Tuesday, March 6, 2012

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 inξp in %
1.65614636619.77100 0.098
0.49684190985.93 30 0.104
0.16561 63661.98 10 0.826
0.04968 19098.59  3 6.250
0.01656  6366.20  125.000
0.00497  1909.86  0.359.172
0.00166   636.62  0.182.645
0.00050   190.99  0.0394.260
0.00017    63.66  0.0198.030

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.

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