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SPECTRAL LINE ANALYSIS: LOG GGiovanni Catanzaro INAF - Osservatorio Astrofisico di Catania

9 april 2013

Spring School of Spectroscopic Data Analyses8-12 April 2013

Astronomical Institute of the University of WroclawWroclaw, Poland

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SPECTRAL LINES

An absorption line is produced in a stellar spectrum whenever photons of energy E=h=hc/=EU-EL are absorbed by an atom or ion that jumps from a lower to an upper energy level.

Lyman

Balmerga

b

a

b

n = R [ 1/nl2 – 1 /nu

2]

R = 3.288 x 1015 Hz

Hydrogen Energy Levels

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9 april 2013SPECTRAL LINES

I

2

Hydrogen Energy Levels

n excitation potential of the level n

2 = 10.2 eV

Ionization Energy for Hydrogen (n to infinity):

I = 13.6 eV ( 912 Ǻ) ionization energy

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9 april 2013SPECTRAL LINES

The intensity of a spectral line is related to the number of absorbers, i.e. atoms or ions of the given elements at the lower level of the transition.

In LTE (Local Thermodynamic Equilibrium) the fraction of ions that can absorb is given by the Saha and Boltzmann equations.

Ionization - Saha

U(T)

eg

N

N kT

χ

n

1

n1

n

Excitation - Boltzmann

Element abundance

Electron pressure gravity

𝑁1

𝑁0

𝑃𝑒=𝑐𝑜𝑛𝑠𝑡𝑇52𝑒

− 1𝐾𝑇

temperature

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9 april 2013SPECTRAL LINES

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0 )-( )(l

Dispersion profile (Lorentzian)

There are several mechanisms that broaden a spectral line, which is never a function (infinitely-narrow feature).

1) Natural atomic absorption

2) Pressure broadening

3) Thermal Doppler broadening

4) Microturbulence

5) Rotation

Etc.

Gaussian profile

200 )exp(

1 )(

DD

gg

m

kT

cD

2

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Different types of pressure broadening

n Type Lines affected

Perturber

2 Linear Stark Hydrogen Protons,electrons

4 Quadratic Stark

Lines in hot stars

Ions, electrons

6 Van der Waals

Lines in cool stars

Neutral hydrogen

Pressure broadening implies a collisional interaction between the atoms absorbing light and other particles: ions, electrons, atoms or molecules (in cool stars).

∆𝐸∝𝑅−𝑛Change in energy of the levels induced by the collisions

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9 april 2013SPECTRAL LINES

Thermal Doppler profiles Dispersion (Lorentzian) profiles

“Weak” lines

Both broadening mechanisms at work the line is shaped by the convolution G()*L(): “Voigt function”

G() L()

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9 april 2013SPECTRAL LINES

Thermal Doppler profiles Dispersion (Lorentzian) profiles

“Strong” lines

When the line intensity increases (e.g. abundance) the optical depth at the line center becomes high the line core reaches a minimum level and the wings get broad. These lines are said “saturated”. Strong lines of abundant elements and ions (HI, NaI, MgII, CaI, CaII, etc.)

G()L()

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9 april 2013SPECTRAL LINES

xlk ρd)( d Optical depth. k , l continuum and line absorption coefficients

SI

I

d

d Line transfer equation

lk

jjS

lc

Source function

j emission coefficients

x

xlkx0

ρd)( )(

tA

EI

cos

normal

To observer

A

Star surface

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9 april 2013SPECTRAL LINES

The solution of transfer equation gives the emerging flux (at the top of the atmosphere =0) as:

0 2 d)()(2 (0) ESF

Numerical solution!

E2 exponential integral of order 2

5.33/)24( 1 For

To a first, rough approximation:

132 ))((

3

4 (0)

SF

)( (0) 1 SF

In LTE the source function at is the Planck function evaluated at T() S is decreasing outwards

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9 april 2013SPECTRAL LINES

The flux at the line center (where l is maximum) comes from the upper atmospheric layers, where the source function is lower.

The larger l the smaller x1 must be to obtain an optical depth 1

1

011 ρd)( )(5.3x

xlkx

x to the star center

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Spring S

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nalysesMEASURING THE GRAVITY IN STARS

9 april 2013

In principle we can measure gravity in a direct way:• Obtain Mass via spectroscopy and the Doppler effect (binary stars for example)• Measure the radius by an independent means (interferometry, lunar occultations, eclipsing binaries)

𝑔=𝐺𝑀𝑅2

In principle this can only be done for very few stars

must rely on spectroscopic determinations

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9 april 2013Increasing of gravity translates into a compression of the photosphere and therefore in an increase of the pressure

𝑃𝑔∝𝑔𝑗 𝑗 ≈

23

𝑃𝑒∝𝑔𝑗 𝑗≈

13

,23

Therefore pressure dependences can be translated in gravity dependences

• Ionization equilibrium• P sensitive to damping constant for strong lines• P dependence of the linear stark broadening in

H

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9 april 2013The strenght of a spectral line is related to the ratio of the line and continuous absorption coefficients:

𝑙𝜈𝜅𝜈

Rewrite the Saha equation in a more schematic form

𝑁𝑟+1

𝑁 𝑟

=Φ (𝑇 )𝑃𝑒

Include all terms not dependent on pressure

Number of atoms/ions in r+1 ionization stages

Number of atoms/ions in the r ionization stages

Remember also that: 𝑙𝜈∝𝑁𝑟

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9 april 2013

Most part of an element in the next higher ionization stage

𝑁𝑟 +1≈𝑁 𝑡𝑜𝑡=𝑐𝑜𝑛𝑠𝑡 𝑁𝑟 ∝𝑐𝑜𝑛𝑠𝑡 𝑃𝑒

Saha

𝑙𝜈∝𝑁𝑟∝𝑐𝑜𝑛𝑠𝑡 𝑃𝑒

In cool stars H- dominates the opacity of the continuos 𝜅𝜈∝𝑐𝑜𝑛𝑠𝑡 𝑃 𝑒

𝑙𝜈𝜅𝜈

=𝑐𝑜𝑛𝑠𝑡These lines are insensitive to gravity, but are useful to set the abundance of the element

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Spring S

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Most part of an element in the same ionization stage

𝑁𝑟 ≈𝑁𝑡𝑜𝑡=𝑐𝑜𝑛𝑠𝑡 𝑁𝑟 ∝𝑐𝑜𝑛𝑠𝑡Saha

𝑙𝜈∝𝑁𝑟∝𝑐𝑜𝑛𝑠𝑡

Again H- dominates the opacity of the continuos 𝜅𝜈∝𝑐𝑜𝑛𝑠𝑡 𝑃 𝑒

𝑙𝜈𝜅𝜈

=𝑐𝑜𝑛𝑠𝑡𝑃 𝑒

≈𝑐𝑜𝑛𝑠𝑡 𝑔− 1

3These lines are sensitive to gravity

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9 april 2013Example: in solar like stars iron is mostly ionized

Fe I lines are insensitive to gravity FeII lines are sensitive to gravity

Fuhrmann et al. (1997) A&A, 323, 909

log g = 3.58

ProcyonTeff = 6500 K

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9 april 2013BROAD WINGS OF NEUTRAL LINES IN COOL STARS

𝑙𝜈∝𝑁𝛾𝛾 6=Φ6 (𝑇 )𝑃𝑔≈𝑐𝑜𝑛𝑠𝑡 𝑔

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𝛾 4=Φ4 (𝑇 ) 𝑃𝑒≈ 𝑐𝑜𝑛𝑠𝑡𝑔13

𝛾𝑛𝑎𝑡

Van der Waals

Quadratic Stark

H- dominates the opacity of the continuos 𝜅𝜈∝𝑐𝑜𝑛𝑠𝑡 𝑃 𝑒

Most part of element is ionized 𝑁𝑟 ∝𝑐𝑜𝑛𝑠𝑡 𝑃𝑒

𝑙𝜈𝜅𝜈

≈𝛾≈𝑐𝑜𝑛𝑠𝑡𝑔23+𝑐𝑜𝑛𝑠𝑡𝑔

13+𝑐𝑜𝑛𝑠𝑡

Depending of the relative size of damping constants we will have different regimes: from no gravity dependence (gnat dominant) to maximum dependence (van der Waals dominant)

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HD100623 K0VHD99322 K0IIIPOP-UVES Database

61 Cyg - K5VTeff=4500 Klogg = 4.57

15 Aql – K1IIITeff=4520 Klogg = 2.65

Courtesy of A. Frasca (Priv. Comm.)

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Teff=7500 K

Catanzaro et al (2013), MNRAS, in press

HD 71297Teff=7500 ± 180 K log g = 4.00 ± 0.10

Fossati et al (2011) MNRAS, 417, 495

Teff=7150 K, log g = 4.20Teff=7380 K, log g = 4.08

Teff=7250 K, log g = 4.20Teff=7670 K, log g = 4.44

log g = 2.0log g = 3.0log g = 4.0log g = 5.0

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9 april 2013BROAD WINGS OF IONIC LINES IN COOL STARS

𝑙𝜈𝜅𝜈

≈𝑐𝑜𝑛𝑠𝑡 𝛾𝑃𝑒

≈𝑐𝑜𝑛𝑠𝑡𝑔13+𝑐𝑜𝑛𝑠𝑡𝑔

− 13+𝑐𝑜𝑛𝑠𝑡

Again, depending we have different regimes: from no gravity dependence (gnat dominant) to maximum dependence (van der Waals dominant)

HD152311Teff=5597 KLog g = 3.97[Fe/H]=0.10Vsin i = 4 km/sRamirez et al., 2007, 465, 271

Log g = 3, 4, 5

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BROAD WINGS OF BALMER LINES IN HOT STARS

Struve (1929): great wings of Balmer lines in early-type stars are due to linear Stark Effect

R Distribution of ions gives a non-zero E at the position of the H

The resulting splitting of atomic levels can be expressed as the Dl of the spectral components:

Δ 𝜆=𝑐𝑜𝑛𝑠𝑡 𝐸=𝑐𝑜𝑛𝑠𝑡𝑒𝑅2

Greater compression of the photosphere results in a greater E,

so ln is proportional to E and than to

Pe

In this stars H absorption dominates

kn than it is proportional to NH

𝑙𝜈𝜅𝜈

∝𝑃𝑒

H lines increase in strength toward lower luminosity classes

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9 april 2013In this case more is the gravity narrow is the line

𝑙𝜈𝜅𝜈

∝𝑃𝑒2

Teff=7000 K

Teff=10000 K

Teff=25000 K

log g = 2.0log g = 3.0log g = 4.0log g = 5.0

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Catanzaro et al. (2004), A&A, 425, 641

Leone & Manfre’ (1997), A&A, 320, 257

Influence of chemical composition

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THE GRAVITY-TEMPERATURE DIAGRAM

9 april 2013

Each curve is computed for a costant iron abundance (fixed using lines not sensitive to g), while varyng the surface gravity for a given temperature (or vice versa) to recover the observed EQWs

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Lehner et al., 2000, MNRAS, 314, 199

Lyumbikov et al., 2002, MNRAS, 333, 9

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Spring S

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9 april 2013

log g = 3.00log g = 4.00log g = 5.00

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9 april 2013Example: g Boo Teff=7600 K, log g = 3.7 (Ventura et al., 2007, MNRAS, 381, 164)

log g = 3.00log g = 4.40

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log g

EMPIRICAL INDICATORS: THE WILSON-BAPPU EFFECT

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Star Teff (K) log g

Arcturus 4158 ± 127

1.89 ± 0.16

x Boo A 5230 ± 115

4.58 ± 0.05

Courtesy of Antonio Frasca

CaII K line

Example: Arcturus (K1.5III) vs x Boo A (G8V)

Allende-Prieto, 2004, A&A, 420, 183

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THANKS FOR YOUR ATTENTION

9 april 2013