Lecture 3: Interstellar Dust, Radiative Transfer and Thermal Radiation

Transcription

1 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: m λ and M λ, respectively m λ = 2.5log (r + m λo M λ = 2.5log (10pc + m λo r m λ = M λ + 5log( 10pc (r: flux at λ, i.e. the energy received in a unit wavelength interval per unit area per time. If dust is present along the line of sight: r m λ = M λ + 5log( 10pc + A λ Extinction at λ Consider two different wavelengths, λ 1 and λ 2 : (m λ1 m λ2 = (M λ1 M λ2 + (A λ1 A λ2 Observed color index, C 12 Intrinsic color index, C 0 12 Color excess, E 12 = C 12 -C 0 12

2 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 λ 3, the ratios A λ3 /E 12 and E 32 /E 12 depend only on intrinsic grain properties. Let the third wavelength to have an arbitrary value: E λ V E B V = A λ E B V R = 3.1 A V E B V = A λ E B V R (in the diffuse interstellar medium; but can be ~5 in dense clouds because grains are larger 2175 Å spike 9.7 µm Rapid rise in the UV Transfer of Radiation: Intensity, Flux, Energy Density Specific intensity, : ΔE = ΔA Δt Δν ΔΩ Specific flux F ν, 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 : F ν µdω Total energy density u ν, per unit frequency at a fixed location (J m -3 Hz -1 or erg cm -3 Hz -1 : u ν 1 c dω Mean intensity J ν, i.e. the average of over all directions: J ν 1 4π dω = c 4π u ν

3 Assume that the radiation field travels along a small distance. 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: Δ1 = ρκ ν κ ν is the opacity (cm 2 g -1, a quantity that depends on the incident frequency ν, the relative number of grains and their intrinsic physical properties. 1/ρκ ν photon mean free path ρκ ν absorption coefficient (cm -1 The optical depth is defined by: Δτ ν = ρκ ν Δ1 = ρκ ν can also increase due to the thermal emission from the dust: Δ 2 = + j ν j ν is the emissivity (W m -3 sr -1 Hz -1 (or erg s -1 cm -3 sr -1 Hz -1, such that j ν ΔνΔΩ is the energy per unit volume per unit time emitted into the direction n.

4 The equation of transfer Δ = Δ1 + Δ 2 = ρκ ν + j ν d ds = ρκ ν + j ν d ds = ρκ I + j ν ν ν Δτ ν = ρκ ν d dτ ν = + S ν Source Function j ν S ν = ρκ ν

5 Optical depth (τ λ and extinction (A λ di λ dτ λ = I λ + j 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 j ν = 0. Integrating the equation along any ray from the stellar surface to P, we obtain: I λ (r = I λ exp( Δτ λ Let s now determine the Flux density (µ 1, and ΔΩ=πR *2 /r 2 : I λ µdω $ (r = πi λ R ' * & % r ( 2 exp( Δτ λ Optical depth (τ λ and extinction (A λ $ (r = πi λ R ' * & % r ( 2 exp( Δτ λ (a Assume now that the same star is located at r 0 from P, with no intervening extinction: Dividing (a by (b and taking the log: $ F * λ (r 0 = πi λ R ' * & % ( $ 2.5log (r = 2.5logF * λ (r 0 + 5log r ' & + 2.5(logeΔτ λ % r 0 ( r 0 2 (b r m λ = M λ + 5log( 10pc + A λ! A λ = 2.5(logeΔτ λ A λ =1.086Δτ λ

6 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: u ν 1 c dω! = cu ν /4π B ν B ν (T = 2hν 3 /c 2 exp(hν /k B T 1 Planck Function B λ (T = $ dν ' B λ = B ν & % dλ ( = $ c ' & % λ B 2 ( ν 2hc 2 /λ 5 exp(hc /λk B T 1

7 Blackbody Radiation Interstellar dust grains ( j ν therm = ρκ ν,abs B ν (T dust Stars are so optically thick at all frequencies that its matter and radiation are very nearly in thermal equilibrium. I λ = B λ (T eff I λ µdω dω = 2πdµ = πb λ (T eff Blackbody Radiation Much of a person s energy is radiated away in the form of infrared light. Wien s displacement law: ν max T = 2.82k B = HzK 1 h λ max T = 0.29cmK

8 Blackbody Radiation = πb λ (T eff B λ (T = 2hc 2 /λ 5 exp(hc /λk B T 1 Integrating over all wavelengths, we obtain the bolometric flux (y hc/λk B T: F bol = 2πk 4 BT 4 c 2 h 3 0 y 3 dy e y 1 π 4 /15 F bol = 2π 5 4 k B 15c 2 h T eff = σ B T eff σ B = ergcm 2 s 1 K 4 Stefan-Boltzmann constant! L bol = 4πR 2 4 * σ B T eff Interstellar Dust: Properties of the Grains 1. Efficiency of extinction d ds = ρκ ν + j ν n d number of dust grains per unit volume (number density σ d geometrical cross section of a typical grain Q ν extinction efficiency factor Δτ ν = ρκ ν A ν =1.086Δτ ν! Q ν A ν /N d The opacity κ ν represent the total extinction cross section per mass of interstellar material. To explore the contribution of each grain: ρκ ν = n d σ d Q ν

9 At far-infrared and millimeter wavelengths, the ISM is generally transparent, so one needs to observe the emission from heated dust clouds to determine A λ and Q λ. From the equation of radiative transfer, ignoring the absorption term and assuming optically thin conditions: I λ = B λ (T d Δτ λ = B λ (T d ΔΩΔτ λ Knowledge of A V and T d can give information on the wavelength dependence of Q λ, but the determination of both A ν and T d is usually problematic. It is conventional to write Q λ λ -β, with β 1-2 between 30µm λ 1mm. β is smaller in the densest 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, a min < a < a max a min = 50 A 0, a max = 0.25µm C = ( cm 2.5 Although most of the mass is in 0.1µm grains, the surface area is mainly in small particles. Useful parameter: Σ d, the total geometric cross section of grains per hydrogen atom: for graphite (silicate Σ d n dσ d n H Mathis, Rumpl & Nordsieck 1977 Weingartner & Draine 2001 PAHs

10 3. The dust-to-gas mass ratio Σ d n dσ d n H = N dσ d N H N d n d L; N H n H L ρκ λ = n d σ d Q λ! Δτ λ = N d σ d Q λ Σ d = Δτ λ Q λ N H = A λ 1.086Q λ N H When λ V, Q V 1 (van der Hulst 1981; Tielens 2005: Σ d = A V A = V / E B V = 2.9 E B V N H N H / E B V N H From Bohlin, Savage & Drake (1978, E B-V /N H = 1.7x10-22 mag cm 2. Σ d cm 2 The mass fraction of the interstellar medium contained in grains, f d : f d = m d n d m gas n gas = 4ρ d a d Σ d 3µm H 0.01 f d Z, the metallicity of the gas! a large portion of heavy elements must be locked up in solid form!

11 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: λ max = 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