The first step is to solve for the orbit of 1997 XF11 by
using only the observations between the discovery, in December 1997,
and March 4, 1998; these were the observations available on March 11,
1998. Before applying the confidence boundary methods discussed in
this paper we need to check the hypothesis, introduced at the
beginning of Section 2, that the least squares fit results in a well
determined solution, with a confidence region small enough to allow
for the use of the confidence ellipsoid as a good approximation to the
confidence region. After solving for the orbit with 98 observations,
from 12 different observatories, the RMS of the residuals is 0.61arc seconds; after discarding 5 outliers in 2 iterations of removal at
the 3 RMS level, the solution with 93 observations has an RMS of
0.50 arc seconds. The conditioning number of the covariance matrix
is
(in non singular equinoctal elements), with a largest
eigenvalue
,
thus the confidence
ellipsoid is quite small and is a good approximation.
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The residuals after this second fit are plotted in Figure 2; the fact that the residuals from each individual observatory are not Gaussian is quite apparent. Nevertheless the errors of the different observatories do not appear to have a common bias/trend, as it reasonable since an asteroid observed near the Earth is not seen always in the same region of the sky, were regional star catalogue errors are expected to be systematic. Adopting the method described in Paper I, Section 2.2, and universally used in the astrodynamics/satellite geodesy applications, we select a weighting scheme such that the non discarded observations are very unlikely to have errors above 3 times the normalized scale (which is the inverse of the weight).
By inspection of the Figure 2, it is clear that a
reasonably prudent choice is to use a scale of 1 arc second, that is
equivalent to using a safety factor about 2 with respect to a naive
application of the Gaussian statistics to non Gaussian errors. Thus
all the normal and covariance matrices used in the following are
computed with weight unity (in arc seconds), and incorporate this
safety factor. We have to acknowledge that this choice of weighting is
not an exact science; e.g. an argument could be proposed to use a
safety factor
;
as we
will see, the results of Figure 7 might be used to argue
that a somewhat higher safety factor would give a better
agreement. The possible values of the weighting scale are anyway not
very different, and the discussion of the following results would not
be significantly affected, apart from the discussion of
Figure 7.
Then we compute the nominal orbit, solution of the least squares fit,
until the year 2028, and find the closest approach position and
velocity, which is at a distance of
AU from the center of
the Earth on October 26, 2028; the planetocentric velocity is 14.6km/s, and the unit vector along this velocity is used to define the
Modified Target Plane. By using the multiple solution formalism
(described in Paper I, section 5) it is possible to find solutions
compatible with the observations passing at about 3 Earth radii
from the surface of the Earth. Note that, the close approach being
very deep, some caution is needed to compute the orbit and to find the
closest approach point with the required accuracy.
For numerical integration we have used the Runge-Kutta-Radau scheme [Everhart 1985], with adaptive stepsize change when stronger perturbations are encountered. This method performs well even in presence of close approaches, but for the deepest ones it sometimes fails to achieve convergence in the standard 2 iterations. We have modified the algorithm allowing for a variable number of iterations.
The dynamic model for a refined close approach analysis must be very
accurate; all the results of this paper about 1997 XF11 are
based upon a model including the general relativistic perturbations
from the Sun, the gravitational perturbations from the 3 largest
asteroids, but not the gravitational perturbations due to the Moon. We
have checked that the inclusion of the gravitational effects of the
Moon would not change significantly the results: e.g. the nominal
closest approach distance for the 1997-98 orbit would be AU, only
km less, due to the displacement of the center of
the Earth from the center of mass of the Earth-Moon system.
The algorithm to determine the closest approach point is based upon an
iterative regula falsi method to find the zero of the radial
derivative (with respect to the approached body); the regula falsi is
activated whenever a change in sign of the radial derivative is
detected, and the following substeps are computed independently from
the prosecution of the orbit integration, with a different algorithm,
Runge-Kutta-Gauss [Butcher 1987], which is known to be especially
stable in hyperbolic orbits. The MTP crossing is detected with the
same method, applied to the derivative of the
coordinate (see
Section 2).