Issue 
A&A
Volume 614, June 2018



Article Number  A67  
Number of page(s)  2  
Section  Celestial mechanics and astrometry  
DOI  https://doi.org/10.1051/00046361/201832575  
Published online  14 June 2018 
Comment on “Outofplane equilibrium points in the restricted threebody problem with oblateness (Research Note)”
^{1}
School of Astronomy and Space Science, Nanjing University,
Nanjing
210046,
PR China
email: zhouly@nju.edu.cn
^{2}
Key Laboratory of Modern Astronomy and Astrophysics in Ministry of Education, Nanjing University,
Nanjing
210046,
PR China
Received:
2
January
2018
Accepted:
26
January
2018
Douskos & Markellos (2006, A&A, 446, 357) first reported the existence of the outofplane equilibrium points in restricted threebody problem with oblateness. This result deviates significantly from the intuitive physical point of view that there is no other force that can balance the combined gravitation in Z direction. In fact, the outofplane equilibrium in that model is illusory and we prove here that such equilibrium points arise from the improper application of the potential function.
Key words: celestial mechanics / minor planets, asteroids: general / methods: analytical
© ESO 2018
1 Introduction
In recent years, several papers by various authors have been devoted to investigating the properties of the outofplane equilibrium points, which are located outside the orbital plane of the oblate primary bodies in the restricted threebody problem. This research is mainly based on Douskos & Markellos (2006), which first reported the existence of outofplane equilibrium points caused by the J_{2} term perturbation.
Intuitively, in the Z direction perpendicular to the orbital plane (O − XY), only gravitational forces from both primaries produce an acceleration towards the O − XY plane and no other force can balance such acceleration. Therefore, the existence of such outofplane equilibrium points is doubtful. We notice that even Douskos and Markellos suspected that the existence of such points is only a mathematical illusion due to the truncation of the potential.
For the case of oblate ellipsoidal primaries, we analytically show in this report that such equilibrium points proposed by Douskos & Markellos (2006) are located just inside the Brillouin sphere, where the potential function adopted therein is invalid. These equilibrium points do not exist in the physical point of view.
2 Estimation of the equilibrium
Just as in Douskos & Markellos (2006), let us consider the circular restricted threebody problem, in which a massless body P moves under the gravitational force exerted by an oblate ellipsoid m_{1} and a point mass m_{2}. We choose the distance between these two primaries as the unit of length, the total mass (m_{1} + m_{2}) as the unit of mass, and the unit of time is chosen so as to make the gravitational constant G = 1. The motion of the massless body in the synodic frame (see Fig. 1) can be described as (1)
where W is the potential function given by (2)
The angular velocity n depends on the oblateness of m_{1} and in this case . The dimensionless quantity A indicates the oblateness of m_{1} which is defined as (3)
where D is the distance between two primaries, while R_{e} (r_{e}) and R_{p} (r_{p}) are the real (normalized) equatorial and polar radii of the ellipsoid m_{1}, respectively. It is worth noting that the potential of m_{1} in W is the truncation up to the J_{2} term of the spherical harmonics expansion of the potential function.
Since we are only interested in the outofplane equilibrium points, we focus on the Z direction, where the equilibrium is attained when ∂W∕∂z = 0, i.e., (4)
Obviously the gravitational force from the point mass m_{2} (the first term) and the main part of m_{1} always pointtowards the XY plane, thus only the component generated by the J_{2} term of m_{1} may balance the acceleration towards the XY plane. For any outofplane point z≠0, Eq. (4) can be possibly satisfied only if the latter term is negative for some r_{1}, i.e., (5)
Actually, z^{2} can take any value from 0 to , and thus the minimum value of f(r_{1}, z) for arbitrary fixed r_{1} can be reached at z = ±r_{1}, i.e., (6)
It is necessary to have f_{min}(r_{1}, z) < 0 to make the inequity Eq. (5) true, which gives the estimation of r_{1}, that is (7)
The inequality r_{1} < r_{e} in Eq. (7) means that any outofplane equilibrium points obtained by solving Eq. (4) must be located inside the Brillouin sphere around m_{1}.
If the rotational ellipsoid satisfies some usual constrains, we can further show that such artificial equilibrium points are even located inside the rotational ellipsoid. Suppose a rotational ellipsoid meets the following conditions: (8)
We first introduce a function g(r_{1}, z) as follows: (9)
where indicates the surface of the ellipsoid. Then, obviously (10)
From Eq. (5), we obtain (11)
Thus, for any fixed r_{1}, the maximum of g(r_{1}, z) is attained when , and thus (12)
Since here g_{max} depends only on r_{1} but not z, the estimation given in Eq. (7) shall be taken into account, resulting in (13)
Because any equilibrium point must satisfy Eqs. (5) and (11), when Eq. (8) is fulfilled, we have for any equilibrium point (14)
that is, the equilibrium points are inside the ellipsoid. Apparently, such rotational ellipsoid satisfying Eq. (8) is ubiquitous. This gives the counterexamples for the existence of outofplane equilibrium because the potential function adopted in Douskos & Markellos (2006) is obviously invalid inside the ellipsoid.
Fig. 1 Illustration of the circular restricted threebody problem with oblateness. The origin of the synodic coordinate is in the barycentre. The primary m_{1} is the rotational ellipsoid whose coordinate is (−μ_{2}, 0, 0), while m_{2} is the point mass located at (μ_{1}, 0, 0). 

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3 Discussion
Base on the force balance in the Z direction, we have analytically shown that the outofplane equilibrium points found in Douskos & Markellos (2006) are located inside the Brillouin sphere of the oblate primary or even below the surface of the ellipsoidal primary body. Since the spherical harmonics expansion of the potential function is not valid at this point, we believe that the existence of these outofplane equilibrium points is caused by the improper application of the potential function. In fact, using the closed form of potential function, wecan also prove that the acceleration produced by an ellipsoidal asteroid will always point towards to the O − XY plane, such that no outofplane balance point can exist. Surely, this conclusion holds for the case that both primaries are oblate.
Acknowledgements
We thank the anonymous referee for very helpful comments. This work has been supported by the National Natural Science Foundation of China (NSFC, Grants No.11473016 & No.11333002).
References
 Douskos, C. N., & Markellos, V. V. 2006, A&A, 446, 357 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
All Figures
Fig. 1 Illustration of the circular restricted threebody problem with oblateness. The origin of the synodic coordinate is in the barycentre. The primary m_{1} is the rotational ellipsoid whose coordinate is (−μ_{2}, 0, 0), while m_{2} is the point mass located at (μ_{1}, 0, 0). 

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