2. CONSTRAINTS FROM THE KRAMERS-KRONIG RELATION

As already mentioned in Section 1, with specific assumptions made concerning the grain size, shape, geometry and composition, in principle one can calculate the absorption cross section Cabs and the opacity abs as a function of wavelength. But this is limited to spherical grains; even for grains with such simple shapes as spheroids and cylinders, the calculation is complicated and limited to small size parameters defined as 2 a / . As a result, astronomers often adopt a simplified formula

 (1)

where 0 and the exponent index are usually treated as free parameters, while 0 is often taken from experimental measurements of cosmic dust analogs or values predicted from interstellar dust models; or

 (2)

where again 0 and are free parameters and (Qabs / a)0 is usually taken from model calculations.

The Kramers-Kronig dispersion relation, originally deduced by Kronig [33] and Kramers [32] from the classical Lorentz theory of dispersion of light, connects the real (m'; dispersive) and imaginary (m''; absorptive) parts of the index of refraction (m[] = m' + i m'') based on the fundamental requirement of causality. Purcell [55] found that the Kramers-Kronig relation can be used to relate the extinction cross section integrated over the entire wavelength range to the dust volume V

 (3)

where Cext is the extinction cross section, and F, a dimensionless factor, is the orientationally-averaged polarizability relative to the polarizability of an equal-volume conducting sphere, depending only upon the grain shape and the static (zero-frequency) dielectric constant 0 of the grain material (Purcell [55]; Draine [18]). Since Cext is the sum of the absorption Cabs and scattering Cabs cross sections both of which are positive numbers at all wavelengths, replacing Cext by Cabs in the left-hand-side of Eq. (3) would give a lower limit on its right-hand-side; therefore we can write Eq. (3) as

 (4)

It is immediately seen in Eq. (4) that should be larger than 1 for since F is a finite number and the integration in the left-hand-side of Eq. (4) should be convergent (also see Emerson [23]), although we cannot rule out 1 over a finite range of wavelengths.

Astronomers often use the opacity abs of the formula described in Eq. (1) to fit the far-IR, submm and mm photometric data and then estimate the dust mass of interstellar clouds:

 (5)

where d is the distance of the cloud, T is the dust temperature, F is the measured flux density at wavelength . By fitting the photometric data points, one first derives the best-fit parameters , 0, and T. For a given 0, one then estimates mdust from Eq. (5). In this way, various groups of astronomers have reported the detection of appreciable amounts of very cold dust (T < 10 K) both in the Milky Way and in external galaxies (Reach et al. [56]; Chini et al. [14]; Krügel et al. [35]; Siebenmorgen, Krügel, & Chini [59]; Boulanger et al. [10]; Popescu et al. [54]; Galliano et al. [24]; Dumke, Krause, & Wielebinski [20]).

As can be seen in Eq. (5), if the dust temperature is very low, one then has to invoke a large amount of dust to account for the measured flux densities. This often leads to too large a dust-to-gas ratio to be consistent with that expected from the metallicity of the region where the very cold dust is detected (e.g. see Dumke et al. [20]), unless the opacity abs is very large. It has been suggested that such a large abs can be achieved by fractal or porous dust (Reach et al. [56]; Dumke et al. [20]). Can abs be really so large for physically realistic grains? At a first glance of Eq. (4), this appears plausible if the dust is sufficiently porous (so that its mass density is sufficiently small). However, one should keep in mind that for a porous grain, the decrease in will be offset by a decrease in F because the effective static dielectric constant 0 becomes smaller when the dust becomes porous, leading to a smaller F factor (see Fig. 1 in Purcell [55] and Fig. 15 in Draine [18]).

Similarly, if one would rather use Qabs / a of Eq. (2) instead of abs of Eq. (1), we can also apply the Kramers-Kronig relation to place (1) a lower limit on - cannot be smaller than or equal to 1 at all wavelengths, and (2) an upper limit on (Qabs / a)0 from

 (6)

Finally, the best-fit parameters , 0, and T should be physically reasonable. This can be checked by comparing the best-fit temperature T with the dust equilibrium temperature Td calculated from the energy balance between absorption and emission

 (7)

where c is the speed of light, and u is the energy density of the radiation field. Alternatively, one can check whether the strength of the radiation field required by Td T is in good agreement with the physical conditions of the environment where the dust is located.