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12: Thermodynamics of HI

James R. Graham

University of California, Berkeley

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Reading

• Tielens, Ch., 8– Overview of the two phase model and

thermal instability

• Observations– Heiles & Troland, 2003 ApJ 586, 1067

– Heiles & Troland, 2005 ApJ 624, 773

• Theory– Wolfire et al. 1995 ApJ 443 152; 2003 ApJ

587 278

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Outline

• Observations of WNM & CNM– The CNM and WNM are physically distinct

components of the ISM, not just results ofobservational selection (emission vs.absorption)

• Detailed models of heating & cooling

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The HI Sky in Emission

Leiden/Dwingeloo HI survey (Hartmann & Burton, 1997)

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HI Absorption

• 21 cm absorptionspectra for 54first Galacticquadrantextragalacticradio sources– Observed

spectrum dividedby the continuumyields exp(-τ)

G31.389-0.383

Kolpak et al. 2002

ApJ 578 868

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Radial Distribution of CNM

• 21 cm optical vs. radius– Dots are Individual

channel measurements

– Various histogramscorrespond to differentchoices for the rotationcurve

– y-axis is cropped

Kolpak et al. 2002

ApJ 578 868

€

In the CNM

τ 0 ≈ 0.6 NH /2 ×1020cm-2

T2S b5

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Evidence for a Two Phase HI?

• In the inner Galaxy there is 21 cm H I emission at allvelocities allowed by Galactic rotation (CNM or WNM)– CNM absorption is less ubiquitous particularly R < 2 kpc

(Garwood & Dickey 1989 ApJ 338 841)

• Recent studies shows that there is significant CNMabsorption between 2-5 kpc– Liszt et al. 1993 AJ 103 2469; Kolpak et al. 2002 ApJ 578 868

• In the outer Galaxy the presence of WNM isreasonably well established (Kulkarni & Heiles 1987)– WNM in absorption features seen toward Cygnus A at R = 9

kpc (6000 K) and 12 kpc (4800 K) (Carilli et al. 1998 ApJL 50279)

• CNM absorption components have been observedfrom R = 14-17 kpc– (Colgan et al. 1988 ApJ 328 275; Kolpak et al. 2002)

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Properties of CNM/WNM

• The CNM & WNM is physically distinct– CNM & WNM components suggests two temperatures

regimes• One low temperature peak and one spread roughly uniformly

over a broad range > 5000 K

– About half of the WNM has 500 < T/K < 5000 (thermallyunstable)

– Median column densities• 0.5x1020 cm-2 per CNM component• 1.3x1020 cm-2 per WNM component

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The CNM is Cold

• Median CNM temperature is Ts ≈ 48 K– About half the temperature usually quoted

• Some very cool gas with Ts ≈ 15 K

– NHI-weighted temperature is Ts ≈ 70 K

solid: b>10o

dotted: b<10o

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Filling Factor

• The WNM is 61% of the total H I column andfills ~ 50% of the volume in the Galactic planeat z = 0

• The WNM– Has ~ 1.5x the mass of the CNM– Occupies ~ 150x more volume than the CNM

• Many lines of sight have no CNM– These form a distinct class and are confined to

particular areas of the sky, including regionsdisturbed by super-shells & super-bubbles

– Column densities depart very markedly from thoseexpected from a plane-parallel distribution

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Kinematics

• The NHI weighted rms velocities are– 7.1 km s-1 for the CNM– 11.4 km s-1 for the WNM

• Most CNM clouds are strongly supersonic– The Mach number of internal macroscopic motions

for CNM clouds is ~ 3, with wide variations• The velocity dispersion of the CNM is

somewhat larger than typical sounds speed inthe WNM– If the CNM consists of clouds moving within the

WNM, then the motion is supersonic

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McKee Ostriker (1977) WNM/CNM

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Morphology• To a good approximation each CNM cloud has an

associated WNM envelope– Too much WNM, and too much of it thermally unstable for

this picture to be generally true• CNM clouds are not isotropic

– CNM features are sometimes large sheets with aspectratios measured in the hundreds

– These sheets contain blobs, which themselves are sheet-like but with smaller aspect ratios

• Heiles & Troland discard the “raisin pudding” modelin favor of the “blobby sheet”– The CNM consists of sheet-like structures with embedded

sheet-like blobs or cloudlets– Each WNM cloud probably contains a few CNM large

sheets

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Cold & Warm HI• The thermodynamics of HI has been discussed

over the last 30 years– Field 1965 ApJ 142 531– Draine 1978 ApJSS 36 595– Shull & Woods 1985 ApJ 288 50– Wolfire et al. 1995 ApJ 443 152; 2003 ApJ 587 278

• Broad range of competing physical processes– Photoelectric heating on grains– Ionization and heating by CR and soft X-rays– Recombination– Cooling

• Fine structure lines (CII, OI) & forbidden lines (OI)• Resonance lines (Lyα)

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HI Phase Equilibrium

• Some simple results & patterns emerge fromthis complex system– Provides a context to think about the observations

• Two thermal phases of HI– Low density phase (= WNM?)

• Dominant at low pressure

– High pressure phase (= CNM?)• Dominant at high pressure

– At intermediate pressure two distinct phases cancoexist

• cf. water in the terrestrial environment, with both phases inequilibrium near sea level

– Expect HI to condense into and evaporate from theCNM in response to pressure changes (weather)

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Thermal Equilibrium & the HI Phase Diagram

• Thermal equilibriumline on the density vs.pressure plane

• Phase diagram forphysical conditions inthe solarneighborhood– P/k ~ 103 - 104 cm-3 K

– WNM T ~ 8000 K

– CNM T ≥ 50 K

log(n)

WNM

CNM

10 K100 K

1000 K

10,000 K

Wolfire et al. 1995 ApJ 443 152; 2003 ApJ 587 278

nΓ=n2Λ

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Heating

• Heating of CNM& WNM isdominated byfar-UV– Photoelectrons

ejected from dust& PAHs dominate

– Contribution fromgas ionization byX-rays & CR

Heating

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Cooling

• Cooling in theCNM isdominated byfine structurelines– C+ 158 µm

• WNM cooling isdominated byLyα

Cooling

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Two Phase Equilibrium

• Gas in thermal equilibrium (heating = cooling)at different (n, T) can coexist at the same P– Dynamical equilibrium– Basis for CNM clouds embedded in WNM inter-

cloud

€

In equilibrium nΓ = n2Λ

or ΓnT

=ΛT

Strongly dependentonly on T forT ≤ 102 K

T ≥ 8000 K

Weaklydependenton T

€

Pressure∝ nT

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Equilibrium & Stability

• In equilibrium

€

ΓnT

=ΛT

which implies PΓ∝TΛ

because P = nkT

€

Heating

nΓ > n2Λ€

PΓ

€

T = const.Λ, Γ = const.∴n2Λ > nΓ

→ cooling

T

€

Equilibrium

nΓ = n2Λ

P = const.

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Stable & Unstable Equilibrium

• F and H are stable; G is unstable– Isobaric temperature perturbations damp out

at F & H

€

Heating

nΓ > n2Λ€

PΓ

€

Cooling

n2Λ > nΓ

T

F

HG

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Stability Condition

• Consider a co-moving volume Vcontaining nV H atoms

– In dt the entropy change, dS, by the first lawis

€

dQ = −n2L dtV = TdS where n2L = n2Λ − nΓ

The entropy per H atom is SH ≡ S /nV

1nV

dSdt

= −nLT

or dSHdt

= −nLT

Define δSH ≡ SH − SH0 d

dtδSH

A

= −nLT−n0L0

T0

A

= −δnLT

A

Net cooling

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(In)stability Condition

• Perturb the system holdingthermodynamic variable A fixed

– Condition for exponential runaway

€

ddtδSH

A

and δSH( )A have the same sign

Given ddtδSH

A

= −δnLT

A

The instability condition −δ nL /T( )AδSH( )A

= −∂ nL /T( )∂SH

A

> 0

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(In)stability Condition

• Condition for exponential runaway

€

If the specific heat cA ≡ T∂SH∂T

A

> 0

−∂ nL /T( )∂SH

A

> 0 can be expressed as

∂ nL /T( )∂T

A

< 0

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Physical Interpretation

• What is the meaning of ?

€

∂ nL /T( )∂T

A

< 0

€

• Consider the cooling time

tc =3/2nkTn2Λ − nΓ

∝TnL

If the cooling shortens (δtc < 0) as T decreases (δT < 0)

∂∂T

tc >0 the system is unstable

∂∂T

1tc

<0 or

∂∂T

nLT

<0

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Physical Interpretation

€

• In equilibrium L = 0

The condition for instability ∂∂T

nLT

<0

becomes ∂∂T

nLT

=

nT∂L∂T

+ L ∂∂T

nT

= 0 when L= 01 2 4 3 4

=nT∂L∂T

< 0

∂L∂T

< 0

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Isobaric Instability

• If l is the perturbation length scale– ts = l/cs time for sound to cross the perturbation

– Provided ts < tcool perturbations will be isobaric• Derivatives can be evaluated for p = const.

€

nΓ > n2Λ→ L < 0€

PΓ

€

n2Λ > nΓ → L > 0

T

F

HG

€

F & H

∂L /∂T( )P > 0

stable

€

G

∂L /∂T( )P < 0

unstable

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Separation of Phases

• Region G evolves into a two phase medium– Cool clouds (F) immersed in

an intercloud medium (H)

• Time scale is the heating or

cooling time at G

• Existence of thermal

instability over a finite range

of T is a necessary and sufficient condition

for a multiphase medium

€

nΓ > n2Λ€

PΓ

€

n2Λ > nΓ

F

HG

T

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Examples

€

• Suppose Λ = Λ0Ta and Γ = const.

L = Λ −Γ /n = Λ0Ta − (Γk /P)T

In equilibrium L = 0

∂∂T

nLT

P

=nT

∂L∂T

P

+ L ∂∂T

nT

P

= 0,L= 01 2 4 3 4

< 0

∂L∂T

P

= aΛ0Ta−1 −Γk /P = aΛ0T

a−1 − Λ0Ta−1

∂L∂T

P

= (a −1)Λ0Ta−1

Unstable for a <1

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Collisional Ionization Equilibrium

• For gas with T >> 104 K ionization is in coronalequilibrium– For T ≥ 105 K cooling is ~ a power law with a < 1

• Unstable

Gae

tz &

Sal

pete

r A

pJS

S 5

2 15

5

Solid: line cooling; dashed: continuum

L∝T

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Non-Equilibrium Isobaric Cooling

€

• Γ = 0 and P = nkT= const.

∂∂T

nLT

P

= nT ∂∂T(L /T 2)

P

= nT ∂∂T(Λ0T

a−2)

P

= (a − 2)nTΛ0Ta−3

Unstable for a < 2 Even more unstable than the L = 0 case

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Cooling Timescale

• The cooling timescale is

– CNM: n ≈ 60 cm-3, nΛ ≈ 6x10-26 erg s-1 H-1

• τth ≈ 3900 yr

– WNM: n ≈ 0.4 cm-3, nΛ ≈ 2x10-26 erg s-1 H-1

• τth ≈ 1.8 Myr

€

τ th = nkT n2Λ = nkT nΓ

≈ 1.6 nT3000 cm-3 K

ncm-3

−1 nΛ

10−26erg s-1 H-1

−1

Myr

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Recombination Timescale

• The recombination timescale is

• For the CNM, xe ~ 10-3, therecombination timescale is 60,000 yr

• For the WNM, xe ~ 10-1, therecombination timescale is 600,000 yr

€

τ rec = 1 neα T( )≈ 200T 0.6 ne yr

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Dynamic Timescale

• A typical dynamical timescale is thesound-crossing time–the time it takes asystem to come into pressure equilibrium

– CNM τs ≈ 1 Myr

– WNM τs ≈ 6 Myr€

τ S = R cS; cS = 0.14T 0.5km/s

≈ 0.7 Rpc

T2

−0.5Myr

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Timescales

• Frequently, it takes longer to establish ionizationequilibrium than thermal equilibrium

• The WNM is more easily perturbed from thermalequilibrium than the CNM

60.61.8WNM

10.060.004CNM

τs(Myr)

τrec(Myr)

τth(Myr)

Phase

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Examples & Counter Examples• Cool WNM (Heiles et al. ApJL 551 105)

• The broad line emission gas in AGN has also been modeled as part of atwo-phase medium (Krolik McKee & Tarter 1981 ApJ 249 422)– The line-emitting gas is located in clouds with a small volume filling factor

• The cooling time of interstellar gas shock-heated by supernova remnantscould be longer than the interval between the passage of successiveshocks (Cox & Smith 1974 ApJL 189 105)– Led to the three-phase model of the ISM in which

– Most of the volume is occupied by shock-heated gas (McKee & Ostriker 1977ApJ 218 148)

– This 106 K gas is an example of a non-equilibrium phase (Spitzer 1956 ApJ124 20)

– Difficult to determine the fate of the hot intercloud medium.• May cool radiatively in a region close to the disk (McKee & Ostriker 1977)

• Vented into the halo through chimneys (McCray & Kafatos 1987 317 190), where itsuffers adiabatic and radiative cooling forming a Galactic fountain (Shapiro & Field1976 ApJ 205 762)

• The hot halo gas may cools sufficiently to form clouds which eventually rain down onthe disk, remain hot enough to drive a galactic wind, or both

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Variation with Metal Abundance• Heating & cooling of the

CNM is dominated byprocess dependent onthe heavy elementabundance, Z– Equilibrium temperature

is weakly dependent on Zif dust content D/G ∝ Z

– Primary effect ofincreasing Z is to lowerthe characteristicpressure of the phasetransition between CNM& WNM

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Variation with FUV Field• FUV powers CNM &

WNM–Equilibrium temperature

is weakly dependent onFUV field for 0.3-3 xlocal value• Photoelectric effect on

dust is quenched bygrain charging

–For > x 3 the local fieldT is increased atconstant pressure• But high-field FUV

regions are also likelyto be regions of overpressure

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