1. The formation of stars and planets
Day 1, Topic 2:
Radiation physics
Lecture by: C.P. Dullemond

2. Astronomical Constants
CGS units used throughout lecture (cm,erg,s...)
AU = Astronomical Unit
= distance earth - sun = 1.49x1013 cm
pc = Parsec = 3.26 lightyear = 3.09x1018 cm
’’ = Arcsec = 4.8x10-6 radian
Definition: ’’ at 1 pc = 1 AU
M = Mass of sun = 1.99x1033 gram
M = Mass of Earth = 5.97x1027 gram
L = 3.85x1033 erg/s

3. Radiative transfer Basic radiation quantity: intensity
Definition of mean intensity
Definition of flux:

4. Radiative transfer Planck function:
In dense isothermal medium, the radiation field is in thermodynamic equilibrium. The intensity of such an equilibrium radiation field is:
(Planck function)
In Rayleigh-Jeans limit (h<<kT) this becomes a power law:
Wien
Rayleigh-Jeans

5. Radiative transfer Blackbody emission:
An opaque surface of a given temperature emits a flux according to the following formula:
Integrated over all frequencies (i.e. total emitted energy):
If you work this out you get:

6. Radiative transfer In vaccuum: intensity is constant along a ray
Example: a star A B
Non-vacuum: emission and absorption change intensity:
(s is path length)
Emission
Extinction

7. Radiative transfer Radiative transfer equation again:
Over length scales larger than 1/ intensity I tends to approach source function S.
Photon mean free path:
Optical depth of a cloud of size L:
In case of local thermodynamic equilibrium: S is Planck function:

8. Radiative transfer

9. Radiative transfer
Observed flux from single-temperature slab:
for
and

10. Radiative transfer Emission/absorption lines: Flux Flux  
Hot surface layer
Cool surface layer
Flux 
Flux 

11. Difficulty of dust radiative transfer
If temperature of dust is given (ignoring scattering for the moment), then radiative transfer is a mere integral along a ray: i.e. easy.
Problem: dust temperature is affected by radiation, even the radiation it emits itself.
Therefore: must solve radiative transfer and thermal balance simultaneously.
Difficulty: each point in cloud can heat (and receive heat from) each other point.

12. Thermal balance of dust grains
Optically thin case:
Heating:
a = radius of grain
= absorption efficiency (=1
for perfect black sphere)
Cooling:
Thermal balance:

13. Optically thick case Additional radiation field:
diffuse infrared radiation from the grains
Intensity obeys tranfer equation along all possible rays:
Thermal balance:

14. Dust opacities. Example: silicate
Opacity of amorphous olivine (silicate) for different grain sizes

15. Crystalline vs. amorphous silicates
Bouwman et al.

16. Rotating molecules Classical case: Quantum case:
I is the moment of inertia
J is the rotational quantum number:
J = 0,1,2,3...
B has the dimension of frequency (Hertz)

17. Rotating molecules Dipole radiative transition: JJ-1:
Quadrupole radiative transition: JJ-2:
Transition energies linear in J

18. Rotating molecules Carbon-monoxide (CO): J = 10 =2.6 mm
CO: I = 1.46E-39
Ground based millimeter dishes. CO emission is brightest molecular rotational emission from space. Often used!
Plateau de Bure
James Clerck Maxwell Telescope

19. Rotating molecules Molecular hydrogen (H2) H2: I = 4.7E-41
J = 20 =28 m
J = 31 =17 m
J = 42 =12 m
Due to symmetry: only quadrupole transitions:
Need space telescopes (atmosphere not transparent). Emission is very weak. Only rarely detected.
Spitzer Space Telescope

20. Rotating molecules
Carbon-monoxide (CO) and Molecular hydrogen (H2)

21. Rotating molecules Molecules with 2 or 3 moments of inertia:
“Symmetric top”: 2 different moments of inertia, e.g. NH3:
“Asymmetric top”: 3 different moments of inertia, e.g. H2O: H N H O
These molecules do not have only J, but also additional quantum numbers.
Water is notorious: very strong transitions + the presence of masers (both nasty for radiative transfer codes)

22. Rotating molecules Symmetric top: NH3
Radiative J to J-1 transitions with K=0 are rapid (~ s). But K0 transitions are ‘forbidden’ (do not exist as dipole transitions).
backbone
Quadrupole K0 transitions are slow (10-9 s) but possible (along backbone)
After book by Stahler & Palla

23. Isotopes
Molecules with atomic isotopes have slightly different moments of inertia, hence different line positions.
Molecules with atoms with non-standard isotopes have lower abundance, hence lines are less optically thick.
Examples:
[12CO]/[13CO] ~
[13CO]/[C18O] ~

24. Vibrating molecules: case of H2
Atomic bonds are flexible: distance between atoms in a molecule can oscillate.
Vibrational frequency for H2:
Vibrating molecule can also rotate. Sum of rot + vib energy:
Selection rules for H2-rovib transitions: from v to any v’, but J=-2,0,2 (quadrupole transitions).
Quadrupole transitions are weak: H2 difficult to detect...

25. Vibrating molecules: case of H2
Transitions from v to v-1 or v-2 etc:
J to J: Q-branch transitions: Pure vibrational transitions (no change in J). All lines roughly at same position.
J to J-2: S-branch transitions: during vibrational transition also downward rotational transition: more energy release. Lines blueward of Q-lines, bluer for higher J.
Example: S(1): v=1-0, J= micron
J to J+2: O-branch transitions: during vibrational transition also upward rotational transition: less energy release. Lines redward of Q-lines, redder for higher J. S Q O

26. Vibrating molecules: case of CO
For CO same mechanism as for H2:
Vibrational frequency for CO:
Often v=0,1,2 of importance
Selection rules for CO-rovib transitions: from v to any v’, but J = -1,0,1.
v=-1 is called fundamental
  4.7 m
v=-2 is called first overtone
  2.4 m

27. Vibrating molecules: case of CO
Transitions from v to v-1 or v-2 etc:
J to J: Q-branch transitions: Pure vibrational transitions (no change in J). All lines roughly at same position.
J to J-1: R-branch transitions: during vibrational transition also downward rotational transition: more energy release. Lines blueward of Q-lines, bluer for higher J.
J to J+1: P-branch transitions: during vibrational transition also upward rotational transition: less energy release. Lines redward of Q-lines, redder for higher J. R Q P

28. Vibrating molecules: case of CO
“Band head”:
Rotational moment of intertia for v=1 slightly larger than for v=0
Therefore rotational energy levels of v=1 slightly less than for v=0
R branch (blue branch): distance between lines decreases for increasing J. Eventually lines reach minimum wavelength (band head) and go back to longer wavelengths.
Calvet et al. 1991
2.294
2.302
 [m]
Band head CO first overtone 2-0

29. Overview of location of molecular lines

30. Ices Freeze-out of gases:
Form ice coatings on dust grains
Rotational lines disappear (molecules are in a solid)
Expect each bond (C-C, C-H, C-O etc) to produce a wide vibr. band (like typical dust features), because molecules can exchange energy and momentum.
Various ices studied: CO (<20 K), CO2(< K), H2O (<90 K), etc. (Note: evaporation temperatures depend on various factors).
Exist far away from star (must be cold enough). But ice bands in near-/mid-IR: ices too cold for emission: ice bands only observed in absorption!

31. Example: solid and gas-phase CO
CO ice
CO ice+gas
CO gas
From lecture
Ewine van
Dishoeck

32. A two-level molecule u LTE level population: d
nu is population of upper level
nd is population of lower level
Emissivity and absorptivity:
extinction
stimulated emission
Einstein relations:

33. A two-level molecule Collisional excitation: LTE / Non-LTE
Rate of change of population of upper level:
Absorbtion
Spontaneous emission
Stimulated emission
Collisional de-excitation
Collisional excitation
“Statistical equilibrium equation”
Collision rates satisfy:

34. A two-level molecule
Critical number density N of the gas defined such that:
Dominates when N<Ncrit. We call this: non-LTE.
Dominates when N>Ncrit. We call this: “local thermodynamic equilibrium” (LTE)

35. Photodissociation of molecules
First and second electronic excited state
Ultraviolet photons can excite an atom in the molecule to a higher electronic state.
The decay of this state can release energy in vibrational continuum, which destroys the molecule.
Electronic ground state

36. Formation of molecules: H2
Due to low radiative efficiency, H+H cannot form H2 in gas phase (energy cannot be lost). Main formation process is on dust grain surfaces:
Once two H meet, they form H2, which has no unpaired e-. So H2 is realeased.
In lattice fault: 0.1 eV, so it stays there.
Binding energy of H = 0.04 eV due to unpaired e-.
Many other molecules also formed in this way (e.g. H2O, though water stays frozen onto dust grain: ice mantel).

37. Formation of molecules: H2
If sufficient free electrons are present, H2 can also form in the gas phase, until all free electrons are used up:
H + e-  H-+h
H- + H  H2+ e-
But these reactions are rare due to limited electron abundance! Main formation is by grain surface reactions.