Stellar Atmospheres II
Stellar Opacity..cont’d
Radiative Transfer
The Transfer Equation
The Profiles of Spectral Lines
Optical Depth
Sources of Opacity
Bound-Bound Transitions
Bound-Free Absorption (photoionization)
•Continuum Opacity and the H- ion
(wavelength < 1640 nm)
Graduate level treatment
Free-Free Absorption
Electron-Electron Scattering
Bound-Bound Transitions
Photons captured by atoms
Hydrogen atom transitions
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= RH ç 2 - 2 ÷
èn m ø
Cross section for photon absorption
Bound-Free Transitions
Photon Ionizes atoms
Free-Free Transitions
Electron scattering
Thomson Scattering Cross-section
Balmer Jump
Continuum Opacity and the H- ion
Hydrogen atoms can “catch” an
additional electron
Binding energy is 0.754 eV
Photons with wavelength of
1640nm or less can ionize this
Hydrogen ion
Total Opacity is sum of
all contributions:
An additional source of opacity for
stars cooler than type F0
Rosseland Mean Opacity
An attempt at estimating the
average opacity over all
Weight by the rate at which
Intensity distribution (blackbody
radiation) varies with
Determine dependence of other
parameters such as temperature
Total Mean Opacity is mean of
the sum of the opacity from the
various sources
– Bound-bound
– Bound-free
– Free -free
– Electron scattering
– H- ion
Kramers-opacity law
k = k 0r
T 3.5
– Bound-bound
– Bound-free
How far do we look into a star?
Looking into a star at any angle,
we always look back to an optical
depth of about t l = 2/3 as
measured straight back along the
line of sight
Photon’s at a distance of less
than 1 mean free path from the
surface are likely to escape
Star’s photosphere is defined to
be the layer from which visible
light originates
Formation of spectral lines
(absorption) occur because
temperature of the material in the
star decreases outwards from the
center of the star
Limb Darkening
When looking at the center of the
sun one can see “deeper” than
when looking at the edge of the
Deeper is hotter
Hotter is brighter…
What’s this stuff about…?
– Stellar atmosphere.
• P(r),T(r),rho(r)
– Radiation Field
• I(r)
• Opacity,Optical Depth
• Limb Darkening
– Spectrum from Star
– Elemental Abundances
Radiative Transfer
Energy Transfer in the form of
Electromagnetic radiation
In an equilibrium steady-state star
there can be no change in the total
energy contained within any layer of
the stellar interior or atmosphere
Implies absorption must be balanced
by emission
Inverse emission processes
Emission processes re-direct photons
from initial path
d=l N
Tortuous path:
100 steps -->
d=10 l
10000 steps --> d=100 l
Random Walk
1000000 steps --> d=1000 l
Optical Depth is roughly the number of
photon mean free paths from that
point to the surface. d = t l l = l N
N = t l2
Radiation Pressure Gradient
Mean free path of photons in the
interior of the star is typically a
few cm.
Center of star is few hundreds of
millions of meters from its surface
How do the photons ever get out?
Temperature of star decreases
outward from center. This causes
the radiation pressure to
decrease outwards as well
Pressure Gradient
Photon “Breeze” from star
Pressure Gradient produces a slight
net movement of photons upward that
carries the radiative flux
This causes a slow upward diffusion of
randomly walking photons
The Transfer Equation
Describes the passage of light through a Star’s Atmosphere
The Emission Coefficient j
– Describes the rate of
emission processes
Source of radiation S=j/
– Ratio of rates of emission to
Equation of Radiative Transport
The Transfer Equation
Case of Blackbody Radiation
Particle and Photons in equilibrium individually and with each other
Every process of absorption is balanced by an inverse process of
Intensity of Radiation described by the Planck Function I = B
Intensity constant throughout box
Þ dIl /ds = 0 Þ Il = Sl
Thermodynamic Equilibrium
Þ Sl = Bl
Source Function is equal to the Planck function
Plane-Parallel Atmosphere
Far from center of star the radius
of curvature is large…consider
plane-parallel (flat) slabs…
Work in units of optical depth
instead of distance…
Z-axis is vertical z=0 is at top of
Vertical Optical Depth
Optical depth for inclined ray
Results in…Transfer Equation
Plane Parallel Atmosphere
Further simplify by using mean
opacity--->”Gray Atmosphere”.
No more dependence on
wavelength. Replace Iand S.
We obtain…
Integrating over solid angles…S
doesn’t depend on angles
And using 9.8 and 4pi.
We obtain
•Multiplying 9.40 by cosand
•The term on the left is the
radiation pressure multiplied by c.
The first term on the right is the
radiative flux the second integral is
•Which is just the radiation
pressure gradient 9.31. Can be
interpreted as the net radiative flux
is driven by differences in radiation
pressure. This equation will be
used in chapter 10 to determine
the tempearture structure in the
interior of a star.
Plane Parallel Atmosphere
In an equilibrium stellar
atmosphere, every process of
absorption is balanced by an
inverse process of emission, no
net energy is subtracted from
or added to the radiation field.
This means that in a planeparallel atmosphere, the radiative
flux must have the same value at
every level in the atmosphere,
including its surface….
Because the flux is a constant
Can integrate 9.42
To obtain the radiation
pressure as a function of
optical depth
Eddington Approximation
Approximate I with Iout and Iin
Evaluate Temperature as a
function of optical depth
Limb Darkening Revisited
Eddington Approximation of Solar Limb
Profiles of Spectral Lines
• Spectral Line conveys
information about the
environment in which it
was formed
• Line width
• Line Shape
– Broadening
– Profile
Optically Thin Spectral Line. Thus termed
because there is now wavelength at which
the Radiant flux has been completely
Sources of Broadening
• Natural Broadening
– From Uncertainty Principle
– ~2-4 x 10-5 nm
• Doppler Broadening
– From thermal motion of atoms
– ~4.27 x 10-2 nm at 5772K
• Pressure and Collisional Broadening
– From atomic orbitals being perturbed from collisions
– ~2-4 x 10-5 nm
Natural Broadening
More Involved calculation…
Doppler Broadening
More in depth analysis…
Tails in M-B dist
Turbulent blobs…
Pressure Broadening
The Voigt Profile
Combination of all broadening
Curve of Growth
Tool to determine the abundances of elements in stellar atmospheres.
Width of spectral line depends on the number of absorbing atoms
Can determine Abundances of elements!!!!
Consider an element not present in stellar atmosphere --> No line
Add some atoms of that element ---> optically thin line
Double concentration -->width doubles
Add more -->saturation
Add even more --> wings of line grows
Add even more ---> pressure broadening…
Curve of Growth
Tool to determine the abundances of elements in stellar atmospheres.
Width of spectral line depends on the number of absorbing atoms
Can determine Abundances of elements!!!!
Curve of Growth for the Sun
Let’s determine the amount of sodium in the Sun!!!
Observe width of spectral line
Read off Na from curve of growth
Computer Modeling of Stellar Atmospheres
Each atmospheric layer is
involved in the formation of line
profiles and contributes to the
spectrum observed for the star.
Many “ingredients”…
– Opacity
– Radiation Pressure
– Transfer equations
– +
• Hydrostatic Equilibrium
• Statistical and Quantum
• Transport of energy by
radiation and
• how temperature, pressure and
density vary with depth.
•Line profiles
•Effective Temperature
•Surface Gravity,….
Worked Problems
Worked Problems
Worked Problems
Worked Problems
Worked Problems
Worked Problems