The Formation of Spectral Lines. I. Line Absorption Coefficient II. Line Transfer Equation

Transcription

1 The Formation of Spectral Lines I. Line Absorption Coefficient II. Line Transfer Equation

2 Line Absorption Coefficient Main processes 1. Natural Atomic Absorption 2. Pressure Broadening 3. Thermal Doppler Broadening

3 Line Absorption Coefficient The classical model of the interaction of light with a photon is a plane electromagnetic wave interacting with a dipole. 2 E t 2 = v 2 2 E x 2 Treat only one frequency since by Fourier composition the total field is a sum of all sine waves. E = E 0 e iω(x/v t) The wave velocity through a medium v=c ( ε 0 µ 0 ε µ ( ½ ε and µ are the electric and magnetic permemability in the medium and free space. For gases µ = µ 0

4 Line Absorption Coefficient 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/ε 0 is just the ratio of the field in the medium to the field in free space ε = E + 4πNqz ε 0 E = 1 + 4πNqz E We need z/e

5 Line Absorption Coefficient For a damped harmonic oscillator where z is the induced separation between the dipole charges d 2 z dz d t 2 + γ 2 + ω dt 0 = e m E 0 eiωt e,m are charge and mass of electron γ is damping constant Solution: z = z 0 e iωt z = e m E 0 e iωt ω 0 ω 2 + iγω 2 = e m E 2 ω 0 ω 2 + iγω

6 Line Absorption Coefficient ε ε = πNe 2 E 1 2 ω 0 ω 2 + iγω For a gas ε ε 0 so second term is small compared to unity The wave velocity can now be written as c v ( ε ε 0 ( ½ πNe2 2 m 1 ω 2 0 ω 2 + iγω ½ Where we have performed a Taylor expansion (1 + x) = 1 + ½ x for small x

7 Line Absorption Coefficient c v 1 + 2πNe2 m 2 ω 0 ω 2 (ω 2 0 ω 2 ) 2 + γ 2 ω 2 i γω (ω 2 0 ω 2 ) 2 + γ 2 ω 2 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 = I 0 e kωx/c = I 0 e l νρx

8 Line Absorption Coefficient l ν ρ = 4πNe 2 mc γω (ω 2 0 ω 2 ) 2 + γ 2 ω 2 This function is sharply peaked giving non-zero values when ω ω ω 0 ω =(ω 0 ω)(ω 0 + ω) (ω 0 ω)2ω 2ωΔω The basic form of the line absorption coefficient: l ν ρ = Nπe 2 mc γω Δω 2 + (γ/2) 2 This is a damping profile or Lorentzian profile, a Cauchy curve, or the Witch of Agnesi

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10 Line Absorption Coefficient Consider the absorption coefficient per atom, α, where l ν ρ = Nα α = 2πe mc γω Δω 2 + (γ/2) 2 α = 2πe mc γ/4π Δν 2 + (γ/4π) 2 α = 2πe mc λ 2 c γλ 2 /4πc Δλ 2 + (γλ 2 /4πc) 2

11 Line Absorption Coefficient α dν = A quantum mechanical treatment 0 πe 2 mc This is energy per unit atom per square radian that the line absorbs from I ν α dν = 0 πe 2 f mc f is the oscillator strength and is related to the transition probability B lu α dν = 0 B lu hν

12 Line Absorption Coefficient mc B f = B lu hν = lu πe 2 λ f = mc 3 2πe 2 ν 2 g u g l A ul There is also an f value for emission g u f em = g l f abs 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.

13 The Damping Constant for Natural Broadening Classical dipole emission theory gives an equation of the form dw 2 e 2 ω 2 d t = γ 3 = mc 3 W W Solution of the form W= W 0 e γt γ = 2e 2 ω 2 3mc 3 = 0.22/λ 2 in cm 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

14 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 E hν 1 l 2. Perturbed energy less than unperturbed 3. Energy greater than unperturbed 2 3 R

15 Pressure Broadening Energy change as a function of R: ΔW = Const/R n 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 = C n /R n

16 Pressure Broadening: The Impact Approximation Duration of encounter typically 10 9 secs Δt j 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 Δt is an infinite sine wave times a box Spectrum is just the Fourier transform of box times sine which is sinc πδt(ν-ν 0 ) and indensity is sinc 2 πδt(ν-ν 0 ). Characteristic width is Δν= 1/Δt

17 Pressure Broadening: The Impact Approximation With collisions, the original box is cut into many shorter boxes of length Δt j < Δt The distribution, P, of Δt j is:. dp(δt j ) = e Δt j /Δt 0 dδt j /Δt 0 Because Δt j < Δt the line is broadened with Δν j = 1/Δt j. The Fourier transform of the sum is the sum of the transforms. t 0 is an average length of an uninterrupted time segment

18 The line absorption coefficient is the weighted average: 0 sinπδt(ν ν Δt 2 0 ) πδt(ν ν 0 ) 2 e Δt/Δt 0 dδt Δt 0 α = C 4π 2 (ν ν 0 ) 2 + (1/Δt 0 ) 2 γ α = C n /4π (ν ν0 ) 2 + (γ n /4π) 2 In other words this is the Lorentzian. To use this in a line profile calculation need to evaluate γ n = 2/Δt 0. This is a function of depth in the stellar atmosphere.

19 Evaluation of γ 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. φ = 2π 0 Δν dt= 2π C n R n dt 0 Δν = C n /R n y v Assume perturber moves past atom in a straight line ρ = R cos θ Atom θ ρ x φ = 2π C cos n θ dt n ρ n 0 R Perturber

20 Evaluation of γ n v = dy/dt = (ρ/cos 2 θ) dθ/dt => dt = (ρ/v)dθ/cos 2 θ φ = 2πC n vρ n 1 π/2 π/2 cos n 2 θ dθ Usually define a limiting impact parameter for a significant phase shift φ = 1 rad n 2 π 3 2 π/2 cos n 2 θ dθ π/2 4 π/2 6 3π/8 ρ 0 = 2πC n v π/2 π/2 1/(n 1) cos n 2 θ dθ ρ 0 is an average impact parameter and we count only those with ρ < ρ 0

21 Evaluation of γ n The number of collisions is πρ 2 0vNT where N is the number of perturbers per unit volume, T is the interval of the collisions. If we set T = Δt 0, the average length of an uninterupted segment a photon will travel. Over this length the number of collisions should be 1. πρ 0 2 vnδt 0 1 γ n = 2/Δt 0 = 2πρ 02 vn

22 Evaluation of γ n : Quadratic Stark In real life you do not have to calculate γ n For quadratic Stark effect (perturbations by charged particles) γ 4 = 39v ⅓ C 4⅔ N Values of the constant C 4 has been measured only for a few lines Na 5890 Å log C 4 = Mg 5172 Å log C 4 = Mg 5552 Å log C 4 = 13.12

23 Evaluation of γ n For van der Waals (n=6) you only have to consider neutral hydrogen and helium log γ log C 6 (H) + log P g 0.7 log T log C 6 = 31.7

24 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.

25 Thermal Broadening Thermal motion results in a component of the thermal motion along the line of sight Δλ λ = Δν ν = v r c v r = radial velocity We can use the Maxwell Boltzmann distribution N dn N = 1 v 0 π ½ exp [ ( v r v 0 ( 2 [ dv r 1.18σ variance v 0 2 = 2kT/m v Velocity

26 Thermal Broadening The Doppler wavelength shift v 0 Δλ D = λ = c λ 2kT ½ ( ( c m c c ( v 0 Δν D = ν = ν 2kT m ( ½ dn N [ = π ½ exp ( Δλ Δλ D ( 2 [ d ( Δλ Δλ D ( The energy removed from the intensity is (πe 2 f/mc)(λ 2 /c) times dn/n α dλ = π ½ e 2 mc f λ2 c [ 1 Δλ exp ( D Δλ Δλ D ( 2 [ dλ

27 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 = γ natural + γ 4 + γ 6 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 /4π 2 Δν 2 + (γ/4π) 2 * 1 π ½ e (Δν/Δν D )2 Lorentzian Gaussian

28 The convolution is the Voigt Function

29 The Combined Absorption Coefficient α = π½ e 2 mc f Δν D H(u,a) H(u,a) is the Hjerting function u = Δν/Δν D = Δλ/Δλ D a = γ 4π 1 Δν D = γ 4π 2 λ 0 c 1 Δλ D H(u,a) = γ /4π 2 (Δν Δν1 ) 2 + (γ/4π) 2 e (Δν 1 /Δν )2 D dν 1 H(u,a) = a π 2 e u 1 (u u1 ) 2 + a 2 du 1

30 The absorption coefficient can be calculated using the series expansion: H(u,a) = H 0 (u) + ah 1 (u) + a 2 H 2 (u) + a 3 H 3 (u) + a 4 H 4 (u) +

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32 Hjerting function tabulated in Gray

33 The Line Transfer Equation dτ ν = (l ν + κ ν )ρdx l ν = line absorption coefficient κ ν = continuum absorption coefficient Source function: S ν = l c j ν + j ν l ν + κ ν l j ν = line emission coefficient c j ν = continuum emission coefficient di ν dτ ν = I ν + S ν This now includes spectral lines

34 Using the Eddington approximation S(τ) = 3F ν 4π (τ + ⅔) At τ ν = (4π 2)/3 = τ 1, S ν (τ 1 ) = F ν (0), the surface flux and source function are equal

35 Across a stellar line l ν 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, l ν decreases and the condition that τ ν = τ 1 is deeper in the atmosphere. An absorption line is formed because the source function decreases outward.

36 Computing the Line Profile In local thermodynamic equilibrium the source function is the Planck function F = 2π B ν (T) E 2 (τ ν )dτ ν 0 = 2π B ν (τ ν ) E 2 (τ ν ) dτ ν dτ 0 dτ 0 0 = 2π B ν (τ ν ) E 2 (τ ν ) l ν + κ ν κ 0 τ 0 dlog τ 0 log e

37 Computing the Line Profile To compute τ ν τ ν (τ 0 ) = log t 0 l ν + κ ν κ 0 t 0 dlog t 0 log e Optical depth without l ν Optical depth with l ν F c F ν F c = S ν (τ c =τ 1 ) S ν (τ ν = τ 1 ) S ν (τ c =τ 1 ) Take the optical depth and divide it into two parts, continuum and line τ ν = 0 τ 0 l ν κ 0 dt τ 0 κ 0 dt 0 τ 0 κν τ ν = τ l + τ c

38 Computing the Line Profile τ l τ c l ν κ 0 τ 0 κ ν κ 0 τ 0 Ignoring the change with depth We need S ν (τ ν = τ 1 ) = S ν (τ l + τ c = τ 1 ) = S ν (τ c = τ 1 τ l ) We are considering only weak lines so τ l << τ c and evaluate S ν at τ 1 τ l using a Taylor expansion around τ c = τ 1 S ν (τ ν = τ 1 ) S ν (τ c = τ ν ) + ds ν dτ ν ( τ l )

39 Computing the Line Profile F c F ν F c = τ l S ν (τ c =τ 1 ) ds ν dτ c = τ l dlns ν dτ c τ 1 l ν τ c dlns ν dτ c κ ν τ 1 = C l ν κ ν Weak lines Mimic shape of l ν i.e. a Voigt profile Strength of spectral line can be increased either by decreasing the continuous absorption or increasing the line strength

40 Contribution Functions F ν = 2π B ν (τ ν ) E 2 (τ ν ) l ν + κ ν κ 0 τ 0 dlog τ 0 log e Contribution function How does this behave with line strength and position in the line?

41 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

42 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. In stellar atmosphere: z + 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

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

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

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

46 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.

47 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

48 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

49 Temperature Dependence Example: Cool star where κ ν behaves like the negative hydrogen ion s boundfree absorption: κ ν = constant T 5/2 P e e 0.75/kT 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

50 Behavior of Spectral Lines: Temperature Dependence Case #1: The number of absorbers in level l is given by : N l = constant N 0 e χ/kt constant e χ/kt The number of neutrals N 0 is approximately constant with temperature until ionization occurs because the number of ions N i is small compared to N 0. Ratio of line to continuous absorption is: R = l ν κ ν = constant T 5/2 P e e (χ+0.75)/kt

51 Behavior of Spectral Lines: Temperature Dependence Recall that P e = constant e ΩT ln R = constant ln T χ kt ΩT 1 R dr 2.5 T + χ dt = kt 2 ΩT

52 Behavior of Spectral Lines: Temperature Dependence Exercise for the reader: Case 2 (neutral line, element ionized): 1 R dr dt = χ I kt 2 Case 3 (ionic line, element neutral): 1 dr 5 = + R dt T χ I kt 2 2ΩT Case 4 (ionic line, element ionized): 1 dr 2.5 = + R dt T χ kt 2 ΩT

53 Behavior of Spectral Lines: Temperature Dependence Exercise for the reader: Case 2 (neutral line, element ionized): 1 R dr dt = χ I kt 2 Case 3 (ionic line, element neutral): 1 dr 5 = + R dt T χ I kt 2 2ΩT Case 4 (ionic line, element ionized): 1 dr 2.5 = + R dt T χ kt 2 ΩT

54 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

55 Behavior of Hydrogen lines with temperature B3IV B9.5V A0 V G0V F0V 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

56 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 P g constant P e 2 P g constant g ⅔ P e constant g ⅓ 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

57 Example of change in line strength with gravity

58 Example of change in wings due to gravity

59 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.

60 Rule #1 Ionization equation: N r+1 N r = Φ j (T) P e By rule one the line is formed in the rth ionization stage, but most of the element is in the N r+1 ionization stage: N r+1 N total N r constant P e l ν constant N r constant P e The line absortion coeffiecient is proportional to the number of absorbers The continous opacity from the negative hydrogen ion dominates: κ ν = constant T 5/2 P e e 0.75/kT l ν κ ν is independent of P e

61 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 N r N total l ν κ ν = constant P e constant g ⅓ Note: this change is not caused by a change in l, but because the continuum opacity of H becomes less as P e decreases Also note: log(l ν /κ ν )/ log g = 0.33 Proof of rule #3 similar. In solar-type stars cases 1) and 2) are mostly encountered

62 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 Δλ D. 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 κ ν. The strength depends on g, but for constant g the equivalent width is proportional to A ½ The graph specifying the change in equivalent width with abundance is called the Curve of Growth

63 Behavior of Spectral Lines: Abundance Dependence Assume that lines are formed in a cool gas above the source of the continuum F ν = F c e τ ν Fc is continuum flux L τ ν = l ν ρdx 0 L = Nα dx L is the thickness of the cool gas. 0 N/ρ = number of absorbers per unit mass N ρ = N N E N E N H N H ρ N/N E is the fraction of element E capable of absorbing, N E /N H is the number abundance A, N H /ρ is the number of hydrogen atoms per unit mass τ ν = A L 0 (N/N E )N h α dx τ ν is proportional to the abundance A and the flux varies exponentially with A

64 Behavior of Spectral Lines: Abundance Dependence For weak lines τ ν << 1 F ν F c (1 τ ν ) F c F ν F c τ ν line depth is proportional τ ν and thus A. The line depth and thus the equivalent width is proportional to A

65 Behavior of Spectral Lines: Abundance Dependence What about strong lines? The wings dominate so α = π½ e 2 mc f Δν D H(u,a) α = πe2 mc γ 4π 2 f Δν D τ ν = A L 0 (N/N E )N h α dx = πe2 mc A f Δν 2 L (N/N E )N H 0 γ 4π 2 dx <γ> A f h Δν 2 <γ> denotes the depth average damping constant, and h is the constants and integral

66 F c F ν F c = 1 e τ ν The equivalent width of the line: W = (1 e τ ν) dν 0 W = (1 e <γ>αf h/δν2 )dν 0 Substituting u 2 = Δν 2 /<γ>a f h W = (<γ>a f h) ½ (1 e 1/ u 2 ) du 0 Equivalent width is proportional to the square root of the abundance

67 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!!!