EDP Sciences
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A&A
Volume 532, August 2011
Article Number L4
Number of page(s) 4
Section Letters
DOI http://dx.doi.org/10.1051/0004-6361/201117504
Published online 14 July 2011

© ESO, 2011

1. Introduction

(1) Ceres, (2) Pallas and (4) Vesta are the largest bodies in the asteroid belt, with respective masses of 4.758, 1.114, and 1.331 × 10-10   M (Fienga et al. 2011), while the mass of the smallest planet, Mercury, is about 1660 × 10-10   M, and even Pluto is already 74 × 10-10   M. These asteroids1 induce some perturbations of several km on the position of Mars (e.g., Kuchynka et al. 2010), and have long been taken into account in the elaboration of high-precision planetary ephemerides (Standish 1998; Fienga et al. 2008, 2009), valid over a few kyr. Meanwhile, they have only recently been introduced in the more simplified models used for the computation of long-term ephemerides, valid over several Myr, and aimed at paleoclimate reconstructions and geological timescale calibrations (Laskar et al. 2004, 2011). The time of validity of these computations is limited by the exponential divergence of the planetary orbits, resulting from their chaotic secular behavior (Laskar 1989, 1990). In practice, this chaotic behavior induces an increase in the initial position error d0 as d = d0 × 10T/10, where the time T is expressed in Myr (Laskar 1989, 1990, 1999). In the La2004 solution (Laskar et al. 2004), the asteroids were not included individually, and only their averaged contribution was considered. The validity of the solution was estimated to be about 40 Myr. In La2010 (Laskar et al. 2011), as in INPOP08 (Fienga et al. 2009), the fivebodies (1) Ceres, (2) Pallas, (4) Vesta, (7) Iris, and (324) Bamberga were included and considered as the planets, with all gravitational interactions, while the numerical algorithm was being improved. In a surprising way, although the solution could be extended to about 50 Myr, its numerical accuracy was not improved, and it appears that the main reason for that is the presence of the asteroids that induce some additional instabilities. In this paper, we analyze the contribution of these asteroids and the instabilities that they generate.

2. Numerical simulations

In the numerical simulations, the full model comprises the eight planets of the Solar System, Pluto, and the Moon, as well asCeres, Pallas, Vesta, Iris, and Bamberga. The model and integrator that are used are the same as in the long-term solution La2010 (Laskar et al. 2011). To evaluate the effects induced by the asteroids, we performed two different sets of integrations. In a first set of solutions, S5a,b, the asteroid interactions were included, while they were omitted in the second S0a,b. The stability of the trajectories were analyzed by comparing two solutions with initial conditions that differ in position in only 10-11 AU (~1.5 m) for the planets and 10-10 AU (~15 m) for the asteroids, which is in both cases much less than the actual uncertainty of these initial conditions.

thumbnail Fig. 1

Evolution of the Earth’s eccentricity uncertainty δe in S0a,b (dotted line) and S5a,b (solid line). In both cases, the initial difference at t = 0 is δe = 1.4 × 10-12. The thin solid line, with slope  −1/10 represents the secular chaotic behavior (Laskar 1989, 1990). The eccentricity solution loses its validity when its uncertainty approaches its averaged value (δe ≈ 0.03).

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The divergence of the two sets of solutions are presented in Fig. 1. In both cases, the differences in the Earth’s eccentricity at t = 0 is 1.4 × 10-12, and thisuncertainty will first grow because of the difference in initial condition, but after 10 Myr, the differences in the S5a,b solutions are about ten time greater than the differences in the S0a,b solutions. As a result, the time of validity of the S5a,b solutions is limited to about 60 Myr, while the (unrealistic) S0a,b solutions are valid over nearly 70 Myr. We note that the secular chaotic regime, with an exponential increase in the error such as 10T/10 where T is in Myr (Laskar 1989, 1990), dominates after about 40 Myr (Fig. 1).

thumbnail Fig. 2

Relative variations in thetotal orbital energy of the Moon and planets for S0a (red), S0b (green), S5a (blue), S5b (purple). The large variations in energy of S5a,b correspond to asteroid close encounters (Table 1).

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The relative change in the total orbital energy of the Moon and planets for all solutions S0a,b and S5a,b is displayed in Fig. 2, after correction from the tidal dissipation of the Moon (Laskar et al. 2004). The energy of S0a,b is preserved up to about 10-13 in relative value, but some large and rapid variations in energy are observed in S5a,b. These are the results of close encounters among the largest asteroids. In Fig. 2, the occurrences of these close encounters appear to be randomly distributed, and differ very much from S5a to S5b. This thus provides a non-compressible error in the energy of the planets, regardless of the accuracy of the initial conditions in the planetary orbits. The rapid variations in energy of S5a,b that are larger than 2.25 × 10-13 in relative value are easily identified as the results of specific asteroids close encounters (Table 1).

Table 1

Close encounters for S5a (top) and S5b (bottom).

3. Stability of asteroids

The stability of the asteroids is estimated by measuring the rate of divergence inmean longitude of S0a,b and S5a,b. At t = 0, the differences in longitude in the solutions a,b for the five considered asteroids are on the order of d0 = 10-10. This error then grows by two means: first, the difference in initial conditions induces a difference in mean motion of about d0, and thus a linear drift d0   t, while the chaotic divergence behaves as d0exp(t/T), where T is the Lyapunov time and 1/T the maximum Lyapunov exponent. The maximum divergence will be given by the linear drift over the  [0,T] time interval, and the exponential drift on  [T,t]. The full rate of divergence is then (1)To determine the Lyapunov time, we solve for T in d(t1) = 1 (Eq. (1)), where t1 is the first time for which d(t) = 1. The Lyapunov exponents (1/T) are given for S0a,b and S5a,b in Table 2. We note that the Lyapunov exponents LE0 computed in the absence of asteroid interactions are close to the values computed in a different way by Knežević & Milani (2003) and Novaković et al. (2011). We note, however, that in our integrations S0a,b, Vesta is much more regular than Ceres, unlike Novaković et al. (2011). When the asteroids are taken into account (S5a,b), the behavior of all asteroids is much more chaotic, and Ceres and Vesta, in particular, become strongly chaotic, with Lyapunov times of only 28 900 and 14 282 yr (Table 2, Fig. 3). We anticipate that numerous asteroids in the main belt will behave in the same way with, as is the case for Ceres and Vesta, much more chaotic behavior than previously thought when the collective effects of the asteroids were not taken into account.

Table 2

Lyapunov exponents (LE) and Lyapunov times (LT).

thumbnail Fig. 3

Decimal logarithm of the differences in longitude for the asteroids in the S0a,b (dotted line) and S5a,b (solid line) solutions. The initial conditions differ by about 1.4 × 10-10 in longitude.

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Table 3

Collision probability (in 10-3 per Gyr) for each asteroid pair (i,j).

In the region of the Vesta family, we can also look more globally at the stability of the asteroids (Fig. 4). In this experiment, the initial conditions were taken on a regular grid in (a,e) with a 0.01 step size in e and 0.001 AU step size in a for a total of 51 × 701 = 35   751 orbits, for which all the other initial elliptical elements (i,ϖ,M,Ω) are set to zero. The orbits were computed over 100 kyr with a step size of 0.01 yr. Frequency analysis was then performed in order to obtain a global view of the stability of the trajectories (Laskar 1993; Dumas & Laskar 1993). The experiment was performed with (Fig. 4a) and without (Fig. 4b) the contribution of the asteroid interactions. The presence of the asteroid interactions makes the orbits much less regular, and in particular, most of the very stable blue zone present in Fig. 4b no longer exists in Fig. 4a.

thumbnail Fig. 4

Global stability of asteroid motion with a) and without b) the mutual interactions of Ceres, Pallas, Vesta, Iris, and Bamberga. The initial conditions are taken on a regular grid of 701 × 51 in (a,e). The color index, obtained through frequency analysis (Laskar 1993), is the measure of the diffusion in mean motion (and thus in semi-major axis through Kepler’s law) over two consecutive time intervals. Blue denotes very regular regions and red highly chaotic ones.

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4. Collision probability

The asteroid orbits present numerous close encounters. We have analyzed all close encounters among asteroids in S5a,S5b, and in an additional solution S5c similar to S5b, but with the opposite initial offset. S5a,S5b,S5c were computed over 121, 117, and 52 Myr, respectively and provide a total of 25 779 (resp. 10 509, 10 359, 4911) asteroid encounters closer than 0.01 AU. Because of the chaotic nature of the solutions, all these close encounters are seen as independent events and are added in a single set of random variables, with each random variable corresponding to the distances of close approach Δi,j for a given asteroid pair (i,j). We then assume that the density probability of this random variable is linear, with zero probability for the exact collision (Δi,j = 0). This is the case for the Rice density (Rice 1945), which represents the density probability of the distance to the origin of a two-dimensional Gaussian random variable, which is linear in the vicinity of the origin. A linear fit is then performed for each pair (i,j) (Fig. 5) in order to provide the probability of collision, that is the probability for Δi,j to be smaller than the sum of the radii of the two bodies (Table 3). This direct method differs somewhat from previous methods that consider large sets of asteroids (Farinella & Davis 1992; Yoshikawa & Nakamura 1994; Vedder 1996, 1998). The probabilities of collision (Table 3) are on the order of 0.1% per Gyr, the highest being the probability of collision of Ceres with Vesta, which reaches 0.2% per Gyr.

5. Conclusion

thumbnail Fig. 5

Probability density of the close encounters for the pairs 1–2, 1–4, 1–7, 1–324. All approaches closer than 0.01 AU are collected in bins of 0.0001 AU. The number of hits in each bin is plotted versus the distance of closest approach Δ (in AU/1000). The solid line is the linear fit to this data, which provides, after normalization, the probability density function for the close approaches (Table 3).

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We show here that when we consider their collective effect, the motions of the main asteroids are much more chaotic than previously thought. This is particularly true for the motions of the largest asteroids, Ceres and Vesta, which can be mistakenly considered as regular when their mutual perturbations are not taken into account. More important, the strong interactions during close encounters also affect the planetary motions, and appear as the main limiting factor for establishing very long-term planetary solution for the Earth eccentricity, beyond 60 Myr, which

would be useful for paleoclimate studies. Moreover, this limit appears to be an absolute limitation, because the horizon of predictability for the motion of Ceres and Vesta is less than 500 kyr (Fig. 3). We also expect that most of the elements of the Vesta family will be perturbed by the main asteroids and their motion is largely chaotic as for the few that are considered here (Fig. 4). It should thus be investigated whether this can induce some additional diffusion in the elements of the Vesta family that has not been considered before (Carruba et al. 2007). Finally, we have obtained a reliable estimate for the collision probability of several asteroid couples, which amounts to 0.2% per Gyr for Ceres and Vesta.


1

We assimilate here the dwarf planet Ceres as an asteroid.

Acknowledgments

This work was supported by GTSnext, ANR-ASTCM, INSU-CNRS, PNP-CNRS, and CS, Paris Observatory. J.L. thanks P. Michel for discussions.

References

All Tables

Table 1

Close encounters for S5a (top) and S5b (bottom).

Table 2

Lyapunov exponents (LE) and Lyapunov times (LT).

Table 3

Collision probability (in 10-3 per Gyr) for each asteroid pair (i,j).

All Figures

thumbnail Fig. 1

Evolution of the Earth’s eccentricity uncertainty δe in S0a,b (dotted line) and S5a,b (solid line). In both cases, the initial difference at t = 0 is δe = 1.4 × 10-12. The thin solid line, with slope  −1/10 represents the secular chaotic behavior (Laskar 1989, 1990). The eccentricity solution loses its validity when its uncertainty approaches its averaged value (δe ≈ 0.03).

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In the text
thumbnail Fig. 2

Relative variations in thetotal orbital energy of the Moon and planets for S0a (red), S0b (green), S5a (blue), S5b (purple). The large variations in energy of S5a,b correspond to asteroid close encounters (Table 1).

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In the text
thumbnail Fig. 3

Decimal logarithm of the differences in longitude for the asteroids in the S0a,b (dotted line) and S5a,b (solid line) solutions. The initial conditions differ by about 1.4 × 10-10 in longitude.

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In the text
thumbnail Fig. 4

Global stability of asteroid motion with a) and without b) the mutual interactions of Ceres, Pallas, Vesta, Iris, and Bamberga. The initial conditions are taken on a regular grid of 701 × 51 in (a,e). The color index, obtained through frequency analysis (Laskar 1993), is the measure of the diffusion in mean motion (and thus in semi-major axis through Kepler’s law) over two consecutive time intervals. Blue denotes very regular regions and red highly chaotic ones.

Open with DEXTER
In the text
thumbnail Fig. 5

Probability density of the close encounters for the pairs 1–2, 1–4, 1–7, 1–324. All approaches closer than 0.01 AU are collected in bins of 0.0001 AU. The number of hits in each bin is plotted versus the distance of closest approach Δ (in AU/1000). The solid line is the linear fit to this data, which provides, after normalization, the probability density function for the close approaches (Table 3).

Open with DEXTER
In the text