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Models of Stellar Atmospheres

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Abstract: a selfconsistent. mathematical model of stellar atmosphere which contains the. information about the distribution ... Selfconsistency in Spectroscopy. Once abundances and other physical properties have been derived. it ...

Models of Stellar Atmospheres
Department of Astronomy
Vienna University
21.11.2007
Models of Stellar Atmospheres
Prerequisites...
In 21th century we still get most of the information via
analysis of electromagnetic radiation
80% of electromagnetic radiation is coming from stars
Analyzing electromagnetic radiation we should understand that:
We directly analyze only the region where the radiation
becomes the one we register via our telescopes. This region is
transparent for the radiation to leave the star and is called
stellar atmosphere.
Everything that is taking place deeper the stellar atmosphere
we can analyze only indirectly
Models of Stellar Atmospheres
Deﬁnition: Stellar Atmosphere
General
Stellar atmosphere: it is a part
of the star which does not have
its own energy sources. Only
redistribution of radiative en
ergy takes place.
Practical
Stellar atmosphere: it is a part
of the star where the line spec
tra is formed (outcoming radia
tion which we observe)
Models of Stellar Atmospheres
Deﬁnition: Model of Stellar Atmosphere
Since the radiation we see interacts with the plasma of stellar
atmosphere, we need a tool to describe this process and to
interpret the observed data we get.
We will call model of stellar atmosphere a selfconsistent
mathematical model of stellar atmosphere which contains the
information about the distribution of main physical quantities
(T,P,...) with geometrical depth counting from some zerolevel.
Selfconsistency means that having a set of free parameters the
obtained solution is unique.
Models of Stellar Atmospheres
Importance of Model Atmospheres
1. Model atmospheres provide a link between theoretical and
observational astrophysics
2. Model atmospheres are the upper boundary conditions for
modeling of stellar structure and evolution
3. Model atmospheres are the intermediate region between
stellar envelopes and interstellar medium
4. Model atmosphere is the only tool now for abundance analysis
in wide range of physical conditions that is important for
studying the chemical evolution of our Galaxy and Universe as
a whole
Models of Stellar Atmospheres
Model Parameters
Energy Conservation
Since there are no energy sources in stellar atmosphere, the total
amount of radiative energy falling at the bottom of the atmosphere
from the inside should be the same at each point throughout the
atmosphere:
div (Frad ) = 0
A blackbody Concept
A useful concept is that of blackbody, as deﬁned as a perfect ab
sorber and emitter of light. It radiates energy at the same rate as
it is being absorbed. The ratio of emitted and absorbed energy at
each frequency depends only on temperature and is known as Plank
function
Models of Stellar Atmospheres
Model Parameters
emission 2hν 3 1
= Bν (T ) = 2 hν/kT
absorption c e −1
The total energy radiated by a blackbody goes as a fourth power
of temperature known as StefanBoltzmann Law:
∞
Bν (T )dν = const × T 4
0
If the radiation at the bottom of the atmosphere is blackbody,
then we can introduce a quantity called eﬀective temperature Teﬀ
so that at each atmospheric layer we have:
∞
Fν dν = const × Teﬀ 4
0
Teﬀ is the temperature of blackbody which radiates (from the unit
surface area) the same amount of energy as the star of interest and
thus is an energetic characteristic of the star
Models of Stellar Atmospheres
Model Parameters
Most of the stars do not change their radii on time scales
compared with their lifetimes. Thus, we assume that the
atmosphere of the star is in hydrostatic equilibrium, i.e. the
gravitational force is balance by kinetic properties of plasma.
GMstar
ratm Rstar , g=
Rstar
Stars with extensive atmospheres
Giants, supergiants, hot stars with strong stellar wind, etc.: R, M
Stars with extended atmospheres
Pulsating stars, supernovas, etc.: should be considered in hydrody
namic (tdependance)
Models of Stellar Atmospheres
Model Parameters
Properties of radiation ﬁeld and plasma in stellar atmosphere
depend upon chemical composition (abundances) used {εi }
1. abundance of each element is given relative to the total
number of atoms of all chemical elements (Ni /Ntotal )
2. abundance of each element is given relative to the number of
atoms of all chemical elements in a volume contains 1012
atoms of hydrogen
3. abundance of each element is given relative to the total mass
of stellar plasma in unit volume
Selfconsistency in spectroscopy
Model atmosphere → abundances → model atmosphere...
Models of Stellar Atmospheres
“Classical” Assumptions
Planeparallel geometry (ratm Rstar ). All the physical
quantities depend only on geometrical depth h
Homogeneous abundances
Hydrostatic equilibrium (no largescale motions)
The atmosphere is timeindependent (statistical equilibrium)
There are no sources or sinks of energy
Energy transport takes place only by radiation and convection
(no heat conduction, acoustic waves, MHD waves, etc.)
The free electrons as well as the free heavier particles obey
the Maxwell distribution with local kinetic temperature Te .
Local Thermodynamical Equilibrium (LTE) is assumed
Models of Stellar Atmospheres
Hydrostatic Equation
dPtotal
= g, dM = −ρdx
dM
Ptotal contains:
Gas pressure Pgas = nkT
4π
Radiation pressure Prad = grad (M)dM, grad = c κν Hν dν
1
Microturbulent pressure Pturb = 2 ρξturb
Convective pressure Pconv
υ2
Pressure due to rotation Prot = R dM
Macroturbulent pressure Pmacro
Magnetic pressure Pmag
Models of Stellar Atmospheres
State Equation
Boltzmann formula for the ratio of occupation numbers:
ni gi Ej − Ei
= exp(− )
nj gj kT
Saha equation for the ratio of ion numbers:
nI 2 gI 2πmkT 3/2 EJ − EI
= ( 2
) exp(− )
nJ ne gJ h kT
Note
ne = f (T , P, i ) and is unknown. However, we know Ntotal and thus
should solve the system of nonlinear equations to ﬁnd ne . Iterative
solution is needed.
Models of Stellar Atmospheres
Equation of Radiative Transfer
Going through gas of the atmosphere, the radiation is absorbed,
reemitted and scattered many time. These deﬁne the way how the
radiation at each frequency is transferred through the atmospheric
region.
dEν
Iν =
cos θdσdνdωdt
[Iν ] = erg cm−2 s −1 Hz −1 sterad −1
Number of photons times hν per dνdωdt
and per cos θdσ
Models of Stellar Atmospheres
Equation of Radiative Transfer
dIν
µ = Iν − Sν
dτν
ην
Source function: Sν =
κν
Optical depth: dτν = −κν ρds
µ = cos θ
Formal solution of RTE is:
τν
dt
Iν (µ) = −e τν /µ Sν e −t/µ
c µ
Models of Stellar Atmospheres
Moments of RTE
In model atmospheres we are meanly interested in integral
characteristics of radiation ﬁeld
1 1 1
Jν = Iν dω, Hν = µIν dω, Kν = µ2 Iν dω
4π 4π 4π
Jν – mean intensity is an energy emitted by unit surface area in
any direction
Hν – radiative ﬂux is an energy emitted by unit surface area in all
directions
Kν – is called Kintegral and does not have physical meaning
(rad) 4π
However: Pν = Kν – radiative pressure acting on unit
c
surface area
Models of Stellar Atmospheres
Moments of RTE
Source Function
It is seen that to solve RTE one needs to know source function
Sν which also depends upon the solution of RTE due to additional
scattering term σν Jν
dIν dHν dKν
µ = Iν − Sν , = Jν − Sν , = Hν
dτν dτν dτν
κν Bν + σν Jν
Sν =
κν + σ ν
∞
1 d n Bν
Diﬀusive approximation (τν 1): Sν (tν ) = n
(tν − τν )n
n! dτν
n=0
1 dBν (τν ) 1
Jν (τν ) ≈ Bν (τν ), Hν (τν ) ≈ , Kν (τν ) ≈ Bν (τν )
3 dτν 3
Models of Stellar Atmospheres
Radiative Equilibrium
dH
= (κν + σν )Jν dν − (κν + σν )Sν dν = Q(M)
dM
Q(M)–function describing the losses of radiative energy via its
transformation to other types
Our experience shows that in most cases Q(M) ≈ 0 so that the
amount of absorbed energy is equal to that of emitted:
(κν + σν )Jν dν = (κν + σν )Sν dν
dH
=0
dM
Models of Stellar Atmospheres
Convective Energy Transport
Schwarzschild instability criteria
γ−1 dlnP dlnT Cp
− < − , γ=
γ dr R
dr R
Cv
Convection take place in the regions with high opacity due to fast
ionization of main plasma components (H, He)
Early type of stars: H is completely ionized so we can expect
only weak convective zones due to He, He+ ionizations
Atype: hydrogen convective zone close to photosphere
Ftype: more strong and deep hydrogen convective zone
Gtype: convection transfers most of the energy in deep layers
and penetrate surface layers
Mtype: convection rules the whole structure of the
atmosphere
Models of Stellar Atmospheres
Convective Energy Transport
Introducing convection in model atmosphere calculations we are
mainly interested in which per sent of the total energy is carried
out by convection
Models of Stellar Atmospheres
Convection Treatment
MixingLength Theory (MLT)
Based on local pressure scale height:
Ptotal 1
Hp = =
ρg α
α has to be tuned for diﬀerent stars
Application: cool stars atmosphere with strong convection regime
CM convection
Canuto & Mazzitelli (1991, 1992) – further improvement of convec
tion model: no need for such free parameter as α any more
Application: atmospheres of A–F stars with weak convection regime
Models of Stellar Atmospheres
Temperature Correction
Problem
T (M) which satisﬁes the conditions of RE is initially unknown
T (M) = T0 (M) + ∆T (M)
(κν + σν )Sν (T0 + ∆T )dν = (κν + σν )Jν dν
∂Sν
Sν (T0 + ∆T ) ≈ Sν (T0 ) + ∆T
∂T
(κν + σν )(Jν + Sν )dν
∆T ≈
∂Sν
(κν + σν ) dν
∂T
Models of Stellar Atmospheres
Convergence Criteria
Optically thick layers
4
(Hrad + Hconv ) − σTeﬀ = 0
Optically thin layers
κν Jν dν − κν Sν dν = 0
Convergence criteria:
1. ε(ﬂux) 1%
2. ε(rad. equilibrium) 1%
3. ∆Ti 1 K at each layer
Models of Stellar Atmospheres
Deﬁnition of the Problem
1. All equations are bound
Sν , Jν , Hν , Kν → f (αν , ην )
αν , ην → f (T , P)
T , P → f (Sν , Jν , Hν , Kν )
2. The system of equations is nonlinear
Jν → Sν → Jν . . .
3. The system of equations is globally dependent throughout the
atmosphere. Scattering processes force the properties of
radiation ﬁeld and plasma at one given point to be dependent
from all other points.
Models of Stellar Atmospheres
Simple Model Atmosphere Calculation Flow Block
Teﬀ , log g (or R, M), { i }, ξturb
First approximation: T (τstd )
Hydrostatic: P(τstd )
State Equation: ne , ni , n0
Model Iteration
ν–integration
cont line
Opacity: αν , αν
Radiation Field: Jν , Hν , Kν , Sν
Convection: Hconv
Tnew = Told + ∆T
Tcorrection: ∆T
Models of Stellar Atmospheres
Interaction Between Matter and Radiation
Absorption
True Absorption
Atom gets photon, goes to exited state and hits another atom
transmitting its energy to the later one.
Plasma heating
Negative Absorption (or Stimulated Emission)
Atom is exited by particle collision and emits the photon of
energy hν in direction ω
Plasma cooling
Models of Stellar Atmospheres
Interaction Between Matter and Radiation
Emission
Atom spontaneously emits a photon due to ﬁnite lifetime. The
atom should be in exited state (whatever was the reason).
Scattering
Direct Scattering
Atom gets photon from the beam, goes to exited state and
then reemits its energy in arbitrary direction. Depending
upon the probability of direction of the reemitted photon we
have isotropic and anisotropic scattering. If the energy of
reemitted photon is equal to that of absorbed one, we have
coherent scattering. Otherwise incoherent.
Reverse Scattering
The photon is added to the beam due to direct scattering of
another beam on a given atom
Models of Stellar Atmospheres
Interaction Between Matter and Radiation
Absorption and Emission processes can take place between the
following energy levels:
discretediscrete (boundbound) ⇒ spectral lines
discretecontinuous (boundfree) ⇒ ionization/recombination
continua
continuouscontinuous (freefree) ⇒ ion required
Models of Stellar Atmospheres
Line Blanketing and Model Atmospheres
Opacity due to atomic lines absorption is one of the most
important eﬀects in stellar atmospheres
Deep layers – backwarming
Surface layers – cooling
Models of Stellar Atmospheres
Line Opacity Treatment
Problem
Hundred of thousands of lines which have to be taken into account
Solution
Several statistical methods have been developed
Opacity Distribution Function (ODF)
Opacity Sampling (OS)
Direct Opacity Sampling
Models of Stellar Atmospheres
Line Opacity Treatment
Models of Stellar Atmospheres
Line Opacity Treatment
Models of Stellar Atmospheres
Energy Redistribution
Models of Stellar Atmospheres
Model Structure
Models of Stellar Atmospheres
Model Atmospheres in Spectroscopy
Fundamental parameters → model atmosphere → synthetic
spectra → physical output
For most spectroscopists model atmosphere is a black box.
Learn how to use existing codes by yourself
Use already calculated public model grids
If you are lazy, ask other people to calculate models for you
(not always work perfectly)
Models of Stellar Atmospheres
Selfconsistency in Spectroscopy
Model atmosphere is consistent with observations if the following
observables are ﬁtted simultaneously:
Energy distribution
Contains information about total energy balance
Hydrogen line proﬁles
Cool stars: most sensitive to temperature changes
Hot stars: most sensitive to pressure changes
Metallic line spectra
Abundances (stratiﬁcation, spots), microturbulence, rotation,
magnetic ﬁelds, etc.
Models of Stellar Atmospheres
Selfconsistency in Spectroscopy
Once abundances and other physical properties have been derived
it is necessary to recalculate the model atmosphere to consistency
consistency
Iterative solution is needed
Models of Stellar Atmospheres
Selfconsistency in Spectroscopy
Models of Stellar Atmospheres
Selfconsistency in Spectroscopy
Models of Stellar Atmospheres
Selfconsistency in Spectroscopy
Models of Stellar Atmospheres
Selfconsistency in Spectroscopy
Models of Stellar Atmospheres
nonLTE
In almost all plasmas (except very thin one with magnetic ﬁelds),
the Maxwell distribution is valid. The validity of Saha and
Boltzmann formulas on the other hand depend on the ratio of
photon number density εphotons to the particle number density
εparticles
4π 4 3
εphotons = Sν dν = σB T 4 , εparticles = nkT
c c 2
εphotons T3
f = = 36.5
εparticles n
f 1: LTE
f ∼ 1: LTE questionable
f 1: nonLTE
Models of Stellar Atmospheres
nonLTE
dni
= −n(i) [R(i→j) + C(i→j) ] + n(j)[R(j→i) + C(j→i) ] = 0
dt
j j
Upward rates, boundbound and boundfree:
Jν
Rup (i → j) = 4παnu dν
hν
Downward rates, boundbound, freebound
nj 4π 8πν 2
Rdown (i → j) = αν e −hν/kT Jν + 2
ni hν c
ni /niLTE
biZwaan ≡ ni /niLTE , biMenzel ≡ LTE
nc /nc
nc –concentration of next ionization stage
Models of Stellar Atmospheres
Validity of LTE
In stellar model atmospheres we are mainly interested in nontrace
elements like H, He, Fe, etc. (most abundant species having many
strong lines in the spectra).
Models of Stellar Atmospheres
Additional Physics
Additional physics have to be included to improve models and our
understanding about outer regions of the stars and to model their
spectra
Spherical symmetry: stars with extended atmospheres
Stellar wind: hot stars (multicomponent stellar wind)
Diﬀusion of chemical elements: CP stars with quite
atmospheres (horizontal and vertical distribution)
Magnetic ﬁelds: CP2 stars with strong magnetic ﬁelds
Rotation: fast rotating hot stars, meridional circulations
Binarity eﬀects: dust disks, xray emission, tidal eﬀects, etc.
...
Models of Stellar Atmospheres
Model Grids
ATLAS9, ATLAS12 (R.L. Kurucz)
http://kurucz.harvard.edu
TLUSTY (Lanz T., Hubeny I.)
http://nova.astro.umd.edu/index.html
MARCS (Gustafsson B. et al.)
http://marcs.astro.uu.se
NEMO grid based on ATLAS9 with diﬀerent convection
treatment
http://ams.astro.univie.ac.at/cgibin/dive.cgi
Models of Stellar Atmospheres
Bibliography
Bender, R. 2002, Astrophysics introductory course
Kurucz R.L. 1970, ATLAS: a computer program for
calculating model stellar atmospheres, SAO Special Report
Mihalas, D. 1978, Stellar atmospheres (2nd edition)
Rutten, R.J. 2003, Radiative transfer in stellar atmospheres
Models of Stellar Atmospheres
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