Lecture 2: Transfer theory. Credit: Kees Dullemond

Transcription

1 Lecture 2: Transfer theory Credit: Kees Dullemond

2 What is radia:ve transfer? Radia:ve transfer is the physical phenomenon of energy transfer in the form of electromagne:c radia:on The propaga:on of radia:on through a medium is affected by absorp:on, emission and scacering processes. The equa:on of radia:ve transfer describes these interac:ons mathema:cally. Applica:ons apart from astrophysics include op:cs, atmospheric science, and remote sensing. Analy:cal solu:ons exist in a few (simple) cases, but more realis:c cases need a numerical treatment.

3 Outline Example: Dust and line radia:on transfer Small recap lecture 1 Radia:ve transfer: Basic concepts: absorp:on, emission, op:cal depth Solu:ons to equa:on of transport Mean free path and random walks Einstein coefficients

4 HL Tau ALMA radio image of protoplanetary disk around young star Rings show up because dust is cleared out by protoplanets Resolu:on 35 microarcsec (penny at 110 km distance)

5 Model of ring system Protoplanets are formed by colliding dust par:cles Icy dust par:cles are s:ckier Models provide loca:on of these ice- lines

6 ProDiMo Example of a model that calculates thermal and chemical structure and the emission signatures. Radia:ve transfer is a crucial step, but this can only be modeled numerically.

7 Radia:ve transfer in the disk Input spectrum of typical T Tauri star Radia:on field throughout the disk

8 Thermal structure: Dust and gas temperature Dust and gas temperatures throughout the disk: Note decoupling of dust and gas

9 Density structure

10 Gas phase species Ionized carbon Neutral carbon Carbon monoxide Note the complex structure of the chemical species, without the detailed structure calcula:on this would not have been recovered

11 Ice chemistry CO2 ice H2O ice Note that the molecules freeze- out at different radii, depending on their freeze- out temperatures

12 Line emission of water Three water reservoirs contribute to line profile. BoCom panel shows exclusion of regions 3

13 Why do we need radia:ve transfer? Crucial to determine radia:on contribu:on throughout the object of interest It is key in determining the hea:ng and cooling processes, and as a result the density, thermal and chemical structure Finally, dust and line radia:ve transfer will provide dust and emission characteris:cs to be observed with telescopes.

14 Previous lecture (1)

15 Previous lecture (2)

16 Radia:ve transfer If a ray passes through a medium, energy can be added or subtracted by emission and absorp:on Therefore: Specific intensity will usually not remain constant when passing through the interstellar medium.

17 Emission (1) The spontaneous emission coefficient j is defined as the energy emiced per unit :me per unit solid angle per unit volume: de = j dv dω dt Or when the emission is monochroma:c: de = j ν dv dω dt dν If the emission is isotropic, we can write: j ν = 1/(4π) P v, with the power per unit volume per unit frequency.

18 Emission (2) In going a distance ds, a beam of cross sec:on da travels through a volume dv = da ds The intensity added to the beam is then: di ν = j ν ds Or compare the specific intensity and the emission coefficient: j ν [erg cm - 3 s - 1 ster - 1 Hz - 1 ] to I v [erg cm - 2 s - 1 ster - 1 Hz - 1 ]

19 Absorp:on The absorp:on coefficient α [cm - 1 ] is defined by the following equa:on: di v = - α ν I ν ds, represen:ng the loss of intensity in a beam as it travels a distance ds. This α can be defined by: α ν =nσ ν or α ν =ρκ ν

20 Radia:on transport The decrement of I ν when passing through a path of length ds: di v = - α ν I ν ds Inside a source, a contribu:on to I ν can be made from emicers. The increment is: di ν = j ν ds The basic equa:on of transport is:

21 Simple solu:ons (1) Emission only: with S the total emission path. The increase in the specific intensity is equal to the emission coefficient integrated along the line of sight

22 Simple solu:ons (2) Absorp:on only: The brightness decreases along the ray by the exponen:al of the absorp:on coefficient integrated along the line of sight

23 Op:cal depth (1) We now introduce the quan:ty op#cal depth:

24 Op:cal depth (2) The intensity decreases as follows: I ν = I ν,0 exp(- τ v ) What follows from this: τ=1 à 1/e (37%) τ>>1 à op:cally thick τ<<1 à op:cally thin

25 Transport equa:on and source func:on The rewricen equa:on of transport becomes We define the source func:on as follows

26 Equa:on of transfer This yields the format solu:on of the EOT: When S ν constant: I ν (τ ν ) = I v,0 exp(- τ ν ) + S ν (1- exp(- τ ν )) τ >>1: I ν à S ν τ <<1: I v à I ν,0 + S ν τ ν

27 A special case When is constant throughout the source, this can be rewricen as: Ques:on: What is the intensity of this source for small and large op:cal depth when it has size R?

28 If then Answer A licle trick. First, we mul:ply by the source size s=r: Op:cally thin (τ << 1): 1 exp(- τ) = τ = τ à Op:cally thick (τ >> 1):

29 The cos(θ) law We oven hear the expression that radia:on from an op:cally thick source comes from its surface We do mean that the emission we see is emiced from a layer with τ = 1. The emiwng volume the observer sees depends on inclina:on.

30 Protoplanetary disk

31 Mean free path (1) The mean free path is the average distance traveled by a photon before being absorbed. The probability of a photon to travel at least and op:cal depth τ ν is exp(- τ ν ). The mean op:cal depth is thus unity: In a homogeneous medium, the average distance traveled is defined as l ν : or

32 Mean free path (2) A source with radius R and total op:cal depth τ > 1 has a mean free path:

33 ScaCering effects: random walks Assume a photon that interacts through scacering inside a source R and op:cal depth τ > 1 How many :mes does it scacer before escaping? How much :me does it take?

34 Random walks The total net displacement aver N scacerings is: R = r 1 + r 2 + r r N = 0 If we want the distance R traveled by a typical photon we need to calculate the square displacement:

35 Random walks The cross products vanish for isotropic scacering Ques:on 1: Ques:on 2:

36 Einstein A coefficients

37 A three level system

38 Transi:on probabili:es A 21 [s - 1 ] = transi:on probability for spontaneous emission per unit :me B 12 J ν = transi:on probability for absorp:on per unit :me. B 21 J ν = transi:on probability for s:mulated emission per unit :me. The last two depend on the strength of the radia:on field:

39 The rela:on between Einstein coefficients (1) In equilibrium: Solve for the radia:on field: The ra:on between n 2 /n 1 :

40 The rela:on between Einstein coefficients Use the previous results to obtain: At equilibrium J ν must be equal to the black body intensity (next lecture), and then the rela:on is:

41 Rela:on to transfer equa:on The emissivity is related to the Einstein coefficients: Total absorp:on coefficient by: