Introduction to Solar Radiative Transfer II Non-LTE Radiative Transfer

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1 Introduction to olar Radiative Transfer II Non-LTE Radiative Transfer Han Uitenbroek National olar Observatory/acramento Peak unspot NM Overview I Basic Radiative Transfer Intensity, emission, absorption, source function, optical depth, transfer equation II Detailed Radiative Processes pectral lines, radiative transitions, collisions, polarization, Non-LTE radiative transfer, molecular concentrations III Observations of olar Radiation olar telescopes, spectroscopy, polarimetry George Ellery Hale CGEP, CU Boulder Lecture, Mar 7, Local Thermodynamic Equilibrium (LTE) Basic Radiative Transfer: Radiation Field Remember: Angle-averaged mean intensity: J ν ( r, t) I ν dω = π π π π I ν sin θ dθ dϕ Radiation field is given by Planck function Velocities are given by Maxwellian distribution Ionization and excitation are given by aha Boltzmann statistics cattering Basic Radiative Transfer: cattering Absorption: di ν σ ν I ν ds Emission: di ν = σj ν ds; (isotropic scattering) catering source function: ν = σ ν J ν /σ ν = J ν In the case of pure scattering the source function is solely determined by the radiation field and, therefore, completely decoupled from local conditions in the atmosphere, resulting in possible departures from Local Thermodynamic Equilibrium (LTE).

2 Radiative Transfer with Absorption and cattering imple olution: Lambda Iteration Operator equation for source function: η ν = α ν B ν + σ ν J ν χ ν = α ν + σ ν Total source function: ν η ν χ ν α ν = σ ν J ν + B ν ; ɛ ν α ν α ν + σ ν α ν + σ ν α ν + σ ν ν =( ɛ ν )Λ ν [ ν ]+ɛ ν B ν imple iterative solution: ν () = B ν [ ν (n) =( ɛ ν )Λ ν ν (n ) ] + ɛ ν B ν =( ɛ ν )J ν + ɛ ν B ν =( ɛ ν )Λ ν [ ν ] + ɛ ν B ν Lamda Iteration: ɛ =.5 Lamda Iteration: ɛ =. B Planck () B Planck () () () () ε = 5.E () ε =.E optical depth τ optical depth τ Lamda Iteration: ɛ =. Lamda Iteration: ɛ =. (). B Planck (). B Planck () (). () (). ε =.E ε =.E optical depth τ optical depth τ

3 Lamda Iteration: ) =., Iterations Accelerated Lambda Iteration BPlanck (). ν = ( )ν )Λν [ν ] + )ν Bν () Auer, in Numerical Radiative Transfer, 97, ed. W. Kalkofen, p. For one wavelength, this is a matrix equation in depth points: k = ( )k )Λν [ν ]k + )k Bk. We could solve this equation easily if the Λ operator where just a multiplication, i.e., if it where a local operator. Use the local part of the operator, i.e, its diagonal Λ (see Olson, Auer & Buchler, 9, JQRT 5, ). ε =.E () optical depth τ Accelerated Lamda Iteration: ) =. Accelerated Lambda Iteration plit off the diagonal part:. Jk = Λν [ν ]k Λ k k + ( J)k Jk = Λ []k Λ k k ource function k = ( )k ) {Λ k k + Jk } + )k Bk New iterative scheme: " # J (n) = Λ(n) (n) Λ (n) (n+) k. (n) = ( )k ) Jk + )k Bk ( )k )Λ k Niter = ε =.E We can invert the diagonal part now directly and only have to lambda iterate the weaker off-diagonal contributions. Accelerated Lamda Iteration: ) =. with convergence extrapolation. Optical depth Image of the un in the light of Hα ource function.. Niter = 7 ε =.E. Optical depth

4 Hα pectral Line Termdiagram and Transitions in Hydrogen Intensity [J m s Hz sr ] Energy [ev] Disk center Wavelength [nm] E = hν = hc λ Termdiagram and Transitions in Hydrogen Termdiagram and Transitions in Hydrogen Energy [ev] Energy [ev] E = hν = hc λ E = hν = hc λ Radiative bound bound transitions Collisional bound bound transitions

5 Radiative bound free transitions Collisional bound free transitions Absorption and emission coefficients for bound-bound transitions ource function of bound bound transition pontaneous emission j i: jν spont = n j (A ji hν ij /π)φ ν timulated emission j i: j stim ν = n j (B ji hν ij /π)φ ν I ν, A ji =(hν /c )B ji Absorption i j: α ν = n i (B ij hν ij /π)ϕ ν, g i B ij = g j B ji Transfer equation: di ν ds = jspont ν + jν stim α ν I ν = n j (A ji hν ij /π)φ ν hν ij /πφ ν (n i B ij n j B ji )I ν ource function: ν = j ν n j A ji = α ν n i B ij n j B ji = hν ij c n j C ji =( ɛ)j + ɛb ν ; ɛ g j /g i n i n j C ji + A ji + B ji J Radiative rates Basic Equation: tatistical Equilibrium Radiative excitation R ij = B i j hν dν dω π hν I νφ ν = B i jj; J π Radiative de-excitation hν dν R ij = Aji + B ji dω π hν I νφ ν = A ji + B ji J dω dνi ν φ ν Consider and atom (or molecule) with levels i =,...,N. tatistical equilibrium for level i: N N n j (Cji + Rji) = n i (Cij + Rij) j= j= Energy [ev] CA III P E CA II P CA II P P E PO DE The set of equations for all levels forms a, generally non-linear, and non-local, setofequationsforthepopulationnumbersn i CA II P D

6 Ca ii source function, J [J m s Hz sr ] When is Non-LTE Transfer Important? total active backgr BPlanck J 7 When density is high and collisions are frequent enough, population numbers are determined by local conditions, and given by the aha-boltzmann relations at the kinetic temperature of the gas λ[nm] The radiation field is then given by the Planck function As densities drop with height, collisions become less frequent, and radiative transitions become relatively more important. Populations are now determined by non-local conditions, namely the radiation field that comes from different places in the atmosphere Column Mass [kg m ]. Need to find a global solution, not only in space, but also in wavelength.. The Zeeman effect in atoms How Do We Extract Physics from Observations? Continuum intensity tokes I tokes Q/I tokes U/I tokes V/I MJ = J, J +,..., J, J H J z = MJ! E = E + g L MJ µ H H J M MJ MJ MJ L P D DLP at the DT, courtesy Alexandra Tritschler (NO/P) Equation of Polarized Radiative Transfer Line Absorption Matrix #! Transfer Equation: di = KI + j ds " $ #! I = (I, Q, U, V ), (tokes vector) j = (jc + jlφ)e, e = (,,, ) K = αc + αlφ, " (Absorption matrix) #! φi φq φq φi Φ= φu ψv φv ψu "! φu φv ψv ψu φi ψq ψq φi φi = φ sin γ + (φ+ + φ ), $ γ $ φ = φq = φ sin γ cos χ + x x φ (φ+ + φ ) y #, φu = φ sin γ sin χ φv = (φ+ φ ) cos γ B z " $

7 Molecular pectral Lines in the olar pectrum Concentration of Molecules Molecules are abundant in the solar atmosphere, in particular in cooler areas like unspot umbrae The G band is one of the most used pass bands in solar high resolution imaging. It its majot source of opacity are lines of the CH molecule. CO lines are a major contributor to radiative cooling of the atmosphere in the infrared. Molecules are sensitive to the Zeeman effect, and have much more diverse sensitivity than atomic lines. This can be used to advantage. Chemical equilibrium: ( ) / ( ) n A n B πmab kt = n AB h e D/kT UA U B Q AB m AB = m Am B m A + m B Non-linear set of coupled equations: n AB n A n B Φ AB (T )= n A + n AB = A A n H n B + n AB = A B n H Use Newton-Raphson Method to olve Molecular Concentrations in the olar Atmosphere et of non-linear chemical equilibrium equations: f( n) = a Iterative solution: f( n (n) + δ n) = a f( n (n) )+δ n f n a ( ) δ n = a f( n (n) ) / f n n (n+) = n (n) + δ n n molecule / n H tot H H+ C N O CH CO CN NH NO OH HO H FALC_.atmos (Thu Apr :7: ) Height [km] CO Concentration in Vertical Cross ection Energy levels of the CO Molecule z [Mm] z [Mm] R x [Mm] R log(n CO / m ) log(temperature / K) Energy [ev] Vibrational level

8 Filter Integrated Intensity High resolution imaging in the G-band.5 Intensity [ J m s Hz sr ] wavelength [nm] y [arcsec] arcsec Filter signal: f = I λ F λ dλ x [arcsec] arcsec z [Mm] Concentration CH Molecules in Magneto-Convection lice.... n CH / n H x [arcsec] Detailed pectra of Granule and Bright Point Intensity [ J m s Hz sr ] Intensity [ J m s Hz sr ] average granule bright point wavelength [nm] wavelength [nm] Planck Function End Part II B λ [J m s nm sr ] K 5 K K wavelength [nm] B λ (T ) = hc λ 5 e hc/λkt

9 gl eff =.5 gl eff =.7 Maxwellian Velocity Distribution aha Boltzmann statistics f(v).5.. C.E+ 5.E+.E+.E+ H.E+ 5.E+.E+.E+ Boltzmann distribution for excitation: [ nj n i ] = g j e E ji/kt LTE g i... Velocity [km s ] aha distribution for ionization: [ nr+, n r, ] LTE = g r+, e χr/kt N e g r, Back ( m ) / f(v)dv = exp( mv /kt)πv dv πkt Formal olution as Operator Equation Voigt Functions J ν = I ν (τ, π l)dω = dω ν (t, π l) e (t τν) dt = Λ ν [ ν ] τ ν Back log H(a, v) / sqrt(π) a =.E.E 5.E.E.E 5 5 v [Doppler units] H(a, v) φ(ν ν )= π νd ν D ν kt c m Γ a = π ν D Back plitting pattern for Fe i.5 nmlevel= and.5 nm. 5 D 5 P. Normalized strength... 5 D 5 P. π component ρ + ρ.5 Wavelength shift [Larmor units] level= Normalized strength..5 Back..5. π component ρ + ρ Wavelength shift [Larmor units]

10 plitting Patterns for atellite Branch ( N J) Branch: O P Branch: Q P. Branch: P Q gl eff =.... gl eff =. gl eff =.99. trength. trength. trength hift [Larmor units] hift [Larmor units] hift [Larmor units]. Branch: R Q gl eff =. Branch: Q R Branch: R Jl = gl eff =.95 gl eff =.9 trength. trength. trength hift [Larmor units] hift [Larmor units] hift [Larmor units]