Up: A catalog of observed comets
Let us consider a dust production rate
(in number of particles s-1).
The dust particles leave the gravitational influence zone of the comet with a
terminal velocity
.
Let us assume that the optically important dust
particles have an average radius
m (Hanner et al.
1985) and a mass density
kg m-3. For a
spherically
symmetrical dust coma with a uniform radial outflow, the number density of
grains at a distance x from the nucleus is
 |
(5) |
The extinction coefficient is then
 |
(6) |
and the equation of radiative transfer is
 |
(7) |
At the nucleus surface (
), the intensity is
.
Integrating
Eq. (6)
through the whole coma to
,
we get
 |
(8) |
and the resulting extinction in magnitudes is
 |
(9) |
The mass of each individual grain is
.
The dust production rate in mass
is then
 |
(10) |
The gas
and dust production rates are related through the dust to
gas ratio
by
.
Assuming that the main gas component is water, the gas production rate
can be computed as:
 |
(11) |
where f is the fraction of active area,
is the gas production
rate per unit area and
is the mass of the water molecule.
is then
 |
(12) |
The dust outflow velocity is given by
m
s-1 (Hanner et al. 1985; Hanner 1985;
Fernández et al.
1999), where
is the
gas production rate per unit area at 1 AU. Introducing the expression
for
and Eq. (12) into Eq. (9), we get an extinction
![\begin{displaymath}\Delta m \simeq \frac{1.8 m_{\rm w}}{{Z_{\rm w0}}^{\frac{1}{2...
...}{2}}}{a \rho_{\rm d}} \ [R_{\rm N}({\rm km})]^{\frac{1}{2}} .
\end{displaymath}](/articles/aas/full/2000/16/ds1835/img96.gif) |
(13) |
As an example, let us consider an active Jupiter family comet near
perihelion (i.e.,
AU). The gas production rate per unit area
at 1 AU is
mol m-2 s-1 and at 1.5 AU is
mol m-2 s-1. If the comet has
a fraction of active area of
10% and assuming a dust to gas
ratio
,
we get
![\begin{displaymath}\Delta m \simeq 0.004 [R_{\rm N}({\rm km})]^{\frac{1}{2}} \cdot
\end{displaymath}](/articles/aas/full/2000/16/ds1835/img101.gif) |
(14) |
For a typical nuclear radius of
km, the coma extinction
should not exceed 10-2 mag, thus too low to affect the estimated
magnitude of the nucleus, considering the other sources of much larger
errors involved in this determination.
One of the weak assumptions of the previous model is the consideration
of a constant outflow velocity
right from the surface. Considering
a simple model where the velocity increases linearly from one tenth of
its final value at the nucleus surface to the final value at a distance
of 100 radii, we get an extinction ten times larger. On this extreme
hypothesis, we only get significant extinction
for very large and/or very active comets.
Up: A catalog of observed comets
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