Transcript Bound-Free Transitions

Chapter 8 – Continuous Absorption
• Physical Processes
• Definitions
• Sources of Opacity
– Hydrogen bf and ff
– H– He
– Scattering
Physical Processes
•
•
•
•
•
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.
Scattering – deflection of a photon from its original path by a
particle, without changing wavelength
– Rayleigh scattering if the photon’s wavelength is greater than
the particle’s resonant wavelength. (Varies as l-4)
– Thomson scattering if the photon’s wavelength is much less
than the particle’s resonant wavelength. (Independent of
wavelength)
– Electron scattering is Thomson scattering off an electron
Photodissociation may occur for molecules
Electron Scattering vs. Free-Free Transition
• Electron scattering – the path of the
photon is altered, but not the energy
• Free-Free transition – the electron
emits or absorbs a photon. A freefree 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 an 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
h
 mv 
c
m0
2
v
v
1 2
c
h  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)
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 dt = krdx. We need to calculate k, the
absorption coefficient per gram of material
• First calculate the atomic absorption coefficient
a (per absorbing atom or ion)
• Multiply by number of absorbing atoms or ions per
gram of stellar material (this depends on
temperature and pressure)
Bound-Bound Transitions
• These 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
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 –
not the statistical weight!)
3
Rg
a
g
l
32 below
e
bf
0 bf
• For the nth
bound
level
the
continuum
and l
a bf (l , n) 

3
5 3
n5
< ln
3 3 h n
• where a0 = 1.044 x 10–26 for l in Angstroms
2
6
Hydrogen Bound-Free Absorption Coefficient
a (cm-2 per atom) x 10^6
3.5E-14
3E-14
2.5E-14
2E-14
Balmer
Absorption
1.5E-14
1E-14
5E-15
Lyman
Absorption n=1
Paschen
Absorption
n=3
n=2
0
100
600
1200 2200 3200 4200 5200 6200 7200 8200 9200
Wavelength (A)
Converting to the MASS Absorption Coefficient
• Multiply by the number of neutral hydrogen atoms
per gram in each excitation state n
• 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

 3


n n
kT
bf
n
3 bf
n0
n0
aN
k (H )  
a
N
l
n g e
Temperature
20000
HI
16000
12000
8000
4000
H fractional ionization
1.2
1
0.8
0.6
0.4
0.2
0
H II
Class Investigation
• Compare kbf at l=5000A and level T=Teff
for the two models provided
• Recall that
a 0 g bf l3
a bf (l , n) 
• and k=1.38x10-16, a0 =1x10-26
• And
n5
r
P  kT

• Use the hydrogen ionization chart provided