Basic Definitions Terms we will need to discuss radiative transfer.

2Basic Definitions Specific Intensity I I ν (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). I ν (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 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 dE ν = I ν (r,n,t) dA cos (θ) dν dω dt dE ν = I ν (r,n,t) dA cos (θ) dν dω dt I ν is measured in erg Hz -1 s -1 cm -2 steradian -1 I ν is measured in erg Hz -1 s -1 cm -2 steradian -1 IνIν

3Basic Definitions Specific Intensity II We shall only consider time independent properties of radiation transfer We shall only consider time independent properties of radiation transfer Drop the t Drop the t We shall restrict ourselves to the case of plane- parallel geometry. 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. 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 The formal requirement for plane-parallel geometry is that d/R ~ 0 For the Sun: d ~ 500 km, D ~ 7(10 5 ) km For the Sun: d ~ 500 km, D ~ 7(10 5 ) km d/D ~ 7(10 -4 ) for the Sun d/D ~ 7(10 -4 ) for the Sun The above ratio is typical for dwarfs, supergiants can be of order 0.3. The above ratio is typical for dwarfs, supergiants can be of order 0.3.

4Basic Definitions Specific Intensity III Go to the geometric description (z,θ,φ) - polar coordinates Go to the geometric description (z,θ,φ) - polar coordinates θ is the polar angle θ is the polar angle φ is the azimuthal angle φ is the azimuthal angle Z is with respect to the stellar boundary (an arbitrary idea if there ever was one). Z is with respect to the stellar boundary (an arbitrary idea if there ever was one). Z is measured positive upwards Z is measured positive upwards + above “surface” + above “surface” - below “surface” - below “surface” Z φ θ IνIν

5Basic Definitions Mean Intensity: J ν (z) Simple Average of I over all solid angles Simple Average of I over all solid angles I ν is generally considered to be independent of φ so I ν is generally considered to be independent of φ so Where µ = cosθ ==> dµ = -sinθdθ Where µ = cosθ ==> dµ = -sinθdθ The lack of azimuthal dependence implies homogeneity in the atmosphere. The lack of azimuthal dependence implies homogeneity in the atmosphere.

6Basic Definitions Physical Flux Flux ≡ Net rate of Energy Flow across a unit area. It is a vector quantity. 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: Go to (z, θ, φ) and ask for the flux through one of the plane surfaces: Note that the azimuthal dependence has been dropped! Note that the azimuthal dependence has been dropped!

7Basic Definitions Astrophysical Flux F ν P 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 df ν = I ν dω where dω = solid angle subtended by a differential area on stellar surface I ν = Specific intensity of that area. To Observer R r θ θ

8Basic Definitions Astrophysical Flux II For this Geometry: dA = 2πrdr but R sinθ = r dr = R cosθ dθ dA = 2R sinθ Rcosθ dθ dA = 2R 2 sinθ cosθ dθ Now μ = cosθ dA = 2R 2 μdμ By definition dω = dA/D 2 dω = 2(R/D) 2 μ dμ To Observer R r θ θ

9Basic Definitions Astrophysical Flux III Now from the annulus the radiation emerges at angle θ so the appropriate value of I ν is I ν (0,μ). Now integrate over the star: Remember df ν =I ν dω NB: We have assumed I(0,-μ) = 0 ==> No incident radiation. We observe f ν for stars: not F ν P Inward μ: -1 < μ < 0 Outward μ: 0 < μ < -1

10Basic Definitions Moments of the Radiation Field

11Basic 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′ dE ν = I ν dAcosθdωdνdt = I ν ′dA′cosθ′dω′dνdt d ω = solid angle subtended by dA′ at P = dA′cos θ ′/r 2 d ω ′ = solid angle subtended by dA at P′ = dAcos θ /r 2 dE ν = I ν dAcosθ dA′cosθ′/r 2 dνdt = I ν ′dA′cosθ′dAcosθ/r 2 dνdt This means I ν = I ν ′ Note that dE ν contains the inverse square law. s s′ dA dA′ PP′ θ θ′θ′

12Basic 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 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 = I ν dAcosθdωdνdt δE = I ν dAcosθdωdνdt 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 δE ν : 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 δE ν : δE ν = I ν (dV/ℓcosθ) cosθ dω dν ℓ/c δE ν = I ν (dV/ℓcosθ) cosθ dω dν ℓ/c δE ν = (1/c) I ν dωdνdV δE ν = (1/c) I ν dωdνdV

13Basic Definitions Energy Density II Now integrate over volume Now integrate over volume E ν dν = [(1/c)∫ V dV  I ν dω]dν E ν dν = [(1/c)∫ V dV  I ν dω]dν Let V → 0: then I ν can be assumed to be independent of position in V. Let V → 0: then I ν can be assumed to be independent of position in V. Define Energy Density U ν ≡ E ν /V Define Energy Density U ν ≡ E ν /V E ν = (V/c)  I ν dω E ν = (V/c)  I ν dω U ν = (1/c)  I ν dω = (4π/c) J ν U ν = (1/c)  I ν dω = (4π/c) J ν J ν = (1/4π)  I ν dω by definition J ν = (1/4π)  I ν dω by definition

14Basic Definitions Photon Momentum Transfer Momentum Per Photon = mc = mc 2 /c = hν/c Momentum Per Photon = mc = mc 2 /c = hν/c Mass of a Photon = hν/c 2 Mass of a Photon = hν/c 2 Momentum of a pencil of radiation with energy dE ν = dE ν /c Momentum of a pencil of radiation with energy dE ν = dE ν /c dp r (ν) = (1/dA) (dE ν cosθ/c) dp r (ν) = (1/dA) (dE ν cosθ/c) = ((I ν dAcosθdω)cosθ) / cdA = I ν cos 2 θdω/c = ((I ν dAcosθdω)cosθ) / cdA = I ν cos 2 θdω/c Now integrate: p r (ν) = (1/c)  I ν cos 2 θdω = (4π/c)K ν Now integrate: p r (ν) = (1/c)  I ν cos 2 θdω = (4π/c)K ν

15Basic Definitions Radiation Pressure It is the momentum rate per unit and per unit solid angle = Photon Flux * (”m”v per photon) * Projection Factor It is the momentum rate per unit and per unit solid angle = Photon Flux * (”m”v per photon) * Projection Factor dp r (ν) = (dE ν /(hνdtdA)) * (hν/c) * cosθ where dE ν = I ν dνdAcosθdωdt so dp r (ν) = (1/c) I ν cos 2 θdωdν dp r (ν) = (dE ν /(hνdtdA)) * (hν/c) * cosθ where dE ν = I ν dνdAcosθdωdt so dp r (ν) = (1/c) I ν cos 2 θdωdν Integrate dp r (ν) over frequency and solid angle: Integrate dp r (ν) over frequency and solid angle: Photon Momentum Transport Redux

16Basic Definitions Isotropic Radiation Field I(μ) ≠ f(μ)