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It is seen in Section 2 that the Kramers-Kronig relation requires beta > 1 for lambda -> infty. For the Milky Way diffuse ISM, the 100 µm-1mm dust emission spectrum obtained by the Diffuse Infrared Background Experiment (DIRBE) instrument on the Cosmic Background Explorer (COBE) satellite is well fitted by the product of a single Planck curve of T approx 17.5 K and an opacity law characterized by beta approx 2 (i.e. kappaabs propto lambda-2; Boulanger et al. [10]), although other sets of T and beta are also able to provide (almost) equally good fits to the observed emission spectrum (e.g. T approx 19.5 K and beta approx 1.7; see Draine [17]). But we should emphasize here that it is never flatter than beta approx 1.65 (Draine [17]).

Smaller beta in the submm and mm wavelength range has been reported for cold molecular cores (e.g. Walker et al. [63]: beta approx 0.9 - 1.8), circumstellar disks around young stars (e.g. Beckwith & Sargent [4]: beta approx - 1-1; Mannings & Emerson [47]: beta approx 0.6; Koerner, Chandler, & Sargent [28]: beta approx 0.6), and circumstellar envelopes around evolved stars (e.g. Knapp, Sandell, & Robson [27]: beta approx 0.9) including the prototypical carbon star IRC+10126 (Campbell et al. [13]: beta approx 1).

However, these results are not unique since the dust temperature and density gradients in the clouds, disks or envelopes have not been taken into account in deriving beta - the beta exponent was usually estimated by fitting the submm and mm spectral energy distribution by a modified black-body lambda-beta Blambda(T) under the "optically-thin" assumption, with beta and T as adjustable parameters. If the dust spatial distribution is not constrained, the very same emission spectrum can be equally well fitted by models with different beta values.

As a matter of fact, an asymptotic value of beta approx 2 (i.e. kappaabs propto lambda-2) is expected for both dielectric and conducting spherical grains: in the Rayleigh regime (where the wavelength is much larger than the grain size)

Equation 8 (8)

where epsilon(lambda) = epsilon1 + i epsilon2 is the complex dielectric function of the grain at wavelength lambda. For dielectric spheres, kappaabs propto lambda-1 epsilon2 propto lambda-2 as lambda -> infty since epsilon1 approaches a constant (>> epsilon2) while epsilon2 propto lambda-1; for metallic spheres with a conductivity of sigma, kappaabs propto lambda-1 epsilon2-1 propto lambda-2 as lambda -> infty since epsilon2 = 2lambda sigma / c propto lambda and epsilon1 << epsilon2. Even for dielectric grains with such an extreme shape as needle-like prolate spheroids (of semiaxes l along the symmetry axis and a perpendicular to the symmetry axis), in the Rayleigh regime we expect kappaabs propto lambda-2:

Equation 9 (9)

where L|| approx (a / l)2 ln(l / a) is the depolarization factor parallels to the symmetry axis; since for dielectric needles epsilon1 -> constant and L||(epsilon1 - 1) + 1 >> L|| epsilon2, therefore kappaabs propto lambda-1 epsilon2 propto lambda-2 (see Li [36] for a detailed discussion). Only for both conducting and extremely-shaped grains kappaabs can still be large at very long wavelengths. But even for those grains, the Kramers-Kronig relation places an upper limit on the wavelength range over which large kappaabs can be attainable (see Li & Dwek [43] for details). The Kramers-Kronig relation has also been applied to interstellar dust models to see if the subsolar interstellar abundance problem can be solved by fluffy dust (Li [40]) and to TiC nanoparticles to relate their UV absorption strength to their quantities in protoplanetary nebulae (Li [37]).

The inverse-square dependence of kappaabs on wavelength derived above applies to both crystalline and amorphous materials (see Tielens & Allamandola [61] for a detailed discussion). Exceptions to this are amorphous layered materials and very small amorphous grains in both of which the phonons are limited to two dimensions and their phonon spectrum is thus proportional to the frequency. Therefore, for both amorphous layered materials and very small amorphous grains the far-IR opacity is in inverse proportional to wavelength, i.e. kappaabs propto lambda-1 (Seki & Yamamoto [58]; Tielens & Allamandola [61]). Indeed, the experimentally measured far-IR absorption spectrum of amorphous carbon shows a kappaabs propto lambda-1 dependence at 5 µm < lambda < 340 µm (Koike, Hasegawa, & Manabe [29]). If there is some degree of cross-linking between the layers in the amorphous layered grains, we would expect 1 < beta < 2 (Tielens & Allamandola [61]). This can explain the experimental far-IR absorption spectra of layer-lattice silicates which were found to have 1.25 < beta < 1.5 at 50 µm < lambda < 300 µm (Day [15]). For very small amorphous grains, if the IR absorption due to internal bulk modes (for which the density of states frequency spectrum is proportional to lambda-2) is not negligible compared to that due to surface vibrational modes (for which the frequency spectrum is proportional to lambda-1), we would also expect 1 < beta < 2 (Seki & Yamamoto [57]).

If there exists a distribution of grain sizes, ranging from small grains in the Rayleigh regime for which beta ~ 2 and very large grains in the geometric optics limit for which beta ~ 0, we would expect beta to be intermediate between 0 and 2. This can explain the small beta values of dense regions such as molecular cloud cores, protostellar nebulae and protoplanetary disks where grain growth occurs (e.g. see Miyake & Nakagawa [50]).

The exponent index beta is temperature-dependent, as measured by Agladze et al. [1] for silicates at T = 1.2-30 K and lambda = 700-2900 µm, and by Mennella et al. [49] for silicates and carbon dust at T = 24-295 K and lambda = 20-2000 µm. Agladze et al. [1] found that at T = 1.2-30 K, beta first increases with increasing T, after reaching a maximum at T ~ 10-15 K it starts to decrease with increasing T. Agladze et al. [1] attributed this to a two-level population effect (Bösch [9]): because of the temperature-dependence of the two-level density of states (i.e. the variation in temperature results in the population change between the two levels), the exponent index beta is also temperature dependent. In contrast, Mennella et al. [49] found that beta increases by ~ 10%-50% from T = 295 K to 24 K, depending on the grain material (e.g. the variation of beta with T for crystalline silicates is not as marked as for amorphous silicates). The increase of beta with decreasing T at T = 24-295 K is due to the weakening of the long wavelength absorption as T decreases because at lower temperatures fewer vibrational modes are activated. Finally, it is noteworthy that the inverse temperature dependence of beta has been reported by Dupac et al. [21] for a variety of regions.

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