Probability of a single impact

The probability that a NEO of a certain dimension will collide with Earth is fundamental to calculate the frequency of impacts. This probability can be calculated in several ways, from experimental observations or from theoretical considerations (both on the observation of the Moon's craters distribution and on the simulation of impact's frequency obtained from computers). Here is presented a very simple estimation based on the concept of dynamical classes of objects. From this probability we will be able to calculate the number of impacts per years for a NEO of a given dimension.

Determination of the probability of an impact
A very intuitive theoretical estimation of this probability, based on the concept of dynamical classes, has been made by the astronomer Ernst Opik in the 50s'.
The orbit of a NEO can only be determined in a statistical way, as a region of uncertainty (click here to know more about how orbits are studied statistically). So it is impossible to calculate for a single NEO the exact evolution of its orbit. Anyway, the characteristics of this orbit (such as eccentricity, and semimajor axis) are almost constant and vary only over long periods of time. In this way, all the asteroids with very similar orbital elements belong to a same dynamical class of asteroids and represent a real statistical sample. Determining a mean evolution of these orbits, it is possible to calculate a probability of impact over long periods of time.

There are two conditions the orbit of an asteroid has to respect to make an impact with a planet possible (click here to see the representation of the orbit of an asteroid, and the definitions of mutual nodes, nodal distance, etc) :

• the two orbits (the Earth's and the asteroid's orbit) must cross, as in the case of NEOs ; this can happen even if the two orbits are not on the same plane ;
• the nodal distance between the Earth's and the asteroid's orbits must be smaller than the Earth dimension. This will happen with a probability named P1;
• Earth and asteroid have to pass contemporaneously by the mutual node (the intersection between the two orbits) . This will happen with a probability called P2

The probability that the last two hypothesis are verified at the same time, has been estimated for a single asteroid with mean values of orbital elements, obtaining :

This value has been determined in very simple, approximate conditions. A first correction of this calculus must be made considering that the real cross section of the Earth is not the geometrical one but it is bigger, making chances of collision higher, due to the phenomena of gravitational focusing. (Click here to know more about gravitational focusing). Considering this phenomenon the probability can be estimated as:

From probability to number of impacts

Once this probability of a single impact has been estimated, it is possible to determine the mean number of impacts per year multiplying this single probability by the number of NEOs of a certain dimension that surround Earth .
We can therefore obtain an estimation of the frequency of impacts with bodies of a certain diameter:

 Diameter d Frequency of the impact d>10 Km every 50 millions years 1Km1Km every 5.000 years 30m

This table can be easily converted to the more notorious representation of the frequency versus the energy of the impact, since there is a direct correspondence between the dimension of the impactor and the energy freed in the impact event.