** **

**ON THE ORBITAL INSTABILITY**

OF THE MAIN CENTAURS

Date: Tue, 28 Aug 2001 23:27:44
-0600

Subject: [centaurresearchproject] On the
orbital stability of centaurs

**Generally
speaking, all orbit-crossers that are not "locked" in mean motion resonance
(e.g., 2 periods of a plutino = 3 periods of Neptune) have very unstable orbits
because they will eventually get very close to the planet they are crossing.
If the physical approaching distance is small enough --say less than 0.5
AU-- the orbit of the crossing minor object will change drastically or "catastrophically"
in a way that makes it impossible to calculate the object's position beyond
that point of time. **

**Such an
orbit is called "chaotic", and most centaurs have by definition chaotic orbits.
It is one of their main characteristics, since they are so small and are crossing
the path of very massive giants. **

**The time-scale
in which this unstable behavior will usually show is probably in the order
of 10,000 to 100,000 years (in the case of the Apollo asteroids it can be
much less than that since they move much faster). For example, we know
that Chiron becomes chaotic (impossible to calculate) before about 600 AD
and after 4700 AD, and Pholus around 3800 BC, because the information is given
in the Swiss Ephemeris documentation, but unfortunately there is no documentation
(that I know) about any of the other numbered centaurs.
**

**The orbital
behavior of these objects in the very far future or past can only be approached
statistically, in terms of probabilities within a sample of "variational orbits"
or orbital clones. **

**In 1993,
after the orbit of Pholus was known well enough thanks to pre-covery observations
going back to 1977, its orbit was integrated forward 800,000 years by statistical
methods (i.e., 27 slightly different test particles of Pholus orbital clones),
in a way similar to the studies on Chiron performed years before. The
whole article can be downloaded from the Nasa astrophysical abstracts service
site: **

**"Orbital
evolution of the large outer Solar system object 5145 Pholus"
**

D. J. Asher & D. I.
Steel

Monthly Notices of the Royal Astronomical
Society, 263:179-190 (1993)

**Since there
is no concrete data available on the other named or numbered centaurs, I decided
to numerically integrate the mutual distances between them and the planets
they cross, for a period of 9900 years, from 4569 BC to 5286 AD. The
program used was "Solex", which is very accurate but does not calculate the
accumulated ephemeris uncertainty that allows other programs to stop after
an uncertainty threshold is reached. **

**However,
we can examine the close physical approaches of centaurs to Saturn, Uranus,
and Neptune, which are responsible for the exponential jump of their ephemeris
uncertainty and the impossibility of calculating their positions beyond those
points. I will use the term "encounter" to refer to minimum distances
between planet and centaurs of less than 1 AU. Roughly speaking, any
approach within 1 AU will significantly change the shape and size of the orbit.
Some recent such encounters are graphically shown in my website in the cases
of Asbolus, SG35/Okyrhoe, and UG5/Elatus?. But when the distance is
less than 0.5 AU, we may expect the orbit to become "chaotic", i.e., it may
become impossible to calculate for us beyond that point in time, it will
be out of our reach. **

**What I will
do is list all the encounters within the range of the integration that are
smaller than 1.0 AU, but I will use the smaller distances as limits or borders.
**

**1- CHIRON
**

**In the case
of Chiron I find: **

` 720
AD 0.2 AU`

`3544 AD 0.9 AU`

`4606 AD 0.7 AU`

**According
to Rique Pottenger, Chiton had an encounter with Saturn in 720 AD, and another
with Uranus in 585/586 AD. As a result of them <<... the position
of Chiron is somewhat uncertain between 586 and 718 AD, and very uncertain
before 585 AD.>>. I did not find any encounter with Uranus during the
whole 9,900 years of my integration. Perhaps this can answer the question
about how uncertain is "somewhat uncertain". **

**In my integration
Chiron approaches Saturn to within only 0.2 AU in 720 AD. This gives
me confidence in the method used. Astrodienst stops calculating Chiron
before 650 AD, and asserts that it cannot be calculated beyond about 4650
AD. This would be consistent with a jump of the uncertainty index after
the encounter with Saturn within 0.7 AU that I find in 4606 AD (there are
no encounters with Uranus), and assures that the calculations done for the
other centaurs with the same method are correct. **

**2- CHARIKLO
**

**Chariklo
never approaches Saturn and only "gently" crosses Uranus occasionally.
In the whole period -4500 to +5200, it never approaches Uranus to less than
2.8 AU, so we may conclude that its orbit is stable within that range of time.
**

**3- ASBOLUS
**

**Asbolus
crosses both Saturn and Uranus. Let's examine the close approaches to
Saturn first: **

` 347
AD 0.4 AU`

`1702 AD 0.6 AU`

**In my site
you will find a detailed examination of the changes suffered by Asbolus as
a consequence of the 1702 encounter: **

**
http://www.expreso.co.cr/centaurs/astronom/8405orbit.html
**

**Before the
1701-1702 encounter, Asbolus period was 81 years, but Saturn shrinked the
orbit during the encounter and now its period is 75-76 years. This didn't
render it chaotic, but the long-range old Swiss Ephemeris file stops in 282
AD. Although the Swiss Ephem file I used does not include the latest
orbital update (there must be a new one available for those who own the CD),
it serves for the purpose of comparison: it clearly stops as a consequence
of the encounter in 347 AD (calculated with the latest updated elements).
**

**There are
no close encounters with Uranus in the 9,900 year period examined.
**

**4- PHOLUS
**

**Pholus crosses
Saturn, Uranus and Neptune (it is a "big sweeper"). Let's examine the
Saturn approaches first: **

`-3974
0.8 AU`

`-3033 0.9`

`-2030 0.7`

` -763 0.9`

(there are no encounters in the future)

**We know,
from the Swiss Ephemeris documentation, that Pholus cannot be calculated beyond
about 3850 BC. The correct year with updated elements should be around
4000 BC, since the encounter with Saturn happens in 3974 BC. However,
we observe that there was a closer encounter in 2031 BC, which apparently
did not render the orbit chaotic. I wonder what the latest Swiss Ephemeris
long-range file says in that respect. (help anyone?).
**

**Pholus never
approaches Uranus closer than 2.7 AU, but it has a close encounter with Neptune
in 803 BC: **

`-802
0.6 AU`

**Both the
Saturn and the Neptune encounters are not as dramatic as those of Chiron with
Saturn so it is surprising to know that they are enough to send the orbit
computation out of our reach. My guess is that the combined action of
several encounters increases the cumulative uncertainties until they add up
and pass beyond a pre-defined margin, after which the integration program
stops. **

**Another
way of examining this is to observe graphically the scale of the changes brought
about by the encounters (I will examine the plots of the orbital elements
later). If the changes are too large, the solution is probably beyond
the chaotic threshold even if the mutual distance doesn't look too radical.
**

**5- HYLONOME
**

**Hylonome
gets close to both Uranus and Neptune, but only near the end of the integration
process. There is one single approach to Uranus within 0.5 AU in the
year 3479 AD, and another single approach to Neptune within a modest 0.9 AU
in 4339 AD. In the past, Hylonome never gets close to Uranus and approaches
Neptune within 1.1 AU in -1585. This means that the changes in the
orbit are relatively slight and it probably doesn't get chaotic within the
range -4500 to +5200 of my exploration, or at least until 3500 AD.
**

**6- NESSUS
**

**Nessus never
gets closer than 3.5 or 4 AU to Neptune. This may be a good example
of orbital resonance avoiding close encounters: it produces large and long
periodic perturbation terms (4 Nessus = 3 Neptunes in this case), but the
orbits are harmonically locked or synchronized --something called "near commensurability"--
so that collision is always avoided in scales of hundreds of thousands or
millions of years. **

**Nessus closest
distance to Uranus is 4.6 AU in the whole 9,900 years period. This
is probably also the result of a near-commensurability, since 2 Nessus =
3 Uranus ( = 1 Pluto). We may conclude that its orbit is safe within
the -4500 + 5000 range... unless of course instability is caused by some other
factor not considered in the hypothesis that the chaotic behavior appears
as a result of very close approaches. **

**7- PYLENOR
**

**Pylenor/1994TA
crosses only Uranus, but never gets closer to it than 1.4 AU. It gets
this close to Uranus only twice, one in the 1960's (closest approach to Uranus
was April 26, 1964, at a distance of 1.57 AU) and another one in 3475 AD.
Such a distance will produce only slight or slow modifications of the orbit.
**

**We need
further orbital updates of Pylenor to see if this situation changes.
**

**Juan
**

**-------------------------------------------------
**

Date: Thu, 20 Dec 2001 18:21:30 -0600

Subject: [Centaurs] 1999UG5 chaotic?

**Everytime
I make a recalculation of UG5 after an orbital update, I get a different
position for the year 622 AD. This suggests that its orbit is probably chaotic
by that time. **

**There are
2 relatively close encounters with Saturn in 1314 (0.67 AU) and in 1471
(0.75 AU). But these approaches are not too impressive. They are no
match for the approach to within 0.46 AU due in May 2002. If any of you has
access to the long-term integration of the new Swiss Ephemeris file
for #31824-UG5, please let me know if it will compute positions around
600 AD. If it does, then its orbit is stable at that time, if it doesn't,
I will remove it from the Riyal "long" file. **

**Juan
**

**-------------------------------------------------
**

Date: Wed, 16 Jan 2002 07:35:48 -0600

Subject: Re: [Centaurs] 1999UG5 chaotic?

**There was
new orbital update of UG5 a few days ago. A new integration back to the year
AD 600 shows --one more time-- a very different result from the previous computation.
This confirms without any doubts that by this time the orbit is chaotic,
since this type of divergent solutions with only tiny variations in the original
conditions is part of the definition of chaotic motion.
**

**Since the
last approach to Saturn was in 1314, this can be considered as the limit in
the past beyond which its position cannot be known.
**

**[...]
**

**Juan
**

**-------------------------------------------------
**

Date: Wed, 16 Jan 2002 10:21:30 -0600

Subject: Re: [Centaurs] 1999UG5 chaotic?

**[...]
**

**In order
for a body to be perturbed by another body in a numerical integration, the
perturbing body must me modelled or included from the start. It is impossible
to model a perturbing body that is unknown. Such unmodelled perturbations
have to be observed first, which would be impossible in the long past.
**

**The known
perturbing body here is Saturn. When UG5 comes very close to it, the effect
can be so strong that it is impossible to calculate its position beyond this
point. The numerical integration can go on and will give results, but the
results will be meaningless the more we go pass some critical point near the
approach. **

**Beyond the
critical point, the orbit is "chaotic", i.e., it cannot be calculated in
any realistic way, usually as a result of an important change in the semimajor
axis of the orbit. The only results possible will be statistical. How do we
know when this point has been reached? **

**One is the
way Astrodienst does it: the orbital integrator written by Steve Moshier is
used by them with several modifications, one of them being that the integration
stops after a control "error" or "uncertainty" quantity passes a certain threshold
beyond which the positions are too uncertain to be considered of any value.
I do not know the specifics of how they calculate this.
**

**The other
way is the one I have shown: if very tiny modification are applied to the
starting conditions of the integration (the osculating elements), it could
happen that after a certain point in time these slightly different orbits
will begin to diverge, i.e., the position of the body using one of the orbits
will begin to diverge from the position using the other slightly different
orbit. When the distance between the two is too large (tens of degrees), this
means that the "variational" orbits are evolving differently: the orbit is
chaotic. **

**A stable
orbit, such as that of Varuna, will show very little difference in the final
position when the slight modification is applied to the starting osculating
elements. The orbit is "recurrent". But the chaotic orbit is non-recurrent,
it disperses away, constantly shrinking or expanding, eventually being ejected
from the solar system. **

**Juan
**

-----------------------------------------------------

Date:
Wed, 07 May 2003 09:36:06 -0600

Subject: [Centaurs] chaotic centaurs

In the past I wrote about the probable chaotic orbit of 199UG5... My approach
then was to track the close encounters between centaurs and the major planets,
and I mentioned that, in absence of an algorithm to measure the "threshold
of uncertainty" after which the calculation of the orbit becomes meaningless,
one could find confirmation of chaotic motion by checking the changes in
the osculating orbital elements, particularly in the semimajor axis.

I have calculated the osculating elements of other centaurs with well-known
orbits back to the year 1830 B.C. The following results are preliminary,
and I am giving only brief comments; they should be complemented by a closer
scrutiny exploring the relationship with the major planets responsible for
the orbits becoming chaotic.

1-) 1999XX143. The orbit goes wild before A.D. 1167. The event that year
looks quite catastrophic. Since it happens during perihelion at 9 A.U., it
is safe to say that it is the result of a close approach to Saturn. Therefore
its positions given by Riyal to A.D. 600 should not be used before A.D. 1167.

2-) 1998SG35. The threshold seems to be around A.D. 850, before which
the semimajor axis expands until the period becomes 33 years (it is 24 years
right now). This comes as a result of an approach to Jupiter.

3-) 1999UG5. As a consequence of approaches to Saturn that I detailed in
my previous research (see above reference), this orbit is probably chaotic
before the year A.D. 1300, so the positions given by Riyal should not be
used before that year.

4-) 2000EC98. This orbit seems to be chaotic before A.D. 1418, when it is
at perihelion, so obviously it is a consequence of an approach to Saturn

5-) 2000GM137. Its orbital period (and semimajor axis) seems to jump
slightly after its perihelion in A.D. 1564, but the orbit looks stable until
the perihelion around the year 85 B.C.

6-) 2001PT13. This centaur reaches perihelion around 1930. This brings it
close to Saturn, and the result is a jump in the semimajor axis before 1925
(the period changes from 34 years today to 38 years), but the change doesn't
make the orbit chaotic.. There is another change in 1132. PT13's instability
is probably very similar to that of UG5, and requires more investigation.

Juan

**----------------------------------------------------**

Date: Wed, 07 May 2003 22:38:12
-0600

Subject: [Centaurs] chaotic centaurs: controlled by Saturn

1-) 1999XX143... The closest approach is in January of A.D. 1165, to a distance
of 0.066 AU. This is extremely close. XX143 "disappears" before that, or
in other words, 1999XX143 "was born" around 1164. What its orbit may have
been before that nobody knows; it may even had been one of the many satellites
of Saturn. Its present diameter is near 80-100 Km.

3-) 1999UG5... UG5 is right now at the time of its closest approach to Saturn
since at least A.D. 360. The closest distance of 0.46 AU was reached around
May 2002. Before that, it had reached a distance of 0.48 AU around July 1314.
Every new integration of UG5 produces very different results for dates before
A.D. 1300, so it is safe to assert that its orbit is definitely chaotic before
that time.

4-) 2000EC98... There is a minimum distance to Saturn of 0.1 A.U. in March
of A.D. 1071, so its orbit is chaotic before that year. EC98 comes close
to Jupiter too, so probably the instability observed in the semimajor axis
before 1418 is a result of an approach to this planet. I will examine the
close approaches to Jupiter in another post.

6-) 2001PT13... A distance to Saturn of 0.51 AU is reached near April 1924.
Before that, the closest distance since A.D 360 is double that of 1924, and
happens in A.D. 1129. So apparently at least to A.D.360 the orbit is
not chaotic. More study is needed in this case.

Juan

----------------------------------------------------
--

Date: Wed, 07 May 2003 23:53:27
-0600

Subject: [Centaurs] Re: chaotic centaurs: controlled by Saturn

2-) 1998SG35... The minimum distance back to A.D. 360 is really moderate,
of 1.4 AU in A.D. 961, so Jupiter is not controlling it. On the other hand,
a minimum distance to Saturn of 0.19 AU is reached on October of A.D. 496.
Before that, one can assume that the orbit is chaotic, and it was already
unstable before at least A.D. 850, when the distance to Saturn reached 0.58
AU.

4-) 2000EC98... The minimum distance to Jupiter is 1.34 AU in A.D. 1144,
so the instability observed before 1418 until it becomes chaotic in AD 1071
must be an approach to Saturn. Since the only approaches other than the fatal
one of 1071 are 0.87 AU in 1265 and 0.94 AU in 1513, it is reasonable to
say that despite the instability observed before 1513, the orbit becomes
chaotic only before A.D. 1071.

5-) 2000GM137... The minimum distance of GM137 to Jupiter back to A.D. 360
is 2.05 AU, so we can discard this planet as source of chaotic behavior during
that interval. The minimum distance to Saturn is 1.06 AU and it happens in
1766. This orbit looks stable during the period 360-2003.

Juan

---------------------------------------------------

Date: Thu, 08 May 2003 08:38:41
-0600

Subject: [Centaurs] chaotic centaurs: summary

Of the 47 bodies that can be considered centaurs discovered so far (including
the quasi-centaurs), there are 33 with reasonably well-established orbits
that can be examined for possible chaotic behavior. I present in this post
all 33 of them with a brief comment about the presence of chaotic motion
in the past.

Keep in mind that, regardless of the instability of the orbit, future close
encounters with the giant outer planets, although they can cause dramatic
changes or "jumps" in the orbit and render it chaotic, the object will not
be lost (i.e. its ephemerides will not cease to be accurate) because the
object will likely be under observational scrutiny.

The -1830 limit represents the limited integration I carried out for the
"other" un-named centaurs. "Uncertain" means "unstable", the orbit
begins to appear to expand or contract; "chaotic" means a clearly observed
infinite expansion (usually) or contraction (less frequent),evidencing a
non-recurrent random orbit; a question mark (?) means "looks like",
"maybe", i.e, it requires closer study.

`Chiron:
becomes chaotic before A.D. 720`

Pholus: uncertain before -2031, chaotic before -3974.

Nessus: stable -4500 to +5200

Chariklo: stable -4500 to +5200

Pelion: stable at least to -1830

1998 TF35: stable at least to -1830

2002 CR46: stable at least to -1830

1999 UG5: chaotic before A.D. 1314

2001 PT13: stable (?) at least to -1830

Hylonome: stable from -4500 to at least A.D. 3500

2000 QC243: stable at least to -1830

Asbolus: chaotic before A.D. 347

1998 BU48: stable at least to -1830

2002 GO9: stable at least to -1830

1999 XX143: becomes chaotic before A.D. 1165

2002 GB10: stable at least to -1830

1999 OX3: stable at least to -1830

2000 QB243: stable at least to -1830

2000 FZ53: stable at least to -1830

2001 BL41: becomes uncertain before -1500

1994 TA: stable -4500 to +5200

2001 KF77: stable at least to -1830

1998 SG35: uncertain in A.D. 850, chaotic before A.D. 496

2000 EC98: uncertain before 1418, chaotic before 1071

2000 GM137: uncertain before -85, chaotic before -716

2000 CO104: stable at least to -1830

2001 XZ255: stable at least to -1830

2001 XA255: stable (?) at least to -1830

2002 PN34: stable at least to -1830

2002 DH5: stable at least to -1830

2001 SQ73: stable at least to -1830

2003 CO1: stable (?) at least to -1830

2002 GZ32: stable at least to -1830

**Juan**

-------------------------------------------------

Date: Thu, 08 May 2003 12:52:24
-0600

Subject: [Centaurs] chaotic centaurs: summary, 2

I extended the numerical integration of the osculating elements of all 33
objects back to -4500. Here are the new results (only the cases affected
are listed).

I would like to emphasize that these results are preliminary and subject
to change, and I am open to input and corrections.

`Pelion:
looks stable to -4500`

1998 TF35: looks stable to -4500

2002 CR46: looks unstable before about -3500

2001 PT13: chaotic before -755

2000 QC243: looks stable to -4500

1998 BU48: looks stable to -4500

2002 GO9: looks stable to -4500

2002 GB10: looks stable to -4500

1999 OX3: looks unstable before -2505

2000 QB243: looks stable to -4500

2000 FZ53: looks stable to -4500

2001 BL41: chaotic before -1263

2001 KF77: looks stable to -4500

2000 CO104: looks stable to -4500

2001 XZ255: looks stable to -4500

2001 XA255: looks unstable before about -2100

2002 PN34: looks stable to -4500

2002 DH5: looks stable to -4500

2001 SQ73: looks stable to -4500

2003 CO1: looks stable to -4500

2002 GZ32: looks stable to -4500

It is interesting to look at the plot of the semimajor axis of 2000EC98 (presently
a Jupiter to almost-Uranus linker) during the fatal Saturn encounter commented
in the previous posts:

1071/06/10 10.91643344

1070/05/06 19.52968683

See how the numerical integration "blasts" the orbit spectacularly. Nothing
can be known about it before A.D. 1070.

Something similar can be seen in the case of the 1999XX143 encounter with
Saturn, also commented already. The numerical integration goes completely
astray and becomes meaningless:

1167/10/24 17.89495666

1166/09/19 18.17457521

1165/08/15 19.52235486

1164/07/11 24.12028140

1163/06/07 22.32088918

Both EC98 (about 60 Km) and XX143 (about 90 Km) may have been satellites
of Saturn, but this is only speculation.

Juan