Motion in the Generalized Restricted Three-body Problem
CHAPTER ONE
OBJECTIVES OF THE STUDY
The objectives of the study include:
- To derive equations of motion of an infinitesimal body under the influence of radiating oblate primaries and gravitational potential from a belt in the restricted three-body
- To examine the effects of these perturbations on the locations of the equilibrium points;
- To investigate the effects of these perturbations on the linear stability of the equilibrium points in restricted three-body
CHAPTER TWO
LITERATURE REVIEW
The three-body problem involves the motion of three celestial bodies under their mutual gravitational attraction. It is an old problem and logically follows from the two-body problem which was solved by Newton in his Principia in 1687. Newton also considered the three-body problem in connection with the motion of the Moon under the influences of the Sun and the Earth, the consequences of which included a headache.
After Newton, Euler (1772) studied the lunar theory using the restricted problem of three bodies; he found that no general closed form solution exists. However, at about the same time, the first special solutions L4 and L5 of the restricted three-body problem were discovered by J. L Lagrange and later, the collinear points L1 , L2
and L3 by L Euler. Poynting (1903) has pointed out that particles, such as small meteors or cosmic dust are comparably affected by gravitation and light radiation force (photogravitational) as they approach luminous celestial bodies.
Radzievskii (1950) formulated the photogravitational restricted three-body problem. This arises from classical problem when one of the interacting masses is an intense emitter of radiation. He discussed it for three specific bodies: the sun, a planet and a dust particle. He studied the equilibrium points of the photogravitational problem and found that their locations depend on the radiation pressure factor.
Szebehely (1967) studied the stability of the equilibrium points of the restricted three body-problem. He established that, in linear sense the collinear points L1 , L2 and L3 are unstable for any value of the mass ratio µ, and the triangular points L4 0 < m < m0 = 0.03852.. and L5 are stable for Bhatnagar and Chawla (1979) investigated the stability of motion around triangular equilibrium points in the photogravitational restricted three body problem. They found that the range of stability decreases due to the radiation pressure.
Sharma (1982) studied the linear stability of the triangular equilibrium points of the restricted three-body problem when the more massive primary is a source of radiation and is an oblate spheroid as well. He found that the eccentricity of the conditional retrograde elliptic periodic orbits around the triangular points at the critical mass µc increases with an increase in the oblateness coefficient and the radiation force and becomes unity when µc
=0. Simmons et al. (1985) obtained a complete solution of the restricted three-body problem. They discussed the existence and linear stability of the equilibrium points for all values of radiation pressures of both luminous bodies and all values of mass ratios.
CHAPTER THREE
EQUATIONS OF MOTION
INTRODUCTION
In this chapter, we shall derive the equations of motion of the infinitesimal body in the gravitational field of radiating oblate primaries (peanut binary stars) together with the influence of gravitational potential from a belt in the restricted three-body problem. We begin with mathematical formulations of the problem, derivation of the equations of motion of the infinitesimal body, and then state the Jacobian integral.
MATHEMATICAL FORMULATIONS OF THE PROBLEM
Let us consider an infinitesimal body (dust particle) of mass m moving under the gravitational influence of the bigger and smaller primaries of masses m1 and m2, respectively. Let us take a coordinate system oxyz with origin at the centre of mass of the primaries and the x-axis is the line joining the primaries; while y-axis is perpendicular to it, the z-axis is perpendicular to the orbital plane of the primaries. The distances between m and the primaries m1 and m2 are r1and r2 respectively, and the distance between the primaries is R. The coordinates of m1, m2 and m are (x1, 0), (x2, 0) and (x, y) respectively as shown in figure (3.1).
CHAPTER FOUR
LOCATIONS OF EQUILIBRIUM POINTS
INTRODUCTION
The equilibrium points are places where the infinitesimal body can sit “motionless” relative to the primaries that are orbiting each other. Hence at the equilibrium positions, all forces acting on the infinitesimal mass in the rotating frame of reference vanish, that is
&x& = &y& = x& = y& = 0.
CHAPTER FIVE
STABILITY OF EQUILIBRIUM POINTS
INTRODUCTION
After determining locations of equilibrium points, it will be interesting to understand the stability properties around these points. In order to arrive at a conclusion about the stability of an equilibrium point, we need to displace the infinitesimal body a little with a small velocity from its equilibrium point under consideration. If the infinitesimal body oscillates considerably around the equilibrium point and returns to the same point as time elapses, the point is said to be stable. If however, motion departs considerably from this point and never returns to it, the point is said to be unstable.
DISCUSSION
The characteristic equation for the triangular points is given by equation (5.12), it differs from that of Singh and Ishwar (1999) only due to the presence of the potential from the belt. If the effect of the belt is neglected, the characteristic equation (5.12) corresponds to that of AbdulRaheem and Singh (2006) in the absence of perturbations in the Coriolis and centrifugal forces.
CONCLUSION
We have found that the collinear equilibrium points remain unstable, while the triangular points are stable in 0 < m < mc and unstable in m £ m £ 1 , c 2 where mc is the critical mass
ratio influenced by the oblateness and radiation of the primaries and potential from the belt. We have established, these perturbations reduce the range of stability (Table 5.1).
CHAPTER 6 SUMMARY AND CONCLUSION
- SUMMARY
We have deduced the equations of motion of an infinitesimal body moving in the gravitational field of radiating oblate primaries, together with the influence of gravitational potential from a belt in the restricted three-body problem. The equations are different from those of the classical case due to these perturbations. We have observed that the equations of motion are affected by the radiation and oblateness factors and the potential from the belt.
We have determined analytically and numerically the locations of equilibrium points. We have found that in addition to the usual five (two triangular, three collinear) equilibrium points, there exist two new collinear equilibrium points due to the potential from the belt, which we call Ln1 and Ln 2 .
CONCLUSION
The equations that govern the motion of an infinitesimal body in the generalized restricted three-body problem have been obtained and they are found to be affected by the radiation and oblateness factors and potential from the belt. We have examined the locations of the equilibrium points and their linear stability and found that:
REFERENCES
- Abdul Raheem, A. and Singh, J. (2006). Combined effects of perturbations, radiation, and oblateness on the stability of equilibrium points in the restricted three-body problem. The Astronomical Journal, 131: 1880–1885.
- Allen, C. W. (1973). Astrophysical Quantites. London; Athlone press.
- Aumann H.H.,Beichman,C.A.,Gillett, F.C.,de Jong, T., Houck,J.R., Low,F.J., Neugebauer,G.,Walker,R.G. and Wesselius,P.R. (1984). Discovery of a shell around Alpha Lyrae. Astrophysical Journal, 278: L23-L27.
- Bhatnagar, K.B.and Chawla, J.M. (1979). A study of the Lagrangian points in the photogravitational restricted three-body problem. Indian J. Pure Appl. Math. 10(11): 1443–1451.
- Corben, H.C.and Stehle, P. (1977). Classical Mechanics, 2nd ed. Dover Publishers, New York.
- Euler, L. (1772). Theoria Motuum Lunae, Typis Academiae Imperialis Scientiarum Petropoli. In: Opera Omnia, Series 2, ed. L. Courvoisier, 22, Lausanne:Orell Fussli Turici, 1958.
- Greaves, J. S., Holland, W. S. , Moriarty-Schieven, G. , Jenness,T., Dent,W. R. F., Zuckerman,B. , Mccarthy, C., Webb, R. A., Butner, H. M., Gear, W. K., And Walker, H. J. (1998). A Dust Ring around Eridani: Analog to the young Solar System. The Astrophysical Journal, 506:L133 L137.
- Ishwar, B.and Elipe, A. (2001). Secular solutions at triangular points in the generalized photogravitational restricted three-body problem. Astrophysics and Space Science. 277: 67–81.
- Jenks E. (2008). Peanut-Stars | New Discovery of Double Star Systems: Oblate Spheroid.
- Retrieved April 1, 2010, from htt: newdiscory/oblate/
- Jiang, I.G. and Yeh, L.C.( 2003). Bifurcation for dynamical systems of planet-belt interaction. Int. J. Bifurcation and Chaos, 13: 534-539
- Jiang, I.G.and Yeh, L.C (2004a). The modified restricted three body problems. RevMex de A A , 21:152–155.
