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.