\begin{table}%t1
\caption{Description of the five element sets used in
the computations of the LOV, and their respective native units and
rescaling parameters.}
%\centerline
 {  \begin{tabular}{l c c c c c c}
  \hline
  \hline
  {\bf Cartesian} & $x$ & $y$ & $z$ & $v_x$ &  $v_y$ & $v_z$ \\
  Units & AU & AU & AU &AU/d&AU/d &AU/d \\
  Scaling & $r$ & $r$ &  $r$ &  $v$ & $v$ & $v$ \\
  \hline
  {\bf Cometary} &  $e$ & $q$ & $t_p$ & $\Omega$ &  $\omega$ & $i$ \\
  Units & -- & AU & d & rad & rad & rad \\
  Scaling & $1$ & $q$ &  $Z$ &  $2 \pi $ & $2 \pi$ & $\pi$ \\
  \hline
  {\bf Keplerian} &  $e$ & $a$ & $M$ & $\Omega$ &  $\omega$ & $i$ \\
  Units & -- & AU & rad & rad & rad & rad \\
  Scaling & $1$ & $a$ &  $2 \pi $ &  $2 \pi $ & $2 \pi$ & $\pi$ \\
  \hline
  {\bf Equinoctial} &  $a$ & $h$ & $k$ & $p$ &  $q$ & $\lambda$ \\
  Units & AU & -- & -- & -- & -- & rad \\
  Scaling & $a$ & $1$ &  $1 $ &  $1 $ & $1$ & $2 \pi$ \\
  \hline
  {\bf Attributable} &  $\alpha$ & $\delta$ & $\dot \alpha$ & $\dot \delta$ &  $r$ & $\dot r$ \\
  Units & rad & rad & rad/d & rad/d & AU &AU/d \\
  Scaling & $2 \pi $ & $ \pi $ & $n_\oplus$ & $n_\oplus$ & $1$ & $n_\oplus$ \\
  \hline
  \end{tabular}  }
\par
%\textsc{note:}
NOTE:  Here $r$ and $v$ are the heliocentric distance and
velocity, respectively, $d$ is one day. The angular rate $n_\oplus =
0.01720209895$ rad/day is approximately the mean motion of the Earth
and is numerically equivalent to the Gaussian gravitational constant,
$k = 0.01720209895~ ({\rm AU}^3/{\rm d}^2)^{-1/2}$. The
quantity $Z=2\pi q^{3/2}n_\oplus^{-1}(1-e)^{-1/2}$ is a characteristic
time for a large eccentricity orbit.
\label{tab:elems}
\end{table}