Interstellar Dust

Question 1

Temperature Gradients in Dust Clouds

Answer 1

• there is a clear temperature gradient
• further to the edges of the cloud, dust is more exposed to the radiation field of the interstellar medium so it is warmer

Question 2

Flux of Radiation for a Blackbody

Answer 2

-the amount of energy emitted from an objects surface per unit area per unit time is called the flux, F
-flux is measured in units of Wm^(-2)
-for a blackbody, the Stefan-Boltzmann law applies:
F = σ T^4
-where σ=5.67*10^(-8) is the Stefan-Boltzmann constant and T is the object’s temperature

Question 3

Luminosity

Answer 3

-multiplying flux by the surface area of the emitting surface we obtain luminosity, L
Lstar = 4π(Rstar)² σ T^4

Question 4

Apparent Magnitude

Answer 4

m_λ = -2.5logFλ(d) + m_λ0

-where F_λ(d) is the flux at wavelength λand distance d in units of parsec and m_λ0 is the magnitude at some reference wavelength

Question 5

Absolute Magnitude

Answer 5

M_λ = -2.5logFλ(10pc) + m_λ0

-where F_λ(10pc) is the flux at wavelength λand distance 10pc and m_λ0 is the magnitude at some reference wavelength

Question 6

Relationship Between Apparent and Absolute Magnitude

Answer 6

m_λ = M_λ + 5log(d/10pc)
-a difference of 1 magnitude corresponds to a difference in brightness by a factor of 2.5 i.e. it is measured on a log scale

Question 7

Extinction Along the Line of Sight

Answer 7

-if there is dust present along the line of sight
m_λ = M_λ + 5log(d/10pc) + A_λ
-where A_λ is the extinction at a wavelength λ
-EXTINCTION IS DEPENDENT ON WAVELENGTH

Question 8

Extinction at Two Wavelengths

Answer 8

-consider two different wavelength λ1 and λ2, subtract the extinction along the line of sight equations for each wavelength:
(m_λ1 - m_λ2) = (M_λ1 - M_λ2) + (A_λ1 - A_λ2)
-where:
m_λ1-m_λ2 = C_12, observed colour index
M_λ1-M_λ2 = C^0_12, intrinsic colour index
A_λ1-A_λ2 = C_12-C^0_12 = E_12, colour excess

Question 9

Extinction, Colour Excess and Density of Dust Grains

Answer 9

-extinction and colour excess are proportional to the column density of dust grains along the line of sight

Question 10

Extinction at Three Wavelengths

Answer 10

-consider another wavelength, λ3, the ratios A_λ3/E_12 and E_32/E_12 depend only on intrinsic grain properties
-let the the third wavelength have some arbitrary value:
E_(λ-V)/E_(B-V) = A_λ/E_(B-V) - A_v/E_(B-V)
= A_λ/E_(B-V) - R_v
-in the diffuse interstellar medium, Rv=3.1, this quantity is the ratio of total to selective extinction
-Rv is determined by the properties of the dust grains, not how many grains there are

Question 11

The Interstellar Extinction Curve

Answer 11

-objects become redder when there is more dust along the line of sight

Question 12

Transfer of Radiation

Description

Answer 12

• assume that the radiation field travels along a small distance Δs
• the radiation can be:
• -absorbed, transformed into internal morion of the grain lattice
• -scattered, a photon is absorbed and then some or all of it is reemitted
• radiation can be added to the beam by:
• -thermal emission, grains in the lattice radiate aas blackbodys
• -scattering into the beam from outside sources

Question 13

Transfer of Radiation

Change in Intensity Dues to Absorption and Scattering

Answer 13

ΔIν1 = - ρκνIν*Δs

-where ρis the mass density, κν is the opacity which is dependent on ν, Iν is the beam’s original specific intensity and Δs is the path length

Question 14

Transfer of Radiation

Photon Mean Free Path

Answer 14

1/ρ*κν

Question 15

Transfer of Radiation

Optical Depth Definition

Answer 15

Δτν = ρκνΔs

Question 16

Transfer of Radiation

Transfer of Radiation Due to Thermal Emission

Answer 16

ΔIν2 = + jν*Δs

-where jν is the emissivity such that jνΔνΔΩ is the energy per unit volume per unit time emitted into the direction n_

Question 17

Transfer of Radiation

Total Transfer of Radiation Equation

Answer 17

ΔIν = ΔIν1 + ΔIν2 = - ρκνIνΔs + jνΔs

Question 18

Transfer of Radiation

The Equation of Radiative Transfer

Answer 18

dIν/ds = - ρκνIν + jν
-changing variable to optical depth:
dIν/dτν = - Iν + Sν
-where Sν is the source function

Question 19

Transfer of Radiation

Source Function Definition

Answer 19

-the ratio of efficiency of emission vs absorption along the line of sight
Sν = jν/ρ*κν

Question 20

Transfer of Radiation

Source Function in the Case of Thermal Emission

Answer 20

Sν = Bν
=>
jν = ρκνBν(Tdust)

Question 21

Transfer of Radiation

General Solution to the Radiative Transfer Equation

Answer 21

Iν(r) = Iν(Rstar)e^(τν) + Sνe^(-τν’) dτν’

Question 22

Transfer of Radiation
Solutions to the Radiative Transfer Equation
Case 1: Zero Absorption & Zero Emission

Answer 22

-zero absorption => τν=0
-zero emission => jν=0
=>
Iν(r) = Iν(Rstar)

Question 23

Transfer of Radiation
Solutions to the Radiative Transfer Equation
Case 2: Zero Emission & Non-Zero Absorption

Answer 23

-zero absorption => τν=0
-non-zero emission => jν≠0
=>
Iν(r) = Iν(Rstar)*e^(-τν)

Question 24

Transfer of Radiation
Solutions to the Radiative Transfer Equation
Case 3: Zero Absorption & Non-Zero Emissivity

Answer 24

-zero absorption => jν=0
-non-zero emission => τν≠0
=>
Iν(r) = Iν(Rstar) - ∫ jν ds
-integral from 0 to r

Question 25

Transfer or Radiation
Solutions to the Radiative Transfer Equation
Case 4: The Optically Thin Case

Answer 25

-we see all photons that are emitted but still have the some absorption of the background radiation field:
Iν(r) ≈ Iν(Rstar) + τνSν

Question 26

Transfer of Radiation

Flux at Point P Distance r from Star for a Purely Absorbing Medium

Answer 26

Fν(r) = π * Iν(Rstar) * (Rstar/r)² * exp(-Δτν)

Question 27

Transfer of Radiation

Flux at Point P Distance r from Star With Nothing Along the Line of Sight

Answer 27

-no extinction or absorption

Fν*(r0) = π * Iν(Rstar) * (Rstar/r)²

Question 28

Relationship Between Extinction and Optical Depth

Answer 28

Aλ = 2.5 log(e) Δτν
Aλ = 1.086 Δτν

Question 29

Opacity

Definition

Answer 29

-the opacity, κν, represents the total extinction cross section per mass of interstellar material:
ρκν = ndσd*Qν
-where:
nd = number density of dust grains
σd = cross sectional area of a typical dust grain
Qv = extinction efficiency factor = Qvabs + Qvscat

Question 30

Equation of Radiative Transfer in Terms of Opacity

Answer 30

dIv/ds = -ρκνIv + jv

Question 31

What is Mie theory?

Answer 31

-a complete analytical solution of Maxwell’s equations for the scattering of electromagnetic radiation off of spherical particles
-Mie demonstrated that when the wavelength of the light is of the order of the size (diameter) of the dust grain, then:
Qλ ∝ ad/λ
–if λ»ad, then Qλ->0
–if λ<2, i.e. constant, independent of λ
-Mie theory works well for all wavelengths between infrared and visible
-there are significant deviations at ultraviolet

Question 32

Relationship Between Efficiency of Extinction, Extinction and Column Density

Answer 32

Qv ∝ Av/Nd

-where Nd=nd*Δs is the column density of dust grains

Question 33

Efficiency and Relative Extinctions

Answer 33

Qλ/Qλo = Aλ/Aλo

-where λo is ant reference wavelength

Question 34

Efficiency of Extinction in the Optical Regime

Answer 34

-in the optical regime:

Qλ ∝ 1/λ

Question 35

How does the efficiency change with wavelength?

Answer 35

-at long wavelengths (FIR and millimetre), the ISM is generally transparent (no absorption) so one needs to observe the emission from heated dust clouds to determine Aλ and Qλ
-recall
Qλ ∝ Av ∝ Δτλ
-thus knowledge of Av and Td give information on the wavelength dependence of Qλ

Question 36

Qλ and β

Answer 36

• typically, Qλ ∝ λ^(-β), with β≈1-2 for 30m ≤ λ ≤ 1mm
• in the densest clouds and circumstellar disks β tends towards the lower end of this range: it lies closer to 2 in more diffuse environments
• once the grain size is larger than wavelength, the opacity and efficiency no longer depend on λ

Question 37

List the Mechanisms That Can Lead Dust to Polarise Light

Answer 37

• dichroic extinction
• scattering
• thermal emission

Question 38

Dichroic Extinction

Answer 38

-there is a correlation between polarisation and extinction:
% polarisation ∝ Aλ
-this is because the grains are elongated and aligned and also they are para-magnetic and spinning in a magnetic field
-the grains tend to rotate about their shortest axis, they have a small electric charge and hence acquire magnetic moment M along the axis of rotation
-interaction with the ambient magnetic field then creates a torque MxB which gradually forces the grains short axis to align with the field
-thus grains tend to line up such that their time-averaged projected lengths are longer in the direction perpendicular to B
-the electric field is most effective in driving charge down the grain’s long axis, this direction becomes the one of maximum absorption of the impinging radiation
-the electric vector of the transmitted radiation lies along the ambient B

Question 39

Polarisation of Scattered Light

Answer 39

• prior to scattering, the incident electric field E oscillates randomly within the plane normal to propagation direction n^
• for the radiation scattered in direction 90’ from n^, the scattered field E only oscillates along the line that is the projections of the new plane and the radiation is linearly polarised
• scattering in other directions results in partial polarisation, E oscillates along two orthogonal lines but with unequal amplitude

Question 40

Polarised Thermal Emission

Answer 40

• light emitted by a star behind a dust cloud is absorbed along the main axis of the dust grains
• the polarisation in the optical is then orthogonal to the grain and parallel to B
• in the sub-mm domain, the dominant component of the radiation is along the main axis of the grain and the radiation is thus polarised orthogonal to the optical polarisation and is perpendicular to B

Question 41

The Zeeman Effect

Answer 41

• atoms have magnetic moments which are proportional to their total angular momentum J
• when a B field is present, it exerts a force on the atom and because the angular momentum is quantised, so are the associated energy levels
• the result is line splitting, the magnitude of which is proportional to the magnetic field