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Modeling Cosmic Dust:. How to Use Optical "Constants". The life-cycle of stardust. Dust and energy flow. Global average temperature. From climatecrocks.com. Diamonds & Graphite. Rubies and sapphires. How light interacts with matter. Reflection Propagation Transmission. Incident light.
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Modeling Cosmic Dust: How to Use Optical "Constants"
Dust and energy flow Global average temperature From climatecrocks.com
How light interacts with matter • Reflection • Propagation • Transmission Incident light Transmitted light Propagation through the medium Reflected light
What phenomena can occur during propagation of light through a medium? • Refraction • Absorption • Luminescence • Scattering • Refraction: reduction in propagation speed; causes light to bend • Absorption: occurs if the frequency of the light is at a resonant frequency of the medium. Transmission is related to absorption, only unabsorbed light is transmitted • Luminescence: spontaneous production of light, due to excitation by absorption of incident light • Scattering: light changes direction Refraction Absorption & luminescence Scattering
Reflectivity and Transmittance • Reflectivity = ratio of reflected flux to incident flux • Transmittivity* = ratio of transmitted flux to incident flux • Both quantities are wavelength dependent • If there is no absorption, R + T = 1 *Transmittivity is not a real word! However, the correct term is Transmittance, which is confusing because reflectance reflectivity; absorbance absorptivity.
dx F Fʹ Opacity k is the opacity ρis the density dx is the thickness of the slab Flux diminishes exponentially with penetration
dx F Fʹ Absorption Coefficient absis the absorption coefficient This is Beer’s Law This assume no reflection
Refraction and absorption • In a real medium light is both refracted and absorbed. • So transmittance is not just T = 1 – R Incident light Transmitted light Propagation through the medium Reflected light • T = (1-R1)e-L(1-R2) • AssumingR1andR2are identical:T = (1-R)2e-L
Terminology AbsorptivityA = Fabs/Finc Absorbance a = absL = kρL A = e-a • T = (1-R)2e-L= (1-R)2A
Refractive index and absorption • For a transparent medium (no absorption), the refractive index is given by: n = c/v. • This is usually wavelength dependent
Complex refractive index • This combines both refraction and absorption into a single physical “quantity” (it is wavelength dependent) • Be careful of notation!! • The real part of m (i.e. n) is just the regular refractive index • The imaginary part, k, is sometime called the absorption coefficient (sometimes the extinction coefficient) • Note that this k abs kλ(opacity)
Optical “constants” • There are two sets of optical functions that are closely interrelated. • the real and imaginary parts of the complex refractive index • the real and imaginary parts of the complex dielectric function (relative permittivity)
What effect does the complex refractive index have? • Once the radiation enters the medium, the velocity of light becomes v = c / m, so that: • The intensity (flux) of the light wave is proportional to H2, and will exponentially decrease with a decay constant of 2k = 4k/. • Assume we have light as an idealized, sinusoidal wave. The incident radiation can be written as: where H0 is the amplitude, is the angular frequency, and is the wavenumber (=2/)
Beer’s Law • Where abs is the absorption coefficient. • T = (1-R)2e-L We can measure T and get k!
Derive n,k from lab spectra (n.b., several alternate methods, depending on spectral & sample type) • Measure absorbance a, specular reflectance R, and sample thickness d. • Solve for ideal absorption coefficient A Solve for imaginary index of refraction (same in every method): OR Back out real index of refraction from k and R 2. Measure absorbance of samples: one thick and one thin. Solve for A and R. For thin films or polished slabs Hofmeister, Pitman, Goncharov, & Speck (2009)
OHM and DL From Draine & Lee (1984)
Calculating interactions of light with small particles… • The total energy depleted from the original beam can be put equal to the incident energy on the area Cext, the extinction cross-section. Cext = Csca + Cabs • Small particles do not behave like blackbodies; how their behaviour deviates from blackbody behave can be defined by efficiency factors – Q-factors. • From the cross sections it is easy to see that the efficiencies of a particle to absorb, scatter and extinguish light are given by: • The Q-factors can be calculated from the complex refractive index (or complex dielectric function)
Emissivity and Q-factors • Q-factors essential describe how a real solid deviates from blackbody behaviour • The flux emitted by a particle is given by: where B(,T) is the Planck blackbody curve. • Emissivity is the ratio of energy radiated by a particular material to energy radiated by a black body at the same temperature = Q! • For metals and covalently bonded solid Q is generally a simple trend with wavelength such that • is sometimes referred to as the emissivity law
How does Q relate to lab data? AbsorptivityA = Iabs/Iinc A = e-a Absorbance a = absL • τλis more or less ≡a • Qabs is more or less equivalent to A
What are we seeing when we look at the spectrum of a dust cloud? the sum of the contributions of particles of different compositions, sizes, crystal structures etc.
Cosmic Silicate New Stuff Black = new!. Blue = Draine (2003) Green = Draine & Lee (1984). Red, yellow = Ossenkopf et al. (1992) Same sample at all l Careful sample prep & analysis to eliminate contributions to n,k from back reflections Cross-checked by comparing many overlapping spectral segments Not grain size dependent Derived from combo of transmission & reflectivity spectra
Absorption Cross Sections UV-VIS-NIR particularly affected New Stuff
Olivine Melilite Pyroxene
Olivine From Pitman et al. (2010)
Speck et al 2011 Dorschner et al. (1995) http://www.astro.uni-jena.de/Laboratory/Database/databases.html • NBO/T vs. Fe/Mg
Conclusions • Beware of vocabulary • Read the paper! • If optical constants are derived from particulates, particle shape effects are embedded in the data. • If optical constants are derived from observations they cannot tell you anything about mineralogy
Case Study: HD 161796 (IRAS 17436+5003) • Simple test environment to model using DUSTY • - Post AGB star + dust BB w/ • twin peaked spectrum • Optically thin • Insignificant photoionization • Typically modeled with • amorphous silicates, • crystalline silicates (Fo, En), • crystalline H2O ice dust Assumptions = central star T = 6750±150 K (Hoozgaad et al. 2002) 1/r2 radial dust density distribution (constant mass loss) MRN size distribution 1 dust species
DUSTY Models: HD 161796 l (mm)
New models τV ≈2.3, Tin ≈ 125 τV ≈1.3, Tin ≈ 120 70-80% Cosmic Silicate 20-30% Metallic Iron
Crystalline Amorphous
History From Gillett et al (1968)
Crystalline olivine Elias 16 Trapezium Amorphous olivine DI Cep Cep
Crystalline olivine Comet Halley Amorphous olivine Comet Hale-Bopp HAe/Be star HD163296
Olivine [M2SiO4] Melilite [M2Si2O7] Pyroxene [M2Si2O6]
Crystalline Amorphous
Crystalline olivine Amorphous olivine
Crystalline olivine Crystalline Amorphous olivine Amorphous