Continuing work initiated in earlier publications [Yamada, Asada, Phys. Rev. D 82, 104019 (2010), 83, 024040 (2011)], we investigate the post-Newtonian effects on Lagrange's equilateral triangular solution for the three-body problem. For three finite masses, it is found that the equilateral triangular configuration satisfies the post-Newtonian equation of motion in general relativity, if and only if all three masses are equal. When a test mass is included, the equilateral configuration is possible for two cases: (1) one mass is finite and the other two are zero, or (2) two of the masses are finite and equal, and the third one is zero, namely a symmetric binary with a test mass. The angular velocity of the post-Newtonian equilateral triangular configuration is always smaller than the Newtonian one, provided that the masses and the side length are the same.
The three-body problem in the Newton gravity represents classical problems in astronomy and physics (e.g, [1][2][3]). In 1765, Euler found a collinear solution for the restricted threebody problem that assumes one of three bodies is a test mass. Soon after, his solution was extended for a general three-body problem by Lagrange, who also found an equilateral triangle solution in 1772. Now, the solutions for the restricted three-body problem are called Lagrange points L 1 , L 2 , L 3 , L 4 and L 5 , which are well known and described in textbooks of classical mechanics [1].
Lagrange points have recently attracted renewed interests for relativistic astrophysics, where they have discussed the gravitational radiation reaction on L 4 and L 5 analytically [4] and by numerical methods [5,6].
As a pioneering work, Nordtvedt pointed out that the location of the triangular points is very sensitive to the ratio of the gravitational mass to the inertial one [7]. Along this course, it is interesting as a gravity experiment to discuss the three-body coupling terms at the post-Newtonian order, because some of the terms are proportional to a product of three masses as M 1 ×M 2 ×M 3 . Such a triple product can appear only for relativistic three (or more) body systems but cannot for a relativistic compact binary nor a Newtonian three-body system.
The relativistic perihelion advance of the Mercury is detected only after much larger shifts due to Newtonian perturbations by other planets such as the Venus and Jupiter are taken into account in the astrometric data analysis. In this sense, effects by the threebody coupling are worthy to investigate. Nevertheless, most of post-Newtonian works have focused on either compact binaries because of our interest in gravitational waves astronomy or N-body equation of motion (and coordinate systems) in the weak field such as the solar system (e.g. [8]). Actually, future space astrometric missions such as Gaia [9, 10] require a general relativistic modeling of the solar system within the accuracy of a micro arc-second [11]. Furthermore, a binary plus a third body have been discussed also for perturbations of gravitational waves induced by the third body [12][13][14][15].
The theory of general relativity is currently the most successful gravitational theory describing the nature of space and time. Hence it is important to take account of general relativistic effects on three-body configurations. The figure-eight configuration that was found decades ago [16,17] has been numerically studied at the first post-Newtonian [18] and also the second post-Newtonian orders [19]. According to their numerical investigations, the solution remains true with a slight change in the figure-eight shape because of relativistic effects.
On the other hand, the post-Newtonian collinear configuration has been recently obtained as a relativistic extension of Euler’s collinear one, where three bodies move around the common center of mass with the same orbital period and always line up [20]. It may offer a useful toy model for relativistic three-body interactions, because it is tractable by hand without numerical simulations. The uniqueness of the collinear configuration has been also proven [21].
Lagrange’s equilateral triangular solution has also a practical importance, since it is stable for some cases. Lagrange’s points L 4 and L 5 for the Sun-Jupiter system are stable and indeed the Trojan asteroids are located there. Clearly it is of greater importance to investigate Lagrange’s equilateral triangular solution in the framework of general relativity. Do the post-Newtonian effects admit such a triangular solution? No one doubts whether the particular configuration is still possible at the post-Newtonian order. We shall study this issue in this paper. The main purpose of this paper is to show that the equilateral triangular configuration can satisfy the post-Newtonian equation of motion, if and only if all three finite masses are equal. Throughout this paper, we take the units of G = c = 1.
First, we consider the Newton gravity among three masses denoted as M I (I = 1, 2, 3).
The location of each mass is written as x I . We choose the origin of the coordinates, so that
We start by seeing whether the Newtonian equation of motion for each body can be satisfied if the configuration is an equilateral triangle. Let us put R 12 = R 23 = R 31 ≡ a, where we define the relative position between masses as
and R IJ ≡ |R IJ | for I, J = 1, 2, 3. Then, the equation of motion for each mass becomes
where M denotes the total mass I M I . Therefore, it is possible that each body moves around the common center of mass with the same orbital period. Eq. ( 3) gives
where ω N denotes the Newtonian angular velocity.
Figure 1 shows an equilateral triangular configuration. Let ℓ I denote the relative position vector of each mass with respect to the common center of mass (but not the geometrical center of the triangle) in the cor
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