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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.

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The Formation of Spectral Lines

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

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

  3. The classical model of the interaction of light with a photon is a plane electromagnetic wave interacting with a dipole. ∂2E = v2 ∂ x2 Treat only one frequency since by Fourier composition the total field is a sum of all sine waves. ∂2E E = E0 e–iw(x/v–t) ∂ t2 The wave velocity through a medium ½ e0 m0 e and m are the electric and magnetic permemability in the medium and free space. For gases m = m0 ( v=c ( e m Line Absorption Coefficient

  4. e E + 4pNqz 4pNqz e0 E E 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 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 = = 1 + We need z/E

  5. d2z Solution: z = z0e–iwt dt2 E e 2 2 w0 – w2 + igw w0 – w2 + igw E0 eiwt e m z = = m Line Absorption Coefficient For a damped harmonic oscillator where z is the induced separation between the dipole charges dz e 2 E0 eiwt + w0 = g + dt m e,m are charge and mass of electron g is damping constant

  6. 1 e e0 4pNe2 The wave velocity can now be written as E ½ 4pNe2 1 e ( c 1 1 + ( ≈ m e0 ≈ 2 2 w0 – w2 + igw w0 – w2 + igw v 2 Where we have performed a Taylor expansion (1 + x) = 1 + ½ x for small x Line Absorption Coefficient 1 + = For a gas e ≈ e0 so second term is small compared to unity ½

  7. 2 w0– w2 2pNe2 m gw 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 c ≈ = I0e–lnrx 2 2 I = I0 e–kwx/c (w0 – w2)2 + g2w2 (w0 – w2)2 + g2w2 v Line Absorption Coefficient 1 +

  8. 4pNe2 gw lnr = mc This function is sharply peaked giving non-zero values when w ≈ w0 w0– w =(w0 – w)(w0 + w) ≈ (w0– w)2w≈ 2wDw The basic form of the line absorption coefficient: 2 (w0 – w2)2 + g2w2 This is a damping profile or Lorentzian profile, a Cauchy curve, or the „Witch of Agnesi“ Npe2 gw lnr = mc Dw2 + (g/2)2 Line Absorption Coefficient 2 2

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

  10. ∞ ∫ = a dn 0 ∞ f ∫ = a dn 0 pe2 pe2 mc mc Line Absorption Coefficient This is energy per unit atom per square radian that the line absorbs from In A quantum mechanical treatment f is the oscillator strength and is related to the transition probability Blu ∞ ∫ Blu hn = a dn 0

  11. Blu 7.484 × 10–7 l gu gl Line Absorption Coefficient mc f = = Blu hn pe2 mc3 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 gfuntil you match the solar line. This value is then good for other stars.

  12. dW e2 w2 –g – W = = W dt mc3 2 3 The Damping Constant for Natural Broadening Classical dipole emission theory gives an equation of the form 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

  13. 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

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

  15. 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

  16. 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

  17. ∞ ∫ 0 sinpDt(n– n0) pDt(n– n0) The line absorption coefficient is the weighted average: 2 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.

  18. ∞ ∞ ∞ ∫ ∫ ∫ 0 0 0 2p = Cn R–n dt dt f 2p = Cn cos q rn Evaluation of gn 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. Dn = Cn /Rn f = 2p Dn dt v y Assume perturber moves past atom in a straight line Atom r r = R cos q x q R n Perturber

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

  20. Evaluation of gn 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

  21. 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

  22. Evaluation of gn For van der Waals (n=6) you only have to consider neutral hydrogen and helium log g6 ≈ 19.6 + 0.4 log C6(H) + log Pg– 0.7 log T log C6 = –31.7

  23. 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.

  24. vr dN Dl Dn = = l n N c vr We can use the Maxwell Boltzmann distribution v0 [ 1 [ dvr = exp v0p½ 2 variance v02 = 2kT/m ( ( – v Thermal Broadening Thermal motion results in a component of the thermal motion along the line of sight vr = radial velocity N 1.18s Velocity

  25. 2kT 2kT l n DlD = l = ( m m c c c dN N v0 ½ ( DnD = n = ( c The energy removed from the intensity is (pe2f/mc)(l2/c) times dN/N 2 [ ( [ p–½ exp – ( = p½e2 Dl 1 ( l2 ( f dl d a dl = DlD DlD mc c Dl Dl 2 [ DlD DlD ( ( [ exp – Thermal Broadening The Doppler wavelength shift v0 ½ (

  26. 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: pe2 • /4p2 1 e–(Dn/DnD)2 a = f * mc p½ Dn2 + (g/4p)2 Lorentzian Gaussian 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)

  27. The convolution is the Voigt Function

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

  29. The absorption coefficient can be calculated using the series expansion: H(u,a) = H0(u) + aH1(u) + a2H2(u) + a3H3(u) + a4H4(u) +

  30. Hjerting function tabulated in Gray

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

  32. 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

  33. 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, lndecreases and the condition that tn = t1 is deeper in the atmosphere. An absorption line is formed because the source function decreases outward.

  34. ∞ ∞ ∞ ∫ ∫ ∫ 0 0 –∞ dtn dt0 dlog t0 ln + kn 2p Bn(tn)E2(tn) = t0 k0 log e Computing the Line Profile In local thermodynamic equilibrium the source function is the Planck function F= 2p Bn(T)E2(tn)dtn dt0 2p Bn(tn)E2(tn) =

  35. Fc– Fn Sn(tc=t1) – Sn(tn = t1) = Fc Sn(tc=t1) dlogt0 ln + kn t0 t0 kn k0 ∫ ln log e ∫ tn = dt0 + dt0 k0 k0 tl tc + 0 0 Computing the Line Profile To compute tn log t0 ∫ tn(t0)= Optical depth without ln t0 –∞ Optical depth with ln Take the optical depth and divide it into two parts, continuum and line t0 tn =

  36. kn ln k0 k0 dSn dtn Computing the Line Profile tl t0 ≈ Ignoring the change with depth 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 Sn(tn = t1) ≈ Sn(tc = tn) + (–tl)

  37. = tl t1 ln ln C = kn kn dSn dlnSn dlnSn dtc dtc dtc Computing the Line Profile Fc– Fn tl = Fc Sn(tc=t1) tc ≈ t1 • 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

  38. ln + kn dlog t0 t0 k0 log e ∞ ∫ –∞ Contribution Functions Bn(tn)E2(tn) Fn = 2p Contribution function How does this behave with line strength and position in the line?

  39. 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

  40. In stellar atmosphere: z + Radial node where amplitude =0 ─ 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. Conclusion: The two lines are formed on opposite sides of a radial node where the amplitude of the pulsations is zero

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

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

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

  44. 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.

  45. 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

  46. 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

  47. Temperature Dependence Example: Cool star where kn behaves like the negative hydrogen ion‘s bound-free absorption: kn = constant T–5/2Pee0.75/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

  48. ln kn T5/2 R = = constant e–(c+0.75)/kT Pe Behavior of Spectral Lines: Temperature Dependence 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:

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