Stable motions of one large and two small bodies under mutual gravitational interactions in closed, stable orbits are numerically investigated in this work.
As known, this is one of the oldest problems of the classical mechanics posed by Newton himself in his attempt to find laws of the Moon movements.
Orbits of the bodies were obtained in this work combining the perturbation technique and the method of the successive approximations. One example was given as an illustration of the exposed procedure. A somewhat intriguing result was obtained: the outcome of the arbitrary chosen initial conditions was the resonant motion of three bodies. Was this a pure chance?
Gravitational forces acting on the bodies meet at one point called center of the gravitational attraction. All the “exact” solutions of the three – body problem are given for the cases when center of attraction and center of mass coincide. However, excepting the restricted three – body problem, all such motions are unstable  and of theoretical interest, only.
Generally, center moves in the invariant plane describing the locus of these points. It was shown in the second part of this work how to find this curve if one large and two small bodies are in question. We propose the short name attractide for this locus.
1. Solution of the Newton’s Problem
1.1. Invariant Plane, Frames of Reference and Equations of Motion
We consider the problem of determining the motion of one large and two small bodies under no influence other than that of their mutual gravitation. They are moving in stable, bounded orbits. Position of the mass centre C of this closed system has to be determined from the condition
For the isolated systems one may formulate the linear momentum conservation law,
so the centre of mass moves with constant velocity
Since the system is closed, the angular momentum conservation law can be formulated also
Vector is orthogonal to the plane passing through the mass center C. This plane is called invariant or Laplace’s plane.
Every velocity consists of two components
= + (k=0,1,2)
The first one is the transport velocity (having direction of ) and the second is relative velocity lying in the invariant plane.
Obviously, only the relative motions in that plane are of interest for our purpose, so the problem can be reduced to the planar one.
Adopting the model in which the bodies and rotate around , we introduce three kinds of reference frames lying in the Laplace’s plane. The first one Cxy is the inertial frame of reference, the second is the transport r.f. 0x’y’ (Ox’, Oy’ parallel with Ox and Oy), while the polar coordinates , and , are the relative reference frames defining positions of the small bodies with respect to the large one (Fig. 1).
Fig. 1 – Reference Frames
Now, the simple laws follow out from (1.1) for the motions in inertial reference frame
Equations of motions of , and in that plane are
1 2. Segregation of Dynamical and Kinematical Quantities
Let’s introduce relations between masses and and mass of the large body 0 as “small numbers”
and write down the known relations following out from (1.5), (16) and (1.7)
and dots over the symbols denote differentiation with respect to time.
Position vectors of the bodies 1 and 2 are now
and similar expressions can be formulated for their velocities and accelerations
In these equations
From the expressions (1.13) – (1.16) it is evident:
- motions, velocities and accelerations of three bodies are entirely defined if relative motions, velocities and accelerations of the small bodies around the large one are known;
- while sizes of the orbits and intensities of the velocities and accelerations of the small bodies are of the order O(), the corresponding quantities defining motions of the large body have to be of the order O().
It’s convenient to adopt , , and () to be variables in the exposed problem. They can be expanded into the power series of small number :
We shall assume that the first terms in these series define initial positions of the bodies
and that the other terms are homogeneous functions of time
1.4. Initial Conditions
If, given the masses of the bodies, one adopts initial positions and velocities arbitrary, the outcome may be either unstable motion with one, or three open orbits, collision of two, or three bodies, either stable motion in closed orbits.
It makes no sense to study the three – body system allowing possibility of its decomposition, so we shut out from consideration unstable movements, taking into account only these initial conditions producing stable motions and bounded orbits.
In order to simplify the problem of initial conditions, it’s convenient to take that the bodies are aligned in the moment t = 0. Conjunctions and oppositions of the bodies 1 and 2 with respect to the body 0 are positions producing extrema in the potential energy field and consequentially, extrema in the related kinematical quantities.
Distances between bodies (0) can be chosen arbitrary. When, by use of (1.5), the center of mass C is found it’s suitable to adopt Cx axis along the aligned bodies.
It’s a little bit more complicated with initial velocities. From the eq. (1.6). it follows out that only two of these vectors are independent. In order that the orbits become bounded these vectors have to be orthogonal to the axis Cx , while their intensities have to be situated between the initial dynamical equilibrium velocities and escape velocities.
1.5. Successive Approximations of the Orbits
Now, one can approach the problem assuming that the small bodies are in circular motions with constant angular velocities around the body 0, that is, taking only leading terms of the series (1.17) into account. The first approximations of the relative motions of the bodies 1 and 2 in the eq. (1.10), (1.13) and (1.14) will be now
Having three orbits (t) on our disposition, we can determine the total gravitational forces (t) acting on the bodies in the moment t and consequentially, the relative accelerations of the bodies 1 and 2.
Obviously, variables in the form (1.20) do not satisfy the right hand sides of the equations (1.21), except in the moment t=0. So we take the second terms of the series (1.17) into account, also
and approximate the derivatives of the variables in the eq. (1.21) by the adequate backward difference quotients in the equally spaced points taking the time intervals t=h.
In the expressions (1.23) – (1.25)
…and so on.
Now we multiply elements of the vector eq. (1.21) with the unit vectors and . Retaining only terms of the order ε in the obtained scalar equations, we come to four equations
from which the unknown quantities and (k=1,2) can be calculated.
and are given by the expressions
The second approximation of the orbits will be obtained when we introduce
together with the angular coordinates obtained by use of the trapezoidal rule
into expressions (1.10), (1.11) and (1.12).
Now, taking the new orbits (t) we calculate the second approximations of the total Newton’s forces (t) (k = 0,1,2) and repeat the procedure (1.21) – (1.29) as many times as it’s needed.
Of course, it’s possible to improve the results introducing the higher power terms of the perturbation series (1.17), or/and parabolic rule for the numerical differentiation and integration, but the essence of the procedure is (1.21) – (1.29).
Consider bodies with masses
initially occupying the aligned position, as follows:
Unit u may be, for instance, mass of the Earth, while the unit u1 may be 1 AU.
In order to simplify the calculus, we will apply nondimensionalisation, removing fundamental units by suitable scaling of these quantities. We shall assume:
where bar over the symbol denotes the real quantity, while G is the gravitational constant. From now on, all kinematical and dynamical quantities (without bar over them) will be dimensionless.
If we denote relation = 0,01 = , masses of the bodies will be now
while distances between bodies in the initial position (Fig. 2) will become
Fig. 2 – Initial Position of the System
First of all, by use of the eq. (1.5), we find the center of mass C and fix the frame of reference xCy
Then, equating accelerations and gravitational forces, we find
now we find
Figure 3 represents four successive approximations of the and orbits, obtained by use of the exposed procedure. The dot lines are the first, while the thickest lines, the fourth approximations. It’s worth of attention that the higher order approximations of the orbit move in the inside, while the higher order orbits of move in the outside direction. Decrease in size of the orbit indicates increase in the angular velocity of the body and vice versa. So, in fact, stable motions require that the body 1 goes faster and the body 2 – slower then it was calculated from the initial equilibrium conditions.
Fourth approximation of the third body orbit is also represented in this figure, but, due to the difference in size, the only recognizable are the orbits of small bodies, while the orbit is nearly a spot.
Fig. 3 – Four Successive Approximations of the Orbits
The mean angular velocities in the fourth approximation are
The obtained result is somewhat intriguing. Arbitrary choosing distances between bodies and assuming initial dynamical stability of the system, we came, from the irrational number representing relation between the orbital time periods of the bodies 1 and 2 in the first approximation,
to the, almost, rational number relating two periods in the fourth approximation
As a matter of fact, these motions became resonante.
Since the motion of the third, massive body is the result of two synchronized motions, it has to be synchronized, also. Its orbital period is equal . Finally, the relation between orbital time periods of three bodies is
It seems that the obtained result is not a pure chance.
The common opinion is that capture into the resonant orbital motion is the result of the dissipative forces’ work. It’s probably early to make a hypothesis, but further investigation may lead to the conclusion that the gravitational field synchronizes stable motions by itself, while the work of the dissipative forces may be responsible for the change in the resonance number.
Approximations of the orbit are represented in the Figure 4. The thinnest line describes the first and the thickest, the fourth approximation. Convergence toward the exact solution is evident.
Fig. 4 – Four Successive Approximations of the Orbit
It’s important to notice that the loops of the fourth approximation are, practically, symmetric with respect to the x – axis: this fact is the consequence of commensurability in the mean angular velocities of small bodies.
Let’s contemplate movements of the movement of the in the the fourth approximation, taking into account a somewhat longer period of time (Fig. 5).
Fig. 5 – The Fourth Approximation of the Orbit
The path of the body 0 consists of two loops (Fig. 6). The large one describes during the conjunctional and the small one, during the oppositional phase of the small bodies motions. The difference in sizes is caused by the fact that, in accordance with the eq. (1.11), greater velocity of the large body corresponds mainly to the conjunctional and the smaller, mainly to the oppositional phase. Simply, the loop sizes are proportionate to the velocity. Conjunctions and oppositions of the bodies 1 and 2 produce maximums and minimums in the curve, while transitions from the large into the small loop and vice versa correspond to the inflection points in this curve (Fig. 6).
Fig. 6 – Large and Small Loop in the Path
2. Attractide: Locus of the Gravitational Attraction Centers
While for the three-body system as a whole gravitational interactions are the inner forces which resultant equals zero, sum of the gravitational forces acting on the isolated body is a non-zero external force, of course. Forces (k = 0,1,2) always meet at one point A which is cold center of the gravitational attraction of the bodies these resultants are acting upon [2, p. 84].
All the “exact” solutions of the three – body problem are given for the cases when center of attraction and center of mass coincide. However, excepting the restricted three – body problem, all such motions are unstable  and of theoretical interest, only.
Actually, during the motions of the bodies, center of the gravitational attraction is a moving point with position vector () describing the curve A(x,y) in the inertial reference plane (Fig. 7). We propose the name attractide for this locus.
It’s important to determine this curve and to make generalization of the “central” motions, because, obviously, displacements of the center must produce perturbations of the orbits.
Fig. 7 – Transient Centers of Gravitational Attraction
A question arises how to find ().
One deceiving plan may be to write down the equations of the force directions passing through the points determining the bodies’ positions at the moment t and to find intersection of these lines, solving their equations simultaneously. The problem lies in the fact that whenever the bodies come near/or in the aligned constellation, the corresponding matrix of the system becomes ill conditioned/or singular.
In order to obtain the correct result, it’s necessary to establish the relationship between positions, velocities and accelerations of the bodies. We propose the following procedure if one large and two small bodies are in question.
It seems that the best way is to take the force = acting on the body 0 in the moment t and to find the center A lying on direction of this force.
In the infinitesimally small time interval t, rotations of directions (1) and (2) around 0 are and . Let’s see how these rotations affect direction of the force (Fig.8). For the present we may neglect displacement of the body 0 because, in accordance with eq. (1.10), no matter how infinitesimally small displacements of the bodies 1 and 2 were, displacement is times smaller.
Fig. 8 – Small Relative Rotations of (1) and (2) and Change in the Direction
Direction of the force is defined by the angle
where is the angle between and , so
Applying the law of sines in the force triangle , , , we obtain
The change of the direction produced by small rotations and is
Now, we take into account the small displacement of the body 0 (Fig. 9).
Fig. 9 – Small Displacement and Position of the Center A
Applying the law of sines in the triangle represented in the Fig. 9 we find
(Of course , , and are defined as the corresponding difference quotient limits when t 0 .)
Knowing the force acting on the body 0 and this body velocity, cosine of the angle can be obtained easily
The formula of the attractide is finally
At the end we expose, in the Fig. 10, the obtained orbit, together with the curve representing the locus of the gravitational attraction centers, obtained by use of the procedure exposed in the previous article.
Fig. 10 – The Orbit and Attractide of the System
There is no doubt that displacement of the center of gravitational attraction along the attractide perturbs the orbits. This fact makes generalization of the central motions concept, for the three body system at least, necessary.
The perturbation technique and the method of successive approximations were combined to obtain the orbits of one large and two small bodies exposed to the mutual gravitational interactions.
One example was given as an illustration of the exposed procedure.
Gravitational forces acting on the bodies meet at one point called center of the gravitational attraction. All the “exact” solutions of the three-body problem are given for the cases when center of attraction and center of mass coincide. However, excepting the restricted three-body problem, all such motions are unstable and of theoretical interest, only.
Generally, center moves in the invariant plane describing the locus of these points. It was shown in the second part of this work how to find this curve for which we propose the short name attractide.
At the end, by use of the exposed procedure, attractide was obtained for the given example.
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