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IAA Transactions, No. 8, ``Celestial Mechanics'', 2002
Migration of asteroids from the 3/1 and 5/2
resonances with Jupiter to the Earth
S. I. Ipatov 1;2
1 Institute of Applied Mathematics, Moscow, Russia
2 NASA/GSFC, Greenbelt, USA
Last years many scientists used symplectic integrators [1], which are much
faster than usual integrators. For asteroids, a step of integration for a symplectic
integrator usually was taken in a range between 7 and 30 days. Some scientists
compared the results obtained with different integration steps, but usually they
did not compare these results with those obtained with a usual integrator. We
have made series of computer runs of the orbital evolution of asteroids for the 3/1
and 5/2 resonances with Jupiter using both the symplectic integrator RMVS3 [1]
and the Bulirsh­Stoer method [2] (BULSTO). The results obtained with different
integrators and different integration steps d s were compared in order to under­
stand what error we usually make if we use a symplectic method for investigations
of orbital evolution of resonant main--belt asteroids. For BULSTO the error '' per
integration step was taken to be less than 10 \Gamma8 or 10 \Gamma9 . For RMVS3, we have
made integrations with d s equal to 3, 10, and 30 days. In each run we considered
the Sun, 7 planets (except Mercury and Pluto) and N asteroids moving in the 3/1
or 5/2 resonances with Jupiter (a ffi = 2:5 or a ffi = 2:823 AU). Initial eccentricities
and inclinations were the same in all runs: e ffi = 0:15 and i ffi = 10 ffi . Initial values
of the mean anomaly and the longitude of the ascending node were different. The
considered time interval T S is equal to several Myr.
Using orbital elements obtained with a step equal to 500 yr, we calculated
the probabilities of collisions of asteroids with the terrestrial planets and obtained
(for all time intervals and all bodies) the total probability P \Sigma of collisions with
a planet and the total time interval T \Sigma during which the perihelion distance q
of asteroids was less than a semi--major axis of the planet. The values of P r =
10 6 P = 10 6 P \Sigma =N and T = T \Sigma =N are presented in the Table together with the
ratio r of the total time interval when orbits were of Apollo type (at a ? 1 AU,
q = a(1 \Gamma e) ! 1:017 AU, e ! 0:999) to that of Amor type (1:017 ! q ! 1:33 AU);
r 2 is the same as r but for Apollo objects with e ! 0:9.
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Table: Values of T (in kyr), P r = 10 6 P , r, r 2 , and r hc for the terrestrial
planets (Venus = V, Earth = E, Mars = M) at N = 144, T S = 10 Myr (except
for the first line for each resonance, for which T S = 50 Myr).
V V E E M M \Gamma \Gamma \Gamma
T P r T P r T P r r r 2 r hc
3=1 10 \Gamma8 739 529 1227 626 2139 116 2:05 1:78 7:41
3=1 10 \Gamma8 628 488 1056 589 1922 114 2:05 1:53 7:67
3=1 10 \Gamma9 699 322 1160 413 2012 69 2:14 1:83 6:9
3=1 10 631 574 1015 675 1736 108 2:48 2:16 0:38
3=1 30 925 3580 1366 2763 2189 167 2:44 2:15 0:84
5=2 10 \Gamma8 109 54:5 223 92:0 516 19:4 1:28 1:15 34:5
5=2 10 \Gamma8 108 54:2 221 91:4 510 19:2 1:29 1:11 33:8
5=2 10 \Gamma9 203 155 334 174 644 32:3 1:68 1:24 16:5
5=2 10 79 50:4 158 73:9 330 15:8 1:66 1:44 9:6
5=2 30 308 2330 475 696 703 56 2:82 2:41 6:1
For the asteroids initially located at the 3/1 resonance with Jupiter, we found
that the ratio r hc of the number of asteroids ejected into hyperbolic orbits to that
collided with the Sun is much larger for BULSTO than for RMVS3. Besides the
values of r hc presented in the Table at N = 144, for the 3/1 resonance at N = 24
we obtained r hc equal to 4.0, 1.7, 0.33, 0.4, and 0.7 at '' = 10 \Gamma8 , 10 \Gamma9 , d s = 3, 10,
and 30 days, respectively. So in some cases a symplectic method can give large
errors. For the 5/2 resonance, the ratio of the values of r hc obtained by BULSTO
and RMVS3 also was not small (? 3). The difference in values of T and P r was
not considerable for RMVS3 at d s = 10 days and for BULSTO. For d s = 30 days
at the 5/2 resonance, 78% of the probability of collisions with the Earth were
caused by 3 asteroids (64%, by two asteroids) and 52% of all collisions with the
Earth were from Aten orbits.
This work was supported by Russian Foundation for Basic Research (01­02­
17540), INTAS (00­240), NASA (NAG5­10776), NRC (0158730), DAAD (referat
325). First runs with a small number of asteroids where made during the author's
visit to Dresden observatory in September 2001, and I am thankful to Prof.
M. Soffel, Andre Noak, Sergei Klioner, and Akmal Vakhidov who helped me
during this visit.
References
1. Levison H. F., Duncan M. J. Icarus, 1994, 108, 18--56.
2. Bulirsh R., Stoer J. Numer. Math., 1966, 8, 1--13.
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