Chaotic Amplification in the Relativistic Restricted Three-body Problem

We first analyse the restricted four-body problem consisting of the Earth, the Moon and the Sun as the primaries and a spacecraft as the planetoid. This scheme allows us to take into account the solar perturbation in the description of the motion of a spacecraft in the vicinity of the stable Earth-Moon libration points L 4 and L 5 both in the classical regime and in the context of effective field theories of gravity. A vehicle initially placed at L 4 or L 5 will not remain near the respective points. In particular, in the classical case the vehicle moves on a trajectory about the libration points for at least 700 days before escaping away. We show that this is true also if the modified long-distance Newtonian potential of effective gravity is employed. We also evaluate the impulse required to cancel out the perturbing force due to the Sun in order to force the spacecraft to stay precisely at L 4 or L 5 . It turns out that this value is slightly modified with respect to the corresponding Newtonian one. In the second part of the paper, we first evaluate the location of all Lagrangian points in the Earth-Moon system within the framework of general relativity. For the points L 4 and L 5 , the corrections of coordinates are of order a few millimeters and describe a tiny departure from the equilateral triangle. After that, we set up a scheme where the theory which is quantum corrected has as its classical counterpart the Einstein theory, instead of the Newtonian one. In other words, we deal with a theory involving quantum corrections to Einstein gravity, rather than to Newtonian gravity. By virtue of the effective-gravity correction to the longdistance form of the potential among two point masses, all terms involving the ratio between the gravitational radius of the primary and its separation from the planetoid get modified. Within this framework, for the Lagrangian points of stable equilibrium, we find quantum corrections of order two millimeters, whereas for Lagrangian points of unstable equilibrium we find quantum corrections below a millimeter. Finally, general relativity corrections to Newtonian position of collinear Lagrangian points turn out to be below the millimiter, whereas on stable equilibrium points they are of order of a few millimiters.

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“…The resulting Lagrangian that describes the planetoid motion in the gravitational field of Earth and Moon reads as [64,65]…”

Section: Quantum Effects On Lagrangian Pointsmentioning

confidence: 99%

Earth-moon Lagrangian points as a test bed for general relativity and effective field theories of gravity

et al. 2015

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“…Moreover, one should bear in mind that three-body systems display chaotic behavior over sufficiently long times [45]. This implies that there might exist critical combinations of some parameters, and in the neighborhood of such critical values, even a very small perturbation could give rise to orbits that differ a lot from each other.…”

Section: Discussionmentioning

confidence: 99%

An effective-gravity perspective on the Sun-Jupiter-comet three-body system

Battista,

Esposito,

Tartaglia

2019

Preprint

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Within the solar system, approximate realizations of the three-body problem occur when a comet approaches a planet which is at a certain distance from the Sun, and this configuration was investigated by Tisserand within the framework of Newtonian gravity. The exact relativistic treatment of the problem is not an easy task, but the present paper develops first an approximate calculational scheme which computes for the first time the tiny effective-gravity correction to the equation of the surface for all points of which it is equally advantageous to regard the heliocentric motion as being perturbed by the attraction of Jupiter, or the jovicentric motion as being perturbed by the attraction of the Sun. In the second part a fully relativistic treatment is instead developed, obtaining the relativistic modifications to the dynamics of comets and displaying their orbits.Eventually, the observational tests of corrections to Newtonian formulas for the Sun-Jupiter-comet system are discussed.

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“…The problem was later revisited by Douskos and Perdios [2002] and Ahmed et al [2006], who found that the relativistic triangular points are linearly stable in the range of mass rations 0 ≤ µ < µ r , where µ r = µ 0 − 17 √ 69/486c 2 [Douskos and Perdios, 2002], and µ r = 0.03840 [Ahmed et al, 2006]. Among different possible applications of the RR3BP as discussed by Brumberg [1991] and Kopeikin et al [2011], let us also mention the calculation of the advance of Mercury's perihelion made by Mandl and Dvorak [1984], the computation of chaotic and non-chaotic trajectories in the Earth-Moon orbital system by Wanex [2003], the relativistic corrections to the Sun-Jupiter libration points computed by Yamada and Asada [2010a], the analysis of stability of circular orbits in the Schwarzschild-de Sitter space-time performed by Palit et al [2009], and the investigation of the GTR effects in a coplanar, non-resonant planetary systems made by Migaszewski and Goździewski [2009].…”

Section: The Restricted Relativistic Three-body Problemmentioning

confidence: 99%

The three-body problem

2014

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Abstract. The three-body problem, which describes three masses interacting through Newtonian gravity without any restrictions imposed on the initial positions and velocities of these masses, has attracted the attention of many scientists for more than 300 years. In this paper, we present a review of the three-body problem in the context of both historical and modern developments. We describe the general and restricted (circular and elliptic) three-body problems, different analytical and numerical methods of finding solutions, methods for performing stability analysis, search for periodic orbits and resonances, and application of the results to some interesting astronomical and space dynamical settings. We also provide a brief presentation of the general and restricted relativistic three-body problems, and discuss their astronomical applications.

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Chaotic Amplification in the Relativistic Restricted Three-body Problem

Cited by 6 publications

References 4 publications

Earth-moon Lagrangian points as a test bed for general relativity and effective field theories of gravity

Earth-moon Lagrangian points as a test bed for general relativity and effective field theories of gravity

An effective-gravity perspective on the Sun-Jupiter-comet three-body system

The three-body problem

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