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Basic Definitions Terms we will need to discuss radiative transfer. r n Iν dA s$ Specific Intensity I Iν(r,n,t) ≡ the amount of energy at position r (vector), traveling in direction n (vector) at time t per unit frequency interval, passing through a unit area oriented normal to the beam, into a unit solid angle in a second (unit time). If θ is the angle between the normal s (unit vector) of the reference surface dA and the direction n then the energy passing through dA is dEν = Iν(r,n,t) dA cos (θ) dν dω dt Iν is measured in erg Hz-1 s-1 cm-2 steradian-1 Basic Definitions 2 Specific Intensity II We shall only consider time independent properties of radiation transfer Drop the t We shall restrict ourselves to the case of planeparallel geometry. Why: the point of interest is d/D where d = depth of atmosphere and D = radius of the star. The formal requirement for plane-parallel geometry is that d/R ~ 0 For the Sun: d ~ 500 km, D ~ 7(105) km d/D ~ 7(10-4) for the Sun The above ratio is typical for dwarfs, supergiants can be of order 0.3. Basic Definitions 3 Specific Intensity III Go to the geometric description (z,θ,φ) - polar coordinates θ is the polar angle φ is the azimuthal angle Z is with respect to the stellar boundary (an arbitrary idea if there ever was one). Z φ Iν θ Z is measured positive upwards + above “surface” - below “surface” Basic Definitions 4 Mean Intensity: Jν(z) Simple Average of I over all solid angles J I ( z, , )d / d but 2 d 0 0 sin d d 4 Therefore 2 J (1/ 4 ) d 0 0 I ( z, , )sin d Iν is generally considered to be independent of φ so J 1/ 2 I ( z, )sin d 0 1 1/ 2 I ( z, )d 1 Where µ = cosθ ==> dµ = -sinθdθ The lack of azimuthal dependence implies homogeneity in the Basic Definitions atmosphere. 5 Physical Flux Flux ≡ Net rate of Energy Flow across a unit area. It is a vector quantity. F P I ( r , n ) nd Go to (z, θ, φ) and ask for the flux through one of the plane surfaces: F P F P kˆ I ( z, , ) cos d 1 2 I ( z, ) d 1 Note that the azimuthal dependence has been dropped! Basic Definitions 6 Astrophysical Flux F (1/ ) F A θ P 1 2 I ( z, ) d R θ r To Observer 1 • FνP is related to the observed flux! • R = Radius of star • D = Distance to Star • D>>R All rays are parallel at the observer • Flux received by an observer is dfν = Iν dω where • dω = solid angle subtended by a differential area on stellar surface • Iν = Specific intensity of that area. Basic Definitions 7 Astrophysical Flux II For this Geometry: dA = 2πrdr but R sinθ = r dr = R cosθ dθ dA = 2R sinθ Rcosθ dθ dA = 2R2 sinθ cosθ dθ Now μ = cosθ dA = 2R2 μdμ By definition dω = dA/D2 dω = 2(R/D)2 μ dμ Basic Definitions θ R θ r To Observer 8 Astrophysical Flux III Now from the annulus the radiation emerges at angle θ so the appropriate value of Iν is Iν(0,μ). Now integrate over the star: Remember dfν=Iνdω R 2 1 f 2 ( ) I (0, ) d D 0 R 2 P R 2 ( ) F ( ) FA D D NB: We have assumed I(0,-μ) = 0 ==> No incident radiation. We observe fν for stars: not FνP Inward μ: -1 < μ < 0 Outward μ: 0 < μ < -1 Basic Definitions 9 Moments of the Radiation Field 1 M ( z, n ) 1/ 2 I ( z, ) n d 1 Order 0 : ( the Mean Intensity ) 1 J ( z ) 1/ 2 I ( z, )d 1 Order 1: ( the Eddington Flux ) 1 H ( z ) 1/ 2 I ( z, ) d 1 Order 1: ( the K Integral ) 1 K ( z ) 1/ 2 I ( z, ) d 2 1 Basic Definitions 10 Invariance of the Specific Intensity • The definition of the specific intensity leads to invariance. P dA θ s s′ θ′ P′ dA′ Consider the ray packet which goes through dA at P and dA′ at P′ dEν = IνdAcosθdωdνdt = Iν′dA′cosθ′dω′dνdt • dω = solid angle subtended by dA′ at P = dA′cosθ′/r2 • dω′ = solid angle subtended by dA at P′ = dAcosθ/r2 • dEν = IνdAcosθ dA′cosθ′/r2 dνdt = Iν′dA′cosθ′dAcosθ/r2 dνdt • This means Iν = Iν′ 11 • Note that dEν contains Basic theDefinitions inverse square law. Energy Density I Consider an infinitesimal volume V into which energy flows from all solid angles. From a specific solid angle dω the energy flow through an element of area dA is δE = IνdAcosθdωdνdt Consider only those photons in flight across V. If their path length while in V is ℓ, then the time in V is dt = ℓ/c. The volume they sweep is dV = ℓdAcosθ. Put these into δEν: δEν = Iν (dV/ℓcosθ) cosθ dω dν ℓ/c δEν = (1/c) Iν dωdνdV Basic Definitions 12 Energy Density II Now integrate over volume Eνdν = [(1/c)∫VdVIνdω]dν Let V → 0: then Iν can be assumed to be independent of position in V. Define Energy Density Uν ≡ Eν/V Eν = (V/c)Iνdω Uν = (1/c) Iνdω = (4π/c) Jν Jν = (1/4π)Iνdω by definition Basic Definitions 13 Photon Momentum Transfer Momentum Per Photon = mc = mc2/c = hν/c Mass of a Photon = hν/c2 Momentum of a pencil of radiation with energy dEν = dEν/c dpr(ν) = (1/dA) (dEνcosθ/c) = ((IνdAcosθdω)cosθ) / cdA = Iνcos2θdω/c Now integrate: pr(ν) = (1/c) Iνcos2θdω = (4π/c)Kν Basic Definitions 14 Radiation Pressure Photon Momentum Transport Redux It is the momentum rate per unit and per unit solid angle = Photon Flux * (”m”v per photon) * Projection Factor dpr(ν) = (dEν/(hνdtdA)) * (hν/c) * cosθ where dEν = IνdνdAcosθdωdt so dpr(ν) = (1/c) Iνcos2θdωdν Integrate dpr(ν) over frequency and solid angle: 1 pr I cos2 d d c 0 4 Or in terms of frequency 1 pr ( ) I cos2 d c 4 4 K c Basic Definitions 15 Isotropic Radiation Field I(μ) ≠ f(μ) 1 1 J ( z ) 1/ 2 I ( z )d K ( z ) 1/ 2 I 2d 1 1/ 2 I ( | ) 1 1 1/ 2 I 2 d 1 1 1 1/ 2 I (1 ( 1)) I 1/ 2 I ( 3 |11 ) 3 1/ 2 I (1/ 3 ( 1/ 3)) I / 3 J / 3 Basic Definitions 16