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(meteorobs) Fireball orbits (From CCNet)
After our discussions on the accuracy of pre-encounter circumstances
determined by visual observations of fireballs, this very lucid discussion
of the uncertainties of determining orbits of small solar system objects
dropped into my mailbox...
(1) WHY IMPACT PREDICTIONS CAN CHANGE DAY BY DAY
>From Andrea Milani and Steve Chesley <chesley@dm.unipidot it>
Dear Benny,
In reference to yesterdays discussion of impact probabilities (CCNet
Digest July 1, 1999), we update on NEODyS
<http://newton.dm.unipidot it/neodys> the orbits of all near-Earth
asteroids as soon as new observations are published by the Minor Planet
Center; with the new orbit, we automatically update the list of close
approaches to the Earth from 1975 to 2075. As an example, you can see
that for 1999 AN10 the new observations by Gladman and Nicholson (from
Mt. Palomar) have been already included in the fit, and the new nominal
(best fit) solution has now a close approach in 2027 at about 250,000
km. However, the nominal solution has no special significance.
When a new orbit becomes available we also recompute the impact
possibilities until 2050 and estimate the probabilities of such events;
this requires many hours of computer time, and is not yet automated,
thus the results normally come in one or more days later. As you know,
1999 AN10 is the "most wanted" asteroid in our impact risk list, being
the only km-size object for which we know an impact is possible,
although unlikely. With yesterday's new data, the least unlikely impact
could occur in August 2044; our order of magnitude estimate of the
probability of such an event is 7 parts in a million.
We need to stress once more that there is no such thing as a unique
impact probability, but the exact value depends upon the statistical
model used to describe the observation errors. In simple terms, if you
know which observation errors are more likely than the others, you also
know which orbits are more likely than the others. Our simple
computations, published on the NEODyS impact risk page at
<http://newton.dm.unipidot it/neodys/risk.html>, ignore this effect, thus
somewhat different values could be obtained by using, e.g., a gaussian
model, but the orders of magnitude would normally not change.
We also would like to stress that the existence of an impact solution,
and the impact probability estimate, will generally not change
dramatically as a result of a single observation, because they result
from the processing of all observations at once. A dramatic change is
only possible if the observation is very far from the others (e.g., at
a different apparition) or the impact is already close to being
excluded. For example, with the addition of the La Palma and Klet
observations our (uniform density) impact probability estimate for
August 2044 changed from 3 to 5 parts in a million. The further
addition of the Palomar observations increases the estimate to 7 parts
in a million.
We realize there are two points of this discussion which may not be
clear to many of your readers, and which are the source of some
discomfort; namely why the impact probabilities can change day by day,
and why the nominal solution is not important for this kind of
computations. This we will try to explain, for the benefit of those who
do not have the know-how to do these computations by themselves, but
wish to understand the process better.
One intuitive way of expressing the uncertainty of the orbit of an
asteroid is to use a finite probabilistic model. You should think to a
large number of orbits, maybe a million, all compatible with the
observations; each of these virtual asteroids has a small uncertainty,
and one of them is the real one, but we do not know which one. In a
simple model, you can think that all of them have the same probability
of being the real one; with this model, if five of the virtual
asteroids have an impact, then the probability is 5 in a million. In a
more refined model, some of the virtual asteroids are more likely than
others, and the exact value of the probability is somewhat different.
The nominal solution is just one of these one million possible
solutions, and it is, according to some statistical models, more likely
than the others, but only slightly; as an example, in the gaussian
model, the nominal solution is just 2.5 times more likely than the
average, but it is not significantly more likely than the neighboring
ones. As new observations come in, the nominal solution can change very
fast, just by jumping from one to some other of the virtual asteroids.
Now, new observations come in. The probability is just a way to
describe our ignorance, thus it is a basic fact that when new
information becomes available the probability changes. (This can
be expressed by speaking of conditional probability of an orbit
given the observations; this is the Bayesian interpretation preferred
by some, including Karri Muinonen.) In our simple model, the new
observations are such that some of the virtual asteroids are
incompatible with them, thus a number of them, say 300,000 out
of a million, are proven not to be the real one, so only 700,000
remain. If, unfortunately, the five impacting solutions are among
those still compatible with the observations, now the impact
probability is 5/700,000, significantly increased from the previous
value. With all virtual asteroids equally likely to be real, the
probability of an impact cannot decrease: either it goes to zero,
because the virtual impactors are now excluded by observation, or it
increases, roughly by the same amount by which the knowledge of the
orbit has been improved. If a more complex model, e.g., gaussian, is
used, then the changes can go both ways, but still it is the case that
both a significant increase and disappearance from the list can take
place. Since the nominal solution is the one in the middle, it is the
least likely to go away with only a few new observations, but it can
indeed go away after many new observations are reported.
As an example, the 2044 virtual impactor of 1999 AN10 is now at 1.44
sigmas, which means it is not any more very close to the nominal, thus
it is possible that it would go away as more observations are obtained
during the favorable observing window of this summer. But, this is by
no means sure, and it is indeed possible that the probabilities will
keep increasing and we shall have to wait until the 2004 radar
observation window to be sure.
We conclude by saying that we do this kind of computation on a regular
basis, as our scientific work but also with some spirit of service. We
do cross check our results with Paul Chodas and Karri Muinonen, and we
can get to an agreement with them after some discussion (sometimes
comparison of the results is not trivial). If other groups were to
acquire the capability to perform this kind of computation (of impact
possibilities and probabilities), we would be very happy to share with
more people the responsibility of such very critical job; however, we
do find it difficult to accept technical criticism from those who do
not do these computations.
Yours,
Andrea Milani and Steve Chesley
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