Manuel Gonzalez (IRAM) September 14 th Radiative transfer basics

Transcription

1 Manuel Gonzalez (IRAM) September 14 th 2013 Radiative transfer basics

2 What is radiation? Introduction

3 Introduction In Astrophysics radiation from the sources can give us very important information: - Spatial distribution of the gas -Temperature and density of the gas - Velocity structure - Presence of certain physical or chemical processes -Mass and column density

4 Introduction Two kind of processes: - Continuum: dust, thermic sources - Lines: atoms, ions and molecules

5 Introduction Three kind of interactions with matter: - Emission - Absorption - Scattering

6 Introduction

7 The radiative transfer equation di ν(x, ν,θ)= κ(x, ν) I νds+ j(x, ν,θ)ds destruction terms creation terms

8 The radiative transfer equation di ν(x, ν,θ)= κ(x, ν)iν ds+ j(x, ν,θ)ds destruction terms creation terms I ν(x, ν,θ): Specific intensity. Units (erg s -1 cm -2 Hz -1 sr -1 ) de=i ν( x,ê) nê da d Ω d νdt E is the Energy received!!

9 The radiative transfer equation The radiative transfer equation di ν(x, ν,θ)= κ(x, ν)iν ds+ j(x, ν,θ)ds destruction terms creation terms I ν(x, ν,θ): Specific intensity. Units (erg s -1 cm -2 Hz -1 sr -1 ) Specific intensity is constant along rays, as long as there is no absorption and emission of matter between emitter and receiver

10 The radiative transfer equation di ν(x, ν,θ)= κ(x, ν) I νds+ j(x, ν,θ)ds destruction terms creation terms κ(x, ν): Absorption coefficient. Units (cm -1 ) j(x, ν,θ): Emission coefficient. Units (erg s -1 cm -3 Hz -1 sr -1 )

11 Let's solve this equation...

12 Solution of the equation di ν(x, ν,θ)= κ(x, ν) I ν ds+ j(x, ν,θ)ds

13 Solution of the equation di ν(x, ν,θ)= κ(x, ν) I ν ds+ j(x, ν,θ)ds ds=dx/cos(θ)

14 Solution of the equation di ν(x, ν,θ)= κ(x, ν) I ν ds+ j(x, ν,θ)ds ds=dx/cos(θ) I ν(x, ν,θ)= κ(x, ν) I ν dx /cos(θ)+ j(x, ν,θ)dx /cos(θ

15 Solution of the equation di ν(x, ν,θ)= κ(x, ν) I ν ds+ j(x, ν,θ)ds ds=dx/cos(θ) I ν(x, ν,θ)= κ(x, ν) I ν dx /cos(θ)+ j(x, ν,θ)dx /cos(θ cos(θ)di ν(x, ν,θ)= κ(x, ν) I ν dx+ j(x, ν,θ)dx

16 Solution of the equation di ν(x, ν,θ)= κ(x, ν) I ν ds+ j(x, ν,θ)ds ds=dx/cos(θ) I ν(x, ν,θ)= κ(x, ν) I ν dx /cos(θ)+ j(x, ν,θ)dx /cos(θ cos(θ)di ν(x, ν,θ)= κ(x, ν) I ν dx+ j(x, ν,θ)dx cos(θ) di ν(x, ν,θ) κ(x, ν)dx = I ν+ j(x, ν,θ)dx κ(x, ν)dx

17 Solution of the equation cos(θ) di ν(x, ν,θ) κ(x, ν)dx = I ν+ j(x, ν,θ) κ(x, ν)

18 Solution of the equation cos(θ) di ν(x, ν,θ) κ(x, ν)dx = I ν+ j(x, ν,θ) κ(x, ν) cos(θ)=μ

19 Solution of the equation cos(θ) di ν(x, ν,θ) κ(x, ν)dx = I ν+ j(x, ν,θ) κ(x, ν) cos(θ)=μ κ(x, ν)dx=d τ Optical depth

20 Solution of the equation cos(θ) di ν(x, ν,θ) κ(x, ν)dx = I ν+ j(x, ν,θ) κ(x, ν) cos(θ)=μ κ(x, ν)dx=d τ Optical depth j(x, ν,θ) κ(x, ν) =S ν Source function

21 Solution of the equation cos(θ) di ν(x, ν,θ) κ(x, ν)dx = I ν+ j(x, ν,θ) κ(x, ν) cos(θ)=μ κ(x, ν)dx=d τ j(x, ν,θ) κ(x, ν) =S ν μ di ν d τ = I ν+ Sν

22 Radiative transfer equation μ di ν d τ = I ν+ Sν

23 Formal solution μ di ν d τ = I ν+ Sν (for the case µ = 1) 1 st step: Multiply everywhere by e τ ν di ν d τ eτν = I νe τ ν + Sν e τν

24 Formal solution μ di ν d τ = I ν+ Sν Formal solution: 1 st step: Multiply everywhere by e τ ν di ν d τ eτν = I νe τ ν + Sν e τν 2 nd step: We pass to the left the term I ν e τ ν di ν d τ eτν + I ν e τν =S νe τν

25 Formal solution di ν d τ eτν + I ν e τν =Sν e τν a '( x)b( x)+ a( x)b' ( x)=(ab)'

26 Formal solution di ν d τ eτν + I ν e τν =Sν e τν a '( x)b( x)+ a( x)b' ( x)=(ab)' 3 rd step: we change the left side by its value di ν + I νe d τ eτν τν = d( I νeτ ν ) d τν

27 Formal solution di ν d τ eτν + I ν e τν =Sν e τν a '( x)b( x)+ a( x)b' ( x)=(ab)' 3 rd step: we change the left side by its value di ν + I νe d τ eτν τν = d( I νeτ ν ) d τ d (I νe τ ν ) d τ =S νe τ ν

28 Formal solution di ν d τ eτν + I ν e τν =Sν e τν a '( x)b( x)+ a( x)b' ( x)=(ab)' 3 rd step: we change the left side by its value di ν + I νe d τ eτν τν = d( I νeτ ν ) d τ d (I νe τ ν ) d τ =S νe τ ν d(i ν e τ ν )=Sν eτ ν d τ

29 Formal solution d(i ν e τ ν )=Sν eτ ν d τ

30 Formal solution d(i ν e τ ν )=Sν eτ ν d τ 4 th step: we integrate that expression between 0 and the maximum optical depth d(i ν e τ ' )= S ν(τ')e τ ' d τ '

31 Formal solution d(i ν e τ ν )=Sν eτ ν d τ 4 th step: we integrate that expression between 0 and the maximum optical depth d(i ν e τ ' )= S ν(τ')e τ ' d τ ' 5 th step: Solve the integral! I νe τ ' 0 τν = S ν(τ ')e τ ' d τ '

32 Formal solution I νe τ ' 0 τν = S ν(τ ')e τ ' d τ '

33 Formal solution I νe τ ' 0 τν = S ν(τ ')e τ ' d τ ' 6 th step: Fixing some details I ν(τν)e τν =I ν(0)e 0 + Sν(τ ')e τ ' d τ '

34 Formal solution I νe τ ' 0 τν = S ν(τ ')e τ ' d τ ' 6 th step: Fixing some details I ν(τν)e τν =I ν(0)e 0 + Sν(τ ')e τ ' d τ ' Multiply everywhere by: e τ ν I ν(τν)=i ν(0)e τ ν + Sν(τ ' τν) ')e(τ

35 Let's come back to physics!!!

36 Physical interpretation I ν(τν)=i ν(0)e τ ν + Sν e( τ ' τ ν) d τ ' Emergent intensity Attenuated background Self attenuated emission by radiatively excited medium

37 Approximate solution: S ν = 0 I ν(τν)=i ν(0)e τ ν + Sν e(τ' τ ν) d τ ' I ν(τν)=i ν(0)e τ ν - Passive medium: there is no emission in the medium - Emergent intensity is the impinging background intensity attenuated by the absorbing medium

38 Approximate solution: S ν = constan I ν(τν)=i ν(0)e τ ν + Sν e( τ ' τ ν) d τ ' I ν(τν)=i ν(0)e τ ν + Sν(1 e τ ν ) - 1 = Attenuated background by the absorbing medium - 2 = Intensity emitted by the medium - 3 = Self attenuation of the medium emission

39 Astronomical observations - Approximate solution S ν is constant - Usually, spectral observations consist in ON (source+background) OFF (background) measurements - ON: I ν = I ν (τ ν ) - OFF I bg = I ν (0) I ν(τν)=i ν(0)e τν + Sν(1 e τν ) I bg =I ν(0) - ON-OFF = I ν - I bg I ν I bg =(Sν I bg )(1 e τ ν )

40 Astronomical observations I ν I bg =(Sν I bg )(1 e τ ν ) - if S ν > I bg => EMISSION LINE - if S ν < I bg => ABSORPTION LINE

41 Astronomical observations I ν I bg =(Sν I bg )(1 e τ ν ) - Optically thin approximation: τ ν << 1 - In that case e τ ~ 1 τ ν I ν I bg τν(sν I bg ) - Optically thick approximation: τ ν >> 1 - In that case e τ ~ 0 I ν I bg (S ν I bg )

42 Brightness temperature - Black body radiation. Planck's law. Bν(T )= 2h ν3 c 2 1 exp(h ν/ K B T ) 1 - h = Planck constant ( erg/s) - c = celerity of light ( cm/s) - Kb = Boltzmann constant ( erg/k)

43 Brightness temperature - Black body radiation. Planck's law. Bν(T )= 2h ν3 c 2 1 exp(h ν/ K B T ) 1 - Wien's limit: hν >> K B T Bν(T )= 2h ν3 c 2 exp( h ν/ K B T ) - Rayleigh-Jeans limit: hν << K B T Bν(T )= 2K B ν2 T c 2

44 Brightness temperature Bν(T )= 2h ν3 c 2 1 exp(h ν/ K B T ) 1 - The brightness temperature T B is a fictitious temperature such that the observed intensity is given by: I ν=bν(t B )

45 Excitation temperature Spontaneous emission Induced emission Collisions

46 Excitation temperature

47 Excitation temperature n u n l = g u g l exp(h ν/ K B T ex ) - The excitation temperature T ex is a fictitious temperature such that the observed population of the levels is given by that expression. - The excitation temperature depends on the transition.

48 Thank you for your attention!