ABSTRACT
The research work examines the existence and linear stability of equilibrium points in the perturbed Robe’s circular restricted three-body problem under the assumption that the hydrostatic equilibrium figure of the first primary is an oblate spheroid. The problem is perturbed in the sense that small perturbations given to the Coriolis and centrifugal forces are being considered. Results of the analysis found two axial equilibrium points on the line joining the centre of both primaries. It is further observed that under certain conditions, points on the circle within the first primary are also equilibrium points. And a special case where the density of the fluid and that of the infinitesimal mass are equal (D = 0) is discussed. The linear stability of this configuration is examined; it is observed that the axial equilibrium points 𝑝1,0,0 and 𝑥11+𝑝2 ,0,0 are conditionally stable, while the circular points 1+r𝑐os𝜃,𝑟𝑠in𝜃,0 are unstable.
TABLE OF CONTENTS
Cover page ………………………………………………………………………………………………… i
Fly leaf …………………………………………………………………………………………………….. ii
Title page …………………………………………………………………………………………………. ii
Declaration ………………………………………………………………………………………………. iii
Certification …………………………………………………………………………………………….. iv
Acknowledgement ……………………………………………………………………………………… v
Dedication ……………………………………………………………………………………………….. vi
Abstract ………………………………………………………………………………………………….. vii
Table of Content ……………………………………………………………………………………… viii
List of Figures ………………………………………………………………………………………….. xi
List of Table …………………………………………………………………………………………….. xi
List of Notations ……………………………………………………………………………………….. xi
CHAPTER ONE: GENERAL INTRODUCTION ……………………………………….. 1
1.1 Introduction ………………………………………………………………………………………. 1
1.2 Statement of the Problem ……………………………………………………………………… 2
1.3 Significance / Justification of the Study ………………………………………………… 2
1.4 Research Methodology ………………………………………………………………………… 3
1.5 Aim and Objectives of the Study ……………………………………………………………. 3
1.6 Preliminary Ideas ……………………………………………………………………………… 4
1.6.1 Circular restricted three-body problem …………………………………………………. 4
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1.6.2 Equations of motion of two-body problem ……………………………………………. 5
1.6.3 Equations of relative motion ………………………………………………………………. 6
1.7 Theoretical Framework ………………………………………………………………………. 7
1.7.1 The buoyancy force ………………………………………………………………………….. 7
1.7.2 Robe’s restricted three-body problem …………………………………………………. 14
1.7.3 Perturbation …………………………………………………………………………………… 16
1.7.4 Coriolis and centrifugal forces…………………………………………………………… 17
1.7.5 Oblate spheroid ………………………………………………………………………………. 18
1.7.6 Moment of Inertia …………………………………………………………………………… 18
1.8 Linear Stability ………………………………………………………………………………… 21
CHAPTER TWO: LITERATURE REVIEW ……………………………………………. 22
CHAPTER THREE: EQUATIONS OF MOTION …………………………………….. 29
3.1 Introduction …………………………………………………………………………………… 29
3.2. Mathematical Formulations of the Problems ……………………………………….. 29
3.3 The Case (𝝆𝟏=𝝆𝟑) ……………………………………………………………………………. 32
3.3 Conclusion …………………………………………………………………………………….. 34
CHAPTER FOUR: LOCATIONS AND LINEAR STABILITY OF EQUILIBRIUM POINTS………………………………………………………………………… 35
4.1 Introduction ……………………………………………………………………………………. 35
4.2 The Equilibrium Points ………………………………………………………………………. 35
4.2.1 Axial equilibrium points………………………………………………………………….. 36
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4.2.2 Locations of circular points ……………………………………………………………. 45
4.3 Linear Stability of The Equilibrium Points …………………………………………….. 46
4.3.1 Variational equations ……………………………………………………………………… 47
4.3.2 Stability of axial equilibrium point (𝑝1,0,0) ……………………………………… 50
4.3.2.1 Critical Mass ……………………………………………………………………………… 65
4.3.2.2 Range of stability ……………………………………………………………………….. 78
4.3.3 Stability of the axial equilibrium point (𝑥11 +𝑝2 ,0,0) ………………………… 82
4.3.4 Stability of the circular equilibrium point ……………………………………………. 91
4.4 Conclusion …………………………………………………………………………………….. 97
CHAPTER FIVE: SUMMARY, CONCLUSION AND RECOMMENDATIONS …………………………………………………………………………. 99
5.1 Introduction …………………………………………………………………………………… 99
5.2 Summary ………………………………………………………………………………………. 99
5.3 Conclusion ……………………………………………………………………………………. 100
5.4 Recommendations ………………………………………………………………………….. 100
REFERENCES …………………………………………………………………………………….. 101
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LIST OF FIGURES
Figure 1.1 A cylinder of height h and radius R fully immersed in a fluid …………….. 8
Figure 1.2 The Robe’s restricted three-body problem ……………………………………… 15
LIST OF TABLE
Table 1: numerical computation of the critical mass 𝜇𝑐 …………………………………… 81
LIST OF NOTATIONS
𝑚1 mass of the first primary…………………………………….…………………29
𝑚2 mass of the second primary ………………………………………………………………. 29
𝑚3 mass of the infinitesimal body …………………………………………………………… 29
𝜌1 density of the fluid …………………………………………………………………………… 29
𝜌3 density of the infinitesimal body ………………………………………………………… 29
𝛼 oblateness coefficient……………………………………….…………………30
𝑛 mean motion…………………………………………………………. ……….30
𝐵 potential due to the first primary………………….………………………….. 30
𝐵’ potential due to the second primary…………………………………. ………30
V potential that explain the combine force upon the infinitesimal mass ………….. 30
ϕ perturbation in Coriolis force ………………………………………………………………. 30
ѱ perturbation in centrifugal force…………………………………………………………… 30
𝜇 mass ratio ………………………………………………………………………………………… 30
ξ, η, ζ, small displacements ……………………………………………………………………….. 48
𝜆 constant …………………………………………………………………………………………… 50
Δ discriminant…………………………………………………………………….
CHAPTER ONE
GENERAL INTRODUCTION
1.1 Introduction
In the scientific exploration of space by man, space dynamics is very important and this later rely on celestial mechanics- which is a branch of science devoted to the movement of artificial celestial bodies such as artificial satellite, while space dynamics is a branch of astronomy which deals with the nature and motions of stellar phenomena such as planets, asteroids and stars. The most celebrated problem of space dynamics is the problem of three bodies, known as the three-body problem (3BP). This is defined in terms of three bodies with arbitrary masses and free to move in space attracting one another according to Newton’s law of gravitation. An example of 3BP is the Sun-Earth-Moon system, when they are considered as point masses; they form the main problem of the lunar theory. (Singh, J 2012) The R3BP describes the motion of an infinitesimal mass moving under the gravitational effects of two finite masses, called primaries, which move in circular orbits around the center of mass on the account of their mutual attraction and the infinitesimal mass not influencing the motion of the primaries. The R3BPs have produced many significant results by well-known mathematicians and scientists in an attempt to understand and predict the motion of natural bodies. A system made up of the Sun and Jupiter as primaries and Trojan asteroid assuming the role of the infinitesimal mass in the Sun-Jupiter system is a typical example of the R3BP.
A special case of R3BP referred to as Robe’s problem was introduced by Robe (1977), where the infinitesimal mass is embedded in the first primary which is filled with
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homogenous incompressible fluid of known density, the second primary is a point mass located outside an orbiting shell. The infinitesimal mass is under the gravitational attraction of the primaries as well as buoyancy force due to the fluid. Robe’s models provides an insight into the study of the oscillations of the Earth’s inner core taking into cognizance the attraction of the Moon; and the motion of an artificial Earth satellite located inside another satellite. This Robe’s problem has been varied by the introduction of perturbing forces by many researchers like Hallan and Rana (2001b), Hallan and Mangang (2007), Singh and Sandah (2012), Singh and Mohammed (2013) to mention a few.
1.2 Statement of the Problem
Hallan and Mangang (2007) studied the existence and linear stability of equilibrium points in the Robe’s circular restricted three-body problem. They assumed one of the primaries of mass 𝑚1 to be a rigid spherical shell filled with homogenous, incompressible fluid of density 𝜌1 in the shape of an oblate spheroid, the second primary 𝑚2 is a point mass located outside the shell and moving around the mass 𝑚1 in a Keplerian orbit; the infinitesimal mass 𝑚3 is a small solid sphere of density 𝜌3 moving inside the shell and is subject to the attraction of 𝑚2 and the buoyancy force due to the fluid of the first primary. It will be interesting to see what happens when perturbations are included in their study. In this present thesis, we examine the effect of small perturbations in Coriolis and centrifugal forces on the existence and linear stability of the equilibrium points. Thus our study will be a generalization of Hallan and Mangang (2007).
1.3 Significance / Justification of the Study
In reality, we found that several heavenly bodies are sufficiently oblate. Earth, Jupiter, Saturn and Ragulus are oblate. The participating bodies in the classical R3BP are assumed to be strictly spherical in shape. The minor planets and meteoroids have
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irregular shapes. Large perturbations from the two-body orbit are caused by the lack of sphericity, oblateness or triaxiality of the celestial bodies; an example of this is the motion of artificial Earth satellites. In recent time, many perturbing forces like oblateness, triaxiality, variation of masses for the primaries and of the infinitesimal mass etc, have been included in the study of the existence and linear stability of the equilibrium points in the Robe’s circular restricted three-body problem.
To this end, we modify the model by considering the effect of small perturbations in the Coriolis and centrifugal forces on the Robe’s circular restricted three-body when the first primary is an oblate spheroid. Taking into account the full buoyancy force and other forces acting on the infinitesimal body when the density parameter is not zero (𝑖.𝑒.,𝜌1≠𝜌3). The Robe’s model under consideration can be applied in space mission design; it also has enormous applications in various astronomical problems. Examples include small oscillations of the earth’s core in the gravitational field of Earth-Moon system.
1.4 Research Methodology
We adopt a rotating coordinate system to derive the equations of motion of the infinitesimal body with respect to the primaries as given in chapter three.
1.5 Aim and Objectives of the Study
The research work is aimed at investigating the motion of an infinitesimal mass in the perturbed Robe’s circular restricted three body problem under an oblate spheroid. In view of the problem stated above, we therefore set the following objectives:
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(a) To derive the equations of motion of the infinitesimal mass
(b) To determine the locations of the axial equilibrium points
(c) To examine the linear stability of the axial equilibrium points
(d) To investigate the model when the density parameter is zero (𝑖.𝑒.,𝜌1=𝜌3)
1.6 Preliminary Ideas
1.6.1 Circular restricted three-body problem
The circular restricted three-body problem (CR3BP) describes the motion of a body of infinitesimal mass under the gravitational influence of two massive bodies, called the primaries, which orbit about their common centre of mass in circular orbits on account of their mutual attraction. The usefulness of studying this problem was verified by the discovery of the Trojan asteroids at the stable Lagrange points of the Sun-Jupiter orbital system.
The classical restricted three-body problem possesses five equilibrium points: three collinear points 𝐿1,2,3 and two triangular points 𝐿4,5, where the gravitational and centrifugal forces just balance each other. The collinear points are unstable, while the triangular points are stable for the mass ratio 𝜇<0.03852… (Szebehely, 1967a). Their stability occurs in spite that the potential energy has a maximum rather than a minimum at the latter points. The stability is actually achieved through the influence of the Coriolis force, because the coordinate system is rotating (Wintner, 1941; Contopoulus, 2012). These equilibrium points are of great astronomical importance. The Solar and Heliospheric observatory (SOHO) and WMAP launched by NASA are in operation at the Sun-Earth 𝐿1and 𝐿2 respectively. The Laser Interferometer Space Antenna (LISA) pathfinder is planned to go to 𝐿1.
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1.6.2 Equations of motion of two-body problem
In the circular restricted three-body problem, the motion of the primaries is governed by the equations of motion of two-body problem with constant masses. Let 𝑚1 𝑎𝑛𝑑 𝑚2 be the masses of the two primaries, 0 be a fixed point in the space of the motion. Let 𝑟 1 𝑎𝑛𝑑 𝑟 2 be their respective vectors from 0 to the respective masses and 𝑟 be the radius vector between the bodies. Then, the force of attraction on the particle of mass 𝑚1 due to another particle of mass 𝑚2 by Newton’s law of gravitation as in Singh and Sandah (2012) is given as 𝐹 1=𝐺𝑚1𝑚2𝑟2𝑟 𝑟 (1.1)
and the force acting on the particle of mass 𝑚2 due to the particle of mass 𝑚1 is 𝐹 2=−𝐺𝑚1𝑚2𝑟2𝑟 𝑟 (1.2)
Where G is the gravitational constant and 𝑟 is the distance between the primaries. By Newton’s second law 𝐹 1=𝑑2𝑟 1𝑑𝑡2𝑚1, 𝐹 2=𝑑2𝑟 2𝑑𝑡2𝑚2. (1.3) Putting the equations of system (1.3) in Eq. (1.1) and Eq. (1.2), we get 𝑑2𝑟 1𝑑𝑡2𝑚1=𝐺𝑚1𝑚2𝑟2𝑟 𝑟, 𝑑2𝑟 2𝑑𝑡2𝑚2=−𝐺𝑚1𝑚2𝑟2𝑟 𝑟 or 𝑟1 =𝐺𝑚2𝑟2𝑟 𝑟, 𝑟2 =−𝐺𝑚1𝑟2𝑟 𝑟 𝑤𝑒𝑟𝑒 𝑟 =𝑟 2−𝑟 1 (1.4)
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Equations of system (1.4) are the vectorial equations of motion of the bodies of masses 𝑚1 𝑎𝑛𝑑 𝑚2.
1.6.3 Equations of relative motion
We assume the velocity of the bodies 𝑚1 𝑎𝑛𝑑 𝑚2 to be 𝑣 1 𝑎𝑛𝑑 𝑣 2 respectively, and 𝑣 R be the velocity vector of the body 𝑚2 𝑤𝑖𝑡 𝑟𝑒𝑠𝑝𝑒𝑐𝑡 𝑡𝑜 𝑚1.
Then, 𝑣 R= 𝑣 2 −𝑣 (1.5) Now the vectorial equations of system (1.4) take the form 𝑑𝑣 1𝑑𝑡=𝐺𝑚2𝑟2𝑟 𝑟 , (1.6) 𝑑𝑣 2𝑑𝑡=−𝐺𝑚1𝑟2𝑟 𝑟. (1.7) Subtracting equation (1.6) from equation (1.7), we get 𝑑𝑣 2𝑑𝑡− 𝑑𝑣 1𝑑𝑡=−𝐺𝑚1𝑟2𝑟 𝑟−𝐺𝑚2𝑟2𝑟 𝑟=−𝐺 𝑚1+ 𝑚2 𝑟2𝑟 𝑟 or 𝑑𝑑𝑡(𝑣 2 –𝑣 1) =−𝐺 𝑚1+ 𝑚2 𝑟2𝑟 𝑟 Using equation (1.5), we have 𝑑𝑣 𝑅𝑑𝑡=−𝐺 𝑚1+ 𝑚2 𝑟2𝑟 𝑟 (1.8) 𝐸𝑞. 1.8 𝑏𝑒𝑐𝑜𝑚𝑒𝑠 𝑑𝑣 𝑅𝑑𝑡=−𝜇𝑟2𝑟 𝑟 , 𝑤𝑒𝑟𝑒 𝜇=𝐺 𝑚1+ 𝑚2 And removing the suffix, we have
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𝑑𝑣 𝑑𝑡=−𝜇𝑟2𝑟 𝑟 (1.9)
Equation (1.9) is the relative motion of the body of mass 𝑚2 with respect to the body of mass 𝑚1, where r is the distance between the bodies.
1.7 Theoretical Framework
The theoretical framework upon which the derivations and finding of this research work is built on, is given in this section
1.7.1 The buoyancy force
The force exerted on an object that is completely or partly immersed in a fluid is known as buoyancy force. It acts upward, generally opposite to the direction of the frame of reference acceleration and its magnitude is equal to the weight of the fluid displaced by the submerged object. This force is caused by the difference between the pressure at the top of the object, which pushes downward and the pressure at the bottom of the object which pushes upward; every submerged object feels upward buoyancy force, because the pressure at the bottom of the object is always greater than the pressure at the top.
Consider a small particle of density d and volume V, falling with a velocity 𝑉 in a fluid of density 𝜌1, the force acting on this particle as in Singh and Sandah ( 2012) are given as
(i) The gravitational force 𝐹𝐴
(ii) The buoyancy force, given by
𝐹𝐵=𝜌1𝑉g (1.10) where, g is the gravity of the fluid.
The pressure distribution on a submerged particle is statistically equivalent to a single point force 𝐹𝐵 then acts at the location called the center of pressure (center of the mass of
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the fluid displaced). Equation (1.10) is derived by relating hydrostatic distribution to an equivalent force – the buoyancy force (𝐹𝐵). Now, consider a completely submerged cylinder of length l and radius R oriented vertically in the fluid basin with depth L (see figure 1.1 below)
Figure 1.1 A cylinder of height h and radius R fully immersed in a fluid
The distance between the top surface of the cylinder and the below surface fluid is 1, the thrust at the top and bottom of the cylinder are 𝑛 T and 𝑛 B respectively acting in opposite direction. Now, from the relation 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒=𝐹𝑜𝑟𝑐𝑒𝐴𝑟𝑒𝑎 (1.11) The force acting on the top of the surface due to hydrostatics is given by 𝐹 𝑡𝑜𝑝 =− 𝑛 𝑇𝑃𝑑𝐴 𝐴𝑇 (1.12) where AT is the area of the top of the cylinder.
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Now, the equation of pressure field in hydrostatics is given by 𝜕𝑝𝜕𝑧=−𝜌 (1.13) here, p is a constant for an incompressible fluid, by integrating the last equation above with respect to z, we have
𝑝=−𝜌𝑔𝑧+𝑐
At 𝑧=𝐿, 𝑝=𝜌0 𝑠𝑜 𝑡𝑎𝑡 𝑐=𝑝0+𝜌𝑔𝐿 𝑎𝑛𝑑 𝑃 𝑧 =𝑝0+𝜌𝑔 𝐿−𝑧 . Thus 𝜕𝑝𝑧1=− 𝜌𝑔𝜕𝑧𝑧1 (1.14)
Evaluating equation (1.14) at 𝑧=𝐿−1, we immediately have
𝑃=𝜌0+𝜌𝑔(𝐿−𝐿+1)
Or 𝑃=𝜌0+𝜌𝑔1 (1.15) Substituting equation (1.15) in (1.12), we have 𝐹 𝑡𝑜𝑝 =− 𝑛 𝑇 𝑃0+𝜌𝑔1 𝑑𝐴 𝐴𝑇 or 𝐹 𝑡𝑜𝑝 =−𝑛 𝑇 𝑃0+𝜌𝑔1 𝑑𝐴 𝐴𝑇
𝑆𝑖𝑛𝑐𝑒 𝑑𝐴 𝐴𝑇 is actually the cross sectional area of the cylinder, therefore the force acting at the top of the solid is
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𝐹 𝑡𝑜𝑝 =−𝑛 𝑇𝜋𝑅2 𝑃0+𝜌𝑔1 (1.16) Similarly, the force acting at the bottom of the cylinder is 𝐹 𝑏𝑜𝑡 =− 𝑛 𝐵𝑃𝑑𝐴 𝐴𝐵=−𝑛 𝐵𝜋𝑅2 𝑃0+𝜌𝑔 1+𝐿 (1.17)
while we have substituted 𝑧=𝐿−(1+𝑙) in equation (1.12).
Now, since 𝑛 𝑇=𝑛 𝐵=𝑘, where 𝑘 is a unit vector along positive z direction and perpendicular to the 𝑥−𝑎𝑥𝑖𝑠, consequently, equation (1.17) may be written as 𝐹 𝑏𝑜𝑡 =𝑛 𝑇𝜋𝑅2 𝑃0+𝑝𝑔 1+𝑙 (1.18) Hence, the net vertical force (i.e., force acting downwards) on the cylinder can be obtained using equations (1.16) and (1.18) as 𝐹 𝑛𝑒𝑡=𝐹 𝑡𝑜𝑝+𝐹 𝑏𝑜𝑡=−𝑛 𝑇𝜋𝑅2 𝑃0+𝜌𝑔1 +𝑛 𝑇𝜋𝑅2 𝑃0+𝜌𝑔 1+𝑙
=𝑛 𝑇𝜋𝑅2 −𝑃0−𝜌𝑔1+𝑃0+𝜌𝑔1+𝜌𝑔𝑙 𝐹 𝑛𝑒𝑡=𝑛 𝑇𝜋𝑅2 𝜌𝑔𝑙 or 𝐹 𝑛𝑒𝑡=𝑘 𝜌𝑔𝜋𝑅2𝑙 (1.19)
Since the volume of cylinder is 𝜋𝑅2𝑙, where 𝑅 is radius and 𝑙 is the height. The net force on the cylinder then takes the form 𝐹 𝑛𝑒𝑡=𝑘 𝜌𝑔𝑉 (1.20) therefore the buoyancy force is 𝐹 𝐵=𝜌𝑔𝑘 𝑉 𝑐𝑦𝑙 (1.21)
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𝑉cyl here refers to the volume of the cylinder, 𝜌 is the density of the fluid and 𝑔 is the gravity of the fluid. Thus, the buoyancy force acting upward (positive z – direction) is equal to the weight of the displaced fluid.
Consider an arbitrary shape body and extend the above analysis using the projected area theorem. Let the submerged body in a fluid be separated into two parts, with the upper surface denoted by 𝑧2 and the lower surface by 𝑧1. Therefore, the force acting on the submerged body is given by 𝐹 =− 𝑛2 𝑃 𝑧2 𝑑𝐴− 𝐴2 𝑛1 𝑃 𝑧1 𝑑𝐴 𝐴1 (1.22)
where 𝑃 𝑧1 𝑎𝑛𝑑 𝑃 𝑧2 are the pressure acting on the lower and upper surface of the body.
The component of this force in the 𝑘 direction gives the buoyancy force. Taking the component of 𝐹 𝑖𝑛 𝑡𝑒 𝑘 direction, the buoyancy force is then 𝐹𝐵=𝐹 .𝑘 =− 𝑛 2.𝑘 𝑃 𝑧2 𝑑𝐴− 𝐴2 𝑛 1.𝑘 𝑃 𝑧1 𝑑𝐴 𝐴1 (1.23) From the projection theorem, we have 𝑛 2.𝑘 𝑑𝐴=𝑑𝐴𝑧, 𝑛 1.𝑘 =−𝑑𝐴𝑧 (1.24)
where 𝑑𝐴𝑧 is the projection of 𝑑𝐴 on the 𝑥𝑦−𝑝𝑙𝑎𝑛𝑒. The use of equation (1.24) in equation (1.23), we have the magnitude of the buoyancy force (the component of the force acting in the vertical direction)
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𝐹𝐵=− 𝑃 𝑧2 𝑑𝐴𝑧− 𝐴2 𝑃 𝑧1 𝑑𝐴𝑧 𝐴1 (1.25) Now, since the hydrostatic pressure at any point on z in the fluid is given by equation (1.14), then 𝐹𝐵= 𝜌𝑔 𝑧2−𝑧1 𝑑𝐴𝑧 𝐴2 or 𝐹𝐵=𝜌𝑔 𝑧2−𝑧1 𝑑𝐴𝑧 (1.26) 𝐴2 From the principal surface of the integral, the equation (1.26) is just the volume of the solid, that is 𝑧2−𝑧1 𝑑𝐴𝑧=𝑉𝑠 𝐴2 Therefore, the buoyancy force is 𝐹𝐵=𝜌𝑔𝑉𝑠 (1.27)
where 𝑉𝑠 is the volume of the solid, this shows that the body is buoyed up by the weight of the displaced fluid.
Consider a solid spherical body of mass 𝑚3 with center of mass 𝑀3, density 𝜌3 and radius b, so that 𝑚3=𝜌343𝜋𝑏3 (1.28) or 𝑏3=3𝑚34𝜋𝜌3 (1.29)
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Using equation (1.29) in (1.27) to get the buoyancy force (in magnitude) in a fluid of 𝜌1 𝐹𝐵=𝜌1𝑔43𝜋×3𝑚34𝜋𝜌3 or 𝐹𝐵=𝜌1𝑔𝑚3𝜌3 this implies that 𝐹 𝐵=−𝑚3𝜌1𝜌3𝑔 (1.30) The weight of the buoyed body is 𝐹 𝑠=𝜌𝑠𝑔 𝑉𝑠 (1.31)
where 𝐹 𝑠 𝑎𝑛𝑑 𝜌𝑠 are the density and volume of the solid body respectively, while 𝑔 is the gravity of the fluid 𝜌1 acting at 𝑀3. Equation (1.31) may be expressed as 𝐹 𝑠=𝜌𝑠𝑔 𝑉𝑠=𝐺𝑚1𝑚3𝑀1𝑀3 𝑅133 (1.32)
Where 𝑀1 is the center of the spherical shell having radius R filled with fluid 𝜌1 and 𝑀3𝑀1 =𝑅31=𝑅13 𝑎𝑛𝑑 𝑚3=𝑉𝑠𝜌𝑠 Putting equation (1.28) in equation (1.32), we have 𝐺𝑚1𝑚3𝑅 13𝑅133=𝜌3𝑔43𝜋𝑏3 (1.33) using equation 1.29 ,we get 𝐺𝑚1𝑚3𝑀1𝑀3 𝑅133=𝜌343𝜋×3𝑚34𝜋𝜌3 which implies 𝑔 = 𝐺𝑚1𝑀1𝑀3 𝑅133. 𝑠𝑖𝑛𝑐𝑒 𝑚1=43𝜋𝑅3𝜌1 (1.34) therefore,
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𝑔 = 43𝐺𝜋𝑅3𝜌1𝑀3𝑀1 𝑅133 𝑏𝑢𝑡 𝑅=𝑅13=𝑅31 𝑀3𝑀1 , the above takes the value 𝑔 = 43𝐺𝜋𝑅3𝜌1𝑀3𝑀1 𝑅133 or 𝑔 = 43𝐺𝜋𝜌1𝑀3𝑀1 (1.35) Putting equation (1.35) in equation (1.30), we have the buoyant force as 𝐹 𝐵=−4𝑚3𝜌123𝜌3𝐺𝜋𝑀3𝑀1 (1.36)
1.7.2 Robe’s restricted three-body problem
Robe (1977) introduced a new kind of restricted three-body problem in which he regarded one of the primaries of mass 𝑚1 as a rigid spherical shell filled with homogenous, incompressible fluid of density 𝜌1; the second one is a point mass located outside the shell and moving around the mass 𝑚1 in a Keplerian orbit; the infinitesimal mass 𝑚3 is a small solid sphere of density 𝜌3 moving inside the shell and is subject to the attraction of 𝑚2 and the buoyancy force due to the fluid of the first primary. He discussed the linear stability of an equilibrium point obtained in two cases; in the first case, the orbit of 𝑚2 around 𝑚1 is circular and in the second case, the orbit is elliptic, while the shell is empty (without fluid) or the densities of 𝑚1 and 𝑚3 are equal.
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Figure 1.2 Robe’s restricted three-body problem
Since we modified Robe’s model in our work .The derivation of the equations of motion of Robe’s problem (1977) is given in this section
Let the orbital plane of 𝑚2 around 𝑚1∗ (the shell with fluid 𝜌1) be taken as 𝑥𝑦−𝑝𝑙𝑎𝑛𝑒, and also let the origin of the coordinate system be at the center of mass, 0, of the two finite masses. Then the forces acting on the third body are;
1) The attraction of 𝑚2
2) The gravitational force 𝐹 𝐴 exerted by the fluid of density 𝜌1 is given by
𝐹 𝐴=𝐺𝑚1𝑚3𝑅 13𝑅133
Substituting the mass of the fluid 𝑚1 in the above gives 𝐹 𝐴=−𝐺4𝜋𝑅3𝑚1𝑚3𝑅 133𝑅133 (1.37)
Since the body of mass 𝑚3 maintains a spherical symmetry about 𝑀1 (center of 𝑚1) due to the pressure of the fluid inside 𝑚1, in this case, the radius R becomes 𝑅=𝑅13. Therefore, the gravitational force equation (1.37) is recast to the form
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𝐹 𝐴=− 43 𝜋𝐺𝜌12𝑚3𝑀1𝑀3 (1.38)
3) The buoyancy force 𝐹 𝐵, is obtained from equation (1.36) as
𝐹 𝐵=−43𝜋𝐺𝜌12𝑚3𝑀1𝑀3 𝜌3 (1.39)
The equation of motion of the infinitesimal mass 𝑚3 under the listed forces above, can be written as, Robe (1977).
𝑥 – 2𝑦 =𝑈𝑥,
𝑦 + 2𝑥 = 𝑈𝑦 , (1.40)
𝑧 = 𝑈𝑧, with 𝑈=12 𝑥2+𝑦2+𝑧2 +𝜇 1−𝜇−𝑥2 2+𝑦2+𝑧2−𝐾2 1−𝜇−𝑥2 2+𝑦2+𝑧2 , 𝜇=𝑚2𝑚1+𝑚2≤1 𝑎𝑛𝑑 𝐾=43𝜋𝜌1 1−𝜌1𝜌3 . System (1.40) is equations of motion for the Robe’s CR3BP.
1.7.3 Perturbation
In astronomy, perturbations is the deviation in the motion of a celestial object caused either by the gravitational force of a passing object or by a collision with it. For example, predicting the Earth’s orbit around the Sun would be rather straightforward if not for the slight perturbations in its orbital motion caused by the gravitational influence of the other planets.
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1.7.4 Coriolis and centrifugal forces
1.7.4.1 Coriolis force: In classical mechanics, the Coriolis force is an inertial force described by the 19th-century French engineer-mathematician Gustave-Gaspard Coriolis in 1835. He described this force in terms of its effect. The effect of the Coriolis force is an apparent deflection of the path of an object that moves within a rotating coordinate system. The object does not actually deviate from its path, but it appears to do so because of the motion of the coordinate system.
The Coriolis effect has great significance in astrophysics and stellar dynamics, in which it is a controlling factor in the directions of rotation of sunspots. It is also an important consideration in ballistics, particularly in the launching and orbiting of space vehicles. (Encyclopedia Britainica Ultimate , 2014)
1.7.4.2 Centrifugal force: A fictitious force, peculiar to a particle moving on a circular path, that has the same magnitude and dimensions as the force that keeps the particle on its circular path (the centripetal force) but points in the opposite direction.
Although it is not a real force according to Newton’s laws, the centrifugal-force concept is a useful one. For example, when analyzing the behaviour of the fluid in a cream separator or a centrifuge, it is convenient to study the fluid’s behaviour relative to the rotating container rather than relative to the Earth; and, in order that Newton’s laws be applicable in such a rotating frame of reference, an inertial force, or a fictitious force (the centrifugal force), equal and opposite to the centripetal force, must be included in the equations of motion. (Encyclopedia Britainica Ultimate 2014)
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