Abstract
Background: The location and stability of the equilibrium points are studied for the Planar Circular Restricted Three-Body Problem where the more massive primary is an oblate spheroid. Methods: The mean motion of the equations of motion is formulated from the secular perturbations as derived by(1) and used in(2–4). The singularities of the equations of motion are found for locating the equilibrium points. Their stability is analysed using the linearized variational equations of motion at the equilibrium points. Findings: As the effect of oblateness in the mean motion expression increases, the location and stability of the equilibrium points are affected by the oblateness of the more massive primary. It is interesting to note that all the three collinear points move towards the more massive primary with oblateness. It is a new result. Among the shifts in the locations of the five equilibrium points, the y–location of the triangular equilibrium points relocate the most. It is very interesting to note that the eccentricities (e) of the orbits around L1 and L3 increase, while it decreases around L2 with the addition of oblateness with the new mean motion. The decrease in e is significant in Saturn-Mimas system from 0.95036 to 0.87558. Similarly, the value of the critical mass ratio mc, which sets the limit for the linear stability of the triangular points, further reduces significantly from 0:285: : :A1 to 0:365: : :A1 with the new mean motion. The mean motion sz in the z-direction increases significantly with the new mean motion from 9A1/4 to 9A1/2.
Highlights
The restricted three-body problem (RTBP) has played an essential role in many different areas of dynamical astronomy and this will continue to do so
From equations (24) and (26) we prove that the critical mass ratio μc further decreases with the new mean motion value
A new mean motion expression for this study is used which is obtained by using the secular perturbation effects of oblateness of the more massive primary on the mean anomaly, argument of perigee and the right ascension of the ascending node (1)
Summary
The restricted three-body problem (RTBP) has played an essential role in many different areas of dynamical astronomy and this will continue to do so. The locations of the equilibrium points in the restricted three-body problem by assuming both the primaries as oblate spheroids with their equatorial planes coincident with the plane of motion was calculated in (5). In (6) the location of the collinear points in the same problem was studied numerically for some systems of astronomical interest These equilibria were shown to be unstable in general, though the existence of conditional infinitesimal (linearized) periodic orbits around them was established. The eccentricities of the orbits at the other equilibrium points increased earlier with oblateness and it increases further with the new mean motion. As the mean motion value increases, the critical mass value (μc) shows further decrease from the critical mass value (μ0) of the unperturbed case Both the angular frequencies of the triangular points increase until μ becomes 0.0266053866. The tadpole orbits around the triangular equilibrium points are studied (Section 5)
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