4. Interstellar Dust Flashcards
What happens to light from a star when it reaches a dust cloud?
Blue - reflected
OR absorbed and reradiated at longer (redder) wavelengths
Red - transmitted more efficiently
What is reddening?
In absorption when red light is transmitted more efficiently than blue
Which type of light is less likely to be scattered: red or blue?
Red
In an image of B68 at visible (red) wavelengths, what do we see?
Black, whereas at IR can see emission
How would you create a temperature map of a dense core?
Treat grains as black bodies, and having multiple wavelengths of emission, can reconstruct the Planck fn
What is Flux F?
The amount of energy emitted from an object’s surface per unit area per unit time
What is flux measured in (metric and cgs)?
W/m^2 or erg cm^-2 s^-1
Which rule applies for a black body regarding flux?
F = σT^4
Where σ constant
In a telescope, what is measured at the aperture: flux or intensity?
Flux
How is luminosity L obtained?
Flux * surface area of the emitting surface
Equation for luminosity?
L* = 4π R*^2 σ T^4
Units of Luminosity L*?
W (metric)
erg s^-1 (cgs)
What are the ways of measuring brightness of a star at wavelength λ?
Apparent or absolute magnitude
Symbol for apparent magnitude?
m_λ
Symbol for absolute magnitude?
M_λ
In the absolute and apparent magnitude equations, what do Fλ(d), d and mλ0 represent?
Fλ(d) = flux at wavelength λ
d = distance d in pc
mλ0 = magnitude at some reference wavelength
Derive the relation between apparent and absolute magnitude
See notes
If there is dust along the line of sight, what is the equation for apparent magnitude?
mλ = Mλ + 5log(d/10pc) + Aλ
What is A_λ?
Extinction at wavelength λ
Is extinction wavelength-dependent?
Yes
(longer λ light, more likely to pass through cloud)
How do we measure apparent magnitude of an object at two different wavelengths?
Observed colour index
What is observed colour index?
The difference in brightness of an object at two different wavelengths (that we see)
What is intrinsic colour index?
The difference in brightness of an object at two different wavelengths (not affected by scattering)
What is colour excess?
Difference in extinction of an object at two different wavelengths
What is the apparent magnitude of an object when measured at two different wavelengths?
(mλ1 - mλ2) = (Mλ1 - Mλ2) + (Aλ1 - Aλ2)
[Observed colour index = Intrinsic colour index + Colour excess]
What are extinction and colour excess proportional to?
The column density of dust grains along the line of sight
What is Rv in extinction?
Extinction in the visible Av / Colour Excess in B-V
i.e. the ratio of total to selective extinction
What is the value of Rv in the Milky Way / diffuse interstellar medium?
3.1
What do Aλ3/E12 and E32/E12 depend on?
Only on intrinsic grain properties
How would you use colour excess at 3 different λs?
Infer extinction another wavelength
Why is Vega an important star?
It’s our reference star for our magnitude system
Calculate the extinction of Vega in the visible when colour excess = 1
See notes
E_(B-V) =1
ANS: Av = 3.1
Calculate the extinction of Vega in the blue wavelengths when colour excess = 1
See notes
ANS: A_B = 4.1 (magnitudes of extinction)
What are the axes for the interstellar extinction curve?
x = inverse wavelength (wavelength decreases as x increases)
y = Aλ/Av ratio of extinction at this λ over in the visible
What wavelength range is 100 nm?
~ UV
What wavelength range is 500 nm?
Visible
Describe interstellar extinction curve
(Log scale)
Decreasing fro visible to UV:
Approx linear increase of extinction through visible wavelengths
Bump at far UV
Towards shorter wavelengths, steep extinction
(Shorter λ of light, more affected by extinction)
How can interstellar extinction curve be changed?
Changing Rv i.e., changing the properties of the dust grains
What happens to extinction as Rv increases?
The higher Rv is above 1, the more effective the dust grains are at scattering and absorbing radiation at short λs vs visible (less extinction)
How can radiation be transferred?
Absorbed (transformed into internal motion of the grain lattice) and
Scattered (a photon with the same energy is reemitted in a different direction)
What is the equation for change in intensity due to absorption and scattering?
ΔIv1 = -ρ * κv * Iv * Δs
where
ρ = gas mass density
κv = opacity (dependent on freq)
ρκv = absorption coefficient (sometimes Xv)
1/ρκv = photon mfp
Equation for optical depth?
Δτv = -ρ * κv * Δs
Which change in intensity equation does optical depth appear?
Due to absorption and scattering
What is the equation for change in intensity due to thermal emission from the dust?
ΔIv2 = + jv * Δs
jv = emissivity
How does emissivity link to energy per unit volume emitted in a certain direction?
E = jv * Δv * ΔQ
Derive dIv/dτv = Iv - Sv
See notes
What is the equation for the source function Sv?
Sv = jv / ρκv
What does the source function become in the case of local thermodynamic equilibrium?
Sv = Bv
If LTE applies, what does emissivity equal?
jv = ρ * κv * Bv(T_dust)
Derive the formal solution of the radiative transfer equation
See notes
What are the 3 conditions to solve the radiative transfer equation?
Absorption only
Emission only
Optically thin emission
How is the solution to the radiative transfer equation calculated for absorption only?
Zero emission: jv = 0, Sv = 0, τv ≠ 0
Derive the solutions to the radiative transfer equation for absorption / emission only, and optically thin emission
See notes
How is the solution to the radiative transfer equation calculated for emission only?
τv = 0, jv ≠ 0, Sv ≠ 0
How is the solution to the radiative transfer equation calculated for optically thin emission?
τv goes to 0, jv ≠ 0, Sv ≠ 0
In what case could you
simplify the radiative transfer equation using emission only conditions?
If you have no background source i.e., Iv(0) = 0
so just add up light emitted from gas and dust
How much light can pass through an optically thin object?
Most light passes through
Derive the relationship between extinction and optical depth
See notes
What is the relationship between extinction and optical depth?
Aλ = 1.086Δτλ
Why is radiation at short wavelengths more strongly scattered and absorbed than that at longer wavelengths?
Extinction
What must be assumed for Luminosity L = 4πR^2σT^4?
Stars radiate as a black body
What scale is apparent / absolute magnitude measured on?
Log
How are apparent and absolute magnitude related?
mλ = Mλ + 5log(d/10pc) + Aλ
What does the equation of radiative transfer describe?
The change in intensity of radiation along a path due to absorption, scattering and emission
What is the equation of radiative transfer?
dIv / dτv = Iv - jv/Xv
where
jv = emissivity
Xv = ρκv (absorption coefficient)
For the case of zero emissivity and scattering, what is the formal solution of the radiative transfer equation?
Iv(r) = Iv(R*)exp(-Δτv)
where
Δτv = ρκvΔs (optical depth)f
What is the opacity κv?
Total extinction cross section per mass of interstellar material
Absorption coefficient ρκv equation?
ρκv = nd * σd * Qv
nd = no. density of dust grains
σd = πa_d^2 = cross-sectional area of a typical dust grain
Qv = extinction efficiency factor = Q_v, abs + Q_v, sca
When does Mie theory apply?
Spherical particles
What are the two limiting factors in Mie theory?
λ»_space; a_d then Qλ goes to 0
λ «_space;a_d then Qλ goes to 2
What is Mie theory?
A complete analytical solution of Maxwell’s equations for the scattering of EM radiation by spherical particles
According to Mie theory, what is Qλ proportional to?
Qλ prop. to a_d / λ
Which wavelength range does Mie theory work well?
Between IR and visible
Where does Mie theory deviate from reality?
UV
Show that extinction efficiency factor Qv ∝ Av/Nd
See notes
In Qv ∝ Av/Nd, what is Nd?
Column density of the dust grains (cm^-2)
[Nd = nd * Δs]
What is Qλ / Qλ0 equal to?
Qλ / Qλ0 = Aλ / Aλ0
How does extinction efficiency change with wavelength?
Long λ (FIR & mm) - ISM transparent, no absorption
Need to observe emission from heated dust clouds
How does Qλ vary with wavelength in the optical regime?
Qλ ∝ λ^-1
How does Qλ vary with wavelength at longer λs?
Qλ ∝ λ^-B
where B = 1-2 for 30µm<λ<1mm
What happens to efficiency and opacity when grain size is larger than λ?
No longer depend on λ
Show how extinction efficiency varies with λ^-B at longer wavelengths
See notes
Which mechanisms can lead dust to polarise light?
Dichroic extinction
Scattering
Thermal emission
What is the correlation between polarisation and extinction?
% extinction ∝ Aλ
Why do dust grains polarise light?
They are elongated and aligned (due to a magnetic field)
What does it mean, that grains are elongated and aligned?
They are paramagnetic and spinning in a magnetic field
How are grains polarised through dichroic extinction?
Grains have a small electric charge and are paramagnetic
So acquire magnetic moment M that points along the axis of rotation
Interaction with ambient M field creates a torque MxB
Torque gradually forces grain’s short axis to align with field
Why do grains undergo dichroic extinction?
Grains are irregular structures that tend to rotate about their shortest axis
How does dichroic extinction allow the grains to see a time-independent magnetic field as they spin?
The torque on the grains forces the short axes to align with the field
For dichroic extinction, in which direction do the grains line up?
So that their time-averaged projected lengths are longer in the direction perpendicular to B
In dichroic extinction, in which direction is the electric field most effective in driving charge? What implication does this have on absorption?
Down the grain’s long axis
This direction is the one of max. absorption of incoming radiation (highest extinction)
In dichroic extinction, which light is let through?
Light which is parallel to the magnetic field
In dichroic extinction, which light is strongly extincted?
Perpendicular to the direction of the magnetic field
(Long grains block the light in this direction)
How does polarised light link to star forming regions
Star forming regions have higher polarisation as this is where dust is concentrated
Why do further away stars appear more polarised?
Have to travel through more material to reach us so are more extinct
What happens if a dust grain scatters light forwards?
No polarisation
For radiation scattered 90º from the normal, what polarisation occurs?
Linearly polarised: Scattered field E only oscillates along the line that is the projections of the new plane
For radiation scattered in non-90º directions, what polarisation occurs?
Partial polarisation: E oscillates along two orthogonal lines but with unequal amplitude
How does polarised thermal emission occur?
Grains are elongated and aligned to m field
Light is absorbed by dust grains
Sub-mm domain, dominant component of radiation is along main axis of the grain
So IR / sub-mm radiation is polarised orthogonal to optical polarisation and perp to B
How does polarisation in optical vs sub-mm differ?
Polarisation in optical is orthogonal to grain and parallel to B (transmitted)
Polarisation in sub-mm / IR is polarised perpendicular to B and in the direction of the main axis of the grain
What is self-scattering?
When large amounts of dust are emitting at long wavelengths, light from dust grains is scattered by other dust grains in the vicinity
Why might polarisation vectors change direction?
Self scattering
What is the Zeeman effect?
Magnetic field exerts force on atom
Atoms have magnetic moments proportional to their total angular momentum J
Since J is quantised, so are associated energy levels
Results in line splitting
In the Zeeman effect, what is the line-splitting proportional to?
The magnitude of line splitting is proportional to the magnetic field
What can give us information on magnetic field?
Dust (polarisation)
and Gas emission lines (Zeeman effect)
What is flux-freezing?
Magnetic field strength increases with density, so an indication that field lines can be compressed along with the gas
[not a general rule]
What theory describes the scattering behaviour of spherical particles?
Mie theory
At sub-mm wavelengths, which direction do grains emit radiation? Why effect does this have on polarisation?
Emit radiation along longer grain axis,
Thus radiation polarised orthogonal to direction of magnetic field
