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.


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.


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.


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:


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.


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.


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.


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.


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.


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.


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.



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.



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.


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.


---------------------------------------------------- --
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.


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



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.