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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
.
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
.
Then we need to identify a
change
in the orbital elements (with respect to X_{j}),
such that

(2) 
To this purpose, we have to consider that DF_{T} is a composition of a
projection
onto a twodimensional subspace
,
followed by an invertible map
:
Here
is the 2dimensional subspace of
spanned
by the two rows of DF_{T}. We further define
to be the
4dimensional subspace of
orthogonal to
,
so
that
.
In simple terms, a change
in the orbital elements along
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
,
by means of an orthogonal
matrix V, in such
a way that
with
and
;
then
the normal matrix C_{X} in the new coordinate system is changed into
in this way the contributions to Eq. (1) from the two
components can be identified.
On the contrary, all changes along
map linearly into
nonzero changes on the target plane; thus there is an inverse map
B_{T}=A_{T}^{1}. The projection
is not invertible,
the preimage of each point being a 4dimensional space. However, we
can select one change in orbital elements
which
has a given displacement as image on the target plane
.
The portion of the preimage contained in the confidence
ellipsoid (1) can be described by the inequality
where
is fixed, uniquely determined by
the MTP displacement .
The above inequality, seen with
only as variable, defines a 4dimensional ellipsoid (with
interior)
;
we are going to select as representative of
the point L_{0}, which is the center of symmetry of this
ellipsoid. L_{0} can be computed in several different ways
[Milani 1999, Sec. 2], e.g., by separating the terms of different
degrees in ,
and is obtained as a function of
as

(3) 
This allows to select

(4) 
in such a way that (2) is satisfied. Then the orbit with
initial conditions
is propagated to the time
of
the close approach under study, and because of the nonlinear effects
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
,
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
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
of the MTP ellipse is
such that impact is possible, a VI exists, although it is not on the
LOV.
Next: 2.5 Preimage of the
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Previous: 2.3 Target plane analysis
Andrea Milani
20000621