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.