Say an asteroid passes in a distance x from Earth, the radius of which we will call r. We now can ask how likely it is that an object that hits a disk of radius r+x also hits a disk of radius r provided that it any point on the larger disk is hit with equal likelihood.
The likelihood then is p = A(r)/A(r+x) where A is the area of a disk of the given radius, In other words p = πr²/π(r+x)² = 1/(1+x/r)². If we now define ξ=x/r (which is the distance in units of Earth's radius we get a quite simple formula: p = 1/(1+ξ)².
Using ξ is advantageous as it is a value you actually find in tables. Let’s try a couple of values; LD means Lunar distance and is the distance in terms of the average distance between Earth and Moon:
| Distance in | ξ | p in % | |
|---|---|---|---|
| LD | km | ||
| 1.65614 | 636619.77 | 100 | 0.098 |
| 0.49684 | 190985.93 | 30 | 0.104 |
| 0.16561 | 63661.98 | 10 | 0.826 |
| 0.04968 | 19098.59 | 3 | 6.250 |
| 0.01656 | 6366.20 | 1 | 25.000 |
| 0.00497 | 1909.86 | 0.3 | 59.172 |
| 0.00166 | 636.62 | 0.1 | 82.645 |
| 0.00050 | 190.99 | 0.03 | 94.260 |
| 0.00017 | 63.66 | 0.01 | 98.030 |
Please note that the closer an encounter is the less meaningless this rough estimate becomes as the asteroid by no means randomly hits the disk of radius r+x but follows a clearly determined path.
