Introduction to the School

Lucio Crivellari Instituto de Astrofísica de Canarias D.pto de Astrofísica, Universidad de La Laguna & INAF Osservatorio Astronomico di Trieste (Italy) Introduction to the School 10/11/17 1

Setting the stage 10/11/17 2

1. The Stellar Atmosphere Physical System A star as a gravitationally bounded open concentration of matter and energy According to thermodynamics a system may be: a open: matter and energy fluxes close: only energy flux isolated: absence of fluxes 10/11/17 3

Fundamental fact: stars emit radiant energy Evidence of mass loss from the observations of solar and stellar winds Hence, definition of Stellar atmosphere: permeable boundary through which both photons and particles can escape outwards Two components: radiation field matter For both either a macroscopic or a microscopic picture 10/11/17 4

fluxes gradients departure from equilibrium TRANSPORT PROCESSES Radiative Transfer The protagonist of this School 10/11/17 5

Stellar winds are the signature of departure from hydrostatic equilibrium hence transport of matter Supersonic velocity fields in early-type stars have important effects on RT in spectral line via Doppler effect Line shifts and P Cyg profiles influence the mechanical structure of the outer layers; provide useful diagnostics tools. 10/11/17 6

Convective transport in late-type stars plays a fundamental role in the energy balance the generation and advection of magnetic field 10/11/17 7

The structure of any physical system is shaped by the the mutal interactions among its components The structure is described by the values at any point of macroscopic (thermodynamic) variables like T, P, the velocity of the (ideal) matter elements physical relations among { variables } constraints internal energy expressed by { conservation } state transport equations 10/11/17 8

Tenet: The physics of the problem dictates the most effective algoritm for its solution The algorithm is a numerical representation of the physical process 10/11/17 9

A visit to the province of Radiative Transfer

The dual nature of light Huygens: wave hypothesis (1690) Newton: corpuscular hypothesis (Opticks, 1704) Experiments by Young (1801) and Fresnel (1817) Maxwell: electromagnetic waves (1873), revealed by Hertz (1888) Einstein: quantum of light (1905) macroscopic representation I 0 = c 8 π E 2 0 I (r, t ;n, ν) = h ν c f (r,t ;n,ν) corpuscular picture wave picture wave particle duality 10/11/17 11

2. Fluid dynamic - like picture 10/11/17 12

Picture based on macroscopic quantities related to the microscopic photon picture (corpuscular model of the radiation field) Analogue with fluid dynamics: flux of photons propagating along the paths of geometrical optics (eikonal equation) that carry on and exchange energy with matter particles 10/11/17 13

Ray : amount of radiant energy of frequency ν carried on along the direction n with speed c per unit time across a unit surface perpendicular to n rays transport of energy 10/11/17 14

Under the assumptions of a weak electromagnetic field and propagation through a diluted medium the energy carried on by rays obeys the empirical laws of radiometry 1. propagation through vacuum along straight lines with speed c 2. all rays through a given point are independent 3. they are linearly additive both in direction and frequency Hypotheses already formulated by Newton in his Opticks (1704) 10/11/17 15

The above laws of photometry warrants that the transport process is intrinsically linear However a single directed quantity (i.e. a vector) is not enough to specify completely the radiation firld: virtually infinite pencil of rays 10/11/17 16

From rays to specific intensity Fundamental physical observable in radiative tyranfer: the energy carried on by a ray Scalar macroscopic local and directed quantity : specific intensity of the radiation field 10/11/17 17

observable: amount of energy δ E ν (n) elements of the measure: oriented surface solid angle δ Ω k δ S around P 1 around n time interval δ t spectral range δ ν δ E ν (n) n k δ S δ Ω δ ν δt (n k ) 1 lim δ S δω δ ν δt 0 δ E ν (n) δ S δω δ ν δt I ( r, t ;n, ν) 10/11/17 18

By definition the specific intensity I (r, t ;n, ν) charactyerized by (n, ν) is the coefficient of proportionality between the observable and the elements of the measurement Dimension: [I ] = (M L 2 T 2 ) L 2 T 1 T = M T 2 i.e. energy flux per unit time and unit frequency band 10/11/17 19

3. The RT equation as a kinetic equation for photons 10/11/17 20

Photon distribution function (microscopic picture) Because of the corpuscular nature of photons, let us define a distribution function such that f ( r,t ;n,ν ) d Ωd ν is equal to the nr. of photons per unit volume at r and t in the band ( ν,ν+d ν ) that propagates along n with speed c into d Ω. f is characterized by the pair (n ; ν) directed and spectral quantity [f ] = L 3 T 10/11/17 21

Transport process in terms of the photon distribution function Specific energy flowing through k n ds d E ν (n) = h ν f (r,t ;n, ν) n k ds c dt dω d ν By direct comparison I ( r,t ; n,ν) = ch ν f (r, t ; n, ν ) 10/11/17 22

A kinetic equation for any transported quantity is formally Total rate of change = Source terms Sink terms (Boltzmann's equation) Total rate of change = Eulerian derivative In our case Sources and sinks determined by: atomic properties of the interaction matter - radiation equation of state of matter (LTE) or SE equations 10/11/17 23

Distribution function p = n hν c Kinetic equation: F ( r, p,t ) ; p = p(n, ν) for photons with momentum d d t F (r, p,t) = [ δ F ] δ t sources [ δ F δ t ] sinks It can be shown that f (r, t ;n, ν) = h3 ν 2 Previusly: I ( r,t ; n,ν ) = ch ν f (r, t ; n, ν ) c 3 F (r, p,t) parameters (n, ν) c2 F (r, p,t ) = I (r, t :n, ν) h 4 3 ν 10/11/17 24

Total derivative: d dt = t + v r + ṗ p ; r = For photons v = c n and ṗ = 0 time space evolution 1 c t I (r, t : n, ν) + n I (r,t ;n, ν) = 1 [ δ I c δt ]sources 1 c [ δ I δ t ] sinks = [ δ I δ l ]sources [ δ I δl ] sinks where δ l = c δ t is a path length along n physical properties 10/11/17 25

The Radiative Transfer equation 1 c t I (r,t : n,ν) + n I (r,t ;n,ν) = [ δ I δ l ]sources [ δ I δl ] sinks A. Schuster (1851-1934) K. Schwarzschild (1873-1916) 10/11/17 26

mathematical formulation of a directional problem in terms of the macroscopic quantity specific intensity For any specific intensity, characterized by the pair of parameters (n ; ν), one specific RT equation Each term in the RT equation has dimension (M L 2 T 2 ) L 2 L 1 = M L 1 T 2 energy flux per unit length 10/11/17 27

5. Macroscopic RT coefficients and the source function 10/11/17 28

Consistently with the macroscopic picture we consider homogeneous volume elements that emit and absorb radiant energy isotropically All the physical information at atomic level is incorporated into a limited number of macroscopic coefficients 10/11/17 29

Thermal emission coefficient Δ E ν th energy emitted along n measurable quantity Δ E ν th lim Δσ ΔΩ Δν Δt 0 by into during Δ V Δ Ω Δ t in (ν, ν+δ ν) parameters of the measure Δ V Δ Ω Δ ν Δ t [η ν th ] = M L 1 T 2 δ E ν th δ V δ Ω δ ν δ t η ν th 10/11/17 30

Decrease of the specific intensity along a path δ l True absorption coefficient δ I (n) I (n) δ l Likewise in the direction n fraction of energy removed converted into internal energy Scattering coefficient a ν (n) : δ I (n) I (n) σ ν (n) : = a ν (n) δ l fraction of energy removed diverted into a different direction 10/11/17 31

Extinction coefficient Global effect of the attenuation, i.e., removal of photons from a given direction n χ ν (n) a ν (n) + σ ν (n) [ χ ν ] = [a ν ] = [ σ ν ] = L 1 Total emission coefficient diverted from n' into n η ν (n) = η th + ν σ ν (n ) p (n, n)d n thermal contribution 10/11/17 32

The source function S ν (n) η ν(n) χ ν (n) [S ν ] = (M L 1 T 2 ) / L 1 = M T 2 = [I ν (n)] = [B ν ] TE (or LTE) η ν = a ν B ν (T ) η ν s isotropic scattering = σ ν 1 4 π I (n )d n = σ ν J ν S ν = a ν B ν (T ) a ν + σ ν + σ ν J ν a ν + σ ν = ε ν B ν (T ) + (1 ε ν )J ν ε ν a ν a ν + σ ν 10/11/17 33

If the sources and sinks are given. the RT equation 1 c t I (r, t : n, ν) + n I (r,t ;n, ν) = [ δ I ] δl sources is a [ δ I δ l ]sinks 1 st order linear ODE 10/11/17 34

From a linear to non-linear problem Mathematical complications arise when the individual specific RT equations are coupled through the source function Non-local problems Moreover transport processes necessarily implies non-local effects 10/11/17 35

Illustrative examples: a ν = 0 pure scattering S ν = J ν = 1 4 π I ν (n)d n internal mechanism σ ν = 0 pure absorption energy balance: S ν = a ν B ν (T ) = S ν (T ) 0 a ν B ν (T )d ν = 0 T a ν J ν d ν external mechanism given by a conservation equation 10/11/17 36

Back to electrodynamics homeland 10/11/17 37

1. Electrodynamic formulation 10/11/17 38

James Clark Maxwell (1831-1879) D = 4 πρ B = 0 E + 1 B c t H + 1 D c t J = ρ v = 0 = 4 π c J A Dynamical Theory of the Electromagnetic Field (1865) Gauss conventional system of units: [E ] = [ D] = [B] = [H ] = M 1/ 2 L 1/ 2 T 1 [J] = M 1/2 L 1/ 2 T 2 ε = μ = 1 dimensionless 10/11/17 39

Following Maxwell: magnetic and electric energy density W mag = 1 8 π H B ; W elec = 1 8 π D E W W mag + W elec energy is localized in the field 10/11/17 40

Poynting's vector: S c 4 π E H [S] = (LT 1 ) (M 1 /2 L 1/ 2 T 1 ) 2 i.e. energy time surface = power flux = ( M L 2 T 2 ) T 1 L 2 By a proper treatment of the last two Maxwell's equations 1 4π H Ḃ + 1 4 π John H. Poynting (1852 19 14 ) E Ḋ +E J + S = 0 Poynting's theorem [each term] = M L 1 T 3 = (M L 2 T 2 ) T 1 L 3 i.e. power density 10/11/17 41

Physical meaning of the Poynting's vector: energy flux per unit time across unit area of the boundary surface of the volume considered Transport of energy of the electromagnetic field The Poynting's vector accounts for the intrinsic directed aspect of the propagation of the electromagnetic field. 10/11/17 42

It can be shown that: The same for Ẇ elec = 1 8 π E Ḋ + 1 8π Ė D = 1 4 π E Ḋ Ẇ mag W J E J Joule heat From the Poynting's theorem: Ẇ + S = W J energy balance of the electromagnetic field By integration over V and Gauss theorem: Σ S n d σ = V [ W t + W J ] conservation equation dv 10/11/17 43

Transport of radiant energy by an e.m. wave monochromatic polarized plane wave propagating along the x axis, specified by 1 c 2 E H : only E y and H z not 0 2 E y t 2 by proper manipulation 2 E y = 0 x 2 solution E y (x,t ) = E 0 cos(k x ωt ) ^x wave equation { 1 [ 1 ( t 8 π k 2 c 2 E y t 2] ) 2 ( + E ) } y x ( x 1 E y 4 π k 2 t E y x ) = 0. e 1 [ 1 ( 8π k 2 c 2 E y t 2] ) 2 + ( E y ) ( f x 1 E y 4 π k 2 t E y ) x 10/11/17 44

From the previuos definitions: wave equation [e] = ( M L 2 T 2 ) L 3 ; energy density e t + f x = 0. equation of continuity => [f ] = (M L 2 T 2 ) T 1 L 2 power flux 2 e(t) = E 0 4 π sin 2 (kx ωt) W (t) = W elec (t) + W mag (t ) = E 0 4 π cos 2 (kx ω t) 2 e <=> W f (t) = E 0 4 π S (t ) = 2 c 4 π E y ω k sin 2 (kx ωt ) = c 4 π E 2 0 sin 2 (kx ω t) 2 (t) ^x = c 4 π E 2 0 cos 2 (kx ωt ) ^x. f ^x <=> S 10/11/17 45

7. Electrodynamical vs. macroscopical picture 10/11/17 46

Energy density of the radiation field In the time interval dt the volume dv = n k ds c dt is filled in by radiant energy Specific energy density: U (r,t ; n,ν ) d E ν (n) directed and spectral dv By definition U (r, t ; n, ν ) d Ω d ν = 1 c I ( r,t ; n,ν ) d Ω d ν By integration over all the directions u ν u(r,t ;ν) 1 c I (r,t ;n,ν)d Ω = 4 π c spectral [u ν ] = ( M L 2 T 2 ) L 3 T J (r,t ; ν) 10/11/17 47

Correspondence between the specific intensity and the electric field strength Monochromatic plane wave of frequency ν 0 = 1/T propagating along n 0 = n 0 (θ 0, ϕ 0 ) n 0 ^x ; ^x E H [E] = [D] = [B] = [H ] 10/11/17 48

The solution of the wave equation is 1 c 2 2 E y t 2 2 E y = 0 x 2 E y (x,t ) = E 0 cos(k x ωt ) From the average over T of W elec 1 8 π E D and => W (t ) T = E 0 2 8 π. W mag 1 8 π H B. 10/11/17 49

Corresponding specific intensity : I (ϑ,ϕ,ν) = I 0 δ (ϑ ϑ 0 ) δ (ϕ ϕ 0 ) δ (ν ν 0 ) [I ] = M T 2 ; [δ (ν ν 0 )] = T ; [I 0 ] = M T 3 From the physical standpoint W (t) T = E 2 0 8 π = u(r,t ) u(r,t) = I 0 c 0 d ν δ(ν ν 0 ) d Ω δ(θ θ 0 ) δ (ϕ ϕ 0 ) = I 0 c => I 0 = c 8 π E 2 0 [I 0 ] = [c E 0 2 ] = M T 3 power flux 10/11/17 50

Moments of the specific intensity 0 th order moment: average mean intensity J (r, t ;ν ) 1 4 π I (r, t ;n, ν) d n = c 4π u ν 1 st order moment: flux of radiation F ν (r, t) I (r,t ;n,ν) n d n scalar vector 2 nd order moment: radiation pressure T ν (r,t) 1 c I (r, t ;n, ν) n n d n tensor (dyadic notation) 10/11/17 51

Electromagnetic counter part of F ν (r, t) F ν (r, t) I (r,t ;n,ν) n d n is the monochromatic power flux of the radiation field ( M L 2 T 2 ) T 1 L 2 T = M T 2 0 E y (x,t ) = E 0 cos(k x ωt ) ; ^x E H ; E 0 = H 0 d ν d n I (r,t ; n, ν) n = I 0 n 0 = c 8 π E 0 bolometric vector flux Poynting vector: (time averaged) 2 n 0 power flux S(t) T = c 8 π E 2 0 n 0 10/11/17 52

Correspondence of the radiative pressure with the Maxwell stress tensor p(n, ν) = ( p x p y p z) = h ν c ( n ) x n y n z moment carried on by a photon(n, ν) radiative pressure tensor: T ν (r,t) 1 c I (r, t ;n, ν) n n d n [T ν ] = (M L T 1 ) L 2 flux of momentum 10/11/17 53

net transport of p : n 1 c F ν = n h ν c f (r,t ;n, ν) cn d n G ν (r, t) 1 c 2 F ν(r,t ) monochromatic momentum density of the radiation field [G ν ] = (M T 2 ) L 2 T 2 = (M L T 1 ) L 3 T G 0 G ν d ν = 1 c 2 0 F ν d ν = 1 c 2 S S bolometric vector flux 10/11/17 54

For each component: (G ν ) j t t 0 = 1 c 2 (F ν ) j t From previous results: G ν d ν = 0 momentum density of the electromagnetic field: = [(G ν ) j c n ] continuity equation T ν d ν => G t = T (M LT 1 ) L 3 T 1 G em t T M (M LT 1 ) L 3 T 1 G <=> G em T = T M 10/11/17 55

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