Download Lecture 3: Interstellar Dust, Radiative Transfer and Thermal Radiation

Lecture 3: Interstellar Dust,
Radiative Transfer and
Thermal Radiation
Interstellar Dust: Extinction and Thermal emission
The brightness of a star near any wavelength λ is measured either by its apparent and
absolute magnitudes: mλ and Mλ, respectively
mλ = −2.5log Fλ (r) + mλo
M λ = −2.5log Fλ (10 pc) + mλo
r
mλ = M λ + 5log(
)
10 pc
Fλ(r): flux at λ, i.e. the energy received in a unit
wavelength interval per unit area per time.
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If dust is present along the line of sight:
r
mλ = M λ + 5log(
) + Aλ
10 pc
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Extinction at λ
Consider two different wavelengths, λ1 and λ2:
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(mλ1 − mλ 2 ) = (M λ1 − M λ 2 ) + (Aλ1 − Aλ 2 )
Observed color
index, C12
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Intrinsic color
index, C012
Color excess,
E12 = C12-C012
The Interstellar Extinction Curve
Both extinction and the color excess are proportional to the column density of dust grains
along the l.o.s. If we consider another wavelength λ3, the ratios Aλ3/E12 and E32/E12 depend
only on intrinsic grain properties. Let the third wavelength to have an arbitrary value:
E λ−V
A
A
A
= λ − V = λ −R
E B −V E B −V E B −V E B −V
R = 3.1 (in the diffuse interstellar medium; but can be ~5 in
dense clouds because grains are larger)
2175 Å spike
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9.7 µm
Rapid rise
in the UV
Transfer of Radiation: Intensity, Flux, Energy Density
•Specific intensity, Iν: ΔE = Iν ΔA Δt Δν ΔΩ
•Specific flux Fν, or flux density, the monochromatic
energy per unit area per unit time passing through a
surface of fixed orientation z (W m-2 Hz-1 or erg s-1
cm-2 Hz-1 or Jansky=10-26 W m-2 Hz-1):
Fν ≡
∫ I µdΩ
ν
•Total energy density uν, per unit frequency at a
fixed location (J m-3 Hz-1 or erg cm-3 Hz-1):
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uν ≡
1
c
∫ I dΩ
ν
•Mean intensity Jν, i.e. the average of Iν over all directions:
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Jν ≡
1
4π
c
∫ I dΩ = 4π u
ν
ν
Assume that the radiation field travels along a small distance Δs. Iν can be absorbed
(radiative energy is transformed to internal motion of the grain lattice) and scattered (a
photon with the same frequency is reemitted in a different direction):
ΔIν 1 = − ρκ ν Iν Δs
κν is the opacity (cm2 g-1), a quantity that depends on the incident frequency ν, the
relative number of grains and their intrinsic physical properties.
1/ρκν≡ photon mean free path
ρκν≡ absorption coefficient (cm-1)
The optical depth is defined by:
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Δτ ν = ρκ ν Δs
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ΔIν 1 = − ρκ ν Iν Δs
Iν can also increase due to the thermal emission from the dust:
ΔIν 2 = + jν Δs
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jν is the emissivity (W m-3 sr-1 Hz-1) (or erg s-1 cm-3 sr-1 Hz-1), such that jνΔνΔΩ is
the energy per unit volume per unit time emitted into the direction n.
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The equation of transfer
ΔIν = ΔIν 1 + ΔIν 2 = − ρκ ν Iν Δs + jν Δs
dIν
= −ρκ ν Iν + jν
ds
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dIν
= −ρκ ν Iν + jν
ds
Δτ ν = ρκ ν Δs
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dIν
= −Iν + Sν
dτ ν
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Source Function
Sν =
jν
ρκ ν
Optical depth (τλ) and extinction (Aλ)
dIλ
j
= −Iλ + λ
dτ λ
ρκ λ
Let’s use the equation of transfer to obtain the specific intensity at a point P located
at a distance r from the center of a star. Assume that jν = 0. Integrating the equation
along any ray from the stellar surface to P, we obtain:
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Iλ (r) = Iλ (R* )exp(−Δτ λ )
Let’s now determine the Flux density (µ≈1, and ΔΩ=πR*2/r2):
Fλ ≡
∫ I µdΩ
λ
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$ R* ' 2
Fλ (r) = πIλ (R* )& ) exp(−Δτ λ )
% r(
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Optical depth (τλ) and extinction (Aλ)
$ R* ' 2
Fλ (r) = πIλ (R* )& ) exp(−Δτ λ )
% r(
(a)
Assume now that the same star is located at r0 from P, with no intervening extinction:
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2
$
'
R
*
Fλ (r0 ) = πIλ (R* )& * )
% r0 (
(b)
Dividing (a) by (b) and taking the log:
$r'
−2.5log Fλ (r) = −2.5log Fλ (r0 ) + 5log& ) + 2.5(loge)Δτ λ
% r0 (
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*
r
mλ = M λ + 5log(
) + Aλ
10 pc
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!
Aλ = 2.5(loge)Δτ λ
Aλ = 1.086Δτ λ
Blackbody Radiation
A black body is a theoretical object that absorbs 100% of the radiation that hits
it. Therefore it reflects no radiation and appears perfectly black.
At a particular temperature the black body would emit the maximum amount of energy possible
for that temperature. This value is known as the black body radiation. It also emits a definite
amount of energy at each wavelength for a particular temperature, so standard black body
radiation curves can be drawn for each temperature, showing the energy radiated at each
wavelength.
Blackbody Radiation
Blackbody radiation is radiation in thermal equilibrium and it is isotropic, i.e. the
specific intensity Iν is independent of direction:
uν ≡
1
c
∫ I dΩ
ν
!
2hν 3 /c 2
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Bν (T) =
€B T) −1
exp(hν /k
$ dν ' $ c '
Bλ = Bν & ) = −& 2 )Bν
% dλ ( % λ (
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2hc 2 / λ5
€Bλ (T) =
exp(hc / λkB T) −1
Iν = cuν /4 π ≡ Bν
Planck Function
Blackbody Radiation
Interstellar dust grains
( jν ) therm = ρκ ν ,absBν (Tdust )
€ that its matter and radiation are
Stars are so optically thick at all frequencies
very nearly in thermal equilibrium.
Iλ (R* ) = Bλ (Teff )
Fλ ≡
∫ I µdΩ
λ
dΩ = 2πdµ
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Fλ (R* ) = πBλ (Teff )
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Blackbody Radiation
Much of a person’s energy is
radiated away in the form of
infrared light.
Wien’s displacement law:
ν max 2.82k B
=
= 5.88 ×1010 Hz K −1
T
h
λmax T = 0.29cm K
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Blackbody Radiation
Fλ (R* ) = πBλ (Teff )
2hc 2 / λ5
Bλ (T) =
exp(hc / λkB T) −1
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Integrating Fλ(R*) over all wavelengths, we obtain the bolometric flux
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(y≡hc/λkBT):
4
Fbol
Fbol
2πkB4 T 4
= 2 3
c h
2π 5 k B4 4
=
T = σ B Teff4
2 3 eff
15c h
∞
∫
0
!
π /15
y 3 dy
e y −1
Lbol = 4 πR*2σ B Teff4
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σ B = 5.67 ×10−5 ergcm−2 s−1 K −4
Stefan-Boltzmann constant
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Interstellar Dust: Properties of the Grains
1. Efficiency of extinction
dIν
= −ρκ ν Iν + jν
ds
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The opacity κν represent the total
extinction cross section per mass
of interstellar material. To explore
the contribution of each grain:
ρκ ν = n d σ d Qν
nd≡ number of dust grains per unit volume (number density)
σd ≡ geometrical cross section of a typical grain
Qν ≡ extinction efficiency factor
Δτ ν = ρκ ν Δs
Aν = 1.086Δτ ν !
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€Qν ∝ Aν /N d
At far-infrared and millimeter wavelengths, the ISM is generally transparent, so
one needs to observe the emission from heated dust clouds to determine Aλ and
Qλ. From the equation of radiative transfer, ignoring the absorption term and
assuming optically thin conditions:
Iλ = Bλ (Td )Δτ λ
Fλ = Bλ (Td )ΔΩΔτ λ
Knowledge of AV and Td can give information on the wavelength dependence of Qλ,
but the determination of both Aν and Td is usually problematic. It is conventional to
write Qλ∝ λ-β, with β ≈ 1-2 between 30µm ≤ λ ≤ 1mm. β is smaller in the densest
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clouds and circumstellar
disks, but closer to 2 in more diffuse environments.
In general, the opacity and efficiency factor of a grain does not depend on λ once
the geometric size becomes larger than λ. Hence, centimeter-size grains have a
small exponent β in the millimeter.
2. Size distribution
dn d = Cn H a−3.5 da, amin < a < amax
0
amin = 50 A, amax = 0.25µm
C = 10−25.13 (10−25.11 ) cm 2.5 for graphite (silicate)
Although most of the mass is in
0.1µm grains, the surface area is
mainly in small particles.
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Useful parameter: Σd, the total
geometric cross section of grains
per hydrogen atom:
Σd ≡
ndσ d
nH
Mathis, Rumpl & Nordsieck 1977
Weingartner & Draine 2001
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PAHs
3. The dust-to-gas mass ratio
Σd ≡
ndσ d N dσ d
=
nH
NH
ρκ λ = n d σ d Qλ
!
N d ≡ n d L; N H ≡ n H L
Δτ λ = N d σ d Qλ
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Δτ λ
Aλ
Σd =
=
Qλ N H 1.086Qλ N H
When λ≡V, QV≈1 (van der Hulst 1981; Tielens 2005):
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Σd =
AV
AV / E B −V
E
=
= 2.9 B −V
1.086 N H 1.086 N H / E B −V
NH
From Bohlin, Savage & Drake (1978), EB-V/NH = 1.7x10-22 mag cm2.
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Σ d ≈ 5 ×10−22 cm 2
The mass fraction of the interstellar medium contained in grains, fd:
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fd =
md n d
4ρ a Σ
= d d d ≅ 0.01
mgasn gas
3µm H
fd ≈ Z, the metallicity of the gas ! a large portion of heavy elements must be
locked €
up in solid form!
Observations at non-visible wavelengths reveal the
shape of the Galaxy
At visible wavelengths (λ), light suffers so much interstellar extinction that
the galactic nucleus is totally obscured from view. But the amount of
interstellar extinction is roughly inversely proportional to λ.
As a result, we can see farther into the plane of the Milky Way at infrared
λ than at visible λ, and radio waves can traverse the Galaxy freely.
Far Infrared emission from dust at 25, 60, and 100 µm (IRAS spacecraft)
Dust lies mostly in the plane of the Galaxy
Stellar emission at 1.2, 2.2 and 3.4 µm (COBE)
Stars lie mostly in the plane of the Galaxy and in the central bulge
Starlight warms the dust grains to T ~ 10 to 90 K. Thus, from Wien’s
Law:
λmax =
0.0029 K m
T
the dust emits predominantly at λ ~ 30 to 300 µm. These are far-infrared
(FIR) λ. At these λ, interstellar dust radiates more strongly than stars, so a
FIR view of
€the sky is principally a view of where the dust is.
Far-Infrared