3.1 The confidence ellipse

Let us assume that the vector of solve for parameters X is the 6-vector of orbital parameters, such as either keplerian elements, or equinoctal elements, or cartesian position and velocity, defining the initial condition for an asteroid/comet orbit at some epoch t₀. At some later time t₁ an observation is either performed or planned. The observation function is a map from the elements space to the celestial sphere, which we shall assume is parametrized by two coordinates (such as right ascension and declination):

$$Y(t_1)=R(X(t_1))\ \ \ ;\ \ \R\;: \; W \longrightarrow \Re^2 \ \ \ ;\ \ \W\subset \Re^6$$

where W is some open set (e.g., the Poincaré domain of the orbits with negative energy; however this choice is restrictive, as we shall see later, in Figures 11-12). X(t₁) is the state vector at time t₁, which is in turn a function of the initial conditions X=X(t₀) through the integral flow $X(t_1)=\Phi_{t_0}^{t_1}[X(t_0)]$.

The composition of the observation function with the integral flow

$$Y=F(X)=R\left( \Phi_{t_0}^{t_1}[X]\right)\ \ \ ;\ \ \F\; : \; W \longrightarrow \Re^2$$

is the prediction function; its Jacobian matrix can be computed by means of the state transition matrix $D\Phi_{t_0}^{t_1}$, by $DF= DR \; D\Phi_{t_0}^{t_1}$.

The prediction function F maps the orbital elements space onto the observations space, therefore it maps the confidence region $\Delta Q \leq \sigma^2$ into a confidence prediction region $Z(\sigma)$in the observation space. The linearised function DF maps the displacement (from the least squares solution X^*) in the orbital elements space $\Delta X = X -X^*$, into linearised deviations from the prediction Y^*=F(X^*):

$$\Delta Y = Y-Y^*= DF(X^*) \; \Delta X$$

and therefore maps the confidence ellipsoid

$$\Delta X\cdot C\,\Delta X\leq\sigma^2$$

onto the confidence ellipse $Z_{lin}(\sigma)$, which is the inside of an ellipse in the observations coordinate plane:

$$\Delta Y \cdot \,C_Y\; \Delta Y \leq \sigma^2\;.$$

The matrix C_Y is the normal matrix for the observations Y (at a given time t₁), and the inverse $\Gamma_Y=C_Y^{-1}$ is the corresponding covariance matrix. By a standard result from the theory of multivariate Gaussian distribution [Jazwinski 1970], the covariance matrix is transformed by

$$\Gamma_Y=DF\; \Gamma\; DF^T \ .$$

This linear prediction formalism is used as a matter of routine in astrodynamics, and it has been proposed to use it systematically for asteroid astrometry [Muinonen and Bowell 1993]. In the latter case, however, the prediction function F is nonlinear, and there is no guarantee that the confidence ellipse is a good approximation of the confidence prediction region; as we shall see later, this is indeed not the case when a poorly determined orbit is used to predict the observations at a time t₁ very far from the last observation used in the orbit determination.