Astron. Astrophys. 353, 797-812 (2000)

5. Long-term integrations and results

The dynamical evolution of the 20 bolides has been studied by integrating the 26 orbits listed in Table 5. We recall that for 5 bolides, two or three different sets of starting orbital elements have been determined (see Sect. 2). The integrations were carried out with a Bulirsch-Stoer variable step-size technique (Stoer & Bulirsch 1980), optimized for dealing accurately with planetary close encounters (cf. Michel et al. 1996a). The dynamical model included all the planets except Pluto and Mercury, with the mass of the latter added to that of the Sun. The integration interval spanned at least 5 Myr backward and forward in time, with a total timespan of 10 Myr (this was extended in some specific cases).

As discussed in several recent papers which deal with long-term integrations of planet-crossing bodies, the results of the numerical integrations cannot be seen as deterministic reconstructions or predictions of the real evolutions. Nevertheless, they are very useful to provide qualitative and/or statistical information on the most common patterns of the orbital behaviours as well as on the efficiency of different dynamical mechanisms and the corresponding lifetimes. Integrating backward and forward in time just provides a simple way of doubling the size of the sample and thus of improving the statistics (note that backward integrations cannot provide information on the sources of the bodies, neither individually nor statistically). We will first consider all the bodies which either have a collision with the Sun or are ejected from the Solar System, and discuss separately the backward and forward integrations. Then, we will describe the evolutions of bodies strongly affected by planetary close approaches. The main results of our integrations are summarized in Table 7.

Table 7. Summary of simulations - Part 1. KLD stands for "Kozai-like dynamics", CE for "close encounter". The time spans over which the different dynamical mechanisms are active are given in brackets, ( Myr).

Table 8. Summary of simulations - Part 2.

Table 9. Summary of simulations - Part 3.

5.1. Backward integrations

A collision with the Sun is recorded for 10 orbits, whereas 4 others are ejected outside Saturn's orbit. Half of these 14 orbits have dynamical lifetimes shorter than 1 Myr (among them 4 collide into the Sun and 3 are ejected).

The and Jovian mean motion resonances are responsible for the ejection of 2 bolides: Abee-1 (1) and Hradec (20) (Fig. 4), respectively. As for Abee-1 (1), a close encounter with the Earth at time Myr (see Table 7) injects it into the resonance, which increases its eccentricity from 0.2 to 0.9. As a consequence, the body gets close to Jupiter's perihelion distance and eventually an approach to this planet ejects it out of Saturn's orbit.

Fig. 4. Orbital evolution of bolide 20 (Hradec) in the time span  Myr. The left-side panels show the semimajor axis a (AU), eccentricity e and inclination i (degrees) vs. time, whereas the right-side panels show the critical arguments for the , and secular resonances. Note that in the backward integration a is locked in the 2/1 mean motion resonance with Jupiter.

The Hradec bolide (20) is located in the resonance during almost all its backward evolution. It is also temporarily located in the , and secular resonances (these are resonances betweeen the average precession rate of the perihelion longitude of the body and the corresponding eigenfrequencies for the secular evolution of the planetary perihelia). The presence of secular resonances inside the mean motion resonance is a well-known source of chaotic motion (Morbidelli & Moons 1993, Moons & Morbidelli 1995). As a consequence, the eccentricity is eventually pumped up to 0.98. Then a close encounter with Jupiter extracts the orbit from the resonance and the bolid is rapidly ejected from the solar system.

The Dobí II (22) and EN270796 (23) bolides have both semimajor axes larger than that of Jupiter, high eccentricities and low orbital inclinations (see Tables 1 and 5). These orbits are very similar to those of many Jupiter-family comets; being close to the orbital plane of the planets, they undergo frequent close encounters with Jupiter. Thus, a close approach to Jupiter ejects them from the Solar System after only 0.1 and 0.04 Myr, respectively.

Different dynamical mechanisms are at the origin of the recorded solar collisions, depending on the starting locations of the small bodies. When the orbits have a semimajor axis AU, the dynamical mechanisms responsible for the collision against the Sun are those described for the first time by Farinella et al. (1994):

• the secular resonance (for bolides 13-Ulm and 19-EN081195B),

• the mean motion resonance with Jupiter (bolides Abee-2 (2) and Polná (9)),

• the overlapping of secular resonances inside mean motion ones (bolides nos. 3-Glanerbrug-1, 21-Ózd, 15-Kouim and 26-Honduras-3). For instance, bolide 3-Glanerbrug-1 is located in the resonance with Jupiter and also in the Kozai resonance, its argument of perihelion librating around (Fig. 5). Bolide Ózd (21), while being in the resonance with Jupiter, between  Myr and  Myr is also affected by the and secular resonances (involving the average precession rates of the perihelion longitude of the Earth and Mars); from Myr to Myr, the orbit is then in , , and . Note that the location of secular resonances involving the orbital frequencies of the terrestrial planets has been determined only recently (Michel 1997). Here we observe for the first time, for a body with  AU, the occurence and effect of the overlapping of a mean motion resonance with the , and secular resonances. Finally, during the interval Myr bolide Kouim (15) is located in the 4/1 mean motion resonance as well as in the and secular resonances; Honduras-3 (26) is also located in these three resonances, but during the timespan Myr.

Fig. 5. Orbital evolution of bolide 3 (Glanerbrug-1) in the time span  Myr. The left-side panels show the semimajor axis a (AU), eccentricity e, inclination i (degrees) and the critical argument of the 2/1 Jovian mean motion resonance, whereas the right-side panels show the critical arguments for the , and secular resonances plus, on the top, the body's argument of perihelion , which shows episodes of libration around due to capture into the Kozai resonance. This orbits is almost always locked in the 2/1 mean motion resonance with Jupiter.

Marshall Islands-1 (orbit 10) over about 0.5 Myr is located in the overlapping region of the and resonances. Such overlapping of two secular resonances with the terrestrial planets (here, the Earth and Mars) has been already analyzed by Michel (1997), but only for orbits with  AU. In the present case, it occurs at  AU but has a similar effect, i.e. it pumps up the eccentricity so that after several close encounters e reaches unity (Fig. 6).

Fig. 6. Orbital evolution of bolide 10 (Marshall Islands-1) in the time span  Myr. The left-side panels show the semimajor axis a (AU) and eccentricity e vs. time, whereas the right-side panels show the critical arguments for the and secular resonances. Note that e reaches unity when the orbit is affected by these secular resonances with the Earth and Mars.

Marshall Islands-2 (orbit 11) hits the Sun while its semimajor axis is smaller than 2 AU. In this case the eccentricity is increased up to 1 due to the fact that the body is located in an overlapping region of two secular resonances: the and resonances, which involve the orbital frequencies of Venus and Jupiter, respectively. The fact that this dynamical mechanism can also lead to a solar collision has recently been pointed out by Gladman et al. (1999).

5.2. Forward integrations

As shown in Table 7, in this sample of integrations 12 bodies hit the Sun and 5 are ejected from the solar system. 8 over 17 objects have a lifetime shorter than 1 Myr ( and , respectively).

While in the backward integration Glanerbrug-1 (3) was driven into the Sun, in the forward one it is ejected outside Saturn's orbit. Fig. 5 shows its evolution. Until Myr it is located in the mean motion resonance. Then it leaves the resonance due to a planetary close encounter. During the whole forward integration, it is also temporarily located in the and secular resonances, the resonant arguments and alternating between circulation and libration (here designates the longitude of perihelion). In addition, the orbit is located in the Kozai resonance, the argument of perihelion librating around . Consequently, the eccentricity evolves in a strongly chaotic manner and undergoes large oscillations between 0.4 and 0.9. Then the bolide is ejected outside Saturn's orbit at Myr, following a close encounter with Jupiter.

The inclination of Marshall Islands-2 (11) remains very low during the entire integration timespan, varying between about 2^o and 5^o. As a consequence, the body suffers frequent planetary close encounters and the eccentricity behaves chaotically, with values ranging between 0.3 and 0.75. Then a close encounter with the Earth injects it in the resonance, where its eccentricity oscillates between 0.6 and 0.9. Finally, a close approach to Jupiter ejects the bolide outside Saturn's orbit at Myr.

Like in the backward integration, the comet-like bolide EN270796 (23) is ejected after only 0.19 Myr by a close encounter with Jupiter. This short lifetime is quite typical for short-period comets (see e.g. Levison & Duncan 1994).

The case of St. Roberts (14) is quite unusual. Since its inclination is very small, it suffers numerous close approaches, especially with Mars. Moreover, between Myr and 3.5 Myr it is located in the overlapping region of the and secular resonances, and its eccentricity increases from 0.5 to 0.9. At this time, although the semimajor axis is approximately 1.8 AU, it undergoes a sequence of very close encounters with both Venus and Mars, which eventually eject it from the Solar System.

It is worthwhile noting that among the 12 solar collisions which have been detected, 7 are caused by dynamical mechanisms which involve secular resonances. For 5 bodies, the solar collision occurs when their semimajor axis is  AU.

As shown in Fig. 7, the orbital evolution of Abee-2 (2) is affected by secular resonances with both the terrestrial and the giant planets during the entire forward integration timespan. The eccentricity at first is increased as an effect of , then due to both and . Finally, the body enters the region where , and are active so that the eccentricity is pumped up to unity. Note that the eccentricity increase is quite regular and its oscillations are coupled with those of the resonant arguments. A similar behaviour is found for EN081195B (19). However, its initial eccentricity is already 0.83, and the orbit lies in both and during the whole forward integration. Then, the eccentricity increases up to 1 in a regular manner.

Fig. 7. Orbital evolution of bolide 2 (Abee-2) in the time span  Myr. The lower left-side panels show the semimajor axis a (AU) and eccentricity e vs. time, whereas the other panels show the critical arguments for the , , , , and secular resonances.

Since its inclination is relatively small, Marshall Islands-1 (10) undergoes many close encounters with Mars and the Earth. The evolutions of the semimajor axis and eccentricity are thus correlated during the first 3 Myr. Then the body undergoes some Kozai-like dynamics - the oscillations of the eccentricity becoming larger, with an amplitude - and is also temporarily located in and , the corresponding resonant arguments alternating between libration and circulation; consequently, the eccentricity is secularly increased up to unity within 2.4 Myr.

As indicated in Table 7, the eccentricity of Dresden (12) is first increased up to 0.7 as an effect of and ; then the orbit enters , which pumps its eccentricity up to 1 within 0.5 Myr, causing a collision with the Sun.

Two other bolides have a collision with the Sun when their semimajor axes are AU. amberk (16) and Tisza (18) have semimajor axes between 1.2 and 1.6 AU. They become Sun-grazing due to their location in the secular resonance (Fig. 8). As already noted, this new route to the Sun has been recently identified by Gladman et al. (1999).

Fig. 8. Orbital evolution of bolide 18 (Tisza) in the time span  Myr. The left-side panels show the semimajor axis a (AU) and eccentricity e vs. time, whereas the right-side panels show the critical arguments for the and secular resonances. The resonance is clearly responsible for the eventual gradual growth of e up to unity.

Since the Dobí II (22) collides with the Sun only after 0.012 Myr, we have been unable to detect any specific transport mechanism. However, its initial conditions imply that this orbit is clearly of a comet-like type.

The dynamical evolution of the last four bodies, namely Abee-1 (1), Polná (9), Kouim (15) and Ózd (21), are affected by both mean motion and secular resonances, as indicated in Table 7. As a result, the evolution of their eccentricity is strongly chaotic. All of them hit the Sun when their semimajor axis is larger than 2 AU.

5.3. Orbits dominated by close approaches

Seven orbits (4, 5, 6, 7, 8, 24, 25) have their semimajor axis strongly affected by close planetary encounters. Indeed, as illustrated in Fig. 9, this parameter undergoes a sort of random walk due to frequent planetary close approaches, both shallow and deep ones. Moreover, six orbits (5, 6, 7, 8, 24, 25), which over the whole integration time (at least 10 Myr) have a semimajor axis AU, are temporarily located in the region where the and nodal secular resonances overlap, causing increases of the inclination. Kozai dynamics is observed for orbits 6, 7, 8 (see Table 7), either temporarily or during the entire timespan (see e.g. Fig. 10). In this regime, the orbits are often protected from close approaches, and therefore their lifetime is lengthened.

Fig. 9. The semimajor axis evolution of bolides 4, 5, 6 and 8 (Glanerbrug-2, EN220991, Lugo-1 and Meuse, respectively) in the time span  Myr. The frequent jumps, resulting in a sort of random walk, are caused by planetary close encounters.

Fig. 10. Evolution in the vs. plane of the orbit of bolide 6 (Lugo-1). This pattern is typical of Kozai-like dynamics.

Let us consider now the evolution of these orbits in the a-e plane. During the entire integration time, bolide Honduras-1 (24) crosses all the region of near-Earth space, being temporarily a ( AU), Aten ( U,  AU), Apollo and Amor-like body (Fig. 11). In the backward integration ( Myr), it is an Amor body with a semimajor axis always larger than 1 AU, and an eccentricity smaller than 0.2. Then it becomes an Apollo, i.e. its trajectory crosses Earth's orbit. Between 0.5 Myr and 2.5 Myr, it enters the region with  AU, defined as the region with AU and aphelion distance AU, and alternates several times between the  AU and Aten states. Finally it goes back in a Apollo-like orbit and then into the Amor region. This evolution shows nicely the continuous interchange, over a time scale of several Myr, among the different sub-populations of near-Earth objects.

Fig. 11. Orbital evolution of bolide 24 (Honduras-1) in the semimajor axis vs. eccentricity plane over the  Myr time span. Dashed and dotted curves correspond to orbits having perihelia and aphelia nearly tangent to the orbits of Mars, the Earth and Venus. While secular resonances at times affect the eccentricity (causing horizontal displacements in this diagram), close encounters with the Earth and Venus move the orbit roughly along lines of constant perihelion or aphelion distance (Michel et al. 1996b), bringing it through the Amor, Apollo, Aten and  AU regions.

A similar behaviour is found in other cases (see Figs. 12 and 13). Bolide Lugo (6, 7), for which 2 different orbits have been integrated, is always a body with  AU or an Aten body (i.e. its semimajor axis is always AU), entering and exiting several times into/from the two regions. Between Myr and Myr, the orbit of bolide 5 interchanges several times between the  AU and Aten states. Then its semimajor axis becomes AU and it becomes an Apollo. Finally, it re-enters the Aten region at Myr. On the other hand, the orbits of bolides 4 and 8 show the same behaviour but in the Amor/Apollo regions. As for bolide 25, it keeps always a semimajor axis  AU and thus remains an Apollo during almost all the integration time, but it makes short visits into the Amor region. Fig. 13 shows that its evolution occurs close to the  AU curve, as expected for a body whose evolution is dominated by Earth encounters.

Fig. 12. Orbital evolution in the semimajor axis vs. eccentricity plane of bolides 5, 6, 7 and 8 (EN220991, Lugo-1, Lugo-2 and Meuse, respectively), throughout the 10 Myr integation time span. Dashed and dotted curves correspond to orbits having perihelia and aphelia nearly tangent to the orbits of Mars, the Earth, Venus and Mercury. These orbits are affected by both resonances, which shift them horizontally, and close encounters, which move them near the lines of constant perihelion or aphelion distance. Thanks to the interplay of these two mechanisms, they wander through different regions of the a-e plane.

Fig. 13. Orbital evolution of bolide 25 (Honduras-2) in the a-e plane during the 10-Myr integration time span. Dashed and dotted curves correspond to orbits having perihelia and aphelia nearly tangent to the orbits of Mars, the Earth and Venus. The dominant role of Earth close encounters is apparent as the orbit keeps its perihelion distance close to 1 AU.

© European Southern Observatory (ESO) 2000

Online publication: December 17, 1999
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