1

6. Stellar spectra

excitation and ionization, Saha’s equation

stellar spectral classification

Balmer jump, H-

2

Occupation numbers: LTE case

Absorption coefficient:

= ni

calculation of occupation numbers needed

LTEeach 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

3

Excitation in LTE

Boltzmann excitation equationnij : 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

= (2S+1) (2L+1) in L-S coupling

The fraction relative to the total number of atoms of in ionization stage j is

nij

nj=

gij

Uj (T)e−Eij/kT Uj (T ) =

Xi

gije−Eij/kT

lognijn1j

= loggijg1j

− Eij(eV)5040T

Uj (T) is called the partition function

4

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

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 = 2dxdy dz dpx dpy dpz

h3

2 spin orientations

n1j+1(v) dxdy dz dpxdpy dpzR R

n1j+1(v )dV d3p = n1j+1

Rdx dy dz =

RdV = ∆V = 1/ne

dpx dpy dpz = 4πp2 dp = 4πm3v2 dv

5

Ionization: Saha’s formula

Sum over all final states: integrate over all phase space

n1j+1 = n1jg1j+1g1j

2∆V

h3e−Ej/kT

∞Z−∞

∞Z−∞

∞Z−∞

e−1

2mkT(p2x+p

2y+p

2z) dpxdpydpz

∞Z−∞

e−x2

dx =√π

(2πmkT )3/2

n1j+1 nen1j

= 2g1j+1g1j

µ2πmkT

h2

¶3/2e−

EjkTSaha 1920

- ionization falls with ne (recombinations)

- ionization grows with T

using Boltzmann formulan1j+1(v)

n1jdV dpxdpydpz =

g1j+1g1j

gEl e−(Ej+1

2mv2)/kT

n1j+1(v)

n1jdV dpxdpydpz =

g1j+1g1j

2dVdpxdpy dpz

h3e− [Ej+

12m(p

2x+p

2y+p

2z)]/kT

6

Ionization: Saha’s formula

Generalize for arbitrary levels (not just ground state):

nj =∞Xi=1

nij nj+1 =∞Xi=1

nij+1

nij+1 = n1j+1gij+1

g1j+1e−Eij+1/kT using Boltzmann’s equation

nj+1 =n1j+1g1j+1

∞Xi=1

gij+1 e−Eij+1/kT

Uj+1(T) : partition function

also nj =n1j

g1jUj(T) nj+1 ne

nj= 2

Uj+1(T)

Uj (T)

µ2πmkT

h2

¶3/2e−

EjkT

7

Ionization: Saha’s formula

Using electron pressure Pe instead of ne (Pe = ne kT)

nj+1nj

Pe = 2Uj+1(T)

Uj(T)

µ2πm

h2

¶3/2(kT)5/2e−

EjkT

which can be written as:

log10nj+1nj

= −0.1761 − log10 Pe+ log10Uj+1(T )

Uj (T )+ 2.5log10 T −

5040

TEj

with Pe in dyne/cm2 and Ej in eV

8

Ionization: Saha’s formula

Example: H at Pe = 10 dyne/cm2 (~ solar pressure at T = Teff )

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

00.20.40.60.8

11.2

40006000800010

00012

00014

00016

00018

00020000

Temperature

H f

ract

iona

l ioniza

tion

H I H II

9

On the partition function

Partition function for neutral H atom

Ei0 ≤

Eion = 13.6 eV

gi0 = 2i2

U0(T) =∞Xi=1

gi0 e−Ei0/kT

divergent

U0(T )≥ 2 e−Eion /kT∞Xi=1

i2

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 rmax4π

3r3max =

1

N=4π

3(r0i

2max)

3 imaxintroduces a pressure dependence of U

infinite number of levels partition function diverges!reason: Hydrogen atom level structurecalculated as if it were alone in the universenot realistic cut-off needed

10

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)

11

The LTE occupation number ni*

From Saha’s equation:

n1j = n1j+1 neg1jg1j+1

1

2

µh2

2πmkT

¶3/2eEjkT

+ Boltzmann:nij

n1j=gij

g1je−E1i/kT

in LTE we can express the bound level occupation numbers as a function of T, neand the ground-state occupation number of the next higher ionization stage.

Note ni* is the occupation number used

to calculate bf-stimulated and bf- spontaneous emission

n∗i := nij = n1j+1 negijg1j+1

1

2

µh2

2πmkT

¶3/2eEijkT

n1j+1 nen1j

= 2g1j+1g1j

µ2πmkT

h2

¶3/2e−

EjkT

12

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 observedLine features as a function of spectral type

13

Stellar classification and temperature: application of Saha – and Boltzmann formulae

The pioneers of stellar spectroscopy…

14

The pioneers of stellar spectroscopy…

at work…

15

Annie Jump Cannon

16

Annie Jump Cannon

17

Stellar classification

Type Approximate Surface Temperature Main Characteristics

O > 25,000 KSingly 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.

B 11,000 - 25,000

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 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.

G 5,000 - 6,000Solar-type spectra. Absorption lines of neutral metallic atoms and ions (e.g. once- ionized calcium) grow in strength. CN 4200 increases with luminosity.

K 3,500 - 5,000Metallic lines dominate, H weak. Weak blue continuum. CN 4200, Sr II 4077 increase with luminosity. Ca I-II.

M < 3,500Molecular bands of titanium oxide TiO noticeable. CN 4200, Sr II 4077 increase with luminosity. Neutral metals.

18

Stellar classification

online Gray atlas at NEDnedwww.ipac.caltech.edu/level5/Gray/frames.html

19

Stellar classification

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

ionization (I.P. 25 eV)

20

O-star spectral types

21

B-star spectral typesSiIV SiIII

SiII

B0Ia

B0.5Ia

B1Ia

B1.5Ia

B2Ia

B3Ia

B5Ia

B8Ia

B9Ia

22

Stellar classification

(I.P. 6 eV)

H & K strongest

T high enough for single ionization, but not further

23

Stellar classification

Are the ionization levels for different elements observed in a given spectral type consistent with a “single” temperature?

Spectral class ion and ionization potential

OHe II C III N III O III Si IV24.6 24.4 29.6 35.1 33.5

BHII C II N II O II Si III Fe III13.6 11.3 14.4 13.6 16.3 16.2

A-MMg II Ca II Ti II Cr II Si II Fe II7.5 6.1 6.8 6.8 8.1 7.9

Balmer lines indicate stellar luminosity

Gravity atmospheric density

line broadening

24

Cecilia Payne-Gaposchkin

25

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.

26

Cecilia Payne-Gaposchkin

27

Stellar continua: opacity sources

28

Stellar continua: the Balmer jump

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

κbf (> 3650)

κbf (< 3650)=

κbf (n = 3) + ...

κbf (n = 2) + κbf (n = 3) + ...' κbf (n = 3)

κbf (n = 2)

Balmer discontinuity (when H^- absorption is negligible T > 9000 K) =

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)

29

Stellar continua: the Balmer jump

κbf (> 3650)

κbf (< 3650)' 8

27e−(E3−E2)/kT

from κbf = σbfn Nn σbfn ∼ 1

n5ν3

and Boltzmann’s equation Nn ∼Ntot gnUn

e−En/kT = n2 Ntot e−En/kT

κbf ∼ 1

n3e−En/kT

if b-f transitions dominate continuous opacity the Balmer discontinuity increases with decreasing T measure T from Balmer jump

e.g. 0.0037 at T = 5,000 K

0.033 at T = 10,000 K

30

Stellar continua: H-

apply Saha’s equation to H- (H- is the ‘atom’ and H0 is the ‘ion’)

N(H0) neN(H−) = 4× 2.411× 1021 T3/2 e−8753/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

log10N(H0)

N(H−) = 0.1248− log10 Pe+ 2.5 log10 T −5040

TEI

31

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 increasesMax 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