Physics of Interstellar Dust

Transcription

1 Physics of Interstellar Dust 1. Optics of Small Particles 2. Inferred Size Distributions for Interstellar Grains 1 B.T. Draine Physics of Interstellar Dust IPMU

2 SRK Lecture in class Review: Importance of C, Si, PAH Mass Absorption Coefficient Dielectric & metals Polarizability (atomic & molecular) Polarization density vector Capacitor with dielectric material Introducing "electrical susceptibilty" and delectric constant Waves in delectric/metal medium Why k is complex? Complex refractive index Example 1: Scattering of X-ray photons by dust particles (Ref: Randal & Dwek 1998, ApJ) Example 2: Rayleigh scattering

3 1.1 What do we seek to calculate? The absorption cross section C abs (λ) at wavelength λ The scattering cross section C sca (λ) The extinction cross section C ext (λ) =C abs + C sca The differential scattering cross section dc sca (θ) dω for incident unpolarized light to be scattered by an angle θ. This is related to the dimensionless Muller matrix element S 11 : The mean value of cos θ for scattered light: The radiation pressure cross section cos θ 1 C sca dc sca (θ) dω π 0 S 11(θ) k 2 cos θ dc sca 2π sin θ dθ dω C pr C abs +(1 cos θ) C sca The degree of polarization P (θ) for incident unpolarized light after scattering through an angle θ. Actually, we would like to calculate the full 4 4 Muller matrix S ij (θ), for scattering of an arbitrary incident Stokes vector. 3 B.T. Draine Physics of Interstellar Dust IPMU

4 1.2 Dielectric Function vs. Refractive Index Response of grain material to applied monochromatic E field is specified by the complex dielectric function (ω) = 1 + i 2. Electrical conductivity σ, if any, can be absorbed within : +4πiσ/ω. Equivalently, can characterize material by complex refractive index m(ω) = Two sign conventions in use: If we choose to write oscillating quantities e ik r iωt, then Im() > 0 and Im(m) > 0 for dissipative materials (propagating waves are attenuated). This is our convention. Alternatively, some choose to use e +iωt. Then Im() < 0 and Im(m) < 0 for dissipative materials. Real materials: 1 1 at X-ray energies 1 O(1) in the optical and UV 1 O(1) for insulators in IR 1 for conductors as λ 4 B.T. Draine Physics of Interstellar Dust IPMU

5 Dielectric Function for Graphite and Astrosilicate from Draine (2003) 5 B.T. Draine Physics of Interstellar Dust IPMU

6 1.3 Electric Dipole Limit: Size λ In this limit, the small grain is subject to an applied E = Re(E 0 e iωt ) that is nearly uniform. Absorption and scattering cross sections can be written C abs = 4πω c Im(α) C sca = 8π ω 4 α 2 3 c where α is the electric polarizability of the grains: electric dipole moment P = αe. Polarizability can be calculated analytically for spheres, spheroids, and ellipsoids: for E principal axis j: α jj = V 1 4π ( 1)L j +1 where L j = shape factor. L 1 + L 2 + L 3 =1. For spheroids, L<1/3for long axis, L>1/3for short axis. Cross sections in electric dipole limit = Rayleigh limit : C abs = V λ f(, shape) f =18π 2 ( 1 +2) for sphere 2 C sca = V 2 g(, shape) g =24π3 1 2 λ4 +2 for sphere Important result: C abs V when particles are small ( λ) power absorbed (or emitted) depends on total mass of dust, but not on particle size. 6 B.T. Draine Physics of Interstellar Dust IPMU

7 Limiting Behavior at Long Wavelengths Insulators: At frequencies ω the lowest frequency resonance in the solid, insulators tend to have 1 0 = const. 2 Aω (A = const) Then Conductors: Then C abs 36π 2 Ac ( 0 +2) 2 V λ 2 C sca 24π 3( 0 1) 2 V 2 ( 0 +2) 2 λ 4 1 const 2 Aω + 4πσ 0 ω C abs 4πc σ 0 V λ 2 C sca 24π 3V 2 λ 4 absorption dominates over scattering note lambda^-2. Important at mm wavelengths absorption dominates over scattering note lambda^-2. ditto 7 B.T. Draine Physics of Interstellar Dust IPMU

8 Spheres with Size Comparable to the Wavelength: Mie Theory Mie theory : analytic solution by Mie (1908) and Debye (1909) for isotropic sphere. Based on separation of variables. Series expansion, number of terms O(2πa/λ), where a = radius of sphere, λ = wavelength. Feasible to evaluate for 2πa/λ < Many public-domain codes. Results depend on x 2πa/λ and complex m(λ). Define dimensionless efficiencies: General behavior: Q x C x(a, λ) πa 2 for x =abs, sca, ext Q abs x for x 1 (electric dipole absorption) Q sca m 1 2 x 4 for x 1 (Rayleigh scattering) conspicuous resonances for 1 < m 1 x < 10 (when Im(m) is small) Q ext > 1 for m 1 x >1 Q ext 2 for m 1 x 1 ( the extinction paradox ) when m 1 x 1, 50% of the extinction is from (small-angle) diffraction around the target ( forward scattering ). Q ext has peak near m 1 x 2, with Q ext 4 ± 1. 8 B.T. Draine Physics of Interstellar Dust IPMU

9 dq sca /d! (sr -1 ) dq sca /d! (sr -1 ) Particles Large Compared to the Wavelength: X-Ray Scattering E=0.5keV a=0.10µm E=0.5keV a=0.20µm "(arcsec) E=1.0keV a=0.10µm E=1.0keV a=0.20µm "(arcsec) X-ray scattering by silicate spheres. Solid curve: Mie theory. Dots: ADT from Draine & Allaf-Akbari (2006). E=2.0keV a=0.10µm E=2.0keV a=0.20µm "(arcsec) X-Ray Regime: Simplifications because m 1 1 and x 2πa/λ 1: Can think in terms of ray optics Reflection and Refraction at interfaces is very small Anomalous Diffraction Theory (ADT) is accurate in the limits x 1 and m 1 1. Calculate change in phase and amplitude of wavefront after traversing target ( shadow function ) ADT has no restriction on m 1 x (unlike Rayleigh-Gans approximation ) ADT has analytic solutions for spheres (van de Hulst 1957) see Draine & Allaf-Akbari (2006). ADT can be applied (numerically) to nonspherical geometry (Draine & Allaf-Akbari 2006; Heng & Draine 2009). 9 B.T. Draine Physics of Interstellar Dust IPMU

10 Q ext for astrosilicate spheres from λ =1Å to 100 µm 0.1 µm grain becomes transparent to X-rays for λ <.002 µm =20Å, or E>0.6keV. silicate grains become opaque in the 10 µm feature for a > 2 µm 10 B.T. Draine Physics of Interstellar Dust IPMU

11 UV-optical-NIR to have dτ/dλ 1 > 0 at V = 0.55 µm: a < 0.2 µm silicate, or a < 0.08 µm graphite. to have dτ/dλ 1 > 0 at B = 0.44 µm: a < 0.15 µm silicate, or a < 0.07 µm graphite. Interstellar extinction curve is characterized by R V A V /E(B V ) A V /(A B A V ) Average extinction curve in diffuse regions of MW has R V 3.1. What size grains have this value of R V? Cannot reproduce extinction curve with a single size dust, but clear that must include grains with a 0.1 µm. 11 B.T. Draine Physics of Interstellar Dust IPMU

12 Size Distribution of Interstellar Grains Observe extinction curve from 2 µm 0.1 µm Mathis et al. (1977) tried to reproduce average interstellar extinction curve using mixture of graphite and silicate spheres. Using non-parametric size distribution with upper and lower cutoffs, a min and a max, they found best-fit size distributions dn/da. Their best-fit size distributions were very close to power-laws!! Therefore MRN proposed using power-laws 1 dn gra,sil n H da = A gra,sil a p for a min <a<a max p = 3.5 a min µm a max 0.25 µm This is the famous MRN size distribution. dn/da a 3.5 has most of mass at large size end, most of area at small size end. dn/da a 3.5 is similar to size distribution of p 3.25 for asteroids with 5 < D < 300 km (Bottke et al. 2005) steady-state coagulation/fragmentation models (Dohnanyi 1969; Weidenschilling 1997; Tanaka et al. 1996, 2005) Problem: Because of PAHs, the MRN distribution can no longer be considered applicable to interstellar dust. Substantial mass in ultrasmall dust grains: 5% of total dust mass is in particles with < 10 3 C atoms. This is much more than MRN extended to very small sizes. PAHs contribute substantially to the UV extinction. Non-PAH extinction not well-fitted by MRN distribution. 16 B.T. Draine Physics of Interstellar Dust IPMU

13 Size Distribution of Interstellar Grains, contd... Start with PAHs, then add size distributions of amorphous silicate and graphite to fit observed extinction: Weingartner & Draine (2001a) (WD01) spheres Zubko et al. (2004) (ZDA04) spheres Draine & Fraisse (2009) (DF09) spheroids, required to reproduce both extinction curve and polarization curve Results differ in detail, but all have similar total volumes of carbonaceous and silicate dust, using as much C, Mg, Si, Fe as allowed by observed depletions 50% of dust mass above/below 0.1 µm tickmarks: 50% of grain mass above and below (from Draine 2011) WD01 and DF09 use observed extinction curve with somewhat more near-ir extinction than for extinction curve assumed by ZDA04. Result: WD01 and DF09 have more mass in a > 0.25 µm dust than ZDA B.T. Draine Physics of Interstellar Dust IPMU

14 Regional Variations in Size Distribution Extinction curves are known to vary from one sightline to another. Denser regions tend to have flatter extinction curves, i.e., higher values of R V A V /E(B V ) increased R V is attributed to tilt in size distribution to decrease numbers of small particles, increase numbers of larger particles. grain growth is presumably due partially to accretion of atoms from gas, but this is only a minor effect (because unless ices can form, most depletable species are already depleted in diffuse clouds) grain growth must be due primarily to coagulation. timescale for dust grain to collide with another dust grain is relatively short: 1 30 cm τ dd = = cm 2 /H 1kms 1 yr n H Σ d ( v) dd dust-dust velocity differences v dd 1kms 1 are expected n H radiation pressure and recoil effects can cause grains to drift through gas with speeds that depend on size and composition ordinary fluid turbulence will give grains random velocities MHD turbulence can pump energy into orbital motions of > 0.1 µm grains in diffuse clouds (Yan et al 2004) It is likely that coagulation modifies the grain size distribution. Presumably balanced by shattering in higher-velocity grain-grain collisions (Yan et al. 2004; Hirashita & Yan 2009; Hirashita et al. 2010) Σ d v dd 18 B.T. Draine Physics of Interstellar Dust IPMU

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