The data the JPL had on the asteroid Apophis back on July 3, 2005 led

to the possibility that the asteroid might, in the near encounter on

Friday, April 13, 2029, go through one of several "keyholes".

One keyhole would have changed its orbital period to 5/4 of a year,

leading to it having a close encounter with the earth in 2034.

Another would have changed its period to 6/5 of a year, making the date

of its return one year later, in 2035. Still another would have made

the period 7/6 of a year, so it would return in 2036, and then another

would make it 8/7 of a year, so it would return in 2037.

And then, yet another one would have made the period 17/15 of a year,

for a return in 2046, and another would have made it 19/17 of a year,

for a return in 2048.

Improved data available on June 20, 2006 left only the 2036 return a

strong contender, with the keyhole for a 2037 return just outside the

3-standard-deviation error ellipse.

What's wrong with this picture?

For one thing, they seem to have left out other fractions. What about

9/8 of a year? Or 11/9, 13/11, or 15/13 of a year?

Just between 7/6 and 8/7, though, there are other possible resonances,

or, at least, rational numbers, besides 15/13, which could lead to the

asteroid hitting the Earth in 2044.

22/19 and 23/20. 29/25 and 31/27. And on and on.

The size of the "keyhole" for a 2036 impact is only 641 meters, less

than 1/5000 of the length of the error ellipse.

For a given small, but finite value epsilon, what proportion of the

points on the real number line are within epsilon of a rational number?

As any mathematician will tell you, all of them are.

However, that implies there is no upper bound to the denominators on

the fractions involved. That Apophis might be deflected into an orbit

which would cause it to collide with the Earth a few billion years

_after_ the Sun left the main sequence would not be a concern.

If, however, the "window" for, say, the 151/131 resonance, leading to a

collision with the Earth in the year 2180, were also around 600 meters

or so in width, which may not be the case, then, with room for about

5000 keyholes between the 7/6 resonance and the 8/7 resonance, the

denominators don't have to grow to billions or even millions.

Unless it is deflected, it therefore is certain to strike the Earth

sometime in the next few thousand years. And, furthermore, even if it

is deflected, so as to avoid an imminent collision with the Earth, in

the relatively simple ways in which it is currently envisaged to

deflect it, it would remain in an Earth-crossing orbit which would

eventually cause it to strike the Earth.

Suitably altering an encounter with the Earth so as to deflect an

asteroid into an orbit which passes close by either Venus or Mars would

be likely to lead to the asteroid ceasing to be an Earth-crossing

asteroid. Still, just delaying an imminent impact by centuries is both

worthwhile, and a good way to get some practice.

John Savard