2.6 Projection from element space onto the sky plane

Once the VIR, with safety margins, X_R has been computed, or at least approximated in the ${\cal X}$ space, we need to be able to compute the image of it onto the sky at any given time in which the observations might take place. Given the approximate representation of X_R as a product, we can separately map the two factors, which can be done in a very efficient way. One observation at time t is mathematically described by a smooth observation function $F_S\; : \; {\cal X} \longrightarrow {\cal S}$, where ${\cal S}$ is the celestial sphere. A point on the celestial sphere is represented by two angular coordinates $S=(\alpha, \delta)$. At time t the virtual impactor has an ephemeris $S_0\in {\cal S}$.

We first map the four corners R_i of R to the four initial conditions $H_T(R_i)=X_{R_i} \in {\cal X}, i=1, \ldots ,4$ according to (4). For these four initial conditions, we compute four displacements with respect to the VI ephemerides $\Delta S_i=F_S(X_{R_i})-S_0$. They mark on the sky a quadrilateral F_s(H_T(R)) which is almost flattened to a segment in the cases of a low probability impact at a very late epoch. (See in the Figures of the next section.)

The space of changes in orbital elements $\Delta X$ is a product ${\cal X}={\cal E}_T\times {\cal L}_T$; we need to map the 4-dimensional ellipsoid $Z(\Delta T_0)\subset {\cal L}_T$ onto the sky plane. To do this we use the same formalism used in [Milani 1999, Sec. 3] to map the confidence ellipsoid onto the sky: the same formulas are used, but they refer to a 4-dimensional space rather than to a 6-dimensional space.

To this purpose, let us consider the differential (linearized map) DF_S of F_S, computed at the VI initial conditions. If we restrict the map F_S to the 4-dimensional subspace ${\cal L}_T$, we can describe DF_S by means of a $2\times 4$ matrix of partial derivatives, with rows $\partial \alpha/\partial L_T$ and $\partial\delta/\partial L_T$. Let ${\cal E}_S$ be the 2-dimensional subspace of ${\cal L}_T$ spanned by these two gradients, and ${\cal L}_S$ the orthogonal space (also 2-dimensional), so that ${\cal L}_T={\cal E}_S\times {\cal L}_S$.

Then DF_S restricted to ${\cal L}_T$ is a composition of an orthogonal projection $\Pi_{{\cal E}_S}$ onto ${\cal E}_S$, followed by an invertible map $A_S:{\cal E}_S \longrightarrow {\cal S}$:

$$DF_S= A_S\circ \Pi_{{\cal E}_S}\ .$$

It is possible to perform a change of coordinates in ${\cal L}_T$, by means of an orthogonal $4\times 4$ matrix V₄, in such a way that

$$\Delta L_T=V_4 \left[\begin{array}{c}{\Delta E_S}\\ {\Delta L_S}\end{array}\right]$$

with $\Delta E_S \in {\cal E_S}$ and $\Delta L_S\in {\cal L}_S$; then the normal matrix C_L in the new coordinate system is changed into

$$V_4^T \; C_L \; V_4 = \left[\begin{array}{cc}{C_{E_S}}&{C_{E_SL_S}}\\ {C_{L_SE_S}}&{C_{L_S}}\end{array}\right]\ .$$

This allows to select a representative in the ${\cal L}_T$ space for every point on the sky near the ephemerides of the virtual impactor:

$$\Delta L_T= V_4\;\left[\begin{array}{c}{\Delta E_S}\\ {-C_{... ...\;C_{L_SE_S}}\end{array}\right]\;B_S\;\Delta S= H_S(\Delta S)$$ (5)

where B_S=A_S^-1. Equation (5) is fundamentally the same as Eq. (4), but for the dimension of the matrices; it can be used by defining the sky ellipse $K\subset \cal S$image of $Z(\Delta T_0)$ by the linearized map DF_S, by computing the pullback H_S(K), sampled by a discrete set of points in the ${\cal L}_T$ space (hence in the ${\cal X}$ space, by adding the VI value of $\Delta E_T$); the image of these points is the semilinear confidence boundary, which is a good approximation of $F_S(Z(\Delta T_0))$.

The last step in the procedure is to represent the image, on the sky, of the Cartesian product $H_T(R)\times Z(\Delta T_0)$ by the Cartesian product of the displacements on the sky, that is

$$F_S\left( H_T(R)\times Z(\Delta T_0)\right) \simeq F_S(H_t(R))\times F_S(Z(\Delta T_0))\ .$$

The left hand side is, by definition, the skyprint of the VI, and the right hand side is computable in practice by means of a moderate number of observations predictions, since only one-dimensional subsets of ${\cal X}$ have to be computed. As discussed in [Milani 1999], conceptually we need anyway to select a finite sample of some region of the ${\cal X}$ space of initial conditions, but in practice to sample a one dimensional line is a small computational load, to sample a 6-dimensional space is computationally very expensive.

Please note that these computations contain a number of approximations which are accurate only provided the skyprint is small. If this is the case, as in the Figures of the next section, then the product structure is clearly visible, since F_S(H_t(R)) is almost a segment and $F_S(Z(\Delta T_0))$ is almost an ellipse, thus the skyprint looks very much like the drawing of an ordinary 3-dimensional cylinder. If the skyprint was large, the linear, semilinear, and Cartesian product approximations used in this section would break down.

On the other hand, if the skyprint was big, to the point of being not well approximated with the linear and semilinear formalisms, then the negative observation campaign would not be worthwhile. We do not have a simple way to predict in which cases the skyprint will be small, in which cases it will be too large to be usable; this depends upon the size of the confidence region (in turn depending upon the number of observations and the length of the observed arc), the date of the possible impact, its probability, the date and the geometry of the observation. Thus we need to apply the computational procedure outlined above case by case; to this purpose we have developed rather efficient software.