A note on commensurability induced by oblateness at L3

Ram Krishan Sharma,Arantza Jency

doi:10.21694/2380-5714.21002

Abstract

t is known that there exists a near one-to-one commensurability ratio between the planar angular frequencies (s1,2,3) and the corresponding angular frequency (sz) in the z-direction at the three collinear points (L1,2,3) in the three-dimensional restricted three-body problem. It is significant for small and practically important values of the mass parameter (μ). In this note we have used a new mean motion expression which includes the secular effects of oblateness on argument of perigee, right ascension of ascending node and mean anomaly[1], which was used in the studies[2,3,4,5]. We establish that there is ono-to-one commensurability at the external points L3 (to the right of the more massive primary), when the more massive primary is considered as an oblate spheroid with its equatorial plane coincident with the plane of motion. This study will be useful in generating the halo orbits at L3. For Saturn-Titan system, the values of the μ and oblateness coefficient (A1) are 0.000236695 and 0.000039653936, respectively [1].It is interesting to note that the value of A1 obtained for one-to-one commensurability is 0.0000344978 for this system, which is very close to the actual value. Thus, halo orbits of small size can be generated in Saturn-Titan system at L3.

Highlights

The locations of the Lagrangian points in the restricted three-body problem (CRTBP) by assuming both the primaries as oblate spheroids with their equatorial planes coincident with the plane of motion was calculated in [6]
We have proved the existence of one-to-one commensurability ratio between the planar angular frequencies (s3) and the corresponding angular frequency in the z-direction at the collinear point (L3)
With the secular perturbations effects of oblateness on argument of perigee, right ascension of ascending node and mean anomaly on the mean motion [1], it is found that the mean motion increases further

Summary

Introduction

The locations of the Lagrangian points in the restricted three-body problem (CRTBP) by assuming both the primaries as oblate spheroids with their equatorial planes coincident with the plane of motion was calculated in [6]. The secular effect of oblateness of the primaries on the motion of the primaries was not included. Later the oblateness of only the more massive primary was considered and the secular effect of oblateness [8] on the mean motion of the primaries was included in [9, 10, 11].

Results

Conclusion