Terms we will need to discuss radiative transfer. Basic Definitions Terms we will need to discuss radiative transfer.

Specific Intensity I s $ r n $ s dA Iν 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

Specific Intensity II We shall only consider time independent properties of radiation transfer Drop the t We shall restrict ourselves to the case of plane- parallel 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, R ~ 7(105) km d/R ~ 7(10-4) for the Sun The above ratio is typical for dwarfs, supergiants can be of order 0.3. Basic Definitions

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 is measured positive upwards + above “surface” - below “surface” Z φ Iν θ Basic Definitions

Mean Intensity: Jν(z) Simple Average of I over all solid angles Iν is generally considered to be independent of φ so Where µ = cosθ ==> dµ = -sinθdθ The lack of azimuthal dependence implies homogeneity in the atmosphere. Basic Definitions

Physical Flux Flux ≡ Net rate of Energy Flow across a unit area. It is a vector quantity. Go to (z, θ, φ) and ask for the flux through one of the plane surfaces: Note that the azimuthal dependence has been dropped! Basic Definitions

Astrophysical Flux FνP is related to the observed flux! To Observer R r θ θ 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

Astrophysical Flux II For this Geometry: dA = 2πrdr but R sinθ = r dr = R cosθ dθ dA = 2πR sinθ Rcosθ dθ dA = 2πR2 sinθ cosθ dθ Now μ = cosθ dA = 2πR2 μdμ By definition dω = dA/D2 dω = 2π (R/D)2 μ dμ To Observer R r θ θ Basic Definitions

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 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

Moments of the Radiation Field Basic Definitions

Invariance of the Specific Intensity The definition of the specific intensity leads to invariance. 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′ Note that dEν contains the inverse square law. s s′ dA dA′ P P′ θ θ′ Basic Definitions

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

Energy Density II Now integrate over volume Eνdν = [(1/c)∫VdVIνdω]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

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

Radiation Pressure Photon Momentum Transport Redux It is the momentum rate per unit time and per unit solid angle = Photon Flux * (“m”v per photon) * Projection Factor dpr(ν) = (dEν/(hνdA)) * (hν/c) * cosθ where dEν = IνdνdAcosθdωdt so dpr(ν) = (1/c) Iνcos2θdωdν Integrate dpr(ν) over frequency and solid angle: Basic Definitions

Isotropic Radiation Field I(μ) ≠ f(μ) Basic Definitions