One Solution Of The Newton’s Three Body Problem

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 [2] 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

for01

For the isolated systems one may formulate the linear momentum conservation law,

for02

so the centre of mass moves with constant velocity

for03

Since the system is closed, the angular momentum conservation law can be formulated also

for04

Vector \vec {L_c} 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

\vec {V_k} = \vec {v_c} + \vec {v_k} (k=0,1,2)

The first one is the transport velocity (having direction of \vec {K} ) 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 m_1 and m_2 rotate around m_0, 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 R_1,\varphi_1 and R_2,\varphi_2 are the relative reference frames defining positions of the small bodies with respect to the large one (Fig. 1).

Reference Frames

Fig. 1 – Reference Frames

Now, the simple laws follow out from (1.1) for the motions in inertial reference frame

for05

Equations of motions of m_0, m_1 and m_2 in that plane are

for06

where

for07

1 2. Segregation of Dynamical and Kinematical Quantities

Let’s introduce relations between masses m_1 and m_2 and mass of the large body 0 as “small numbers”

for08

and write down the known relations following out from (1.5), (16) and (1.7)

for09

where

for10

and dots over the symbols denote differentiation with respect to time.

Position vectors of the bodies 1 and 2 are now

for11

and similar expressions can be formulated for their velocities and accelerations

for12

In these equations

for13

and

for14

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(\varepsilon^0), the corresponding quantities defining motions of the large body have to be of the order O(\varepsilon).

1.3. Variables and their Expansion into the Power Series of  epsilon

It’s convenient to adopt R_1, R_2, \omega_1 and \omega_2 (\omega_{1,2} = \dot\varphi_{1,2}) to be variables in the exposed problem. They can be expanded into the power series of small number \varepsilon:

for17

We shall assume that the first terms in these series define initial positions of the bodies

for18

and that the other terms are homogeneous functions of time

for19

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 R_k(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 [2], 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

for20

Having three orbits \vec{r_k}(t) on our disposition, we can determine the total gravitational forces \vec{F_k}(t) acting on the bodies in the moment t and consequentially, the relative accelerations of the bodies 1 and 2.

for21

Obviously, variables R_k,\omega_k 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

for22

and approximate the derivatives of the variables R_k,\omega_k in the eq. (1.21) by the adequate backward difference quotients in the equally spaced points taking the time intervals \trianglet=h.

for232425

In the expressions (1.23) – (1.25)

for25a

…and so on.

Now we multiply elements of the vector eq. (1.21) with the unit vectors \vec e_{Rk} and \vec e_{\varphi k}. Retaining only terms of the order ε in the obtained scalar equations, we come to four equations

for26

from which the unknown quantities R_{k1} and \omega_{k1} (k=1,2) can be calculated.

A_k and B_k are given by the expressions

for27

The second approximation of the orbits will be obtained when we introduce

for28

together with the angular coordinates \varphi_k obtained by use of the trapezoidal rule

for29

into expressions (1.10), (1.11) and (1.12).

Now, taking the new orbits \vec{r_k}(t) we calculate the second approximations of the total Newton’s forces \vec{F_k}(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).

1.6. Example

Consider bodies with masses

1.6.1

initially occupying the aligned position, as follows:

1.6.2

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:

1.6.3

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 \frac {m_1} {m_0} = 0,01 = \varepsilon , masses of the bodies will be now

1.6.4

while distances between bodies in the initial position (Fig. 2) will become

1.6.5

Initial Position of the System

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

1.6.6

Then, equating accelerations and gravitational forces, we find

1.6.7

Taking

1.6.8

now we find

1.6.9

Figure 3 represents four successive approximations of the m_1 and m_2 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 m_1 orbit move in the inside, while the higher order orbits of m_2 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 m_0 orbit is nearly a spot.

Four Successive Approximations of the Orbits

Fig. 3 – Four Successive Approximations of the Orbits

The mean angular velocities in the fourth approximation are

16a

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,

16b

to the, almost, rational number relating two periods in the fourth approximation

16c

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 T_2. Finally, the relation between orbital time periods of three bodies is

16d

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 m_0 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.

Four Successive Approximations of the m0 Orbit

Fig. 4 – Four Successive Approximations of the m_0 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 m_0 in the the fourth approximation, taking into account a somewhat longer period of time (Fig. 5).

The Fourth Approximation of the m0 Orbit

Fig. 5 – The Fourth Approximation of the m_0 Orbit

The path of the body 0 consists of two loops (Fig. 6). The large one m_0 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 m_0 velocity. Conjunctions and oppositions of the bodies 1 and 2 produce maximums and minimums in the m_0 curve, while transitions from the large into the small loop and vice versa correspond to the inflection points in this curve (Fig. 6).

Large and Small Loop in the m0 Path

Fig. 6 – Large and Small Loop in the m_0 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 \vec F_k (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 [2] and of theoretical interest, only.

Actually, during the motions of the bodies, center of the gravitational attraction is a moving point with position vector \vec r_A (x_A,y_A) 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.

Transient Centers of Gravitational Attraction

Fig. 7 – Transient Centers of Gravitational Attraction

A question arises how to find \vec r_A (x_A,y_A).

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 \vec F_0 = \vec F_{01} + \vec F_{02} 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 \trianglet, rotations of directions (1) and (2) around 0 are \triangle \varphi_1 and \triangle \varphi_2. Let’s see how these rotations affect direction of the force \vec F_0 (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 \triangle \vec {r_0} is \varepsilon times smaller.

Small Relative Rotations of (1) and (2) and Change in the F0 Direction

Fig. 8 – Small Relative Rotations of (1) and (2) and Change in the \vec F_0 Direction

Direction of the force \vec F_0 is defined by the angle

for20-1

where \alpha is the angle between \vec F_{01} and \vec F_0 , so

for20-2

Applying the law of sines in the force triangle F_0, F_{01}, F_{02} , we obtain

for20-3

The change of the \vec F_0 direction produced by small rotations \triangle \varphi_1 and \triangle \varphi_2 is

for20-4

Now, we take into account the small displacement \triangle \vec {r_0} of the body 0 (Fig. 9).

Small Displacement and Position of the Center A

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

for20-5

(Of course v_0, \dot{\theta}, \omega_1 and \omega_2 are defined as the corresponding difference quotient limits when \trianglet \to 0 .)

Knowing the force acting on the body 0 and this body velocity, cosine of the angle \beta can be obtained easily

for20-6

The formula of the attractide is finally

for20-7

At the end we expose, in the Fig. 10, the obtained m_0 orbit, together with the curve representing the locus of the gravitational attraction centers, obtained by use of the procedure exposed in the previous article.

The m0 Orbit and Attractide of the System

Fig. 10 – The m_0 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.

Conclusion

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.

References

[1] M. Milanković: Osnovi nebeske mehanike, Naučna knjiga, Beograd, 1988.
[2] T. Anđelić: Uvod u astrodinamiku, Matematički institut, Beograd, 1983.
[3] Hawking S., Israel W. 1989. 300 Years of Gravitation, Cambridge University Press.
[4] Pars L. A. 1956. A Treatise of Analytical Dynamics, Heinemann, London.
[5] Marjanov M. 2004. Homogeneous and Inhomogeneous Gravitational Fields, Zbornik radova Gradjevinskog fakulteta u Subotici.

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