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Lecture 2: How do we learn about the ISM? Dr Graham M. Harper

Room 3.03a SNIAM

Office Hours: TBD

PY4A04 Senior Sophister

Physics of the ISM and IGM

- the stuff between the stars and galaxies

Different types of spectra

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3

Fraunhofer lines (absorption spectra)

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Emission spectrum

Herschel PACS spectrum of HH46 on Spitzer IR image

Doppler-shifted jet emission from a young sun

How do we glean information about the ISM?

Observable Electromagnetic (EM) radiation

radiative transfer

images and spectra

Need to understand how EM radiation

is created, e.g., particle collisions, thermal blackbody, non-thermal

(synchrotron)

changes as it propagates through the source region

e.g., scattering in a dusty nebulae

changes as it propagates through the ISM

e.g., pulsar signal dispersions, Faraday rotation

changes as it propagates through the Earth’s atmosphere

e.g., telluric features – ozone, cloud, mosquitos

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Atoms and ions - electronic transitions

Schematic energy level diagram

Discrete bound-bound energy levels

Continuum of unbound energy levels

Transitions

Bound-bound: spectral line

Free-bound: broad spectral features

Free-free: continuum features

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Going back into the ISM?

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Herschel SPIRE

M supergiant

Is this where the

molecules in the ISM

come from?

To completely describe macroscopically the radiation field at point r, at a

time t, travelling in direction, n, we define the

Specific Intensity Iν(n,r,t) so that the amount of energy which during a

time interval, dt, passes through area, cosϴ dσ, into a solid angle dω,

and whose frequency lies within dν about ν is

E=Iν dν dω (cosϴ dσ) dt

Equation of Radiation Transfer The Transport Equation for Radiation

dσ dω

I(n,r,t)

k

ϴ

Iν is Energy per Everything! Iν=E /(dν dω cosϴ dσ dt)

Iν is the fundamental description of unpolarized

radiation

From Iν other important quantities are derived

Equivalent to Surface Brightness (in some forms)

Fν = E /(dν cosϴ dσ dt) (flux density, aka flux)

F = E /(cosϴ dσ dt) (flux, aka integrated flux)

Specific Intensity [wb: flux]

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Constancy of Specific Intensity

Energy is conserved in absence of sinks and sources

E1=Iν1 dω1 cosϴ1dσ1 dν dt and E2=Iν2 dω2 cosϴ2dσ2 dν dt

For given: dt, dν and conservation of energy: E1 = E2

therefore Iν1 = Iν2

dσ1

dσ2

r

Iν

ϴ1

ϴ2

E1

E2

2

221

cos

r

dd

2

112

cos

r

dd

k1

k2

Iν

Vacuum: no emission and no absorption

Iν

Contribution of Energy added/removed by element: dE = 0

Note that volume element is dV=ds dσ

ds

dσ

0ds

dI

The change of Iν along a pencil of beams is described by the flow of

energy through the end surfaces of a cylinder of length ds

Iν +dIν Iν

Contribution added by element: dE = jν dν dω dσ ds dt

Sources: thermal creation, scattering into sightline, fluorescence

Emission – no absorption

ds

dσ

Emission coefficient: jν

j

ds

dI

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Iν - dIν Iν

Energy added by element: dE =-κν Iν dν dω dσ ds dt

Sources: destruction, fluorescence, scattering out of sightline

Dimensions of κν are cm-1 and 1/ κν is a measure of the photon

mean-free path

Absorption (with no emission)

ds

dσ

Absorption coefficient: κν

vIds

dI

scaabs

jI

ds

dI

The equation of radiative transfer is then

dsd

Now define the optical depth backward along the ray is

jI

d

dI

τ=0 at the observer, and if κν is +ive τ increases towards the source.

and

Combining terms

Finally defining the Source Function:

jS

SI

d

dI

Blackbody: Planck Function

Raleigh-Jeans Tail

Exponential tail

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122

3

kThec

hjBS

In thermal equilibrium Kirchoff’s Law

where Bν is the Planck Function (below)

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The Formal Solution

)()( xgyxfy

exp)(exp dxxf

Linear 1st-order differential equation of form

Integrating factor

)()(

)()(

1)(

Ixy

Sxg

xf

x

SI

d

dI

dtttSeII

2

1

12

121 exp

44

1dIJ vMean Intensity =

Some simple solutions [wb: ex]

Generally need perform the formal solution numerically

Good simple robust techniques are available

When Sν=constant (sometimes a good assumption)

If there is no emission, Sν=0, or Sν is very small (typical in the

ISM) then

eSeII bck 1

eII bck

Opacity and stimulated emission

Opacity: κν=nσν(1-fstim) where fstim is the correction for

stimulated emission. n=density of particles (cm-3) and σν cross

section (cm+2) .

Stimulated emission – can be regarded as “negative absorption”

For lines κν=nσtot φν (1-fstim) where φν is the normalized line

profile

0

1 d

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Lorentz profile (absorption & emission)

Kramers-Heisenberg formula for elastic scattering

This gives essentially the same result near line centre as the

classical damped oscillator (derivation is in text):

Lorentzian

22

0

2

4

4

ui li

liuiul AA

2

1111

422

0

2

44 ....

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ii

isi

i

iis DDDD

ch

e

d

d

Line profiles (absorption & emission)

Each atom with Lorentz profile is moving with a Maxwellian velocity

distribution so the total profile is the convolution of Lorentz and Doppler

shifts resulting from a Gaussian velocity distribution

Convolution

D

2

D

2

v

dvvvexp

1dvvf

-14

D skm )10(85.122v ATmkT

vvv dfcLorentz

Most probable

speed (velocity)

Voigt Profile

Combined = Voigt function (convolution of the two)

normally computed with fast numerical algorithm

For lines of low optical depth can use the Gaussian Doppler core

profile.

In terms of line centre optical depth, we can write

Line shape -> observed line profile and broadening parameter

dy

ay

yaV

22

2exp,

2

0expD

0expexp

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22

I

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I

I

I

I

C II

Observables

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I

v = projected velocity

C II

Observables

v “center”

b = line width

v = projected velocity

C II

Observables

v “center”

b “width”

b = line width

v = projected velocity

N = column density**

C II

Observables

v “center”

b “width”

N “depth”

**and inherent quantum mechanical properties (e.g., cross-section - oscillator strength).

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I

I

I

1 ion, 1 sightline Velocity, Column Density

Redfield & Linsky (2004a)

Observational Diagnostics

b = line width

v = projected velocity

N = column density

C II (60%)

Observables

v “center”

b “width”

N “depth”

C III (30%)

C I (10%)

Physical Properties

ionization

Other ions same element

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b = line width

v = projected velocity

N = column density

C II (m = 12.0)

Observables

v “center”

b “width”

N “depth”

Physical Properties

ionization

temperature

turbulence

Fe II (m = 55.8)

D I (m = 2.0)

Other elements

Redfield & Linsky (2004)

Redfield & Linsky (2004b)

T and Structure Temperature and turbulence

Annual Reviews

Solar Abundances: a local reference for cosmic values

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b = line width

v = projected velocity

N = column density

C II

Observables

v “center”

b “width”

N “depth”

Physical Properties

ionization

temperature dust

turbulence

N I, O I, Mg II, Al III, Si II, Fe II, … Other elements

missing material

Wood, Redfield, et al. (2003)

• Compare abundances observed in the gas with the cosmic standard (solar abundance)

• Typically there are deficiencies in the abundances and it is assumed that these ions are trapped on dust grains

Dust Composition

1 ion, 1 sightline Velocity, Column Density

multiple ions,

1 sightline

multiple ions,

multiple sightlines

Temperature, Turbulence, Volume Density,

Abundances, Depletion onto Dust Grains,

Ionization Fraction

Global Morphology, Global Kinematics,

Intercloud Variation

Observational Diagnostics

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