6 Stellar spectra - UH Institute for Astronomy · 2020. 4. 8. · Solar-type spectra. Absorption...

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6. Stellar Spectra

Excitation and ionization, Saha’s equation Stellar spectral types Luminosity effects on stellar spectra Balmer jump, H-

Occupation numbers: LTE caseAbsorption coefficient: κν = niσν

! calculation of occupation numbers needed!

Recap: Basic assumptions for LTE are - 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

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Excitation in LTEBoltzmann 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 (gij = 2i2 for hydrogen)

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

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Eij excitation energy relative to ground state

Uj (T) is called the partition function

Boltzmann factor

Recap from last lecture:Thomson & Rayleigh Scattering

κsc = ne σe

Boltzmann Distribution & Partition Function

�(!) ! �T

✓!

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Recap Quiz

!5

•You are considering a line with a resonant frequency of 731GHz. What will be the frequency dependence for scattering of photons at 100MHz?

•You are using the Boltzmann equation to calculate occupation numbers for two stars with effective temperatures T1 = 2*T2. Which star will have a larger number of neutral iron atoms in the ground state? Which star will have a larger number of singly ionized iron atoms in the ground state?

Ionization: Saha’s formula

Saha 1920

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➔ ionization decreases with increasing ne (due to recombinations) ➔ ionization increases with increasing T

Compare to Boltzmann formula:

Meghnad Saha (1893-1956)

Ionization: Saha’s formulaGeneralize for arbitrary levels (not just ground state):

using Boltzmann’s equation

also

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Ionization: Saha’s formulaUsing electron pressure Pe instead of ne (Pe = nekT)

which can be written as:

with Pe in dyne/cm2 and Ej in eV

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Ionization: Saha’s formula

Hydrogen at Pe = 10 dyne/cm2

Gray 2005

H fr

actio

nal i

oniz

atio

n0

0.25

0.5

0.75

1

Temperature6000 10000 14000 18000

Fe at Pe = 1 dyne/cm2

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On the partition function

Partition function for neutral H atom:

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On the partition function

Partition function for neutral H atom:

Infinite number of levels ! partition function diverges!

Reason: Hydrogen atom level structure calculated as if it were alone in the universe. Not realistic ! cut-off needed!

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On the partition functionSolution: define cut-off based on perturbation by neighbording particles that can unbind electron (pressure ionization)

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consider perturbers with number density np, occupying the volume:

radius for hydrogenic atom with atomic number Z is:

4

3⇡r30 ⇠ 1/np

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setting rn = r0 (i.e. electron bound if rn < r0) and using typical photospheric particle density np = 1016 cm-3

Partition function U is pressure dependent

Bohr radius

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imax

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An example: pure hydrogen atmosphere in LTE

Given Temperature T and total particle density N: use Saha equation to calculate electron density ne

But what is n1? Using Saha’s equation and ne = np (only for pure H plasma):

!13

α(T)

!14

+An example: pure hydrogen atmosphere in LTE

only for pure Hydrogen plasma!

The LTE occupation number ni*

From Saha’s equation:

+ Boltzmann:

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 (Saha-Boltzmann distribution)

!15

ni* is the occupation number used to calculate b-f emission

Recap from last lecture:

Saha Equation

Partition Function

Recap Quiz

!17

• Consider the following two ionization fraction curves below. What can you conclude about the two elements?

What can you conclude about these elements?

• Consider two stars with identical temperatures and surface gravities log(g) = 2.0 and log(g) = 4.5. Which star will have a higher hydrogen ionization fraction and why?

Stellar Spectroscopy: Some History

Harvard College Observatory, circa 1899

!18Digital Access to a Sky Century @ Harvard: http://dasch.rc.fas.harvard.edu/project.php

Stellar Spectroscopy: Some History

The pioneers of stellar spectroscopy

!19

Annie Jump Cannon (1863 - 1941)

!22

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

temperature (spectral type) & pressure (luminosity class) variations + chemical abundance changes. !23

Stellar classificationType Approximate Surface

Temperature Main Characteristics

O > 25,000 K

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.

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 - 6,000

Solar-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.

!24

Stellar classification

!25

Main Sequence O4-O9

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.

A DIGITAL SPECTRAL CLASSIFICATION ATLAS R. O. Gray http://nedwww.ipac.caltech.edu/level5/Gray/Gray_contents.html

ionization (I.P. 25 eV)

!26

Main Sequence O9-B5

Break between the O and 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.

!27A DIGITAL SPECTRAL CLASSIFICATION ATLAS R. O. Gray http://nedwww.ipac.caltech.edu/level5/Gray/Gray_contents.html

Main Sequence B5-A5


Break between the B and A-type stars is the absence of Helium. Ca II K-Line becomes notable in A stars, and metallic lines start to emerge. Very deep hydrogen lines!

!28

Main Sequence A5-G0


H lines begin to weaken, Ca II K-line continues to strengthen and saturated by late F. Metallic lines grow dramatically. G-band (thousands of closely spaced lines due to the diatomic molecule CH) appears around F2

!29

Main Sequence G0-K5


H continues to fade while the metallic-line spectrum continues to increase. The G-band continues to increase in strength until ~K2, and then begins to fade. The Ca I 4227 line grows gradually in strength until early K, and then becomes dramatically stronger by mid-K.

!30


In the K-type dwarfs, the spectral type may be estimated from the ratio of Ca I 4227 to Fe I 4383, in the sense that Ca I/Fe I grows toward later types. By M0, bands due to TiO become visible in the spectrum, and these strengthen quite dramatically toward later types; by M4.5 they dominate the spectrum.

!31

Main Sequence K5-M4.5

Stellar classification: Luminosity effects


Balmer lines indicate stellar luminosity

!32



Primary luminosity discriminant at G0 is the strength of the Sr II 4077 line. Of use as well are the blends of Ti II and Fe II which were used in the F-type stars to distinguish luminosity types. Generally, luminosity/gravity effects are more subtle than for early-type stars

!33



Ca I 4227 line is the most striking luminosity indicator in the M2 stars. The morphology of the MgH/TiO blend near 4770 can be used as well to distinguish luminosity classes; notice that the MgH band dominates this blend in the dwarf star, producing a tooth-shaped feature.

!34

Stellar classification

!35

Boltzmann: fraction in excited level

What causes spectral types? Example of Hydrogen

!37


Saha: fraction ionized


!38


Saha: fraction ionized

Strength of Bound Transition



Alexey Belyanin

Cecilia Payne-Gaposchkin (1900-1979)

!40

Stellar classificationSaha’s equation ! stellar classification (C. Payne’s thesis, Harvard 1925)

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 against temperature T (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.

!41

Russell 1929 !42

Russell 1929 !43

Recap from last lecture:

!44

+

Recap Quiz

!45

•You are predicting the line strength of Hδ and Ca II in an atmosphere with nH/nCa=1e6. The fraction of singly ionized to neutral Ca is 0.99, and most Ca is in the ground state. The fraction of neutral H in the first excited state to the ground state is 1e-9. Which line will be stronger and why?

•You are measuring the equivalent width of the Hδ line for two stars with identical effective temperature and determine that EW1>EW2. What can you conclude about the physical properties of the star?

!46

?

?

?

?

?

!47

K0V

F0V

K5V

B5III

B5V

He I

He IHe I

He I

Ca II H&K

Ca II

H HH

HHH

!48

?

?

?

!49

White Dwarf

A0VspSiSrCr

B2Ve

Stellar continua: opacity sources

!50

Stellar continua: the Balmer jump

ionization changes are reflected in changes in the continuum flux for λ > λ(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

From the ground we can measure part of Balmer (λ < 3646 A) and Bracket (λ > 8207 A) continua, and complete Paschen continuum (3647 – 8206 A)

!51

Stellar continua: the Balmer jump

from

and Boltzmann’s equation

if H b-f transitions dominate continuous opacity ! 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

!52

Stellar continua: H-

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

under solar conditions: N(H-) / N(H0) ~ 4 x 10-8

at the same time: N2 / N(H0) ~ 1 x 10-8 N3 / N(H0) ~ 6 x 10-10

N(H-)/ N(H0): > N3 / N(H0): b-f from H- more important than H b-f in the visible

!53

Stellar continua: H-

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 (gravity) in A-F stars

!54

Balmer Jump: Teff and Surface Gravity

Grey 2005!55

S=Teff/5776

Hot Stars Cool Stars

6 Stellar spectra - UH Institute for Astronomy · 2020. 4. 8. · Solar-type spectra. Absorption...

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