COMPLETE SCHOOL PROJECT TOPICS & MATERIALS :
CHAPTERS: Chapter 1-5
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ABSTRACT
This thesis investigates motion in the generalized restricted three-body problem. It is generalized in the sense that both the primaries are radiating oblate bodies, together with the effect of gravitational potential from a belt. It derives the equations of motion, locates the positions of the equilibrium points and examines their linear stability. It has been found that in addition to the usual five equilibrium points, there appear two new collinear points
Ln1 , Ln2
due to the potential from the belt, and in the presence of all these perturbations,
the equilibrium points L1 , L3 ,
L4 ,
L5 come nearer to the primaries; while
L2 , Ln2 move
towards the bigger primary and
Ln1 moves away from it. The collinear equilibrium points
remain unstable, while the triangular points are stable in 0 < μ < μc
and unstable in
μ £ μ £ 1 , where
c 2
μc is the critical mass ratio influenced by the oblateness and radiation of
the primaries and potential from the belt. This model can be applied in the study of binary systems, especially motion near oblate, radiating binary stars.
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CHAPTER 1
1.0 INTRODUCTION
The restricted three-body problem is a famous model of classical mechanics. It describes the motion of an infinitesimal mass moving under the gravitational effects of the two finite masses, called primaries, which move in circular orbits around their center of mass on account of their mutual attraction and the infinitesimal mass not influencing the motion of the primaries. The approximate circular motion of the planets around the sun and the small masses of asteroids and the satellites of planets compared to the planet’s masses, originally suggested the formulation of the restricted problem.
In certain stellar dynamics problems it is altogether inadequate to consider solely the gravitational interaction force. For example, when a star acts upon a particle in a cloud of gas and dust, the dominant factor is by no means gravity, but the repulsive force of the radiation pressure (Poynting, 1903). Since a large fraction of all stars belong to binary systems Allen (1973), the particle motion in the field of a double star offers special interest. There are disks of dust with various masses in the extra solar planetary systems, which are regarded as young analogues of the kuiper belt in our solar system. The gravitational potential due to these belts also have great influence on the infinitesimal body (Jiang and Yeh, 2004a).
The participating bodies in the classical restricted three-body problem are strictly spherical in shape, but in actual situations we find that, some of the natural and artificial bodies moving in the space are not point masses or spherical rather they are oblate bodies. For instance, Saturn, Jupiter, Regulus (star) and Peanut binary stars are sufficiently oblate. Singh and Ishwar (1999), pointed out that, lack of sphericity, or oblateness of the primaries
affects the motion of an infinitesimal body. The motions of artificial Earth satellites are examples.
In the classical problem, the effects of the gravitational attraction of the infinitesimal body and other perturbations have been ignored. Thus, the classical restricted three- body problem is inadequate to explain the motion of the infinitesimal body in the presence of any perturbing forces such as radiation pressure, oblateness of a body and gravitational potential from a belt.
Hence, it becomes imperative to modify the classical restricted three-body problem by including some of these perturbing forces.
1.1 STATEMENT OF THE PROBLEM
Consider an infinitesimal mass (e.g. dust particle) moving in the orbital plane of oblate binary stars. Then, the problem is to study its motion in the generalized restricted three- body problem. The problem is generalized in the sense that both the primaries are radiating oblate bodies, together with the effect of gravitational potential from the belt. Thus we are to study the combined effect of radiation and oblateness of the primaries and gravitational potential from a belt on the stability of equilibrium points in restricted three-body problem.
1.2 JUSTIFICATION/SIGNIFICANCE OF THE STUDY
The human species stands on the edge of a new frontier, the transition from a planet-bound to a space-faring civilization. Just as the transition from hunter-gatherer to farmer necessitated new approaches to solve new problems, so the expansion into the space, in terms of dynamics of artificial satellite , requires the formulation of new models that include the effects of some of the perturbing forces on the satellite. Motion in the
generalized restricted three-body problem is one of such models that considered the effects of radiation and oblateness of the primaries and gravitational potential from a belt on the satellite. Thus, this model will be very helpful in the study of binary stars, especially dynamics near oblate binary stars. We choose the primaries as oblate spheroid: peanut- stars, the two stars appear to be nearly identical, each 15 to 20 times the mass of our sun (Jenks, 2008).
1.3 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
1.4 THEORETICAL FRAME WORK
The outline of the theoretical bases on which the problem is built, are given here:
1.4.1 Circular Restricted Three-body Problem
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. Unlike the two body problem, there is no closed form analytical solution for the differential equations governing the motion in the three-body problem. However, it is still possible, although not easy, to gain insight into the qualitative nature of the solutions in this system.
This task is more tractable if several simplifying assumptions are introduced. In reducing the general three-body equations, the first assumption is that the mass of one of the bodies is infinitesimal, that is, it does not affect the motion of the other two bodies. Thus, the two massive bodies, or primaries, move in Keplerian orbits about their common center of mass. This reduced model is called the restricted three-body problem, and was formalized by Euler in the late 18th century (Szebehely, 1967). The problem is further simplified by constraining the primaries to move in circular orbits about their center of mass and are kept fixed on the x– axis, the (x, y), plane is the plane of motion of the primaries, and the z-axis is orthogonal to the (x, y) plane. These coordinates are sometimes called synodical. The resulting simplified model is usually labeled the circular restricted three-body problem (CR3BP). Although a less complex dynamical model than the general problem (in terms of the number of equations and the number of dependent variables), analysis in the circular problem offers further understanding of the motion in a regime that is of increasing interest to space science.
In the restricted circular three body problem, the units are usually chosen in such a way that the properties of the system depend on a single parameter.
- The sum of the masses of the primaries is taken as the unit of mass. The mass of the smaller primary is denoted by µ, whence the mass of the bigger primary 1-µ,
where μ = m2 is the mass parameter and m , m are the masses of the bigger
m + m 1 2
1 2
and smaller primaries, respectively.
- The distance between the primaries is the unit of distance. The distances of the smaller primary and bigger primary from the centre of mass are then 1 − µ and µ,
- The unit of time is chosen so that the mean motion of the primaries is n = 1. From these it follows that the gravitational constant is The only remaining
parameter is µ.
The equations of motion of the infinitesimal mass m with the coordinates ( x, y, z )
in the
circular restricted three body problem, relative to a frame that rotates with the primaries Figure 1.1 are:
x- 2 y = Wx ,
y + 2x = W y ,
z = Wz
where W( x, y, z ) is
W = 1 ( x2 + y2 ) + 1- μ + μ
2 r1 r2
and
r 2 = ( x + μ )2 + y2 + z 2 , r 2 = ( x + μ -1)2 + y2 + z 2 , where r , r are the distances of the
1 2 1 2
infinitesimal mass m from the bigger primary m1 and smaller primary m2 , respectively.
y¢
x¢
Unfortunately, even with the simplifying assumptions, no general closed form solution to the CR3BP is available. However, particular solutions can be determined. Of notable interest are the equilibrium solutions, first identified by Lagrange in 1772 (Szebehely,1967), which represent locations where the infinitesimal particle remains fixed, relative to a frame of reference that rotates with the primaries.
If the restricted problem is formulated in terms of a coordinate frame that rotates with the primaries, it is possible to identify five equilibrium solutions, also known as libration or
Lagrange points. Three of these points L1 , L2 , L3
are collinear Figure 1.2, and lie along the
line joining the primaries and are unstable equilibrium points. The other two points L4 , L5
form equilateral triangles with the primaries, in the plane of primary motion and are
conditionally stable equilibrium points.
L4 and
L5 are located at
æ 1 – μ, ± 3 ö
ç 2 2 ÷
è ø
The focus of most investigations into the circular restricted three-body problem is the motion of the infinitesimal particle in the vicinity of the equilibrium points.
y
x
1.4.2 Radiation
For our investigation we adopt Radzievskii’s simplifying theory that has been used by many investigators (Bhatnagar and Chawla, 1979; Kumar and Choudhry 1988, 1989; Ragos and Zafiropoulos 1995; Papadakis 2006; Kalvouridis and Hadjifotinou, 2008; etc.) and is based on some assumptions that can be summarized as follows:
- The radiation emitted from the primaries (stars) influences the motion of the small body but does not affect the motion of the
The mass and the temperature of the radiating stars are almost constant.
- The particle is assumed spherical and homogeneous with constant density. Its surface has a uniform reflection coefficient and its size must exceed the wavelength of the incident
- Light has no fluctuations, propagates in straight lines and screening or shadowing effects produced by other bodies of the system are
- The force of light pressure is inversely proportional to the square of the distance between the particle and the illuminating
Then the effect of radiation pressure of a source on a small particle, is expressed by means of a reduction factor q = 1 − b. Parameter b, that is often mentioned as the radiation
coefficient, is the ratio of force Fr which is caused by radiation to force Fg
which results
from gravitation, that is
b = Fr
Fg
. It depends on the physical properties of the radiating
bodies, as well as those of the particle. Corben and Stehle (1977) proposed the expression
b= 3L
16π cGMρs