Double-Antiprism Central Conﬁgurations of the 3 n -Body Problem

In this paper we study numerically a new type of central conﬁgurations of the 3 n -body problem with equal masses which consist of three n -gons contained in three planes z = 0 and z = ± β (cid:2)= 0. The n -gon on z = 0 is scaled by a factor α and it is rotated by an angle of π/ n with respect to the ones on z = ± β . In this kind of conﬁgurations, the masses on the planes z = 0 and z = β are at the vertices of an antiprism with bases of different size. The same occurs with the masses on z = 0 and z = − β . We call this kind of central conﬁgurations double-antiprism central conﬁgurations . We will show the existence of central conﬁgurations of this type.


Introduction
We consider the spatial N -body problem . . , N , where q k ∈ R 3 is the position vector of the punctual mass m k in an inertial coordinate system, and G is the gravitational constant which can be taken equal to one by choosing conveniently the unit of time. We fix the center of mass for k = 1, . . . , N and N + 1 unknowns q k for k = 1, . . . , N plus the unknown λ > 0. The simplest spatial central configurations of the N -body problem are the ones with N equal masses at the vertices of a regular polyhedron of N vertices, see for instance [1]. There are also some papers studying central configurations consisting of nested regular polyhedra, see for instance [2,3] and the references therein.
In [1] the authors study the spatial central configurations of the N -body problem with equal masses such that the set of positions {q 1 , . . . , q N } is an orbit by the action of a finite subgroup of O (3). Under this condition, they find that the antiprism with N = 2k > 6 vertices, the prism with regular bases with N = 2k > 4 vertices are central configurations. Moreover they find some additional classes of symmetric central configurations for N = 4, 6,8,12,20,24,30,48,60,120, between them the regular polyhedra.
In addition to regular polyhedron central configurations, the simplest spatial central configurations of the N -body problem are the ones known as pyramidal central configurations, which consists of N = n + 1 masses, n of which are coplanar and the (n + 1)-th being off the plane (see for instance [5] and [9]). The n positions of the coplanar masses are called the base of the pyramidal central configuration. There are also the central configurations known as double pyramidal central configurations. Such configurations consist of N = n + 2 masses, n of which are coplanar and the other two being off the plane and positioned symmetrically above and below that plane. In the literature we can find some papers related with double pyramidal central configurations with different shapes of bases. For instance, in [10] the authors studied for all n ≥ 4 the double pyramidal central configurations such that the n equal coplanar masses are at the vertices of a regular n-gon, and the two masses outside the plane determined by the n-gon are equal. In [4] the authors also consider spatial central configurations consisting of n ≥ 2 equal masses at the vertices of a regular n-gon and two masses outside the plane determined by the n-gon but they do not impose conditions on the two masses being outside the plane, neither on the positions, nor on the values of these masses (this problem has also been studied in [6,7] for n = 3).
In [8] the authors find spatial central configurations that bifurcate from planar ones consisting of two regular n-gons, lying in horizontal planes, centered on a common vertical axis, and aligned so that corresponding vertices lie in the same vertical halfplane, i.e. the masses are at the vertices of a truncated pyramid.
In this paper we show the existence of spatial central configurations of the N = 3n body problem with equal masses consisting of three regular n-gons with n ≥ 2, lying in three horizontal equi-spaced planes centered on a common vertical axis, and such that the n-gons on the upper and the bottom plane are identical whereas the n-gon on the middle plane is scaled by a factor α and rotated by an angle π/n with respect to the other two, see Fig. 1. Note that in these configurations the masses at the vertices located on the upper and the bottom plane form a prism, whereas the masses at the vertices located on the upper and the middle one (respectively, the bottom and the middle one) form an antiprism whose bases have different sizes. For this reason we call this kind of central configurations double-antiprism central configurations. Notice that these central configurations are neither a prism nor an antiprism, so in general they are not the ones obtained in [1].
As we shall see in this paper, we have found numerically double-antiprism central configurations for n = 2, 3, . . . , 100000. We also provide some conjectures related to the existence and the uniqueness for each integer n ≥ 2 of the double-antiprism central configurations.

Equations of the Double-Antiprism Central Configurations
We consider 3n equal masses m 1 = · · · = m 3n = 1. Let α, β > 0 and let θ j = 2π j/n and ϕ j = θ j + π/n for all j = 1, . . . , n. Assume that the masses m 1 , . . . , m n are at the vertices of a regular n-gon on the plane z = 0 and that their positions are q j = (α e iθ j , 0) ∈ C × R ≡ R 3 with α > 0. The masses m n+1 , . . . , m 2n are at the vertices of a regular n-gon on the plane z = β > 0 and their positions are q n+j = (e iϕ j , β). And finally the masses m 2n+1 , . . . , m 3n are at the vertices of a regular n-gon on the plane z = −β and their positions are q 2n+j = (e iϕ j , −β). It is easy to check that the center of mass of these configurations is at the origin of coordinates. Then Eq. (1) become The equations for the complex coordinate of the vectorial equations (2) are n j=1 e iϕ k − α e iθ j r 3 for k = 1, . . . , n. The equations for the real coordinate of the vectorial equations (2) are for k = 1, . . . , n. The explicit expressions of the denominators that appear in (2) are for j, k = 1, . . . , n. Note that for all k = 1, . . . , n Eqs. (4) and (5) are the same, and Eqs. (6) and (7) only differ in the sign. Therefore system (2) is equivalent to the system formed by the Eqs. (3), (4) and (6) for k = 1, . . . , n. By dividing the kth equation of (3) by e iθ k , the kth equation of (4) by e iϕ k , and the kth equation of (6) by β we get that system (2) is equivalent to system Playing with the real and imaginary part of the previous equations and by using the properties of the trigonometric functions we can see that all the summations in Eq. (8) are independent of k. Moreover, by choosing conveniently the value of k in each summation, after some tedious computations these equations can be written as where Substituting λ from the third equation of (9) into the first two equations of (9) we get the system of two equations with the two unknowns α and β.
In short we have proved the following result.

Lemma 1 If n is even, then there exists a curve β = h(α) solution of G(α, β)
Moreover this is the unique curve solution of G(α, β) = 0 satisfying these properties.
Assume now that n is odd, that is n = 2k + 1 for some k ∈ N, n j=1 a j = 2 k j=1 a j + a k+1 .
So b 1 > |b 2 |, b 3 > |b 4 |, and so on. Therefore k j=1 b j > 0 for all α, β > 0. We have numerical evidences that for all k ≥ 1. So, under this assumption n j=1 a j > 0 for all α, β > 0. Therefore we do the following conjecture.
Proof The proof follows immediately from simple computations. From Lemma 2(e) the curve H does not intersect the β axis with β > 0, and from Lemma 2(f) it cannot go to β = +∞ when α → 0. Therefore H must tend to the origin when α → 0. Moreover, since the limit when (α, β) → (0, 0) on the straight lines of the form β = b α (see Lemma 2(h)) equals zero when b = (4 A) −1/3 , we have that the curve H tends to the origin tangent to the straight line β = (4 A) −1/3 α when α → 0.
From Lemma 2(c), the curve H cannot go to β = +∞ with a finite value of α > 0, from Lemma 2(a) it cannot cross the α axis with α > 0, from Lemma 2(b) it cannot tend to α = +∞ with β → 0, and from Lemma 2(g) it cannot go to α = +∞ for a finite value of β > 0. Therefore, either the curve H is defined for all α > 0 and tends to β = +∞ when α → +∞, or it is a curve that starts and ends at the point (0, 0) tangent to the straight line β = (4 A) −1/3 α (i.e. H satisfies statement (a)).
Applying Lemma 3 again we know the existence of β = H 1 (β) that is a solution of F(α, H 1 (α)) = 0 for α sufficiently large. Let H 1 the curve defined by this solution. By using the previous arguments this curve must tend to β = +∞ when α → +∞. Therefore, either the curve H 1 coincides with the curve H , or it is a curve that starts and ends at (+∞, +∞). Now we analyze how the curve H 1 tends to β = +∞ when α → +∞. By equating the limit when (α, β) → (+∞, +∞) on the straight lines of the form β = a α to zero, we get that equation L(a) = 0 is equivalent to the equation (see Lemma 2(d)). We can see easily that is defined for all a > 0, lim a→0 + (a) = +∞, lim a→+∞ (a) = 0, and (a) = 0 has a unique real solution a = a 0 ∼ 1.16161 which correspond to a minimum of with (a 0 ) < 0, see Fig. 2. Since A/n is positive for all integer n ≥ 2, then equation L(a) = 0 has a unique solution a. Therefore the curve H 1 tends to (+∞, +∞) tangent to the line β = a α.
In short, either H 1 starts and ends at (+∞, +∞) tangent to the line β = a α (i.e. it satisfies statement (b)), or H 1 coincides with H and consequently it is a curve defined for all α > 0 that tends to the origin tangent to the straight line β = b α when α → 0 We have plotted, with the help of Mathematica, the curves defined implicitly by the equations F(α, β) = 0 and (d F/dβ)(α, β) = 0 for a large number of values of n (in Fig. 3 there is a sample of them). We observe that for all these values of n the curve (d F/dβ)(α, β) = 0 is upper the curve F(α, β) = 0, thus the derivative d F/dβ is different from zero on the curve solution of F(α, β) = 0. Moreover we see that the behaviour of these curves is essentially the same for all those values of n, so we conjecture that the derivative d F/dβ evaluated at the solutions of F(α, β) = 0 is different from zero for all n ≥ 2. In short, we have the following.  For n ≥ 2 odd if Conjectures 1 and 2 hold, it follows the existence of doubleantiprism central configurations of the 3n-body problem.
We note that in both previous paragraphs we only get the existence but not the uniqueness of the double-antiprism central configurations of the 3n-body problem for n ≥ 2.
We have plotted, with the help of Mathematica, the curves defined implicitly by F(α, β) = 0 and G(α, β) = 0 for a large number of values of n (see Fig. 4 for a sample of them). We observe that the behaviour of these curves the curves β = h(α) and β = H (α) when we vary n is essentially the same, and that they intersect at a unique point. In short, we can do the following conjecture.

Conjecture 3
For all integer n ≥ 2 there exist a unique double-antiprism central configuration of the 3n-body problem.

Some numerical values of the solutions of system (11)
With the help of Mathematica, we have computed numerically the solution of system (11) for all n = 2, . . . , 100000. We have seen that for all these values of n system (11) has a unique solution, which gives more numerical evidence that Conjecture 3 holds. In Table 1 we show the solutions of this system for some values of n. We note that the solution for n = 2 can also be found analytically.
We note that for some values of n the central configurations provided by the solutions of Table 1 are already known. For instance, if n = 2, then the configuration corresponds to the octahedron, if n = 4, then it corresponds to the cuboctahedron (both studied in [1]). But in general they are new.