UDC 521.13

Astronomy

V. G. GOLUBEV

ON REGIONS OF IMPOSSIBILITY OF MOTIONS IN THE THREE-BODY PROBLEM

(Presented by Academician A. N. Kolmogorov on 11 VII 1966)

§ 1. We shall consider the motion of three bodies \(P_1, P_2, P_3\), with masses \(m_1, m_2, m_3\), attracting one another according to Newton’s law, in a barycentric coordinate system \(Oxyz\), oriented so that the direction of the axis \(Oz\) coincides with the direction of the vector of the angular momentum of the system, which, as is known, is a constant quantity. We shall consider systems in which the modulus of this vector \(C \ne 0\). In such systems, by Sundman’s theorem, a triple collision of the bodies is impossible \([2]\).

The energy integral has the form \(T = U + h\), where \(T\) is the kinetic energy of the system; \(U = f(m_1m_2r_{12}^{-1} + m_1m_3r_{13}^{-1} + m_2m_3r_{23}^{-1})\) is the force function (\(f\) is the gravitational constant, \(r_{ij}\) is the distance between \(P_i\) and \(P_j\)); \(h\) is the energy constant. From the qualitative point of view, the most difficult case (in view of the variety of possible motions) is \(h < 0\), for which it is convenient to introduce the constant \(h' = -h > 0\). From the integral \(T = U - h'\) it is readily obtained that in this case complete disintegration of the system is impossible, since

\[ \min_{i \ne j} r_{ij} \leqslant \delta = \frac{f}{h'}(m_1m_2 + m_1m_3 + m_2m_3). \tag{1} \]

In the present paper, for the case \(C \ne 0,\ h < 0\) (\(h' > 0\)), regions of impossibility of motion are found for each of the three bodies, as well as the conditions for their existence and their basic properties.

§ 2. We shall rely on the inequality relating the moment of inertia \(I_\zeta\) of a system of \(n\) bodies with masses \(m_i\) relative to a certain axis \(O\zeta\)

\[ \left(I_\zeta = \sum_{i=1}^{n} m_i(\xi_i^2 + \eta_i^2)\ \text{in the rectangular coordinate system } O\xi\eta\zeta\right), \]

the kinetic energy \(T_\zeta\) of the motion of the system in projection onto the plane perpendicular to the axis \(O\zeta\)

\[ \left(T_\zeta = \frac{1}{2}\sum_{i=1}^{n} m_i(\dot{\xi}_i^2 + \dot{\eta}_i^2)\right), \]

and the angular momentum \(L_\zeta\) of the system relative to the axis \(O\zeta\)

\[ \left(L_\zeta = \sum_{i=1}^{n} m_i(\xi_i\dot{\eta}_i - \eta_i\dot{\xi}_i)\right), \]

namely

\[ I_\zeta T_\zeta \geqslant {1}/{2}L_\zeta^2. \tag{2} \]

We give a scheme of the proof of (2). Make the change of variables
\[ \xi = \xi' \cos \omega t - \eta' \sin \omega t,\quad \eta = \xi' \sin \omega t + \eta' \cos \omega t,\quad \zeta = \zeta', \]
where \(\omega\) is an arbitrary but constant angular velocity of rotation of the system \(O\xi'\eta'\zeta'\) about the axis \(O\zeta' = O\zeta\). Express \(I_\zeta, L_\zeta\), and \(T_\zeta\) in terms of the new variables. After this it turns out that

\[ T_\zeta = \frac{1}{2}\sum_{i=1}^{n} m_i(\dot{\xi}_i'^2 + \dot{\eta}_i'^2) + \omega L_\zeta - \frac{1}{2}\omega^2 I_\zeta \geqslant \omega L_\zeta - \frac{1}{2}\omega^2 I_\zeta . \]

Thus, \(I_\xi\omega^2-2L_\xi\omega+2T_\xi\geqslant 0\) for any \(\omega\). For \(I_\xi=0\) also \(L_\xi=0\), so that (2) is satisfied. Therefore one may assume that \(I_\xi>0\). From the inequality for a quadratic trinomial it is clear that it cannot have distinct real roots, i.e., its discriminant is nonpositive, which leads to (2).

Since \(T\geqslant T_\xi\), \(I_\xi T\geqslant \frac12 L_\xi^2\). Applied to the three-body problem, it follows from this that

\[ I_\xi(U-h')\geqslant \frac12 C^2\sin^2\varphi, \tag{3} \]

if the axis \(O\xi\) makes an angle \(\varphi\) with the plane \(Oxy\). The plane \(Oxy\), by virtue of the choice of the coordinate system, is called the Laplace plane \((^1)\).

It is easy to infer from (3) that the location of the three bodies on one straight line and, in particular, a double collision are possible only in the case when this straight line belongs to the Laplace plane.

§ 3. We now turn to the main purpose of the paper. We shall denote by the indices \(i,j,k\) the numbers \(1,2,3\), taken in an arbitrary but fixed order. We shall call the body \(P_i\) distant (and the bodies \(P_j\) and \(P_k\) close) if \(r_{ij}\geqslant r_{jk}\), \(r_{ik}\geqslant r_{jk}\) (in particular cases two, or each of the three, bodies may turn out to be distant). In this section we shall find a restriction on the spatial motion of the body \(P_i\) under the assumption that it is distant. After this it will be easy to consider the cases when \(P_i\) is close to \(P_j\) or \(P_k\). The region of impossibility of motion of the body \(P_i\), common to all three cases, will be an unconditional region of impossibility of motion of the body \(P_i\). Since the index \(i\) may take any of the values \(1,2,3\), we shall find these regions for each of the three bodies.

Thus, let \(P_i\) be the distant body. Then \(r_{jk}\leqslant\delta\), where \(\delta\) is given by (1). Denote \(OP_i\) by \(r_i\), and the angle between \(OP_i\) and the plane \(Oxy\) by \(\varphi_i\). We apply (3) to the axis \(OP_i\), assuming that \(\varphi_i\ne0\) (for \(\varphi_i=0\) we do not obtain a restriction on \(r_i\)). The center of mass \(O_{(jk)}\) of the bodies \(P_j\) and \(P_k\) lies on the straight line \(OP_i\) in the direction opposite to the body \(P_i\), at a distance from it equal to \(\rho_i=mr_i/(m_j+m_k)\), where \(m\) is the mass of the entire system. The distances of \(P_j\) from \(O_{(jk)}\) and of \(P_k\) from \(O_{(jk)}\) are:
\(r_{j(jk)}=m_k r_{jk}/(m_j+m_k)\leqslant m_k\delta/(m_j+m_k)\), \(r_{k(jk)}=m_j r_{jk}/(m_j+m_k)\leqslant m_j\delta/(m_j+m_k)\). Obviously,
\(r_{ij}\geqslant \rho_i-r_{j(jk)}\geqslant (mr_i-m_k\delta)/(m_j+m_k)\),
\(r_{ik}\geqslant \rho_i-r_{k(jk)}\geqslant (mr_i-m_j\delta)/(m_j+m_k)\).
The right-hand sides of these estimates are positive for \(r_i>A_i\), where \(A_i=\delta\max(m_j,m_k)/m\). Only these values of \(r_i\) will be considered (the opposite inequality \(r_i\leqslant A_i\) already constitutes a restriction on \(r_i\)). For \(\varphi_i\ne0\) the bodies do not lie on one straight line. In the plane \(OP_iP_jP_k\) draw through the point \(O_{(jk)}\) a straight line perpendicular to \(O_{(jk)}OP_i\). Denote the projection of the segment \(P_jP_k\) onto this straight line by \(l_{jk}\), so that \(r_{jk}\geqslant l_{jk}>0\). The moment of inertia of the system with respect to the axis \(OP_i\) is equal to \(I_i=m_jm_k l_{jk}^2/(m_j+m_k)\). For the force function we obtain the estimate
\(U=f(m_jm_k r_{jk}^{-1}+m_im_j r_{ij}^{-1}+m_im_k r_{ik}^{-1})<f\{m_jm_k l_{jk}^{-1}+m_i(m_j+m_k)[m_j/(mr_i-m_k\delta)+m_k/(mr_i-m_j\delta)]\}\).
As a result, from (3) we obtain:

\[ \frac{m_jm_k}{m_j+m_k} \left[ h'-fm_i(m_j+m_k) \left( \frac{m_j}{mr_i-m_k\delta}+ \frac{m_k}{mr_i-m_j\delta} \right) \right]l_{jk}^2 - \frac{fm_j^2m_k^2}{m_j+m_k}l_{jk} + \frac{C^2}{2}\sin^2\varphi_i<0. \tag{4} \]

Denote
\(F_i(r_i)=m_j/(mr_i-m_k\delta)+m_k/(mr_i-m_j\delta)\). As \(r_i\) varies from \(A_i\) to \(+\infty\), \(F_i\) decreases monotonically from \(+\infty\) to zero. Therefore there is a unique value \(A_i'\), \(A_i<A_i'<+\infty\), such that \(F_i(A_i')=h'/fm_i(m_j+m_k)\). We shall consider only \(r_i>A_i'\). Then the previous condition \(r_i>A_i\) is automatically satisfied, and the coefficient of \(l_{jk}^2\) in (4) is positive. In view of inequality (4), the quadratic trinomial has two distinct real roots, i.e., a positive discriminant. This leads to the desired inequality
\(F_i(r_i)>\Phi_i(\sin\varphi_i)\), where
\(\Phi_i(u)=[h'-f^2m_j^3m_k^3/2(m_j+m_k)C^2u^2]/fm_i(m_j+m_k)\).
As \(|\varphi_i|\) varies from

from \(0\) to \(\pi/2\) \(\Phi_i(\sin\varphi_i)\) increases monotonically from \(-\infty\) to \(\Phi_i(1)<F_i(A_i')\). If \(h'C^2>f^2m_j^3m_k^3/2(m_j+m_k)\), then \(\Phi_i(\sin\varphi_i)\) passes through zero at
\[ |\varphi_i|=\varphi_i^*=\arcsin fm_jm_k\left[m_jm_k/2(m_j+m_k)h'C^2\right]^{1/2}. \]
\(0<\varphi_i^*<\pi/2\). It is clear that for \(0\le |\varphi_i|\le \varphi_i^*\) there will be no restriction on \(r_i\). For \(\varphi_i^*<|\varphi_i|\le \pi/2\), on the contrary, an upper restriction on \(r_i\) appears: \(r_i<R_i(\varphi_i)\) for \(r_i>A_i'\), \(\varphi_i^*<|\varphi_i|\le \pi/2\). From the properties of \(F_i(r_i)\) and \(\Phi_i(\sin\varphi_i)\) it follows that, as \(|\varphi_i|\) varies from \(\varphi_i^*\) to \(\pi/2\), \(R_i(\varphi_i)\) decreases monotonically from \(+\infty\) to \(R_i(\pi/2)\). Since \(F_i(A_i')>\Phi_i(1)\), \(R_i(\pi/2)>A_i'\). Consequently, in the result obtained by us
\[ r_i<R_i(\varphi_i),\quad \varphi_i^*<|\varphi_i|\le \pi/2, \tag{5} \]
the condition \(r_i>A_i'\) can be removed (from \(r_i\le A_i'\), (5) follows all the more).

It remains to give an explicit expression for (5). From \(F_i(r_i)>\Phi_i(\sin\varphi_i)\), under the condition \(r_i>A_i\), follows the inequality:
\[ r_i^2-\frac{m_i+m_k}{m}\left[\delta+\frac{1}{\Phi_i(\sin\varphi_i)}\right]r_i+ \frac{\delta}{m^2}\left[m_jm_k\delta+\frac{m_j^2+m_k^2}{\Phi_i(\sin\varphi_i)}\right]<0. \tag{6} \]
The quadratic trinomial (6) has two unequal positive roots: \(\widetilde R_i(\varphi_i)\) and \(R_i(\varphi_i)\), \(\widetilde R_i<R_i\). The solution of inequality (6) has the form \(\widetilde R_i<r_i<R_i\). From the properties of \(F_i(r_i)\) it follows that always \(\widetilde R_i<A_i\). Therefore \(A_i<r_i<R_i\), since (6) was derived under the assumption \(r_i>A_i\). But now it can be removed. Hence (5) is valid, where
\[ R_i(\varphi_i)=\frac{m_j+m_k}{2m}\left[\delta+\frac{1}{\Phi_i(\sin\varphi_i)}\right]+ \]
\[ +\left\{\frac{(m_j+m_k)^2}{4m^2} \left[\delta+\frac{1}{\Phi_i(\sin\varphi_i)}\right]^2 -\frac{\delta}{m^2}\left[m_jm_k\delta+\frac{m_j^2+m_k^2}{\Phi_i(\sin\varphi_i)}\right]\right\}^{1/2}. \tag{7} \]
For the practical construction of the boundary (5), however, it is more convenient to put the inequality \(F_i(r_i)>\Phi_i(\sin\varphi_i)\) in the form
\[ |\sin\varphi_i|<fm_jm_k \left\{\frac{m_jm_k}{2(m_j+m_k)C^2\left[h'-fm_i(m_j+m_k)F_i(r_i)\right]}\right\}^{1/2}. \tag{8} \]

§ 4. Let now \(P_i\) be close to \(P_j\), so that \(r_{ij}\le\delta\). The distance of \(P_i\) from \(O_{(ij)}\) is equal to \(r_{i(ij)}=m_jr_{ij}/(m_i+m_j)\le m_j\delta/(m_i+m_j)\). Since \(P_k\) is a distant body, \(r_k<R_k(\varphi_k)\), \(\varphi_k^*<|\varphi_k|\le\pi/2\). Hence, for \(O_{(ij)}\):
\[ r_{(ij)}<m_kR_k|\varphi_{(ij)}|/(m_i+m_j),\quad \varphi_k^*<|\varphi_{ij}|\le\pi/2. \]
Thus,
\[ r_i<R_i^j(\varphi_i),\quad \varphi_k^*<|\varphi_i|\le\pi/2, \tag{9} \]
where \(R_i^j(\varphi_i)\) passes farther than the boundary for \(O_{(ij)}\) and parallel to it at a distance \(m_j\delta/(m_i+m_j)\). Similarly, if \(P_i\) is close to \(P_k\),
\[ r_i<R_i^k(\varphi_i),\quad \varphi_j^*<|\varphi_i|\le\pi/2, \tag{10} \]
where \(R_i^k(\varphi_i)\) passes farther than the boundary for \(O_{(ij)}\) and parallel to it at a distance \(m_k\delta/(m_i+m_k)\).

Choosing, for a given \(|\varphi_i|\), the largest of the right-hand sides of (5), (9), (10), one can construct an unconditional boundary of the region of impossibility of motion of the body \(P_i\). It exists for \(\varphi^*<|\varphi_i|\le\pi/2\), where \(\varphi^*=\max(\varphi_i^*,\varphi_j^*,\varphi_k^*)=\max(\varphi_1^*,\varphi_2^*,\varphi_3^*)\). If the masses of the bodies are numbered in decreasing order: \(m_1\ge m_2\ge m_3\), then from the expressions for the angles \(\varphi_i^*\) it follows that
\[ \varphi^*=\arc\sin fm_1m_2\left[m_1m_2/2(m_1+m_2)h'C^2\right]^{1/2}. \]
Thus, an unconditional boundary of the region of impossibility of motion for each of the three bodies will exist if
\[ h'C^2>f^2m_1^3m_2^3/2(m_1+m_2), \tag{11} \]

where \(m_1, m_2\) are the two largest masses. It is easy to construct examples in which condition (11) is indeed satisfied. For lack of space we shall confine ourselves to the simplest example. For Lagrange’s triangular solution with equal masses,

\[ \varphi^* = \arcsin \frac{1}{3\sqrt{2}} \simeq 13^\circ 38'. \]

By slightly changing the initial conditions, one can obtain from the planar motion a spatial motion for which the angle \(\varphi^*\) will differ little from the value given above.

Received
15 VI 1966

REFERENCES

1. G. N. Duboshin, Celestial Mechanics. Basic Problems and Methods, Moscow, 1963.
2. J. D. Birkhoff, Dynamical Systems, Moscow, 1941.