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Next: 2.4 Finding virtual impactors Up: 2. Computational methods Previous: 2.2 Scanning for close

2.3 Target plane analysis

Let us assume that the orbit with initial conditions Xi undergoes a close approach to the Earth at a time $t_i\gg t_0$; the Modified Target Plane (MTP) for that encounter is the plane perpendicular to the geocentric velocity at closest approach. Let this plane be ${\cal
T}$ and the points on it be $T\in {\cal T}$; then there is a differentiable map

\begin{displaymath}F_T:{\cal X} \longrightarrow {\cal T}

which can be explicitly computed by orbit propagation from epoch t0to a time near ti when the orbit crosses the MTP; the matrix DFTof the partial derivatives of FT is computable by means of the state transition matrix, the solution of the variational equation. If $\Delta X= X-X_i$ is a small change in the orbital elements with respect to the reference solution Xi, then the mapping on the target plane can be approximated by its differential

\begin{displaymath}\Delta T = DF_T \; \Delta X\ .

By this linear map, the ellipsoid of Eq. (1) is mapped onto the target plane as an elliptical disk with equations [Milani and Valsecchi 1999]

\begin{eqnarray*}\sigma^2 &\geq& \Delta T\cdot C_T \; \Delta T\\
\Gamma_T&=&DF_T\; \Gamma_X\; \left(DF_T\right)^T\ .

By computing the eigenvalues $\lambda_1>\lambda_2>0$ of $\Gamma_T$, we find that there is again a weak direction corresponding to the long axis of the MTP ellipse. As the time elapsed from t0increases, the confidence region becomes longer and longer in the phase space, and simultaneously thinner and thinner (this follows from Liouville's theorem, by which the phase space 6-dimensional volume is invariant). Thus, when a close approach takes place decades after the initial epoch t0, the two eigenvalues of $\Gamma_T$ have a very large ratio. An orbital solution that moves the MTP intersection along the weak direction results in a negligible change in the value of the target function, that is a negligible increase in the RMS of the observation residuals. On the contrary, a comparatively small change in the orthogonal direction, along the minor axis of the ellipse, would result in a significant increase of the residual RMS.

Thus the points on the MTP which can be reached with a negligible increase in the RMS are the points of the straight line which is the eigenspace of the larger eigenvalue $\lambda_1$; let Tmin be the point along that line that is closest to the center of the Earth, provided it is not too far along the line (with respect to $\sqrt{\lambda_1}$). The distance dmin of Tmin provides an estimate of the closest approach distance possible within the confidence ellipsoid; this estimate involves two approximations. First, the width $\sqrt{\lambda_2}$ of the ellipse is neglected; second, the nonlinear map FT is replaced by its linearization DFT. Both approximations can be removed, e.g., by the method of semilinear confidence boundaries described in [Milani and Valsecchi 1999]. However, for the purpose of finding VIs another approach is more efficient.

next up previous
Next: 2.4 Finding virtual impactors Up: 2. Computational methods Previous: 2.2 Scanning for close
Andrea Milani