Astro 305 Lecture Notes Wayne Hu

Astro 305 Lecture Notes Wayne Hu

Set 1: Radiative Transfer

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

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

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

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)

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

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

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

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

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

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

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

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

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

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

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.

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

Random Walk Each of N steps has length L in a random direction r i = Lˆr i Total distance R = r 1 + r 2... + r N R 2 = R R = r 1 r 1... + 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 )

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 )

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

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!

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 25-50 moments to solve for S ν and then 3000-6000 moments to describe I ν ]

Fluid or Eddington Approximation Angular moments of specific intensity: J ν = 1 2 H ν = 1 2 K ν = 1 2 +1 1 +1 1 +1 1 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

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

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

Set 1: Statistical Mechanics

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

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

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

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)