Unstable orbits: orthogonality condition

1. Introduction

If three bodies move in closed orbits, exposed only to the mutual gravitational interactions, they, as a rule, gradually enter into a gravitational resonance: periods of their rotations become related as the rational fractions [12]. This proportionality is the consequence of the fact that, over a time, such a system assumes dynamical configuration which provides minimal potential energy to every individual body. That phenomenon may lead to, either and for the most part, stabilization of the orbits or, for some period ratios, may also generate chaotic motions.

This paper offers an explanation for the genesis of the chaotic motion. It is based on the Newton’s three-body model consisting of one massive and two considerably smaller bodies turning around it (m_0 \gg m_1,m_2), so that it can be applied to the Sun and two planets,  as well as to the Sun, planet and asteroid (m_0 \gg m_1,m_2 \approx 0).

In the work M. Marjanov: “Three – Body System: Stable and Chaotic Orbits” (2011, lecture on the Math. In. SANU, Department of Mechanics), based on the same model, it was shown, by use of the perturbation technique, that the chaotic motions zones are situated around the T_1:T_2 \sim 3:1 and 1:3 resonances. Obtained outcome was in accordance with the Scholl & Froeshle’s (1974) and the J. Wisdom’s (1983) results that the most prominent (Kirkwood) gap in the Main Belt coincides with the orbit of the body rotating three times faster than Jupiter around the Sun.

2. Coordinate systems

For simplicity we assume that the small bodies 1 and 2 circulate around the body 0.

Let us introduce three kinds of reference frames lying in the invariable plane. The first one Cxy is the inertial frame of reference (C is the common center of masses), the second – 0x^'y^' (0x^', 0y^' parallel with Cx and Cy) represents transport reference frame, while the polar coordinates R_1,\alpha and R_2,\beta, defining positions of the small bodies with respect to the large one, are the relative frames of reference 1R_1,\alpha and 2R_2,\beta (Fig. 1). Reference frames

Fig. 1 – Reference frames

3. Initial conditions

If, given the masses of the bodies, one adopts initial positions and velocities arbitrary, the outcome may be either motions with one, or three open orbits, collision of two, or three bodies, either motions in closed orbits.

It makes no sense to study the three-body system allowing possibility of its imminent decomposition, so we are going to consider only these initial conditions producing bounded orbits, without presumption about their stability.

In order to simplify the initial condition problem we shall take that the bodies are aligned in the moment t(0) = t^0. We choose conjunction of the small bodies with respect to the large one (Fig.2) for the initial (“capture”) constellation.

Accordingly, it was assumed that the bodies lie on the Cx axis of the inertial reference frame Cxy.

Distances from the massive body R_i(0) , i = 1,2 can be chosen arbitrary. On the other hand, in order that the orbits become bounded, the velocities \vec{v}(0) = \vec{v}^{\,0} have to be orthogonal to the Cx axis, while their intensities have to be smaller than the escape (parabolic [6]) velocities.

Aiming to obtain the orbits, which are close to the regular paths of m_1 and m_2, we shall take that in the assumed capture constellation motions of the small bodies are dynamically balanced with the massive body attraction force, only. Consequently, initial angular velocities of these bodies can be easily calculated using the Newton’s law of gravitation. Initial Constellation, Conjunctions and Oppositions

Fig. 2 – Initial Constellation, Conjunctions and Oppositions

4. Chaotic motion generating resonances

It is obvious that in the case of the three bodies’ conjunction the attraction force acting on the outer small body is maximal and the one, acting on the inner body is minimal. On the other hand, when the small bodies are in opposition with respect to the large one, the outer body is exposed to the minimal attraction, while the inner body is exposed to the maximal attraction force (Fig. 2). As said, we assume that conjunction \beta(0) – \alpha(0) = 0 represents the initial position the bodies. Dynamical and therefore the kinematical characteristics of the system have extremes when three bodies are aligned (Fig. 2)

\beta(t^k) – \alpha(t^k) = n\pi,   k,n = 0,1,2,3,…,

i.e., when the small bodies are eider in conjunctions (n = 0,2,4,…,) or in oppositions (n = 1,3,5,…,) with respect to the large one. Index k designs the number of alignment and t^k represents the corresponding time.

The law of the angular momentum conservation, valid for the closed system in consideration, can take place only due to the redistributions of the angular momentums between single bodies during movements. The change of rate of body’s angular momentum around the point 0, between two successive linear constellations is equal to the impulse of the small bodies’ interaction moment around 0 (Fig. 3) in the corresponding interval. In the case of repeatedly coincidences of the same linear constellations with the same two directions in the orbital plane, the cumulative effect of these impulses produce changes in the angular momentums, generating deformations of the orbits.

That is only possible if these directions are mutually orthogonal.

Therefore, orthogonal directions of conjunction and opposition represent a condition for the occurrence of instability of orbits.

It is, of course, related to all three orbits. We shall see later, that, even the orbit of the massive body must be distorted to some extent.

Let w_1^a and w_2^a be average orbital velocities of small bodies and the time interval between two successive alignments h^k = t^kt^{k-1}. Conjunctions of the small bodies will always fall on the line from which the movements started and the opposition at the orthogonal direction under following conditions

    \[ $\alpha$($t^k$ - $t^{k-1}$) = $w_1^a$$h^k$ = $\frac{m\pi} {2}$; $\beta$($t^k$ - $t^{k-1}$) = $w_2^a$$h^k$ = $\frac{(m+2)\pi} {2}$; \]

k = 1,2,3,…; m = 1,3,5 …

If masses m_1 and m_2 are of equivalent order, these two directions will eventually, due to the orbital precessions, become apsides of the orbits (Fig. 4). When the body comes to apogee, it will be exposed to the minimal gravitational force.

So, the resonances T_1:T_2 = w_2^a:w_1^a producing alignments in two orthogonal directions are:

T_1:T_2 = (m + 2):m = 3:1, 5:3, 7:5, …;m = 1,3,5 …

The lower-order resonances correspond, obviously, to the T_1:T_2 decrease. When T_2 \rightarrow T_1, the bodies will either enter into the same orbit or they will form a binary system.

Apparently, the 3:1 resonance, providing alignments whenever the more distant, slower rotating body hits one of the orthogonal directions, more than any other, affects deformations of the orbits [8], [9]. In the case of 5:3 resonance, for instance, the outer body passes twice the mentioned directions before alignment with the inner, faster rotating body (passing, simultaneously, the alignment directions four times). Doubtless, the alignments frequency has an important role when the appearance of the chaotic motion is in question. Let us call 3:1 relationship the critical resonance.

It should be noted that a sequence of inverse resonances T_2:T_1 = m:(m + 2) = 1:3, 3:5, 5:7, … also produces unstable orbits, if the body 1 is the inner one. Resonance 1:3 is also critical. Both small bodies are associated with an internal and an external domain of unstable orbits which correspond to these resonances.

As said, in the case of orthogonal conjunction and opposition directions the cumulative effect of the impulse of the small bodies’ interaction moment, around 0, generates distortion of their orbits. It is important to emphasize that a long-term dynamics is in question: a noticeable deformation of an orbit sometimes requires millions of the circulations around the massive body. Clearly, the rate of changes of the orbit’s form is inversely proportionate to the size of the mass: shape of the smaller body’s orbit is more affected by the gravitational interaction than the shape of the more massive one.

It should be noted that, in fact, all three bodies move in resonance. Period of orbital rotation of the massive body corresponds to longer period of two small bodies. Accordingly, it should be consistent to write T_0:T_1:T_2 = 3:3:1, 5:5:3, 7:7:5, etc. (see Fig. 5). Gravitational interaction between small bodies

Fig. 3 – Gravitational interaction between small bodies, circular relative rotations, resonance 3:1

Orbits of the bodies 1 and 2, resonance 3:1

Fig. 4 – Orbits of the bodies 1 and 2, resonance 3:1

We point out the following fact. As a general rule, orbits of the small bodies are assumed to be Kepler’s ellipses with the massive body in their foci. In fact, ellipses (circles, in this paper) represent only approximation of their real paths. They have such forms only in the transport reference frame 0x’y’ (Fig. 1), that is, if the massive body does not move in the Laplace’s plane. In the inertial coordinate system, these trajectories are the results of superposition of one ellipse and the massive body’s miniature orbit, which, on the other hand, represents the sum of two reduced ellipses and it is not elliptical, at all. Although orbits of the small bodies look like two ellipses, the tiny large body’s orbit even does not have such a shape. Sometimes, depending on the relationships between masses, it even may form the loops along which this body moves ostensibly retrograde with respect to the small bodies. For example, in the case of, for our purpose significant 3:1 resonance, with m_1:m_0 = 0,001 and m_2:m_0 = 0,0015, orbit of the massive body will, approximately, have the following form (Fig. 5).

Orbit of the body 0

Fig.5 – Orbit of the body 0: resonance 3:1 (3:3:1), m_1:m_0 = 0,001 , m_2:m_0 = 0,0015

5. Domains of the unstable orbits in the solar system

Neglecting all the other influences and assuming that bodies in the Solar System are points with appropriate masses, moving in the same, invariable plane, the Sun, together with any couple (not necessarily mutually closest) heavenly bodies, provided that they do not form the binary system, correspond to our model. It goes without saying, we always consider that these three bodies were isolated from the gravitational influences of the other bodies.

In such a way, combining the Sun and two planets, or the Sun, planet and asteroid into Newton’s three body system, it was possible to point out, roughly, some of the Solar System regions in which the orbits may become chaotic. Orbital eccentricities of the numerous meteoroids, comets and asteroids in these zones steadily increase and their paths began to cross the solar system planets’ orbits.

Let us take that R_i represents the distance between Sun and the planet i (i = 1, 2…8). Then, radii of the unstable orbits corresponding to the resonances 3:1, 5:3, 7:5…, approximately, are R_i and 0,48R_i; R_i and 0,71R_i; R_i and 0,80R_i; … while those corresponding to the resonances 1:3, 3:5, 5:7… are R_i and 2,08R_i; R_i and 1,41R_i; R_i and 1,25R_i; … For consistency, we shall call first of these pairs the critical orbits.

Since the problem belongs to the domain of nonlinear dynamics, to each critical resonance corresponds, of course, a range of an infinite number of initial conditions generating bundles of infinitely many orbits that lie in zones of bifurcations and zones of chaotic motions.

In the work [13] perturbation technique was used to establish whether, for the given initial relations R_2(0):R_1(0) stable or chaotic orbits arise. By use of the second-order perturbations, and, inevitably, the finite number iterations, it has been found that for the initial conditions which generate orbits in the belts around 3:1 and around 1:3 resonances, the results become divergent. Out of these zones the solutions converged toward the stable solutions. It took only a few iterations in order to find the domains of stable and chaotic orbits, but it was more difficult to establish the interspace i.e., domains of bifurcation. Thousands of iterations are required to establish whether specific initial conditions provide stable or unstable orbits in these zones. One approximate criterion was adopted for that purpose. Extensions of the planets unstable orbits zones mainly depend on the mass relations m_1:m_0 and m_2:m_0.

Somehow, this procedure resembles the check of system dynamic sensitivity to initial conditions by use of the Lyapunov characteristic exponent (LE), applied in the Wisdom’s work [9].

Taking into account only critical resonances in the work [13] were roughly delineated zones of the stable orbits (LE < 0), zones of bifurcations (LE ~ 0) and zones of chaotic orbits (LE > 0) in the solar system.

Application of the higher-order perturbations, as well as the higher-order difference approximations, would produce zones of chaotic orbits around 5:3, 7:5… and 3:5, 5:7… resonances, also.

There are a lot of the small Solar System bodies orbiting the Sun. Most of these bodies inhabit two asteroid belts. The first one is between Mars and Jupiter, the Main Belt, situated on 2,1 to 3,3 AU distance from the Sun, while the second one, the Kuiper Belt, lies in the region beyond the planets extending from the orbit of Neptune, at 30 AU, to, approximately, 55 AU from the Sun.

The beginning and the end of the Main Belt correspond to the outer critical orbital radii of the Sun, Earth Asteroid (~2, 08 AU) and Sun-Mars-A (~3, 16 AU) systems, while the most prominent Kirkwood gap (~2, 50 AU) corresponds to the inner critical orbital radius of the Sun-Jupiter-A system [8],[9],[13].

It is also worth mentioning that there are, practically, no small bodies in the Kuiper belt moving around the Sun three times slower than Uranus (~40 AU).

As known, Saturn is surrounded by numerous rings primarily composed of objects ranging in size from microscopic to hundreds of meters; each moves in its own orbit about the planet. Also, 62 known moons and at least 150 moonlets turn around this planet.The gaps between Saturn’s rings could be explained by the chaotic motions causing resonances in the Saturn-moon-A systems. Namely, the empty spaces between the rings probably coincide with the internal and external instability domains corresponding to the bigger bodies.

6. Conclusion

This paper offers a possible explanation for the genesis of the chaotic motion. Orbital resonances which cause chaotic motions in the Newton’s three-body system were determined. Since the Sun, together with any couple (not necessarily mutually closest) heavenly bodies correspond to this model, it was possible to point out, roughly, some of the Solar System regions in which the orbits may become chaotic.


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