Numerical simulation of orbits

Once experimental
measures of the position of the asteroid (or the comet) have been
made and its region of uncertainty has been
determined, the next step is to evaluate the orbit of the object.

Obviously, being the
real position of the asteroid inside the region unknown, this
orbit cannot be simply calculated. In fact, every single virtual
asteroid of the region of uncertainty will follow a different
orbit that needs to be calculated. Furthermore, these orbits
cannot be determined by simple calculations, being the solutions
of a two body problem (represented by the Sun and the virtual
asteroid) perturbed by the presence of the planets. These planets
are moving along orbits that can be considered perfectly
determined, on the period of time considered when simulating an
asteroid's motion. Therefore, the planets perturb the keplerian
orbit of the virtual asteroids in a way that can be solved by*
numerical simulation* with computers. On the basis of these
numerical simulations of orbits, it is possible to foresee
encounters with the Earth or other planets (click here to go
to the impact-previsions page).

How is an orbit simulated
with computers?

Characteristics of numerical
simulations

The simulation of an orbit explained
above, has an interesting characteristic: the error of the
determination can be somehow measured and controlled during the
calculus. In fact, this error is strictly bound to the choice of
the time step *dt*: the bigger is *dt*, the faster
is the calculus, but the bigger is the error on the orbit
determination.

Anyhow, such a simulation can be calculated
over short period of times for chaotic orbits. In fact, this
determination is normally limited over periods of 50 years (50
years is in fact the classical Lyapounov time for
asteroids).

What about the calculation time? How much time does a PC need to simulate the orbit of an asteroid? The simulation of the orbit has to be made for many virtual asteroids, covering the region of uncertainty. Normally, in the best hypothesis, this calculus has to be repeated for at least 1200 virtual asteroids. In this case, the orbit of an asteroid over a period of 50 years can be calculated with a "normal " PC (of about 500 MHz) in a period that ranges from 1 hour to one day, in the most complicated cases.