Thermal Instability Theory1. Observations Relevant to Phases 2. Linear Instability Theory3. Updated FGH Model4. Challenges

ReferencesField ApJ 142 5311965Field, Goldsmith & Habing, ApJ 15 L15 1969Draine ApJS 36 595 1978 (photoelectric heating) Shull & Woods ApJ 288 50 1985 (clear presentation)Shull in “Interstellar Processes”, pp. 225-234

(short introduction c.1987)Wolfire et al. ApJ 443 152 1995 + ApJ 587 278 2003

1. Observations Relevant for PhasesObservations indicate that gas has a range of physical properties, highly simplified in the following table.

Phase Observations T nCNM HI etc. abs. lines 80 40WNM HI emission lines 0.08 4,000WIM DM, EM, Hα em. 0.03 8,000HIM UV abs., soft X-rays 0.003 106

The numbers are only meant to be suggestive. Poorly known filling factors have been ignored. For example, Heiles & Troland find T for the WNM to be in the range 500-5,000 K.

HI Phase EquilibriumObservations suggest two phases of HI:Low density phase (WNM) - dominant at low pressureHigh density phase (CNM - dominant at high pressureAt intermediate pressures these phases can coexist.

The motivating analogy is terrestrial water, where phase equilibrium is observed at sea level.

One might expect HI to condense into (and evaporate from) the CNM in response to pressure changes – a kind of interstellar weather. But we know that, although water vapor is key role, meteorology is more than just phase diagrams – but of course we can still try.

HI Phase Diagram

log(n)

WNMCNM

10 K100 K

1000 K10,000 K

CNMPhase diagram (solid curve)near the SunWolfire et al. 1995

Coexistence for T(WNM) ~ 8,000 K P/k ~ 103 – 104 cm-3K T(CNM) ≥ 50 K

2. Linear Instability Theory

Recall from Lecture 4 the equations for a single fluid

)(),()(

0

0

1 ΛΓΛTΛdtdsT

pdtvd

vdtdρ

mm −=∇⋅∇=+

=∇+

=⋅∇+

−ρκρ

ρ

ρr

r

RR

where Λm is the net heating rate for unit mass, and the entropy change for a polytropic gas is

)(1

1 ρρ

γγ

ρ dpdpTds −−

=

Ionization, gravity, and magnetic fields are ignored.

Linear instability analysis (Field 1965)As in the elementary theory of sound propagation, the hydro equations are linearized (around a steady state):

TTTvvv δδδρρρ +=+=+= 000 ,, vr

with the condition0),( 00 =TΛm ρ

When δρ, δv, and δT are assumed to vary as expi(kz - ωt), i.e., as waves traveling in the direction z, a cubiccharacteristic equation is obtained for ω. The condition (on Λm) for this equation to have roots corresponding to growing modes gives Field’s thermal instability condition (thermalbecause it is a condition on the cooling function Λm).

Heuristic Derivation

δs)sΛ

(δΛ Am

m ∂∂

=

Return to the heat equation and consider the time evolution (dt) of a small fluctuation (δT, δρ, etc.) for a co-moving fluid element:

With one thermodynamic variable A fixed (e.g., the pressure)

dtsTd mΛ−= δδ )(this equation becomes

grA

m

τ)

sΛ(

Tδs

dtd

δs11)(1

≡∂∂

−=

where τgr is a characteristic timescale

Heuristic Derivation (cont’d)From the last equation we conclude that

0<⎟⎠⎞

⎜⎝⎛∂∂

A

m

sΛ

is the condition for exponential growth (or thermal instability),and that τgr is the growth rate of the instability. With the aid of thermodynamics, the derivative can be expressed in terms of T and ρ (e.g., using Tdsρ = cρdt and Tdsp= cpdt etc.,), we can get these more specific conditions:

0

0

<⎟⎠⎞

⎜⎝⎛∂Λ∂

−⎟⎠⎞

⎜⎝⎛∂Λ∂

=⎟⎠⎞

⎜⎝⎛∂Λ∂

<⎟⎠⎞

⎜⎝⎛∂Λ∂

T

mm

p

m

m

TTTT

T

ρ

ρ

ρ

isochoric

isobaric

FGH: Two Phase EquilibriumBasic Idea: Gas in thermal balance (Λ=Γ) can coexist at the same p with two (n,T) combinations. The two sets would correspond to cool clouds embedded in a warmintercloud medium, now referred to as CNM and WNM. Thermal Balance:

.λγnΓΛ

,γnΓλnΛ

Γ),(ΛρΛm

=⇒=

==

−= −

H

H2

H and

1

λ is sensitive and γ insensitive to T; Λ ~ nH2 (cooling mainly

comes from low-density, sub-thermal collisions) & Γ ~ nH(heating almost always involves energy transfer from an external “reservoir” -- radiation field, cosmic rays, turbulence, shocks, etc).

FGH - 2Thermal balance is described by equations which express the density or the pressure as a function of T :

,using)(orH nkTpkTnnpnH

===λγλ

γ

NB: n/nH = 1.425 (solar abundances); λ has dimensions erg cm3 s-1.

FGH used cosmic ray heating and CII fine-structure heating:

)K91()/K91exp( eV20ands10 with

C

115CRCR

kTCx lu −=≈≈= −−

λεςςεγ

3/2P -------------CII ground state 157.7 µm

1/2P -------------Clu is the abundance weighted sum ofe and H collisional rate coefficients

FGH - 3Schematic plots (backwards van der Waals): p vs.1/v

cooling

heating

F = stable WNMG = thermally unstableH = CNM

Equivalent version:

nT

nTnkT

n−<

∂∂

⇒<∂∂ 0)(

Recall that positive compressibility is a thermodynamic condition for stability.

→←

FG-4

cooling dominates

heating dominates

G is unstable to isobaric perturbations: Displacing it makes it go to F or G, whereas similar perturbations restore F & G. Thus F and G can be in pressure equilibrium, and are pictured as cool clouds immersed in a warm (“intercloud”) medium..

Timescales for Growth of InstabilitiesEarlier we found that the growth time for thermal instability is

AAAgr Tm

csm

⎟⎠⎞

⎜⎝⎛∂Λ∂

−=⎟⎠⎞

⎜⎝⎛∂Λ∂

−=11

τ

The usual heat capacities per unit mass are

nmmkc

mkc p

ρρ === ,

25and

23

where m = ρ/n is the mass per particle (not equal to the mass per H nucleus, m’ = ρ/n = 1.425). In this notation,the cooling function per unit mass is

( )γλ −=Λ H'1 nmm

Timescales (cont’d)

⎟⎠⎞

⎜⎝⎛ +−=⎟

⎠⎞

⎜⎝⎛−=

TdTd

kn

mm

dTd

kn

mm

p

λλτ

λτ ρ 2

5'1

23'

1 HH

Isochoric fluctuations are unstable for λ decreasingwith T.

Isobaric fluctuations are unstable for λ either (i) decreasing with T or (ii)increasing less rapidly than λ /T.

Equivalently, the isobaric condition is:or λ increasing less rapidly than T. 1

lnln

<Td

d λ

It is now straightforward to evaluate the growth times and instability conditions for the case γ constant and λ a functionof T only:

Cooling Time and Growth TimesThe above instability growth times are closely related tothe more familiar cooling time:

Tkn

mm

nkTλ

τ 23

H

23

th '1

=Λ

=

This provides a lower limit to the isobaric growth time.

We estimate it for an unstable state (G in FGH) using the results of Wolfire et al. (1995) to be discussed below:

Myr4.3Myr94.0005.0cm5.1K2000

recth

e3

H

===== −

ττxnT

These timescales may not be short compared to dynamical events in the ISM. The recombination time is also longer than the thermal time, raising some questions about the theory.

3. Updated FGH Models“The Neutral Atomic Phases of the ISM”

Wolfire, Hollenbach, McKee, TielensApJ 443 152 1995 (with Bakes) & ApJ 587 278 2003

Steady state thermal and chemical balance for the 10 mostabundant elements plus grains & PAHs subject to:

process ionization heating cooling

FUV ♪ ♪CRs & X-rays ♪ ♪recombination ♪ ♪line emission ♪

Heating and Cooling

Dominant heating: PAH photoelectric effect.Dominant cooling: Fine structure lines for CNM

H and forbidden lines for WNM

dashed lines: heating solid lines: cooling

nvsn .and γλ

Variation 1: Metal Abundance• Heating & cooling of

CNM is dominated by processes dependent on heavy element abundance Z

• Weak dependence on Z when D/G ∝ Z

• Increasing Z reduces the range of allowed stable pressures

D/G= dust/gas ratio relative to local SM)

For the standard choice D/G =1 (local value), theallowed pressure range is quite narrow (30%)

Variation 2: FUV Field

UV powers CNM & WNMbut T responds weakly to small FUV changes.

Photoelectric heatingis quenched by PAH and grain charging.

Large FUV increaseswipe out instability butlead to over-pressuredregions.

See Wolfire et al. 1995 for other variations and Wolfire et al. 2003 for changes with galactic radius.

Summary of the Two-Phase Model The brief for the two-phase model is laid out in detail by Wolfire et al. (1995 & 2003). The models are complex (it is important that you read these papers yourselves), and the observations are limited and difficult to interpret.

Our first concern has to be whether steady state theory applies.

1. A clear observational warning comes Heiles & Troland who find that ½ of the warm HI is in the unstable part of the phase diagram.

2. The long thermal and recombination timescales may begreater than the timescales for mechanical perturbations.

4. Further Considerations of the Phase Concept

The observations discussed so far indicate at leastfour phases (not all necessarily in phase equilibrium),even without including the densest gas in the midplane.

McKee & Ostriker (ApJ 218 148 1977) were the first to conceive of a theory of the ISM which recognized the existence of phases other than the CNM & WNM and also gave a central role to supernovae as the primary source of injection of mechanical energy into the ISM. Their ideas are the background for the furtherdiscussion (next two weeks) of the diffuse ISM.

McKee-Ostriker Picture

SNR

SNR3. 2 pc 2. 30x40 pc

1. 600x800 pcSupernovaSupernova Supernova!

Numerical Simulations: The Future?

Despite the 1977 McKee-Ostriker revolution, the steadyfinite-phase theory persists, partly because it allows forthe full complexity of physical processes in the ISM. Numerical simulations, on the other hand, require highly simplified versions of the physics & chemistry,but they are the wave of the future. For a preview, see:

K Wada & CA Norman, ApJ 516 L13 1999ApJ 547 172 2001

AG Kritsuk & ML Norman ApJ 569 L127 2002ApJ 580 L51 2002

http://akpc.ucsd.edu/ThermalInstability