5. Atomic radiation processes

Transcription

1 5. Atomic radiation processes Einstein coefficients for absorption and emission oscillator strength line profiles: damping profile, collisional broadening, Doppler broadening continuous absorption and scattering 1

2 2

3 A. Line transitions Einstein coefficients probability that a photon in frequency interval in the solid angle range is absorbed by an atom in the energy level E l with a resulting transition E l à E u per second: atomic property ~ no. of incident photons probability for absorption of photon with probability for with probability for transition l à u absorption profile B lu : Einstein coefficient for absorption 3

4 Einstein coefficients similarly for stimulated emission B ul : Einstein coefficient for stimulated emission and for spontaneous emission A ul : Einstein coefficient for spontaneous emission 4

5 Einstein coefficients, absorption and emission coefficients I ν df dv = df ds Number of absorptions & stimulated emissions in dv per second: ds absorbed energy in dv per second: and also (using definition of intensity): stimulated emission counted as negative absorption Absorption and emission coefficients are a function of Einstein coefficients, occupation numbers and line broadening for the spontaneously emitted energy: 5

6 Relations between Einstein coefficients Einstein coefficients are atomic properties à do not depend on thermodynamic state of matter We can assume TE: From the Boltzmann formula: for hν/kt << 1: for T à 0 6

7 Relations between Einstein coefficients for T à Note: Einstein coefficients atomic quantities. That means any relationship that holds in a special thermodynamic situation (such as T very large) must be generally valid. only one Einstein coefficient needed 7

8 Oscillator strength Quantum mechanics The Einstein coefficients can be calculated by quantum mechanics + classical electrodynamics calculation. Eigenvalue problem using using wave function: Consider a time-dependent perturbation such as an external electromagnetic field (light wave) E(t) = E 0 e iωt. The potential of the time dependent perturbation on the atom is: d: dipol operator with transition probability 8

9 Oscillator strength The result is f lu : oscillator strength (dimensionless) classical result from electrodynamics = cm 2 /s Classical electrodynamics electron quasi-elastically bound to nucleus and oscillates within outer electric field as E. Equation of motion (damped harmonic oscillator): ma = damping force + restoring force + EM force damping constant resonant (natural) frequency the electron oscillates preferentially at resonance (incoming radiation ν = ν 0 ) The damping is caused, because the de- and accelerated electron radiates 9

10 Classical cross section and oscillator strength Calculating the power absorbed by the oscillator, the integrated classical absorption coefficient and cross section, and the absorption line profile are found: Z integrated over the line profile L,cl º dº = n l ¼e 2 m e c = n l ¾ cl tot n l : number density of absorbers σ tot cl : classical cross section (cm 2 /s) '(º)dº = 1 ¼ /4¼ (º º 0 ) 2 + ( /4¼) 2 [Lorentz (damping) line profile] oscillator strength f lu is quantum mechanical correction to classical result (effective number of classical oscillators, 1 for strong resonance lines) From (neglecting stimulated emission) 10

11 Oscillator strength absorption cross section; dimension is cm 2 Oscillator strength (f-value) is different for each atomic transition Values are determined empirically in the laboratory or by elaborate numerical atomic physics calculations Semi-analytical calculations possible in simplest cases, e.g. hydrogen g: Gaunt factor Hα: f= Hβ: f= Hγ: f=

12 Line profiles line profiles contain information on physical conditions of the gas and chemical abundances analysis of line profiles requires knowledge of distribution of opacity with frequency several mechanisms lead to line broadening (no infinitely sharp lines) - natural damping: finite lifetime of atomic levels - collisional (pressure) broadening: impact vs quasi-static approximation - Doppler broadening: convolution of velocity distribution with atomic profiles 12

13 1. Natural damping profile finite lifetime of atomic levels à line width NATURAL LINE BROADENING OR RADIATION DAMPING t = 1 / A ul Δ E t h/2π (¼ 10-8 s in H atom 2 à 1): finite lifetime with respect to spontaneous emission uncertainty principle '(º) = 1 ¼ line broadening /4¼ (º º 0 ) 2 + ( /4¼) 2 Δν 1/2 = Γ / 2π Δλ 1/2 = Δν 1/2 λ 2 /c e.g. Ly α: Δλ 1/2 = A Hα: Δλ 1/2 = A Lorentzian profile 13

14 Natural damping profile resonance line excited line natural line broadening is important for strong lines (resonance lines) at low densities (no additional broadening mechanisms) e.g. Ly α in interstellar medium but also in stellar atmospheres 14

15 2. Collisional broadening radiating atoms are perturbed by the electromagnetic field of neighbour atoms, ions, electrons, molecules energy levels are temporarily modified through the Stark - effect: perturbation is a function of separation absorber-perturber energy levels affected à line shifts, asymmetries & broadening ΔE(t) = h Δν = C/r n (t) r: distance to perturbing atom a) impact approximation: radiating atoms are perturbed by passing particles at distance r(t). Duration of collision << lifetime in level à lifetime shortened à line broader in all cases a Lorentzian profile is obtained (but with larger total Γ than only natural damping) b) quasi-static approximation: applied when duration of collisions >> life time in level à consider stationary distribution of perturbers 15

16 Collisional broadening n = 2 linear Stark effect ΔE ~ F for levels with degenerate angular momentum (e.g. HI, HeII) field strength F ~ 1/r 2 à ΔE ~ 1/r 2 important for H I lines, in particular in hot stars (high number density of free electrons and ions). However, for ion collisional broadening the quasi-static broadening is also important for strong lines (see below) à Γ e ~ n e n = 3 resonance broadening atom A perturbed by atom A of same species important in cool stars, e.g. Balmer lines in the Sun à ΔE ~ 1/r 3 à Γ e ~ n e 16

17 Collisional broadening n = 4 quadratic Stark effect ΔE ~ F 2 field strength F ~ 1/r 2 à ΔE ~ 1/r 4 (no dipole moment) important for metal ions broadened by e - in hot stars à Lorentz profile with Γ e ~ n e n = 6 van der Waals broadening atom A perturbed by atom B important in cool stars, e.g. Na perturbed by H in the Sun à Γ e ~ n e 17

18 Quasi-static approximation t perturbation >> τ = 1/A ul à perturbation practically constant during emission or absorption atom radiates in a statistically fluctuating field produced by quasi-static perturbers, e.g. slow-moving ions given a distribution of perturbers à field at location of absorbing or emitting atom statistical frequency of particle distribution à probability of fields of different strength (each producing an energy shift ΔE = h Δν) à field strength distribution function à line broadening Linear Stark effect of H lines can be approximated to 0 st order in this way 18

19 Quasi-static approximation for hydrogen line broadening Line broadening profile function determined by probability function for electric field caused by all other particles as seen by the radiating atom. W(F) df: probability that field seen by radiating atom is F For calculating W(F)dF we use as a first step the nearest-neighbor approximation: main effect from nearest particle 19

20 Quasi-static approximation nearest neighbor approximation assumption: main effect from nearest particle (F ~ 1/r 2 ) we need to calculate the probability that nearest neighbor is in the distance range (r,r+dr) = probability that none is at distance < r and one is in (r,r+dr) Integral equation for W(r) differentiating à differential equation probability for no particle in (0,r) relative probability for particle in shell (r,r+dr) N: particle density Differential equation Normalized solution 20

21 Linear Stark effect: Quasi-static approximation mean interparticle distance: normal field strength: define: = F F 0 = { r 0 r }2 note: at high particle density à large F 0 à stronger broadening from W(r) dr à W(β) dβ: Stark broadened line profile in the wings, not Δλ -2 as for natural or impact broadening 21

22 Quasi-static approximation advanced theory complete treatment of an ensemble of particles: Holtsmark theory + interaction among perturbers (Debye shielding of the potential at distances > Debye length) Holtsmark (1919), Chandrasekhar (1943, Phys. Rev. 15, 1) W( ) = 2 ¼ Z 1 0 e y3/2 y sin( y)dy number of particles inside Debye sphere Ecker (1957, Zeitschrift f. Physik, 148, 593 & 149,245) W( ) = 2 ±4/3 ¼ Z 1 0 e ±g(y) y sin(± 2/3 y)dy g(y) = 2 Z 1 3 y3/2 (1 z 1 sinz)z 5/2 dz y D = 4.8 T n e 1/2cm Debye length, field of ion vanishes beyond D Mihalas, 78! ± = 4¼ 3 D3 N number of particles inside Debye sphere 22

23 3. Doppler broadening radiating atoms have thermal velocity Maxwellian distribution: Doppler effect: atom with velocity v emitting at frequency frequency :, observed at 23

24 Doppler broadening Define the velocity component along the line of sight: The Maxwellian distribution for this component is: thermal velocity if we observe at v, an atom with velocity component ξ absorbs at ν in its frame if v/c <<1 à line center 24

25 Doppler broadening line profile for v = 0 à profile for v 0 New line profile: convolution profile function in rest frame velocity distribution function 25

26 Doppler broadening: sharp line approximation : Doppler width of the line thermal velocity Approximation 1: assume a sharp line half width of profile function << delta function 26

27 Doppler broadening: sharp line approximation Gaussian profile valid in the line core 27

28 Doppler broadening: Voigt function Approximation 2: assume a Lorentzian profile half width of profile function > '(º) = 1 ¼ /4¼ (º º 0 ) 2 + ( /4¼) 2 Voigt function (Lorentzian * Gaussian): calculated numerically 28

29 Voigt function: core Gaussian, wings Lorentzian normalization: usually α <<1 max at v=0: H(α,v=0) 1-α ~ Gaussian ~ Lorentzian Unsoeld, 68! 29

30 Doppler broadening: Voigt function Approximate representation of Voigt function: line core: Doppler broadening line wings: damping profile only visible for strong lines General case: for any intrinsic profile function (Lorentz, or Holtsmark, etc.) the observed profile is obtained from numerical convolution with the different broadening functions and finally with Doppler broadening 30

31 General case: two broadening mechanisms two broadening functions representing two broadening mechanisms Resulting broadening function is convolution of the two individual broadening functions 31

32 4. Microturbulence broadening In addition to thermal velocity: Macroscopic turbulent motion of stellar atmosphere gas within optically thin volume elements. This is approximated by an additional Maxwellian velocity distribution. Additional Gaussian broadening function in absorption coefficient 32

33 blueshift 5. Rotational broadening If the star rotates, some surface elements move towards the observer and some away redshift blueshift redshift Gray, 1992 to observer Rotational velocity at equator: v rot = Ω R Observer sees v rot sini = v rotsin(i) x =+ v rotsin(i) x c R c R stripes of constant wavelength shift Intensity shifts in wavelength along stripe 33

34 Rotation and observed line profile stellar spectroscopy uses mostly normalized spectra à F λ / F continuum integral over stellar disk towards observer, also integral over all solid angles for one surface element observed stellar line profile 34

35 Spring 2013 F λ /F continuum M33 A supergiant Keck (ESI) U, Urbaneja, Kudritzki, 2009, ApJ 740,

36 Rotation changes integral over stellar surface stellar spectroscopy uses mostly normalized spectra à F λ / F continuum Gray, 1992 integral over stellar disk towards observer = v rotsin(i) c x R =+ v rotsin(i) c x R observed stellar line profile Correct treatment by numerical integral over stellar surface with intensity calculated by model atmosphere 36

37 Approximation: assume intrinsic P λ const. over surface independent of cosθ P ( )=I(,cos )/I c (,cos ) integral of Doppler shifted profile over stellar disk and weighted by continuum intensity towards obs. = v rotsin(i) c x R = v rotsin(i) c x R =+ v rotsin(i) c x R continuum limb darkening 37

38 P rot ( )= F ( ) F c P rot ( )= R 1 1 P [ ( x)]a( x)d x R 1 1 A( x)d x A( x) = Z p 1 x 2 0 (1 + p 1 x 2 +ỹ 2 )dỹ P rot ( )= G[ ( x] = Z 1 1 P [ ( x)]g[ ( x)]d x p 2 1 x2 + 2 (1 x 2 ) using Z pa 2 x 2 dx = 1 2 (xp a 2 x 2 + a 2 arcsin x a ) and normalization à rotational line broadening profile function 38

39 Rotational line broadening G[ ( x] = 2 p 1 x2 + 2 (1 x 2 ) = v rotsin(i) c x R convolution of original profile with rotational broadening function Rotationally broadened line profile 39

40 Rotational broadening profile function G(x) Unsoeld,

41 Rotationally broadened line profiles Note: rotation does not change equivalent width!!! v sini km/s Gray,

42 observed stellar line profiles v sini large v sini small Gray,

43 Rotationally broadened line profiles Note: rotation does not change equivalent width!!! v sini km/s Gray,

44 observed stellar line profiles θ Car, B0V vsini~250 km/s Schoenberner, Kudritzki et al

45 6. Macro-turbulence The macroscopic motion of optically thick surface volume elements is approximated by a Maxwellian velocity distribution. This broadens the emergent line profiles in addition to rotation 45

46 Rotation and macro-turbulence rotation rotation + macro-turb. 5 Å Gray,

47 B. Continuous transitions 1. Bound-free and free-free processes absorption hν + E l à E u emission spontaneous E u à E l + hν (isotropic) photoionization - recombination! hν lk bound-bound: spectral lines stimulated hν + E u à E l +2hν (non-isotropic) bound-free free-free κ ν cont consider photon hν hν lk (energy > threshold): extra-energy to free electron e.g. Hydrogen R = Rydberg constant = cm -1 47

48 b-f and f-f processes Hydrogen l à continuum Wavelength (A) Edge 1 à continuum 912 Lyman 2 à continuum 3646 Balmer 3 à continuum 8204 Paschen 4 à continuum Brackett 5 à continuum Pfund 48

49 b-f and f-f processes - for a single transition - for all transitions: Gaunt factor 1 for hydrogenic ions Kramers 1923 Gray, 92! for H: σ 0 = n cm 2 ν l = / n 2 Hz absorption per particle à 49

50 b-f and f-f processes Gray, 92! late A peaks increase with n: late B for non-h-like atoms no ν -3 dependence peaks at resonant frequencies free-free absorption much smaller 50

51 b-f and f-f processes Hydrogen dominant continuous absorber in B, A & F stars (later stars H - ) Energy distribution strongly modulated at the edges: Balmer Paschen Vega Brackett 51

52 b-f and f-f processes : Einstein-Milne relations Generalize Einstein relations to bound-free processes relating photoionizations and radiative recombinations line transitions b-f transitions stimulated b-f emission 52

53 2. Scattering In scattering events photons are not destroyed, but redirected and perhaps shifted in frequency. In free-free process photon interacts with electron in the presence of ion s potential. For scattering there is no influence of ion s presence. in general: κ sc = n e σ e Calculation of cross sections for scattering by free electrons: - very high energy (several MeV s): Klein-Nishina formula (Q.E.D.) - high energy photons (electrons): Compton (inverse Compton) scattering - low energy (< 12.4 kev ' 1 Angstrom): Thomson scattering 53

54 Thomson scattering THOMSON SCATTERING: important source of opacity in hot OB stars independent of frequency, isotropic Approximations: - angle averaging done, in reality: σ e à σ e (1+cos 2 θ) for single scattering - neglected velocity distribution and Doppler shift (frequency-dependency) 54

55 Simple example: hot star -pure hydrogen atmosphere total opacity Total opacity line absorption bound-free free-free Thomson scattering 55

56 Simple example: hot star -pure hydrogen atmosphere total emissivity Total emissivity line emission bound-free free-free Thomson scattering 56

57 Rayleigh scattering important in cool stars RAYLEIGH SCATTERING: line absorption/emission of atoms and molecules far from resonance frequency: ν << ν 0 from classical expression of cross section for oscillators: important in cool G-K stars for strong lines (e.g. Lyman series when H is neutral) the λ -4 decrease in the far wings can be important in the optical 57

58 What are the dominant elements for the continuum opacity? - hot stars (B,A,F) : H, He I, He II - cool stars (G,K): the bound state of the H - ion (1 proton + 2 electrons) only way to explain solar continuum (Wildt 1939) The H - ion - important in cool stars ionization potential = ev à λ ion = Angstroms H - b-f peaks around 8500 A H - f-f ~ λ 3 (IR important) He - b-f negligible, He - f-f can be important in cool stars in IR requires metals to provide source of electrons dominant source of b-f opacity in the Sun Gray, 92! 58

59 Additional absorbers Hydrogen molecules in cool stars H 2 molecules more numerous than atomic H in M stars H + 2 absorption in UV H - 2 f-f absorption in IR Helium molecules He - f-f absorption for cool stars Metal atoms in cool and hot stars (lines and b-f) C,N,O, Si, Al, Mg, Fe, Ti,. Molecules in cool stars TiO, CO, H 2 O, FeH, CH 4, NH 3, 59

60 Examples of continuous absorption coefficients Unsoeld, 68! T eff = 5040 K B0: T eff = 28,000 K 60

61 Modern model atmospheres include millions of spectral lines (atoms and ions) all bf- and ff-transitions of hydrogen helium and metals contributions of all important negative ions molecular opacities (lines and continua) 61

62 Concepcion 2007 complex atomic models for O-stars (Pauldrach et al., 2001) 62

63 NLTE Atomic Models in modern model atmosphere codes lines, collisions, ionization, recombination Essential for occupation numbers, line blocking, line force Accurate atomic models have been included 26 elements 149 ionization stages 5,000 levels ( + 100,000 ) 20,000diel. rec. transitions b-b line transitions Auger-ionization recently improved models are based on Superstructure Eisner et al., 1974, CPC 8,270 AWAP 05/19/05 63

64 Concepcion

65 Recent Improvements on Atomic Data requires solution of Schrödinger equation for N-electron system efficient technique: R-matrix method in CC approximation Opacity Project Seaton et al. 1994, MNRAS, 266, 805 IRON Project Hummer et al. 1993, A&A, 279, 298 accurate radiative/collisional data to 10% on the mean 65

66 Confrontation with Reality Photoionization Electron Collision! Nahar 2003, ASP Conf. Proc.Ser. 288, in press Williams 1999, Rep. Prog. Phys., 62, 1431! ü high-precision atomic data ü 66

67 Improved Modelling for Astrophysics e.g. photoionization cross-sections for carbon model atom! Przybilla, Butler & Kudritzki 2001b, A&A, 379, 936! 67

68 Red supergiants, NLTE model atom for FeI! Bergemann, Kudritzki et al.,

69 Red supergiants, NLTE model atom for TiI! Bergemann, Kudritzki et al.,

70 TiII Bergemann, Kudritzki et al., 2012 TiI 70

71 USM 2011 TiO in red supergiants! MARCS model atmospheres, Gustafsson et al., 2009! 71

72 USM 2011 TiO in red supergiants! A small change in! carbon abundances!! MARCS model! atmospheres! 72

73 CO molecule in red supergiants The Scutum RSG clusters Davies, Origilia, Kudritzki et al.,

74 Rayner, Cushing, Vacca, 2009: molecules in Brown Dwarfs 74

75 Exploring the substellar temperature regime down to ~550K Burningham et al. (2009) Jupiter T9.0 ~ 550K T8.5 ~ 700K 75