1. Collision rate =  coll -1 = n e  coll v ~ (k B T/m H ) 1/2 n e  coll ~ 9 x10 -12 n e T 1/2 s -1 collisions of electron against atoms Spontaneous emission rate for electric dipole of hydrogen - first excited state to ground state has A 21 ~ 10 8 s -1 (Einstein’s coefficient), much smaller rates for transitions such as forbidden lines. Forbidden lines are commonly observed in the ISM because thanks to very low densities excited metastable states (that can decay via low probability forbidden lines) are not rapidly collisionally de-excited like in Earth’s atmosphere Clearly  coll -1 << A 21 for typical ISM densities and temperatures v ~ v s where v s = gas sound speed In steady state: collisional excitation rate=collisional de-excitation rate + radiative rate Consider two energy levels E 1 (ground state) and E 2 (excited state) with gap E 12 = ½ m e (v 1 2 – v 2 2 ) (v = velocity of the electron colliding with atom) Then in steady state one has: n e n 1 R 12 =n e n 2 R 21 + n 2 A 21 (1) (collision rates R 21 and R 12 measured per unit volume)

2. Recall that the energy levels of distinguishable particles (any classical gas) obey the Maxwell-Boltzmann statistics. For the principle of detailed balance (eg Einstein’s relations)  process being in equilibrium with its inverse at equilibrium n 1 R 12 =n 2 R 21 where n 1 =g 1 e -  E 1 n 2 =g 2 e -  E 2  = 1/k B T Therefore R 12 = R 21 g 2 /g 1 e -  E 12 Solving the steady state equation (1) for n 2 /n 1 and recalling the above expressions for n  and n 2 one obtains: n 2 /n 1 = g 2 /g 1 e -  E 12 ( 1 + A 21 /n e R 21 ) -1 (2) If n e >> n c, n c = critical (particle) density= A 21 /R 21, then (2) yields the result expected in thermodynamical equilibrium. If n e << n c (low density limit) then the excited state n 2 is depleted compared to the thermodynamical equilibrium and the depletion depends on the spontaneous radiative emission rate A 21 The conditions in the ISM are typically in the second regime and this is another way to see that radiative emission is crucial in determining T ISM

3. The cooling rate from spontaneous decay (radiative cooling rate) will be given by C ine =  A 21 n 2 x  E = n 2 A 21 E 21 The cooling rates in the two different regimes will be: (a) C line (ne >> nc) ~ n 2 A 21 h 21 ~ n 1 (g 2 /g 1 )A 21 h 21 e -  E 12 (b) C line (ne <<nc) ~ n 2 A 21 h 21 ~ n 1 n e R 12 h 12 (from the fact that (2) in this limit yields n 2 /n 1 ~ n e R 12 /A 21 ), which is independent of A 21. Example of importance of regime (b) - line cooling efficient at low densities: Gas in which cooling is provided by OIII line (forbidden line at 5007 A, E 12 ~ 2.5 ev) and hydrogen recombination. One can show that OIII line cooling wins over recombination cooling even though abundance of heavy ions (O, C, N, Ne) is low in ISM (n ion /n H ~ 10 -4 ) for gas at T ~ 10 4 K, where: C line ~ n ion n e E 12 exp(-  E 12 ) (3) (from (b) + detailed collisional balance) Similar expression for C hy (hydrogen recombination rate to ground state), then take the ratio finding that it is ~ (n ion /n H ) (E 12 /E 0 )e  (E 0 – E 12 ), E 0 =13.6 ev For regime (b) to apply n e << n c at T ~ 10 4 K – but this is always satisfied because n c ~ 2 x 10 7 cm -3 for OIII line (similar high densities can be inferred for other lines)

4. (ii)Radiative cooling is the reason why T ism < 10 5 K Many ions in the ISM get excited to metastable states and have forbidden transitions with energy gaps of only a few ev. Assume line cooling of a ion (transition energy E 12 ) and that C line is in the low density regime. Assume steady state Heating rate by ionization = radiative cooling rate via line emission and detailed balancing for ionization: Recombination rate=ionization rate. Then, if E 0 = 13.6 ev (ionization potential of hydrogen ground state)  ion E 12 e - E 12 / k B T ~  (T) E 0, which solving for T yields T ~ 10 4 K From (3) plus the fact that in (3) the cross section is effectively the cross section of the electron in collisional excitation (  prop.  (h/2  *1/m e v) 2, where v ~ v s ) one obtains the dependence of the cooling rate on T: C line prop. to T -1/2 exp(-E 12 /k b T) (this for any radiative transition in ISM conditions) From which it appears that line cooling is maximum when T neither too large or too small, k B T ~ E 12 (both at high and low T C line tends to zero) At different temperatures different transitions/lines will be most effective thermal de Broglie wavelength of ideal electron gas

5. Cooling in high density/low T ISM (molecular phase) (2)Dust cooling. Dust grains mixed with gas (atomic, moelcular) in ISM of galaxies. Mdust/Mgas ~ 0.01 at least in solar neighborhood, proportional to mass density ratio between “metals” and H/He). Dust can cool via thermal emission -  if Tdust < Tgas dust can act as a coolant for the gas in the ISM (see next slide) Collisions with molecules/atoms lead to lattice vibration of dust grains or dust heated through absorption of optical and UV photons grain goes to excited energy level and decays through emission of infrared photon. Tdust in general different from Tgas since both their cooling and heating rates different. To calculate radiative emission from dust one typically assumes that dust grains emit as blackbodies (for photons moving inside them grains are really optically thick media given their high densities!). Peak of blackbody dust emission at long wavelength (100  m - infrared) for typical dust temperature Td ~ 30 K (recall Wien’s law) -  wavelength longer that typical dust grain size (1  m) so no absorption by neighboring grains. Dust cooling rate =====  d = 1 x 10 -10 (n H /10 -3 cm -3 )(Td/10 K) 6 eVcm -3 s -1 (equation assumes all grains have same temperature, range of T modest enough) (1)Molecules produce line cooling via both radiative and rotovibrational transitions. Some molecules, e.g. CO, NH 3, H 2 0, HCN, can cool the gas to T << 100 K even if they are much less abundant than H 2 (H 2 cooling only efficient at ~ 1000 K). Note that “metals” (high Z atoms/ions) enough to cool ISM down to T ~ 100 K (e.g. CII or OIII line) + in gas at T ~ 50-200 K fine-structure transitions in ions also important.

6. Cooling of gas via gas-dust collisions Important in densest regions of molecular clouds (in the “cores” that collapse and form stars). These regions are dense enough that optical and UV photons do not penetrate (absorbed in outer envelope) --  dust is cold, Tdust < Tgas (gas is efficiently heated by cosmic rays at high densities,  > 100 cm -3 ). Cooling rate depends on collision rate between dust grains and gas molecules. Consider hydrogen molecule sticking on grain and releasing translation kinetic energy = 3/2k B Tg. If molecule has time to reach thermal equilibrium with grain lattice it leaves grain with energy (3/2)k B Td. The net cooling rate for gas is thus:  g->d = 3/2 k B (Tg – Td) n d /t coll = [use n d  d =  d n H and tcoll= (n H  d V therm ) -1, V therm ~ v s ~ (Tg) 1/2 ] = = 2 x 10 -14 (n H /10 3 cm -3 ) 2 (Tg/10 K) 1/2 ((Tg-Td)/10K) eV cm -3 s -1 Note small pre-factor (smaller than e.g. in dust cooling) and that after many collisions the condition Tg > Td can still hold if molecule sticks for time longer than it takes for the grain to release the energy gained in collisions via infrared photon emission This is only significant cooling channel for gas when T < 50 K (while in the range 50 – 1000 K cooling via molecular lines (e.g. CO, NH 3, less H 2 ) dominates).

7. HEATING PROCESSES IN ISM HEATING AGENTS (make energy available) (1) ENERGETIC PARTICLES = COSMIC RAYS, MOSTLY RELATIVISTIC PROTONS, ELECTRONS, HEAVY ATOMS/IONS. HIGH ENERGIES ~ 10-10 14 MeV. Not easily absorbed because of high energies – -  can penetrate to the highest density regions of molecular clouds -  most important heating source at  > 100 cm -3 (e.g. in protostellar cores) (2) INTERSTELLAR RADIATION FIELD PHOTONS PRODUCED BY STARS. MOST EFFICIENT HEATING FROM UV PHOTONS and X-rays (10 -4 L for main sequence star, more in T Tauri stars) because of energetic photons) (3) DUST-GAS COLLISIONS (important at the highest densities and lowest temp  cores of molecular clouds) HEATING MECHANISMS (transfer energy to gas) IONIZATION of ATOMS/MOLECULES OR PHOTOELECTRIC HEATING OF GRAINS (e.g. p + H 2  H 2 + + e- + p or p+ H  H + + e- +p) ELECTRON IS LIBERATED AND THEN TRANSFERS THERMAL ENERGY TO SURROUNDING GAS VIA SCATTERING AGAINST ATOMS/MOLECULES + H 2 DISSOCIATION

8. -- In H 2 dissociation e- + H 2  H + H + e- and H atoms have kinetic energy that can be transferred to the gas via collisions -- With cosmic rays electrons produced via ionizing H 2 or HI still very energetic  secondary electrons trigger cascade of ionization events (will lose energy via collisions once they cannot ionize anymore) - in ISM Carbon Ionization very important (lower ionization potential than hydrogen and most abundant among “metals”. n C/ n H ~ 3 x 10 -4 ). -Typical grains have “work functions” (equivalent to ionization potential for atoms) ~ 6 eV, so heating more efficient than for H. Small grains (e.g. PAHs) have even lower ionization potentials and have higher number densities  contribute most of the heating. (2) HEATING OF GRAIN LATTICE ELECTRONS THAT DO NOT GAIN ENOUGH ENERGY TO LEAVE THE GRAIN (energy less than work function of grain i.e. its global electrostatic potential) HEAT THE GRAIN LATTICE AS THEY BOUNCE WITHIN IT --  THE GRAIN RAISES ITS TEMPERATURE. Grains eventually lose its kinetic energy by radiating as blackbodies (see cooling by dust grain irradiation in previois slides). NOTE: optical photons also efficient at heating grains, less energetic but higher flux than UV and X-rays in normal ISM conditions (and grains have high cross section)

9. Pressure equilibrium in the ISM Most phases of the interstellar medium are in pressure equilibrium:  1 T 1 =  2 T 2 =…  N T N for N phases This is well established for CNM and WNM (P/k B = nT ~ 3 x 10 3 cm -3 K) Pressure equilibrium can exist because thermally stable points of T  exist, i.e. points such that  P/P << 1 (  P ~  T) if a small perturbation  is applied. Stable and unstable points, and in general the shape of T(  ), are the result of the balance between heating and cooling processes in the ISM Exception 1: HII regions transient phase of ISM, equilibrium concepts do not make sense Exception 2: cold molecular phase not in pressure equilbrium ---> outer, lower density envelope of molecular clouds probably in pressure equilibrium but inner, dense core has much higher pressure because (1) is strongly compressed by outer envelope and (2) undergoes gravitational collapse, becoming even denser and more pressurized (gravity decouples dense region of molecular clouds from rest of ISM) WNM CNM

10. Thermal balance and molecular cloud formation Molecular clouds (MCs) are surrounded by HI envelopes detected with 21 cm (part of CNM (T ~ 50-100 K,  ~ 10-50 cm -3 )). Suggests HI clouds somehow turn into MCs (1) Dense HI clouds probably form from the WNM as it is compressed in spiral arms. As density increases line cooling by forbidden CII and OIII lines (again, ions important!) becomes high and not counteracted by heating until temperature has been reduced by a lot. Gas thermally unstable in this regime (in this context spiral arms provide density perturbation) ---> change of phase from WNM to CNM. (2) Thermal balance from  PE =  CII for atomic gas at the density of HI clouds, yields Tg=Teq ~ 54 K. Now the gas is thermally stable but density high enough (> 10 cm -3 ) to have efficient H 2 formation. Molecules form as atoms stick onto grains and then combine on their surface (grain is catalyst, puts H atoms in close contact). H 2 is dissociated by UV photons, but dissociation becomes inefficient at  > 100 cm -3 (cloud interior) because photons absorbed by molecules in lower density envelope (3) Cloud interior becomes thus fully molecular (phase transition from CNM to molecular phase), CO also foms via ion-molecules reactions in gas phase (no grain catalysis) and provides most of the cooling together with gas-dust collisions. Thermal balance condition is now  CR =  CO +  g->d with Td determined by  d->g =  d. Temperatures obtained in range 5-12 K (depending on density).