A statistical vision of orbits

Our knowledge of orbits cannot be perfect: as all physical quantities measured, the determination of the position of a celestial body in the sky (and consequently the determination of its orbit) is affected by an error. In the case of the orbits of planets, these little errors don't lead to enormous differences in the future, and the parameters that describe their orbits are almost constant, since the planets' orbits are stable. In the case of minor bodies such as asteroids or comets, this error can lead in the future to very different evolutions of the orbit, since these orbits are chaotic.
In the case of minor bodies, a new statistical vision of orbit has therefore been developed, replacing the classical concept of orbit by a region of space, where the body is like to be, called region of uncertainty.

How errors affect the position's determination
The position of an asteroid can't be perfectly known from a single measure. First because, from a single optical measure (made with photographs or CCD) it isn't possible to determine the distance between the observer and the object. Second because of the errors of measure that affect the determination of the real position of the discovered object. For these two reasons the position of the object isn't perfectly known. What we will know from a single measure is that the object lies inside a cone with an angular dimension given by the error that affects the position's determination (see the animation) .

Definition of the region of uncertainty

This region of space where the object (for example, an asteroid) is like to be, is called region of uncertainty. Each point inside this region is a possible position of the asteroid and is therefore called virtual asteroid . To make this region smaller, we need to determine better the position of the NEO making refinements of the position determination or other different kinds of measures .
The nominal position is the most central of the virtual asteroids. In this way, the region can be seen as the most probable postion of the asteroid (the nominal position) affected by an error that gives an information on the dimension of the region. This vision of the region of uncertainty is however quite imprecise, since it doesn't give information about the shape of the region, that can be very irregular.


On the left, a representation of the region of uncertainty and the nominal position of the asteroid. Initially this region is an ellipsoid. After some time, the region will result deformed (on the right.)

Deformation of the region of uncertainty

What happerns when time passes by? All the virtual asteroids inside the region of uncertainty will move with a different speed, following different orbits (click here to know why). For this reason, the region of uncertinty will move in space and its initial shape will result deformed. In other words, the region of uncertainty will evolve in time, including all orbits that are compatible with the initial measurements.
To know the exact shape and position of the region, we could follow all these orbits with numerical simulations. But in reality, most of the orbits will diverge, being chaotic, making it impossible to foresee all possible positions of the object over long periods of time. What is actually done is a successive refinements of the orbit determination with a follow up of the object or a numerical prevision of impact's probability over short periods of time.