The Formation of Spectral Lines - tls-tautenburg.de · The Formation of Spectral Lines I. Line Absorption Coefficient II. Line Transfer Equation

The Formation of Spectral Lines

I.  Line Absorption Coefficient

II.  Line Transfer Equation

Line Absorption Coefficient

Main processes

1.  Natural Atomic Absorption

2.  Pressure Broadening

3.  Thermal Doppler Broadening


The classical model of the interaction of light with a photon is a plane electromagnetic wave interacting with a dipole.

∂2 E ∂ t2 = v2 ∂2 E

∂ x2

Treat only one frequency since by Fourier composition the total field is a sum of all sine waves.

E = E0 e–iω(x/v – t)

The wave velocity through a medium

v=c ( ( ε0 µ0 ε µ

½ ε and µ are the electric and magnetic permemability in the medium and free space. For gases µ = µ0


The total electric field is the sum of the electric field E and the field of the separated charges induced by the electric field which is 4πNqz where z is the separation of the charges and N the number of dipoles per unit volume

The ratio of ε/ε0 is just the ratio of the field in the medium to the field in free space

ε ε0

= E + 4πNqz E

4πNqz E

1 + =

We need z/E

Line Absorption Coefficient For a damped harmonic oscillator where z is the induced

separation between the dipole charges

d2 z d t2 + γ

dz dt + ω0

2 = e m E0 eiωt

e,m are charge and mass of electron

γ is damping constant

Solution: z = z0e–iωt

z = e m

E0 eiωt

ω0 – ω2 + iγω 2 = e m

E

ω0 – ω2 + iγω 2


= ε

ε0 1 +

ω0 – ω2 + iγω 2

1 4πNe2

E

For a gas ε ≈ ε0 so second term is small compared to unity

The wave velocity can now be written as

c v ≈ ( ( ε

ε0

½ ≈ 1 + 1

2

4πNe2

m ω0 – ω2 + iγω 2 1

Where we have performed a Taylor expansion (1 + x) = 1 + ½ x for small x

½


c v ≈ 1 +

2πNe2

m (ω0 – ω2)2 + γ2ω2 2 ω0 – ω2 2

(ω0 – ω2)2 + γ2ω2 2

γω

– i

This can be written as a complex refractive index c/v = n – ik. When it is combined with iωx/v it produces an exponential extinction e–kωx/c . Recall that the intensity is EE* where E* is the complex conjugate. The light extinction can be expressed as:

I = I0 e–kωx/c = I0e–lνρx


lνρ = 4πNe2

mc (ω0 – ω2)2 + γ2ω2 2

γω

This function is sharply peaked giving non-zero values when ω ≈ ω0

ω0 – ω =(ω0 – ω)(ω0 + ω) ≈ (ω0 – ω)2ω ≈ 2ωΔω

The basic form of the line absorption coefficient:

lνρ = Nπe2

mc Δω2 + (γ/2)2

γω This is a damping profile or Lorentzian profile, a Cauchy curve, or the „Witch of Agnesi“

2 2


α = 2πe

mc Δω2 + (γ/2)2

γω

Consider the absorption coefficient per atom, α, where lνρ = Nα

α = 2πe

mc Δν2 + (γ/4π)2

γ/4π

α = 2πe

mc Δλ2 + (γλ2/4πc)2

γλ2/4πc λ2

c


A quantum mechanical treatment

∫ = πe2

mc α dν 0

∞

∫ = α dν 0

∞ πe2

mc f

f is the oscillator strength and is related to the transition probability Blu

∫ = α dν 0

∞ Blu hν

This is energy per unit atom per square radian that the line absorbs from Iν


There is also an f value for emission gu fem = gl fabs

= πe2

mc f = Blu hν 7.484 × 10–7 Blu λ

2πe2ν2

mc3 f = Aul gu

gl

Most f values are determined from laboratory measurements and most tables list gf values. Often the gf values are not well known. Changing the gf value changes the line strength, which is like changing the abundance. Standard procedure is you take a gf value for a line, fit it to the solar spectrum, and change gf until you match the solar line. This value is then good for other stars.

The Damping Constant for Natural Broadening

dW d t = –γ 2

3 = – e2 ω2

mc3 W W

Classical dipole emission theory gives an equation of the form

Solution of the form

= 0.22/λ2 in cm γ = 2e2 ω2

3mc3

W= W0e–γt

The quantum mechanical radiation damping is an order of magnitude larger which is consistent with observations. However, the observed widths of spectral lines are dominated by other broadening mechanisms

Pressure Broadening

Pressure broadening involves an interaction between the atoms absorbing the light and other particles (electrons, ions, atoms). The atomic levels of the transition of interest are perturbed and the energy altered.

•  Distortion is a function of separation R, between absorber and perturber

•  Upper level is more strongly altered than the lower level

hν

1

2

3

l

u 1: unperturbed energy

2. Perturbed energy less than unperturbed

3. Energy greater than unperturbed

R

E

Pressure Broadening

Energy change as a function of R:

ΔW = Const/Rn

n Type Lines affected Perturber 2 Linear Stark Hydrogen Protons, electrons

4 Quadratic Stark Most lines, especially in hot stars Ions, electrons

6 Van der Waals Most lines, especially in cool stars Neutral hydrogen

Δν = Cn /Rn

Pressure Broadening: The Impact Approximation

Photon of duration Δt is an infinite sine wave times a box

Spectrum is just the Fourier transform of box times sine which is sinc πΔt(ν-ν0) and indensity is sinc2πΔt(ν-ν0). Characteristic width is Δν= 1/Δt

Δtj

First formulated by Lorentz in 1906 who assumed that the electromagnetic wave was terminated by the impact and with the remaining photon energy converted to kinetic energy

Duration of encounter typically 10–9 secs

Pressure Broadening: The Impact Approximation

With collisions, the original box is cut into many shorter boxes of length Δtj < Δt

Because Δtj < Δt the line is broadened with Δνj = 1/Δtj. The Fourier transform of the sum is the sum of the transforms.

The distribution, P, of Δtj is:.

dP(Δtj) = e–Δtj/Δt0 dΔtj/Δt0 t0 is an average length of an uninterrupted time segment

The line absorption coefficient is the weighted average:

Δt2 sinπΔt(ν – ν0) πΔt(ν – ν0)

2

e–Δt/Δt0 dΔt Δt0 ∫

0

∞

α = C 4π2(ν – ν0)2 + (1/Δt0)2

α = C (ν – ν0)2 + (γn/4π)2 γn/4π

In other words this is the Lorentzian. To use this in a line profile calculation need to evaluate γn = 2/Δt0. This is a function of depth in the stellar atmosphere.

Evaluation of γn Simplest approach is to assume that all encounters are in one of two groups depending on the strength of the encounter. If phase shift is too small ignore it. The cumulative effect of the change in frequency is the phase shift.

φ = 2π ∫ 0

∞

Δν dt = 2π ∫ 0

∞

Cn R–n dt

Assume perturber moves past atom in a straight line

y

x

v

R

θ

ρ ρ = R cos θ

= 2π ∫ 0

∞

Cn cos θ dt φ ρn

Atom

Perturber

Δν = Cn /Rn

n

Evaluation of γn

v = dy/dt = (ρ/cos2θ) dθ/dt => dt = (ρ/v)dθ/cos2θ

φ = cosn–2 θ dθ ∫ –π/2

2π Cn

vρn–1

π/2

cosn–2 θ dθ ∫ –π/2

π/2 n

2 π

3 2

4 π/2

6 3π/8

Usually define a limiting impact parameter for a significant phase shift φ = 1 rad

cosn–2 θ dθ ∫ –π/2

2π Cn

v

π/2 1/(n–1)

ρ0 =

ρ0 is an average impact parameter and we count only those with ρ < ρ0

Evaluation of γn

The number of collisions is πρ20vNT where N is the number of perturbers per unit

volume, T is the interval of the collisions. If we set T = Δt0, the average length of an uninterupted segment a photon will travel. Over this length the number of collisions should be ≈ 1.

γn = 2/Δt0 = 2πρ02vN

πρ02 vNΔt0 ≈ 1

Evaluation of γn : Quadratic Stark

In real life you do not have to calculate γn

For quadratic Stark effect (perturbations by charged particles)

γ4 = 39v⅓C4⅔N

Values of the constant C4 has been measured only for a few lines

Na 5890 Å log C4 = –15.17

Mg 5172 Å log C4 = –14.52

Mg 5552 Å log C4 = –13.12

Evaluation of γn

For van der Waals (n=6) you only have to consider neutral hydrogen and helium

log γ6 ≈ 19.6 + 0.4 log C6(H) + log Pg – 0.7 log T

log C6 = –31.7

Linear Stark in Hydrogen

Struve (1929) was the first to note that the great widths of hydrogen lines in early type stars are due to the linear Stark effect. This is induced by ions near the hydrogen atom. Above are the Balmer profiles for an A0 V star.

Thermal Broadening

Thermal motion results in a component of the thermal motion along the line of sight

Δλ λ

= Δν ν

= vr

c vr = radial velocity

We can use the Maxwell Boltzmann distribution

dN N

= 1 v0π½ exp ( vr

v0 –

( 2 [ [

dvr

variance v02 = 2kT/m

N

1.18σ

Velocity v

Thermal Broadening

( ½

The Doppler wavelength shift

v0 (

ΔλD = λ = c

2kT m

λ c ( ( ½

ΔνD = ν = v0

c 2kT m

ν c

dN N

= π–½ exp ( – ( 2

[ [ Δλ ΔλD

Δλ ΔλD

d ( (

The energy removed from the intensity is (πe2f/mc)(λ2/c) times dN/N

α dλ = π½e2

mc f λ2

c 1

ΔλD exp ( –

( 2

[ [ Δλ ΔλD

dλ

The Combined Absorption Coefficient

The Combined absorption coefficient is a convolution of all processes

α(total) = α(natural)*α(Stark)*α(v.d.Waals)*α(thermal)

The first three are easy as they can be defined as a single dispersion profile with γ:

γ = γnatural + γ4 + γ6

The last term is a Gaussian so we are left with the convolution of a Gaussian with the Dispersion (Lorentzian) profile:

α = πe2

mc f γ /4π2

Δν2 + (γ/4π)2 * 1

π½ e–(Δν/ΔνD)2

Lorentzian Gaussian

The convolution is the Voigt Function

The Combined Absorption Coefficient

α = π½e2

mc f

H(u,a) ΔνD

H(u,a) is the Hjerting function

u = Δν/ΔνD = Δλ/ΔλD a = γ

4π

1 ΔνD

= λ0

c

1 ΔλD

2 γ 4π

dν1

γ /4π2

(Δν – Δν1)2 + (γ/4π)2 e–(Δν1/ΔνD)2 ∫ – ∞

∞

H(u,a) =

du1 (u – u1)2 + a2

2 e–u1 ∫ – ∞

∞

H(u,a) = a π

The absorption coefficient can be calculated using the series expansion:

H(u,a) = H0(u) + aH1(u) + a2H2(u) + a3H3(u) + a4H4(u) +

Hjerting function tabulated in Gray

The Line Transfer Equation

dτν = (lν + κν)ρdx lν= line absorption coefficient

κν= continuum absorption coefficient

Source function:

Sν = jν + jν l c jν = line emission coefficient

l

jν = continuum emission coefficient c lν + κν

= –Iν + Sν dIν dτν

This now includes spectral lines

S(τ) =

3Fν

4π (τ + ⅔)

Using the Eddington approximation

At τν = (4π – 2)/3 = τ1 , Sν(τ1) = Fν(0), the surface flux and source function are equal

Across a stellar line lν changes being larger towards the center of the line. This means at line center the optical depth is larger, thus we see higher up in the atmosphere. As one goes farther from line center, lν decreases and the condition that τν = τ1 is deeper in the atmosphere. An absorption line is formed because the source function decreases outward.

F = 2π ∫ 0

∞

Bν(T) E2(τν)dτν

Computing the Line Profile

In local thermodynamic equilibrium the source function is the Planck function

2π ∫ 0

∞

Bν(τν) E2(τν) dτν dτ0

dτ0 =

2π ∫ –∞

∞

Bν(τν) E2(τν) τ0 lν + κν

κ0

dlog τ0

log e =

τν(τ0) =


To compute τν

t0 ∫ –∞

log t0

lν + κν κ0

dlog t0

log e

Fc – Fν

Fc = Sν(τc=τ1) – Sν(τν = τ1)

Sν(τc=τ1) Take the optical depth and divide it into two parts, continuum and line

τν = dt0 ∫ 0

τ0

lν κ0

dt0 ∫ 0

τ0

κν κ0

+

τν =

τ0

τl + τc

Optical depth without lν

Optical depth with lν


τl ≈ lν κ0

τ0

τc ≈ κν κ0

τ0

We need Sν(τν = τ1) = Sν(τl + τc = τ1) = Sν(τc = τ1 – τl)

We are considering only weak lines so τl << τc and evaluate Sν at τ1 – τl using a Taylor expansion around τc = τ1

Sν(τν = τ1) ≈ Sν(τc = τν) + dSν dτν

(–τl)

Ignoring the change with depth


Fc – Fν

Fc = τl

Sν(τc=τ1) dSν dτc

τl dlnSν dτc

= τ1

dlnSν dτc

≈ τc lν κν τ1

C = lν κν

⇒ Weak lines

•  Mimic shape of lν •  Strength of spectral line can be increased either by decreasing the continuous absorption or increasing the line strength

i.e. a Voigt profile

2π –∞ ∫ ∞

Bν(τν) E2(τν) lν + κν

κ0 τ0

dlog τ0

log e = Fν Contribution

function

Contribution Functions

How does this behave with line strength and position in the line?

Sample Contribution Functions Strong lines

Weak line On average weaker lines are formed deeper in the atmosphere than stronger lines.

For a given line the contribution to the line center comes from deeper in the atmosphere from the wings

The fact that lines of different strength come from different depths in the atmosphere is often useful for interpreting observations. The rapidly oscillating Ap stars (roAp) pulsate with periods of 5–15 min. Radial velocity measurements show that weak lines of some elements pulsate 180 degres out-of-phase with strong lines.

z

+

─

In stellar atmosphere:

Conclusion: The two lines are formed on opposite sides of a radial node where the amplitude of the pulsations is zero

Radial node where amplitude =0

The mean amplitude versus mean equivalent width (line strength) of pulsations in the rapidly oscillating Ap star HD 101065

The mean amplitude versus phase of pulsations of the Balmer lines in the rapidly oscillating Ap star HD 101065

Ca II line

The Ca II emission core in solar type active stars

Δλ (Å)

Strong absorption lines are formed higher up in the stellar atmosphere. The core of the lines are formed even higher up (wings are formed deeper). Ca II is formed very high up in the atmospheres of solar type stars.

Behavior of Spectral Lines

The strength of a spectral line depends on:

•  Width of the absorption coefficient which is a function of thermal and microturbulent velocities

•  Number of absorbers (i.e. abundance)

-  Temperature

-  Electron Pressure

-  Atomic Constants

Behavior of Spectral Lines: Temperature Dependence

Temperature is the variable that most strongly controls the line strength because of the excitation and power dependences with T on the ionization and excitation processes

Most lines go through a maximum

•  Increase with temperature is due to increase in excitation

•  Decrease beyond maximum can be due to an increase in continous opacity of negative hydrogen atom (increase in electron pressure)

•  With strong lines atomic absorption coefficient is proportional to γ

•  Hydrogen lines have an absorption coefficient that is temperature sensitive through the stark effect

Temperature Dependence

Example: Cool star where κν behaves like the negative hydrogen ion‘s bound-free absorption:

Four cases

1.  Weak line of a neutral species with the element mostly neutral

2.  Weak line of a neutral species with the element mostly ionized

3.  Weak line of an ion with the element mostly neutral

4.  Weak line of an ion with the element mostly ionized

κν = constant T–5/2 Pee0.75/kT


The number of absorbers in level l is given by :

Nl = constant N0 e–χ/kT ≈ constant e–χ/kT

The number of neutrals N0 is approximately constant with temperature until ionization occurs because the number of ions Ni is small compared to N0.

Ratio of line to continuous absorption is:

R = lν κν

= constant T5/2

Pe e–(χ+0.75)/kT

Case #1:


Recall that Pe = constant eΩT

ln R = constant + 5

2 ln T – χ + 0.75 kT

– ΩT

dR 2.5

T + χ + 0.75

kT2 – ΩT

dT 1 R =


Exercise for the reader:

dR χ + 0.75 – I

kT2 dT 1 R =

Case 2 (neutral line, element ionized):

Case 3 (ionic line, element neutral):

dR 5T

+ χ + 0.75 + I

kT2 – 2ΩT

dT 1 R =

dR 2.5

T +

χ + 0.75

kT2 – ΩT

dT 1 R =

Case 4 (ionic line, element ionized):


Exercise for the reader:

dR χ + 0.75 – I

kT2 dT 1 R =

Case 2 (neutral line, element ionized):

Case 3 (ionic line, element neutral):

dR 5T

+ χ + 0.75 + I

kT2 – 2ΩT

dT 1 R =

dR 2.5

T +

χ + 0.75

kT2 – ΩT

dT 1 R =

Case 4 (ionic line, element ionized):

The Behavior of Sodium D with Temperature

The strength of Na D decreases with increasing temperature. In this case the absorption coeffiecent is proportional to γ, which is a function of temperature

Behavior of Hydrogen lines with temperature

The atomic absorption coefficient of hydrogen is temperature sensitve through the Stark effect. Because of the high excitation of the Balmer series (10.2 eV) this excitation growth continues to a maximum T = 9000 K

A0 V B9.5V

B3IV

F0V

G0V

Behavior of Spectral Lines: Pressure Dependence

Pressure effects the lines in three ways

1.  Ratio of line absorbers to the continous opacity (ionization equilibrium)

2.  Pressure sensitivity of γ for strong lines

3.  Pressure dependence of Stark Broadening for hydrogen

For cool stars Pg ≈ constant Pe 2

Pg ≈ constant g⅔

Pe ≈ constant g⅓

In other words, for F, G, and K stars the pressure dependencies are translated into gravity dependencies

Gravity can influence both the line wings and the line strength

Example of change in line strength with gravity

Example of change in wings due to gravity

Rules:

1. weak lines formed by any ion or atom where most of the element is in the next higher ionization stage are insenstive to pressure changes.

Pressure dependence can be estimated by considering the ratio of line to continuous absorption coefficients

3. weak lines formed by any ion or atom where most of the element is in the next lower ionization stage are very pressure sensitive: lower pressure causes a greater line strength.

2. weak lines formed by any ion or atom where most of the element is in that same stage of ionization are presssure sensitive: lower pressure causes a greater line strength

Rule #1

Ionization equation: Φj(T)

Pe =

Nr+1

Nr ≈ constant Pe

By rule one the line is formed in the rth ionization stage, but most of the element is in the Nr+1 ionization stage: Nr+1 ≈ Ntotal

lν ≈ constant Nr ≈ constant Pe The line absortion coeffiecient is proportional to the number of absorbers

The continous opacity from the negative hydrogen ion dominates:

κν = constant T–5/2 Pee0.75/kT

lν κν

is independent of Pe

Nr

Rule #2

If the line is formed by an element in the r ionization stage and most of this element is in the same stage, then Nr ≈ Ntotal

lν κν

≈ constant g–⅓ = constant

Pe Note: this change is not caused by a change in l, but because the continuum opacity of H– becomes less as Pe decreases

Also note:

∂ log(lν/κν)/∂ log g = –0.33

Proof of rule #3 similar.

In solar-type stars cases 1) and 2) are mostly encountered

Behavior of Spectral Lines: Abundance Dependence The line strength should also depend on the abundance of the absorber, but

the change in strength (equivalent width) is not a simple proportionality as it depends on the optical depth.

Weak lines: the Doppler core dominates and the width is set by the thermal broadening ΔλD. Depth of the line grows in proportion to abundance A

3 phases:

Saturation: central depth approches maximum value and line saturates towards a constant value

Strong lines: the optical depth in the wings become significant compared to κν. The strength depends on g, but for constant g the equivalent width is proportional to A½

The graph specifying the change in equivalent width with abundance is called the Curve of Growth

Behavior of Spectral Lines: Abundance Dependence

Assume that lines are formed in a cool gas above the source of the continuum

Fν = Fce–τν Fc is continuum flux

τν = lνρdx ∫ 0

L

= L is the thickness of the cool gas. Nα dx ∫ 0

L

N/ρ = number of absorbers per unit mass

Nρ

NNE NH

NE

ρ

NH = N/NE is the fraction of element E capable of absorbing, NE/NH is the number abundance A, NH/ρ is the number of hydrogen atoms per unit mass

τν = (N/NE)Nhα dx ∫ 0

L

A τν is proportional to the abundance A and the flux varies exponentially with A


Fν ≈ Fc(1 – τν)

For weak lines τν << 1

Fc – Fν

Fc ≈ τν

→ line depth is proportional τν and thus A. The line depth and thus the equivalent width is proportional to A


What about strong lines?

α = π½e2

mc f

H(u,a) ΔνD

The wings dominate so

f ΔνD

α = πe2

mc γ

4π2

τν = (N/NE)Nhα dx ∫ 0

L

A = πe2

mc γ

4π2

A f Δν2 dx (N/NE)NH ∫

0

L

≈ <γ> A f h Δν2 <γ> denotes the depth average

damping constant, and h is the constants and integral

Fc – Fν

Fc = 1 – e–τν

The equivalent width of the line:

W = ∫ 0

∞

(1 – e–τν) dν

W = ∫ 0

∞

(1 – e–<γ>Αf h/Δν2) dν

Substituting u2 = Δν2/<γ>A f h

W = (<γ>A f h)½ ∫ ∞

(1 – e–1/u2) du

0

Equivalent width is proportional to the square root of the abundance

A bit of History

Cecilia Payne-Gaposchkin (1900-1979).

At Harvard in her Ph.D thesis on Stellar Atmospheres she:

•  Realized that Saha‘s theory of ionization could be used to determine the temperature and chemical composition of stars

•  Identified the spectral sequence as a temperature sequence and correctly concluded that the large variations in absorption lines seen in stars is due to ionization and not abundances

•  Found abundances of silicon, carbon, etc on sun similar to earth

•  Concluded that the sun, stars, and thus most of the universe is made of hydrogen and helium.

Otto Struve: „undoubtedly the most brilliant Ph.D thesis ever written in Astronomy“

Youngest scientist to be listed in American Men of Science !!!

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The Formation of Spectral Lines - tls-tautenburg.de · The Formation of Spectral Lines I. Line Absorption Coefficient II. Line Transfer Equation