Line Profiles

Note - Figure obtained from www.physics.utoledo.edu/~lsa.atnos/SAElp05.htm

Chapter 13 – Behavior of Spectral Lines

• Formalism of radiative transfer in spectral lines– Transfer equation for lines– The line source function

• Computing the line profile in LTE• Depth of formation• Temperature and pressure dependence

of line strength• The curve of growth

We began with line absorption coefficients which give the shapes of spectral lines. Now we move into thecalculation of line strength from a stellar atmosphere.

The Line Transfer Equation• We can add the continuous absorption coefficient and the line

absorption coefficient to get the total absorption coefficient:d = (l+)dx

• And the source function is the sum of the line and continuous emission coefficients divided by the sum of the line and continuous absorption coefficients.

• Or define the line and continuum source functions separately:– Sl=jl/l– Sc=jc/

• In either case, we still have the basic transfer equation:

l

jjS

cl

/1

)/(l

SSl clS

SI

d

dI

deSI sec)0( sec

0

The Line Source Function

• The basic problem is still how to obtain the source function to solve the transfer equation.

• But the line source function depends on the atomic level populations, which themselves depend on the continuum intensity and the continuum source function. This coupling complicates the solution of the transfer equation for lines.

• Recall that in the case of LTE the continuum source function is just B(T), the Planck Function.

• The assumption of LTE simplifies the line case in the same way, and allows us to describe the energy level populations strictly by the temperature without coupling to the radiation field.

• This approximation works when the excitation states of the gas are defined primarily by collisions and not radiative excitation or de-excitation.

The Gray Atmosphere

• Recall for the gray atmosphere,

• So, at = (4-2)/3, S() = F(0)

• This is about =3.5 – and gives us a “mapping” between the source function and the line profile

• The center of a line is formed higher in the atmosphere than the wings because the opacity is higher in the center

3243 )0()( FS

Mapping the Line Source Function

• The line source function with depth maps into the line profile

• The center of the line is formed at shallower optical depth, and maps to the source function at smaller

• The wings of the line are formed in progressively deeper layers

Depth of Formation

• It’s straightforward to determine approximately where in the atmosphere (in terms of the optical depth of the continuum) each part of the line profile is formed

• But even at a specific , a range of optical depths contributes to the absorption at that wavelength

• It’s not straightforward to characterize the depth of formation of an entire line

• The cores of strong lines are formed at very shallow optical depths.

The Strength of Spectral Lines

• The strengths of spectral lines depend on– The number of absorbers

• Temperature• Electron pressure or luminosity• Atomic constants

– The line absorption coefficient– The ratio of the line/continuous absorption

coefficient– Thermal and microturbulent velocities – In strong lines – collisional line broadening

affected by the gas and electron pressures

Computing the Line Profile• The line profile results from the solution of the transfer equation

at each through the line.• The line profile will depend on the number of absorbers at each

depth in the atmosphere• The simplifying assumptions are

– LTE, collisions dominage– Pure absorption (no scattering)

• How well does this work?• To know for sure we must compute the line profile in the

general case and compare it to what we get with simplifying assumptions

• Generally, it’s pretty good

• Start with the assumed T() relation and model atmosphere• Recompute the flux using the line+continuous opacity at each

wavelength around the line• For blended lines, just add the line absorption coefficients

appropriate at each wavelength

The Effect of Temperature• Temperature is the

main factor affecting line strength

• Exponential and power of T in excitation and ionization

• Line strength increases with T due to increase in excitation

• Decrease beyond maximum– an increase in

the opacity – drop in

population from ionization

The Effect of Temperature

on Weak Lines

1. neutral line, mostly neutral species2. neutral line, mostly ionized species3. ionic line, mostly neutral species4. ionic line, mostly ionized element

Neutral lines from a neutral species

• Number of absorbers proportional to N0exp(-/kT)

• Number of neutrals independent of temperature (why?)• If H- is the dominant opacity, the ratio of line to

continuous absorption coefficient is given by

• But Pe is ~ proportional to exp(T/1000), so… (Ch. 9)

kT

e

eP

TlR

)75.0(25

001.075.05.212

kTTdT

dR

R

001.0

75.02

5.2

kTT

EQW

EQWT

Neutral Lines of a Neutral Species

• Oxygen triplet lines at 7770A.– Excitation potential = 8 eV– Ionization potential = 13.6 eV

• Oxygen resonance line [O I] at 6300A• By what factor will each of these lines

change in strength from 5000 to 6000K?

001.0)75.0(1016.15.2

2

4

T

x

TT

EQW

EQW

Neutral Lines of an Ionized Species

• How much would you have to change the temperature of a 6000K star to decrease the equivalent width of the Li I 6707 resonance line by a factor of two?

• Ionization potential = 5.4 eV

2

4 )75.0(1016.1

T

IxT

EQW

EQW

Ionic Lines of a Neutral Element

• Fe II lines in giants are often used to determine the spectroscopic gravity.

• How sensitive to temperature is a 2.5eV Fe II line (I=7.9 eV) in a star with Teff=4500K? (Estimate for T=100K)

002.0

)75.0(1016.152

4

T

Ix

TT

EQW

EQW

Ionic Lines of Ionized Species

• How strong is a Ba II line (at 0 eV) in a 6000K star compared to a 5000K star?

• How do the strengths of a 5 eV Fe II line compare in the same two stars?

• For Ba II, EQW decreases by 25%• For Fe II, EQW is almost x3 larger

001.0

)75.0(1016.15.22

4

T

x

TT

EQW

EQW

Line Strength Depends on Pressure

• For metal lines, pressure (gravity) affects line strength in two ways:– Changing the line-to-

continuous opacity ratio (by changing the ionization equilibrium)

– Pressure broadening

• Pressure effects are much weaker than temperature effects

Rules of Thumb for Weak Lines

• When most of the atoms of an element are in the next higher state of ionization, lines are insensitive to pressure – When H- opacity dominates, the line and the continuous

absorption coefficients are both proportional to the electron pressure

– Hence the ratio line/continuous opacity is independent of pressure

• When most of the atoms of an element are in the same or a lower state of ionization, lines are sensitive to pressure– For lines from species in the dominant ionization state, the

continuous opacity (if H-) depends on electron pressure but the line opacity is independent of electron pressure

• Lines from a higher ionization state than the dominant state are highly pressure dependent

– H- continuous opacity depends on Pe

– Degree of ionization depends on 1/Pe

Examples of Pressure Dependence

• Sr II resonance lines in solar-type stars

• 7770 O I triplet lines in solar-type stars

• [O I] in K giants• Fe I and Fe II lines in solar-type

stars• Fe I and Fe II lines in K giants• Li I lines in K giants

The Mg I b lines

• Why are the Mg I b lines sensitive to pressure?

H- Profiles

• H lines are sensitive to temperature because of the Stark effect

The high excitation of the Balmer series (10.2 eV) means excitation continues to increase to high temperature (max at ~ 9000K).

Most metal lines have disappeared by this temperature. Why?

Pressure Effects on Hydrogen Lines

• When H- opacity dominates, the continuous opacity is proportional to pressure, but so is the line abs. coef. in the wings – so Balmer lines in cool stars are not sensitive to pressure

• When Hbf opacity dominates, is independent of Pe, while the line absorption coefficient is proportional to Pe, so line strength is too

• In hotter stars (with electron scattering) is nearly independent of pressure while the number of neutral H atoms is proportional to Pe

2. Balmer profiles are very pressure dependent

What Is Equivalent Width?

• The equivalent width is a measure of the strength of a spectral line

• Area equal to a rectangle with 100% depth

• Triangle approximation: half the base times the width

• Integral of a fitted line profile (Gaussian, Voigt fn.)

• Measured in Angstroms or milli-Angstroms

• How is equivalent width defined for emission lines?

The Curve of Growth• The curve of growth is a mathematical relation between the chemical

abundance of an element and the line equivalent width• The equivalent width is expressed independent of wavelength as log W/

Wrubel COG from Aller and Chamberlin 1956

Curves of

Growth

• When abundance is small - "linear part," line strength increases linearly When abundance is mid-range - "flat part," absorption begins to saturate

• When abundance is large - "damping part," optical depth in the wings becomes large

Note - Figures obtained from www.physics.utoledo.edu/~lsa.atnos/SAElp05.htm

Curves of Growth Traditionally, curves of growth

are described in three sections• The linear part:

– The width is set by the thermal width

– Eqw is proportional to abundance

• The “flat” part:– The central depth approaches

its maximum value– Line strength grows

asymptotically towards a constant value

• The “damping” part:– Line width and strength

depends on the damping constant

– The line opacity in the wings is significant compared to

– Line strength depends (approximately) on the square root of the abundance

Effect of Pressure on the COG• The higher the damping constant, the stronger the lines

get at the same abundance. • The damping parts of the COG will look different for

different lines