

Astronomical Applications Department, U.S. Naval Observatory  lyap2 (Page 9)Wodocs >> Science : Astronomy >> Astronomical Applications Department, U.S. Naval Observatory lyap2 Page 9
, and the rest
lie well within 2
. This distribution is consis
tent with the notion that in the outerbelt aster oids we are seeing the remnants of a larger original distribution. The bulk of that popula tion has been cleared out by Jupiter, leaving the longT
e
tail members to survive to the present.
The observed value of
fluctuates with
small changes of the initial conditions. For example, Figure 2 shows the Lyapunov times cale as a function of initial semimajor axis for (522) Helga. The value of T
L
is greater than
32,000 yr (corresponding to T
e
~ 22 Myr),
except in a narrow interval around the 12:7 resonance, a
0
[0.69770, 0.69815], where T
L
averages about 5800 yr (T
e
~ 0.95 Myr). The
J
). The 12:7 mean motion
resonance is at a=0.697922 a
J
, marked by the
symbol in the second panel of Figure 2. We noted variations in T
L
of roughly 12 percent
within the resonance. The chaos exhibited by the motion of (522) Helga is apparently associ ated with the 12:7 resonance, and the semima jor axis width of this resonance is clearly delineated by the behavior of T
L
, the mean
5
T
J
integrations.
Those with short Lyapunov times are, like
(522) Helga, possibly associated with the cor responding resonance for the particular initial conditions we used. The range in semimajor axis for (522) Helga was more than a factor of 7 larger than the width of the resonance as evi denced in Figure 2. Asteroid (1390) Abastu mani, falling between 15:8 and 13:7, was the only one whose semimajor axis did not at any time in our numerical integration cross a reso nance. (This asteroid exhibited behavior con sistent with quasiperiodic motion, cf. Table II.) We are not certain to what extent, if any, the asteroids are affected by the corresponding resonances, but the association is suggestive (see below).
An Interpretation of Short Lyapunov Times
We propose the following picture for the
dynamics leading to an "event." First we review briefly the relevant dynamics of a two degree of freedom system, then we conjecture that analogous behavior is occurring in the much more complicated, many degree of free dom, Hamiltonian system represented by the outerbelt asteroids.
It is wellknown that, in a two degree of
freedom Hamiltonian system, invariant surfaces (KAM tori) divide the phase space. Under the influence of a sufficiently strong perturbation, the "outermost" invariant surfaces are destroyed, giving rise to a global sea of chaos surrounding an inner stable region, where invariant surfaces still exist. At the core of the stable region is a stable period one orbit. Fur ther out, past the outermost intact surface and embedded in the chaotic sea, are secondary invariant surfaces surrounding elliptic period n
Chaotic Motion in the Outer Asteroid Belt
page 8
<< Previous 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Next >>
Other Documents:
NAO 150, Rotate Vector, lyapcalc, wedges, Scifull, Precession Memo, Precession Maple, Curves 3 D, NOVAS 2004 Overview, NOVAS 2006 Overview, thesis, Ranson, 





WODocs
New Docs
Documents Category:
Arts
(Design, Movies, Music, Radio, Television)
Automotive
(Cars, Marine, Motorcycle, ATV, Snowmobiles)
Business
(Biotechnology and Pharmaceuticals, Chemicals, Construction and Maintenance, Materials, Real Estate, Services)
Electronics
(Computers, Motion Control, Power Supply)
Games
(Board, Family, Party, Card, Construction, RC Toys)
Health
(Animalm, Beauty, Healthcare, Medicine, Pharmacy, Surgery, Weight Loss)
Home
(Accessories, Cooking, Decor and Design, Electrical, Family, Pets)
News
(Newspapers, Sports, Television)
Recreation
(Collecting, Hiking, Scouting, Survival, Travel)
Reference
(Education, Libraries and Archives, Museums)
Science
(Agriculture, Astronomy, Biology, Chemistry)
Shopping
(Antiques and Collectibles, Clothing, Flowers, Food, Home and Garden)
Sports
(Bicycle, Snowboard, Skiing, Other)




