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 C_{abs} 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 (Q_{abs} / 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 C_{ext} 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 C_{ext} is the sum of the absorption C_{abs} and scattering C_{abs} cross sections both of which are positive numbers at all wavelengths, replacing C_{ext} by C_{abs} 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 m_{dust} 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 Q_{abs} / 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 (Q_{abs} / 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 T_{d} 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 T_{d} T is in good agreement with the physical conditions of the environment where the dust is located.