chapter 8 – continuous absorption

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Chapter 8 – Continuous Absorption • Physical Processes • Definitions Sources of Opacity Hydrogen bf and ff –H - H 2 – 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

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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 Presentation

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Page 1: Chapter 8 – Continuous Absorption

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

Page 2: Chapter 8 – Continuous Absorption

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

Page 3: Chapter 8 – Continuous Absorption

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

Page 4: Chapter 8 – Continuous Absorption

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

Page 5: Chapter 8 – Continuous Absorption

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:

Page 6: Chapter 8 – Continuous Absorption

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

Page 7: Chapter 8 – Continuous Absorption

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

)(

Page 8: Chapter 8 – Continuous Absorption

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

Page 9: Chapter 8 – Continuous Absorption
Page 10: Chapter 8 – Continuous Absorption

Model Flux Model Flux DistributionsDistributions• Sharp edges

are the result of sudden drop in bound-free opacities due to ionization

Page 11: Chapter 8 – Continuous Absorption

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

Page 12: Chapter 8 – Continuous Absorption

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?

Page 13: Chapter 8 – Continuous Absorption

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)

Page 14: Chapter 8 – Continuous Absorption

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

Page 15: Chapter 8 – Continuous Absorption

H- Bound-Free Absorption CoefficientH- Bound-Free Absorption Coefficient

• Two theoretical calculations

• Important in the optical and near infrared

• Peaks at 8500Å

Page 16: Chapter 8 – Continuous Absorption

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

Page 17: Chapter 8 – Continuous Absorption

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

Page 18: Chapter 8 – Continuous Absorption

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

Page 19: Chapter 8 – Continuous Absorption

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

Page 20: Chapter 8 – Continuous Absorption

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)

Page 21: Chapter 8 – Continuous Absorption

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.

Page 22: Chapter 8 – Continuous Absorption

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

Page 23: Chapter 8 – Continuous Absorption

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 )()()(

Page 24: Chapter 8 – Continuous Absorption

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

Page 25: Chapter 8 – Continuous Absorption

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

Page 26: Chapter 8 – Continuous Absorption

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-

Page 27: Chapter 8 – Continuous Absorption

Opacity Sources at 5143KOpacity Sources at 5143K

Page 28: Chapter 8 – Continuous Absorption

Opacity at 6429 KOpacity at 6429 K

Page 29: Chapter 8 – Continuous Absorption

Opacity at 7715 KOpacity at 7715 K

Page 30: Chapter 8 – Continuous Absorption

Opacity at 11600 KOpacity at 11600 K

Page 31: Chapter 8 – Continuous Absorption

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

Page 32: Chapter 8 – Continuous Absorption

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

Page 33: Chapter 8 – Continuous Absorption

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

Page 34: Chapter 8 – Continuous Absorption

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