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Date: 07 Oct 2006 19:45:10
From:
Subject: Friday the 13th is CERTAIN to be unlucky
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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
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Date: 08 Oct 2006 07:41:34
From:
Subject: Re: Friday the 13th is CERTAIN to be unlucky
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jsavard@ecn.ab.ca wrote:
> Henry Spencer wrote:
> > The most likely end result is for perihelion to be
> > pumped down far enough that it goes into the Sun, and the next likeliest
> > is for aphelion to be pumped up enough to encounter Jupiter, which almost
> > invariably expels the object from the solar system.
> That's interesting. I would have thought that *any* encounter with
> planet A always, invariably, leads to the body being perturbed from one
> A-crossing orbit to another A-crossing orbit - since when it gets far
> enough away from the planet, it stops being deflected by it in a major
> way.
Oh, silly me. Of course the new orbit still has to cross the orbit of
Jupiter. But it can be a hyperbola instead of an ellipse.
John Savard
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Date: 08 Oct 2006 04:32:17
From:
Subject: Re: Friday the 13th is CERTAIN to be unlucky
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Henry Spencer wrote:
> It's not. In general, the larger the denominator -- that is, the more
> years elapsed before the next encounter -- the more time there is for
> small errors to build up to a miss, and hence the narrower the keyhole is.
Ah, of course.
> Note also, if I recall correctly, that the 641-m keyhole is for a 2036
> *close encounter* of some kind, not a 2036 *impact* -- only a very small
> part of that keyhole steers the asteroid to an actual impact.
You're quite right.
> As above, it's not that simple. Moreover, near misses are much more
> likely than direct hits, and any near miss will change the orbit yet
> again. This process can easily pump the orbit up or down enough that the
> object encounters other planets.
True, but each near miss is another shot at a hit.
> The most likely end result is for perihelion to be
> pumped down far enough that it goes into the Sun, and the next likeliest
> is for aphelion to be pumped up enough to encounter Jupiter, which almost
> invariably expels the object from the solar system.
That's interesting. I would have thought that *any* encounter with
planet A always, invariably, leads to the body being perturbed from one
A-crossing orbit to another A-crossing orbit - since when it gets far
enough away from the planet, it stops being deflected by it in a major
way.
Thus, an encounter with Jupiter might lead to the body being, almost
invariably, perturbed into a cometary orbit with a period over 50,000
years, but as far as acquiring solar system escape velocity, that
should not happen. Of course, this may well be based on some simplistic
assumptions.
John Savard
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Date: 08 Oct 2006 18:52:32
From: Henry Spencer
Subject: Re: Friday the 13th is CERTAIN to be unlucky
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In article <1160307137.800175.4890@k70g2000cwa.googlegroups.com >,
<jsavard@ecn.ab.ca > wrote:
>> As above, it's not that simple. Moreover, near misses are much more
>> likely than direct hits, and any near miss will change the orbit yet
>> again. This process can easily pump the orbit up or down enough that the
>> object encounters other planets.
>
>True, but each near miss is another shot at a hit.
Correct, but *on average*, you get many near misses before a direct hit.
The "near miss" zone around Earth is far larger than Earth itself.
>> ...the next likeliest
>> is for aphelion to be pumped up enough to encounter Jupiter, which almost
>> invariably expels the object from the solar system.
>
>That's interesting. I would have thought that *any* encounter with
>planet A always, invariably, leads to the body being perturbed from one
>A-crossing orbit to another A-crossing orbit...
True, but as you note in your followup, the new orbit can be hyperbolic,
so it's not coming back. Jupiter is massive enough and moving fast enough
that the chances of that happening on a random encounter are substantial.
(New Horizons's Jupiter gravity assist never takes the spacecraft closer
than 2.2 million km, and nevertheless boosts it well past solar escape.)
And if a particular Jupiter encounter doesn't do that, then the new orbit
still intersects Jupiter's, and there'll probably be another encounter
fairly soon. Jupiter has such clout that, by the standards of the sort of
orbit evolution we're talking about, this happens very quickly.
--
spsystems.net is temporarily off the air;
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Date: 09 Oct 2006 15:00:53
From: Chris Jones
Subject: Re: Friday the 13th is CERTAIN to be unlucky
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henry@spsystems.net (Henry Spencer) writes:
> (New Horizons's Jupiter gravity assist never takes the spacecraft closer
> than 2.2 million km, and nevertheless boosts it well past solar escape.)
Depends on what you mean by "boosts it ... past" since NH has been
moving faster than solar escape velocity since its launcher stages
finished their job.
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Date: 08 Oct 2006 02:07:21
From: William Elliot
Subject: Re: Friday the 13th is CERTAIN to be unlucky
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On Sat, 7 2006 jsavard@ecn.ab.ca wrote:
> 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".
>
After 2029? No, that can't be. By the time it get's here to wipe US
(including UN etc) out, it'll be too late. US (and UN etc) will have
already wiped US (including UN etc) out. Damn, and to think if it wasn't
for Bush accelerating the date, I would have resting peacefully, missed
the whole event.
> 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.
>
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Date: 08 Oct 2006 06:49:22
From: Henry Spencer
Subject: Re: Friday the 13th is CERTAIN to be unlucky
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In article <1160275510.112627.206830@m73g2000cwd.googlegroups.com >,
<jsavard@ecn.ab.ca > wrote:
>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...
It's not. In general, the larger the denominator -- that is, the more
years elapsed before the next encounter -- the more time there is for
small errors to build up to a miss, and hence the narrower the keyhole is.
Note also, if I recall correctly, that the 641-m keyhole is for a 2036
*close encounter* of some kind, not a 2036 *impact* -- only a very small
part of that keyhole steers the asteroid to an actual impact.
Moreover, remember that the story is more complex than just hitting one of
those resonances. There are perturbations present, notably from other
planets. For many denominators, there may *be* no plausible numerator
that actually produces a close encounter, because perturbations may chance
to affect the close-encounter cases more than other cases. (The flip side
of that is that you can get focusing effects, which widen the keyholes for
specific cases.) This is probably why they seem concerned about only a
few specific resonances.
>Unless it is deflected, it therefore is certain to strike the Earth
>sometime in the next few thousand years.
As above, it's not that simple. Moreover, near misses are much more
likely than direct hits, and any near miss will change the orbit yet
again. This process can easily pump the orbit up or down enough that the
object encounters other planets. If memory serves, for a random case, the
long-term probability of impact on *any* of the inner planets is only
something like 20%. The most likely end result is for perihelion to be
pumped down far enough that it goes into the Sun, and the next likeliest
is for aphelion to be pumped up enough to encounter Jupiter, which almost
invariably expels the object from the solar system.
--
spsystems.net is temporarily off the air;
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Date: 09 Oct 2006 09:27:48
From: Hop David
Subject: Re: Friday the 13th is CERTAIN to be unlucky
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jsavard@ecn.ab.ca wrote:
>
> 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.
Of interest here is the "height" of a rational number.
"Every rational number has the form p / q, and we may assume that q is
always positive. Define the height of a rational number p / q as
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