Once we have established that the intercept of the MTP is an
explicitly computable differentiable function, the mathematical theory
is the same, whatever the astronomical interpretation of the function
Y=F(X). Therefore we can use the same theory developed in Paper I
for the computation of confidence boundaries in celestial coordinates
[right ascension, declination]. We will not
repeat the derivation, which can be found in Paper I, Section 3, but
only summarize hereafter the algorithmic path. We refer in the
following to the target space of the Y variables, with the
understanding that both the
and the
interpretations are possible.
In the linear approximation, the confidence ellipsoid
in
the space of orbital elements is mapped onto an elliptic disk (the
ellipse plus its inside) in the target space: let the disk
defined by
The easily computed elliptic disks
are good
approximations whenever the nonlinearity of the function F is
small. Unfortunately, this is not the case when the orbits have to be
propagated for a long time, and especially when close approaches take
place. A good compromise between computational efficiency and accurate
representation of the nonlinear effects is obtained by drawing in the
target space the semilinear confidence boundaries
,
defined as follows.
The boundary ellipse
of the confidence disk
is the image, by the linear map DF, of an ellipse
in the orbital elements space, which lies on the surface
of the ellipsoid
.
The semilinear confidence boundary
is by definition the nonlinear image
in
the target space of the ellipse
.
In the Y plane, the closed curve
is the boundary of
some subset
.
We use
as an approximation to
,
which is the set of all possible predictions on the
target space compatible with the observations.
To explicitly compute
a couple of additional steps are
required. The rows of the Jacobian matrix DF span a subspace in the
orbital elements space. Let us assume that an orthogonal coordinate
system is used in the X space, such that
Whatever the method of representation of the confidence region
,
in the end we can only explore it by computing a
finite number of orbits. To increase the level of resolution of this
representation, however, the dimensionality of the space being sampled
matters.
is 6-dimensional, and to increase the
resolution by a factor 10 the number of orbits grows by a factor
;
the semilinear confidence boundary
is a one
dimensional curve, and the resolving power increases linearly with the
number of orbits computed. In practice, even very complicated and
strongly nonlinear examples can be dealt with only a few ten to a few
hundred orbit propagations.