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