Friday, January 11, 2008

Why Asteroid Will Miss Mars

Last month, it was reported that a very small asteroid had a 1-in-75 chance of hitting Mars, which was very exciting.  It would be awesome to see the effects of such a collision.  Then the number was 1-in-25, which was even more exciting.  Now the number has dropped to 1-in-10,000, so it is very unlikely to happen.  How could the numbers change that much?

[see also next post, More on Why Asteroid Will Miss Mars]

Suppose it was your job to calculate the probability the asteroid would hit.  You would take the most accurate measurements of the asteroid, extrapolate its position, and come up with your best estimate of the path for the asteroid.  Now there would be some uncertainty in your estimate for the path.  Let's call s the distance of closest approach to Mars of your best guess for the path.  The plot of  probabilities is given by this bell curve (also called a Gaussian curve):

If you calculated that s=0, that the most likely path just grazes the surface of Mars, then all paths to the right of s=0 would hit Mars, and you'd say that the probability of hitting was 1/2 (half the area under the curve is to the right of s=0).  If you calculated that s=2.2 (which they did in December), then only paths more than 2.2 standard deviations from the most likely path would hit Mars, a chance of 75 to 1 (less than 2.1%).  And if your calculation shifted just a little, so that s=3.7 (the value now), then only the paths more than 3.7 standard deviations from the most likely path would hit Mars, a chance of 10,000 to 1.  It takes only a little shift out on a bell curve to make the probability plummet.   

And so a small refinement in measurements of the asteroid positions made the impact probability... crash.
[Notice that I did not put any units on s, because s is really distance/error-in-path-estimation, so that s=1 corresponds to whatever 1 standard deviation is in this case.  We don't need the actual distances in km because we are taking a ratio.]

[image from here, arrows and text added by me (feel free to use)]

[confidence: likely, my qualifications: informed]

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