dynamic pro cesses and the mo deling pro cedure
is sketched in Fig A uid element in the CSE of a pulsating RCB star is
considered In phase the element is hit by a propagating sho ck wave where it is
heated and compressed V During phase it co ols down and further compresses
due to fast approximately isobaric radiative co oling V According to the
min
p erio dicity in these envelopes the gas element nally re expands during the rest of
each p erio dical cycle in phase V These three phases rep eat p erio dically
thermodynamic processes model
1) shock solution of the transition jump conditions
2) post−shock cooling solution of dp ^ dh = + Q dt V dt rad for given p=p(t) 3) re−expansion } : shock heating : radiative heating / cooling
: adiabatic cooling
Figure The three p erio dically rep eating phases of sho ck transition p ost
sho ck co oling and re expansion for a uid element in the circumstellar envelope
of a pulsating star Wiggled arrows indicate net radiative co oling in phase or
heating in phase The theoretical description of the pro cesses is outlined on
the right hand side
THE MODEL
The mo del simulates these pro cesses by solving the sho ck jump conditions at the
instants of time where the sho ck waves hit the gas element t f P P g and
by calculating the rst law of thermo dynamics in the meantime The calculations
are continued until the variations of the thermo dynamic quantities in the gas ele
ment b ecome p erio dically Usually to p erio ds are required in order to achieve
p erio dicity In detail the calculations pro ceed as follows
D E
dv
Cho ose a xed radiation eld J a xed velocity gradient a xed
dl
sho ck velocity v and at xed pre sho ck gas pressure p Start with an
arbitrary initial enthalpy h
At every full p erio d solve the jump conditions Eq for the p ost
sho ck state p h
Consider the time variation of the gas pressure during the forthcoming
p erio d as explicitly given according to Eq and calculate the en
thalpy according to Eq yielding p h at the end of this p erio d
Go back to step unless all variations have b ecome p erio dic
Overview of Introduced Parameters
The nal p erio dic results of the mo del dep end on the following parameters
Two parameters for the description of the background continuous ra
diation eld T and r R cf Eq
e
Two parameters for the strength and the frequency of the propagating
sho ck waves v and P
Parameters for the comp osition and the overall density of the considered
gas element and p
i
Two additional parameters whose e ects on the results are small that
D E
dv
is the lo cal mean velocity gradient and the p ower index for the
dl
explicit pressure time dep endence during re expansion cf Eq
Examined Range of Parameters
T and r R The e ective temp erature of the central RCB star is assumed to
e
b e K throughout this chapter which apparently is a representative value for
this class of stars Lambert Rao The variation of the radial p osition of the
uid element as sketched in Fig is assumed to b e small compared to R so
that r R for simplicity is xed Thereby the mean intensity J is assumed to
b e constant during the calculations In contrast a signi cant radial motion of the
uid element would imply an additional time dep endence of J which is regarded as
further complication of the mo del of minor imp ortance Radial distances of R
are considered
CHAPTER SHOCK INDUCED CONDENSATION IN RCB STARS
P The pulsation p erio d is assumed to b e days which is the value suggested by
Fernie et al for R CrB Since other RCB stars show very similar values for P
cf App endix A this parameter is also xed for the calculations
v The sho ck velocity is uncertain may b e di erent for di erent RCB stars and
will furthermore dep end on the considered radial distance Time dep endent mo dels
for the circumstellar envelopes of long p erio d variables Bowen Fleischer et al
Winters et al Feuchtinger et al indicate that the sho cks b egin to
develop somewhere b elow the photosphere where the velocity variation is usually a
few km s The sho cks considerably steep en up according to the exp onential density
gradient in the outer atmosphere and so on reach sho ck velocities of km s the
sho ck velocity can approximately b e identi ed with the amplitude of velocity jumps
o ccurring in these mo dels At larger radial distances the density gradient b ecomes
smaller and the sho ck velocity usually tends to decrease again leading to km s
or even almost zero dep ending on the mo del It is unclear whether these results
can b e adopted to RCB stars Large photospheric velocity variations of km
implying sho ck velocities of km s have b een observed for RY Sgr which is
the strongest known pulsating RCB star cf App endix A These measurements
refer to the line formation region i e to the photosphere of the star Considerably
stronger sho cks can b e exp ected in the CSE However RY Sgr is an exceptional
case Other RCB stars show radial velocity variations of ab out to km s
but no line splitting which is an indicator for sho ck activity in the photosphere
App endix A Sho ck activity in the CSE is probably not directly detectable at
least not at maximum light apart from the decline events when the star is to o
bright Therefore the questions of the existence of sho ck waves in the CSEs of
RCB stars and their velocities cannot b e decided by observations yet This work
presupp oses the presence of sho ck waves in RCB star envelopes since i RCB stars
show considerable radial velocity variations at the photosphere and ii even small
amplitude waves are known to steep en up to considerable sho ck waves in the CSE
from theory Sho ck velocities of km s are considered
The elemental abundances of the prototype star R CrB are adopted from Cottrell
i
Lambert cf Fig
p The pre sho ck pressure of the uid elements is varied indep endently of r R
Although the mean gas pressure can b e exp ected to monotonically decrease with
increasing radial distance the actual density structure of the circumstellar envelopes
The following theoretical consideration p oints to larger sho ck velocities in RCB star envelopes
compared to AGB star envelopes For sho ck levitated CSEs as sketched in Fig v g P
e
is a go o d approximation g is the gravitational deceleration corrected for radiative acceleration
e
The gravitational force at the photosphere of an RCB star is ab out times larger than that of
an AGB star roughly assuming equal stellar masses and luminosities and the pulsation p erio d
ab out a factor of smaller yielding ab out times larger sho ck velocities compared to AGB stars
If km s is considered as a typical value for AGB stars values of ab out km s are deduced
for the envelopes of RCB stars
RESULTS
of RCB stars is not known Pre sho ck gas pressures of dyn cm are
considered
The mean lo cal velocity gradient is no crucial parameter to the mo del and is assumed
E D
dv
v R The stellar radius is assumed to b e R R in this to b e given by
dl
context Fernie The p ower index for the prescription of the time dep endence
of the gas pressure is set to
Results
Cyclic Variations in the Periodically Sho cked Fluid Elements
An example of the results for the cyclic variations of the thermo dynamic state
variables in xed p erio dically sho cked circumstellar uid elements is depicted in
Fig The rst p erio ds after p erio dicity has b een achieved are plotted on a
linear time scale The p ost sho ck gas temp erature is found to b e K in the
considered case which is out of the depicted range During the rst of the
p erio d hours the gas e ciently co ols down to K which causes further
compression The sho ck compression factor is and the p ost sho ck compression
factor is An attempt to depict the di erence b etween the sho ck and the p ost
sho ck compression is made in the middle panel but on this linear scale the total
sho ck p ost sho ck compression phase app ears like a single almost instantaneous
pro cess The gas approximately re expands adiabatically after this compression A
RE
saw tooth like b ehavior of the volume results The temp erature reaches T after
g
RE
ab out of the p erio d and is clearly b elow T afterwards Here and in the
g
RE
remainder of this chapter T denotes the rst stable high temp erature solution
g
RE
of radiative equilibrium as de ned in Chapter T is density dependent and
g
hence not constant
RE
In fact the value of T is practically meaningless for the gas in the depicted case
g
of p dyn cm which corresp onds to a density variation of n
He
cm The time dep endent temp erature of the gas is essentially
determined by the sho ck transition and the e ciency of the radiative co oling at high
temp eratures during the p ost sho ck co oling phase These two phases determine the
start temp erature and the total de compression factor for the forthcoming phase
of re expansion which pro ceeds approximately adiabatically RE is never realized
and cannot b e used to determine the temp erature of the gas
Further details are shown in Fig where the same setting of the parameters
is investigated except for a larger sho ck velocity of km s In this case the
p ost sho ck temp erature is found to b e K and the sho ck and p ost sho ck
compression factors are and resp ectively All radiative pro cesses cause
net co oling b ehind the sho ck and since the co oling time scale is as short as initially
s the uid element very quickly co ols down due to radiative losses Within
the rst of the p erio d hours the temp erature drops to K
CHAPTER SHOCK INDUCED CONDENSATION IN RCB STARS
Figure Time variations in a xed p erio dically sho cked circumstellar uid
element of an RCB star The gas pressure assumed upp er panel and the
sp eci c volume and the gas temp erature calculated middle and lower panel is
plotted for distance r R sho ck velocity v km s and pre sho ck pressure
p dyn cm The dotted lines in the middle panel indicate the pre
sho ck p ost sho ck and minimum volume The dashed line in the lower panel
depicts the radiative equilibrium gas temp erature
RESULTS
During this phase which is plotted on a logarithmic time scale in Fig the
uid element compresses as can b e seen from the increasing density in the upp er
RE
panel The temp erature reaches T after of the p erio d hours During
g
the remaining time of the p erio d the uid element re expands by a total factor of
This re expansion causes intense adiabatic co oling as indicated by the co oling
rate Q V dp dt in Fig which is the concurring rate for dh dt in Eq
adb
RE
Consequently the gas temp erature decreases b elow T and the total net radiative
g
heating function Q changes its sign note the twofold logarithmic y axis in the
rad
lower panel of Fig
The decisive p oint for the thermal b ehavior of this uid element is reached now The
p oint is related to the rst intermediate maximum of Q T depicted in Fig
rad g
at a temp erature of K The question is whether or not the adiabatic co oling
of the gas is su cient in order to overcome this maximum If the answer is no the
adiabatic co oling of the gas is comp ensated by net radiative heating the co oling of
the gas is stopp ed and the re expansion pro ceeds more or less isothermally with
RE
If the answer is yes the adiabatic rate dominates during the remaining T T
g
g
time of the p erio d at lower gas temp eratures jQ j is usually smaller compared to
rad
the rst maximum and the re expansion approximately pro ceeds adiabatically
The character of the re expansion pro cess b eing either isothermal or adi
abatic is decided by the e ciency of the radiative heating in the predom
inantly neutral atomic phase of the gas caused by line and b ound free
transitions prior to molecule formation
In the gure the adiabatic co oling exceeds the net radiative heating rate jQ j
adb
b
jQ j and thus the co oling of the gas continues Subsequently the gas b ecomes
rad
co ol enough in order to allow for considerable molecule formation While so far the
line heating co oling rate has b een dominating now the vibrational and rotational
heating co oling functions enter into comp etition and so on b ecome more imp ortant
than Q Since the molecular rates cause net radiative co oling for the present
Lines
Q again changes its sign and the adiabatic co oling of the gas is nally even
rad
supp orted by net radiative co oling However Q remains the most imp ortant rate
adb
during the re expansion which is plotted on a linear scale in Fig Consequently
the re expansion which takes ab out of the p erio d approximately is an adiabatic
pro cess
Thus the gas temp erature is lower than in radiative equilibrium almost all the time
The gas temp erature nally reaches a minimum value of K and is b elow K
for ab out of the p erio d at densities n cm These
He
are thermo dynamic conditions favorable for e ective carb on nucleation as will b e
discussed in the Sect
This statement refers to cases where the gas rst of all is capable to quickly radiate away the
excess internal energy dissipated by a sho ck cf Sect
CHAPTER SHOCK INDUCED CONDENSATION IN RCB STARS
Figure Details of the time variations in a xed p erio dically sho cked circumstellar
uid element of an RCB star The x axis is broken in this plot The rst of the
p erio d are depicted on a logarithmic scale whereas the other are plotted linearly
The upp er panel shows the gas temp erature full line the RE temp erature dashed line
and the total helium particle density dotted line The middle and lower panel depict
the heating and co oling rates resp ectively The thick full line shows the total net radiative
rate and the other thin dotted and dashed lines depict partial rates theses are the net
free free b ound free atomic line vibrational and rotational rates The latter rates are the
sums of the radiative gains losses caused by the indicated transition type i e all b ound
free transitions all vibrational transitions etc The adiabatic co oling rate is depicted by
the thick dotted line Parameters r R v km s and p dyn cm
RESULTS
Figure Cyclic variations of density and temp erature in p erio dically sho cked
uid elements at r R The elements di er by di erent values of the pre sho ck
gas pressure The gray and black cycles depict the results for sho ck velocities
v km s and v km s resp ectively The short dashed lines indicate
the sho ck transitions The long dashed line shows the radiative equilibrium gas
temp erature
Dep endence on Density
The results discussed so far have b een calculated for a particular pre sho ck gas
pressure p which xes the mean density of the gas during the p erio dic variations
Since the e ciency of the radiative heating co oling is strongly a ected by the den
sity cf Chapter the thermal b ehavior of the gas in resp onse to the p erio dic
sho cks is quite di erent for other densities This density dependence is depicted
in Fig where the p erio dically rep eating thermo dynamic pro cesses app ear as
counterclockwise cycles
Concerning very large densities the three phases sketched in Fig are well sep
arated and an almost triangular cycle results see l h s of the gure Beginning
with the p ost sho ck state upp er corner the gas element reaches RE left corner
CHAPTER SHOCK INDUCED CONDENSATION IN RCB STARS
within of the p erio d due to e cient approximately isobaric radiative co ol
ing The slight departure from a straight line on this co oling track is related to
the recombination of He where the mean gas particle weight changes by a factor
of Since the coupling to RE is strong at these densities the adiabatic co oling
rates are negligible compared to the radiative rates in the subsequent phase of re
RE
expansion Therefore the gas temp erature stays close to T during this pro cess
g
leading to the right corner Finally the uid element is sho cked and jumps to the
upp er corner again etc The triangular cycles are typical results for the limiting
case of isothermal sho cks For ab out of the p erio d the element is close to
RE Therefore the condition of RE can b e used to determine the gas temp erature
during the overwhelming part of the p erio d However this pro cedure is only feasi
ble for large densities Roughly sp eaking RE is a reliable criterion for temp erature
determination for densities n cm
He
On the other extreme considering the case of very small densities the excess internal
energy dissipated by one sho ck cannot b e radiated away during one p erio d and
consequently the gas never approaches RE On the contrary a corona like situation
results where the gas is heated up to extremely high temp eratures due to the energy
dissipation of waves For example in the cycle on the r h s for v km s the gas
is predominantly ionized and always hotter than K This b ehavior is typical
for the limiting case of adiabatic sho cks The resulting p erio dic tracks see r h s of
Fig consist of sho ck transition He recombination and adiabatic re expansion
Once the gas has recombined its co oling time scale b ecomes much larger than the
p erio d
Concerning an intermediate range of densities the radiative energy exchange is
e cient enough in order to cause a fast relaxation of the gas towards RE after the
sho cks but is not to o e cient in order to b e maintain balance with the adiabatic
co oling rates during the phases of re expansion In this case the uid element co ols
RE
down far b elow T as discussed in Sect The small kinks on the almost
g
adiabatic tracks at the lower part of Fig are caused by molecule formation
mainly CO and C where the further co oling of the gas is delayed by the lib eration
of molecule disso ciation energy
A sup erco oling of the gas o ccurs within a distinct density interval caused
by a two step pro cess of radiative co oling at high temp eratures followed
by adiabatic co oling at low temp eratures
The density dependence depicted in Fig is a natural consequence of the density
dep endent co oling heating e ciency of the gas Regarding the broad sp ectrum of
densities encountered in CSEs it seems inevitable that somewhere in the envelope
of a pulsating star the density is just appropriate for this e ect
RESULTS
Dep endence on Sho ck Velocity
Larger sho ck velocities v pro duce higher maximum temp eratures b ehind the sho cks
but do also allow for lower minimum temp eratures At rst sight this dep endence
might b e surprising but it is actually straightforward The total compression ratios
are larger for strong sho cks implying larger adiabatic co oling rates during the phases
of re expansion The sho ck velocity can b e regarded as a measure for the amplitude
of the p erturbation causing b oth up and downward deviations from RE
Preconditions for Carb on Nucleation
In the following the p ossibility of e ective carb on nucleation to take place in these
p erio dically sho cked uid elements is investigated Considering the densities en
countered in circumstellar envelopes the size of the critical cluster usually is as small
as atoms Therefore the chemical reactions involved in the formation pro cess of
such seeds are assumed to b e controlled by the gas temp erature rather than the RE
temp erature of macroscopic dust particles the dust temp erature According to
The formation of macroscopic dust grains is not discussed in this work According to the
assumed radiation eld macroscopic grains strictly sp eaking graphite grains in the small particle
limit of Mie theory de nitely evaporate at the small radial distances under investigation b ecause
their internal temp eratures are much to o high Fadeyev In contrast large molecules might
b e stable provided that their optical and UV absorption prop erties are comparable smaller
Apparently the formation of dust close to the star must b e accompanied by some kind of shielding
Absorption by the dust itself is a promising candidate in order to blo ck o the radiation eld and
to cause a lo cal reduction of the dust temp erature The phase transition from gas to dust can
easily increase the absorption co e cient of the gas dust mixture by a factor of Therefore
once a dust cloud has formed the radiation ows around the optically thick region and new dust
particles may condense and grow in the shadow of this cloud whereas the grains at the inner edge
of the cloud towards the stars will evaporate A quasi stable situation might b e conceivable where
the dust cloud survives the strong radiation eld via self shielding in a dynamical sense
In contrast the formation of a spherical dust shell seems to b e absolutely imp ossible close to the
star Spherical dust formation in a distinct radial layer causes an increase of the dust temp eratures
in the layers within the shell via back warming and has almost no e ect on the dust temp erature
in the outer layers b ecause the radiation ux is not blo cked but just transmitted
Thus an instability caused by dust formation p ossibly exists which favors dust cloud formation
rather than dust shell formation in cases where the gas is su ciently dense and co ol for nucleation
but a strong radiation eld hinders the seed particles to grow further
Di cult questions are raised by these considerations which may b e imp ortant not only for the
dust formation in RCB stars but for any harsh radiation eld environments e g in Wolf Rayet
star envelopes In order to clarify these questions at least D mo del calculations are required
which must include radiative transfer and time dep endent dust formation destruction a very
challenging problem which go es far b eyond the scop e of this work Therefore I will concentrate
on the rst necessary step concerning the transition from the gas phase to dust particles which is
the formation of seed particles and leave aside the problem of the thermal stability of macroscopic
dust grains Dust formation close to the star in any case must pro ceed via this rst step
CHAPTER SHOCK INDUCED CONDENSATION IN RCB STARS
this assumption the sup ersaturation ratio S is calculated as
n k T
C g
S
p T
sat g
where n is the particle density of neutral carb on atoms in the gas phase and p
C sat
is the vapor pressure of carb on atoms over the bulk material graphite at gas
temp erature A necessary condition for carb on nucleation to take place is S
Figure shows this condition and gives an overview of all results concerning the
p erio dically sho cked uid elements at r R The minimum gas temp erature
o ccurring in one p erio dic cycle is depicted as a function of the mean helium particle
density during the cycle which is de ned as
Z
P
n dt n
He He
P
Figure demonstrates that the conditions appropriate for e ective carb on nucle
ation are temp orarily present in the p erio dically sho cked uid elements concerning
a distinct density interval bracketed by n cm and cm In con
He
trast dust formation is thermo dynamically imp ossible at r R if the temp erature
RE
which is the high temp erature solution of radiative equi of the gas is given by T
g
librium cf Chapter
The nucleation rate J which is the number of seed particles forming p er volume and
p er second is calculated by applying classical nucleation theory Gail et al
The nucleation rates are plotted as contour lines in Fig Considering the inter
esting density interval nucleation rates of J n s o ccur
He
which are large values compared to those exp erienced from time dep endent mo d
els for the envelopes of long p erio d variables Fleischer et al indicating that
e cient carb on nucleation may take place
The total growth time for a seed particle to reach a macroscopic size say m
can b e estimated by considering the three dimensional growth by accretion taking
into account all thermally impinging carb on b earing sp ecies except for the amount
of carb on lo cked in CO The dust temp erature is assumed to b e su ciently low so
that dust growth is p ossible cf fo otnote
m
tot
gr
V n n v
C O th
V a is the monomer volume a cm for graphite Gail et al
q
k T m the thermal velocity which is a bit smaller if molecules are con v
g C th
sidered and the sticking probability Assuming a lower limit is calculated
for the actual total growth time
In order to cause a RCB decline event the total growth time in the dust forming
region should not exceed the time scale of the initial drop of the light curve which
is of the order of a few weeks Feast In any case the total growth time must
RESULTS
Figure Minimum gas temp eratures and the p ossibility of carb on nucleation
to o ccur at a radial distance of r R The full lines and p oints depict
the minimum temp eratures o ccurring in one p erio dic cycle as function of the
mean gas density during the cycle for two di erent sho ck velocities as indicated
The dashed line is the radiative equilibrium gas temp erature The lower part
of the gure sketches the condensation regime i e the region of favorable
thermo dynamic conditions for carb on nucleation Only b elow the S limit
the gas is sup ersaturated with resp ect to graphite Contour lines of the logarithm
of the classical nucleation rate J cm s are plotted On the right edge the
growth time for a seed particle to reach the macroscopic size of m exceeds
one pulsation p erio d
CHAPTER SHOCK INDUCED CONDENSATION IN RCB STARS
not exceed the pulsation p erio d of the star This condition is additionally shown
in Fig constituting an absolute lower limit for the density in the dust forming
region which can b e resp onsible for an RCB decline event As depicted in Fig
this condition is just ful lled within the density interval where the sup erco oling of
the gas o ccurs
Dep endence on Radial Distance
The predictions of the mo del concerning the minimum radial distance required for
e cient carb on nucleation are of sp ecial interest Previous mo deling of dust forma
tion in the CSEs of RCB stars has su ered from the necessity to consider rather large
radial distances in order to obtain su ciently low temp eratures whereas observa
tions tell us that dust formation probably o ccurs much closer to the star cf Sect
and App endix A
The dep endence of the results of this mo del on the parameter r R is depicted in
Table where the minimum of the T curve cf Fig is stated in the rst
g min
row and the interval of mean helium particle densities with T n K
g min He
is stated in the second row
Table Results as function of radial distance and sho ck velocity
v km s v km s
K K
r R
cm
K K
r R
cm
K K
r R
cm cm
K K
r R
cm cm
The general tendency of the results is as exp ected the larger the radial distance to
the star the easier low temp eratures appropriate for carb on nucleation are achiev
able However in the examined case of time dep endent non RE this dep endence
is much less distinctive than exp erienced from RE A change of the sho ck velocity
for example can easily cause very di erent conditions Considering the km s
sho cks gas temp eratures lower than K are pro duced for r R whereas con
cerning the km s sho cks even lower gas temp eratures o ccur for all considered
radial distances
Compared to the in uence of r R the density dependence is very selective The
temp oral sup erco oling of the gas b ehind sho ck waves is only p ossible within a sp ecial
narrow density interval The deep er the temp erature minimum the wider this
interval centered around a few cm in all considered cases The particular
DISCUSSION
density range is in go o d agreement with the estimates presented by Go eres
for the density of the dust forming regions in the envelopes of RCB stars
Therefore the in uence of r R on the results is less pronounced than the in uence
of v and n The predicted values for the condensation distance are hence
He
not very distinct The essential outcome of this mo del it that sho ck waves are
principally capable to pro duce low temp eratures appropriate for carb on nucleation
at radial distances as small as R Strictly sp eaking this statement refers to
the investigated case of p erio dic sho cks However since the basic pro cesses of sho ck
heating and compression followed by re expansion is supp osed to b e an inevitable
straightforward consequence of circumstellar sho ck waves I conclude
Favorable thermo dynamic conditions for carb on nucleation o ccur when
ever a su ciently strong sho ck wave encounters those parts of the cir
cumstellar envelope where the gas density is just appropriate for the
two step co oling pro cess describ ed in Sect
Discussion
The thermo dynamic b ehavior of p erio dically sho cked uid elements in the CSEs of
pulsating RCB stars has b een investigated The complex interplay b etween sho ck
heating radiative heating and co oling and adiabatic co oling has b een examined
Large time variations of the thermo dynamic conditions in xed uid elements are
found to o ccur in this situation comprising orders of magnitudes for b oth the
gas density and the gas temp erature dep ending on the sho ck velocity
The calculations provide a hypothesis for the physical cause of the onset of dust
formation close to a pulsating RCB star connected with the question of the trigger
of the RCB type decline events As a consequence of the presence of sho ck waves
the gas is usually not in RE In the time dep endent non RE situation favorable
conditions for carb on nucleation are found to b e temp orarily present close to the
star despite of the high e ective temp eratures of the RCB stars The following
two basic conditions are required in order to allow for e ective condensation close
to the star The gas density must b e bracketed by ab out cm and the
sho ck velocity must b e larger than ab out km s The results can b e generalized
to arbitrary sho ck waves no matter how the sho ck was created
In the following I will brie y summarize the advantages of this mo del on the one
hand and the main p oints of criticism on the other hand For comparison an
overview of previously published mo dels can b e found in App endix A A short
discussion of the p ossible links to observations completes this chapter
Advantages of the Mo del
The obvious attraction of the mo del is that dust formation close to the star
is explained from physics The mo del predicts that temp eratures as low as
CHAPTER SHOCK INDUCED CONDENSATION IN RCB STARS
K can b e present at radial distances as small as R According to the
calculations these conditions last for more than half of the p erio d which in
fact means favorable conditions for carb on nucleation
The condensation distances are found to b e as small as R in agree
ment with the values inferred from observations e g Clayton et al
cf Sect In contrast none of the published mo dels can explain this
fundamental feature of the dust formation in RCB envelopes in a quantitative
way
The narrow density interval necessary for the two step co oling pro cess causing
the low temp eratures agrees with previous estimates of the density in the nu
cleation zone of RCB stars Go eres This agreement is not self evident
The density dependence of the mo del is caused by a completely di erent phys
ical prop erty of the gas which is the radiative heating co oling e ciency This
e ciency decreases with decreasing gas density due to increasing non LTE
e ects
The dep endency of the mo del on the e ective temp erature of the star is small
Preliminary test calculations with T K yield similar results
e
as depicted ab ove Even for T K gas temp eratures b elow K
e
o ccur at R for a sho ck velocity of km s This insensibility of the
mo del with resp ect to T apparently agrees with observations since the RCB
e
phenomenon is rep orted for a variety of stars comprising e ective temp eratures
of K In contrast all other prop osed physical mo dels exhibit a
pronounced T dep endence
e
Criticism
Sho ck activity in the photosphere of RCB stars is only con rmed for one
exceptional ob ject which is RY Sgr Other RCB stars show considerable radial
velocity variations but no sho ck activity in the photosphere as inferred from
absorption line splitting cf App endix A Therefore the presence of sho ck
waves in RCB envelopes is generally doubted
Comments The basic problem of the ab ove argument is that the observations refer
to the photosphere of the star whereas informations ab out the conditions in the
circumstellar envelope are required A direct observation of circumstellar sho ck
waves is very di cult due to contrast e ects with regard to the bright star A
chance to observe circumstellar prop erties may b e present during the early phases
of the decline events At the present state of observations no precise informations
ab out circumstellar sho ck activity have b een deduced at least one cannot rule out
the p ossibility that sho ck waves are in fact present in all RCB stars envelopes From
theory even small amplitude subsonic waves in the photosphere of the star are
known to b e capable to steep en up to considerably strong sho ck waves in the CSE
dep ending on the photospheric density gradient Just in those cases where the
initial radial pulsation is small the density gradient turns out to b e large which
ampli es the waves
DISCUSSION
RCB stars show similar decline events with resp ect to decline frequency time
scales and decline amplitude regardless of their sp ecial pulsation prop erties
e g the radial velocity amplitudes Therefore a causal connection b etween
pulsation and dust formation seems susp ect
Comments This is the most serious ob jection to the prop osed mo del Of course
the comments on the upp er p oint may b e rep eated but in fact this criticism is more
substantial In any case the pulsation of the star should b e resp onsible for the cir
cumstellar sho ck waves so that some correlations are exp ected From observations
the often claimed correlation b etween the b egin of a decline and the pulsation phase
of the star cf App endix A would contradict the ab ove argument but observational
evidence is p o or concerning this correlation cf App endix A
The mo del at rst sight seems to suggest dust shell formation rather than
dust cloud formation as far as a spherically symmetric pulsation of the star is
considered Dust shell formation however can b e ruled out from observations
cf App endix A Feast argues that the prop osed instability which
might b e resp onsible for a fragmentation of the forming dust shell into dust
clouds cf fo otnote is not very convincing at least cannot explain why
always only a very few probably just one dust cloud p er pulsation p erio d
survives the dust destroying radiation eld
Comments This is certainly a weak p oint of the mo del However the mo del in
fact do es not make any predictions of what happ ens after the onset of nucleation
It only intends to show how the onset of nucleation is p ossible near to the star
which in any case must b e the starting p oint of the decline events Once the gas
b ecomes optically thick due to dust formation the basic assumption of optical thin
ness breaks down and the radiation eld must b e calculated by means of the solution
of a non lo cal radiative transfer It is principally not p ossible to mo del the for
mation of a cloud without taking into account the imp ortant physical interactions
in a more dimensional way All mo dels published so far su er from this inconsis
tency A trivial way out is the prescription of a non spherical situation prior to
dust formation One could for example consider a sup erp osition of non spherical
sho ck waves due to non radial pulsations or one could prescrib e the existence of
inhomogeneities In b oth cases the conditions for dust formation are di erent in
neighboring uid elements which might lead to the formation of dust clouds The
presented thermo dynamic metho ds are of course applicable to such prescrib ed sit
uations However in my p ersonal opinion such assumptions do not really explain
anything What is necessary is the mo deling of the physical pro cess of cloud forma
tion from a previously homogeneous situation no matter whether this pro cess takes
place prior to dust formation or whether the cause of cloud formation is related to
the pro cess of dust formation itself The problem of the survival of the dust close
to the star is serious and is not restricted to the prop osed mo del
Interpretations of Observations with Regard to the Mo del
If the RCB decline events are in fact caused by sho ck induced condensation
similar initial thermo dynamic conditions of the gas would b e present in the
nucleation zone at the b eginning of all declines concerning for instance the gas
density This could to some extent explain the principal similarity of the light
CHAPTER SHOCK INDUCED CONDENSATION IN RCB STARS
curves concerning light amplitudes and time scales involved irresp ective of the
wide range of stellar prop erties of RCB stars such as e ective temp erature
element abundances and pulsation prop erties The mo del suggests that the
decline events are caused by a distinct physical pro cess of the gas indep endent
and apart from the star
According to the mo del a forming dust cloud in the line of sight would al
ways b e lo cated b ehind a sho ck wave whose hot p ost sho ck region might b e
resp onsible for some chromospheric line emissions as observed during the
early declines If then the entire complex of sho ck wave and dust cloud moves
farther out the sho ck encounters less and less dense parts of the circumstellar
envelope probably causing a fading of the line emissions as the decline pro
gresses Additionally the observed blue shift of the emission lines of typically
km s agrees with the prop osed scenario b ecause the sho ck propagates
outwards leaving the p ost sho ck gas with an outward directed velocity of
ab out v The sho ck induced emission lines are exp ected to b e sharp and
unp olarized in agreement with observations b ecause the gas emits undis
turb ed in front of the dust Therefore the mo del seems to generally agree
with the sp ectral prop erties and the time evolution of the observed narrow
emissions lines if interpreted as sho ck activity rather than as activity from a
static chromosphere
The prop osed mechanism constitutes a causal connection b etween sho ck waves
and dust formation in circumstellar envelopes which might naturally explain
the observed correlation b etween the b egin of a decline and the pulsation phase
of the star at maximum light e g Lawson et al supp osed that the small
range of the gas densities necessary for this mechanism is solely present at a
particular radial distance to the star and supp osed that the sho cks always
take a particular time to reach this distance
In summary the prop osed mo del provides a solution to the central problem how dust
condensation may o ccur close to the star but is certainly not capable to provide a
complete explanation of the puzzle of the RCB decline events so far More complex
mo del calculations are required in order to achieve this aim The present mo del
yields ab out the right conditions for dust formation as inferred from theory and
from observations temp eratures densities radial distances and time scales It
apparently bridges a gap b etween the theory of dust formation on the one hand and
the observations of RCB stars on the other hand cf Sect which is manifested
by the controversy ab out the condensation distances Therefore it seems promising
to include the developed thermo dynamic metho ds into more complex calculations
as a kind of starting p oint
Other results which are worth to b e mentioned are as follows The low temperature
solutions of RE found in Chapter are never found to b e realized or to have any
e ect on the results in the p erio dically sho cked situation Once the adiabatic co oling
rates are su ciently strong in order to destabilize the high temp erature solution
they are de nitely stronger than the remaining heating co oling rates around the
DISCUSSION
low temperature solutions Similarly the details of the chemistry the radiative
heating and co oling rates and the sp ectral p eculiarities of the background radiation
eld do not cause principal changes in the cyclic thermo dynamic pro cesses What
happ ens for example if another imp ortant heating co oling rate is included is that
the density interval appropriate for the two step co oling pro cess shifts a bit Hence
in contrast to the results of the thermal bifurcations discussed in Chapter the
results of this chapter have a much more general meaning
In principle the e ect discussed in this work is exp ected to o ccur in all circumstel
lar envelopes of pulsating stars not only in RCB envelopes The in uence of the
stellar parameters must b e further investigated Esp ecially the dep endences on the
e ective temp erature the pulsation prop erties of the star and the elemental abun
dances e g the H de ciency might provide an explanation for the fact that the
RCB phenomenon is restricted to a sp ecial class of ob jects Time dep endent hydro
dynamic mo del calculations are required in order to allow for a more realistic mo d
eling of the circumstellar envelopes of RCB stars Higher dimensional calculations
including radiative transfer and a time dep endent treatment of the dust complex
would b e required in order to mo del the formation and destruction of macroscopic
dust grains close to the star closely related to the self shielding in dust clouds
Chapter
Conclusions
The thermal state of diluted gases b eing sub ject to stellar radiation elds has b een
investigated Radiative heating and co oling rates have b een calculated consider
ing the typical p T range and the radiation elds present in the circumstellar
envelopes CSEs of co ol and warm stars
These studies intend to lay the foundations for theoretical metho ds to determine the
temp erature of gases under static as well as dynamic conditions As an imp ortant
ingredient such metho ds must b e part of any fundamental mo deling of CSEs esp e
cially with regard to the simulation of the chemical and dust formation pro cesses
which are known to b e strongly temp erature dep endent
The results of this work show that a non LTE treatment of the atoms and molecules
is essential in order to calculate the e ciency of the radiative heating and co oling
pro cesses in CSEs The p ossibility to include the calculated heating and co oling
rates into more complex calculations e g time dep endent hydrodynamic mo dels
is also regarded as essential A prop er coupling however can b e achieved only if
the basic assumptions are compatible Therefore a compromise metho d has b een
prop osed where the state of the gas is calculated by means of the assumption of a
steady state On one hand this metho d accounts for non LTE ionization non LTE
p opulation of the excited electronic vibrational and rotational states and optical
depths e ects of sp ectral lines in Sob olev approximation On the other hand all
macroscopic prop erties of the gas do not dep end on history and can b e calculated
as function of lo cal instantaneous physical quantities which are available in such
mo dels
Thus a thermo dynamic description has b een developed where the state of the gas is
determined by two indep endent state variables e g and T as usual in LTE plus
g
two external parameters which are the radiation eld J and the lo cal mean velocity
D E
dv
gradient The metho d go es one step b eyond LTE but do es not represent a
dl
full time dep endent non LTE approach It includes LTE as a limiting case which
o ccurs at large densities The latter is achieved by strictly including all reverse
pro cesses by means of detailed balance considerations
Three applications of this metho d have b een presented
First the top ology of the solutions of radiative equilibrium RE has b een examined
considering the CSEs of R Coronae Borealis RCB stars The results show that
the condition of RE i e the equality of radiative gains and losses can have two
or more stable temp erature solutions Two di erent types of solutions have b een
identi ed high temp erature predominantly atomic states and low temperature
CHAPTER CONCLUSIONS
predominantly molecular states The molecule rich states are found to b e substan
tially co oler than a black b o dy in RE This result is straightforward inferred from
the large sensitivity of the molecules in the infrared sp ectral region It is exp ected
that this result is valid in co ol stellar envelopes as well p ossibly with imp ortant con
sequences for the chemistry and the dust formation in these envelopes Concerning
the CSEs of warm stars with T K the high temp erature atomic solutions
e
additionally come into play which means that in principle a spatial co existence of
high temp erature and low temperature gas phases is conceivable b oth in RE and
in pressure balance with each other thermal bifurcations
Second the time scales of radiative relaxation pro cesses towards RE have b een stud
ied for the case of C stars envelopes Comparison to the other time scales involved
in the pro cess to b e mo deled yields a criterion for the applicability of the metho ds
of temp erature determination which are based on RE If the radiative co oling time
scale is much shorter than the others the character of the thermo dynamic pro cess is
approximately isothermal and the temp erature can b e calculated by means of RE In
the opp osite case the gas b ehaves more or less adiabatically The thermal relaxation
of the gas b ehind circumstellar sho ck waves has b een discussed accordingly The gas
density has b een identi ed to b e the key quantity which decides up on the character
of such relaxation With deceasing density increasing non LTE e ects lead to a de
crease of the e ciency of the radiative heating and co oling pro cesses Consequently
a gradual change of the nature of the sho cks is exp ected to o ccur around cm
changing from predominantly isothermal to predominantly adiabatic These results
strongly suggest to include time dep endent metho ds for temp erature determination
into the mo dels of the envelopes of pulsating stars
Third a mo del for p erio dically sho cked uid elements has b een developed applicable
to sho ck levitated atmospheres of pulsating stars Large time variations of the
thermo dynamic conditions are found to o ccur in such uid elements comprising
orders of magnitudes in the state variables dep endent on the sho ck velocity
With regard to RCB stars the following e ect has come to light In certain cases
after the heating and compression by a sho ck wave the gas rst radiates away the
excess internal energy dissipated by a sho ck wave and then re expands adiabatically
This two step co oling pro cess can pro duce temp eratures substantially lower than in
RE within a distinct density interval Temperatures as low as K are found to
temp orarily o ccur for sho ck velocities km s at radial distances as small as
R despite of the high e ective temp eratures of these stars Such conditions
are favorable for carb on nucleation Thus the present work states the hypothesis
that the onset of dust condensation close to the star is caused by sho ck waves which
might trigger the sp ectacular RCB type decline events
In conclusion basic studies of the thermo dynamic b ehavior of gases in circumstel
lar envelopes have b een undertaken providing new insights and new ideas on the
pro cesses leading to dust formation
App endix A
Current Status of RCB Research
This app endix intends to give a brief overview on the current status of RCB research
providing an imp ortant background for the investigations in Chapter and Since
this class of stars shows so many interesting asp ects in various elds only the topics
which provide clues on the dust formation and the decline events are summarized
The reader can nd further informations in the diploma thesis of S Friedrich
and in the recent reviews of Lambert Rao and Clayton
A General Observations
Classi cation The class of RCB stars to day comprises known ob jects in our
Galaxy Lambert Rao This number varies as a consequence of recent ob
servations and the classi cation of some ob jects is still under discussion Clayton
A certain veri cation requires at least the successful observation of one de
cline event which is quite a di cult observational task Therefore the true number
of RCB stars is undoubtfully much larger probably b etween and in the
Galaxy Lawson et al For example the third brightest RCB star on the sky
V Cen was not discovered b efore The main criterion for classi cation is
the o ccurrence of the RCB type decline events cf A Light Curves Additional
criteria are a carb on overabundance C He to and a clear but strongly
varying hydrogen de ciency among the ob jects of log H He to and
also the o ccurrence of small scale visual brightness variations RCB stars are single
stars of typical sp ectral type F G Ib with absolute brightnesses M to
V
Feast suggesting luminosities of ab out L L The stellar masses can b e
determined by pulsational mo dels yielding M Wei The e ective
temp eratures typically are T K Lambert Rao However
e
also extreme values from ab out K for WX CrA and S Aps up to ab out K
for DY Cen and V Sgr o ccur Compared to this large spread of e ective temp er
atures the decline events of the individual RCB stars show a remarkable similarity
in their decline light curves e g the decline amplitudes and time scales involved
Therefore a unique physical mechanism seems to b e resp onsible for all the events
involving dust formation which is apparently not very sensitive to the e ective
temp erature of the star
This is b ecause this ob ject is in decline most of the time
App endix A
Pulsations Besides the decline events all thoroughly observed RCB stars show
more or less p erio dical visual brightness variations with amplitudes ranging from
to mag for p erio ds of typically days Feast Lawson et al
Lawson Kilkenny For a considerable number of RCB stars radial velocity
variations have additionally b een deduced from the Doppler shift of photospheric
absorption lines yielding ab out to km s and ab out km s for the ex
ceptional case of RY Sgr The variations of velocity and brightness usually o ccur in
phase suggesting the RCB stars to b e irregular radial pulsators Lawson et al
There are several clues that the pulsations have a direct feedback on the dynam
ics and the chemistry of the outer atmospheres of these stars The most extreme
pulsator RY Sgr shows phase correlated line splitting which is interpreted as the
propagation of a sho ck wave through the atmosphere of the star Lawson
Lawson et al Furthermore b esides some p ermanent emission lines probably
of chromospheric character e g CI I A there o ccur phase correlated emission
features in the UV which might b e caused by sho ck heating The equivalent widths
of electronic absorption bands of C Swan and CN violet asso ciated with the
outer atmosphere show a clear correlation with the pulsational phase of the star in
case of R CrB Clayton et al
Mass Loss The question of whether or not the RCB stars b esides the o ccasional
mass loss events due to dust cloud formation undergo an underlying p ermanent
mass loss is under controversial discussion The narrow emission lines seen during
the declines cf A Sp ectroscopy are blue shifted by typically km s which
according to Feast suggests a p ermanent radiation driven mass loss
However these e ects can hardly b e distinguished from the dust and gas clouds
which apparently are present out of the line of sight at any time near the star
cf A Dust Shells For the theoretical understanding of the dust formation in
RCB envelopes a clari cation of this question would b e decisively imp ortant In
a massive stellar wind which is not driven by the dust itself the dust formation
might b e a secondary pro cess and could o ccur rather distant from the star In
contrast if there is no massive wind the densities are to o low and dust cannot form
at large distances Estimates for the mean mass loss rates inferred from the sum of
dust cloud formation events range from M yr Feast to M yr
Clayton et al
Dust Shells The infrared photometry of RCB stars show a clear excess at ab out
m irresp ective whether the star is in decline or not The excess can b e tted
by blackbo dy curves of characteristic temp eratures K Kilkenny Whittet
This thermal emission is obviously caused by the total amount of dust in the
vicinity of the star which has b een formed during the former decline events or out
of the line of sight The mean radial distances of these dust shells are estimated
This interpretation is strongly supp orted from the observation of the very rst declines of FG
Sge Jurcsik showing a more or less p ermanent IR excess after those declines but no excess
b efore
Since these dust clouds have proven to b e optically thin in late decline it would actually b e
more appropriate to t the excess by B T
A OBSERVATIONS DURING THE DECLINE EVENTS
to b e stellar radii Walker Recently Feast et al published ex
tensive long term infrared photometry data for RCB stars and concluded that
there is evidence for a spread in dust temp eratures in each RCB shell where the
hottest comp onents are always limited by ab out K Feast argues that
the mean dust temp erature is increasing if the L ux representing the total dust
mass increases This is an argument in favor of the formation of hot dust close to
the star Since the limiting value of K is constant for all RCB stars observed
the condensation temp erature of carb on p ossibly is the controlling factor in all RCB
envelopes Furthermore another excess in the far IR m can b e observed
for several ob jects e g R CrB and SU Tau which p oints to distant fossil dust and
gas shells probably connected with the former evolution of the star According to
Gillett et al the linear size of the shell of R CrB is ab out arc min p c
Stellar Evolution The p opularity of the RCB stars is also caused by their mys
terious origin Since the RCB stars are so rare they must b e either a manifestation
of a p eculiar side path of stellar evolution or a common but rapidly evolving stage
Two ma jor evolutionary scenarios have b een worked out during the last years
onberner Renzini indicating that the RCB stars are Ib en Sch
in fact a topic of recent research on stellar evolution The merging of two white
dwarfs Double Degenerate DD scenario and the re birth of an p ost AGB star
as a consequence of a last thermal pulse Final helium shell Flash FF scenario
These mo dels make di erent predictions ab out the surface elemental comp osition
the lifetimes of RCB stars and their spatial distribution in the Galaxy However
agreement with observations is still rather p o or Last but not least the very fast
evolution of FG Sge Kipp er Jurcsik across the HR diagram during the
last century suggests that the birth of a new RCB star has actually b een observed
A Observations During the Decline Events
Light Curves The individual light curves of the sp ectacular decline events are
quite di erent in app earance concerning b oth the di erent events and the di erent
stars Nevertheless their eye catching shap e is so typical that they essentially de ne
this class of ob jects The light curves of the RCB stars start with a sudden drop in
the visual brightness of typically mag within a few weeks whereas the recovery
from deep declines usually takes months or years Between the initial decrease and
the nal recovery phase there usually is a phase of low level chaotic light variation
lasting b etween zero and several years Go eres Sedlmayr Multiple minima
sup erimp osing each other are often observed suggesting multiple dust formation
events
Color Variations The decline events of RCB stars are accompanied by complex
color variations The initial decrease in light always app ears slightly reddened The
In go o d agreement with the carb on condensation temp erature cf A Go eres Sedlmayr
As huge as the angular size of the mo on
App endix A
light then may b ecome bluish a blue decline according to Cottrell et al or
remains reddened a red decline As the decline progresses a strong reddening o c
curs until the light nally increases again and the star slowly reaches its usual bright
ness and color These variations result in typical lo ops in the V B V diagram
Alexander et al Cottrell et al The nal light increase apparently pro
ceeds on a unique line for all RCB stars with slop e in the V B V diagram
which is an unusually large value compared to interstellar reddening Cottrell
providing clues on the nature of the dust The cause of the bluing in early decline
has b een prop osed to b e an additional radial extended hotter light source than the
star itself Pugach Clayton et al emitting at A mainly in
form of line emission Asplund This chromosphere initially is not or is at
least much less eclipsed by the dust cloud The di erence b etween red and blue
declines may b e caused by di erent cloud geometries during the declines Varying
cloud radii during the initial formation and or varying distances from the line of
sight might pro duce red or blue declines
IR Observations The light variations in the infrared regions during the declines
have smaller amplitudes compared to the optical region As a rule the amplitudes
decrease with increasing wavelength and vanishes at ab out the L band m
where the light is already dominated by thermal dust emission e g Feast et al
No anti correlation b etween the optical and IR brightness has b een found which
would b e exp ected if the dust was present in a spherical symmetric fashion This
is the main observational argument for dust cloud formation rather than dust shell
formation Forrest et al Furthermore the dust mass pro duced in one decline
apparently is small compared to the total mass of the dust present in the vicinity of
the star
Extinction of the Dust Particles The p ossibility to observe the RCB stars
twice uncovered and covered by dust allows for a direct determination of the ex
tinction curve of the material resp onsible for the decline events The results clearly
indicate the carb onaceous character of the dust material However the
p osition of the well known graphite bump at A is shifted to A in
case of RCB stars e g Hecht et al The general shap e of the extinction curve
as well as the sp ecial app earance of this feature is discussed in various publications
concerning the nature of the dust grains in RCB envelopes e g Holm et al
onberner Hecht Je ery Maron Wright Drilling Sch
Zubko Many interpretations are p ossible Unusual size distributions no
hydrogen at the surface or unusual lattice or microscopic structures glassy car
b on onion like structures amorphous carb on cores covered by graphite mantles
fullerenes The only clear unique tendency in these pap ers seems to b e the unusual
small radii of the dust particles typically A to maximum values of ab out A
Sp ectroscopy So far only two decline events of RCB stars have b een completely
monitored as function of time by optical sp ectroscopy The decline of RY Sgr
Alexander et al and the decline of R CrB Cottrell et al How
ever fragmentary sp ectral data is available for several events of the three brightest
Current Status of RCB Research
RCB stars R CrB RY Sgr and V Cen covering certain phases of the declines
e g Lambert et al Lawson et al Clayton et al Rao Lambert
Asplund The sp ectra indicate a sp ecial time evolution Until the b eginning of
a decline no sp ectral changes have b een rep orted so far Cottrell et al Lawson
As the intensity of the photospheric absorption line sp ectrum decreases a
rich chromospheric emission line sp ectrum comes to light Alexander et al
distinguish b etween three comp onents named E E and BL Most of the emission
line b elong to the class of narrow km s E lines of high excitation energy
eV of neutral or singly ionized metal atoms which are blue shifted by typically
km s and disapp ear after some weeks A smaller number of narrow E emission
lines of low excitation energies eV mainly multipletts of Sc I I and Ti I I remain
visible for days As the decline progresses the optical sp ectrum mainly
consists of ve broad km s unshifted BL emission lines Ca I I H K
Na I D and a line at A probably He I Feast Finally the photospheric
absorption line sp ectrum re app ears and so on dominates the light from the remain
ing emission lines The physical nature of the emission lines is usually describ ed by
chromospheric although they do not lo ok like the chromospheric emissions of any
other stars Clayton
The BL lines usually show a multi component structure Di erent emission and
absorption comp onents can b e observed esp ecially blue shifted features with typical
velocities of km s towards the observer These comp onents are supp osed to
originate from the gas dragged along with the dust clouds b eing accelerated by
radiation pressure Hence these velocities can b e asso ciated with the velocities of
the dust clouds
Polarization The light during the declines generally is strongly p olarized Serkow
ski Kruszewski Coyne Shawl Standford et al E mov In
the continuum degrees of p olarization up to esp ecially in the blue sp ectral
region have b een rep orted In contrast the emission lines remain more or less unp o
larized Whitney et al The physical e ect causing the p olarization is mainly
the scattering of light at the surfaces of dust grains Therefore these observations
allow for imp ortant conclusions First the dust is distributed non spherically and
second the dust cloud causing the decline do es not eclipse the regions resp onsible
for the line emissions at least much less than the photosphere Consequently the
dust seems to form b elow the line emission regions suggesting dust formation o ccurs
rather close to the photosphere
Further Observations Several observational clues can b e found which p oint to
a causal connection b etween the pulsations and the decline events For at least
two ob jects RY Sgr and V Cen there is some evidence for the onset of the
declines to o ccur at particular pulsational phases of the star e g Lawson et al
Furthermore the multiple drops of the light curve in the b eginning of the decline
To catch a star just b efore a decline however needs a very lucky moment since no predictions
can b e made
Possibly at the surfaces of other distant dust clouds out of the line of sight
App endix A
events seem to o ccur at time intervals corresp onding to the p erio d of the star Feast
According to Jurcsik the decline activity i e the inverse of the
mean time b etween the declines is increasing with increasing hydrogen abundance
Finally long time variations of the decline activity have b een rep orted acting on
time scales of a few thousand years Menzies Feast According to the
p ersonal opinion of the author the observations reviewed in this last paragraph are
less striking than those outlined b efore still leaving enough ro om for interpretation
A Mo dels
Compared to the number and the quality of observations only a few theoretical ap
proaches to mo del the RCB decline events have b een carried out so far The mo dest
activity of the theoreticians is p ossibly caused by the complexity of the pro cesses
and the somewhat troublesome geometry involved These obstacles prevent simple
theoretical approaches and solutions A consistent physical description of the prob
lem obviously must contain i a detailed calculation of thermo dynamics chemistry
and dust formation ii a solution of radiative transfer and iii a mo deling of the
hydrodynamics for the dust enriched gas Due to the cloudy geometry all these
investigations have to b e worked out in more than one spatial dimension
None of the published mo dels including this work satisfy these requirements
Presently there are on one hand a few theoretical works which fo cus on certain
key problems of the declines e g on the trigger for the sudden onset of dust for
mation On the other hand several empirical mo dels prescrib e the existence the
geometry and the movement of the dust in front of the star calculate the observable
consequences and argue in favor or against certain scenarios
A Historical Mo dels
Loreta and O Keefe Loreta assumes that dust forma
tion o ccasionally o ccurs in a massive spherical stellar out ow which causes the
declines O Keefe agrees with Loreta s hypothesis but prop oses that the
dust forms in ejected blobs of gas similar to solar protub erances The solid matter
prop osed to condense is b elieved to b e principally carb on Both mo dels assume
that dust formation takes place rather distant from the star where the temp erature
is low enough to allow for the phase transition Based on some fundamental ther
mo dynamic considerations O Keefe derives condensation temp eratures of K
densities cm and distances of ab out stellar radii Reviewing these early
statements which up to date provide the basic idea for the explanation of the RCB
declines the progress since then has apparently b een rather slow
A Mo del Calculations
Wdowiak Giant convection cells are prop osed to b e present at the surfaces
of RCB stars Scaling the observations of the granulation and the sup er granulation
Current Status of RCB Research
of the sun to giant star dimensions Wdowiak argues for considerably lower temp er
atures over certain restricted areas of the star Following his ideas this favors dust
formation over these areas followed by dust cloud ejection Feast to ok over
this picture and argued that even the semi regular visual light variations might b e
caused by this e ect rather than by stellar pulsation Problems remain as even a
few thousand degrees less may not b e su cient for dust formation in the photo
sphere esp ecially for hot RCB stars as Wdowiak stated himself His argument is
only qualitatively as no calculations of the sup er granulation have b een carried out
and the formation of dust has not b een calculated
Fadeyev Y Fadeyev was the rst who applied classical nucleation
theory based on the bulk material data for graphite to the circumstellar envelopes
of RCB stars In his latest most advanced work Fadeyev the temp erature
is prescrib ed as T T r and a radially expanding uid element is followed starting
at the sonic p oint with a given initial velocity According to the assumptions of an
optically thin radiation eld grey gas opacities radiative equilibrium in the gas and
including the greenhouse e ect for amorphous carb on dust formation is p ossible
outside of ab out R Strong correlations with the e ective temp erature of the
star and its prescrib ed mass loss rate are found The acceleration time scale of the
gas due to radiation pressure on dust grains yields ab out days In his earlier
works a temp oral enhancement of the gas density caused by propagating sho ck
waves are also considered The mo del principally has di culties to explain i the
dust formation in RCB stars of di erent e ective temp eratures ii the pre existence
of a massive stellar wind and iii the o ccurrence of high velocity features so on after
the b eginning of a decline
Go eres Sedlmayr Go eres and Go eres Sedlmayr have
thoroughly investigated the carb on chemistry and the nucleation under the prevail
ing conditions in RCB envelopes The chemistry is dominated by a mixture of pure
carb on molecules in an inert helium gas similarly to recent lab oratory exp eriments
concerning the formation of bucky balls C However the main chemical path
way to the formation of so ot particles involves small carb on chains mono cyclic rings
and larger dehydrogenized curved but not closed p olyaromatic carb on molecules
PACs Fullerenes are prop osed to form as minor by products of this pathway Gas
temp eratures roughly b elow K are inevitably necessary for carb on nucleation
The main growth sp ecies is the abundant C radical Molecule drift is prop osed to
trigger the further growth to larger molecules The declines are caused by density
enhancements due to sup erimp osing sho ck waves which originate from non radial
pulsations The descending and the ascending branches of the light curve are ex
plained by hydrostatic dust growth and radial dilution at a constant out ow velocity
resp ectively The gas temp erature is prescrib ed as in the mo del of Fadeyev causing
the same principal problems as ab ove
Asplund Gustafsson Gustafsson Asplund have worked out
detailed atmosphere calculations for hydrogen de cient stars static planeparal
and published his results
App endix A
lel LTE using accurately calculated line blanketed absorption co e cients which
yield go o d agreement with the observed sp ectra According to these mo dels the
surfaces of the stars are b elow the so de ned Eddington limit g g
rad grav
However radiative instabilities are present in the deep er photospheric layers at the
helium ionization zone at In these layers the radiative acceleration
Ross
exceeds the gravitational deceleration which according to their mo dels is
balanced by pressure inversions Asplund Gustafsson recognize that such
layers are unstable against compression and outward acceleration of gas blobs As
they put forward themselves the reason for a decline event is prop osed to b e the
acceleration of such a gas blob in the deep photosphere of the star followed by a
sup ersonic injection through the atmosphere radiative co oling and dust formation
Thus the cause of the RCB declines might b e found in the radial atmospheric struc
ture of the star itself The mo del seems to b e promising but so far the investigations
are restricted to hydrostatic considerations Hydro dynamical mo dels for the pro cess
of blob injection have not b een p erformed Dust formation has not b een calculated
The mo del do es not explain the reason for dust formation close to the star it only
provides the necessary density conditions
A Empirical Mo dels
Humphreys Ney A secondary co ol star with an optically thick dust
envelope causes the decline events Such binary mo dels have principal problems to
explain the asymmetry and the true randomness of the light curves Furthermore
no observational evidence for binary RCB stars have b een rep orted so far
Wing et al and Grinin Orbiting dust clouds from time to time
obscure the star the problems are the same as ab ove Moreover the dust clouds
should b e driven away from the star due to radiation pressure rather than doing
Kepler orbits
Pugach Pugach and co workers have developed a comprehensive
mo del for the dust cloud evolution which causes the declines Over the years the
approach has varied a bit but the main idea remains the expansion of a dust cloud of
constant mass at a xed lo cation in front of the star Radiative transfer calculations
for the di erent colors have b een p erformed for the following geometry A massive
initially in nitesimal small spherical dust cloud with a Gaussian density pro le
homologously expands v r t at a xed place with a certain o set from the
line of sight in front of the star The mo del introduces three parameters the
total dust mass the o set from the line of sight and the intensity of scattered or
additionally emitted radiation which is not a ected by the dust cloud but dep ends
on wavelength Pugach showed with his work that the shap e of the light curve
and the color variations can b e repro duced by this scenario No hydrodynamical
movement of the cloud is needed no dust formation must b e considered It can all b e
explained by pure geometry Estimates for the total dust cloud masses yield values
of g Pugach Koval chuk The shortcoming of this mo del is of
Current Status of RCB Research
course that it do es not really explain anything The existence of the dust cloud is
prescrib ed and the reason for the homologous cloud expansion remains mysterious
E mov Sp ontaneous changes of the absorption prop erties of a pre existing
dust shell cause the declines e g via sp ontaneous alignment of non spherical dust
particles The mo del can to some extent describ e the shap e of the light curve and the
color variations but a reason for the sp ontaneous changes as well as the existence
of such sp ecial dust is not provided
Further Mo dels Many further comments and estimates are stated in the lit
erature e g Feast Alexander et al Forrest et al
However these publications mainly present observations and discuss the re
sults in view of some ad ho c assumed scenarios Therefore they do not app ear
as extra mo dels in this App endix Nevertheless imp ortant conclusions can b e
drawn from these considerations The standard mo del in these publications clearly
is the formation of dust clouds near to the star followed by radial expansion and
dilution From the IR observations of R CrB Forrest et al concluded that the
dust cloud causing the decline only covered ab out of the solid angle corre
sp onding to a semi cone angle of ab out According to Feast the angular
sizes of the dust clouds and the decline activities of RCB stars are in agreement with
the formation of one dust cloud p er pulsational p erio d
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Meinen Dank : : :
achst Herrn Prof Dr Sedlmayr aussprechen Von seiner unverwechel ochte ich zun m
baren Art zu denken hab e ich und werde ich ho entlich auch no ch in Zukunft viel lernen
aumte Ich danke ihm fur die Freiheit die er mir auf meinem wissenschaftlichen Weg einr
und das Vertrauen das er mir trotz zwischenzeitlicher Di erenzen schenkte Er war es
oglichte an das Institut fur Astonomie der es mir durch unburokratische Ma nahmen erm
und Astrophysik zuruckzukehren und hier meine Dissertation zu b eenden
Weiterhin danke ich Herrn Priv Doz Dr Kaufmann fur die Erstellung des Zweitgutachtens
art hat den Prufungsvorsitz zu sowie Herrn Prof Dr Zimmermann der sich b ereit erkl
ub ernehmen
Fur die Toleranz die Hilfb ereitschaft und die fruchtbaren Diskussionen b ei der Erstellung
ochte ich allen Mitgliedern des Institutes danken insb esondere b edanke ich der Arb eit m
mich b ei Holger Beck Christiane Helling Jan Martin Winters und b ei Peter Cottrell die
aftig aten tatkr mir b ei der Korrektur der Arb eit und der Erledigung der Prufungsformalit
o tenteils zur Seite gestanden hab en Weiterhin danke ich Uwe Bolick der durch seine gr
oglich machte unentgeltliche Arb eit an den Rechnern des Institutes diese Arb eit erst m
Daniel Kruger hat ganz wesentlich b ei der naturwissenschaftlichen Konzeption dieser Ar
b eit mitgewirkt Die Grundidee erwuchs aus seiner Diplomarb eit auf die wiederum Andre
ochte mich b ei Dir Daniel fur die zahlreichen Diskussionen as Gauger Ein u hatte Ich m
und Deine Korrekturen b esonders b edanken vielleicht kann ich mich dafur bald revan
chieren
Achim Go eres danke ich von Herzen fur die Beratung nicht nur in fachlichen Fragen Neb en
einer Art innerer Seelenverwandschaft fuhle ich b ei ihm stets das aufrichtige Bestreb en
meine Arb eit o der vielmehr meine Person zu unterstutzen Ich werde nie vergessen wie
er auf der internationalen Tagung in Bamberg seinen eingeladenen Vortrag dafur
hergab um meinen darauf folgenden Kurzvortrag einzuleiten
ackigkeit schier un ort jedo ch Dietrich Ewert von dessen Hartn Mein innigster Dank geh
endlicher Energie und lieb evoller Zielstrebigkeit ich no ch viel lernen werde In einer Zeit
als ich mit der Astrophysik innerlich fast schon abgeschlossen hatte hat er mich taglich
ermuntert meine Arb eit fortzusetzen
Leb enslauf
Peter Woitke
onliche Daten Pers
Alter Jahre
Geburtsort Berlin Spandau
Familienstand ledig
Anschrift Schwendyweg Berlin
Schule und Studium
Sept Juli Astrid Lindgren Grundschule in Berlin Staaken
Sept Dez Freiherr vom Stein Gymnasium in Berlin Spandau
Apr Okt at Berlin Studium der Physik an der Technischen Universit
Thema der Diplomarb eit Staubbildung in der Sup ernova
A Abschlu als Diplom Physiker
atigkeiten Studienbegleitende T
arz Jan M Industriepraktikum b ei der Firma Siemens
Apr Feb Kurse und Praktika in Bionik und Evolutionsstrategie
arz Apr M Tutor im physikalischen Grundpraktikum Pro jektlab or
Beru icher Werdegang
atigkeit als wissenschaftlicher Mitarb eiter am Institut ab Nov T
fur Astronomie und Astrophysik der TU Berlin b ei
Prof Dr E Sedlmayr mit den Aufgab enbereichen
agen und Diplomarb eiten Betreuung von Seminarvortr
Arb eit und Mitarb eit an wissenschaftlichen Publikationen
Teilnahme an internationalen Tagungen z B St Louis
atigkeit als Netzwerk Administrator und Informatiker b ei T Okt Jan
der Firma BBJ Servis gGmbH
Besondere Kenntnisse
osisch Englisch sicher in Wort und Schrift Franz Sprachen
Schulkenntnisse
EDV diverse Erfahrungen mit PCs work stations und Gro rech
nern unter DOS Windows und UNIX Computersprachen
C C Fortran GFA Basic Pascal und Assembler Erfah
rungen mit PC Netzwerken unter Novel Datenbank
Programmen Paradox sowie mit Standardsoftware
Pro dukten wie MS Word und Excel
atigkeit als Trainer von Herren und Damenmannschaften T Sp ort
im Volleyball
Berlin den Juni