The closest approach point on the nominal orbit with initial
conditions X* (solution of the least squares fit) belongs by
definition to the MTP. The closest approach points with nearby initial
conditions X, however, do not in general lie on the MTP, although
they are nearby (for small ). Thus the intercept with the
MTP in general is not at the minimum distance; the fact that some
confidence boundary does not intersect the disk with radius r=RP(where RP is the radius of the approached planet) does not
guarantee that there is no impact in a point outside the MTP.
Although the set of possible close approach points is not a linear
subspace of the phase space, it can be represented on the MTP. We have
chosen a representation such that the closest approach distance is the
planetocentric distance of the representative point on the MTP: if
are planetocentric polar coordinates on the
MTP, and rCL is the closest approach distance on the same orbit,
we use the point with polar coordinates
.
In
practice, the MTP intercept and the close approach manifold
representative point are very close, with the exception of very deep
and/or very slow encounters.
![]() |
Figure 1 shows the target plane analysis outlined in this
Section for the lost asteroid 1978 CA, which can have had a
close approach to the Earth on September 3, 1997. In the plot we have
drawn the intercept on the MTP of the nominal orbit (cross), the
linear confidence ellipse (dotted line), the semilinear confidence
boundary on the MTP, and the representation of the close approach
points of the same orbits, all for .
For small enough
displacements from the nominal orbit the linear approximation is good
enough; for larger displacements the differences between linear and
semilinear approximation, and between the MTP and the close approach
manifold, are visible. The maximum deviation of the semilinear
boundary on the MTP from the linear ellipse is as large as the
Earth-Moon distance.
In this case the semilinear approximations show only moderate deviations from the linear approximation, essentially because 1978 CA could not have a very close approach (the MOID is 0.15AU). Nevertheless, this lost asteroid could have passed the Earth at a quarter of the distance predicted using the nominal orbit. Any attempt to predict in which direction in the sky astronomers should have looked to observe this asteroid when it was apparently brightest would have failed, because there are solutions compatible with the 1978 observations arriving from opposite directions. Indeed this asteroid has not been recovered, although it could have been quite bright.