Fundamental Stellar Parameters

Transcription

1 Fundamental Stellar Parameters Radiative Transfer Specific Intensity, Radiative Flux and Stellar Luminosity Observed Flux, Emission and Absorption of Radiation Radiative Transfer Equation, Solution and Boundary Conditions Diffusion Approximation and Radiative Equilibrium Stellar Atmospheres Equations of Stellar Structure Nuclear Reactions in Stellar Interiors

2 Radiative Transfer Introduction Relative abundances of chemical elements are adjusted until synthetic spectra agree with observation; requirements are: Accurate laboratory determinations of atomic and molecular data, primarily oscillator strengths. T eff and log g from some combination of energy distributions, Balmer line profiles, ionisation ratios and molecular association ratios. Temperature, gas pressure and electron pressure dependence on geometric or optical depth in the stellar atmosphere. Broadening theory for lines used in abundance determinations. Radiation transfer theory to calculate the wavelength-dependent emergent flux Dependent on radiative transfer; the subject of this lecture.

3 Specific Intensity normal de ν θ dω da de ν = I ν cos θ da cos θ dω dν dt I λ = c λ 2I ν

4 Stellar Evolution Radiative Transfer Specific Intensity, Radiative Flux and Stellar Luminosity Specific Intensity Specific Intensity normal θ da dω deν deν = Iν cos θdacos θdω dνdt Consider an infinitesimal area da which radiates energy de ν in time dt, in frequency interval dν, into solid angle dω and in a direction inclined at an angle θ to the normal to da. The projected infinitesmal area in the direction of de ν is da cos θ. Clearly de ν is proportional to dt, dν, dω and da cos θ. I ν cos θ is adopted as the constant of proportionality, the cos θ dependence arises because de ν = when θ = π/2 and attains a maximum when θ =. Iλ = c λ 2Iν 1

5 Specific Intensity Invariance Emitter ˆn θ ˆn θ Receiver da da dω de dω ν dω = de ν = I ν (cos θ) da(cos θ) dω dν dt = I ν (cos θ ) da (cos θ ) dω dν dt projected area = da (cos θ ) da(cos θ), dω = distance 2 d 2 d 2 I ν = I ν

6 Stellar Evolution Radiative Transfer Specific Intensity, Radiative Flux and Stellar Luminosity Specific Intensity Invariance Specific Intensity Invariance da Emitter ˆn θ dω = dω deν dω deν = Iν(cos θ) da(cos θ) dω dν dt = Iν (cos θ ) da (cos θ ) dω dν dt projected area = da (cos θ ), dω = distance 2 d 2 Iν = Iν An infinitesimal area da acts as a receiver of an infinitesimal amount of energy de ν emitted by another infinitesimal area da, at which da subtends an infinitesimal solid angle dω. Similarly, da subtends an infinitesimal solid angle dω at da. As energy is conserved in the absence of any absorption or scattering along the line of sight, the de ν emitted by da into dω is also received by da from dω. After substituting for dω and dω in terms of the projected area divided by the squared distance between da and da, it follows that the specific intensity I ν at the receiver is the same as the specific intensity I ν emitted by the emitter. 1 ˆn θ Receiver da da(cos θ) d 2

7 Spherical Coordinates dω = da r 2 da = (r dθ)(r sin θ dφ) θ+dθ θ r sin θ dφ r da r dθ r sin θ dφ dω = sin θ dθ dφ Let µ = cos θ dµ = sin θ dθ dω = dµ dφ

8 r sin θ dφ r da r dθ r sin θ dφ Stellar Evolution Radiative Transfer Specific Intensity, Radiative Flux and Stellar Luminosity Spherical Coordinates Spherical Coordinates dω = da r 2 da = (r dθ)(r sin θ dφ) dω = sin θ dθ dφ Let µ = cos θ dµ = sin θ dθ dω = dµ dφ θ+dθ θ Since it is usual to consider an infinitesimal area da to reside on the surface of a sphere, it is customary to express it and an infinitesimal solid angle dω in terms of spherical polar coordinates (r, θ, φ). Here r is the radius of a sphere, θ the altitudinal angle and φ the azimuthal angle. 1

9 Radiative Flux Energy flowing through element of area da in unit time, per unit frequency interval, per unit area is the monochromatic physical flux F ν : a F ν = π F ν = I ν (cos θ) cos θ dω = = 2π π 2π π/2 = F ν + + F ν I ν (cos θ) cos θ sin θ dθ dφ I ν (cos θ) cos θ sin θ dθ dφ + 2π π π/2 I ν (cos θ) cos θ sin θ dθ dφ

10 π/ Stellar Evolution Radiative Transfer Specific Intensity, Radiative Flux and Stellar Luminosity Radiative Flux Radiative Flux Energy flowing through element of area da in unit time, per unit frequency interval, per unit area is the monochromatic physical flux Fν: Fν = π Fν a = Iν(cos θ) cos θ dω 2π π = Iν(cos θ) cos θ sin θ dθ dφ 2π π/2 = Iν(cos θ) cos θ sin θ dθ dφ + = Fν + + Fν 2π π Iν(cos θ) cos θ sin θ dθ dφ For historical reasons, a monochromatic astrophysical flux F ν a is defined such that when this quantity is multiplied by π, the monochromatic physical flux is obtained. The latter is defined as the energy flowing through an element of area da at frequency ν in unit time, per unit frequency interval, per unit area; this is the specific intensity, with projection effects taken into account, integrated over all solid angles. Expressing the dω in terms of dθ and dφ leads to a double integral over θ and φ. Since φ 2π it follows that θ π. The interval θ π/2 corresponds to an emergent monochromatic flux (F ν + ) whereas the interval π/2 θ π corresponds to an incident monochromatic flux (F ν ).

11 Stellar Luminosity Energy flowing through element of area da in unit time, per unit area is the total physical flux F rad : F rad = F ν dν In general, no incident flux at outer boundary of atmosphere (r = R ): F rad = = F ν + dν 2π π/2 I ν (cos θ) cos θ sin θ dθ dφ dν Stellar luminosity is energy per unit time from the entire stellar surface: L = 4π R 2 F rad

12 Stellar Evolution Radiative Transfer Specific Intensity, Radiative Flux and Stellar Luminosity Stellar Luminosity Stellar Luminosity Energy flowing through element of area da in unit time, per unit area is the total physical flux Frad: Frad = Fν dν In general, no incident flux at outer boundary of atmosphere (r = R): Frad = = Fν + dν 2π π/2 Iν(cos θ) cos θ sin θ dθ dφ dν Stellar luminosity is energy per unit time from the entire stellar surface: L = 4π R 2 Frad The total physical flux passing through an element of area is the integral over all frequencies of the monochromatic flux; this gives the energy per unit time and per unit area emerging from the stellar surface, in the usual case where there is no incident flux. Multiplication by the surface area gives the stellar luminosity.

13 Observed Flux - I da da θ θ To Observer R For telescope aperture area da at distance d: dω = da d 2 Contribution of da to energy received by observer at frequency ν in interval dν, in time dt (note da = R 2 sin θ dθ dφ): de ν = I ν (cos θ) cos θ da dω dν dt

14 Stellar Evolution Radiative Transfer Observed Flux, Emission and Absorption of Radiation Observed Flux - I Observed Flux - I da da R θ θ To Observer For telescope aperture area da at distance d: dω = da d 2 Contribution of da to energy received by observer at frequency ν in interval dν, in time dt (note da = R 2 sin θ dθ dφ): deν = Iν(cos θ) cos θ dadω dν dt The aperture of an observer s telescope, and distance from the star, determine the solid angle subtended at the observer s telescope by any element of area on the stellar surface. 1

15 Observed Flux - II de ν = I ν (cos θ) cos θ da dω dν dt = da d R 2 2 I ν (cos θ) cos θ sin θ dθ dφ dν dt Integrating over the half-sphere facing the observer: E ν = da d R 2 2 dν dt = da d 2 R 2 dν dt F ν + 2π π/2 I ν (cos θ) cos θ sin θ dθ dφ Monochromatic flux received by observer in unit time, per unit area per unit frequency: f ν = R 2 d F 2 ν +

16 Fν Stellar Evolution Radiative Transfer Observed Flux, Emission and Absorption of Radiation Observed Flux - II Observed Flux - II deν = Iν(cos θ) cos θ dadω dν dt = da d 2 R2 Iν(cos θ) cos θ sin θ dθ dφ dν dt Integrating over the half-sphere facing the observer: Eν = da d 2 2π π/2 R2 dν dt Iν(cos θ) cos θ sin θ dθ dφ = da d 2 R2 dν dt Fν + Monochromatic flux received by observer in unit time, per unit area per unit frequency: fν = R 2 d 2 Substituting for that solid angle (dω) and expressing the element of area da in terms of spherical polar coordinates, yields an expression which can be integrated over the hemisphere facing the observer to give the monochromatic flux received by the observer in unit time per unit area per unit frequency. The result deduced in the previous lecture is recovered, namely that the observer s flux is the corresponding flux emergent at the stellar surface scaled by the squared ratio of the stellar radius to the stellar distance.

17 Specific Intensity Moments J ν = 1 4π I ν dω = 1 4π 2π dφ +1 1 I ν dµ = I ν dµ H ν = 1 4π K ν = 1 4π I ν cos θ dω = 1 2 I ν cos 2 θ dω = I ν µ dµ = F ν a 1 I ν µ 2 dµ 4

18 Stellar Evolution Radiative Transfer Observed Flux, Emission and Absorption of Radiation Specific Intensity Moments Specific Intensity Moments Jν = 1 4π Hν = 1 Iν dω = 1 4π Kν = 1 4π 4π 2π +1 dφ 1 Iν cos θ dω = 1 2 Iν cos 2 θ dω = Iν dµ = Iν µ dµ = Fνa +1 1 Iν µ 2 dµ 4 Iν dµ Weighted averages of the specific intensity are introduced for further development of radiative transfer theory. General expressions are used to derive special cases, valid in spherical geometry, by expressing the infinitesimal solid angle in terms of the spherical polar coordinates θ and φ remembering that µ = cos θ. J ν, H ν, K ν are respectively the zeroth, first and second moments of the specific intensity I ν. J ν is also referred to as the mean intensity and H ν as the Eddington Flux.

19 Energy Density & Radiation Pressure Energy Density: radiation energy u ν = volume Radiation Pressure: = 1 c 4π I ν dω p ν = force area d momentum (= E/c) 1 = dt area = 1 c 4π I ν cos 2 θ dω

20 Stellar Evolution Radiative Transfer Observed Flux, Emission and Absorption of Radiation Energy Density & Radiation Pressure Energy Density & Radiation Pressure Energy Density: Radiation Pressure: uν = = 1 c pν = force area = = 1 c radiation energy 4π volume Iν dω d momentum (= E/c) 4π dt Iν cos 2 θ dω 1 area The monochromatic energy density is the radiation energy at frequency ν per unit frequency interval, per unit time, per unit volume. For radiation flow through unit area, the volume traversed in unit time is determined by the velocity of light. The required energy density per unit frequency is therefore the specific intensity integrated over all solid angles and divided by the speed of light. Since the radiation flow is normal to the unit area, cos θ = 1. The monochromatic radiation pressure due to radiation emerging at an angle θ with respect to the normal to an infinitesimal area is the rate of change of radiation field momentum in the direction normal to the element of area, divided by its area, which is (de ν /c) cos θ/da.

21 Absorption of Radiation I ν ds I ν + di ν di ν = κ ν I ν ds κ ν : absorption coefficient [ cm 1] Microscopically: κ ν = n σ ν Over a distance s: s di ν = I ν s κ ν ds = τ ν, dτ ν = κ ν ds I ν (s) = I ν () exp( τ ν ) By convention, τ ν = at top of atmosphere and increases inwards.

22 Stellar Evolution Radiative Transfer Observed Flux, Emission and Absorption of Radiation Absorption of Radiation Absorption of Radiation Iν ds Over a distance s: s diν Iν Iν + diν s = κν ds = τν, diν = κν Iν ds κν : absorption coefficient [ cm 1] Microscopically: κν = nσν Iν(s) = Iν() exp( τν) dτν = κν ds By convention, τν = at top of atmosphere and increases inwards. An absorption coefficient κ ν is expressed in units of inverse length so as to express the decrease in specific intensity (I ν ) on passing through a slab of material of thickness ds. Obviously the absorption is directly proportional to the incident radiation in contrast with emission which is independent of it. Microscopically κ ν will be given by the absorption cross-section (in appropriate units of area) of each absorber, multiplied by the number of absorbers per unit volume (also in appropriate units). 1

23 Optical Depth Optical thickness of a layer determines the specific intensity fraction passing through it. If τ ν = 1, I ν (s) = I ν ()/e.37i ν (). We can see through an atmosphere to the point where τ ν 1. Optically thick(thin) medium: τ ν >(<)1. τ ν = 1 has a geometrical interpretation in terms of the mean free path of photons s. τ ν = 1 = s κ ν ds Photons travel on average for a distance s before absorption.

24 Radiative Acceleration Infinitesimal energy absorbed : de ν a = di ν a cos θ da dω dt dν = κ ν I ν cos θ da dω dt dν ds If κ ν is isotropic, total energy absorbed is: E a = = κ ν I ν cos θ dω dν da ds dt κ ν F ν dν da ds dt Consequent rate of change of photon momentum (E/c)/dt leads to the radiative acceleration (g rad ): κ 1 ν F ν dν da dt ds = g rad dm c dt g rad = 1 κ ν F ν dν dm = ρ da ds cρ

25 κν Stellar Evolution Radiative Transfer Observed Flux, Emission and Absorption of Radiation Radiative Acceleration Radiative Acceleration Infinitesimal energy absorbed : deν a = diν a cos θ da dω dt dν = κν Iν cos θdadω dtdνds If κν is isotropic, total energy absorbed is: E a = = Iν cos θ dω dν da ds dt κν Fν dνdadsdt Consequent rate of change of photon momentum (E/c)/dt leads to the radiative acceleration (grad): κν Fν dν 1 da dt ds = grad dm c dt grad = 1 κν Fν dν dm = ρ da ds cρ If the absorption coefficient is isotropic the total energy absorbed by a slab of thickness ds, of area da in time dt is obtained by integrating over frequency and solid angle. The corresponding rate of change of photon momentum gives the radiative force causing radiative acceleration.

26 Emission of Radiation dv = da cos θ ds ds da cos θ dω di ν e Energy radiated into dω by da cos θ due to emission processes in dv : de ν e = di ν e da cos θ dω dν dt = ɛ ν da dω cos θ dν dt ds = ɛ ν dv dω dν dt ɛ ν is defined as the emission coefficient and has dimensions [ erg cm 3 sr 1 Hz 1 s 1]

27 Stellar Evolution Radiative Transfer Observed Flux, Emission and Absorption of Radiation Emission of Radiation Emission of Radiation dv = da cos θ ds ds da cos θ dω diν e Energy radiated into dω by da cos θ due to emission processes in dv : deν e = diν e da cos θ dω dν dt = ɛν dadω cos θdνdtds = ɛν dv dω dν dt ɛν is defined as the emission coefficient and has dimensions [ erg cm 3 sr 1 Hz 1 s 1] The infinitesimal increment to the monochromatic specific intensity (de ν ) contributed by emission in a slab of thickness ds is ɛ ν ds. Note that unlike absorption processes this is independent of the incident specific intensity (I ν ).

28 Equation of Radiative Transfer Combine emission and absorption de ν a = di ν a da cos θ dω dt dν = κ ν I ν cos θ da dω dt dν ds de ν e = di ν e da cos θ dω dt dν = ɛ ν cos θ da dω dt dν ds de ν a + de ν e = (di ν a + di ν e ) da cos θ dω dν dt Writing = ( κ ν I ν + ɛ ν ) da cos θ dω dν dt ds di ν = (di ν a + di ν e ) gives the differential equation (the equation of radiative transfer) di ν ds = κ ν I ν + ɛ ν describing the flow of radiation through matter.

29 Stellar Evolution Radiative Transfer Radiative Transfer Equation, Solution and Boundary Conditions Equation of Radiative Transfer Equation of Radiative Transfer Combine emission and absorption Writing deν a = diν a da cos θ dω dt dν = κν Iν cos θ da dω dt dν ds deν e = diν e da cos θ dω dt dν = ɛν cos θ da dω dt dν ds deν a + deν e = (diν a + diν e ) da cos θ dω dν dt = ( κν Iν + ɛν) da cos θdω dνdtds diν = (diν a + diν e ) gives the differential equation (the equation of radiative transfer) diν = κν Iν + ɛν ds describing the flow of radiation through matter. The equation of radiative transfer is a differential equation giving the rate of change of monochromatic specific intensity (I ν ) with distance (s), due to emission and absorption processes, as radiation is propagated through a slab of gas. Solving the equation of radiative transfer gives I ν (s). As presented here, the equation of radiative transfer is one-dimensional and time-independent.

30 Equation of Radiative Transfer - Plane Parallel lines of constant ρ, T dx = cos θ ds = µ ds ds dx θ ray µ di ν(µ, x) dx d ds = µ d dx = κ ν I ν (µ, x) + ɛ ν R i R o x 1

31 ds dx θ ray Stellar Evolution Radiative Transfer Radiative Transfer Equation, Solution and Boundary Conditions Equation of Radiative Transfer - Plane Parallel Equation of Radiative Transfer - Plane Parallel Ri lines of constant ρ, T Ro x 1 µ dx = cos θ ds = µ ds diν(µ, x) dx d ds = µ d dx = κν Iν(µ, x) + ɛν In plane parallel geometry, density and temperature (for example) are constant at any given depth. The thickness of the stellar atmosphere in this approximation is taken to be negligible in comparison with the radius of the star. All quantities are specified in terms of the depth coordinate (x) and a simple transformation allows a monochromatic specific intensity increment to be expressed in terms of x and µ = cos θ.

32 Equation of Radiative Transfer - Spherical d ds = dr ds r + dθ ds θ lines of constant ρ, T Angle θ between ray and radial direction is not constant. θ+dθ ray dr = cos θ ds = µ ds r dθ = sin θ ds dr ds = cos θ dθ ds = sin θ r θ = µ θ µ = sin θ µ dθ R i ds θ dr r (dθ < ) r dθ R o d ds =µ r + sin2 θ r µ =µ r + 1 µ2 r µ µ I ν(µ, r) r + 1 µ2 r I ν (µ, r) = κ ν I ν (µ, r) + ɛ ν µ

33 dθ Ri ds θ dr r r dθ θ+dθ (dθ < ) ray Ro Stellar Evolution Radiative Transfer Radiative Transfer Equation, Solution and Boundary Conditions Equation of Radiative Transfer - Spherical Equation of Radiative Transfer - Spherical lines of constant ρ, T Angle θ between ray and radial direction is not constant. d ds = dr ds r + dθ ds θ dr = cos θ ds = µ ds r dθ = sin θ ds θ = µ θ Iν(µ, r) µ + 1 µ2 r r dr ds = cos θ dθ ds = sin θ r µ = sin θ µ d ds =µ r + sin2 θ r =µ r + 1 µ2 r Iν(µ, r) µ µ µ = κν Iν(µ, r) + ɛν In spherical geometry the angle θ between a ray and the radial direction is not constant. As a consequence, the equation of radiative transfer is more complicated than in the plane-parallel case.

34 Optical Depth & Source Function Optical depth increasing towards interior. τ ν x p τ ν = κ ν dx = dτ ν τ ν = xp For a plane-parallel atmosphere gives where R o κ ν dx µ di ν(µ, x) = κ ν (x) I ν (µ, x) + ɛ ν dx µ di ν(µ, τ ν ) dτ ν = I ν (µ, τ ν ) S ν (τ ν ) S ν = ɛ ν /κ ν is the SourceFunction. R i R o x κ ν = dτ ν ds τ ν s 1 s S ν = ɛ ν /κ ν ɛ ν s Since photon mean free path is τ ν = 1, S ν corresponds to intensity emitted over this distance. 1

35 Ri τν xp Ro κν dx = dτν dτν Sν = ɛν/κν Ro Stellar Evolution Radiative Transfer Radiative Transfer Equation, Solution and Boundary Conditions Optical Depth & Source Function Optical Depth & Source Function Optical depth increasing towards interior. τν = xp τν = κν dx For a plane-parallel atmosphere gives where diν(µ, x) µ dx diν(µ, τν) µ = κν(x)iν(µ, x) + ɛν = Iν(µ, τν) Sν(τν) is the SourceFunction. x κν = dτν ds τν s 1 s Sν = ɛν/κν ɛν s Since photon mean free path is τν = 1, Sν corresponds to intensity emitted over this distance. 1 Monochromatic optical depth (τ ν ) is introduced as minus the integral of the appropriate absorption coefficient over the corresponding interval in atmospheric height, where the minus sign is needed because optical depth increases as height in the atmosphere decreases. Then it is natural to express the ratio ɛ ν /κ ν as the source function (S ν ) and reformulate the equation of radiative transfer accordingly.

36 Radiative Transfer Equation Formal Solution ray τ/µ τ θ ν τ 2 τ 1 For a plane-parallel atmosphere: µ di ν dτ ν = I ν S ν e τν/µ µ di ν dτ ν = I ν e τν/µ S ν e τν/µ d (I ν e τν/µ ) = S ν e τν/µ dτ ν µ Integrate between τ 1 (outside) to τ 2 (inside, τ 1 < τ 2 ): [I ν e τν/µ ] τ2 τ 1 = τ2 τ 1 S ν e τν/µ dτ ν µ I ν (τ 1, µ) = I ν (τ 2, µ) e (τ 2 τ 1 )/µ + τ2 τ 1 S ν (t) e (t τ 1)/µ dt µ In general, S ν depends on I ν and an actual solution is challenging.

37 τν τ2 τ1 τ1 τ1 τ Stellar Evolution Radiative Transfer Radiative Transfer Equation, Solution and Boundary Conditions Radiative Transfer Equation Formal Solution Radiative Transfer Equation Formal Solution τ/µ θ ray For a plane-parallel atmosphere: µ diν dτν e τν/µ µ diν dτν = Iν Sν = Iν e τν/µ Sν e τν/µ d (Iν e τν/µ Sν e τν/µ ) = dτν µ Integrate between τ1 (outside) to τ2 (inside, τ1 < τ2): [Iν e τν/µ ] τ2 τ2 dτν τν/µ = Sν e µ τ2 Iν(τ1, µ) = Iν(τ2, µ) e (τ2 τ1)/µ dt (t τ1)/µ + Sν(t) e µ In general, Sν depends on Iν and an actual solution is challenging. A formal solution of the radiative transfer equation for plane parallel geometry, using optical depth as the independent variable, may be obtained by multiplying by e τν/µ and collecting terms in I ν on the left-hand side and S ν on the right-hand side. The left-hand side is easily integrated between two optical depths to give the monochromatic specific intensity at one optical depth in terms of the monochromatic specific intensity at the other optical depth and some integral involving the monochromatic source function. In general S ν is dependent on I ν and further progress is a challenge

38 Radiative Transfer Equation Boundary Conditions For incoming radiation µ < and inward radiation from outside is usually neglected: I ν (τ ν =, µ < ) = I in ν (τ ν, µ) = τ ν dt (t τν)/µ S ν (t) e µ For outgoing radiation µ > and we have either a finite slab or shell for which I ν (τ max, µ) = I + ν (µ) or a semi-infinite (planar or spherical) case where The second applies in the case of stellar atmospheres and so I out ν (τ ν, µ) = τ ν dt (t τν)/µ S ν (t) e µ lim I ν(τ ν, µ) e τν/µ = τ ν For the case where τ ν =, the above equation gives the emergent intensity: I ν (, µ) = S ν (t) e t/µ dt µ

39 τν τν Stellar Evolution Radiative Transfer Radiative Transfer Equation, Solution and Boundary Conditions Radiative Transfer Equation Boundary Conditions From any point in a stellar atmosphere, the gas becomes opaque Radiative Transfer Equation Boundary Conditions For incoming radiation µ < and inward radiation from outside is usually neglected: Iν(τν =, µ < ) = Iν in (τν, µ) = dt (t τν)/µ Sν(t) e µ For outgoing radiation µ > and we have either a finite slab or shell for which Iν(τmax, µ) = I + ν (µ) or a semi-infinite (planar or spherical) case where The second applies in the case of stellar atmospheres and so Iν out (τν, µ) = dt (t τν)/µ Sν(t) e µ lim Iν(τν, µ) τν e τν/µ = For the case where τν =, the above equation gives the emergent intensity: (τ ν at all frequencies in the direction of the stellar centre. As a result, radiation is only received from the surface layers; this provides the boundary condition in so far as the specific intensity at any finite optical depth is determined by an integral involving the source function between that optical depth and infinity. Iν(, µ) = dt t/µ Sν(t) e µ

40 Source Function Simple Cases I In Local Thermodynamic Equilibrium, photons are absorbed and re-emitted at the local temperature (T ) (Kirchhoff s Law) S ν = ɛ ν κ ν = B ν (T ). For coherent isotropic scattering, absorption is characterised by the scattering coefficient σ ν, analogous to κ ν : di ν = σ ν I ν ds de em ν = ɛ sc ν dω de abs ν = σ sc ν I ν dω. 4π 4π At each ν, de em ν = de abs ν : ɛ sc ν dω = σ ν I ν dω 4π 4π sc ɛ ν dω = σ ν I ν dω 4π 4π sc ɛ ν = 1 κ ν 4π S ν = J ν. 4π I ν dω Source function is completely dependent on radiation field and independent of T.

41 Source Function Simple Cases II Mixed case: S ν = ɛ ν + ɛ ν sc κ ν + σ ν = κ ν κ ν + σ ν ɛ ν κ ν + σ sc ν σ ν κ ν + σ ν ɛ ν = κ ν κ ν + σ ν B ν + σ ν κ ν + σ ν J ν

42 Stellar Evolution Radiative Transfer Radiative Transfer Equation, Solution and Boundary Conditions Source Function Simple Cases II Source Function Simple Cases II Mixed case: Sν = ɛν + ɛνsc κν + σν = κν κν + σν = κν κν + σν ɛν κν + Bν + σν κν + σν σν κν + σν ɛν sc σν Jν Simple solutions to the radiative transfer equation arise when photons are absorbed and re-emitted at the local temperature so that the source function (S ν ) is equal to the Planck Function (B ν ). A second simple case is where absorption is absent and photons are only scattered isotropically and coherently; in this case the source function is the mean intensity (J ν ). In the mixed case, the contributions to S ν from absorption and isotropic coherent scattering are separated. Because J ν depends on I ν, some iterative scheme is needed to solve the transfer equation in all but the simplest case when S ν = B ν.

43 Diffusion Approximation I At large optical depths (τ ν 1) and photons are local so that S ν B ν. Expanding as a power series about τ ν : S ν (t) = B ν (t) = d n B ν (τ ν ) (t τ ν ) n /n! n= dτ ν n Photons are local and therefore (t τ ν ), justifying the retention of only the first order term (Diffusion Approximation): I out ν (τ ν, µ) = = B ν (t) = B ν (τ ν ) + db ν dτ ν (t τ ν ) τ ν τ ν dt (t τν)/µ S ν (t) e µ [ B ν (τ ν ) + db ν dτ ν (t τ ν ) ] e (t τν)/µ dt µ

44 Diffusion Approximation II Let and since u = t τ ν µ u k e u du = k! dt = µ du [ Iν out (τ ν, µ) = B ν (t) + db ] ν µ u e u du τ ν/µ dτ ν I in ν (τ ν, µ) = = B ν (t) + µ db ν dτ ν τν/µ [ B ν (t) + db ] ν µ u e u du dτ ν Eddington-Barbier relation for observed emergent intensity obtained for τ ν = ; it depends linearly on µ.

45 τν/µ Stellar Evolution Radiative Transfer Diffusion Approximation and Radiative Equilibrium Diffusion Approximation II Diffusion Approximation II Let and since u = t τν µ Iν out (τν, µ) = u k e u du = k! [ Bν(t) + dbν = Bν(t) + µ dbν dτν dt = µ du dτν τν/µ [ Iν in (τν, µ) = Bν(t) + dbν dτν ] µ u e u du ] µ u e u du Eddington-Barbier relation for observed emergent intensity obtained for τν = ; it depends linearly on µ. The diffusion approximation provides a robust solution to the radiative transfer equation at large optical depths, as appropriate for stellar interiors, when photons are necessarily local. It is supposed that temperature is monotonically increasing with optical depth (τ ν ) and so the Planck Function (B ν ) may, in this context, be regarded as a function of τ ν. The approach is then to expand B ν in a Taylor s Series about the local τ ν and as the optical depth range is small, only the first order term need be retained. The standard integral is Equation in Gradshteyn & Ryzhik.

46 Eddington Approximation In a planar atmosphere with we have: J ν = 1 2 H ν = F ν a K ν = 1 2 = = 1 2 I ν dµ = db ν dτ ν = I ν µ 2 dµ = 1 2 I ν = B ν + µ db ν dτ ν [ µ B ν + µ2 2 I ν µ dµ = κ ν db ν dx = 1 3 [ µ 3 [ µ 2 3 B ν + µ4 4 db ν dτ ν ] B ν + µ3 3 1 db ν dt κ ν dt dx db ν dτ ν ] +1 1 = B ν (τ ν ) db ν dτ ν ] +1 1 = 1 3 B ν(τ ν )

47 Stellar Evolution Radiative Transfer Diffusion Approximation and Radiative Equilibrium Eddington Approximation Eddington Approximation In a planar atmosphere with Iν = Bν + µ dbν dτν we have: Jν = 1 +1 Iν dµ = 1 [ µ Bν + µ ] +1 dbν dτν = Bν(τν) Hν = Fνa 4 = 1 +1 Iν µ dµ = 1 [ ] µ 2 +1 µ3 dbν Bν dτν = 1 dbν = 1 1 dbν 3 dτν 3 κν dx = 1 1 dbν dt 3 κν dt dx Kν = 1 2 [ +1 µ 3 Iν µ 2 dµ = ] +1 µ4 dbν Bν + 4 dτν = 1 3 Bν(τν) Using the Diffusion Approximation, simple expressions follow for J ν, H ν and K ν.

48 Schwarzschild-Milne Equations I J ν = 1 I ν dµ = 1 I out ν dµ + 1 I in ν dµ = 1 [ 1 dt τν ] (t τν)/µ S ν (t) e 2 τ ν µ dµ dt (t τν)/µ S ν (t) e 1 µ dµ = 1 [ S ν (t) e (t τν)w dt dw τν 2 1 τ ν w + S ν (t) e (τν t)w dt dw ] 1 w = 1 [ S ν (t) e (t τν)w dw τν 2 w dt + S ν (t) e (τν t)w dw ] w dt τ ν 1 Where w = 1/µ and w = 1/µ for left and right double integrals respectively; in both cases dw/w = dµ/µ. Since both exponents are greater than zero: J ν = 1 2 = 1 2 S ν (t) 1 1 e w t τν dw w dt S ν (t) E 1 ( t τ ν ) dt

49 Schwarzschild-Milne Equations II Introducing the Λ-Operator: Λ τν = 1 2 f(t) E 1 ( t τ ν ) dt J ν (τ ν ) = Λ τν [S ν (t)] Similarly for the other two specific intensity moments: H ν (τ ν ) = 1 S ν (t) E 2 ( t τ ν ) dt 1 2 τ ν 2 = Φ τν [S ν (t)] K ν (τ ν ) = 1 2 = X τν [S ν (t)] S ν (t) E 3 ( t τ ν ) dt τν S ν (t) E 2 ( τ ν t ) dt J ν, H ν and K ν are depth-weighted means of S ν, the largest contribution being when t τ ν =. E 1, E 2 and E 3 are the first, second and third exponential integrals.

50 τν Stellar Evolution Radiative Transfer Diffusion Approximation and Radiative Equilibrium Schwarzschild-Milne Equations II Schwarzschild-Milne Equations II Introducing the Λ-Operator: Λτν = 1 f(t) E1( t τν ) dt 2 Jν(τν) = Λτν [Sν(t)] Similarly for the other two specific intensity moments: Hν(τν) = 1 Sν(t) E2( t τν ) dt 1 τν Sν(t) E2( τν t ) dt 2 2 = Φτν [Sν(t)] Kν(τν) = 1 Sν(t) E3( t τν ) dt 2 = Xτν [Sν(t)] Jν, Hν and Kν are depth-weighted means of Sν, the largest contribution being when t τν =. E1, E2 and E3 are the first, second and third exponential integrals. In plane parallel geometry, the zeroth, first and second moments of the radiation field may be expressed as integrals over the source function multiplied respectively by the first, second and third exponential integrals. The first exponential integral can be obtained from a published Chebyshev series; from this the second and higher order exponential integrals may be obtained using the usual recurrence formula.

51 Lecture 2: Summary Essential points covered in second lecture: Specific intensity defined and its invariance, in the absence of absorption, verified. It was shown how specific intensity is related to radiative flux, luminosity and observed flux. Energy density, radiation pressure and the absorption of radiation were discussed. The equation of radiative transfer, optical depth and source function were introduced. Simple special case solutions of the transfer equation were presented with formal solution and boundary conditions. Diffusion approximation needed for stellar structure and evolution calculations was derived. Schwarzschild-Milne equations were also derived as these are needed for stellar atmosphere and synthetic spectrum calculations. Stellar evolution depends on initial photospheric abundances and their determination from spectra are to be discussed in the next lecture.

52 Acknowledgement Material presented in this lecture on radiative transfer is based almost entirely on slides prepared by R.-P. Kudritzki (University of Hawaii).