ISM Lecture 7

H I Regions I:

Observational probes

H I versus H II regions

T and nH are different in H I regions Different processes play a role Different observational techniques

H II regions Mostly emission lines at optical wavelengths

H I regions H I radio line at 21 cm Optical absorption lines

Ref: Kulkarni & Heiles 1987, in Interstellar Processes, p.87 Burton 1992, in Galactic ISM

7.1 Review of radiation transport Radiative transfer equation

or

with the optical depth

General solution of radiative transport equation

with r = total opacity from s=0 to s=r

d d dI I s j s

d d s

d d dI Ij

FHG

IKJ

I I e ej

r

r

FHG

IKJ

z( )00

d

Source function

Special case: Thermodynamic Equilibrium

In thermodynamic equilibrium, Kirchhoff’s Law applies:

I=Bν(T) is the Planck function

The radiative transfer equation then becomes

j T T B T ( ) ( ) ( )

B Th

c eh kT

( )

/

2 1

1

3

2

I I e B T e

( ) ( ) ( )( ) 0 1

General ISM case (non-TE)

Suppose line has Gaussian profile

Vone-dimensional velocity dispersion

V2=kT/M for thermal velocities

( )

( ) /

/

/ FHG

IKJ

F

HGIKJ1

2 1 2

1

2

2

ce

o

co o

V

V

( )d z 1

Absorption and emission coefficients

jn A

hu ul

4

( )

FHG

IKJ

LNM

OQP

LNM

OQP

LNM

OQP

LNM

OQP

n n

nn

n

g

g

ne

mcf

n

n

g

g

ng

g

A n

n

g

g

l lu u ul

lu

l

l

ulu

l luu

l

l

u

lu

l

ul u

l

l

u

1

1

81

2

2

2

( )

( )

Excitation temperature Tex

with excitation temperature

j n

n

g

g

h

nn

gg

h

c nn

gg

h

c e

u

l

l

u u

l

l

u

l

u

u

l

h kTex

LNM

OQP

LNM

OQP

2 1

1

2 1

1

2 1

1

2

3

2

3

2 /

Th k

nn

gg

ex

u

l

l

u

FHG

IKJ

/

ln

General radiation transport

=>

Rayleigh-Jeans limit: h<<kT

jB Tex

( )

I I e B T eex

( ) ( )( )0 1

BkT

c

kT

2 22

2 2

Antenna temperature The antenna temperature TA is defined by

Advantage TA is a linear function of I For h/kT<<1, B(TA)=I

=> Rayleigh-Jeans limit, which is a good approximation at radio wavelengths Disadvantage

For h/kT1, B(TA)<I

=> e.g. at sub-mm wavelengths TA does not correspond to a physical temperature, even if emission is thermal

TI

kc

I

kA( )

2 22 2

2

7.2 H I 21 cm line emission

H atom consists of 1 proton + 1 electron Electron: spin S=1/2 Proton: nuclear spin I=1/2 Total spin: F = S + I = 0, 1

Hyperfine interaction leads to splitting of ground level: F = 1 gu = 2F+1 = 3 E = 5.8710–6 eV F = 0 gl = 2F+1 = 1 E = 0 eV

H I 21 cm line emission

Transition between F = 0 and F = 1: ν = 1420 MHz, λ = 21.11 cm ΔE / k = 0.0682 K Aul = 2.86910–15 s–1 = 1/(1.1107 yr) (very small!)

flu=5.7510-12

For all practical purposes kTex >> hν

Tex for H I is called “spin temperature” TS n

n

g

ge n n

n n

u

l

u

l

h kTu

l

ex

/ ( )

( )

33

4

1

4

HI

HI

Spin temperature and kinetic temperature

Often excitation is dominated by collisions TS = Tkin (e.g., in cold clouds with n 0.05 cm–3)

In warm, tenuous clouds (T 300 K): TS < Tkin

In some regions: upper level pumped by Lyα radiation TS > Tkin

Optical depth of H I line

Consider uniform cloud of length L and Gaussian line with FWHM Δν = ν/c ΔV (see eq. 7.1, with V=22V )

N(HI) = 4 nl L is the total HI column density V=FWHM in km s-1

o

Sc g

gA

n L ce

n L

T

N

T

o

u

lul

l

o

h kT

l

S

S

2

2

18

19

8

0 939441

2 20 101

5 50 101

.[ ]

.

.( )

/

V

V

HI

V

Example of optical depth H I line

TS = 100 K, ΔV = 3 km/s τ << 1 for N(HI) << 5.5 1020 cm–2

For τ << 1

or N(H I)1.8181018TAV

=> TA proportional to N(H I), independent of TS

In most cases N(H I) << 5.5 1020 cm–2

=> 21 cm emission usually gives information on column density of H I, but not on temperature

T T T e T

N

A S bg S

( )( )

.( )

1

5 5 10 19

H I

V

7.3 H I emission-absorption studies

Study extended cloud in front of extragalactic radio source

Observe two positions (on-source and off-source = blank) Assume that cloud is uniform properties of H I are the same in source and

blank positions

H I emission-absorption studies (cont’d)

Measure on-line and off-line at each position

)1()(

)(

)1()(

)(

eTeTsrcT

TsrcT

eTeTblankT

TblankT

Ssrcon

srcoff

Sbgon

bgoff

ebgsrc

bgsrc

offoff

ononTT

eTT

blankTsrcTblankTsrcT

)()()()(

SbgbgSblankoffe

blankTblankTTTTTToffon

)(

1

)()(

H I emission-absorption studies (cont’d)

Both τ and TS can be measured both N(H I) and TS can be determined!

Holds only over small regions need small beam size (3 for Arecibo 330 m, 30 for VLA and Westerbork interferometers)

Recall

τ very small if TS large

warm H I regions cannot be measured in absorption

5 5 10119.

( )N

V TS

HI

Two clouds along the line of sight: apparent TS

Assume Tbg<<TS =>definition of “naïvely-derived” spin temperature at velocity V

If cloud homogeneous, TS = TN

Often, there are two H I clouds along the line of sight with overlapping V. Assume cloud 1 closer than cloud 2

=> TN depends on V and lies between T1 and T2

T T e TS A N( ) ( ) / [ ] ( )V V V 1

T VT e T e e

eNS S( )

[ ] [ ]

[ ], ,

1 21 1

1

1 2 1

1 2

Consider 3 examples

Optically thick foreground: >>1 => TN(V)=TS,1

no information on cloud 2 Optically thick background, thin foreground:

TN(V)=TS,11(V)+TS,2

=> TN is larger than TS,2

Both clouds optically thin (usual case):

TN is weighted harmonic mean of T1 and T2

TN N

N T N TNS S

( )( ) ( )

( ) / ( ) /, ,

VV V

V V

1 2

1 1 2 2

Effect of foreground cloud on observed TS

Example: T1 = 8,000 K, T2 = 80 K, N2/N1 = 0.1

Small cold cloud can reduce “naïvely derived” spin temperature of warm background cloud from 8,000 K to 800 K

TN

11

1 8000 0 1 80800

.

/ . /K

7.4 Evidence for two-phase ISM

Good agreement between narrow absorption lines and narrow peak emission lines: V3 km/s

There is emission outside region over which absorption occurs (dashed lines): V9 km/s

H I emission and absorption spectrum

Note that the absorption features are sharper than the corresponding emission spectrum

=> Observations indicate that H I consists of 2 components

1. Cold Neutral Medium Cold diffuse clouds with T 80 K => narrow

absorption + emission components Every velocity component corresponds to an

individual cloud CNM occurs in clumps throughout the disk of

the Milky Way with z 100 pc Typically in disk: N(HI)full thickness 61020 cm-2

Locally: N(HI) 41020 cm-2 => We live in a “H I hole”

2. Warm Neutral Medium Broad emission component => temperature

difficult to estimate V9 km s-1 => T<10000 K Limits on => T>3000 K

WNM is distributed throughout Milky Way with substantial filling factor “raisin-pudding” model of ISM

Large scale height (Gaussian z 250 pc or exponential z 500 pc >> z of CNM) => warm H I halo?

=> T8000 K

T – relation?

Clouds with higher optical depth tend to have lower temperatures

‘Luke-warm’ H I in Milky Way?

In general, absorption lines narrower than corresponding emission features => do cold clouds have a warm (T500 K) envelope? T lower if larger

Maps indicate Clumps are responsible for H I absorption

T30-80 K, n20-50 cm-3

Filaments/sheets have T500 K and are responsible for 80% of the H I emission not seen in absorption

7.5 Optical absorption lines:Voigt profiles and equivalent widths

Traditional way of studying H I clouds: mostly Na I and Ca II lines

Optical lines => ΔE >> kT for T 80 K neglect (stimulated) emission

where =Nl with Nl=column density in level l

dI

dI I I e

( )0

Line broadening mechanisms

Upper level has finite radiative lifetime Lorentzian profile with damping width αL

Thermal and random / turbulent broadening Gaussian profile with HWHM αD

L o

L

L oL ul

l

A( )/

( )

2 2

1

4 with Hz

D o

D

o

D

( ) exp ln F

HGIKJ

LNMM

OQPP

12

2

Do D

o D

c

kT

M

T M

LNM

OQP

22

3 5825 10

1 2

7 1 2

ln

. ( / )

/

/amu Hz

Gaussian vs. Lorentzian profiles

Measures of Doppler width

αD is HWHM by definition; units are Hz

FWHM in velocity units is Frequently the width of the Gaussian is

given in terms of the “Doppler parameter”

V 2c

oD

b V V

2 2 1 6651 2(ln ) ./

( ) . maxv v Vd z 1 065

Voigt profile

Convolution of Lorentzian and Gaussian profiles

Define

Voigt profile: with

Here

x yo

D

L

D

(ln ) ; (ln )/ /2 21 2 1 2

N K x yl o ( , )

K x yy t

y x tt K x y x( , )

exp( )

( ); ( , ) /

z z

2

2 21 2d d

e

m cf K x y

e

m c

f f

elu o

oe

lu

D

lu

D

2

2 1 220 012466

FH IK

zz( ) ( , )

ln.

/

d d

cm 2

Equivalent width of spectral lines

In practice, resolution at optical wavelengths often insufficient to resolve line measure only line strength or equivalent width

Definition of equivalent width of line:

Wν is the width of a rectangular profile from 0 to Iν(0) that has the same area as actual line

Wν measures line strength, but units are Hz In wavelength units

W eI I

I

L

NMOQP

z z10

0c hd d Hz

( )

( )

W Wc

W

d

d cm

2

Schematic drawing of equivalent width of line

Curve of growth analysis

Goal: relate equivalent width Wν or W to column density Nl

Relation is monotonic, but non-linear Classical theory developed in context of stellar

atmospheres, but equally applicable to ISM Three regimes, depending on τ at line center: τ0 << 1, linear regime

τ0 large, flat regime

τ0 very large, square-root (damping) regime

Curve of growth (schematic)

D

D

L

L

Universal curve of growth

Curve of growth: linear regime

Weak lines, τ0 << 1

Linear regime: WNl

If Wλ and λ in Å,

W Ne

m cN fl

el lu

zzd d( )2

NW

fllu

113 10202

2.

cm

Curve of growth: flat regime

Large τ0: all background light near line center o is absorbed, line is “saturated”

Far from line center there is partial absorption because σ is smaller

=> Wν grows very slowly with Nl: flat part of the curve of growth

Onset if deviation from linear >10%, depends on Doppler parameter:

N f bl lu / . 187 1014 s 1

Curve of growth: square root regime

Very large τ0: Lorentzian wings of profile dominate the absorption

Asymptotic form

Square-root or damping regime: WNl1/2

x l oN

yx

1 22

/

Examples of interstellar Na absorption lines

Linear and flat regimes

Flat regime

Square-root regime

Hobbs 1969

UV absorption lines in ISM towards ζ Oph

7.6 Optical absorption line observations

Technique limited to bright background sources Mostly local (< 1 kpc), mostly AV < 1 corresponding

to N(H) < 51020 cm–2 Strong Na I lines in every direction, same clouds as

seen in H I emission and absorption, also seen in IRAS 100 μm cirrus CNM

H I column densities from Lyα observations in UV at 1215 Å

Information about T, nH from excitation C II, C I lines (see later): T80 K, nH100 cm-3

Depletions Absorption line studies of various atoms

abundances w.r.t. H information on depletions In diffuse clouds many abundances are much

smaller than solar depletion onto grains log D = log abundance meas – log abundance cosmic

Ca: log D –4 10,000 times less than solar Plot log D as a function of condensation

temperature Tc strong correlation elements with large Tc condensed onto grains when formed in circumstellar envelopes

Depletions log D versus condensation temperature

Jenkins 1987, in Interstellar Processes