Collision rate = tcoll-1 = nescollv ~ (kBT/mH)1/2 nescoll ~ 9 x10-12 neT1/2s-1 collisions of

1
v vs where vs gas sound speed
Collision rate tcoll-1 nescollv (kBT/mH)1/2
nescoll 9 x10-12 neT1/2s-1 collisions of

electron
against atoms
Spontaneous emission rate for electric dipole of
hydrogen - first excited state to ground state
has A21 108 s-1 (Einsteins 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 Earths
atmosphere Clearly tcoll-1 ltlt A21 for typical ISM
densities and temperatures
In steady state collisional excitation
ratecollisional de-excitation rate radiative
rate Consider two energy levels E1 (ground
state) and E2 (excited state) with gap E12 ½
me(v12 v22) (v velocity of the electron
colliding with atom) Then in steady state one
has nen1R12nen2R21 n2A21 (1)
(collision rates R21 and R12 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 Einsteins relations)
? process being in equilibrium with its inverse
at equilibrium n1R12n2R21 where
n1g1e-bE1 n2g2e-bE2 b
1/kBT Therefore R12 R21 g2/g1e-bE12 Solving
the steady state equation (1) for n2/n1 and
recalling the above expressions for n1 and n2 one
obtains n2/n1 g2/g1 e-bE12(1 A21/neR21)-1
(2) If ne gtgt nc, nc critical (particle)
density A21/R21, then (2) yields the result
expected in thermodynamical equilibrium. If ne
ltlt nc (low density limit) then the excited state
n2 is depleted compared to the thermodynamical
equilibrium and the depletion depends on the
spontaneous radiative emission rate A21 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
TISM
3
The cooling rate from spontaneous decay
(radiative cooling rate) will be given by C ine
A21n2 x DE n2A21E21 The cooling rates in
the two different regimes will be (a) Cline
(ne gtgt nc) n2A21hn21 n1(g2/g1)A21hn21e-bE12 (
b) Cline (ne ltltnc) n2A21hn21 n1neR12hn12
(from the fact that (2) in this limit yields
n2/n1 neR12/A21), which is independent of
A21. 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, E12 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 (nion/nH 10-4) for gas at T 104 K,
where Cline nionnelts21vgt E12 exp(-bE12)
(3) (from (b) detailed collisional
balance) Similar expression for Chy (hydrogen
recombination rate to ground state), then
take the ratio finding that it is (nion/nH)
(E12/E0)e b(E0 E12) , E0 13.6 ev For regime
(b) to apply ne ltlt nc at T 104 K but this
is always satisfied because nc 2 x 107 cm-3 for
OIII line (similar high densities can be inferred
for other lines)
4
(ii)Radiative cooling is the
reason why Tism lt 105 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 E12) and that Cline
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
rateionization rate. Then, if E0 13.6 ev
(ionization potential of hydrogen ground
state) aionE12ltsvgt e -E12/kBT a(T) E0, which
solving for T yields T 104 K From (3) plus
the fact that in (3) the cross section is
effectively the cross section of the electron in
collisional excitation (s prop. p(h/2p1/mev)2,
where v vs) one obtains the dependence of the
cooling rate on T Cline prop. to T-1/2
exp(-E12/kbT) (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, kBT E12 (both
at high and low T Cline 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)

• Molecules produce line cooling via both radiative
  and rotovibrational
• transitions. Some molecules, e.g. CO, NH3, H20,
  HCN, can cool the gas to T ltlt 100 K even
• if they are much less abundant than H2 (H2
  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.

• 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
  lt 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 mm - infrared) for typical
• dust temperature Td 30 K (recall Wiens law) -?
  wavelength longer that typical
• dust grain size (1 mm) so no absorption by
  neighboring grains.
• Dust cooling rate Ld 1 x 10-10 (nH/10-3
  cm-3)(Td/10 K)6 eVcm-3s-1
• (equation assumes all grains have same
  temperature, range of T modest enough)

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 lt Tgas (gas is
efficiently heated by cosmic rays at high
densities, r gt 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
3/2kBTg. If molecule has time to reach thermal
equilibrium with grain lattice it leaves grain
with energy (3/2)kBTd. The net cooling rate for
gas is thus Lg-gtd 3/2 kB(Tg Td) nd/tcoll
use ndsd SdnH and tcoll (nHsdVtherm)-1, Vther
m vs (Tg)1/2 2 x 10-14 (nH/103 cm-3)2
(Tg/10 K)1/2 ((Tg-Td)/10K) eV cm-3s-1 Note small
pre-factor (smaller than e.g. in dust cooling)
and that after many collisions the condition Tg gt
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 lt 50 K (while in the
range 50 1000 K cooling via molecular lines
(e.g. CO, NH3, less H2) 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 10-1014 MeV. Not easily absorbed
  because of high energies
• -? can penetrate to the highest density regions
  of molecular clouds -? most
• important heating source at r gt 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 H2 ? H2 e- p or p
  H ? H e- p)

8
-- In H2 dissociation e- H2 ? 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 H2 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. nC/nH 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 GRAINRAISES 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 r1T1 r2T2 rNTN
for N phases This is well established for CNM
and WNM (P/kB nT 3 x 103 cm-3 K) Pressure
equilibrium can exist because thermally stable
points of T(r) exist, i.e. points such that dP/P
ltlt 1 (dP drdT) if a small perturbation dr/r
is applied. Stable and unstable points, and in
general the shape of T(r), 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 ---gt 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, r 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) ---gt change of phase from WNM to
CNM. (2) Thermal balance from GPELCII for
atomic gas at the density of HI clouds, yields
TgTeq 54 K. Now the gas is thermally stable
but density high enough (gt 10 cm-3) to have
efficient H2 formation. Molecules form as atoms
stick onto grains and then combine on their
surface (grain is catalyst, puts H atoms in close
contact). H2 is dissociated by UV photons, but
dissociation becomes inefficient at r gt 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 GCR LCO Lg-gtd with
Td determined by Gd-gtg Ld. Temperatures
obtained in range 5-12 K (depending on density).