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:
= n i
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
n ij : number density of atoms in excited level i of ionization stage j
(ground level: i=1 neutral: j=0)
n ij
n 1j
= g ij
g 1j
e −E ij/kT
g ij : statistical weight of level i = number of degenerate states
E ij excitation energy relative to ground state
g ij = 2i 2 for hydrogen
= (2S+1) (2L+1) in L-S coupling
log n ij
=log g ij
− E ij (eV) 5040
n 1j g 1j T
The fraction relative to the total number of atoms of in ionization stage j is
n ij
n j
= g ij
U j (T) e−E ij/kT
U j (T )= X i
g ij e −E ij/kT
U j
(T) is called the
partition function
3

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:
final state:
n 1j & statistical weight g 1j
n 1j+1 + free electron & statistical weight g 1j+1 g El
n 1j+1 (v) dx dy dz dp x dp y dp z
R R
n1j+1 (v)dV d 3 p = n 1j+1
number of ions in groundstate with free electron with velocity in (v,v+dv) in phase space
g El : volume in phase space normalized to smallest possible volume (h 3 ) for electron:
g El =2 dx dy dz dp x dp y dp z
h 3
2 spin orientations
R
dx dy dz =
R
dV = ∆V =1/ne
dp x dp y dp z =4πp 2 dp =4πm 3 v 2 dv
4

Ionization: Saha’s formula
n 1j+1 (v)
dV dp x dp y dp z = g 1j+1
g El e −(E j + 1 2 mv 2 )/kT
n 1j g 1j
using Boltzmann formula
n 1j+1 (v)
dV dp x dp y dp z = g 1j+1
2dV dp x dp y dp z
e −[E j+ 1
n 1j g 1j h 3 2m (p 2 x +p2 y +p2 z )]/kT
Sum over all final states: integrate over all phase space
n 1j+1 = n 1j
g 1j+1
g 1j
2 ∆V
h 3
e −E j/kT
Z ∞
Z ∞
Z ∞
−∞ −∞ −∞
e − 1
2mkT (p2 x +p2 y +p2 z ) dp x dp y dp z
Z∞
−∞
e −x2 dx = √ π
(2πmkT ) 3/ 2
Saha 1920
n 1j+1 n e
n 1j
=2 g 1j+1
g 1j
µ 2π mkT
h 2
3/ 2
e − E j
kT
- ionization falls with n e
(recombinations)
- ionization grows with T
5

Ionization: Saha’s formula
Generalize for arbitrary levels (not just ground state):
n j =
∞X
n ij n j+1 =
i=1
∞X
i=1
n ij+1
n ij+1 = n 1j+1
g ij+1
g 1j+1
e −E ij+1/kT
using Boltzmann’s equation
n j+1 = n 1j+1
g 1j+1
∞ X
i=1
g ij+1 e −E ij+1/kT
U j+1 (T) :
partition function
also
n j = n 1j
g 1j
U j (T) n j+1 n e
n j
=2 U j+1(T)
U j (T)
µ 2πmkT
h 2
3/2
e − E j
kT
6

Ionization: Saha’s formula
Using electron pressure P e instead of n e (P e = n e kT)
n j+1
n j
P e =2 U j+1(T)
U j (T)
µ 2πm
h 2
3/ 2
(kT) 5/2 e − E j
kT
which can be written as:
n
log j+1
U
10 = −0.1761 −log 10 P e +log j+1 (T )
10
n j U j (T )
+2.5log 10 T − 5040
T
E j
with P e in dyne/cm 2 and E j in eV
7

Ionization: Saha’s formula
Example: H at P e = 10 dyne/cm 2 (~ solar pressure at T = T eff )
T (K) 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
H fractional ionization
1.2
1
0.8
0.6
0.4
0.2
0
4000
6000
H I
8000
10000
H II
Temperature
12000
14000
16000
18000
20000
8

On the partition function
Partition function for neutral H atom
E i0 ≤ E ion = 13.6 eV
U 0 (T)=
∞X
g i0 e −E i0/kT
i=1
g i0 = 2i 2
U 0 (T ) ≥ 2 e −E ion /kT
infinite number of levels partition function diverges!
reason: Hydrogen atom level structure
∞X
i 2 calculated as if it were alone in the universe
not realistic cut-off needed
i=1
divergent
idea: orbit radius r = a 0 i 2 (i is main quantum number)
there must be a max i corresponding
to the finite spatial extent of atom r max
4π
3 r3 max = 1 N = 4π 3 (r 0i 2 max) 3 i max
introduces a pressure
dependence of U
9

An example: pure hydrogen atmosphere in LTE
Temperature T and total particle density N given: calculate n e , n p , n i
N = n e + n p +
iX
max
i=1
iX
max
n i = n e + n p + n 1
i=1
g i
g 1
e −E i,1/kT
From Saha’s equation and n e = n p (only for pure H plasma):
n p n e
n 1
=2 g µ 3/2
p 2πmkT
e − E ion
g 1 h 2
kT
g p =1 g 1 =2
n 1 = n 2 e
µ
h
2
2πmkT
3/2
e E ion
kT
N =2n e + n 2 e
µ h
2
2πmkT
3/2
e E ion
kT
iX
max
i=1
i 2 e −E i,1/kT =2n e + n 2 e α(T)
(T)
n e = − 1
α(T) + r
1
α 2 + N =⇒ n p = n e =⇒ n 1 → n i
10

The LTE occupation number n i
*
From Saha’s equation:
n 1j+1 n e
n 1j
=2 g 1j+1
g 1j
µ 2πmkT
h 2
3/2
e − E j
kT
n 1j = n 1j+1 n e
g 1j 1
g 1j+1 2
µ
h
2
2πmkT
3/2
e E j
kT
+ Boltzmann:
n ∗ i := n ij = n 1j+1 n e
g ij 1
g 1j+1 2
n ij
n 1j
= g ij
g 1j
µ h
2
2πmkT
e −E 1i/kT
3/2
e E ij
kT
in LTE we can express the bound level
occupation numbers as a function of T, n e
and the ground-state occupation number
of the next higher ionization stage.
Note n i
*
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

Stellar classification
Type
O
Approximate Surface
Temperature
> 25,000 K
B 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.
A 7,500 - 11,000
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.
F 6,000 - 7,500
G 5,000 - 6,000
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.
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.
14

online Gray atlas at NED
nedwww.ipac.caltech.edu/level5/Gray/frames.html
Stellar classification
15

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
16

O-star spectral types
17

B0Ia
SiIV
SiIII
B-star spectral types
B0.5Ia
B1Ia
B1.5Ia
SiII
B2Ia
B3Ia
B5Ia
B8Ia
B9Ia
18

Stellar classification
(I.P. 6 eV)
H & K strongest
T high enough for
single ionization, but
not further
19

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
O
B
A-M
ion and ionization potential
He II C III N III O III Si IV
24.6 24.4 29.6 35.1 33.5
HII C II N II O II Si III Fe III
13.6 11.3 14.4 13.6 16.3 16.2
Mg II Ca II Ti II Cr II Si II Fe II
7.5 6.1 6.8 6.8 8.1 7.9
20

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
N r,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 P e = N e 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.
21

Stellar continua: opacity sources
22

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) =
κ bf (> 3650)
κ bf (< 3650) = κ bf (n =3)+...
κ bf (n =2)+κ bf (n =3)+... ' κbf (n =3)
κ bf (n =2)
23

Stellar continua: the Balmer jump
from
κ bf = σ bf
n N n σ bf
n ∼ 1
n 5 ν 3
and Boltzmann’s equation N n ∼ N tot
g n
U n
e −E n/kT = n 2 N tot e −E n/kT
κ bf ∼ 1
n 3 e−E n/kT
κ bf (> 3650)
κ bf (< 3650) ' 8 27 e−(E 3−E 2 ) /kT
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
24

Stellar continua: H -
apply Saha’s equation to H - (H - is the ‘atom’ and H 0 is the ‘ion’)
N(H 0 ) n e
N(H − )
=4× 2.411 × 10 21 T 3/2 e −8753/T
N(H 0 )
log 10
N(H − ) =0.1248 − log 10 P e +2.5log 10 T − 5040
T
under solar conditions: N(H - ) / N(H 0 ) ' 4 × 10 -8
E I
at the same time: N 2 / N(H 0 ) ' 1 × 10 -8
N 3 / N(H 0 ) ' 6 × 10 -10 (Paschen continuum)
N(H - )/ N(H 0 ): > N 3 / N(H 0 ): b-f from H - more important than H b-f in the visible
25

Stellar continua
at solar T H - b-f dominates from Balmer limit up to H - threshold (16500 A). H 0
b-f dominates in the visible for T > 7,500 K.
Balmer jump smaller than in the case of pure H 0
absorption: instead of increasing at low T,
decreases as H - absorption increases
Max of Balmer jump: ∼ 10,000 K (A0 type)
H - opacity ∝ n e higher in dwarfs than
supergiants
Balmer jump sensitive to both T and P e in A-F
stars
26