talk start talk files all talks SUMMARY OF SOLAR SPECTRUM FORMATION
advertisement
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 examples: local – nonlocal – converted photons white light corona 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 radiative transfer in stellar atmospheres: scattering thick transfer Boltzmann-Saha simple absorption line VALIIIC model [RTSA 3.4; 4.1 – 4.2] [RTSA 4.3] NLTE solar radiation escape bb processes bb rates solar radiation processes VAL3C continuum formation VAL3C radiation budget realistic absorption line Na D1 Ca II H from RADYN chromosphere summary RTSA rap coronium lines bb equilibria radiative cooling Na D1 blends talk start 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 ρ2 = x 2 + y 2 I(x) = 2 j(ρ) dy = 2 dρ ρ2 − x 2 0 x 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 530.29 nm yellow line 569.45 red line 637.45 Coronal sky at Dome C [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 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 σlu f (v) v dv ≈ nu Aul hν ∝ nion Ne dz = Ne dz v0 talk start talk files BRIGHT AND DARK IN EUV IMAGES all talks • 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 ) jνc + jνl Sνc + ην Sνl = ανc + ανl 1 + ην stimulated emission negatively into αν , κν Transport equation along the beam with τν as optical thickness dIν dIν = jν − αν Iν = Sν − Iν dτν ≡ αν ds ds αν ds Plane-parallel transport equation with τν as radial Z optical depth z0 dτν ≡ −κν ρ dz ην ≡ ανl /ανc τν (z0 ) = − κν ρ dz ∞ µ dIν = Sν − Iν dτν dIν = Iν − Sν dτν talk start talk files CONSTANT SOURCE FUNCTION all talks Transport equation along the beam (τν = optical thickness)Z τν dIν = Sν − Iν 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) Iν (0) = 0 : Iν (D) ≈ τν (D) Sν = jν D talk start talk files all talks THICK TRANSFER RTSA 2.2.2; Stix 4.1.1 Radial optical depth dτν = −κν ρ dr r radial αν cm−1 = cm2 /cm3 κν cm2 /gram Hubený τνµ Transport equation µ dIν = Iν − Sν dτν Integral form −(τ0−τν )/µ Iν (τν , µ) = Iν (τ0 , µ) e Z τ0 + (1/µ) 0 Sν (τν0 ) e−(τν−τν )/µ dτν0 τν “formal solution” both directions Emergent intensity Z Iν (0, µ) = (1/µ) pm: Doppler anisotropy Sν ∞ Sν (τν ) e−τν /µ dτν 0 Eddington-Barbier approximation Iν (0, µ) ≈ Sν (τν = µ) exact for linear Sν (τν ) σν cm2 /particle talk start talk files PLANCK RTSA Chapt. 2; SSA 3; Stix 4.5 2hν 3 1 Planck: Bν (T ) = 2 hν/kT c e −1 2hν 3 −hν/kT Wien: exp(hν/kT ) 1 : Bν (T ) ≈ 2 e c 2ν 2 kT Rayleigh-Jeans: exp(hν/kT ) 1 : Bν (T ) ≈ c2 all talks talk start talk files SOLAR VISIBLE AND INFRARED RTSA Chapt. 8; SSB 2 all talks talk start talk files SOLAR VISIBLE AND INFRARED RTSA Chapt. 8; SSB 2 all talks talk start talk files SOLAR VISIBLE AND INFRARED RTSA Chapt. 8; SSB 2 all talks talk start talk files SOLAR VISIBLE AND INFRARED RTSA Chapt. 8; SSB 2 all talks talk start talk files SOLAR VISIBLE AND INFRARED RTSA Chapt. 8; SSB 2 all talks talk start talk files SOLAR VISIBLE AND INFRARED RTSA Chapt. 8; SSB 2 all talks 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 Saha–Boltzmann Schadeenium all talks talk start talk files all talks 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 ∞ Iν (0, 1) = Bν exp(−τν ) dτν = 0 ∞ τC Z (1 + ην ) Bν exp − 0 (1 + ην ) dτC0 dτC 0 Eddington-Barbier: Iν (0, 1) ≈ Bν [τν = 1] = Bν [τC = 1/(1+ην )] extinction coefficient shape = Maxwell Gauss ⊗ damping Lorentz r Z +∞ γ exp(−(ν −ν 0 )2 /∆νD2 ) 0 1 ν0 2RT 2 H(a, v) dν = √ + ξt φ(ν) = √ ∆νD ≡ 0 2 2 c A π∆νD π∆νD −∞ [2π(ν −ν0 )] + γ /4 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 nLTE ρ (1−e−hν/kT ) = l 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) gu 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 Scheffler & Elsässer, courtesy Karin Muglach all talks talk start talk files VALIIIC MODEL Vernazza, Avrett, Loeser 1981ApJS...45..635V all talks talk start talk files SOLAR SPECTRUM FORMATION Avrett 1990IAUS..138....3A all talks 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) Sl = nu Aul 2hν 3 1 = 2 gu nl nl Blu − nu Bul c −1 gl 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 4πZ ∞ Iν φ(ν) dν dΩ 0 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 files VAL3C CONTINUUM FORMATION all talks talk start talk files VAL3C CONTINUUM FORMATION all talks talk start talk files VAL3C CONTINUUM FORMATION all talks talk start talk files VAL3C CONTINUUM FORMATION all talks talk start talk files VAL3C CONTINUUM FORMATION all talks talk start talk files VAL3C CONTINUUM FORMATION all talks talk start talk files RADIATIVE COOLING all talks RTSA 7.3.2 Radiative equilibrium condition Φtot (z) ≡ dFrad (z) =0 Zdz ∞ αν (z) [Sν (z) − Jν (z)] dν Z ∞Z +1 = 2π [jνµ (z) − ανµ (z) Iνµ (z)] dµ dν = 4π 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 files VAL3C RADIATION BUDGET all talks talk start talk files REALISTIC ABSORPTION LINE all talks • 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 SOLAR NaI D2 all talks 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 SHOCK GRAIN DIAGNOSIS Carlsson & Stein, ApJ 481, 500, 1997 Z Iν (0) = ∞ Sν e 0 −τν Z dτν = 0 ∞ Sν τν e−τν d ln τν dz dz all talks talk start 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 radial optical depth αν cm−1 σν cm2 part−1 P P Sν = jν / αν R∞ τν (z0 ) = z0 αν dz plane-parallel transport µ dIν /dτν = Iν − Sν source function κν cm2 g−1 Iν = I0 + (Sν − I0 ) τν R∞ thick emergent intensity Iν+ (0, µ) = 0 Sν (τν ) e−τν /µ dτν /µ thin cloud mean mean intensity Iν+ (0, µ) ≈ Sν (τν = µ) R ∞ R +1 ϕ J ν0 = 12 0 −1 Iν ϕ(ν −ν0 ) dµ dν photon destruction εν = ανa /(ανa + ανs ) ≈ Cul /(Aul + Cul ) complete redistribution Sνl 0 = (1 − εν0 ) J ν0 + εν0 Bν0 √ Sν0 (0) = εν0 Bν0 Eddington-Barbier isothermal atmosphere ϕ
