Download 6. Stellar Spectra

6. Stellar spectra
excitation and ionization, Saha’s equation
stellar spectral classification
Balmer jump, H-
1
Occupation numbers: LTE case
Absorption coefficient:  = ni
 calculation of occupation numbers needed
LTE
each volume element in thermodynamic equilibrium at temperature T(r)
hypothesis: electron-ion collisions adjust equilibrium
difficulty: interaction with non-local photons
LTE is valid if effect of photons is small or radiation field is described by
Planck function at T(r)
otherwise: non-LTE
2
Excitation in LTE
Boltzmann excitation equation
nij: number density of atoms in excited level i of ionization stage j
(ground level: i=1 neutral: j=0)
gij: statistical weight of level i = number of degenerate states
Eij excitation energy relative to ground state
gij = 2i2 for hydrogen
log
= (2S+1) (2L+1) in L-S coupling
nij
g
5040
= log ij − E ij (eV )
n1j
g1j
T
The fraction relative to the total number of atoms of in ionization stage j is
nij
gij
=
e−Eij /kT
nj
Uj (T )
Uj (T ) =
X
i
gij e−Eij /kT
Uj (T) is called the
partition function
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Ionization in LTE: Saha’s formula
Generalize Boltzmann equation for ratio of two contiguous ionic species j and j+1
Consider ionization process j  j+1
initial state:
n1j & statistical weight g1j
final state:
n1j+1 + free electron & statistical weight g1j+1 gEl
RR
n1j+1 (v) dx dy dz dp x dpy dpz
n1j+1(v )dV d3p = n1j+1
number of ions in groundstate with free electron with velocity in (v,v+dv) in phase space
gEl: volume in phase space normalized to smallest possible volume (h3) for electron:
gEl = 2
dx dy dz dpx dpy dpz
h3
R
dx dy dz =
R
dV = ∆V = 1/ne
dpx dpy dpz = 4πp2 dp = 4πm3v2 dv
2 spin orientations
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Ionization: Saha’s formula
n1j +1(v)
1
2
g
dV dpx dpy dpz = 1j+1 gEl e −(Ej + 2 mv )/kT
n1j
g1j
using Boltzmann formula
n1j +1 (v)
1
g1j +1
dp x dpy dpz − [Ej+ 2m
(p 2x+p2y +p2z )]/k T
dV dpx dpy dpz =
2 dV
e
n1j
g1j
h3
Sum over all final states: integrate over all phase space
n1j+1
∆V
g
= n1j 1j +1 2 3 e− Ej/k T
g1j
h
Z∞ Z∞ Z∞
e
1
− 2 mkT
(p2x +p2y +p2z )
dpx dpy dpz
−∞ −∞ −∞
Z∞
2
e−x dx =
√
π
−∞
(2πmkT )3/ 2
Saha 1920
g
n1j +1 ne
= 2 1j+1
n1j
g1j
µ
2π mkT
h2
¶3/ 2
E
e
− kTj
- ionization falls with ne (recombinations)
- ionization grows with T
5
Ionization: Saha’s formula
Generalize for arbitrary levels (not just ground state):
nj =
∞
X
nij
nj+1 =
i=1
nij+1 = n1j +1
nj +1
∞
X
nij+1
i=1
gij+1 − Eij+1 /k T
e
g1j+1
using Boltzmann’s equation
∞
n1j +1 X
=
gij+1 e− Eij+1 /k T
g1j +1
i=1
Uj +1 (T ) :
also
nj =
n1j
Uj (T )
g1j
partition function
U (T )
nj +1 ne
= 2 j +1
nj
Uj (T )
µ
2πmkT
h2
¶3/2
E
− kTj
e
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Ionization: Saha’s formula
Using electron pressure Pe instead of ne (Pe = nekT)
U (T )
nj +1
Pe = 2 j+1
nj
Uj (T )
µ
2πm
h2
¶ 3/ 2
Ej
(kT )5/2e − kT
which can be written as:
log10
nj +1
U (T )
5040
= −0.1761 − log10 P e + log10 j +1
+ 2.5 log10 T −
Ej
nj
Uj (T )
T
with Pe in dyne/cm2 and Ej in eV
7
Ionization: Saha’s formula
n(H+) / n(H)
n(H) /[n(H) + n(H+)]
n(H+) /[n(H) + n(H+)]
4,000
2.46E-10
1.000
0.246E-9
6,000
3.50E-4
1.000
0.350E-3
8,000
5.15E-1
0.660
0.340
10,000
4.66E+1
0.021
0.979
12,000
1.02E+3
0.000978
0.999
14,000
9.82E+3
0.000102
1.000
16,000
5.61E+4
0.178E-4
1.000
1.2
1
0.8
0.6
0.4
0.2
0
HI
H II
40
0
0
60
0
0
80
0
10 0
00
12 0
0
0
14 0
00
16 0
00
18 0
0
0
20 0
0
00
T (K)
H fractional ionization
Example: H at Pe = 10 dyne/cm2 (~ solar pressure at T = Teff)
Temperature
8
On the partition function
Partition function for neutral H atom
U0 (T ) =
∞
X
gi0 e− Ei0 /k T
i=1
Ei0 ≤ Eion = 13.6 eV
gi0 = 2i2
U0 (T ) ≥ 2 e −Eion /kT
infinite number of levels  partition function diverges!
reason: Hydrogen atom level structure
∞
X
calculated as if it were alone in the universe
i2
not realistic  cut-off needed
i=1
divergent
idea: orbit radius r = a0 i2
(i is main quantum number)
 there must be a max i corresponding
to the finite spatial extent of atom rmax
4π 3
1
4π
rmax =
(r0 i2max)3
=
3
N
3
imax
introduces a pressure
dependence of U
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An example: pure hydrogen atmosphere in LTE
Temperature T and total particle density N given: calculate ne, np, ni
From Saha’s equation and ne = np (only for pure H plasma):
(T)
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The LTE occupation number ni*
g1j+1
n1j+1 ne
=2
n1j
g1j
From Saha’s equation:
n1j
g
1
= n1j +1 ne 1j
g1j +1 2
µ
+ Boltzmann:
gij 1
n∗i := nij = n1j+1 ne
g1j+1 2
h2
2πmkT
¶3/2
e
h2
2πmkT
2πmkT
h2
¶3/2
E
− kTj
e
Ej
kT
nij
gij −E1 i /kT
=
e
n1j
g1j
µ
µ
¶3/2
e
Eij
kT
in LTE we can express the bound level
occupation numbers as a function of T, ne
and the ground-state occupation number
of the next higher ionization stage.
Note ni* is the occupation number used
to calculate bf-stimulated and bfspontaneous emission
11
Stellar classification and temperature:
application of Saha – and Boltzmann formulae
temperature (spectral type) & pressure
(luminosity class) variations
+ chemical abundance changes.
Qualitative plot of strength of observed
Line features as a function of spectral type
12
Stellar classification and temperature:
application of Saha – and Boltzmann formulae
The pioneers of
stellar
spectroscopy…
13
The pioneers of
stellar
spectroscopy…
at work…
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Annie Jump Cannon
15
Annie Jump Cannon
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Stellar classification
Type
O
B
Approximate Surface
Temperature
> 25,000 K
11,000 - 25,000
Main Characteristics
Singly ionized helium lines either in emission or absorption. Strong ultraviolet
continuum. He I 4471/He II 4541 increases with type. H and He lines weaken with
increasing luminosity. H weak, He I, He II, C III, N III, O III, Si IV.
Neutral helium lines in absorption (max at B2). H lines increase with type. Ca II K starts
at B8. H and He lines weaken with increasing luminosity. C II, N II, O II, Si II-III-IV, Mg
II, Fe III.
Hydrogen lines at maximum strength for A0 stars, decreasing thereafter. Neutral metals
stronger. Fe II prominent A0-A5. H and He lines weaken with increasing luminosity. O I,
Si II, Mg II, Ca II, Ti II, Mn I, Fe I-II.
A
7,500 - 11,000
F
6,000 - 7,500
G
5,000 - 6,000
Solar-type spectra. Absorption lines of neutral metallic atoms and ions (e.g. onceionized calcium) grow in strength. CN 4200 increases with luminosity.
K
3,500 - 5,000
Metallic lines dominate, H weak. Weak blue continuum. CN 4200, Sr II 4077 increase
with luminosity. Ca I-II.
M
< 3,500
Molecular bands of titanium oxide TiO noticeable. CN 4200, Sr II 4077 increase with
luminosity. Neutral metals.
Metallic lines become noticeable. G-band starts at F2. H lines decrease. CN 4200
increases with luminosity. Ca II, Cr I-II, Fe I-II, Sr II.
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online Gray atlas at NED
nedwww.ipac.caltech.edu/level5/Gray/frames.html
Stellar classification
18
Stellar classification
ionization (I.P. 25 eV)
The spectral type can be judged easily by the ratio of the
strengths of lines of He I to He II; He I tends to increase in
strength with decreasing temperature while He II decreases
in strength. The ratio He I 4471 to He II 4542 shows this trend
clearly.
The definition of the break between the O-type stars
and the B-type stars is the absence of lines of ionized
helium (He II) in the spectra of B-type stars. The lines
of He I pass through a maximum at approximately B2,
and then decrease in strength towards later (cooler)
types. A useful ratio to judge the spectral type is the
ratio of He I 4471/Mg II 4481.
A DIGITAL SPECTRAL CLASSIFICATION ATLAS
R. O. Gray
http://nedwww.ipac.caltech.edu/level5/Gray/Gray_contents.html
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O-star spectral types
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SiIII
SiIV
B-star spectral types
B0Ia
B0.5Ia
B1Ia
B1.5Ia
SiII
B2Ia
B3Ia
B5Ia
B8Ia
B9Ia
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Stellar classification
(I.P. 6 eV)
H & K strongest
T high enough for
single ionization, but
not further
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Stellar classification
Balmer lines indicate stellar
luminosity
Gravity  atmospheric density
 line broadening
Are the ionization levels for different elements
observed in a given spectral type consistent with a
“single” temperature?
Spectral class
ion and ionization potential
O
He II C III N III O III Si IV
24.6 24.4 29.6 35.1 33.5
B
HII C II N II O II Si III Fe III
13.6 11.3 14.4 13.6 16.3 16.2
A-M
Mg II Ca II Ti II Cr II Si II Fe II
7.5 6.1 6.8 6.8 8.1 7.9
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Cecilia Payne-Gaposchkin
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Stellar classification
Saha’s equation  stellar classification (C. Payne’s thesis, Harvard 1925)
The strengths of selected lines along the spectral sequence.
variations of observed line strengths with spectral type in
the Harvard sequence.
Saha-Boltzmann predictions of the fractional concentration
Nr,s/N of the lower level of the lines indicated in the upper
panel against temperature T (given in units of 1000 K along
the top). The pressure was taken constant at Pe = Ne k T =
131 dyne cm-2. The T-axis is adjusted to the abscissa of the
upper diagram in order to obtain a correspondence between
the observed and computed peaks.
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Cecilia Payne-Gaposchkin
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Stellar continua: opacity sources
27
Stellar continua: the Balmer jump
From the ground we can measure part of Balmer ( < 3646 A) and
Bracket ( > 8207 A) continua, and complete Paschen continuum
(3647 – 8206 A)
Provide information on T, P. Spectrophotometric measurements in
UV (hot stars), visible and IR (cool stars)
ionization changes are reflected in changes in the
continuum flux
for  > (Balmer limit) no n=2 b-f transitions
possible  drop in absorption
 atmosphere is more transparent
 observed flux comes from deeper hotter layers
 higher flux BALMER JUMP
Balmer discontinuity (when H^- absorption is negligible T > 9000 K) =
κbf (> 3650)
κbf (n = 3) + ...
κbf (n = 3)
= bf
' bf
κbf (< 3650)
κ (n = 2) + κbf (n = 3) + ...
κ (n = 2)
28
Stellar continua: the Balmer jump
from
κbf = σbf
n Nn
and Boltzmann’s equation
κbf ∼
σnbf ∼
1
n5ν 3
Nn ∼ Ntot
gn −En /kT
e
= n2 Ntot e −En /k T
Un
1 − En /k T
e
n3
κbf (> 3650)
8 − (E3 −E2) /kT
'
e
κbf (< 3650)
27
e.g. 0.0037 at T = 5,000 K
0.033 at T = 10,000 K
if b-f transitions dominate continuous opacity  the Balmer
discontinuity increases with decreasing T  measure T
from Balmer jump
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Stellar continua: H-
apply Saha’s equation to H- (H- is the ‘atom’ and H0 is the ‘ion’)
N(H 0) ne
= 4 × 2.411 × 1021 T 3/2 e −8753/T
−
N(H )
log10
N(H 0)
5040
=
0.1248
−
log
EI
P
+
2.5
log
T
−
e
10
10
N(H − )
T
under solar conditions: N(H-) / N(H0) ' 4 × 10-8
at the same time: N2 / N(H0) ' 1 × 10-8
N3 / N(H0) ' 6 × 10-10 (Paschen continuum)
N(H-)/ N(H0): > N3 / N(H0): b-f from H- more important than H b-f in the visible
30
Stellar continua
at solar T H- b-f dominates from Balmer limit up to H- threshold (16500 A). H0
b-f dominates in the visible for T > 7,500 K.
Balmer jump smaller than in the case of pure H0
absorption: instead of increasing at low T,
decreases as H- absorption increases
Max of Balmer jump: ∼ 10,000 K (A0 type)
H- opacity ∝ ne  higher in dwarfs than
supergiants
Balmer jump sensitive to both T and Pe in A-F
stars
31