Asteroids Missing Earth: What Means “Close Call”?

Every now and then you read about asteroids passing Earth in a certain distance but nobody gives you a feeling of how close such an encounter is. Allow me to fill in this gap.

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.