Abstract
Background/Objectives: This study deals wit h the stationary solutions of the planar circular restricted three-body problem when the more massive primary is a source of radiation and the smaller primary is an oblate spheroid with its equatorial plane coincident with the plane of motion. The objective is to study the location of the Lagrangian points and to find the values of critical mass. Also, to study the periodic orbits around the Lagrangian points. Methods: A new mean motion expression by including the secular perturbation due to oblateness utilized by(1,2) is used in the present studies. The characteristic roots are obtained by linearizing the equation of the motion around the Lagrangian points. Findings: The critical mass parameter µcrit(3,4) , which decreases radiation force, whereas it increases with oblateness when we consider the value of new mean motion. Through special choice of initial conditions, retrograde elliptical periodic orbits exist for the case µ = µcrit, whose eccentricity increases with oblateness and decreases with radiation force for non-zero oblateness, although there is slight variation in L2 location. Keywords: Restricted three body problem; Lagrangian points; Eccentricity; Oblateness; Critical mass; Radiation force; Mean motion.
Highlights
The three-body problem, in general, is about the study of three massive bodies (m1, m2, m3) in space which affect one another by their gravitational forces
The value of the Critical mass that we got from the Mean motion (n2) = 1 + 6A2 is μcrit = 0.0385209 − 0.0089174ε + 0.0005816ε2 + A2(0.119406935 + 0.011202827ε + 0.6131487216ε2) And the value of the Critical mass that we got from the mean motion expression given by (n2) = 1 + 3/2A2 (3) is μcrit = 0.0385209 − 0.0089174ε + 0.0005816ε2 − A2(0.0627795 − 0.0292011ε + 0.003436104ε2) First value of oblateness in or Critical mass is been found out for different values of mean motions (n2)
When the secular effect of the oblateness on the mean motion is considered, argument of perigee and right ascension of the ascending node (1,2), the resulting mean motion ‘n’ of the primaries is given by n2=1+6A2 is included in the present studies
Summary
The three-body problem, in general, is about the study of three massive bodies (m1, m2, m3) in space which affect one another by their gravitational forces. An oblate spheroid is obtained by rotating an ellipse about its minor axis i.e., the equatorial radius becomes longer than the polar radius This oblateness of the planets render a change to the mean motion of orbit of the primaries, because of the variation of gravitation. In (5) the authors study the periodic orbits numerically for fixed values of the mass parameter and oblateness coefficient of the smaller primary and by changing the radiation pressure and the energy constant. In(7) a version of the relativistic restricted three-body problem which includes the effects of oblateness of the secondary and radiation of the primary was considered to determine the positions and analyse the stability of the triangular points. In(8) interior resonance periodic orbits around the Sun in the Sun-Jupiter photogravitational restricted three-body problem by including the oblateness of Jupiter was studied using the method of Poincaré surface of section. It is found to decrease with radiation force of the more massive primary and increase with oblateness of the smaller primary
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