Download Chapter 8 – Continuous Absorption

Chapter 8 – Continuous Absorption
• Physical Processes
• Definitions
• Sources of Opacity
–
–
–
–
–
Hydrogen bf and ff
HH2
He
Scattering
• How does kn affect the
spectrum?
– More continuous
absorption, less
continuum light at that
wavelength
– More continuous
absorption, lines must
form in shallower layers,
at lower optical depth
– Need kn to determine
T(t) relation
Many physical processes
contribute to opacity
• Bound-Bound Transitions – absorption or emission of
radiation from electrons moving between bound energy levels.
• Bound-Free Transitions – the energy of the higher level
electron state lies in the continuum or is unbound.
• Free-Free Transitions – change the motion of an electron
from one free state to another.
• Electron Scattering – deflection of a photon from its
original path by a particle, without changing its wavelength
– Rayleigh scattering – photons scatter off bound
electrons. (Varies as l-4)
– Thomson scattering –photons scatter off free electrons
(Independent of wavelength)
• Photodissociation may occur for molecules
What can various particles do?
• Free electrons – Thomson scattering
• Atoms and Ions –
– Bound-bound transitions
– Bound-free transitions
– Free-free transitions
• Molecules –
– BB, BF, FF transitions
– Photodissociation
• Most continuous opacity is due to hydrogen in one
form or another
Monochromatic Absorption Coefficient
• Recall dtn = knrdx. We need to calculate kn, the
absorption coefficient per gram of material
• First calculate the atomic absorption coefficient
an (per absorbing atom or ion)
• Multiply by number of absorbing atoms or ions per
gram of stellar material (this depends on
temperature and pressure)
MOSTLY HYDROGEN
Bound-Bound Transitions
• Bound-bound transitions produce spectral lines
• At high temperatures (as in a stellar interior) these may
often be neglected.
• But even at T~106K, the line absorption coefficient can
exceed the continuous absorption coefficient at some
densities
Remember the hydrogen atom:
1 
1
 R 2  2 
l
n m 
1
R is the Rydberg Constant,
R = 1.1 x 10-3 Å-1
As m > ∞, the transition approaches a boundfree condition. For photons of higher energy,
the hydrogen atom is ionized
Bound Free Transitions
• An expression for the bound-free coefficient was
derived by Kramers (1923) using classical physics.
• A quantum mechanical correction was introduced
by Gaunt (1930), known as the Gaunt factor (gbf is
not the statistical weight!)
3
2
6
Rg
a
g
l
32 e
bf
0 bf
a bf (l , n) 

3
5 3
n5
3 3 h nn
(for the nth bound level below the continuum and l < ln)
• where a0 = 1.044 x 10–26 for l in angstroms and gbf
is of order 1
• The atomic absorption coefficient abf(H) has units
of cm2 per neutral H atom
Must also consider level populations
• Back to Boltzman and Saha!

Nn
g n  kT

e
N u0 (T )
• gn = 2n2 is the statistical weight
• u0(T) = 2 is the partition function
• So, the abs. coef. per neutral H atom is (summing
over all levels n):
an Nn
l

k ( H bf )  
 a n  3 g bf e kT
N
n0
n0 n


3
One more step
• Terms with n > n0+2 can be replaced with an integral
(according to Unsöld)
• Plus a little manipulation, gives
n0  2


gbf  log e
  3
3
I
k (Hbf )  a 0 l   3 10 
(10
 10 )
2I
 n0 n

• This is the absorption coefficient per neutral
hydrogen atom
• Here, I is the ionization potential, NOT the
intensity!
Model Flux
Distributions
• Sharp edges
are the result
of sudden drop
in bound-free
opacities due
to ionization
Free-Free Absorption from H I
• Much less than bound free absorption
• Kramers (1923) + Gaunt (1930) again
• Absorption coefficient depends on the
speed of the electron (slower electrons are
more likely to absorb a photon because
their encounters with H atoms take longer)
• Adopt a Maxwell-Boltzman distribution for
the speed of electrons
• Again multiply by the number of neutral
hydrogen atoms:
log e I
3
k ( H ff )  a 0 l g f
2I
10
Opacity from Neutral Hydrogen
• Neutral hydrogen (bf and ff) is the
dominant source of opacity in stars of
B, A, and F spectral type
• Discussion Questions:
– Why is neutral hydrogen not a dominant
source of opacity in O stars:
– Why not in G, K, and M stars?
Opacity from the H- Ion
• Bound–free and free-free
• Only one known bound state for bound-free
absorption
• 0.754 eV binding energy
• So l < 16,500A = 1.65 microns
• Requires a source of free electrons
(ionized metals)
• Major source of opacity in the Sun’s
photosphere
• Not a source of opacity at higher
temperatures because H- becomes too
ionized (average e- energy too high)
More H- Bound-Free Opacity
• Per atom absorption coefficient for H- can
be parameterized as a polynomial in l:
abf  a0  a1l  a2l  ...
2
N (H )
5040
log
  log Pe 
I  2.5 log T  0.1248

N (H )
T

5
2
k (Hbf )  4.158x10 abf Pe 10
10
0.754
• Units of cm2 per neutral hydrogen atom
H- Bound-Free Absorption Coefficient
• Two
theoretical
calculations
• Important
in the
optical and
near
infrared
• Peaks at
8500Å
H- Free-Free Absorption Coefficient
• The free-free H- absorption
coefficient depends on the speed of
the electron
• Possible because of the imperfect
shielding of the hydrogen nucleus by
one electron
• Proportional to l3
• Small at optical wavelengths
• Comparable to H- bf at 1.6 microns
• Increases to the infrared
H- Free Free Absorption Coefficient
• H- ff is important in the
infrared
• combining H- bf and ff
gives an opacity minimum
at 1.6 microns
• H- ff parameterized as
k ff (H )  10 Pe10

ff
26
f0  f1 log  f2 log2 
• the f’s are functions of
logl and  is 5040/T
• Units are cm2 per neutral
H atom
Molecular H2, H2+, H2- Opacities
• H2 is more common than H in stars cooler
than mid-M spectral type (think brown
dwarfs!!)
• Recall that these are important in L and T
dwarfs! Also in cool white dwarfs…
• Not important in optical region (H2+ less
than 10% of H- in the optical)
• H2 in the infrared
• H2+ in the UV,
• H2- has no stable bound state, but ff
absorption is important in cooler stars
Linsky/JILA
Collision induced
opacity of
molecular
hydrogen
• H2 has no dipole moment - no rotation or vibration-rotation
spectrum
• Collisions with (H2, He, H) can induce transient dipole moments
• Fundamental VR band at 4162 cm-1 (2.4 microns).
• First overtone VR band at 8089 cm-1 (1.2 microns).
• Second overtone VR band at 11786 cm-1 (0.2 microns).
• Collisions are fast - individual spectral lines broad and overlap
• H2CIO is important for computing the temperature structure of
brown dwarfs because it is a near-continuous opacity source that
fills in the opacity gaps between the molecular absorption lines.
Helium Absorption
• He in hot stars only, O and early B stars –
1=19.7eV, I1=24.6 eV, I2=54.4 eV
– He I absorption mimics H
– He II also mimics H, but x4 in energy, ¼ in l
• Bound-free He- absorption is negligible
(excitation potential of 19 eV!)
• Free-free He- can be important in cool
stars in the IR
• BF and FF absorption by He is important in
the hottest stars (O and early B)
Electron Scattering vs. Free-Free Transition
• Electron scattering (Thomson scattering) – the
path of the photon is altered, but not the energy
• Free-Free transition – the electron emits or
absorbs a photon. A free-free transition can only
occur in the presence of an associated nucleus. An
electron in free space cannot gain the energy of a
photon.
Why Can’t a Lone Electron Absorb a Photon?
• Consider an electron at rest that is encountered by a photon,
and let it absorb the photon….
• Conservation of momentum says
• Conservation of energy says
hn
 mv 
c
m0
2
v
v
1 2
c
hn  m0 c 2  (m  m0 )c 2  m0 c 2
• Combining these equations gives
1  ( v ) 2  (1  v ) 2
c
c
• So v=0 (the photon isn’t absorbed) or v=c (not allowed)
Electron Scattering
• Thomson scattering (photons scatters off a free
electron, no change in l, just direction):
8
e2 2
a e   ( 2 )  6.654x1025 cm2e 1
3 mc
Ne
Pe
k (e )  a (e )
 a (e )
r
PH
• Independent of wavelength
• In hot stars (O and early B) where hydrogen is
ionized (Pe~0.5Pg), k(e)/Pe is small unless Pe is small
• In cool stars, e- scattering is small compared to
other absorbers for main sequence star but is
more important for higher luminosity stars
Rayleigh Scattering
• Photons scatter off bound electrons
(varies as l-4)
• Generally can be neglected
• But – since it depends on l4, it is
important as a UV opacity source in
cool stars with molecules in their
atmospheres.
• H2 can be an important scattering
agent
Other Sources
• Metals: C, Si, Al, Mg, Fe produce boundfree opacity in the UV
• Line Opacity: Combined effect of
millions of weak lines
– Detailed tabulation of lines
– Opacity distribution functions
– Statistical sampling of the absorption
• Molecules: CN-, C2-, H20- , CH3, TiO are
important in late and/or very late stars
100%
80%
H(BF)
60%
H2+
40%
Mg+Al+Si
20%
H-
Optical Depth
10
1
0.1
0.01
0%
0
Fraction of Opacity
Sources of Opacity for Teff=4500 Log g = 1.5
Opacity Sources at 5143K
Opacity at 6429 K
Opacity at 7715 K
Opacity at 11600 K
Dominant Opacity vs. Spectra Type
Low
Electron scattering
(H and He are too
highly ionized)
He+ He
Low pressure –
less H-, lower
opacity
Neutral H H-
H-
High
(high pressure forces more H-)
O
B
A
F
G
K
M