Lecture 2 - Trinity College Dublin€¦ · Lecture 2: How do we learn about the ISM? Dr Graham M....
Transcript of Lecture 2 - Trinity College Dublin€¦ · Lecture 2: How do we learn about the ISM? Dr Graham M....
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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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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
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cos
r
dd
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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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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
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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
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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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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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