2.4 Finding virtual impactors

Around some times t_i many of the virtual asteroids X_i 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 X_j 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 X_j), such that

$$DF_T\; \Delta X = \Delta T\ .$$ (2)

To this purpose, we have to consider that DF_T 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 $A_T:{\cal E}_T \longrightarrow {\cal T}$:

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

Here ${\cal E}_T$ is the 2-dimensional subspace of ${\cal X}$ spanned by the two rows of DF_T. 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

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

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

$$V^T \; C_X \; V = \left[\begin{array}{cc}{C_{E_T}}&{C_{E_TL_T}}\\ {C_{L_TE_T}}&{C_{L_T}}\end{array}\right]\;$$

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 B_T=A_T^-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

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

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

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

This allows to select

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

$$F_T(X_j+\delta X)\neq F_T(X_j)+\Delta T$$

but the distance between the new MTP point and T_min 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 DF_T, find a new weak direction, a new minimum distance point T_min 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.