Introduction to Solar Radiative Transfer IBasicRadiativeTransfer

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1 Introduction to Solar Radiative Transfer IBasicRadiativeTransfer Han Uitenbroek National Solar Observatory/Sacramento Peak Sunspot NM George Ellery Hale CGEP, CU Boulder Lecture 13, Mar 5 13 Wy Radiative Transfer? In general we cannot visit te astronomical objects we are interested in, and tus cannot take in-situ measurements Instead, to determine te object s properties, we ave to rely on te information carried to us by te electromagnetic radiation emitted and/or reflected by te object. Multi-wavelengt (spectroscopic) observations and analysis are te only available means to determine te pysical conditions of astronomical objects. To analyze spectroscopic data meaningfully we need to understand ow pysical information is encoded in te radiation (Radiative Transfer). We need to understand ow te radiative signal is modified as it travels to our instruments and is detected wit tem. Te Solar Spectrum and Surface Temperature Te Solar Spectrum and Surface Temperature [K] [K] [J m s 1 Hz 1 sr 1 ] [J m s 1 Hz 1 sr 1 ] [K] Te Solar Spectrum and Surface Temperature Overview [J m s 1 Hz 1 sr 1 ] [K] [K] [K] I Basic Radiative Transfer, emission, absorption, source function, optical dept, transfer equation, line formation II Detailed Radiative Processes Spectral lines, radiative transitions, collisions, polarization, Non-LTE radiative transfer III Observations of Solar Radiation Solar telescopes, spectroscopy, polarimetry

2 Bibliograpy Sort History Rutten: Radiative Transfer in Stellar Atmosperes (ttp://esoads.eso.org/abs/3rtsa.book...r) 1 Wollaston First to observe dark gaps in spectrum: spectral lines 114 Fraunofer rediscovers lines. Assigns names e.g., C (Hα), D (Na i), G (CH molecules), F (Hβ), and H (Ca ii) Rybicki and Ligtman: Radiative Processes in Astropysics Mialas: Stellar Atmosperes 13 Herscel realized spectra contain information on composition of source from flame spectra Su: Te Pysics of Astropysics. I. Radiation 14 Becquerel potograps spectra, discovers lines in te UV, beyond te visible Gray: Observation and Analysis of Stellar Potosperes del Toro Iniesta: Introduction to Spectropolarimetry 15 Bunsen and Kircoff discover wavelengt correspondence between brigt flame emission and dark solar absorption lines. Start of quantitative spectroscopy. Allen: Astropysical Quantities Solar atmospere is also very strongly time dependent Te Sun Courtesy: Mats Carlsson A vertical cross section troug a 3-D Convection Simulation Te Visible Solar Spectrum x [arcsec] 4 4 x [arcsec] 1 T [13 K] z [km] 4 4 x [arcsec]

3 Solar Spectrum in te Blue and Red Spatially Resolved Spectral Lines [J m s 1 Hz 1 sr 1 ] Spectrum..4.. Polarization [J m s 1 Hz 1 sr 1 ] Basic Radiative Transfer: Radiation Field Basic Radiative Transfer: Mean Angle-averaged Mean intensity: J ( r, t) 1 I λ dω = 1 4π 4π π π I λ sin θ dθ dϕ Specific intensity I is te radiative energy tat flows, at te location r, per second, per wavelengt interval, and per solid angle, in te direction l troug te surface area da perpendicular to l.isconserved wit distance in te absence of emission and absorption or scattering processes. Specific : deλ rad I λ ( r, l, t)dtda dλ dω = I λ ( r, l, t)dt cos θ da dλ dω Units: Js 1 m nm 1 ster 1 Unlike te Specific te Angle-averaged Mean is not conserved wit distance θ z ϕ r sin θ ϕ r θ y Units: Js 1 m nm 1 ster 1 x Basic Radiative Transfer: Flux Basic Radiative Transfer: Absorption Flow of radiative energy troug a surface. Flux: π π F λ ( r, n, t) I λ cos θ dω = I λ cos θ sin θ dθ dϕ (1) Units: Js 1 m nm 1 Flux in radial direction: F λ (z) = = π π π π I λ cos θ sin θ dθ dϕ + I λ cos θ sin θ dθ dϕ π π π π F + λ (z) F λ (z) π I λ cos θ sin θ dθ dϕ () I λ (π θ) cos θ sin θ dθ dϕ Absorption α λ : I λ (s +ds) =I λ (s)+ = I λ α λ I λ ds Units: m 1

4 Basic Radiative Transfer: Emission Basic Radiative Transfer: Source Function Emission j λ : I λ (s +ds) =I λ (s)+ = I λ + j λ (s)ds Source function: S λ j λ /α λ Units: Jm 3 s 1 nm 1 ster 1 Units: Js 1 m nm 1 ster 1 For multiple proceses active at te same wavelengt: Sλ tot = j λ / α λ Sλ tot = jc λ + jl λ αλ c + = Sc λ + η λsλ l, η λ α αl λ 1+η λ/α l λ c λ Basic Radiative Transfer: Transport Equation Basic Radiative Transfer: Transport Equation Transport along a ray: (s) =I λ (s +ds) I λ (s) =j λ (s)ds α λ (s)i λ (s)ds (3) ds = j λ α λ I λ α λ ds = = S λ I λ dτ λ Optical lengt and tickness: dτ λ α λ (s)ds (4) τ λ (D) = D α λ (s)ds Transport along a ray: = S λ I λ (5) dτ λ I λ (τ λ )=I λ ()e τ λ + Homogeneous medium: τλ S λ (t)e (τ λ t) dt I λ (D) =I λ ()e τ λ(d) + S λ (1 e τ λ(d) ) () Optically tick: I λ (D) S λ Optically tin: I λ (D) I λ () + [S λ I λ ()] τ λ (D) Basic Radiative Transfer: Troug an Atmospere Optical pat: dτ µλ = α λ ds α λ dz µ Basic Radiative Transfer: Eddington Barbier Emergent intensity at te surface: I + λ (τ λ =,µ)= Substitute power series: S λ (t)e t/µ dt/µ Standard plane parallel transport equation: N S λ (τ λ )= a n τλ n (using : n= I + λ (τ λ =,µ)=a + a 1 µ+a µ n!a N µ N e t t n dt =!n) dτ µλ = µ dτ λ = I λ S λ Eddington Barbier relation: I + λ (τ λ =,µ) S λ (τ λ = µ)

5 Eddingtomn-Barbier approximation Basic Radiative Transfer: Limb Darkening b θ a S a I a b b sin θ 1 r/r =sinθ Absorption lines in te solar spectrum Wy do we get spectral lines in absoption? 1. α total τ I S wavelengt [nm] Optical dept unity in te Na i D line In te UltraViolet te Spectral Lines are in Emission 1. z [km] [nm] λ[nm] Source Function [J m s 1 Hz 1 sr 1 ] x [km]

6 Wy do we get spectral lines in emission? Wy do we get spectral lines in emission and absorption? total α τ α τ I S 1 Eddington Barbier is an approximation!! Eddington Barbier is an approximation!!.4 µ =.93 T [1 3 K] 1 radiation temperature [1 3 K] 7. T rad x [Mm] T form y [Mm].. z [Mm] T [1 3 K] x [Mm] x [Mm] Continuum processes Tere is a lot of information in spectral lines Outside spectral lines te solar plasma as significant opacity in so called continuum processes. Tey are called tis way because teir opacity varies very slowly wit wavelengt. Atomic Bound free and free free transitions H bound free and free-free Tomson scattering αe T π qe 4 = N e σ e = N e 3m 4 ec (4πɛ ) Rayleig scattering Uitenbroek & Tritscler, IBIS DST α R (ω) =σ e f ij ω 4 /(ω ij ω )

7 Molecular Oxygen in te Eart Atmospere End Part I.4.. Fe I Fe I O O Differences in spectral lines Invariance of Specific along Rays Specific as been defined in suc a way as to be independent of te source and te observer. [J m s 1 Hz 1 sr 1 ] de λ = I λ cos θ dt da dλ dω = I λ cos θ dt da dλ dω dω = da cos θ /R dω =da cos θ/r I λ = I λ H Opacity Bound Free Cross Sections of Hydrogen 1 1 Hydrogen H opacity T = 1 9 α λ [m 5 /J] H b f H f f Wavelengt [µm] E bf =.754 ev