Astro 305 Lecture Notes Wayne Hu

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1 Astro 305 Lecture Notes Wayne Hu

2 Set 1: Radiative Transfer

3 Radiation Observables From an empiricist s point of view there are 4 observables for radiation Energy Flux Direction Color Polarization Energy Flux (diagram: da and flux lines) F = de dtda Direction: columate in an acceptance angle dω normal to da surface brightness S(Ω) = de dtdadω

4 Radiation Observables Color: filter in a band of frequency dν specific intensity I ν = de dtdadωdν which is the fundamental quantity for radiative processes (energy per unit everything) Polarization: filter in linear (1,2) [or 45 degrees rotated (1,2 )] or circular (+,-) polarization states Stokes parameters I ν = I ν1 + I ν2 Q ν = I ν1 I ν2 U ν = I ν1 I ν2 V ν = I ν+ I ν

5 Radiative Transfer Radiative Transfer = change in I ν as radiation propagates Simple example: how does the specific intensity of sunlight change as it propagates to the earth Energy conservation says F (r 1 )4πr1 2 = F (r 2 )4πr2 2 F r 2 But the angular size subtended by the sun Ω r 2 S = F Ω = const And frequency doesn t change so that I ν = const

6 Radiative Transfer More generally: I ν changes due to Direction (elastic scattering) Changes in frequency (redshift, inelastic scattering) Absorption Emission If frequency changes due to redshift ν (1 + z) 1, photon conservation implies I ν /ν 3 = const and so surface brightness S = I ν dν scales as the redshift of (1 + z) 4 (cosmological surface brightness dimming, conversely relativistic Doppler boost) Liouville s theorem (conservation of I ν /ν 3 ) in absence of interactions; Boltzmann equation (radiative transfer equation)

7 Auxiliary Quantities Specific intensity is defined as energy flux per unit area, per unit solid angle (normal to the area), per unit frequency So specific flux through a detector not oriented normal to source has a lowered cross section due to projection F ν = I ν cos θdω, F = dνf ν so that if I ν is constant in angle, there is no net energy flux through a surface F ν = 0 = F Angle averaged intensity J ν = dω 4π I ν

8 Auxiliary Quantities Specific energy density: de = I ν dtdadωdν since light travels at c, de is the energy in the volume dv = cdtda de = I ν c dv dωdν so specific energy density u ν (Ω) I ν /c Energy density: u = dν dωi ν /c = dν4πj ν /c

9 Auxiliary Quantities Momentum flux: momentum q = E/c ê r, momentum flux dq/dtda = (F/c)ê r dω; the component normal to surface is the pressure ẑ ê r = cos θ p ν,abs = 1 F ν cos θdω = 1 I ν cos 2 θdω c c and is also the pressure on a surface that absorbs the radiation If surface reflects radiation then p ν = 2p ν,abs = 2 I ν cos 2 θdω c

10 Auxiliary Quantities Equation of state of isotropic radiation I ν = J ν p ν = 2 c J ν cos 2 θdω = 4π 3c J ν p = 4π dνj ν = 1 3c 3 u

11 Radiative Processes Spontaneous Emission: matter spontaneously emits a photon/radiation Absorption: radiation absorbed by matter Stimulated Emission: passing radiation stimulates matter to emit in the same frequency and direction. Stimulated emission is mathematically the same as negative absorption - both proportional to the incoming radiation Scattering: absorption followed immediately by emission Coherent or elastic: emission at the same frequency Isotropic: radiates equally in all directions All can be related to the matrix element for interaction - interrelated by Einstein relations

12 Spontaneous Emission Given a source isotropically radiating power into a band dν at a rate power volume = dνp ν The amount of energy radiated per unit solid angle is de em = P ν dνdv dt dω 4π Generalize to a non-isotropically emitting source P ν 4π j ν(ω) called the monocromatic emission coefficient

13 Spontaneous Emission Emission coefficient is alternately given per unit mass ɛ ν = 4πj ν /ρ where ρ is the mass density de em = ɛ ν 4π dνdmdt, dm = ρdv Effect on specific intensity across a path length ds di ν dtdadωdν = de em Radiative transfer equation = j ν dv dωdνdt di ν = j ν ds, di ν ds = j ν, I ν (s) = I ν (s 0 ) + dv = dsda s s 0 j ν (s)ds

14 Absorption Travelling through a material a fraction α ν of the radiation is absorbed per unit length di ν I ν = α ν ds where α ν is called the absorption coefficient If material does not emit, solution is an exponential suppression ln I ν = dsα ν + C I ν (s) = I ν (s 0 )e R α ν ds If α ν < 0 (stimulated emission) then there is an exponential growth

15 Scattering Scattering can be viewed as absorption and emission where the emission is directly proportional to the absorption Isotropic scattering: fraction α ν I ν absorbed and reradiated into 4π j ν = α ν dω 4π I ν = α ν J ν Thus the source function S ν = j ν /α ν = J ν and at high optical depth, the specific intensity approaches its angle averaged value J ν But despite this simple interpretation, the radiative transfer function is an integro-differential equation di ν = I ν + J ν dτ ν di ν dω = I ν + dτ ν 4π I ν

16 Optical Depth Useful to measure length in units of the typical path length to absorption or interaction L = 1 α ν so that the total path length in units of L becomes ds α ν ds = L τ ν, I ν (s) = I ν (s 0 )e τ ν Radiative transfer equation di ν ds = α νi ν + j ν di ν dτ ν = I ν + S ν, where S ν is the source function. S ν j ν /α ν

17 Cross Section Particle description of absorbers and light Absorption coefficient related to cross section for interaction Example: if medium is full of opaque disks each of area σ and with a number density n then the fraction of specific intensity lost is simply the covering fraction da abs = σdn = σndv = σndads di ν I ν = σnds So for a particle description of the absorption coefficient α ν = nσ.

18 Cross Section Applies to a generalized version of σ (e.g. free electrons are point particles but Thomson cross section is finite), cross section for interaction, can depend on frequency. Mean free path of a photon L = 1/α ν = 1/nσ, where n is the spatial number density of interacting particles (c.f. consider Thomson scattering in air) But the total distance travelled by a photon is typically much greater than s = τl - an individual photon propagates through the medium as a random walk

19 Random Walk Each of N steps has length L in a random direction r i = Lˆr i Total distance R = r 1 + r r N R 2 = R R = r 1 r r N r N + 2 r 1 r 2... Uncorrelated cross terms r i r j = δ ij L 2 Average distance R 2 = NL 2 or R rms = NL How many scatterings before escaping a distance R? N = R 2 /L 2 = τ 2 ν for optically thick For optically thin τ ν 1, a typical photon does not scatter and so by definition a fraction τ ν will interact once, the rest zero and so the average N = τ ν Quick estimate N = max(τ ν, τ 2 ν )

20 Multiple Processes Combining processes: differential elements add, e.g. total opacity is sum of individual opacities so highest opacity process is most important for blocking I ν But given multiple frequency or spatial channels, energy escapes in the channel with the lowest opacity Example: a transition line vs continuum scattering photons will wander in frequency out of line and escape through lower opacity scattering lines are often dark (sun) Scattering (α νs ) and absorption (α νa ) di ν ds = α νa(i ν S νa ) α νs (I ν J ν )

21 Multiple Processes Collect terms di ν ds = (α νa + α νs )I ν + (α νa S νa + α νs J ν ) Combined source function and absorption α ν = α νa + α νs, S ν = j ν α ν = α νas νa + α νs J ν α νa + α νs Mean free path L = 1/α ν = 1/(α νa + α νs ), typical length before absorption or scattering. Fraction that ends in absorption ɛ ν = α νa α νa + α νs, 1 ɛ ν = α νs α νa + α νs single scattering albedo S ν = ɛ ν S νa + (1 ɛ ν )J ν

22 Formal Solution to Radiative Transfer Formal solution I ν (τ ν ) = I ν (0)e τ ν + τν 0 dτ νs(τ ν)e (τ ν τ ν) Interpretation: initial specific intensity I ν attenuated by absorption and replaced by source function, attenuated by absorption from foreground absorption Special case S ν independent of τ ν take out of integral I ν (τ ν ) = I ν (0)e τ ν + S ν τν = I ν (0)e τ ν + S ν [1 e τ ν ] 0 dτ νe (τ ν τ ν ) at low optical depth I ν unchanged, high optical depth I ν S ν Integration is along the path of the radiation (direction dependent); source function can depend on I ν in a different direction!

23 Fluid or Eddington Approximation Often a good approximation that radiation is nearly isotropic: consider the fact that scattering, absorption and emission randomizes the direction of radiation Eddington (or fluid) approximation: radiation has a dipole structure at most Plane parallel symmetry (e.g. star, Fourier expansion): specific intensity depends only on polar angle from normal to plane and vertical spatial coordinate z. Path length ds = dz/ cos θ = dz/µ. Now approximate I ν as a linear function of µ (isotropic + dipole energy density + bulk momentum density: I ν = a + bµ [More generally: a Legendre polynomial expansion of I ν - e.g. CMB typically keeps moments to solve for S ν and then moments to describe I ν ]

24 Fluid or Eddington Approximation Angular moments of specific intensity: J ν = 1 2 H ν = 1 2 K ν = I ν dµ Radiative transfer equation: (energy density) µi ν dµ (momentum density) µ 2 I ν dµ = 1 3 J ν (pressure) di ν ds = µdi ν dz = α ν(i ν S ν ) µ di ν dτ = (I ν S ν ), dτ = α ν dz

25 Fluid or Eddington Approximation Angular moments of specific intensity: zeroth moment 1 2 (for simplicity assume S ν is isotropic) dµ... dh ν dτ = (J ν S ν ) In fluid mechanics this is the Euler equation: local imbalance generates a flow First moment 1 2 µdµ... dk ν dτ = 1 3 dj ν dτ = H ν This is the continuity equation, a flow generates a change in energy density

26 Fluid or Eddington Approximation Take derivative and combine 1 3 d 2 J ν dτ 2 = dh ν dτ = J ν S ν Explicit equation for J ν (z) if S ν considered an external source Consider scattering + absorption/emission 1 3 d 2 J ν dτ 2 = J ν [ɛ ν S νa + (1 ɛ ν )J ν ] = ɛ ν (S νa J ν ) Explicitly solve with two boundary conditions: e.g. J ν (0) = J ν0 and J ν ( ) = S νa. If S νa constant J ν = S νa + [J ν0 S νa ]e τ 3ɛ ν So relaxation to source at τ 1/ 3ɛ z/l

27 Set 1: Statistical Mechanics

28 How Many Particles Fit in a Box? Counting momentum states due to the wave nature of particles with momentum p and de Broglie wavelength λ = h p = 2π h p

29 Distribution Function The distribution function f gives the number of particles per unit phase space d 3 xd 3 q where q is the momentum (conventional to work in physical coordinates) Consider a box of volume V = L 3. Periodicity implies that the allowed momentum states are given by q i = n i 2π/L so that the density of states is dn s = g V (2π) 3 d3 q where g is the degeneracy factor (spin/polarization states) The distribution function f(x, q, t) describes the particle occupancy of these states, i.e. N = dn s f = gv d 3 q (2π) 3 f 28-1

30 Bulk Properties Integrals over the distribution function define the bulk properties of the collection of particles Number density Energy density where E 2 = q 2 + m 2 n(x, t) N/V = g ρ(x, t) = g d 3 q (2π) 3 f d 3 q (2π) 3 E(q)f

31 Bulk Properties Pressure: particles bouncing off a surface of area A in a volume spanned by L x : per momentum state p q = F A = N part q A t ( q = 2 q x, t = 2L x /v x ) = N part V q x v x = f q v = f q2 3 3E so that summed over states d 3 q q 2 p(x, t) = g (2π) 3 3E(q) f Likewise anisotropic stress (vanishes in the background) π i d 3 q 3q i q j q 2 δ i j j(x, t) = g f (2π) 3 3E(q)

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