The Formation of Spectral Lines

The Formation of Spectral Lines
Line Absorption Coefficient Line Transfer Equation

Line Absorption Coefficient
Main processes Natural Atomic Absorption Pressure Broadening 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 ∂ x2 Treat only one frequency since by Fourier composition the total field is a sum of all sine waves. E = E0 e–iw(x/v – t) The wave velocity through a medium v=c ( e0 m0 e m e and m are the electric and magnetic permemability in the medium and free space. For gases m = m0

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 4pNqz where z is the separation of the charges and N the number of dipoles per unit volume The ratio of e/e0 is just the ratio of the field in the medium to the field in free space e e0 E + 4pNqz E 4pNqz E = = 1 + We need z/E

For a damped harmonic oscillator where z is the induced separation between the dipole charges d2 z d t2 dz e g + w0 2 = E0 eiwt + dt m e,m are charge and mass of electron g is damping constant Solution: z = z0e–iwt z = e m E0 eiwt w0 – w2 + igw 2 = E

w0 – w2 + igw 2 1 4pNe2 E 1 + = For a gas e ≈ e0 so second term is small compared to unity The wave velocity can now be written as c v ≈ ( e e0 1 + 1 2 4pNe2 m w0 – w2 + igw Where we have performed a Taylor expansion (1 + x) = 1 + ½ x for small x

2pNe2 m (w0 – w2)2 + g2w2 2 gw – i c v ≈ (w0 – w2)2 + g2w2 2 w0 – w2 1 + This can be written as a complex refractive index c/v = n – ik. When it is combined with iwx/v it produces an exponential extinction e–kwx/c . Recall that the intensity is EE* where E* is the complex conjugate. The light extinction can be expressed as: I = I0 e–kwx/c = I0e–lnrx

lnr = 4pNe2 mc (w0 – w2)2 + g2w2 2 gw This function is sharply peaked giving non-zero values when w ≈ w0 w0 – w =(w0 – w)(w0 + w) ≈ (w0 – w)2w ≈ 2wDw 2 2 The basic form of the line absorption coefficient: lnr = Npe2 mc Dw2 + (g/2)2 gw This is a damping profile or Lorentzian profile, a Cauchy curve, or the „Witch of Agnesi“

Consider the absorption coefficient per atom, a, where lnr = Na 2pe gw a = mc Dw2 + (g/2)2 2pe g/4p a = mc Dn2 + (g/4p)2 2pe l2 c Dl2 + (gl2/4pc)2 gl2/4pc a = mc

∫ = pe2 mc a dn ∞ This is energy per unit atom per square radian that the line absorbs from In A quantum mechanical treatment ∫ = a dn ∞ pe2 mc f f is the oscillator strength and is related to the transition probability Blu ∞ ∫ a dn = Blu hn

mc 7.484 × 10–7 Blu l f = = Blu hn pe2 mc3 gu gl Aul f = 2pe2n2 There is also an f value for emission gu fem = gl fabs 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
Classical dipole emission theory gives an equation of the form dW d t = –g 2 3 – e2 w2 mc3 W Solution of the form W= W0e–gt 2e2 w2 g = = 0.22/l2 in cm 3mc3 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 u 1: unperturbed energy 2. Perturbed energy less than unperturbed 3. Energy greater than unperturbed E hn l 1 2 3 R

Pressure Broadening DW = Const/Rn Dn = Cn /Rn
Energy change as a function of R: DW = 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 Dn = Cn /Rn

Pressure Broadening: The Impact Approximation
Duration of encounter typically 10–9 secs Dtj 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 Photon of duration Dt is an infinite sine wave times a box Spectrum is just the Fourier transform of box times sine which is sinc pDt(n-n0) and indensity is sinc2pDt(n-n0). Characteristic width is Dn= 1/Dt

Pressure Broadening: The Impact Approximation
With collisions, the original box is cut into many shorter boxes of length Dtj < Dt Because Dtj < Dt the line is broadened with Dnj = 1/Dtj. The Fourier transform of the sum is the sum of the transforms. The distribution, P, of Dtj is:. dP(Dtj) = e–Dtj/Dt0 dDtj/Dt0 t0 is an average length of an uninterrupted time segment

∫ The line absorption coefficient is the weighted average:
2 ∫ ∞ sinpDt(n – n0) pDt(n – n0) dDt Dt2 e–Dt/Dt0 Dt0 C a = 4p2(n – n0)2 + (1/Dt0)2 gn/4p a = C (n – n0)2 + (gn/4p)2 In other words this is the Lorentzian. To use this in a line profile calculation need to evaluate gn = 2/Dt0. This is a function of depth in the stellar atmosphere.

∫ ∫ ∫ Evaluation of gn = 2p Cn R–n dt Dn = Cn /Rn f = 2p Dn dt
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. ∫ ∞ = 2p ∫ ∞ Cn R–n dt Dn = Cn /Rn f = 2p Dn dt v y Assume perturber moves past atom in a straight line Atom r = R cos q r x q = 2p ∫ ∞ Cn cos q dt f rn R n Perturber

∫ ∫ Evaluation of gn cosn–2 q dq 2p Cn vrn–1 f = cosn–2 q dq 2p Cn v
v = dy/dt = (r/cos2q) dq/dt => dt = (r/v)dq/cos2q cosn–2 q dq ∫ –p/2 2p Cn vrn–1 p/2 cosn–2 q dq ∫ –p/2 p/2 f = n 2 p 3 4 p/2 6 3p/8 Usually define a limiting impact parameter for a significant phase shift f = 1 rad cosn–2 q dq ∫ –p/2 2p Cn v p/2 1/(n–1) r0 = r0 is an average impact parameter and we count only those with r < r0

Evaluation of gn gn = 2/Dt0 = 2pr02vN pr02 vNDt0 ≈ 1
The number of collisions is pr20vNT where N is the number of perturbers per unit volume, T is the interval of the collisions. If we set T = Dt0, the average length of an uninterupted segment a photon will travel. Over this length the number of collisions should be ≈ 1. pr02 vNDt0 ≈ 1 gn = 2/Dt0 = 2pr02vN

Evaluation of gn : Quadratic Stark
In real life you do not have to calculate gn For quadratic Stark effect (perturbations by charged particles) g4 = 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 gn log g6 ≈ 19.6 + 0.4 log C6(H) + log Pg – 0.7 log T
For van der Waals (n=6) you only have to consider neutral hydrogen and helium log g6 ≈ 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 Dl l = Dn n vr c vr = radial velocity N dN N =
Thermal motion results in a component of the thermal motion along the line of sight Dl l = Dn n vr c vr = radial velocity N We can use the Maxwell Boltzmann distribution dN N = 1 v0p½ exp ( vr v0 – 2 [ dvr variance v02 = 2kT/m 1.18s v Velocity

[ [ ( ( ( ( ( Thermal Broadening v0 DnD = n = v0 c 2kT m DlD = l = c
The Doppler wavelength shift v0 ( DnD = n = v0 c 2kT m ( DlD = l = c 2kT m l c ( dN N = p–½ exp ( – 2 [ Dl DlD d The energy removed from the intensity is (pe2f/mc)(l2/c) times dN/N a dl = p½e2 mc f l2 c 1 DlD exp ( – 2 [ Dl dl

The Combined Absorption Coefficient
The Combined absorption coefficient is a convolution of all processes a(total) = a(natural)*a(Stark)*a(v.d.Waals)*a(thermal) The first three are easy as they can be defined as a single dispersion profile with g: = gnatural + g4 + g6 The last term is a Gaussian so we are left with the convolution of a Gaussian with the Dispersion (Lorentzian) profile: a = pe2 mc f /4p2 Dn2 + (g/4p)2 * 1 p½ e–(Dn/DnD)2 Lorentzian Gaussian

The convolution is the Voigt Function

The Combined Absorption Coefficient
p½e2 mc f H(u,a) DnD H(u,a) is the Hjerting function = l0 c 1 DlD 2 g 4p a = g 4p 1 DnD u = Dn/DnD = Dl/DlD dn1 /4p2 (Dn – Dn1)2 + (g/4p)2 e–(Dn1/DnD)2 ∫ – ∞ ∞ H(u,a) = du1 (u – u1)2 + a2 2 e–u1 a p

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
dtn = (ln + kn)rdx ln= line absorption coefficient kn= continuum absorption coefficient Source function: jn + jn l c jn = line emission coefficient l Sn = ln + kn jn = continuum emission coefficient c dIn This now includes spectral lines = –In + Sn dtn

3Fn (t + ⅔) S(t) = 4p Using the Eddington approximation
At tn = (4p – 2)/3 = t1 , Sn(t1) = Fn(0), the surface flux and source function are equal

Across a stellar line ln 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 tn = t1 is deeper in the atmosphere. An absorption line is formed because the source function decreases outward.

Computing the Line Profile
In local thermodynamic equilibrium the source function is the Planck function ∫ ∞ F = 2p Bn(T) E2(tn)dtn ∫ ∞ dtn dt0 = 2p Bn(tn) E2(tn) dt0 2p ∫ –∞ ∞ Bn(tn) E2(tn) t0 ln + kn k0 dlog t0 log e =

To compute tn log t0 ln + kn k0 dlog t0 log e ∫ tn(t0) = t0 Optical depth without ln –∞ Optical depth with ln Fc – Fn Fc = Sn(tc=t1) – Sn(tn = t1) Sn(tc=t1) Take the optical depth and divide it into two parts, continuum and line tn = dt0 ∫ t0 ln k0 kn + t0 tn = tl + tc

ln k0 tl ≈ t0 Ignoring the change with depth kn k0 tc ≈ t0 We need Sn(tn = t1) = Sn(tl + tc = t1) = Sn(tc = t1 – tl) We are considering only weak lines so tl << tc and evaluate Sn at t1 – tl using a Taylor expansion around tc = t1 dSn dtn Sn(tn = t1) ≈ Sn(tc = tn) + (–tl)

Fc – Fn tl dSn dtc = Fc Sn(tc=t1) tl dlnSn dtc = t1 ln kn dlnSn dtc tc ≈ t1 C = ln kn Weak lines Mimic shape of ln Strength of spectral line can be increased either by decreasing the continuous absorption or increasing the line strength i.e. a Voigt profile

Contribution Functions
–∞ ∫ ∞ ln + kn k0 t0 dlog t0 log e Fn = 2p Bn(tn) E2(tn) Contribution function 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: Radial node where amplitude =0 Conclusion: The two lines are formed on opposite sides of a radial node where the amplitude of the pulsations is zero

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

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

Ca II line The Ca II emission core in solar type active stars

Dl (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 g Hydrogen lines have an absorption coefficient that is temperature sensitive through the stark effect

Temperature Dependence
Example: Cool star where kn behaves like the negative hydrogen ion‘s bound-free absorption: kn = constant T–5/2 Pee0.75/kT Four cases Weak line of a neutral species with the element mostly neutral Weak line of a neutral species with the element mostly ionized Weak line of an ion with the element mostly neutral Weak line of an ion with the element mostly ionized

Case #1: The number of absorbers in level l is given by : Nl = constant N0 e–c/kT ≈ constant e–c/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 = ln kn = constant T5/2 Pe e–(c+0.75)/kT

Recall that Pe = constant eWT 5 c kT ln R = constant + ln T – – WT 2 1 R dR c 2.5 – WT = + dT T kT2

Exercise for the reader: Case 2 (neutral line, element ionized): dR c – I kT2 dT 1 R = Case 3 (ionic line, element neutral): dR 5 T + c I kT2 – 2WT dT 1 R = Case 4 (ionic line, element ionized): 1 R dR c 2.5 – WT = + dT T kT2

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 g, which is a function of temperature

Behavior of Hydrogen lines with temperature
B9.5V B3IV F0V G0V 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

Behavior of Spectral Lines: Pressure Dependence
Pressure effects the lines in three ways Ratio of line absorbers to the continous opacity (ionization equilibrium) Pressure sensitivity of g for strong lines Pressure dependence of Stark Broadening for hydrogen For cool stars Pg ≈ constant Pe 2 In other words, for F, G, and K stars the pressure dependencies are translated into gravity dependencies Pg ≈ constant g⅔ Pe ≈ constant g⅓ 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

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

ln ≈ constant Nr ≈ constant Pe
Rule #1 Nr+1 Fj(T) Ionization equation: = Nr Pe 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 ln ≈ constant Nr ≈ constant Pe The line absortion coeffiecient is proportional to the number of absorbers The continous opacity from the negative hydrogen ion dominates: kn = constant T–5/2 Pee0.75/kT ln kn is independent of Pe

Rule #2 ln kn constant = ≈ constant g–⅓ Pe
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 ln kn constant = ≈ constant g–⅓ 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(ln/kn)/∂ 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. 3 phases: Weak lines: the Doppler core dominates and the width is set by the thermal broadening DlD. Depth of the line grows in proportion to abundance A 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 kn. 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 Fn = Fce–tn Fc is continuum flux L Na dx ∫ L ∫ tn = lnrdx = L is the thickness of the cool gas. N/r = number of absorbers per unit mass N r NE NH = N/NE is the fraction of element E capable of absorbing, NE/NH is the number abundance A, NH/r is the number of hydrogen atoms per unit mass tn = (N/NE)Nha dx ∫ L A tn is proportional to the abundance A and the flux varies exponentially with A

For weak lines tn << 1 Fc – Fn Fc ≈ tn Fn ≈ Fc(1 – tn) → line depth is proportional tn and thus A. The line depth and thus the equivalent width is proportional to A

What about strong lines? a = p½e2 mc f H(u,a) DnD The wings dominate so a = pe2 mc g 4p2 f DnD tn = (N/NE)Nha dx ∫ L A ∫ L pe2 mc A f g 4p2 = (N/NE)NH dx Dn2 ≈ <g> A f h Dn2 <g> denotes the depth average damping constant, and h is the constants and integral

∫ ∫ ∫ Fc – Fn Fc = 1 – e–tn W = (1 – e–tn) dn W =
The equivalent width of the line: W = ∫ ∞ (1 – e–tn) dn ∞ ∫ W = (1 – e–<g>Af h/Dn2) dn Substituting u2 = Dn2/<g>A f h W = (<g>A f h)½ ∫ ∞ (1 – e–1/u2) du Equivalent width is proportional to the square root of the abundance

A bit of History Cecilia Payne-Gaposchkin ( ). 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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