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Radiative Transfer

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Radiative Transfer Astro 305 Lecture Notes Wayne Hu Set 1: Radiative Transfer Radiation Observables • From an empiricist’s point of view there are 4 observables for radiation • Energy Flux • Direction • Frequency • Polarization Observables: Flux . • Energy Flux dE F = dtdA • Units: erg s−1 cm−2 • Radiation can hit detector from all angles dA flux=energy/time/area detector Observables: Surface Brightness . • Direction: columate (e.g. pinhole) in an acceptance dΩ angle dΩ normal to dA ! surface brightness dE surface brightness= S(Ω) = dA energy/time/area/solid-angle dtdAdΩ (normal to detector) detector • Units: erg s−1 cm−2 sr−1 Observables: Specific intensity . • Frequency: filter in a band of frequency dΩ dν ! specific intensity filter dE bandwidth dν I = ν dtdAdΩdν which specific intensity= dA energy/time/area/solid-angle/ frequency is the fundamental quantity detector (normal to detector) for radiative processes • Units: erg s−1 cm−2 sr−1 Hz−1 • Astro-lingo: color is the difference between frequency bands Observables: Polarization . • Polarization: filter in linear (1,2) [or 45 dΩ degrees rotated (1’,2’) or filter circular (+,-) polarization bandwidth dν states] – Stokes parameters polarizer x2 x1 Stokes parameter= Iν = Iν1 + Iν2 energy/time/area/solid-angle/ dA frequency [sum+difference in polarization states] Qν = Iν1 − Iν2 detector Uν = Iν10 − Iν20 Vν = Iν+ − Iν− • Units: erg s−1 cm−2 sr−1 Hz−1 Radiative Transfer . • Radiative Transfer = change in Iν as radiation propagates • Simple example: how does the specific intensity of sunlight r1 change as it propagates to the earth ∆Ω r2 • Energy conservation says . 2 2 F (r1)4πr1 = F (r2)4πr2 F / r−2 Radiative Transfer • But the angular size subtended by the sun ∆Ω / r−2 F S = = const ∆Ω • And frequency doesn’t change so that Iν = const Radiative Transfer • More generally: Iν changes due to • Scattering (directional change) • Doppler or Red-shift (frequency change) • Absorption • Emission • If frequency changes due to redshift ν / (1 + z)−1, photon conservation implies 3 Iν/ν = const R and so surface brightness S = Iνdν scales as the redshift of (1 + z)−4 (cosmological surface brightness dimming, conversely relativistic Doppler boost) 3 • Liouville’s theorem (conservation of Iν/ν ) in absence of interactions; Boltzmann equation (radiative transfer equation) Auxiliary Quantities . • Specific intensity is defined as energy flux per unit area, per unit solid angle (normal Iν to the area), per unit frequency • So specific flux cosθ dA through a detector not oriented θ dA normal to source has a lowered detector cross section due to projection . Z Z Fν = Iν cos θdΩ;F = dνFν if Iν is constant in angle, no net energy flux Fν = 0 = F Auxiliary Quantities . • Angle averaged intensity dΩ Z dΩ J = I ν 4π ν • Specific energy density: start with energy detected dV=cdtdA dE = I dtdAdΩdν dA ν dE = Iν dt dA dΩ dν Auxiliary Quantities • Light travels at c, dE is the energy in the volume dV = cdtdA I dE = ν dV dΩdν c • Specific energy density uν(Ω) ≡ Iν=c • Energy density: Z Z Z u = dν dΩIν=c = dν4πJν=c Auxiliary Quantities . • Momentum q = E=c e^r Iν • Change in momentum per area: e3 er dq 1 = Iν cos θdΩdνe^r dtdA c cosθ dA • Pressure: the component θ normal to surface e^3 · e^r = cos θ dA • If surface absorbs radiation 1 Z 1 Z p = F cos θdΩ = I cos2 θdΩ ν;abs c ν c ν Auxiliary Quantities • If surface reflects radiation pν = 2pν;abs 2 Z = I cos2 θdΩ c ν • Equation of state of isotropic radiation Iν = Jν 2 Z 4π p = J cos2 θdΩ = J ν c ν 3c ν 4π Z 1 p = dνJ = u 3c ν 3 Radiative Processes • Spontaneous Emission: matter spontaneously emits a photon/radiation • Absorption: radiation absorbed by matter • Stimulated Emission: passing radiation stimulates matter to emit in the same frequency and direction. Stimulated emission is mathematically the same as negative absorption - both proportional to the incoming radiation • Scattering: absorption followed immediately by emission Coherent or elastic: emission at the same frequency Isotropic: radiates equally in all directions • All can be related to the matrix element for interaction - interrelated by Einstein relations Spontaneous Emission jν dA Iν spontaneous emission ds • Given a source isotropically radiating power into a band dν at a rate power = dνP volume ν • The amount of energy radiated per unit solid angle is dΩ dE = P dνdV dt em ν 4π Spontaneous Emission • Generalize to a non-isotropically emitting source P ν ! j (Ω) 4π ν called the monocromatic emission coefficient • Emission coefficient is alternately given per unit mass ν = 4πjν/ρ where ρ is the mass density dE = ν dνdmdt; dm = ρdV em 4π Spontaneous Emission • Effect on specific intensity across a path length ds dIνdtdAdΩdν = dEem = jνdV dΩdνdt dIν = jνds; dV = dsdA • Radiative transfer equation Z s dIν = jν;Iν(s) = Iν(s0) + jν(s)ds ds s0 Absorption αν dA Iν absorption ds • Travelling through a material a fraction αν of the radiation is absorbed per unit length dIν = −ανds Iν where αν is called the absorption coefficient Absorption • If material does not emit, solution is an exponential suppression Z ln Iν = − dsαν + C R − αν ds Iν(s) = Iν(s0)e • If αν < 0 (stimulated emission) then there is an exponential growth αν<0 dA Iν stimulated ds emission Optical Depth • Useful to measure length in units of the typical path length to absorption or interaction 1 L = αν so that the total path length in units of L becomes Z Z ds α ds = ≡ τ ;I (s) = I (s )e−τν ν L ν ν ν 0 • Radiative transfer equation dI ν = −α I + j ds ν ν ν dIν = −Iν + Sν;Sν ≡ jν/αν dτν where Sν is the source function. Cross Section αν σ dA Iν absorption ds • Particle description of absorbers and light • Absorption coefficient related to cross section for interaction • Example: if medium is full of opaque disks each of area σ • Given number density n, the covering fraction dAabs = σdN = σndV = σndAds Cross Section • Radiative transfer equation dAabs = σdN = σndV = σndAds dI ν = −σnds Iν • So for a particle description of the absorption coefficient αν = nσ. Cross Section • Applies to a generalized version of σ (e.g. free electrons are point particles but Thomson cross section is finite), cross section for interaction, can depend on frequency. • Mean free path of a photon L = 1/αν = 1/nσ, where n is the spatial number density of interacting particles (c.f. consider Thomson scattering in air) • But the total distance travelled by a photon is typically much greater than s = τL - an individual photon propagates through the medium via scattering as a random walk Scattering jν Iν dA scattering ds • Scattering can be viewed as absorption and emission where the emission is directly proportional to the absorption • Isotropic scattering: fraction ανIν absorbed and reradiated into 4π Z dΩ j = α I = α J ν ν 4π ν ν ν Scattering • Thus the source function Sν = jν/αν = Jν and at high optical depth, the specific intensity approaches its angle averaged value Jν • But despite this simple interpretation, the radiative transfer equation is an integro-differential equation dIν = −Iν + Jν dτν Z dIν dΩ = −Iν + Iν dτν 4π • that depends on Iν not only in the observation direction but all directions. Random Walk N L interactions R • Each of N steps has length L in a random direction ri = L^ri R = r1 + r2 ::: + rN • Total distance 2 hR i = hR · Ri = hr1 · r1i ::: + hrN · rN i + 2hr1 · r2i ::: 2 • Uncorrelated cross terms hri · rji = δijL Random Walk p 2 2 • Average distance hR i = NL or Rrms = NL • How many scatterings before escaping a distance R? 2 2 2 N = R =L = τν for optically thick • For optically thin τν 1, a typical photon does not scatter and so by definition a fraction τν will interact once, the rest zero and so the average N = τν 2 • Quick estimate N = max(τν; τν ) Multiple Processes • Combining processes: differential elements add, e.g. total opacity is sum of individual opacities – so highest opacity process is most important for blocking Iν • But given multiple frequency or spatial channels, energy escapes in the channel with the lowest opacity Example: a transition line vs continuum scattering – photons will wander in frequency out of line and escape through lower opacity scattering ! lines are often dark (sun) • Scattering (ανs) and absorption (ανa) dI ν = −α (I − S ) − α (I − J ) ds νa ν νa νs ν ν Multiple Processes • Collect terms dI ν = −(α + α )I + (α S + α J ) ds νa νs ν νa νa νs ν • Combined source function and absorption jν ανaSνa + ανsJν αν = ανa + ανs;Sν = = αν ανa + ανs • Mean free path L = 1/αν = 1=(ανa + ανs), typical length before absorption or scattering. Fraction that ends in absorption ανa ανs ν = ; 1 − ν = single scattering albedo ανa + ανs ανa + ανs Sν = νSνa + (1 − ν)Jν Formal Solution to Radiative Transfer Sν Iν(0) Iν(τν) 0 τν' τν • Formal solution τν Z 0 −τν 0 0 −(τν −τν ) Iν(τν) = Iν(0)e + dτνSν(τν)e 0 • Interpretation: initial specific intensity Iν attenuated by absorption and replaced by source function, attenuated by absorption from foreground matter Formal Solution to Radiative Transfer • Special case Sν independent of τν take out of integral τν Z 0 −τν 0 −(τν −τν ) Iν(τν) = Iν(0)e + Sν dτνe 0 −τν −τν = Iν(0)e + Sν[1 − e ] at low optical depth Iν unchanged, high optical depth Iν ! Sν • Integration is along the path of the radiation (direction dependent); source function can depend on Iν in a different direction! Fluid or Eddington Approximation .
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