5. Qualitative dynamical behavior

The well known asteroid classification in Apollo, Amor and Athen, summarized in the following table, is useful as a classification of instantaneous, osculating orbital elements, but is by no means stable for time spans of $100, 1\,000$ years or more [Milani et al. 1989].

  

Figure: In the phase space (current $e$, $a$) we have plotted 987 asteroids drawing with a cross the 94 not Earth crossing Apollo, with a circle the 11 Earth crossing Amor and with a circled cross the 3 not Earth crossing Athen. The two curves plotted represent $a' = a(1-e)$and $a' = a(1+e)$ so that the Earth crossing area is between them.

  

Figure: In the phase space (minimum $e$, $a$) some Apollo asteroids have moved in the non crossing area; these non Earth crossing Apollo are marked with a cross.

  

Figure: In the phase space (maximum $e$, $a$ five Amor asteroids have moved in the Earth crossing region; these Earth crossing Amor are marked with a circle.

APOLLO	$a > 1$	perihelion $<$ 1.017
AMOR	$a > 1$	perihelion $>$ 1.017
ATHEN	$a < 1$	aphelion $>$ 1

Over a longer time span, conditions of the form

$$a(1-e)<a'<a(1+e)$$

cannot assure the existence of crossings, unless we use the simplistic assumption that the argument of perihelion $\omega$ precesses without changes in eccentricity and inclination; therefore the Apollo + Athen class is not equivalent to the class of Earth crossing asteroids, that are by definition the asteroids whose orbit crosses the Earth's during their secular evolution.

As a result of the full secular evolution, we find more complicated behaviors than the simple quadruple crossing Apollo/Athen and the simple non-crossing Amor. This results, e.g., from the $e-\omega$coupling (the minimum of $e$, occurring when $\omega=0,\pi$, is such that $a(1-e)>a'$). There is also the case in which the argument of perihelion $\omega$ librates around either $\omega=\pi/2$ or $\omega=3\pi/2$; in this case an asteroid can be an Apollo during a fraction of the $\omega$ cycle, and still never cross the orbit of the Earth (Kozai class dynamics, [Milani et al. 1989]). In other cases, the number of crossings can be 8 per period of $\omega$; for the Athen asteroids, libration around $\omega=0,\pi$ can occur [Michel and Froeschlé 1997].

  

Figure: Time evolution of Near Earth Asteroids whose perihelion argument is circulating; the complete evolution of these objects can be obtained from the partial one, plotted here, employing the symmetries of the averaged perturbing function in the plane $(e,\omega )$. The node crossing lines with the Earth (E), Mars (M) and in some cases with Venus (V) are also shown.

  

Figure 8: Time evolution of Near Earth Asteroids whose perihelion argument is librating.

In figure 4 we have plotted in the phase space (current $e$, $a$) the orbital elements of the 987 objects examined and we have enhanced the ones that, during their time evolutions, do not behave like their initial classification in Apollo, Amor and Athen should suggest: indeed, we have found 94 Apollo (out of 504) which are not Earth crossing (crosses), 11 Amor (out of 408) which are Earth crossing (circles) and only 3 Athen (out of 75) not Earth crossing (circled cross).

We can observe that in figure 5 the non Earth crossing Apollo have moved outside the Earth crossing area while in figure 6 the Earth crossing Amor have moved inside it.

In our integration we have detected two of the main qualitative behaviors in the $(e,\omega )$ plane [Gronchi and Milani 1999]: direct circulation and counter-clockwise symmetric libration of perihelion argument. No asymmetric libration (that is a libration whose center is not an integer multiple of $\omega=\pi/2$) has been found in this sample of orbits, although we know it could occur for comet/Centaur type orbits.

In the following figures there is a sample of the secular evolution of the NEA's in the $(e,\omega )$ plane; the level lines of the averaged perturbing function are plotted and the time evolution of the objects considered is enhanced by crosses along the curve representing the constant value of their perturbing function. Only a portion of the evolution is plotted here, but this is enough to compute by symmetry [Gronchi and Milani 1999] the evolution at any time (within the validity of this theory).

In figure 7 there are four examples of circulation of $\omega$: the first object, 1991TT is an Apollo asteroid that is preserved from Earth crossings by a $e-\omega$ coupling mechanism, that is the eccentricity oscillates and reaches its minimum value at the integer multiples of $\pi$ (when the absidal line corresponds to the mutual node line); the second asteroid, 1991BB, is an ordinary quadruple crosser (crosses the Earth four times for period of revolution of $\omega$) while the third, 1994JX, is one of the 11 Amor asteroids that become crosser during their evolution (it is even an octuple crosser). The last case, 1998FG12, is peculiar because it crosses the Earth orbit in a tangent way, staying in a neighborhood of the crossing curve for a very long time; this object will have a comparatively high probability of impacting the Earth, but this will happen during the fifth millennium AD.

In figure 8 there are four examples of librating objects: the first is one of the 3 Athen asteroid preserved from crossing the Earth orbit, and it is just its symmetric libration around $\omega = 2\pi$ that provides to its safety. On the contrary, the libration in the second example, (2102) Tantalus, forces it to cross the orbits of Venus, the Earth and Mars, each one four times for period of the libration argument.

The dynamics of the asteroid 1999 AN10 has been extensively studied because of its very small distance to the orbit of the Earth at both nodes. When fewer observations were available, it was not possible to exclude an impact with our planet in the next century. As figure 8 shows, the secular evolution is now very close to a double crossing at both the ascending and descending node with the Earth; the period of libration of $\omega$ is very long, this implies that the hazardous proximity of this orbit to the orbit of our planet will be maintained for centuries. In fact precisely the interest for this case forced us to revise our algorithm to make it efficient and stable also in the double crossing case, as discussed in Section 2.3.

The last asteroid of figure 8, 1997NC1, has a very strange averaged dynamics: the level curve of the averaged Hamiltonian, to which its motion is bounded, is very close to the separatrix between libration and circulation, so that, while its perihelion argument in the long run circulates, its orbit draws an arc of libration near the integer multiples of $\pi$.