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2.4 Finding virtual impactors

Around some times ti many of the virtual asteroids Xi experience a close approach to the Earth; this we call a virtual shower. A shower can be decomposed into separate returns, which are continuous strings of solutions having the same close approach; they are represented by sequences of n consecutive solutions $X_i,\;
i=k,k+1, \ldots ,k+(n-1)$. It is often the case that the shower at a given time contains several different returns [Milani et al. 1999, Table 1]. For each return we want to identify the closest possible approach in order to decide if an impact is possible. Starting from the solution Xj that has the closest approach among the ones of the return, we apply corrections to it to push the approach towards a closer one; the method is a variant of Newton's method, also called differential corrections in the context of orbit determination. It is similar, but not identical, to one method used in [Muinonen 1999].

We begin by defining the correction we would like to apply on the MTP, which is $\Delta T= T_{min}-F_T(X_j)$. Then we need to identify a change $\Delta X$ in the orbital elements (with respect to Xj), such that

 \begin{displaymath}DF_T\; \Delta X = \Delta T\ .
\end{displaymath} (2)

To this purpose, we have to consider that DFT is a composition of a projection $\Pi_{{\cal E}_T}$ onto a two-dimensional subspace ${\cal
E}_T\in {\cal X}$, followed by an invertible map $A_T:{\cal E}_T
\longrightarrow {\cal T}$:

\begin{displaymath}DF_T= A_T\circ \Pi_{{\cal E}_T}\ .

Here ${\cal E}_T$ is the 2-dimensional subspace of ${\cal X}$ spanned by the two rows of DFT. We further define ${\cal L}_T$ to be the 4-dimensional subspace of ${\cal X}$ orthogonal to ${\cal E}_T$, so that ${\cal E_T}\times {\cal L}_T={\cal X}$. In simple terms, a change in the orbital elements along ${\cal L}_T$ does not change the position on the target plane, as far as the linear approximation is applicable. It is possible to perform a change of coordinates in ${\cal X}$, by means of an orthogonal $6\times 6$ matrix V, in such a way that

\begin{displaymath}\Delta X=V \left[\begin{array}{c}{\Delta E_T}\\
{\Delta L_T}\end{array}\right]

with $\Delta E_T \in {\cal E_T}$ and $\Delta L_T\in {\cal L}_T$; then the normal matrix CX in the new coordinate system is changed into

\begin{displaymath}V^T \; C_X \; V = \left[\begin{array}{cc}{C_{E_T}}&{C_{E_TL_T}}\\

in this way the contributions to Eq. (1) from the two components can be identified.

On the contrary, all changes along ${\cal E}_T$ map linearly into nonzero changes on the target plane; thus there is an inverse map BT=AT-1. The projection $\Pi_{{\cal E}_T}$ is not invertible, the preimage of each point being a 4-dimensional space. However, we can select one change in orbital elements $\Delta X\in {\cal X}$ which has a given displacement as image on the target plane $\Delta T\in
{\cal T}$. The portion of the preimage contained in the confidence ellipsoid (1) can be described by the inequality

\begin{eqnarray*}\sigma^2 &\geq& \Delta X \cdot C_X \;\Delta X\\
&=& \Delta L_...
&=& (\Delta L_T-L_0)\cdot C_{L_T}\; (\Delta L_T-L_0) +const

where $\Delta E_T= B_T\; \Delta T$ is fixed, uniquely determined by the MTP displacement $\Delta T$. The above inequality, seen with $\Delta L_T$ only as variable, defines a 4-dimensional ellipsoid (with interior) $Z(\Delta T)$; we are going to select as representative of $Z(\Delta T)$ the point L0, which is the center of symmetry of this ellipsoid. L0 can be computed in several different ways [Milani 1999, Sec. 2], e.g., by separating the terms of different degrees in $\Delta L$, and is obtained as a function of $\Delta E_T$as

 \begin{displaymath}L_0=-C_{L_T}^{-1}\;C_{L_TE_T}\;\Delta E_T\ .
\end{displaymath} (3)

This allows to select

 \begin{displaymath}\Delta X= V\;\left[\begin{array}{c}{\Delta E_T}\\
...}\;C_{L_TE_T}}\end{array}\right]\;B_T\;\Delta T= H_T(\Delta T)
\end{displaymath} (4)

in such a way that (2) is satisfied. Then the orbit with initial conditions $X_j+\Delta X$ is propagated to the time $\simeq t_i$ of the close approach under study, and because of the nonlinear effects

\begin{displaymath}F_T(X_j+\delta X)\neq F_T(X_j)+\Delta T

but the distance between the new MTP point and Tmin is now smaller, and the procedure can be iterated. This means we reset the `nominal' orbit to $X_j+\Delta X$, we compute its MTP and its partial derivatives matrix DFT, find a new weak direction, a new minimum distance point Tmin along the weak direction, then select a new $\Delta X$ satisfying the new version of Eq. (2), and so on until convergence. If the close approach distance at convergence is less than one Earth radius, a virtual impactor has been found. If the close approach distance at convergence of Newton's method is above one Earth radius, but the width $\sqrt{\lambda_2}$ of the MTP ellipse is such that impact is possible, a VI exists, although it is not on the LOV.

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Next: 2.5 Preimage of the Up: 2. Computational methods Previous: 2.3 Target plane analysis
Andrea Milani