STELLAR ATMOSPHERES
A VERY SHORT INTRODUCTION
PART I
Ewa Niemczura
Astronomical Institute, UWr
eniem@astro.uni.wroc.pl
Stellar spectra
Stellar spectra
One picture is worth 1000 words, but
one spectrum is worth 1000 pictures!
Ivan Hubeny
What is a Stellar Atmosphere?
Stellar atmosphere:
• medium connected physically to a star; from this medium
photons escape to the surrounding space
• region where the radiation observable by a distant
observer originates
What is a Stellar Atmosphere?
Stellar atmosphere:
• usually, a very thin layer on the surface of the star
• late stars: photosphere, chromosphere, corona
What is a Stellar Atmosphere?
Stellar atmosphere:
• usually, a very thin layer on the surface of the star
• late stars: photosphere, chromosphere, corona
• early stars: photosphere, expanding regions
Why study stellar atmospheres?
„Why in the world would anyone want to study stellar
atmospheres? They contain only 10-10 of the mass of a
typical star! Surely such a negligible fraction of a star mass
cannot possibly affect its overall structure and evolution!”
Question to D. Mihalas, about 50 years ago
From the lecture of Ivan Hubeny
Why study stellar atmospheres?
Atmospheres are all we see;
we have to use this information in the fullest.
Stars:
Stellar atmospheres:
Determination of atmospheric parameters.
Stellar spectra
What can we obtain?
• Spectral classification
• Atmospheric parameters
• Effective temperature Teff
• Surface gravity logg
• Chemical abundances
• Metallicity [m/H]
• Microturbulence, macroturbulence
• Chemical peculiarities
• Stratification of elements
• Rotation velocity Vsini
• Stellar wind parameters
• Magnetic field parameters
Stellar spectra
What can we obtain?
• Stellar classification
• Atmospheric parameters
• Chemical peculiarities
• Stratification of elements
• Rotation velocity
• Stellar wind parameters
• Magnetic field parameters
• Multiple systems
• Variability in spectral lines
• Radial velocities
• Orbit determination
• Cluster membership
• Pulsations
• …
Why study stellar atmospheres?
Stars:
Stellar atmospheres:
Determination of primary (Teff, logg, chemical composition) and
secondary atmospheric parameters (rotation velocity, turbulence etc.)
Stellar structure and evolution:
Determination of basic stellar parameters (M, R, L)
4
𝑇𝑒𝑓𝑓 → 𝐿 = 4𝜋𝑅2 𝜎𝑇𝑒𝑓𝑓
𝐺𝑀
𝑔→ 2
𝑅
Determination of the detailed physical state;
Boundry for the stellar structure/evolution models;
Atmospheres do influence the stellar evolution after all (mass loss from the
atmosphere).
Why study stellar atmospheres?
Global context:
Galaxies are made of stars (special case: very bright stars in distant galaxies);
Sources of chemical species;
(…)
Why study stellar atmospheres?
Methodological importance:
Radiation determines the physical structure of the atmosphere, and this
structure is probed only by the radiation;
Sophisticated modeling approach needed – stellar atmospheres are guides
for modeling other astronomical objects (e.g. accretion discs, planetary
nebulae, planetary atmospheres etc.).
Models: typical assumptions
Geometry
• Plane-parallel symetry ∆𝑟/𝑅 ≪ 1, very small curvature
(e.g. main-sequence stars);
• Typically for stellar photospheres: ∆𝑟/𝑅 ≪ 1
• Sun: 𝑅 = 700,000 km
• Photosphere: ∆𝑟 = 300 km;
∆𝑟
𝑅
= 4 × 10−4
• Chromosphere: ∆𝑟 = 3000 km;
• Corona:
∆𝑟
𝑅
≈3
∆𝑟
𝑅
= 4 × 10−3
Models: typical assumptions
Geometry
• Plane-parallel symetry ∆𝑟/𝑅 ≪ 1, very small curvature
(𝛼~𝛽, e.g. main-sequence stars);
• Spherical symetry ∆𝑟/𝑅 ≈ 1, significant curvature
(𝛼 ≠ 𝛽, e.g. giants, supergiants);
• …
Models: typical assumptions
Homogeneity
• We assume the atmosphere
to be homogeneous.
• But it’s not always the case, e.g.
sunspots, granulations, non-radial
pulsations, magnetic Ap-stars (stellar
spots), clumps and shocks in hot star
winds etc.
Models: typical assumptions
Stationarity
• We assume the atmosphere to be stationary
• In most cases this assumption can be accepted
• Exceptions: pulsating stars, supernovae, mass transfer in
close binaries etc.
Models: typical assumptions
Conservation of momentum and mass
• We assume hydrostatic equilibrium;
• plane-parallel geometry:
• spherical geometry:
𝒅𝑷
𝒅𝒓
𝒅𝑷
𝒅𝒓
= −𝝆 𝒓 𝒈 𝒓
= −𝝆 𝒓 (𝒈 𝒓 − 𝒈𝒓𝒂𝒅 )
• Exceptions: effects of magnetic fields, interaction in binary
systems etc.
• no hydrostatic equilibrium:
𝒅𝑷
𝒅𝒓
= −𝝆 𝒓 (𝒈 𝒓 − 𝒈𝒓𝒂𝒅 + 𝒗
𝒅𝒗
)
𝒅𝒓
Models: typical assumptions
Conservation of energy
• Nuclear reactions and production of energy: stellar
interiors
• Stellar atmospheres: negligible production of energy
• We assume that the energy flux is conserved at any
radius:
• 𝑓𝑙𝑢𝑥 = 𝑒𝑛𝑒𝑟𝑔𝑦/𝑎𝑟𝑒𝑎 ∙ 𝑡𝑖𝑚𝑒
• 4𝜋𝑟 2 𝐹 𝑟 = 𝑐𝑜𝑛𝑠𝑡 = 𝑙𝑢𝑚𝑖𝑛𝑜𝑠𝑖𝑡𝑦 𝐿
Different stars – different atmospheres
Temperature:
• MS stars, T~2000 – 60,000K
• Brown dwarfs, T < 2000K
• Hot, degenerate objects, T~104 – 108 K
• White dwarfs, T < 100,000K
• Neutron stars, T~107K
Density:
• MS stars, N~1010 – 1015 cm-3
• WD, N~1021 – 1026 cm-3
Basic Structural Equations
Stellar atmosphere:
plasma composed of particles (atoms, ions, free electrons,
molecules, dust grains) and photons.
Conditions:
temperatures: ~103 – ~105 K;
densities: 106 – 1016 cm-3.
Starting point for physical description: kinetic theory
Distribution function (most general quantity which describes the
system):
𝑓𝑖 𝑟, 𝑝, 𝑡 𝑑 𝑟𝑑 𝑝 - number of particles 𝑖 in a volume of the phase space
at position 𝑟, momentum 𝑝, and time t.
Basic Structural Equations
Kinetic (Boltzmann) equation (describes a development of the
distribution function):
𝜕𝑓𝑖
𝐷𝑓𝑖
+ 𝑢 ∙ 𝛻 𝑓𝑖 + 𝐹 ∙ 𝛻𝑝 𝑓𝑖 =
𝜕𝑡
𝐷𝑡
𝑐𝑜𝑙𝑙
𝛻, 𝛻𝑝 – nabla differential operators with respect to the position and momentum
components
𝑢 – particle velocity
𝐹 – external force
𝐷𝑓𝑖
𝐷𝑡 𝑐𝑜𝑙𝑙
– collisional term (describes creations and destructions of particles of type 𝑖
with the position (𝑟, 𝑟 + 𝑑 𝑟) and momentum (𝑝, 𝑝 + 𝑑𝑝).
Kinetic equation – complete description of the system
Problem – number of unknowns (e.g. different excitation states of atoms etc.)
Simplification – moments of the distribution function – integrals over momentum
weighted by various powers of 𝑝
Basic Structural Equations
Moment equations:
(moment equations of the kinetic equation, summed over all kinds of particles;
hydrodynamic equations):
Continuity equation (1):
Momentum equation (2):
𝜕𝜌
𝜕𝑡
+ 𝛻 ∙ 𝜌𝑣 = 0
𝜕(𝜌𝑣)
+
𝜕𝑡
𝛻 ∙ 𝜌𝑣𝑣 = −𝛻𝑃 + 𝑓
Energy balance equation (3):
𝜕 1
2
𝜌𝑣
𝜕𝑡 2
+ 𝜌𝜖 + 𝛻 ∙
1
2
𝜌𝑣
2
+ 𝜌𝜖 + 𝑃 𝑣 = 𝑓 ∙ 𝑣 − 𝛻 ∙ 𝐹𝑟𝑎𝑑 + 𝐹𝑐𝑜𝑛
𝑣 – macroscopic velocity
𝜌 – total mass density
𝑃 – pressure
𝑓 – external force
𝜖 – internal energy
𝐹𝑟𝑎𝑑 , 𝐹𝑐𝑜𝑛 – radiation and conductive flux
Basic Structural Equations
Additional equation (zeroth-order moment equation):
conservation equation for particles of type 𝑖:
𝜕𝑛𝑖
𝐷𝑛𝑖
+ 𝛻 ∙ 𝑛𝑖 𝑣 =
𝜕𝑡
𝐷𝑡
𝑐𝑜𝑙𝑙
𝑛𝑖 – number density (or occupation number, or population) of particles of type 𝑖.
Basic Structural Equations
Significant simplification of the system:
Stationary
𝜕
𝜕𝑡
= 0 , static medium (𝑣 = 0), 1-D (all quantities depend
on one coordinate):
Statistical equilibrium equation (0):
𝐷𝑛𝑖
𝐷𝑡
𝑐𝑜𝑙𝑙
=0
Hydrostatic equilibrium equation (2; assumption):
𝑑𝑃
𝛻𝑃 = 𝑓 ⟹
= −𝜌𝑔
𝑑𝑧
Radiative equilibrium equation (3; assumption):
4
𝛻𝐹𝑟𝑎𝑑 = 0 ⟹ 𝐹𝑟𝑎𝑑 = 𝑐𝑜𝑛𝑠𝑡 ≡ 𝜎𝑇𝑒𝑓𝑓
Basic Structural Equations
Convection:
transport of energy by rising and falling bubbles of material with
properties different from the local medium; non-stationary and nonhomogeneous process.
In 1-D stationary atmosphere, simplification – mixing-length
theory;
Radiative equilibrium equation with convection:
4
𝐹𝑟𝑎𝑑 + 𝐹𝑐𝑜𝑛𝑣 = 𝜎𝑇𝑒𝑓𝑓
𝐹𝑐𝑜𝑛𝑣 – convective flux; specified function of basic state parameters
TE and LTE
• TE – thermodynamic equilibrium:
• simplification; particle velocity distribution and the distribution of
atoms over excitation and ionisation states are specified by two
thermodynamic variables: absolute temperature 𝑇 and total particle
number density 𝑁 (or the electron number density 𝑛𝑒 ).
• stellar atmosphere is not in TE – we see a star – photons are
escaping – there are gradients of state parameters.
TE and LTE
• LTE – local thermodynamical equilibrium:
• simplification; standard thermodynamical relations are employed
locally for local values of 𝑇, 𝑁 or 𝑛𝑒 , despite of the gradients that
exist in the atmosphere.
• equilibrium values of distribution functions are assigned to massive
particles, the radiation field can depart from equilibrium (Planckian)
distribution function.
LTE
LTE is characterised by three
distributions:
1. Maxwellian velocity distribution of
particles:
𝑚
𝑓 𝑣 𝑑𝑣 =
2𝜋𝑘𝑇
3
2
2
𝑚𝑣
4𝜋𝑣 2 exp −
𝑑𝑣
2𝑘𝑇
𝑚, 𝑣 – particle mass and velocity
𝑘 – Boltzmann constant
LTE
2. Boltzmann excitation equation
𝑛𝑗
𝑔𝑗
(𝐸𝑗 −𝐸𝑖 )
=
exp −
𝑛𝑖
𝑔𝑖
𝑘𝑇
𝑔𝑖 , 𝑔𝑗 − statistical weight of levels 𝑖, 𝑗
𝐸𝑖 , 𝐸𝑗 – level energies (measured from the
ground state)
LTE
3. Saha ionisation equation
𝑁𝐼
𝑈𝐼
𝜒𝐼
−3 2
= 𝑛𝑒
𝐶𝑇
exp
𝑁𝐼+1
𝑈𝐼+1
𝑘𝑇
𝑁 – total number density of ionisation stage 𝐼
𝜒𝐼 – ionisation potential of the ion 𝐼
𝑈 – partition function defined by
∞
𝑈=
𝐶=
ℎ2
2𝜋𝑚𝑘
1
3
2
𝐸𝑖
𝑔𝑖 exp −
𝑘𝑇
= 2.07 × 10−16 (cgs)
In LTE the same temperature applies to all kind of particles and to
all kinds of distributions.
LTE
• Maxwell, Saha, Boltzmann equations – LTE from macroscopic
point of view
• Microscopically – LTE is hold if all atomic processes are in
detailed balance
• the number of processed 𝑎 → 𝑏 is exactly balanced by the number
of inverse processes 𝑏 → 𝑎,
• 𝑎, 𝑏 – any particle state between which there exists a physically
reasonable transition.
• e.g. – 𝑎 is an atom in an excited state 𝑖, 𝑏 the same atom in
another state 𝑗, etc.
LTE vs. non-LTE
• non-LTE (NLTE) – any state that departs from LTE
(usually it means that populations of some selected energy levels of
some selected atoms/ions are allowed to depart from their LTE
value, but velocity distributions of all particles are assumed to be
Maxwellian, with the same kinetic temperature 𝑇).
LTE vs. non-LTE
When we have to take non-LTE into account?
• LTE breaks down if the detailed balance of at least one transition
𝑎 → 𝑏 breaks down
• Radiative transitions – interaction involves particles and photones
• Collisional transitions – interactions between two or more
massive particles
• Collisions tend to maintain LTE (their velocities are Maxwellian)
• Validity of LTE depends on whether the radiative transitions are in
detailed balance or not.
LTE vs. non-LTE
Departures from LTE:
1.
Radiative rates in an important atomic transition dominate over
the collisional rates and
2.
Radiation is not in equilibrium (intensity does not have Planckian
distribution)
Collisional rates are proportional to the particle density – in high
densities departures from LTE will be small.
Deep in the atmosphere photons do not escape and intensity is close
to the equilibrium value – departures from LTE are small even if the
radiative rates dominate over the collisional rates.
LTE vs. non-LTE
non-LTE if:
rate of photon absorptions >> rate of electron collisions
LTE if: low temperatures and high densities
non-LTE if: high temperatures and low densities
LTE vs. non-LTE
LTE if: low temperatures and high densities
non-LTE if: high temperatures and low densities
R. Kudritzki
lecture
Transport of energy
Mechanisms of energy transport:
a. radiation: 𝐹𝑟𝑎𝑑 (most important in all stars)
b. convection: 𝐹𝑐𝑜𝑛𝑣 (important especially in cool stars)
c. conduction: e.g. in the transition between solar
chromosphere and corona
d. radial flow of matter: corona and stellar wind
e. sound and MHD waves: chromosphere and corona
Intensity of Radiation
𝑰(𝒓, 𝒏, 𝝂, 𝒕) specific intensity of
radiation at position 𝑟, travelling in
direction 𝑛, with frequency 𝜈 at time 𝑡
– the amount of energy 𝑑𝐸 transported by
radiation in the frequency range (𝜈, 𝜈 + 𝑑𝜈),
across an elementary area 𝑑𝑆 into a solid
angle 𝑑𝜔 in a time interval 𝑑𝑡:
𝑑𝐸 = 𝐼 𝑟, 𝑛, 𝜈, 𝑡 𝑑𝑆 cos 𝜃 𝑑𝜔𝑑𝜈𝑑𝑡
dS
𝜃 – angle between 𝑛 and the normal to the surface 𝑑𝑆.
𝐼 − specific intensity, proportionality factor;
dimention: erg cm-2 sec-1 hz-1 sr-1
Intensity of Radiation
Photon distribution function 𝒇:
𝑓 𝑟, 𝑛, 𝜈, 𝑡 𝑑𝜔𝑑𝜈 – number of photons per unit volume at location
𝑟 and time 𝑡, with frequencies in the range (𝜈, 𝜈 +
Intensity of Radiation
From the comparison of energy of photons:
𝑓 𝑐 ∙ 𝑑𝑡 𝑑𝑆 cos 𝜃 𝑑𝜔𝑑𝜈 ℎ𝜈
with defined before:
𝑑𝐸 = 𝐼 𝑟, 𝑛, 𝜈, 𝑡 𝑑𝑆 cos 𝜃 𝑑𝜔𝑑𝜈𝑑𝑡
Relation between specific intensity and photon distribution function:
𝐼 = 𝑐ℎ𝜈 𝑓
Using this relation we define moments of the distribution function
(specific intensity): energy density, flux and stress tensor.
Intensity of Radiation
Energy density of the radiation (𝑓 is the number of photons in an elementary volume,
ℎ𝜈 is the energy of photon):
1
𝐸 = ℎ𝜈 𝑓𝑑𝜔 =
𝐼𝑑𝜔
𝑐
Energy flux of the radiation (𝑐𝑛 is the vector velocity); how much energy flows trough
the surface element:
𝐹=
ℎ𝜈 ∙ 𝑐𝑛 𝑓𝑑𝜔 =
𝑛𝐼𝑑𝜔
Ratiation stress tensor:
𝑃=
(ℎ𝜈) 𝑛𝑛𝑓𝑑𝜔 =
1
𝑐
𝑛𝑛 𝐼𝑑𝜔
Photon momentum density (momentum of an individual photon is
ℎ𝜈
1
𝐺=
𝑛𝑓𝑑𝜔 = 2 𝐹
𝑐
𝑐
ℎ𝜈
𝑐
𝑛):
Absorption and Emission Coefficient
The radiative transfer equation describes the changes of the radiation
field due to its interaction with the matter.
Absorption coefficient – removal of energy from the radiation field by
matter:
𝑑𝐸𝑎𝑏𝑠 = 𝝌 𝒓, 𝒏, 𝝂, 𝒕 𝑰 𝒓, 𝒏, 𝝂, 𝒕 𝒅𝑺𝒅𝒔𝒅𝝎𝒅𝝂𝒅𝒕
Element of matterial of cross-section 𝑑𝑆 and length 𝑑𝑠 remove from a beam of
specific intensity 𝐼 an amount of energy 𝑑𝐸𝑎𝑏𝑠 .
The dimention of 𝜒 is cm-1
Absorption and Emission Coefficient
The radiative transfer equation describes the changes of the radiation
field due to its interaction with the matter.
Absorption coefficient – removal of energy from the radiation field by
matter:
𝑑𝐸𝑎𝑏𝑠 = 𝜒 𝑟, 𝑛, 𝜈, 𝑡 𝐼 𝑟, 𝑛, 𝜈, 𝑡 𝑑𝑆𝑑𝑠𝑑𝜔𝑑𝜈𝑑𝑡
1/𝜒 – dimention of length; measures a characteristic distance a photon can
travel before it is absorbed – a photon mean free path.
Absorption and Emission Coefficient
Emission coefficient – the energy released by the material in
the form of radiation.
Elementary amount of material of cross-section 𝑑𝑆 and length 𝑑𝑠
releases into a solid angle 𝑑𝜔 in direction 𝑛 within a frequency
band 𝑑𝜈 an amount of energy:
𝑑𝐸𝑒𝑚 = 𝜂 𝑟, 𝑛, 𝜈, 𝑡 𝑑𝑆𝑑𝑠𝑑𝜔𝑑𝜈𝑑𝑡
The dimention is erg cm-3 hz-1 sec-1 sr-1
dS
Absorption and Emission Coefficient
Microscopic physics – all contributions from microscopic
processes that give rise to an absorption or emission of
photons with specified properties.
True absorption and scattering:
True (thermal) absorption – photon is removed from a beam and is
destroyed.
Scattering – photon is removed from a beam and immediately reemitted in a different direction with slightly different frequency.
𝜒 𝑟, 𝑛, 𝜈, 𝑡 = 𝜅 𝑟, 𝑛, 𝜈, 𝑡 + 𝜎 𝑟, 𝑛, 𝜈, 𝑡
Absorption and Emission Coefficient
Quantum theory of radiation – processes that give rise to an
absorption or emission of a photon:
• Induced absorption – an absorption of a photon and transition of an
atom/ion to a higher energy state;
• Spontaneous emission – an emission of a photon and a
spontaneous transition of an atom/ion to a lower energy state;
• Induced emission – an interaction of an atom/ion with a photon and
an emission of another photon with identical properties (negative
absorption).
Radiative Transfer Equation
Basic
Radiative transfer equation:
We express conservation of the total photon energy when a radiation
beam passes through an elementary volume of matter of crosssection 𝑑𝑆 and length 𝑑𝑠 (measuread along the direction of
propagation):
𝐼 𝑟 + Δ𝑟, 𝑛, 𝜈, 𝑡 + Δ𝑡 − 𝐼 𝑟, 𝑛, 𝜈, 𝑡 𝑑𝑆𝑑𝜔𝑑𝜈𝑑𝑡
= 𝜂 𝑟, 𝑛, 𝜈, 𝑡 − 𝜒 𝑟, 𝑛, 𝜈, 𝑡 𝐼 𝑟, 𝑛, 𝜈, 𝑡 𝑑𝑆𝑑𝑠𝑑𝜔𝑑𝜈𝑑𝑡
The difference between spectific intensity before and after passing
through the elementary volume of path length 𝑑𝑠 is equal to the
difference of the energy emitted and absorbed in the volume.
Iν + ΔIν
Radiative Transfer Equation
Basic
𝐼 𝑟 + Δ𝑟, 𝑛, 𝜈, 𝑡 + Δ𝑡 − 𝐼 𝑟, 𝑛, 𝜈, 𝑡 𝑑𝑆𝑑𝜔𝑑𝜈𝑑𝑡
= 𝜂 𝑟, 𝑛, 𝜈, 𝑡 − 𝜒 𝑟, 𝑛, 𝜈, 𝑡 𝐼 𝑟, 𝑛, 𝜈, 𝑡 𝑑𝑆𝑑𝑠𝑑𝜔𝑑𝜈𝑑𝑡
The differences of intensities:
𝜕𝐼
𝜕𝐼
𝜕𝐼 1 𝜕𝐼
diff 𝐼 = 𝑑𝑠 + 𝑑𝑡 =
+
𝑑𝑠
𝜕𝑠
𝜕𝑡
𝜕𝑠 𝑐 𝜕𝑡
General form of the radiative transfer equation:
1 𝜕
𝑐 𝜕𝑡
+ 𝑛 ∙ 𝛻 𝐼 𝑟, 𝑛, 𝜈, 𝑡 = 𝜂 𝑟, 𝑛, 𝜈, 𝑡 − 𝜒 𝑟, 𝑛, 𝜈, 𝑡 𝐼 𝑟, 𝑛, 𝜈, 𝑡
Radiative Transfer Equation
Basic
Special case: one dimentional planar
atmosphere:
𝑛𝑧 =
𝑑𝑧
= 𝑐𝑜𝑠𝜃 ≡ 𝜇
𝑑𝑠
𝜃 – angle between direction of
propagation of radiation and the normal
to the surface
dz
Time independent RTE:
𝑑𝐼 𝜈,𝜇,𝑧
𝜇
𝑑𝑧
= 𝜂 𝑟, 𝑛, 𝜈, 𝑡 −
𝜒 𝑟, 𝑛, 𝜈, 𝑡 𝐼 𝑟, 𝑛, 𝜈, 𝑡
z
Radiative Transfer Equation
Basic
Special case: spherical coordinates
𝜕
𝜕
1 − 𝜇2 𝜕
=𝜇
+
𝜕𝑠
𝜕𝑟
𝑟
𝜕𝜇
Time independent:
𝜕𝐼 𝜈, 𝜇, 𝑟
1 − 𝜇2 𝜕𝐼(𝜈, 𝜇, 𝑟)
𝜇
+
𝜕𝑟
𝑟
𝜕𝜇
= 𝜂 𝜈, 𝜇, 𝑟 − 𝐼 𝜈, 𝜇, 𝑟 𝜒 𝜈, 𝜇, 𝑟
Radiative Transfer Equation
Optical Depth, Source function
1-D transfer equation:
𝜇
𝑑𝐼𝜈
= 𝜂𝜈 − 𝜒𝜈 𝐼𝜈
𝑑𝑧
Elementary optical depth:
𝑑𝜏𝜈 ≡ −𝜒𝜈 𝑑𝑧
Source function:
𝑆𝜈 ≡
𝜂𝜈
𝜒𝜈
The emission and absorption coefficients are local quantities, so the definition of
source function is good for all geometries.
𝜇
𝑑𝐼𝜈
= 𝐼𝜈 − 𝑆𝜈
𝑑𝜏𝜈
z
Radiative Transfer Equation
Optical Depth, Source function
Physical meaning of optical depth:
RTE in the absence of emission:
𝑑𝐼
=𝐼
𝑑𝜏
With the solution:
𝐼 𝜏 = 𝐼(𝜏 + Δ𝜏)exp−Δ𝜏
⇒ the optical depth is the e-folding distance for attenuation of the
specific intensity due to absorption;
⇒ the probability that a photon will travel an optical distance 𝜏 is:
𝑝 𝜏 = exp(−𝜏)
Radiative Transfer Equation
Optical Depth, Source function
Physical meaning of source function:
𝜂𝜈
𝑆𝜈 ≡
𝜒𝜈
𝑁𝑒𝑚 = 𝜂𝑑𝑠
4𝜋
ℎ𝜈
𝑑𝜈𝑑𝑡𝑑𝑆
number of photons emitted in an elementary volume to all direction
(4𝜋 comes from integration over the all solid angles, and ℎ𝜈 transforms energy to
the number of photons).
𝑁𝑒𝑚
4𝜋
= 𝑆 𝜏 𝑑𝜏
𝑑𝜈𝑑𝑡𝑑𝑆
ℎ𝜈
⟹ 𝐭he source function is proportional to the number of photons
emitted per unit optical depth interval.
Radiative Transfer Equation
Elementary Solutions
1. No absorption, no emission, 𝜼 = 𝝌 = 𝟎
𝐼 = 𝑐𝑜𝑛𝑠𝑡
⟹ Radiation intensity remains constant.
2. No absorption, only emission, 𝜼 > 𝟎, 𝝌 = 𝟎
𝑧
𝐼 𝑧, 𝜇 = 𝐼 0, 𝜇 +
𝜂 𝑧 ′ 𝑑𝑧 ′ /𝜇
0
Outcoming radiation from an optically thin radiative slab (e.g. forbidden
line radiation from planetary nebulae, radiation from solar transition
region or/and corona).
3. No emission, only absorption 𝜼 = 𝟎, 𝝌 > 𝟎
𝐼 0, 𝜇 = 𝐼 𝜏, 𝜇 exp − 𝜏 𝜇
Radiative Transfer Equation
Elementary Solutions
4. Absorption and emission, 𝜼 > 𝟎, 𝝌 > 𝟎
General formal solution of the transfer equation – formal because it is
assumed that both 𝜒 and 𝜂 are specified functions of position and
frequency:
𝜏2 − 𝜏1
𝐼 𝜏1 , 𝜇 = 𝐼 𝜏2 , 𝜇 exp −
+
𝜇
𝜏2
𝜏1
𝑡 − 𝜏1
𝑆 𝑡 exp −
𝑑𝑡/𝜇
𝜇
5. Semi-infinite atmosphere
– emergent radiation (𝜏1 = 0, 𝜏2 = ∞)
∞
𝐼 0, 𝜇 =
0
𝑡
𝑆 𝑡 exp − 𝑑𝑡/𝜇
𝜇
observed intensity is a weighted average of the source function along
the line of sight.
Radiadive Transfer Equation
Moments
Moments of specific intensity (of the photon distribution function):
photon energy density, radiation flux, radiation stress tensor.
𝑐𝐹𝜈
𝐹𝜈
𝑐𝑃𝜈
=
1
𝑛 𝐼𝜈 𝑑𝜔
𝑛𝑛
Integration of transfer equation (kinetic equation) – relations between these
moments:
𝜕𝐸𝜈
+ 𝛻 ∙ 𝐹𝜈 = 𝜂𝜈 − 𝜒𝜈 𝑐𝐸𝜈
𝜕𝑡
1 𝜕𝐹
+ 𝑐𝛻 ∙ 𝑃𝜈 = −𝜒𝜈 𝐹𝜈
𝑐 𝜕𝑡
Structure of the moment equations of the kinetic equation:
𝝏
𝒅𝒆𝒏𝒔𝒊𝒕𝒚 𝒐𝒇 𝒒𝒖𝒂𝒏𝒕𝒊𝒕𝒚 + 𝒈𝒓𝒂𝒅𝒊𝒆𝒏𝒕 𝒐𝒇 𝒊𝒕𝒔 𝒇𝒍𝒖𝒙 = 𝒔𝒐𝒖𝒓𝒄𝒆𝒔 − 𝒔𝒊𝒏𝒌𝒔
𝝏𝒕
Radiadive Transfer Equation
Moments
In astrophysics: moments are angle averaged, not angle-integrated
quantities:
𝐽𝜈
𝐻𝜈
𝐾𝜈
1
=
4𝜋
𝑐𝐸𝜈
𝐹𝜈
𝑐𝑃𝜈
1
=
4𝜋
1
𝑛 𝐼𝜈 𝑑𝜔
𝑛𝑛
In the plane-parallel approximation moments of specific intensity are
scalar quantities:
1 1
𝐽𝜈 =
𝐼 𝜇 𝑑𝜇
2 −1 𝜈
1 1
𝐻𝜈 =
𝜇𝐼 𝜇 𝑑𝜇
2 −1 𝜈
1 1 2
𝐾𝜈 =
𝜇 𝐼 𝜇 𝑑𝜇
2 −1 𝜈
Radiative transfer equation
Moments
The moment equations of RTE:
𝑑𝐻𝜈
= 𝐽𝜈 − 𝑆𝜈
𝑑𝜏𝜈
𝑑𝐾𝜈
= 𝐻𝜈
𝑑𝜏𝜈
Eddington factor:
𝑓𝜈𝐾 ≡
𝐾𝜈
𝐽𝜈
Combination of two moment equation:
𝑑 2 𝑓𝜈𝐾 𝐽𝜈
= 𝐽𝜈 − 𝑆𝜈
𝑑𝜏𝜈2
There is no dependence on the angle; useful in numerical solution.
Model atmosphere
Definition and terminology
Model atmosphere –
• Specification of all atmospheric state parameters as function of depth.
• Table of values of the state parameters in the discrete depth points.
• State parameters: depend on the type of the model (on the basic
assumption under which the model is constructed)
• Massive particle state parameters – from this we can determine the
radiation field by a formal solution of the transfer equation.
Basic equations
Classic stellar atmosphere problem
Radiative transfer equation
First order form:
𝜇
𝑑𝐼𝜈
= 𝐼𝜈 − 𝑆𝜈
𝑑𝜏
Second order form:
𝑑 2 (𝑓𝜈𝐾 𝐽𝜈 )
= 𝐽𝜈 − 𝑆𝜈
2
𝑑𝜏𝜈
NA x NF vs. NF
NA – number of angle points
NF – number of frequency points
Basic equations
Classic stellar atmosphere problem
Hydrostatic equilibrium equation:
𝑑𝑃
= −𝑔𝜌
𝑑𝑧
𝒅𝑷
𝑑𝑚 = −𝜌𝑑𝑧 ⟹
=𝒈
𝒅𝒎
𝑚 – mass in the column of a cross-section of 1 cm2 above a given point
in the atmosphere
Basic equations
Classic stellar atmosphere problem
Total pressure:
𝑃 = 𝑃𝑔𝑎𝑠 + 𝑃𝑟𝑎𝑑 + 𝑃𝑡𝑢𝑟𝑏
4𝜋
= 𝑁𝑘𝑇 +
𝑐
∞
0
1 2
𝐾𝜈 𝑑𝜈 + 𝜌𝜐𝑡𝑢𝑟𝑏
2
The hydrostatic equilibrium equation – effective gravity acceleration:
𝑑𝑃𝑔𝑎𝑠
4𝜋
=𝑔−
𝑑𝑚
𝑐
∞
0
𝑑𝐾𝜈
4𝜋
=𝑔−
𝑑𝑚
𝑐
∞
0
𝜒𝜈
𝐻 𝑑𝜈
𝜌 𝜈
true gravity acceleration (acting downward) – the radiative acceleration
(acting upward).
Basic equations
Classic stellar atmosphere problem
Radiative equlibrium equation:
• the total radiation flux is conserved
∞
0
𝜎 4
𝐻𝜈 𝑑𝜈 = 𝑐𝑜𝑛𝑠𝑡 =
𝑇
4𝜋 𝑒𝑓𝑓
• using the radiative transfer equation:
∞
0
∞
𝜅𝜈 𝐽𝜈 − 𝜂𝜈 𝑑𝜈 =
0
𝜅𝜈 𝐽𝜈 − 𝑆𝜈 𝑑𝜈 = 0
Basic equations
Classic stellar atmosphere problem
Statistical equilibrium eqations = rate equations
The conservation equation for a particle 𝑖:
𝐷𝑛𝑖
𝐷𝑡
𝑐𝑜𝑙𝑙
=0
Explicitly:
𝑛𝑖
𝑅𝑖𝑗 + 𝐶𝑖𝑗 =
𝑗≠𝑖
𝑛𝑗 (𝑅𝑗𝑖 + 𝐶𝑗𝑖 )
𝑗≠𝑖
𝑅 and 𝐶 – radiative and collisional rate;
Total number of transitions out of level 𝑖 = total number of transitions into level 𝑖;
Radiative rates depend on radiation intensity;
Collisional rates depend on temperature and electron density.
Basic equations
Classic stellar atmosphere problem
Total number conservation equation (or abundance
definition equation):
𝑛𝑖 = 𝑁𝑎𝑡𝑜𝑚
𝑖
Only a limited number of levels of an atom/ion is treated explicitly (the
rate equation is written and solved, low-lying levels); remaining levels –
approximations, and the abundance definition:
𝑛𝑖 +
𝑒𝑥𝑝𝑙𝑖𝑐𝑖𝑡
𝑛𝑖 = 𝑁𝑎𝑡𝑜𝑚
𝑎𝑝𝑝𝑟𝑜𝑥𝑖𝑚𝑎𝑡𝑖𝑜𝑛
Basic equations
Classic stellar atmosphere problem
Charge conservation equation:
• global electric neutrality of the medium
𝑛𝑖 𝑍𝑖 − 𝑛𝑒 = 0
𝑖
𝑍𝑖 is the charge associated with level 𝑖 (0 for levels of neutral atoms, 1
for once ionised atoms etc.);
summation extends over all levels of all ions of all species.
Additional equations
Classical stellar atmosphere problem
Definition equations of the absorption and emission coefficients:
𝜒𝜈 =
𝑛𝑖 −
𝑖
𝑗>𝑖
𝑔𝑖
𝑛 𝜎 𝜈 +
𝑔𝑗 𝑗 𝑖𝑗
𝜂𝜈 =
•
•
•
•
𝑐2
𝑖
𝑛𝑒 𝑛𝜅 𝜎𝜅𝜅 𝜈, 𝑇 1 − 𝑒 −ℎ𝜈
+
2ℎ𝜈 3
(𝑛𝑖 − 𝑛𝑖∗ 𝑒 −ℎ𝜈/𝑘𝑇 )𝜎𝑖𝑘 𝜈
𝑘𝑇
+ 𝑛𝑒 𝜎𝑒
𝜅
[
𝑛𝑗
𝑖
𝑗>𝑖
𝑔𝑖
𝜎 𝜈 +
𝑔𝑗 𝑖𝑗
𝑛𝑖∗ 𝜎𝑖𝜅
𝜈
𝑒 −ℎ𝜈/𝑘𝑇
𝑖
ℎ𝜈
−𝑘𝑇
𝑛𝑒 𝑛𝜅 𝜎𝜅𝜅 (𝜈, 𝑇)𝑒
]
+
𝜅
Bound-bound transitions (i.e. spectral lines)
Bound-free transitions (continua)
Free-free absorption (inverse brehmstrahlung)
Electron scattering
Another equations: relevant cross-sections, definition of LTE populations, etc.
Basic equations: summary
Equation
State parameter
Radiative transfer
Mean intensities, 𝐽𝜈
Radiative equilibrium
Temperature, 𝑇
Hydrostatic equilibrium
Total particle density, 𝑁
Statistical equilibrium
Populations, 𝑛𝑖
Charge conservation
Electron density, 𝑛𝑒
Types of model atmosphere
Static models:
• assumption of hydrodynamical equilibrium
• applies only to atmospheric layers that are close to
hydrodynamic equilibrium, i.e. the macroscopic velocity is
small compared to the thermal velocity of atoms – stellar
photospheres
• basic input parameters: effective temperature, surface
gravity and chemical composition,
• additional parameters: microturbulence, and in case of
convective models: mixing length.
Types of model atmosphere
Static LTE models:
• LTE assumption, two state parameters, temperature and
density (or electron density) describe the physical state of the
atmosphere at any given depth.
• standard models
• Example: Kurucz ATLAS9 code
Types of model atmosphere
Static non-LTE models: model which takes into account
some kind of departure from LTE:
• models solving for the full structure (TLUSTY)
• restricted non-LTE problem:
• the atmospheric structure is assumed to be known from previous
calculations (LTE or simplified non-LTE),
• radiative transfer and statistical equilibrium for a chosen atom/ion is solved
simultaneously (DETAIL/SURFACE).
• non-LTE line-blanketed models:
• non-LTE is considered in most energy levels and transitions between them
– lines and continua – that influence the atmospheric structure.
• number of such lines may go to millions.
non-LTE models
When departures from LTE may be important in stellar
atmospheres?
non-LTE models
For a star of any spectral type, there is always a
wavelength range, and of course a layer in the
atmosphere, where non-LTE effects are important.
„important non-LTE effects” –
• arbitrary – it depends what precision do we want;
• e.g. for B stars, visual part 10% - LTE will be OK, 2-5% non-LTE is necessary;
• EUV, the same star – 10-20% requires non-LTE effects.
Types of model atmosphere
Unified models:
• no assumption of hydrostatic equilibrium in the whole
atmosphere ranging from a static photosphere to a dynamical
outher parts.
Literature
• D. Mihalas: „Stellar Atmospheres”
• I. Hubeny: „Stellar atmospheres theory: an introduction”
in: Stellar atmospheres: Theory and Observations,
Lecture note in physics, J.P. De Greeve, R.Blomme, H.
Hensberg (Eds.), Springer
• D. Gray: „The observations and analysis of stellar
photospheres”