The Equation of Transfer

Equation of Transfer2 Basic Ideas Change = Source - Sink Change = Source - Sink Change In Intensity = Emitted Energy -“Absorbed” Energy Change In Intensity = Emitted Energy -“Absorbed” Energy dI ν dνdμdtdA = j ν (ρdAds)dνdμdt -Κ ν I ν (ρdAds)dνdμdt dI ν dνdμdtdA = j ν (ρdAds)dνdμdt -Κ ν I ν (ρdAds)dνdμdt dI ν = ρj ν ds - ρΚ ν I ν ds dI ν = ρj ν ds - ρΚ ν I ν ds The geometry gives: dz = μds Z dz θ ds dI ν = j ν (ρ/μ) dz - Κ ν I ν (ρ/μ) dz μ dIν / ρdz = j ν - Κ ν I ν Remember: dτ ν = -ρΚ ν dz μ(dI ν /dτ ν ) = I ν – S ν Where S ν ≡ j ν / Κ ν

Equation of Transfer3 The Source Function S ν Κ ν is the total absorption coefficient Κ ν is the total absorption coefficient Κ ν = κ ν + σ ν Κ ν = κ ν + σ ν The equation of transfer is μ(dI ν /dτ ν ) = I ν - S ν The equation of transfer is μ(dI ν /dτ ν ) = I ν - S ν The Emission Coefficient j ν = j ν t + j ν s The Emission Coefficient j ν = j ν t + j ν s If there is no scattering: j ν s = 0 and σ ν = 0 If there is no scattering: j ν s = 0 and σ ν = 0 Then j ν = j ν t and Κ ν = κ ν Then j ν = j ν t and Κ ν = κ ν Thus S ν = j ν / Κ ν = j ν t / κ ν = B ν (T) Thus S ν = j ν / Κ ν = j ν t / κ ν = B ν (T) In the case of no scattering the source function is the Planck Function! In the case of no scattering the source function is the Planck Function!

Equation of Transfer4 The Slab Redux Slab: No Emission but absorption Slab: No Emission but absorption μdI ν /ρdz = - Κ ν I ν μdI ν /ρdz = - Κ ν I ν μdI = -ρΚ ν dz I ν μdI = -ρΚ ν dz I ν dI ν /I ν = dτ ν /μ dI ν /I ν = dτ ν /μ I(0,μ) = I(τ ν,μ)e -τ/μ I(0,μ) = I(τ ν,μ)e -τ/μ Note that the previous solution was for μ = 1 which is θ = 0. Note that the previous solution was for μ = 1 which is θ = 0.

Equation of Transfer5 Formal Solution μdI/dτ = I - S : Suppress the frequency dependence μdI/dτ = I - S : Suppress the frequency dependence It is a linear first-order differential equation with constant coefficients and thus must have an integrating factor: e -τ/μ It is a linear first-order differential equation with constant coefficients and thus must have an integrating factor: e -τ/μ d/dτ (Ie -τ/μ ) = -S e -τ/μ / μ d/dτ (Ie -τ/μ ) = -S e -τ/μ / μ dI/dτ e -τ/μ + I e -τ/μ (-1/μ) = -S e -τ/μ / μ dI/dτ e -τ/μ + I e -τ/μ (-1/μ) = -S e -τ/μ / μ dI/dτ + I/-μ = -S/μ dI/dτ + I/-μ = -S/μ μdI/dτ = -S + I μdI/dτ = -S + I

Equation of Transfer6 Boundary Conditions For the optical depths: τ 1 = 0 and τ 2 → ∞ For the optical depths: τ 1 = 0 and τ 2 → ∞ We Require: We Require: The above is the lower boundary “boundedness” condition. The upper boundary condition is that there be no incoming radiation. The above is the lower boundary “boundedness” condition. The upper boundary condition is that there be no incoming radiation.

Equation of Transfer7 Physical Meaning The physical meaning of the solution is that the emergent intensity is the weighted mean of the source function with the weights proportional to the fraction of energy getting to the surface at each wavelength / frequency. The solution depends on the form of S: The physical meaning of the solution is that the emergent intensity is the weighted mean of the source function with the weights proportional to the fraction of energy getting to the surface at each wavelength / frequency. The solution depends on the form of S: Let S = a+bt Let S = a+bt Then I(0,μ) = a + bμ Then I(0,μ) = a + bμ

Separate τ Into Domains Incoming and Outgoing at an Arbitrary Depth

Equation of Transfer9 What Is the Difficulty? Consider the Source Function S Consider the Source Function S What is the “Usual” form for S? What is the “Usual” form for S? For coherent isotropic scattering (a very common case): For coherent isotropic scattering (a very common case): S ν = (κ ν /(κ ν + σ ν )) B ν + (σ ν /(κ ν + σ ν )) J ν S ν = (κ ν /(κ ν + σ ν )) B ν + (σ ν /(κ ν + σ ν )) J ν So the solution of the equation of transfer is: So the solution of the equation of transfer is: But J depends on I: J ν = (1/4π)  I ν dμ But J depends on I: J ν = (1/4π)  I ν dμ Therefore S depends on I Therefore S depends on I This is not an easy problem in general! This is not an easy problem in general! These appear simple enough but …

Equation of Transfer10 Schwarzschild-Milne Equations Let us look at the Mean Intensity J ν Let us look at the Mean Intensity J ν In the second integral we have 1) reversed the order of integration so μ => -μ and 2) changed the sign on both parts of the exponent. (See Slide 8). In the second integral we have 1) reversed the order of integration so μ => -μ and 2) changed the sign on both parts of the exponent. (See Slide 8).

Equation of Transfer11 Continuing... For integrand 1: w = μ -1 => dw = -μ -2 dμ so dw/w = - dμ/μ. For the second w = -μ -1 for which dw/w = - dμ/μ also. In the first integral we also changed the order of integration (which cancelled the minus sign on dw/w). For integrand 1: w = μ -1 => dw = -μ -2 dμ so dw/w = - dμ/μ. For the second w = -μ -1 for which dw/w = - dμ/μ also. In the first integral we also changed the order of integration (which cancelled the minus sign on dw/w).

Equation of Transfer12 The Exponential Integral The Integrals in J(τ) are Called the Exponential Integrals These integrals are recursive!

Equation of Transfer13 Schwarzschild-Milne Equations The Mean Intensity Equation is called the Schwarzschild Equation and the flux equation is called the Milne equation. The surface flux is: