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Appendix A: The problem of coma extinction

Let us consider a dust production rate $Q_{\rm d}$ (in number of particles s-1). The dust particles leave the gravitational influence zone of the comet with a terminal velocity $v_{\rm d}$. Let us assume that the optically important dust particles have an average radius $a \sim 0.5$ ${\mu}$m (Hanner et al. 1985) and a mass density $\rho_{\rm d} \sim 10^3$ 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

\begin{displaymath}n = \frac{Q_{\rm d}}{4 \pi x^2 v_{\rm d}}\cdot
\end{displaymath} (5)

The extinction coefficient is then

\begin{displaymath}\tau = n \pi a^2 = \frac{Q_{\rm d} a^2}{4 x^2 v_{\rm d}}
\end{displaymath} (6)

and the equation of radiative transfer is

\begin{displaymath}\frac{{\rm d}I}{I} = - \frac{Q_{\rm d} a^2}{4 v_{\rm d} x^2} {\rm d}x.
\end{displaymath} (7)

At the nucleus surface ( $x = R_{\rm N}$), the intensity is $I_{\rm o}$. Integrating Eq. (6) through the whole coma to $ x \to \infty$, we get

\begin{displaymath}\frac{I}{I_{\rm o}} = \exp\left(- \frac{Q_{\rm d} a^2}{4 v_{\rm d} R_{\rm N}}\right)
\end{displaymath} (8)

and the resulting extinction in magnitudes is

 \begin{displaymath}
\Delta m = \frac{2.5\log{e}\ Q_{\rm d} a^2}{4 v_{\rm d} R_{\...
...}}
\simeq 0.27 \frac{Q_{\rm d} a^2}{v_{\rm d} R_{\rm N}}\cdot
\end{displaymath} (9)

The mass of each individual grain is $m_{\rm d} = \frac{4}{3} \pi a^3 \rho_{\rm d}$. The dust production rate in mass $Q_{\rm d}'$ is then

\begin{displaymath}Q_{\rm d}' = \frac{4}{3} \pi a^3 \rho_{\rm d} Q_{\rm d}.
\end{displaymath} (10)

The gas $Q_{\rm g}$ and dust production rates are related through the dust to gas ratio $\alpha$ by $Q_{\rm d}' = \alpha Q_{\rm g}$.

Assuming that the main gas component is water, the gas production rate can be computed as:

\begin{displaymath}Q_{\rm g} = 4 \pi R_{\rm N}^{\ 2} f Z_{\rm w} m_{\rm w}
\end{displaymath} (11)

where f is the fraction of active area, $Z_{\rm w}$ is the gas production rate per unit area and $m_{\rm w}$ is the mass of the water molecule. $Q_{\rm d}$ is then


 \begin{displaymath}
Q_{\rm d} = \frac{3 \alpha R_{\rm N}^{\ 2} f Z_{\rm w} m_{\rm w}}{a^3 \rho_{\rm d}}\cdot
\end{displaymath} (12)

The dust outflow velocity is given by $v_{\rm d} = 450 \ [Z_{\rm w}/Z_{w0} \ \ R_{\rm N}({\rm km})]^{1/2}$ m s-1 (Hanner et al. 1985; Hanner 1985; Fernández et al. 1999), where $Z_{\rm w0}$ is the gas production rate per unit area at 1 AU. Introducing the expression for $v_{\rm d}$ 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} (13)

As an example, let us consider an active Jupiter family comet near perihelion (i.e., $q \sim 1.5$ AU). The gas production rate per unit area at 1 AU is $\sim 3.2 \ 10^{21}$ mol m-2 s-1 and at 1.5 AU is $\sim 1.2 \ 10^{21}$ mol m-2 s-1. If the comet has a fraction of active area of $\sim$10% and assuming a dust to gas ratio $\alpha = 0.5$, we get

\begin{displaymath}\Delta m \simeq 0.004 [R_{\rm N}({\rm km})]^{\frac{1}{2}} \cdot
\end{displaymath} (14)

For a typical nuclear radius of $R_{\rm N} \sim 1$ 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 $v_{\rm d}$ 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.


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