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For dust with sizes much smaller than the wavelength of the incident radiation, analytic solutions to the light scattering problem exist for certain shapes. Let a be the characteristic length of the dust, and x ident 2pi a / lambda be the dimensionless size parameter. Under the condition of x << 1 and |mx| << 1 (i.e. in the "Rayleigh" regime), Cabs = 4pi k Im{alpha}, Csca = (8pi/3) k4 |alpha|2, where alpha is the complex electric polarizability of the dust. Apparently, Csca << Cabs and Cext approx Cabs. In general, alpha is a diagonalized tensor; 7 for homogeneous spheres composed of an isotropic material, it is independent of direction

Equation 32 (32)

where V is the dust volume. 8 For a homogeneous, isotropic ellipsoid, the polarizability for electric field vector parallel to its principal axis j is

Equation 35 (35)

where Lj is the "depolarization factor" along principal axis j (see [13]). The electric polarizability alpha is also known for concentric core-mantle spheres [69], confocal core-mantle ellipsoids [13, 19], and multi-layered ellipsoids of equal-eccentricity [18]. For a thin conducting cylindrical rod with length 2l and radius ra << l, the polarizability along the axis of the rod is [38]

Equation 36 (36)

In astronomical modeling, the most commonly invoked grain shapes are spheres and spheroids (oblates or prolates). 9 In the Rayleigh regime, their absorption and scattering properties are readily obtained from Eqs. (32,35). For both dielectric and conducting spheres (as long as x << 1 and |mx| << 1)

Equation 37 (37)

At long wavelengths, for dielectric dust epsilon" propto omega while epsilon' approaches a constant much larger than epsilon" (see Eqs. 18,19), we see Cabs propto omega epsilon" propto omega2; for metallic dust, epsilon" propto 1/omega while epsilon' approaches a constant much smaller than epsilon" (see Eqs. 22,23), we see Cabs propto omega / epsilon" propto omega2; therefore, for both dielectric and metallic dust Cabs propto lambda-2 at long wavelengths! 10

It is also seen from Eq. (37) that for spherical dust in the Rayleigh regime the albedo alpha approx 0, and the radiation cross section Cpr approx Cabs. This has an interesting implication. Let betapr(a) be the ratio of the radiation pressure force to the gravity of a spherical grain of radius a in the solar system or in debris disks illuminated by stars (of radius R* and mass M*) with a stellar flux of Flambda* at the top of the atmosphere [53],

Equation 37 (38)

where G is the gravitational constant, and rhodust is the mass density of the dust. For grains in the Rayleigh regime (g approx 0; Csca << Cabs; Cabs propto a3), we see betapr propto Cabs / a3 is independent of the grain size a (see Fig. 4)!

Figure 4

Figure 4. betapr - Ratio of radiation pressure to gravity for compact silicate grains in the debris disk around the Sun-like star BD+20 307 (G0, age ~ 300 Myr). We note for nano-sized grains betapr approx 0.12, independent of grain size; for grains larger than ~ 0.3 µm, betapr is inverse proportional to grain size. Song et al. [68] attributed the dust in this disk to recent extreme collisions between asteroids. Taken from [51].

Spheroids are often invoked to model the interstellar polarization. In the Rayleigh approximation, their absorption cross sections for light polarized parallel (||) or perpendicular (perp) to the grain symmetry axis are 11

Equation 39 (39)

where the depolarization factors parallel (L||) or perpendicular (Lperp) to the grain symmetry axis are not independent, but related to each other through L|| + 2Lperp = 1, with

Equation 40 (40)

for prolates (ra > rb) where xie is the eccentricity, and

Equation 41 (41)

for oblates (ra < rb). For spheres L|| = Lperp = 1/3 and xie = 0. For extremely elongated prolates or "needles" (ra >> rb), it is apparent Cabsperp << Cabs||, we thus obtain

Equation 42 (42)

where L|| approx (rb / ra)2 ln(ra / rb). For dielectric needles, Cabs propto omega epsilon" propto lambda-2 at long wavelengths since L||(epsilon' - 1) + 1 >> L||epsilon" (see [45]); for metallic needles, for a given value of epsilon" one can always find a sufficiently long needle with L|| epsilon" < 1 and L||(epsilon' - 1) << 1 so that Cabs propto omega epsilon" propto sigma which can be very large (see [45]). Because of their unique optical properties, metallic needles with high electrical conductivities (e.g. iron needles, graphite whiskers) are resorted to explain a wide variety of astrophysical phenomena: (1) as a source of starlight opacity to create a non-cosmological microwave background by the thermalization of starlight in a steady-state cosmology [27]; (2) as a source of the grey opacity needed to explain the observed redshift-magnitude relation of Type Ia supernovae without invoking a positive cosmological constant [1]; (3) as the source for the submm excess observed in the Cas A supernova remnant [16]; and (4) as an explanation for the flat 3-8 µm extinction observed for lines of sight toward the Galactic Center and in the Galactic plane [15]. However, caution should be taken in using Eq. 42 (i.e. the Rayleigh approximation) since the Rayleigh criterion 2pi ra |m| / lambda << 1 is often not satisfied for highly conducting needles (see [45]). 12

In astronomical spectroscopy modeling, the continuous distribution of ellipsoid (CDE) shapes has been widely used to approximate the spectra of irregular dust grains by averaging over all ellipsoidal shape parameters [6]. In the Rayleigh limit, this approach, assuming that all ellipsoidal shapes are equally probable, has a simple expression for the average cross section

Equation 43 (43)

where Log epsilon is the principal value of the logarithm of epsilon. The CDE approach, resulting in a significantly-broadened spectral band (but with its maximum reduced), seems to fit the experimental absorption spectra of solids better than Mie theory. Although the CDE may indeed represent a distribution of shape factors caused either by highly irregular dust shapes or by clustering of spherical grains into irregular agglomerates, one should caution that the shape distribution of cosmic dust does not seem likely to resemble the CDE, which assumes that extreme shapes like needles and disks are equally probable. A more reasonable shape distribution function would be like dP / dL|| = 12 L|| (1 - L||)2 which peaks at spheres (L|| = 1/3). This function is symmetric about spheres with respect to eccentricity e and drops to zero for the extreme cases: infinitely thin needles (e -> 1, L|| -> 0) or infinitely flattened pancakes (e -> infty, L|| -> 1). Averaging over the shape distribution, the resultant absorption cross section is Cabs = integ01 dL|| dP / dL|| Cabs(L||) where Cabs(L||) is the absorption cross section of a particular shape L|| [62, 49, 48]. Alternatively, Fabian et al. [17] proposed a quadratic weighting for the shape distribution, "with near-spherical shapes being most probable".

When a dust grain is very large compared with the wavelength, the electromagnetic radiation may be treated by geometric optics: Qext ident Cext / Cgeo -> 2 if x ident 2pi a / lambda >> 1 and |m - 1| x >> 1. 13 For these grains (g approx 1; Cabs approx Cgeo), the ratio of the radiation pressure to gravity betapr propto Cabs / a3 propto 1 / a. This is demonstrated in Figure 4. For dust with x >> 1 and |m - 1| x << 1, one can use the "anomalous diffraction" theory [69]. For dust with |m - 1| x << 1 and |m - 1| << 1, one can use the Rayleigh-Gans approximation 14 to obtain the absorption and scattering cross sections [6 39 69]:

Equation 44 (44)

It is important to note that the Rayleigh-Gans approximation is invalid for modeling the X-ray scattering by interstellar dust at energies below 1 keV. This approximation systematically and substantially overestimates the intensity of the X-ray halo below 1 keV [66].

7 alpha can be diagonalized by appropriate choice of Cartesian coordinate system. It describes the linear response of a dust grain to applied electric field E: p = alphaE where p is the induced electric dipole moment. Back.

8 For a dielectric sphere with dielectric function given in Eq. (17), in the Rayleigh regime the absorption cross section is

Equation 33 (33)

Similarly, for a metallic sphere with dielectric function given in Eq. (21),

Equation 34 (34)

It is seen that the frequency-dependent absorption cross section for both dielectric and metallic spheres is a Drude function. This is also true for ellipsoids. Back.

9 Spheroids are a special class of ellipsoids. Let ra, rb, and rc be the semi-axes of an ellipsoid. For spheroids, rb = rc. Prolates with ra > rb are generated by rotating an ellipse (of semi-major axis ra and semi-minor axis rb) about its major axis; oblates with ra < rb are generated by rotating an ellipse (of semi-minor axis ra and semi-major axis rb) about its minor axis. Back.

10 However, various astronomical data suggest a flatter wavelength-dependence (i.e. Cabs propto lambda-beta with beta < 2): beta < 2 in the far-IR/submm wavelength range has been reported for interstellar molecular clouds, circumstellar disks around young stars, and circumstellar envelopes around evolved stars. Laboratory measurements have also found beta < 2 for certain cosmic dust analogues. In literature, the flatter (beta < 2) long-wavelength opacity law is commonly attributed to grain growth by coagulation of small dust into large fluffy aggregates (see [47] and references therein). However, as shown in Eq. (31), the Kramers-Kronig relation requires that beta should be larger than 1 for lambda -> infty since F is a finite number and the integration in the left-hand-side of Eq. (31) should be convergent although we cannot rule out beta < 1 over a finite range of wavelengths. Back.

11 For grains spinning around the principal axis of the largest moment of inertia, the polarization cross sections are Cpol = (Cabs|| - Cabsperp) / 2 for prolates, and Cpol = (Cabsperp - Cabs||) for oblates; the absorption cross sections for randomly-oriented spheroids are Cabs = (Cabs|| + 2Cabsperp) / 3 [42]. Back.

12 The "antenna theory" has been applied for conducting needle-like dust to estimate its absorption cross sections [75]. Let it be represented by a circular cylinder of radius ra and length l (ra << l). Let rhoR be its resistivity. The absorption cross section is given by Cabs = (4pi / 3c)(pi ra2 l / rhoR), with a long wavelength cutoff of lambdao = rhoR c (l / ra)2 / ln(l / ra)2, and a short-wavelength cutoff of lambdamin approx (2pi c me) / (rhoR ne e2), where me, e, and ne are respectively the mass, charge, and number density of the charge-carrying electrons. Back.

13 At a first glance, Qext -> 2 appears to contradict "common sense" by implying that a large grain removes twice the energy that is incident on it! This actually can be readily understood in terms of basic optics principles: (1) on one hand, all rays impinging on the dust are either scattered or absorbed. This gives rise to a contribution of Cgeo to the extinction cross section. (2) On the other hand, all the rays in the field which do not hit the dust give rise to a diffraction pattern that is, by Babinet's principle, identical to the diffraction through a hole of area Cgeo. If the detection excludes this diffracted light then an additional contribution of Cgeo is made to the total extinction cross section [6]. Back.

14 The conditions for the Rayleigh-Gans approximation to be valid are |m - 1| << 1 and |m - 1| x << 1. The former ensures that the reflection from the surface of the dust is negligible (i.e. the impinging light enters the dust instead of being reflected); the latter ensures that the phase of the incident wave is not shifted inside the dust. For sufficiently small scattering angles, it is therefore possible for the waves scattered throughout the dust to add coherently. The intensity (I) of the scattered waves is proportional to the number (N) of scattering sites squared: I propto N2 propto rho2 a6 (where rho is the mass density of the dust). This is why the X-ray halos (usually within ~ 1° surrounding a distant X-ray point source; [63]) created by the small-angle scattering of X-rays by interstellar dust are often used to probe the size (particularly the large size end; [14, 67, 73]), morphology (compact or porous; [58, 66]), composition, and spatial distribution of dust. Back.

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