talk start talk files all talks SUMMARY OF SOLAR SPECTRUM FORMATION
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talk start talk files all talks SUMMARY OF SOLAR SPECTRUM FORMATION Rob Rutten, Utrecht University & Oslo University Jun 21 2007, Oslo Rob Rutten: “Radiative Transfer in Stellar Atmospheres” (RTSA) Michael Stix: “The Sun”, 2nd edition, Springer 2004 white light corona examples: local – nonlocal – converted photons EUV corona EUV bright/dark [Zanstra & Bowen PN lines] radiative transfer basics basic quantities LTE solar radiation escape Schadeenium in LTE simple emission line solar spectrum formation constant Sν Planck H-minus in LTE LTE lines solar ultraviolet spectrum Ca II H versus Halpha radiative transfer in stellar atmospheres: math thick transfer Boltzmann-Saha simple absorption line VALIIIC model [RTSA 3.4; 4.1 – 4.2] radiative transfer in stellar atmospheres: scattering [RTSA 4.3] bb processes bb rates NLTE solar radiation escape solar radiation processes VAL3C continuum formation VAL3C radiation budget realistic absorption line Na D1 Ca II H from RADYN chromosphere summary talk start coronium lines bb equilibria radiative cooling Na D1 blends RTSA rap talk files WHITE LIGHT CORONA all talks Stix section 9.1.3 Grotrian (1931): Thomson scattering 8000 Z km s−1 electrons (S 9.2–9.3) Z ∞ ∞ ρ j(ρ) p j(ρ) dy = 2 dρ ρ2 = x 2 + y 2 I(x) = 2 ρ2 − x 2 x 0 Ne from inverse Abel transform Z= isotropically scattered irradiation (S 9.4–9.5) Z 1 ∞ dI/dx 1 p j(ρ) = − dx = σT Ne I (θ) dΩ π ρ 4π x 2 − ρ2 talk start talk files “CORONIUM” LINES Stix section 9.1.3 all talks http://laserstars.org/spectra/Coronium.html Grotrian, Edlén 1942: forbidden lines high ionization stages (Stix Table 9.2 p. 398) name wavelength identification ∆λD v Aul previous ion green line yellow line red line 530.29 nm 569.45 637.45 [Fe XIV] [Ca XV] [Fe XI] 0.051 nm 0.087 0.049 29 km/s 60 s−1 46 95 23 69 Fe XIII Ca XIV Fe IX χion 355 eV 820 235 Coronal sky at Dome C talk start talk files EUV CORONA all talks Stix section 9.1.3 bf equilibria: collisional ionization = radiative recombination ⇒ only f (T ), not f (Ne ) bb equilibria: collisional excitation = spontaneous deexcitation ⇒ f (T, Ne ) (S 9.9–9.10) −1 Z ∞ Z Z X dT 2 dT ≡ EM nl Clu = nl Ne hν ∝ nion Ne dz = Ne σlu f (v) v dv ≈ nu Aul dz v0 talk start talk files all talks BRIGHT AND DARK IN EUV IMAGES • iron lines – Fe IX/X 171 Å: about 1.0 MK – Fe XII 195 Å: about 1.5 MK – Fe XIV 284 Å: about 2 MK • bright – collision up, radiation down – thermal photon creation, NLTE equilibrium – 171 Å: selected loops = special trees in forest • dark – radiation up, re-radiation at bound-free edge – matter containing He+ , He, H: 104 – 105 K – large opacity talk start talk files all talks BASIC QUANTITIES RTSA 2.1–2.2, Stix 4.2.5 Monochromatic emissivity (Stix uses ε) dEν ≡ jν dV dt dν dΩ units jν : erg cm−3 s−1 Hz−1 ster−1 dIν (s) = jν (s) ds Iν : erg cm−2 s−1 Hz−1 ster−1 Monochromatic extinction coefficient dIν ≡ −σν n Iν ds dIν ≡ −αν Iν ds units: per particle (physics) per cm (RJR lectures) dIν ≡ −κν ρ Iν ds per gram (astronomy) Monochromatic source function Sν ≡ jν /αν = jν /κν ρ Sνtot P jν =P αν Sνtot = thick: (κν , Sν ) more independent than (κν , jν ) Sνc + ην Sνl jνc + jνl = ανc + ανl 1 + ην stimulated emission negatively into αν , κν Transport equation along the beam with τν as optical thickness dIν dIν = jν − α ν I ν = Sν − I ν dτν ≡ αν ds ds αν ds Plane-parallel transport equation with τν as radial Z optical depth z0 dτν ≡ −κν ρ dz τν (z0 ) = − ην ≡ ανl /ανc κν ρ dz ∞ µ dIν = Sν − I ν dτν dIν = I ν − Sν dτν talk start talk files all talks CONSTANT SOURCE FUNCTION Transport equation along the beam (τν = optical thickness)Z τν dIν = Sν − I ν Iν (τν ) = Iν (0) e−τν + Sν (tν ) e−(τν −tν ) dtν dτν 0 Invariant Sν Iν (D) = Iν (0) e−τν (D) + Sν 1 − e−τν (D) example: Sν = Bν for all continuum and line processes in an isothermal cloud Thick object Iν (D) ≈ Sν Thin object Iν (D) ≈ Iν (0) + [Sν − Iν (0)] τν (D) talk start Iν (0) = 0 : Iν (D) ≈ τν (D) Sν = jν D talk files all talks THICK TRANSFER RTSA 2.2.2; Stix 4.1.1 Radial optical depth dτν = −κν ρ dr r radial κν cm2 /gram Hubený τνµ Transport equation µ αν cm−1 = cm2 /cm3 dIν = I ν − Sν dτν Integral form −(τ0−τν )/µ Iν (τν , µ) = Iν (τ0 , µ) e “formal solution” + (1/µ) both directions Emergent intensity Iν (0, µ) = (1/µ) Eddington-Barbier approximation Z Z τ0 0 τν Sν (τν0 ) e−(τν−τν )/µ dτν0 pm: Doppler anisotropy Sν ∞ Sν (τν ) e−τν /µ dτν 0 Iν (0, µ) ≈ Sν (τν = µ) exact for linear Sν (τν ) σν cm2 /particle talk start talk files PLANCK all talks RTSA Chapt. 2; SSA 3; Stix 4.5 Planck: Bν (T ) = Wien: 2hν 3 −hν/kT e c2 2ν 2 kT exp(hν/kT ) 1 : Bν (T ) ≈ c2 exp(hν/kT ) 1 : Bν (T ) ≈ Rayleigh-Jeans: talk start 2hν 3 1 2 hν/kT c e −1 talk files SOLAR VISIBLE AND INFRARED RTSA Chapt. 8; SSB 2 all talks talk start talk files SOLAR VISIBLE AND INFRARED all talks RTSA Chapt. 8; SSB 2 talk start talk files SOLAR VISIBLE AND INFRARED RTSA Chapt. 8; SSB 2 all talks talk start talk files SOLAR VISIBLE AND INFRARED all talks RTSA Chapt. 8; SSB 2 talk start talk files SOLAR VISIBLE AND INFRARED RTSA Chapt. 8; SSB 2 all talks talk start talk files SOLAR VISIBLE AND INFRARED all talks RTSA Chapt. 8; SSB 2 talk start talk files Saha–Boltzmann level populations Boltzmann distribution per ionization stage: partition function: Ur ≡ X all talks nr,s gr,s −χr,s /kT = e Nr Ur gr,s e−χr,s /kT s Saha distribution over ionization stages: 3/2 1 2 Ur+1 2πme kT Nr+1 e−χr /kT = 2 Nr Ne Ur h talk start talk files all talks talk files all talks Saha–Boltzmann Schadeenium talk start LTE LINES RTSA Chapt. 2; Stix 4.1.3 continuum optical depth scale ην ≡ κl /κC dτν = dτC + dτl = (1 + ην ) dτC emergent intensity Z at disk center in LTE Z ∞ ∞ Bν exp(−τν ) dτν = Iν (0, 1) = 0 0 Z (1 + ην ) Bν exp − τC (1 + ην ) 0 dτC0 dτC Eddington-Barbier: Iν (0, 1) ≈ Bν [τν = 1] = Bν [τC = 1/(1+ην )] extinction coeffi ⊗ damping Lorentz r cient shape = Maxwell Gauss Z +∞ γ 1 exp(−(ν −ν 0 )2 /∆νD2 ) 0 ν0 2RT φ(ν) = √ + ξt2 dν = √ H(a, v) ∆νD ≡ 0 2 2 c A π∆νD −∞ [2π(ν −ν0 )] + γ /4 π∆νD Voigt function ν −ν0 v≡ ∆νD γ a≡ 4π∆νD a H(a, v) ≡ π Z +∞ −∞ line extinction coefficient hν πe2 e2 fl √ σl = Blu φ(ν) = fl φ(ν) = H(a, v) 4π me c 40 me c π∆νD κl = σl nlLTE ρ (1−e−hν/kT ) = 2 e−y dy (v − y)2 + a2 σl n nij nijk Pi (1−e−hν/kT ) µmH ni ni nij Aul = 6.67×1013 area √ π gl flu −1 s (λ nm) g u λ2 i, j, k : species, stage, lower level talk start talk files SIMPLE ABSORPTION LINE all talks • extinction: bb process gives peak in κtotal = κC + κl = (1 + ην ) κC • optical depth: height-invariant κtotal ⇒ linear (1 + ην ) τC • source function: same for line (bb) and continous (bf, ff, electron scattering) processes • intensity: Eddington-Barbier (nearly) exact talk start talk files SIMPLE EMISSION LINE all talks • extinction: bb process gives peak in κtotal = κC + κl = (1 + ην ) κC • optical depth: height-invariant κtotal ⇒ linear (1 + ην ) τC • source function: same for line (bb) and continous (bf, ff, electron scattering) processes • intensity: Eddington-Barbier (nearly) exact talk start talk files SOLAR ULTRAVIOLET SPECTRUM all talks Scheffler & Elsässer, courtesy Karin Muglach talk start talk files VALIIIC MODEL Vernazza, Avrett, Loeser 1981ApJS...45..635V all talks talk start talk files SOLAR SPECTRUM FORMATION all talks Avrett 1990IAUS..138....3A talk start talk files CaÍI H & Hα in LTE all talks Leenaarts et al. 2006A&A...449.1209L http://dot.astro.uu.nl talk start talk files all talks BOUND-BOUND PROCESSES AND EINSTEIN COEFFICIENTS Spontaneous deexcitation Aul ≡ transition probability for spontaneous deexcitation from state u to state l per sec per particle in state u Radiative excitation ϕ Blu J ν0 ≡ number of radiative excitations from state l to state u per sec per particle in state l Induced deexcitation χ Bul J ν0 ≡ number of induced radiative deexcitations from state u to state l per sec per particle in state u Collisional excitation and deexcitation Clu ≡ number of collisional excitations from state l to state u per sec per particle in state l Cul ≡ number of collisional deexcitations from state u to state l per sec per particle in state u talk start talk files all talks BOUND-BOUND RATES RTSA 2.3.1, 2.3.2, 2.6.1; Stix 4.2 Monochromatic bb rates expressed in Einstein coefficients (intensity units) nu Aul χ(ν)/4π nu Bul Iν ψ(ν)/4π nl Blu Iν φ(ν)/4π nu Cul spontaneous emission Einstein relations gu Bul = gl Blu stimulated emission radiative excitation (gu /gl )Aul = (2hν 3 /c2 ) Blu nl Clu collisional (de-)excitation Cul /Clu = (gl /gu ) exp(Eul /kT ) required for TE detailed balancing with Iν = Bν , but hold universally General line source function hν hν nu Aul χ(ν) αν = [nl Blu φ(ν) − nu Bul ψ(ν)] jν = 4π 4π Sl = nu Aul χ(ν) nl Blu φ(ν) − nu Bul ψ(ν) Simplified line source function CRD: χ(ν) = ψ(ν) = φ(ν) 1 2hν 3 nu Aul = 2 g u nl Sl = nl Blu − nu Bul c −1 g l nu Statistical equilibrium equations for level j N N X X nj Rji = nj Rij Rji = Aji + Bji Jji + Cji j6=i j6=i time-independent population bb rates per particle in j 1 Jji ≡ 4π Boltzmann: Sl = Bν (T ) Z 0 4πZ ∞ Iν φ(ν) dν dΩ 0 total (= mean) mean intensity talk start talk files BOUND-BOUND EQUILIBRIA all talks RTSA missing; Stix 4.1.2 • LTE = large collision frequency – interior, low photosphere – up: mostly collisional = thermal creation (d + e) – down: mostly collisional = large destruction probability (a) – photon travel: “honorary gas particles” or negligible leak • NLTE = statistical equilibrium or time-dependent – chromosphere, TR – photon travel: non-local impinging (pumping), loss (suction) – two-level scattering: coherent/complete/partial redistribution – multi-level: photon conversion, sensitivity transcription • coronal equilibrium = hot tenuous – coronal EUV – up: only collisional = thermal creation (only d) – down: only spontaneous (only d) – photon travel: escape / drown / scatter bf H I, He I, He II talk start talk files all talks SOLAR ATMOSPHERE RADIATIVE PROCESSES RTSA Chapt. 8; Stix Fig. 4.5 • bound-bound – Sν , κν NLTE? PRD? – neutral atom transitions – ion transitions – molecule transitions • bound-free – Sν , κν NLTE? always CRD – H− optical, near-infrared – H I Balmer, Lyman; He I, He II – Fe I, Si I, Mg I, Al I electron donors • free-free – Sν always LTE, κν NLTE – H− infrared, sub-mm – H I radio • electron scattering – always NLTE, Doppler? – Thomson scattering – Rayleigh scattering • collective – p.m. – cyclotron, synchrotron radiation – plasma radiation talk start talk start talk files all talks talk files all talks VAL3C CONTINUUM FORMATION VAL3C CONTINUUM FORMATION talk start talk start talk files all talks talk files all talks VAL3C CONTINUUM FORMATION VAL3C CONTINUUM FORMATION talk start talk start talk files all talks talk files all talks VAL3C CONTINUUM FORMATION VAL3C CONTINUUM FORMATION talk start talk files RADIATIVE COOLING all talks RTSA 7.3.2 Radiative equilibrium condition Φtot (z) ≡ dFrad (z) =0 Zdz ∞ = 4π αν (z) [Sν (z) − Jν (z)] dν Z ∞Z +1 = 2π [jνµ (z) − ανµ (z) Iνµ (z)] dµ dν 0 0 −1 Net radiative cooling in a two-level atom gas Φul = 4πανl 0 (Sνl 0 − J ν0 ) = 4πjνl 0 − 4πανl 0 J ν0 = hν0 nu (Aul + Bul J ν0 ) − nl Blu J ν0 = hν0 [nu Rul − nl Rlu ] Net radiative cooling in a one-level-plus-continuum gas Z ∞ bi bc −hν/kT LTE −hν/kT Φci = 4π ni bc σic (ν) Bν 1 − e − Jν 1 − e dν bc bi ν0 talk start talk files VAL3C RADIATION BUDGET all talks talk start talk start talk files all talks talk files all talks VAL3C RADIATION BUDGET REALISTIC ABSORPTION LINE • extinction: bb peak lower and narrower at larger height • optical depth: near-log-linear inward increases • source function: split for line (bb) and continous (bf, ff, electron scattering) processes • intensity: Eddington-Barbier for Sνtotal = (κC SC + κl Sl )/(κC + κl ) = (SC + ην Sl )/(1 + ην ) talk start talk files all talks SOLAR NaI D2 Uitenbroek & Bruls 1992A&A...265..268U Na I D2 is a good example of two-level scattering with complete redistribution: very dark Eddington-Barbier approximation: line-center τ = 1 at h ≈ 600 km chromospheric velocity response but photospheric brightness response Can you use the Eddington-Barbier approximation to estimate the formation height of the blend lines? talk start talk files all talks SOLAR NaI D2 Eddington-Barbier for the blends? Moore, Minnaert, Houtgast 1966sst..book.....M: 5888.703 5889.637 5889.756 * 5889.973M * 5890.203 * 5890.495 5891.178 5891.178 10. 14. 752. 752. 752. 5. 17. 17. 2. 2. 4. 120.SS 3. 1.S" 3.S 3.S ATM H2O ATM H2O ATM H2O NA 1(D2) ATM H2O FE 1P ATM H2O FE 1P R4 R4 R3 0.00 R4 5.06 R3 4.65 302 401 401 1 302 1313 401 1179 26 26 26 26 17,26 talk start talk files Ca II H2V GRAIN SIMULATION all talks • observation Lites, Rutten & Kalkofen 1993 – sawtooth line-center shift – triangular whiskers – H2V grains • simulation Carlsson & Stein 1997 – 1D radiation hydrodynamics – subsurface piston from Fe I blend – observer’s diagnostics • analysis (RR radiative transfer course) – source function breakdown – dynamical chromosphere – H2V grains = acoustic shocks talk start talk files DYNAMIC LOWER CHROMOSPHERE Carlsson & Stein, ApJ 440, L29, 1995 all talks talk start talk files all talks SHOCK GRAIN DIAGNOSIS Carlsson & Stein, ApJ 481, 500, 1997 Iν (0) = talk start Z ∞ Sν e 0 −τν dτν = Z ∞ Sν τν e−τν 0 d ln τν dz dz talk files BASIC RADIATIVE TRANSFER EQUATIONS all talks RTSA last page specific intensity Iν (~r, ~l, t) erg cm−2 s−1 Hz−1 ster−1 emissivity jν erg cm−3 s−1 Hz−1 ster−1 extinction coefficient αν cm−1 σν cm2 part−1 P P Sν = jν / α ν R∞ τν (z0 ) = z0 αν dz source function radial optical depth plane-parallel transport κν cm2 g−1 µ dIν /dτν = Iν − Sν Iν = I0 + (Sν − I0 ) τν R∞ thick emergent intensity Iν+ (0, µ) = 0 Sν (τν ) e−τν /µ dτν /µ thin cloud Eddington-Barbier mean mean intensity photon destruction complete redistribution isothermal atmosphere Iν+ (0, µ) ≈ Sν (τν = µ) R ∞ R +1 ϕ J ν0 = 12 0 −1 Iν ϕ(ν −ν0 ) dµ dν εν = ανa /(ανa + ανs ) ≈ Cul /(Aul + Cul ) ϕ Sνl 0 = (1 − εν0 ) J ν0 + εν0 Bν0 √ Sν0 (0) = εν0 Bν0
