\\
        1950 DA       & 3.69  & 3.76  & 3.76   \\
        Encke         & 3.36  & 3.42  & 3.43   \\
        Brorsen       & 0.93  & 0.94  & 0.94   \\
        Grigg-Mellish & 0.022 & 0.022 & 0.022  \\
        Temple-Tuttle & 0.62  & 0.62  & 0.60   \\
        Neujmin       & 0.94  & 0.93  & 0.93   \\
        Schaumasse    & 1.13  & 1.15  & 1.15   \\
        Pons Brooks   & 0.041 & 0.041 & 0.041  \\
        \hline
    \end{tabular}
    \label{tab:tab1}
  \end{center}
\end{table}

To carry out such a comparison we also need to modify the calculation to
derive the more typically quoted quantity of the intrinsic collision
probability, $P_i$, which is the probability of impact per unit time
divided by $\tau^2$ where $\tau=(D+D_{\rm{im}})/2$ (Wetherill 1967).
Our method included the $\tau^2$ factor from the outset
(see equation~\ref{eq:f}) and integrated the collision rate over
the size distribution capable of causing catastrophic impacts, whereas
this was accounted for at a later stage by other methods. 
Using our notation, intrinsic probability in $10^{-18}$ km$^{-2}$ yr$^{-1}$
is given by 
\begin{equation}
  P_i = 0.030 \sum_{r,\phi} 
     \bar{\sigma}(r,\phi)(q_1,Q_1)
     \bar{\sigma}_{\rm{v}}(r,\phi)(q_2,Q_2)
    \langle v_{\rm{rel[1,2]}} \rangle (r,\phi),
  \label{eq:pi}
\end{equation}
where the constant includes both a conversion between units and an
extra factor $\pi$ (since our calculation considers cross-sectional
area rather than $\tau^2$).
In Table~\ref{tab:tab1} we compare the intrinsic collision probabilities
we find using the Monte-Carlo approach with 200,000 planetesimals in each
population with those of Dell'Oro \& Paolicchi (1998) and Bottke \& Greenberg (1993)
showing excellent agreement between the methods.

As noted by Bottke et al. (1994) (their point I.2), to utilise intrinsic collision
probabilities requires knowledge of the velocity probability distribution
and it is not sufficient to assume a single mean relative velocity of collisions.
This is automatically included in our method, since we keep track of the
velocities encountered by a planetesimal at each location and these are
used to work out the amount of cross-sectional area of impactors at that location
that are able to cause a catastrophic collision. 
To illustrate that our Monte-Carlo method computes an accurate velocity probability
distribution, here we reproduce Figure 1 of Bottke et al. (1994) (see also
Fig.~1 of Dell'Oro \& Paolicchi 1998) in our
Figure~\ref{fig:bottke94}.

\begin{figure}
  \begin{center}
    \vspace{-0.2in}
    \begin{tabular}{c}
      \psfig{figure=wyattfig4.ps,height=2.4in}
    \end{tabular}
    \caption{Contribution of different relative velocities to the
    intrinsic collision probability for two planetesimals, one with
    orbital elements $a=3.42$AU, $e=0.578$, $I=0.435$rad and the other
    with $a=1.59$AU, $e=0.056$, $I=0.466$rad for comparison with
    Fig.~1 of Bottke et al. (1994) and Fig.~1 of Dell'Oro \& Paolicchi (1998).}
   \label{fig:bottke94}
  \end{center}
\end{figure}

Similarly we find that we can reproduce Figs~5 and 6 of Krivov et al. (2005)
(not shown here) noting that their $\Delta$ and $\bar{v}_{\rm{imp}}$ are given by
\begin{eqnarray}
  \Delta(a_1,e_1,a_2,e_2) & = & \sum_{r,\phi} \bar{\sigma}(r,\phi)(q_1,Q_1)
    \bar{\sigma}_{\rm{v}}(r,\phi)(q_2,Q_2), \\
  \bar{v}_{\rm{imp}} & = & \sum_{r,\phi} \bar{\sigma}(r,\phi)(q_1,Q_1)
    \bar{\sigma}_{\rm{v}}(r,\phi)(q_2,Q_2)
  \times \nonumber \\ & & 
    \langle v_{\rm{rel[1,2]}} \rangle (r,\phi) / \Delta(a_1,e_1,a_2,e_2),
\end{eqnarray}
in our notation, although we should point out again that such a mean collision
velocity must be used with care as a range of velocities contribute to
the collision rate (see Fig.~\ref{fig:bottke94}).

In summary we conclude that, for the same assumptions, our method for
calculating collision rates produces results that are in agreement with the
most accurate methods available in the literature, and that although this
method was derived with the intention of studying high eccentricity orbits,
it is also applicable regardless of the magnitude of the eccentricity.


\subsection{Implementation of evolution}
\label{ss:implem}
The factor $\bar{R}_{\rm{cc}}(D_{\rm{max}},q_1,Q_1;q_2,Q_2)$ (equation 
\ref{eq:rcolbar12}) depends only on the way in which the parameter space
is divided up.
Thus an approach was implemented in which this factor was worked out ahead of time,
and then used to work out the catastrophic collision rate for the largest
planetesimals in cell $(q,Q)$ from planetesimals in all other cells:
\begin{equation}
  R_{\rm{cc}}(D_{\rm{max}},q,Q) = 
  \sum_{q_2}
  \sum_{Q_2}
    M_{\rm{tot}}(q_2,Q_2)
    \bar{R}_{\rm{cc}}(D_{\rm{max}},q,Q;q_2,Q_2).
\end{equation}

Using the assumptions about the evolution of the size distribution
described in \S \ref{ss:evolsd}, this collision rate can then be used to
work out the mass which would be removed in a timestep $dt$, and the mass
distribution stepped forward in time.
\begin{equation}
  M_{\rm{tot}}(q,Q,t+dt) = M_{\rm{tot}}(q,Q,t)[1 - dt*R_{\rm{cc}}(D_{\rm{max}},q,Q)].
\end{equation}
The timestep $dt$ is chosen so that some fraction, say 0.1\%, of the mass is removed
from the most rapidly evolving cell in that timestep, i.e.,
\begin{equation}
  dt = 10^{-3}/\rm{max}[R_{\rm{cc}}(D_{\rm{max}},q,Q)].
\end{equation}

\subsection{Application to scattered disk-like distribution}
\label{ss:sd}
One of the simplest examples of this evolution is its application
to a scattered disk-like distribution wherein planetesimals have
a common pericentre distance, $q$, but a range of apocentre distances.
Such a distribution might arise from planetesimals scattered by a planet
on a circular orbit at a radius just inside $q$.

\begin{figure}
  \begin{center}
    \vspace{-0.2in}
    \begin{tabular}{cc}
      \textbf{(a)} & \hspace{-0.5in} \psfig{figure=wyattfig5a.ps,height=2.4in} \\
      \textbf{(b)} & \hspace{-0.5in} \psfig{figure=wyattfig5b.ps,height=2.4in} \\
      \textbf{(c)} & \hspace{-0.5in} \psfig{figure=wyattfig5c.ps,height=2.4in}
    \end{tabular}
    \caption{Collisional evolution of a scattered disk-like distribution.
    \textbf{(a)} The function $T(e_1,e_2