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

∂ E∂ x2

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

E = E0 e–ix/v– t)

The wave velocity through a medium

v=c (

(0 0

½ and are the electric and magnetic permemability in the medium and free space. For gases = 0


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

The ratio of /0 is just the ratio of the field in the medium to the field in free space

0

= E + 4NqzE

4NqzE

1 + =

We need z/E

Line Absorption CoefficientFor a damped harmonic oscillator where z is the induced

separation between the dipole charges

d2 zd t2 +

dzdt

+ 0

2=

em

E0 eit

e,m are charge and mass of electron

is damping constant

Solution: z = z0e–it

z = em

E0 eit

0 – 2 + i2 = em

E

0 – 2 + i2


= 0

1 + 0 – 2 + i2

1 4Ne2

E

For a gas ≈ 0

The wave velocity can now be written as

cv ≈ ( (

½≈ 1 +

1

2

4Ne2

m 0 – 2 + i2

1

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


cv ≈ 1 +

2Ne2

m 0 – 2)2 + 20 – 2

0 – 2)2 + 2

– i

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

I = I0 e–kx/c = I0e–lx


l = 4Ne2

mc 0 – 2)2 + 2

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

0 – =(0 – )(0 + ) ≈ (0 – )2 ≈ 2

The basic form of the line absorption coefficient:

l = Ne2

mc 2 + (

This is a damping profile or Lorentzian profile

2 2


= 2e

mc 2 + (

Consider the absorption coefficient per atom, , where l = N

= 2e

mc 2 + (

= 2e

mc 2 + (cc

c


A quantum mechanical treatment

∫ = e2

mcd

0

∞

∫ = d0

∞ e2

mcf

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

∫ = d0

∞Blu h

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


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

= e2

mcf = Blu h 7.484 × 10–7

Blu

2e2mc3

f = Aulgu 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 = –23 =–

e2

mc3 W W

Classical dipole emission theory gives an equation of the form

Solution of the form

= 0.22/2 in cm =2e2 3mc3

W= W0e–t

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

h

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:

W = 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

= Cn /Rn

Pressure Broadening: The Impact Approximation

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

Spectrum is just the Fourier transform of box times sine which is sinc t(-0) and indensity is sinc2t(-0). Characteristic width is = 1/t

tj

Pressure Broadening: The Impact Approximation

With collisions, the original box is cut into many shorter boxes of length tj < t

Because tj < t the line is broadened with j = 1/tj. The Fourier transform of the sum is the sum of the transforms.

The distribution, P, of tj is:.

dP(tj) = e–tj/t0 dtj/t0

The line absorption coefficient:

t2sint( – 0)

t( – 0)

2

e–t/t0dtt0∫

0

∞

= C42( – 0)2 + (1/t0)2

= C ( – 0)2 + (n/4)2

n/4

In other words this is the Lorentzian. To use this in a line profile calculation need to evaluate n = 2/t0. This is a function of depth in the stellar atmosphere.

Evaluation of n

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.

= 2∫0

∞

dt

= 2∫0

∞

Cn R–n dt

Assume perturber moves past atom in a straight line

y

x

v

R

R cos

= 2∫0

∞

Cn cos dtn

Atom

Perturber

Evaluation of n

v = dy/dt = (/cos2) d/dt => dt = (/v)d/cos2

=

cosn–2d∫–/2

2Cn

vn–1

/2

cosn–2d∫–/2

/2n

2

3 2

4 /2

6 3/8

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

cosn–2d∫–/2

2Cn

v

/2 1/(n–1)

=

The number of collisions is 0vNT where N is the number of perturbers per unit volume, T is the interval of the collisions

n = 202vN

Evaluation of n : Quadratic Stark

In real life you do not have to calculate n

For quadratic Stark effect

4 = 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 n

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

log 6 ≈ 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

=

=vr

cvr = radial velocity

We can use the Maxwell Boltzmann distribution

dNN

=1

v0½ exp ( vr

v0–

(2[

[

dvr

variance v0 = 2kT/m

N

1.18

Velocity

v

Thermal Broadening

( ½

The Doppler wavelength shift

v0 (D = =c

2kTm

c ( (½

D = =v0

c2kTm

c

dNN

= –½ exp (–(2[ [D

D

d(

(

The energy removed from the intensity is (e2f/mc)(2/c) times dN/N

d =½e2

mcf2

cD

exp (–

(2[ [D

d

The Combined Absorption Coefficient

The Combined absorption coefficient is a convolution of all processes

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

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

= natural + + 6

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

=e2

mcf

/42

2 + (2 *½ e–(/

D)2

Lorentzian Gaussian

The Combined Absorption Coefficient

=½e2

mc

fH(u,a)

D

H(u,a) is the Hjerting function

u = /D = /D a =

1D

= c

1D

2

d1

/42

– 1)2 + (/4)2 e–(/D)2∫

– ∞

∞

H(u,a) =

du1u – u1)2 + a2

2e–u1

∫– ∞

∞

H(u,a) =a

Hjerting function tabulated in Gray

The Line Transfer Equation

d = (l + )dx l= line absorption coefficient

= continuum absorption coefficient

Source function:

S = j + jl c j = line emission coefficient

l

j = continuum emission coefficientcl +

= –I + SdId

This now includes spectral lines

S() =3F4

( + ⅔)

Using the Eddington approximation

At = (4 – 2)/3 = 1 , S(1) = F(0), the surface flux and source function are equal

Across a stellar line l 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, ln decreases and the condition that = is deeper in the atmosphere. An absorption line is formed because the source function decreases outward.

F= 2 ∫

∞

BTE2)d

Computing the Line Profile

In local thermodynamic equilibrium the source function is the Planck function

2 ∫

∞

BE2) d d

d=

2 ∫–∞

∞

BE2) l+

dlog log e

=

=


To compute

t∫–∞

log t0

l+

dlogt log e

Fc – F Fc

=S(c=1) – S

S(c=)Take the optical depth and divide it into two parts, continuum and line

= dt∫0

0

l

dt∫0

0

+

=

0

l + c


l ≈ l0

0

c ≈ 0

0

We need S( = 1) = S(l + c = 1) = S(c = 1 – l)

We are considering only weak lines so l << c and evaluate S at 1 – l using a Taylor expansion around c = 1

S( =1) ≈ S(c = ) +dSd

(–l)


Fc – F Fc

=l

S(c=)dSdc

ldlnSdc

=1

dlnSdc

≈ 0l 1

C=l

Weak lines

• Mimic shape of l

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

2 –∞∫∞

BE2) l+

dlog

log e=F Contribution

function

Contribution Functions

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

Sample Contribution Functions

Strong lines

Weak lineOn 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

Ca II line

(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

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

Temperature Dependence

Example: Cool star where behaves line 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

constant T–5/2 Pee0.75/kT


The number of absorbers in level l is given by :

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

= constantT5/2

Pe

e–(/kT

Case #1:


Recall that Pe = constant eT

ln R = constant +52

ln T – + 0.75kT

– T

dR 2.5T

+ + 0.75

kT2– T

dT 1 R =


Exercise for the reader:

dR + 0.75 – I

kT2 dT

1

R =

Case 2 (neutral line, element ionized):

Case 3 (ionic line, element neutral):

dR 5

T+ + 0.75 + I

kT2– 2T

dT

1

R =

dR 2.5

T+ + 0.75

kT2– T

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 , 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 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:j(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

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

The continous opacity from the negative hydrogen ion dominates:

= constant T–5/2 Pee0.75/kT

l

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

l

≈ 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(l/)/∂ 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 D. 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 . 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


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

F = Fce–Fc is continuum flux

= ldx ∫0

L

= L is the thickness of the cool gas.Ndx ∫0

L

N/ = number of absorbers per unit mass

N

NNE NH

NE

NH=

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

= (N/NE)Nhdx∫0

L

A is proportinal to the abundance A and the flux varies exponentially with A


F ≈ Fc(1 – )

For weak lines << 1

Fc – FFc

≈

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


What about strong lines?

=½e2

mc

fH(u,a)

D

The wings dominate so

f

D

=e2

mc

= (N/NE)Nhdx∫0

L

A =e2

mc

A f

2dx(N/NE)NH∫

0

L

≈<>A f h

2 <> denotes the depth average damping constant, and h is the constants and integral

Fc – FFc

= 1 – e–

The equivalent width of the line:

W = ∫0

∞

(1 – e– d

W = ∫0

∞

(1 – e–f h d

Substituting u2 = 2/<>A f h

W = (<>A f h)½ ∫∞

(1 – e–1/u2 du0

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 !!!

the formation of spectral lines i.line absorption coefficient ii.line transfer equation

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