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.