2.2 Semilinear confidence boundary

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 $[\alpha, \delta]=\;$[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 $Y=[\eta,\zeta]^T$ and the $Y=[\alpha, \delta]^T$ interpretations are possible.

In the linear approximation, the confidence ellipsoid $Z_X(\sigma)$ in the space of orbital elements is mapped onto an elliptic disk (the ellipse plus its inside) in the target space: let the disk $Z_{lin}(\sigma)$ defined by

$$\Delta Y^T \, C_Y\,\Delta Y\leq \sigma^2$$

be the image of $Z_X(\sigma)$, that is $\Delta X^T \, C_X\,\Delta X\leq \sigma^2$, by the linear map $\Delta Y= DF(X^*)\, \Delta X$. As is well known from the theory of Gaussian probability distributions [Jazwinski 1970], the covariance matrices of the variables X and Y are related by

$$C_Y^{-1}=\Gamma_Y=DF\; \Gamma_X\; DF^T \ .$$

The easily computed elliptic disks $Z_{lin}(\sigma)$ 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 $K_N(\sigma)$, defined as follows.

The boundary ellipse $K_{lin}(\sigma)$ of the confidence disk $Z_{lin}(\sigma)$ is the image, by the linear map DF, of an ellipse $K_X(\sigma)$ in the orbital elements space, which lies on the surface of the ellipsoid $Z_X(\sigma)$. The semilinear confidence boundary $K_N(\sigma)$ is by definition the nonlinear image $F(K_X(\sigma))$ in the target space of the ellipse $K_X(\sigma)$. In the Y plane, the closed curve $K_N(\sigma)$ is the boundary of some subset $Z_N(\sigma)$. We use $Z_N(\sigma)$ as an approximation to $F(Z_X(\sigma))$, which is the set of all possible predictions on the target space compatible with the observations.

To explicitly compute $K_N(\sigma)$ 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

$$\Delta X = \left[\begin{array}{c}{\Delta L}\\ {\Delta E}\end{array}\right] \ ;$$

where $\Delta E$ are two coordinates in the space spanned by the rows of DF, and $\Delta L$ four coordinates in the orthogonal space. Let the normal matrix C_X, in this coordinate system, decompose in this way:

$$C_X = \left[\begin{array}{cc}{C_{LL}}&{C_{LE}}\\ {C_{EL}}&{C_{EE}}\end{array}\right] \ .$$

The linear map DF can be described as the composition of the orthogonal projection $X\to E$ and of an invertible linear map $A: E \to Y$. Then $A^{-1}(K_{lin}(\sigma))$ is an ellipse in the E space, and the ellipse $K_X(\sigma)$ on the surface of the ellipsoid can be computed as its image by the map

$$\Delta E \to \left[\begin{array}{c}{\Delta E}\\ {\Delta L(\... ...\\ Delta L = -C_L^{-1}\; C_{LE}^{\phantom{1}} \; \Delta E \ .$$

Whatever the method of representation of the confidence region $F(Z_X(\sigma))$, 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. $Z_X(\sigma)$ is 6-dimensional, and to increase the resolution by a factor 10 the number of orbits grows by a factor $1\,000\,000$; the semilinear confidence boundary $K_N(\sigma)$ 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.