chapter 8 – continuous absorption
DESCRIPTION
Physical Processes Definitions Sources of Opacity Hydrogen bf and ff H - H 2 He Scattering. How does k n 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 - PowerPoint PPT PresentationTRANSCRIPT
Chapter 8 – Continuous Absorption
• Physical Processes• Definitions• Sources of Opacity
– Hydrogen bf and ff– H-
– H2
– He– Scattering
• How does 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 to determine 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 -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 d = dx. We need to calculate , the absorption coefficient per gram of material
• First calculate the atomic absorption coefficient (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
As m > ∞, the transition approaches a bound-free condition. For photons of higher energy, the hydrogen atom is ionized
22
111
mnR
R is the Rydberg Constant, R = 1.1 x 10-3 Å-1
Remember the hydrogen atom:
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!)
(for the nth bound level below the continuum and < n)
• where0 = 1.044 x 10–26 for in angstroms and gbf is of order 1
• The atomic absorption coefficient bf(H) has units of cm2 per neutral H atom
5
30
353
62
33
32),(
n
g
n
Rg
h
en bfbf
bf
Must also consider level populations
• Back to Boltzman and Saha!
• 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):
kTnn eTu
g
N
N
)(0
kTbf
nn
n
nnbf eg
nN
NH
00
3
3
)(
One more step
• Terms with n > n0+2 can be replaced with an integral (according to Unsöld)
• Plus a little manipulation, gives
• This is the absorption coefficient per neutral hydrogen atom
• Here, I is the ionization potential, NOT the intensity!
2
33
0
0
0
3 )1010(2
log10)(
n
n
Ibfbf I
en
gH
Model Flux Model Flux DistributionsDistributions• Sharp edges
are the result of sudden drop in bound-free opacities due to ionization
Free-Free Absorption from H IFree-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:
Ifff I
egH
10
2
log)( 3
0
Opacity from Neutral HydrogenOpacity 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 HOpacity from the H-- Ion Ion
• Bound–free and free-free• Only one known bound state for bound-
free absorption • 0.754 eV binding energy• So < 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 HMore H-- Bound-Free Opacity Bound-Free Opacity
• Per atom absorption coefficient for H- can be parameterized as a polynomial in :
• Units of cm2 per neutral hydrogen atom
...2210 aaabf
1248.0log5.25040
log)(
)(log TI
TP
HN
HNe
754.02
510 1010158.4)( ebfbf PxH
H- Bound-Free Absorption CoefficientH- Bound-Free Absorption Coefficient
• Two theoretical calculations
• Important in the optical and near infrared
• Peaks at 8500Å
HH-- Free-Free Absorption Coefficient 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 to3
• Small at optical wavelengths• Comparable to H- bf at 1.6 microns• Increases to the infrared
HH-- Free Free Absorption Coefficient 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
• the f’s are functions of log and is 5040/T
• Units are cm2 per neutral H atom
2
210 loglog26 1010)( fffeffff PH
Molecular HMolecular H22, H, H22++, H, H22
-- OpacitiesOpacities
• 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
Collision Collision induced induced
opacity of opacity of molecular molecular hydrogenhydrogen
• 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.
Linsky/JILA
Helium AbsorptionHelium 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
• 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
• Combining these equations gives
• So v=0 (the photon isn’t absorbed) or v=c (not allowed)
v
cv
mmv
c
h
2
2
0
1
20
20
20 )( cmcmmcmh
22 )1()(1 cv
cv
Electron ScatteringElectron Scattering• Thomson scattering (photons scatters off a free
electron, no change in , just direction):
• Independent of wavelength• In hot stars (O and early B) where hydrogen is
ionized (Pe~0.5Pg), (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
122522
2
10654.6)(3
8 ecmxmce
e
H
ee
PP
eN
ee )()()(
Rayleigh ScatteringRayleigh Scattering
• Photons scatter off bound electrons (varies as -4)
• Generally can be neglected• But – since it depends on 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 SourcesOther Sources
• Metals: C, Si, Al, Mg, Fe produce bound-free 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
Sources of Opacity for Teff=4500 Log g = Sources of Opacity for Teff=4500 Log g = 1.51.5
0%
20%
40%
60%
80%
100%
0
0.0
1
0.1 1
10
Optical Depth
Fra
ctio
n of
Opa
city
H(BF)
H2+
Mg+Al+Si
H-
Opacity Sources at 5143KOpacity Sources at 5143K
Opacity at 6429 KOpacity at 6429 K
Opacity at 7715 KOpacity at 7715 K
Opacity at 11600 KOpacity at 11600 K
Opacity vs. Spectral TypeOpacity vs. Spectral Type
0
10
20
30
40
5500
6500
8000
12000
25000
Temperature
Kap
pa R
osse
land
Main Sequence
Supergiants
Dominant Opacity vs. Spectra Dominant Opacity vs. Spectra TypeType
O B A F G K M
H-Neutral H
H-
Electron scattering(H and He are too highly ionized)
He+ He
Ele
ctr
on
Pre
ssu
r e
High
Low
(high pressure forces more H-)
Low pressure –less H-, loweropacity
Class Exercise – Electron Class Exercise – Electron ScatteringScattering
• Estimate the absorption coefficient for electron scattering for the models provided at a level where T=Teff
• Recall that
• and
• with in AMU and k=1.38x10-16
• How does e compare to Rosseland
ee
Nx 25106.6
kTP
Class Investigation
• Compare bf at =5000A and level T=Teff for the two models provided
• Recall that
• and k=1.38x10-16, a0 =1x10-26
• And
• Use the hydrogen ionization chart from your homework.
kTP
5
30),(n
gn bf
bf