Previsions of impacts

Once an asteroid (or a comet) is discovered, it is impossible to determine precisely its future orbit over long periods of time. But often, what we hear are statistics on its probability of colliding with the Earth. How are such previsions made ?

Seeing orbits from the Earth's point of view
As already said, errors of measure affect the determination of the real position of the discovered object. The orbit in this case can't be perfectly determined; what is known is a region in space called region of uncertainty. Each point inside this region is a possible position of the asteroid and is therefore called virtual asteroid (click here to know more about this statistical vision of orbits). The nominal orbit is the most central of this virtual asteroids. As time passes by, all the virtual asteroids will move along different and divergent orbits, with different speed, and this region will consequently move in space resulting deformed.
To make a prevision of collision with a planet, for example Earth, the simplest thing is to consider the relative motion of the asteroid as it could be seen by an observer on Earth (this can be done because the motion of all the planets is known with a great precision, and so any motion can be expressed as a relative one).

Definition of the target plane and of the virtual impactor
Let's consider the nominal orbit and the point where the asteroid comes closer to Earth. This is called the point of maximum approach (or point of close encounter with Earth). The period of the close encounter between the planet and the asteroid is very brief compared to the overall orbital period and so, during the close encounter, the orbit can be approximated with a straight one. In this way it is possible to define the target plane, as the plane orthogonal to the trajectory of the object, containing both the point of maximum approach and the position of Earth.
Since all virtual asteroids will all have orbits that can be approximated as parallel and straight, all of them will intersecate the target plane in a different point of maximum approach. In this way, on the target plane is formed a 2 dimensional figure that represents the area where the asteroid could make its maximum approach with the planet. In this way it is possible to make a simple prevision of impacts, verifying if the planet lies inside of this area or not. If the collision with a certain a virtual asteroid is possible, this asteroid will be called virtual impactor.

In this image is represented the region of uncertainty of an asteroid approaching Earth .The target plane is also represented with the projection of this region. In this case the planet lies outside the region and so an encounter is impossible.

In the case the planet lies inside this area, the impact is possible and a first determination of the probability of impact is given by the ratio between the area formed by the planet's cross section and the region of the target plane:

Probability of impact = Cross section of the planet /Area on the target plane

Follow up and refinement of a probability

It is clear that the dimension of this area formed on the target plane depends on the uncertainty of the determination of the asteroid's orbit: if we make ulterior measurements of the position of the NEO (and therefore we refine the orbit determination), the area of the intersection between the target plane and the region of uncertainty will diminish. In this case the precision of the impact determination will grow.
This will lead to a "strange" phenomenon (explained in the drawing below): after having found a virtual impactor, it is in fact possible for the probability of its impact to grow up at first (after a first refinement of the determination of its position) and then fall back to zero (after a second measurement)!.

In this drawing, a virtual impactor is represented on the target plane of a planet. From a first determination of the asteroid position, we get an area A1 (which is the intersection between the target plane and the region of uncertainty). In this case, the asteroid is a virtual impactor with a probability that is inversely proportional to area A1. To refine the determination of the orbit we will make a second determination of the orbit of the asteroid, getting the area A2. The are A2 is smaller than A1 because the precision of the determination has augmented. In the case of the picture, the impact can't be excluded from this second measurement: on the contrary, this probability has grown (since this probability is inversely proportional to the area. If the third time, we get the area A3, we can exclude the possibility of impact, since the planet falls outside this area.