ABSTRACT
We investigate in the elliptic framework of the restricted problem of three-bodies, the motion of an oblate infinitesimal particle in the vicinity of a luminous primary and an oblate secondary. The locations and stability of the equilibrium points are found to be affected by the eccentricity, oblateness and radiation pressure parameters. We highlights the effects of the said parameters on the locations of the triangular points and stability using CEN X-3 model. The triangular points are also found to be stable for 0 < m < mC ; where m is the mass ratio (m £ ^{1}/_{2}). Further analysis indicates that collinear points remain unstable in spite of the introduction of these parameters.
TABLE OF CONTENTS
Cover page i
Tittle page iii
Declaration iii
Certification iii
Acknowledgement Err
or! Bookmark not defined.
Dedication vii
Abstract vii
Table of contents xi
List of figures xii
List of Tables xiii
List of Symbols xiv
CHAPTER ONE 1
INTRODUCTION 1
1.1 Historical Background 1
1.2 Statement of the Problem 3
1.3 Aim and Objectives 3
1.4 Significance/Justification of the Study 3
1.5 Research Methodology 5
1.6 Theoretical Frame Work 6
1.6.1 Three Body Problem (3BP) 6
1.6.2 RESTRICTED THREE BODY-PROBLEMS (R3BP) 6
1.6.3 Circular Restricted Three Body-Problem (CR3BP) 6
1.6.4 Elliptical Restricted Three Body-Problem (ER3BP) 7
1.6.5 Photo-gravitational Restricted Three Body-Problem 7
1.6.6 Radiation and Radiation Pressure 7
1.6.7 Ellipsoid 9
1.6.8 Oblateness 9
1.6.10 Eccentricity 11
1.6.11 Equilibrium points 11
1.6.12 Stability of Equilibrium Points 13
1.6.13 Stability of a Linear System 13
1.6.14 Kepler’s Laws of Planetary Motion 14
1.6.15 Newton’s Laws of Motion 14
1.6.16 Perturbation 15
1.6.17 Stability in the sense of Lyapunov 16
CHAPTER TWO 17
2.0 LITERATURE REVIEW 17
CHAPTER THREE 22
LOCATIONS AND LINEAR STABILITY OF THE TRIANGULAR
EQUILIBRIUM POINTS 22
3.1 Introduction 22
3.2 Mathematical Model 22
3.3 Variational Equation 23
3.4 Characteristic Equation 24
3.5 Jacobian Integral 25
3.6 Locations of the Equilibrium Points 26
3.6.1 Locations of Triangular Equilibrium points 26
3.7 Linear Stability of the Equilibrium Points 33
3.7.1 Stability of the triangular Equilibrium points 33
CHAPTER FOUR 54
LOCATIONS AND LINEAR STABILITY OF THE COLLINEAR EQUILIBRIUM POINTS 49
4.1 Introduction 49
4.2 Locations of the Collinear Equilibrium Points 50
4.3 Stability of the collinear Equilibrium points 57
4.3.1 Stability of L_{1} (x > x_{2} )
4.3.2 Stability of L2 (x1 < x < x2 )
4.3.3 Stability of L_{3} (x_{1} > x )
CHAPTER FIVE 56
RESULTS AND DISCUSSION 56
5.1 Introduction 56
5.2 Numerical Results of the Triangular Equilibrium Points 57
5.2.1 Discussion of the Triangular Equilibrium Points 67
5.3 Numerical Results of the Collinear Equilibrium Points 68
5.3.1 Discussion of the Collinear Equilibrium Points 75
CHAPTER SIX 76
SUMMARY, CONCLUSION AND RECOMMENDATION 76
6.1 Summary 76
6.2 Conclusion 75
6.3 Recommendations 76
6.4 Contribution to knowledge 81
REFERENCES 82
LIST OF TABLES
Table 5.1: Relevant Numerical data
Table 5.2: Effect of oblateness of the Secondary on𝐿4,5 of CEN X-3 for e=0.3 and a=0.9 A=0.01
Table 5.3: Effect of oblateness of the third body on𝐿4,5 of CEN X-3 for e=0.3 a=0.9 and A2=0.I
Table 5.4: Effect of eccentricity on𝐿4,5 of CEN X-3 for a=0.9
Table 5.5: Effect of radiation pressure on𝐿4,5 of CEN X-3 for a=0.9 and e=0.3
Table 5.6: Effect of oblateness of the third body for𝐴2 = 0.1 𝑎𝑛𝑑 𝑒 = 0.3 on L1 of CEN X-3
Table 5.7: Effect of oblateness of the secondary for𝐴 = 0.1 𝑎𝑛𝑑 𝑒 = 0.3 on L1 of CEN X-3
Table 5.8: Effect of eccentricity for𝐴 = 0.01, 𝐴2 = 0.01 𝑎𝑛𝑑 𝑒 = 0.3 on L1 of CEN X-3
Table 5.9: Effect of radiation for𝐴 = 0.01, 𝐴2 = 0.01 𝑎𝑛𝑑 𝑒 = 0.3 on L1 of CEN X-3
Table 5.10: Effect of semi major axis for𝐴 = 0.01, 𝐴2 = 0.01 𝑎𝑛𝑑 𝑒 = 0.3 on L1 of CEN X-3
Table 5.11: Effect of oblateness of the third body for𝐴2 = 0.1 𝑎𝑛𝑑 𝑒 = 0.3 on L2 of CEN X-3
Table 5.12: Effect of oblateness of the secondary for𝐴 = 0.1 𝑎𝑛𝑑 𝑒 = 0.3 on L2 of CEN X-3
Table 5.13: Effect of eccentricity for𝐴 = 0.01, 𝐴2 = 0.01 𝑎𝑛𝑑 𝑒 = 0.3 on L2 of CEN X-3
Table 5.14: Effect of radiation for𝐴 = 0.01, 𝐴2 = 0.01 𝑎𝑛𝑑 𝑒 = 0.3 on L2 of CEN X- 3
Table 5.15: Effect of semi major axis for𝐴 = 0.01, 𝐴2 = 0.01 𝑎𝑛𝑑 𝑒 = 0.3 on L2 of CEN X-3
Table 5.16: Effect of oblateness of the third body for𝐴2 = 0.1 𝑎𝑛𝑑 𝑒 = 0.3 on L3 of CEN X-3
Table 5.17: Effect of oblateness of the secondary for𝐴 = 0.1 𝑎𝑛𝑑 𝑒 = 0.3 on L3 of CEN X-3
Table 5.18: Effect of eccentricity for𝐴 = 0.01, 𝐴2 = 0.01 𝑎𝑛𝑑 𝑒 = 0.3 on L3 of CEN X-3
Table 5.19: Effect of radiation for𝐴 = 0.01, 𝐴2 = 0.01 𝑎𝑛𝑑 𝑒 = 0.3 on L3 of CEN X- 3
Table 5.20: Effect of semi major axis for𝐴 = 0.01, 𝐴2 = 0.01 𝑎𝑛𝑑 𝑒 = 0.3 on L3 of CEN X-3
LIST OF FIGURES
Figure 1.1: Circular restricted three body problem Figure 1.2: Shape of an ellipsoid
Figure 1.3: The five Equilibrium points of the Sun-Earth-Moon System Figure 1.4: Lyapunov Stability Region
Figure 4.1: Position of the collinear equilibrium point L Figure 4.2: Position of the collinear equilibrium point L Figure 4.3: Position of the collinear equilibrium point L
Figure 5.1: Effect of radiation pressure on𝐿4,5 𝑜𝑓 Cen X-3 for𝑎 = 0.9, 𝐴 = 0.01 𝑎𝑛𝑑, 𝐴2 = 0.01
Figure 5.2: Effect of eccentricity on𝐿4,5 𝑜𝑓 Cen X-3 for𝑎 = 0.9, 𝐴 = 0.01 𝑎𝑛𝑑, 𝐴2 = 0.01
Figure 5.3: Effect of oblateness of the third body on𝐿4,5 𝑜𝑓 Cen X-3 for 𝐴 = 0.01 𝑎𝑛𝑑, 𝐴2 = 0.01
Figure 5.4: Effect of semi-major axis (a) on the stability region for A=0.001,𝑞 = 0.99998, 𝐴 = 0.001 𝑎𝑛𝑑 𝐴2 = 0.002
Figure 5.5: Effect of semi-major axis (a) on the stability region for A=0.001,𝑞 = 0.99998, 𝐴 = 0.001 𝑎𝑛𝑑 𝐴2 = 0.00225
Figure 5.6: Effect of eccentricity on the stability region for A=0.00,𝑎 = 0.9, 𝑎𝑛𝑑 𝐴2 = 0.01
Figure 5.7: Effect of oblateness of the third bodyon the stability region for e=0.3a=0.9𝑞 = 0.99998, 𝑎𝑛𝑑 𝐴2 = 0.01
Figure 5.8: Effect of eccentricity on the stability region for a=0.9,𝑞 = 0.99998, 𝐴 = 0.001 𝑎𝑛𝑑 𝐴2 = 0.01
Figure 5.9: Effect of oblateness of the third body on the collinear points 𝐿𝑖(1,2,3)of CEN X-3
Figure 5.10: Effect of oblateness of the smaller primary on the collinear points 𝐿𝑖(1,2,3)of CEN X-3
Figure 5.11: Effect of eccentricity on the collinear points 𝐿𝑖(1,2,3)of CEN X-3 Figure 5.12: Effect of radiation pressure on the collinear points 𝐿𝑖(1,2,3)of CEN X-3
Figure 5.13: Effect of semi major axis on the collinear points 𝐿𝑖(1,2,3)of CEN X-3
LIST OF SYMBOLS
m Mass ratio
mc Critical value of the mass parameter
m1 Mass of the bigger primary
m2 Mass of the smaller primary
W Potential force function
A Oblateness of the third body
𝐴_{2} Oblateness of the smaller primary body
r1 Distance from the third body to the bigger primary
r2 Distance from the third body to the smaller primary
a = 1-a Semi-major axis of the system
e Eccentricity of the system
q = 1- b Radiation coefficient of the bigger primary.
n Mean motion
CHAPTER ONE
INTRODUCTION
1.1 Historical Background
Classical mechanics may be defined as that branch of mechanics which deals with the deterministic motion of bodies. This is the oldest branch of mechanics. It dates back to the scientific evolution and is said to have originated in Galileo's verification of the heliocentric theory by means of telescope. It is also connected to Galileo’s study of falling bodies although some of the ideas are found in the early works of Oresume, who studied the motion of a uniformly accelerated body. The law of inertia was already known to Davinci while Kepler discovered several laws of planetary motion. ( Amuda, 2014)
The origin of classical mechanics goes back to Koperniks theory of the solar system. This theory opened up the possibility that celestial and terrestrial matter might be of the same nature, hence, governed by the same law of mechanics using Hugyens results on centripetal acceleration. Hooke and Wren realized that this force diminishes as the inverse square of the distance. Newton identified this force as the same force, which makes objects fall near the surface of the earth and succeeded in computing the orbits of celestial bodies using the inverse square law. Based on these laws, it was possible to solve what appeared complicated, the two body problem which describes the motion of two bodies of finite masses moving under a mutual gravitational attraction. The motion of any of the planets around the sun constitutes a two –body problem. Example is the solar system, which consists of the Sun and its nine planets. (Laraba, 2012)
The most celebrated problem of space dynamics is the problem of three-body. The three-body problem is defined in terms of three bodies with arbitrary masses attracting one another according to the Newton law of gravitation which are free to move in space in any given manner. (Hussain, 2017)
A classical example of the 3BP is the Sun- Planet -Planet system. When they are considered as point masses, they form what is known as the main problem of the lunar theory. The degenerated case of the 3BP is the restricted three body problem (R3BP) which describes the motion of an infinitesimal mass moving under the gravitational effects of two finite masses called the primaries moving in circular or elliptic orbits around their common center of mass on account of their mutual gravitational attraction and the infinitesimal mass not influencing the motion of the primaries.
The R3BP is one of the most widely studied areas in space dynamics as well as celestial mechanics. Significant results have been produced by well-known mathematicians and scientists in an attempt to understand and predict the motion of natural bodies. The application of R3BP spans solar system dynamics, lunar theory, motion of space crafts and stellar dynamics. A typical example of the R3BP is seen in a system made up of the Sun and Jupiter as primaries and then a Trojan asteroid assuming the role of the infinitesimal mass in the Sun –Jupiter system. This problem began with Euler in 1767, in connection with his lunar theories which brought about his major accomplishment in the introduction of synodic (rotating) coordinate system. This led to the discovery of the Jacobian integral by Jacobi in 1836.(Amuda, 2014)
In recent times, many properties such as shape, surface area, light, perturbing forces are taken into consideration in describing the motion of satellites (both artificial and natural), meteorites, asteroid and their stability.
1.2 Statement of the Problem
Singh and Umar (2013b) investigated in the elliptic framework of the R3BP the application of double pulsars to the axisymmetric R3BP in which both pulsars are oblate. They found that the triangular and collinear points are unstable.Their study however did not consider one of the primaries to be luminous. In this study we shall consider the motion of an oblate third body (satellite, test particle, asteroids, comet or circumbinary planet) around a luminous primary and an oblate secondary. This is applicable to both the solar system (Sun-Planet-Moon-system) and stellar system(Cen X-3).
1.3 Aim and Objectives
The aim of this research work is to investigate and examine the motion of an oblate third body in the frame work of ER3BP when the primary is radiating and the secondary is an oblate spheroid. In line with this idea, the objectives of the study are to:
1. Locate the collinear and triangular equilibrium points in the neighbourhood of binary System CEN X-3.
2. Examine the linear stability of the equilibrium points in the neighbourhood of the binary system CEN X-3.
1.4 Significance/Justification of the Study
The study of the restricted three-body problem over the last 200 years has produced significant results and is the backbone of space technology. Man-made satellites are modelled and built as test particles in orbits of celestial bodies. Results considering the shape of the Earth and radiation effect of the Sun, in the Sun-Earth-Satellite system are examples. The equilibrium points of the R3BP are of enormous importance to space application in the past, present and future. The equilibrium solutions of these systems are widely used in many branches of astronomy, both for constant and variable masses as in the case of the Roche model for binary star system (Lyapunov, 1956).
The collinear equilibrium points are good spots for space-based observatories given their location and accessibility and they provide easy access to orbits in the case of lunar and Earth orbits; though all of them are unstable. This means a spacecraft to be kept at or orbiting around them will require correction maneuvers typically to be performed at the expenses of a propellant mass. The triangular equilibrium points are also very interesting from the point of view of astronomical objects; possible location of interplanetary dust and asteroids, hence they have been suggested as convenient sites to locate future space colonies. As such scientists and astronomers have been devoted to set up a colony at one of the two triangular points of the Earth-Moon system.
This research exposes us to the behavior of an oblate test particle when perturbations such as radiation pressure force and oblateness of the primaries act on it. The participating bodies in the classical R3BP are assumed to be strictly spherical in shape, but in reality, it has been proved that several heavenly bodies are sufficiently oblate or triaxial, for instance, Earth, Jupiter and Saturn, are oblate while the moon of the Earth, Pluto and its moon Charon are triaxial. Also, the minor planets and meteoroids have irregular shapes. In these cases, on account of the small dimensions of the bodies in comparison with their distances from the primaries, they are considered to be point masses, but in many cases the dimension of the bodies are larger than the distances from their respective primaries. Thus, the results obtained are far from realistic. The lack of sphericity, (triaxiality, oblateness and so on) of the celestial bodies causes large perturbation from two- body orbit. The motions of artificial Earth satellites are examples of this.
These foresaid perturbations can actually cause a large deviation in the location of libration points and may change the entire pattern of their stability which is very vital in the lunch of artificial satellites and various outer space probes. The motions of artificial Earth satellites are examples of this and also several studies that have considered one or both primaries as oblate spheroids. These include among many others SubbaRao and Sharma (1975), Sharma (1987), Khanna and Bhatnagar (1999), Singh and Ishwar (1999) Douskos and Markellos (2006) Abdurraheem and Singh (2006) Kushvah (2008), Singh and Begha (2011), Singh and Leke (2012) and Singh and Umar (2015).
Hence in connection to the above studies, it is reasonable to study the case when the bigger primary is a source of radiation and the smaller one an oblate spheroid. This model will have enormous applications in various astronomical problems such as space mission design.
1.5 Research Methodology
The effects of the oblateness of an artificial satellite on the orbits around the triangular points of the Earth –Moon system was studied by Singh and Umar (2013a). Thus following Singh and Umar (2013a), we wish to modify the equations of motion of the infinitesimal particle with respect to the primaries, locate the equilibrium points and study their linear stability when the primary is radiating and the secondary is an oblate spheroid in the vicinity of the binary system CEN X-3.Lastly, we intend to use theMathematicasoftware 10.3 for numerical computations and plotting of graphs where necessary. We shall show both numerically and graphically the effects of the perturbing forces (radiation pressure of the primary, oblateness of the secondary body and third body (test particle) as well, and the eccentricity of the orbit of the primaries on the position and stability of the equilibrium points.
1.6 Theoretical Frame Work
1.6.1 Three Body Problem (3BP)
The study of the motion of three particles which are free to move in space under their mutual gravitational influence and initially move in any given manner is called the three-body problem (3BP).
1.6.2 RESTRICTED THREE BODY-PROBLEMS (R3BP)
The restricted three-body problem (R3BP) is a simplified form of the general 3BP, in which one of the bodies is of infinitesimal mass, and therefore does not influence the motion of the remaining two massive bodies called the primaries.
1.6.3 Circular Restricted Three Body-Problem (CR3BP)
At the Newtonian level, the CR3BP deals with the motion of the infinitesimal mass in the gravitational field of the primaries, which revolve in circular orbit around their center of mass. Consider an isolated dynamical system consisting of three gravitationally interacting point masses, m1, m2, and m3. Suppose, however, that the third mass, m3, is so much smaller than the other two that it has a negligible effect on their motion. Suppose, further, that the first two masses, m1 and m2, execute circular orbits about their common center of mass. Later, we shall investigate this simplified problem, which is generally known as the circular restricted three-body problem.
Figure 1.1:The circular restricted three-body problem.
1.6.4 Elliptical Restricted Three Body-Problem (ER3BP)
The ER3BP deals with the motion of the infinitesimal mass in the gravitational field of the primaries, which revolve in elliptic orbits around their center of mass.
1.6.5 Photo-gravitational Restricted Three Body-Problem
An R3BP is called photogravitational when one or both of the primaries is /are a source of light.
1.6.6 Radiation and Radiation Pressure
Radiation pressure implies an interaction between electromagnetic radiation and bodies of various types, including clouds of particles or gases. The interaction can be absorption, reflection, or both (the common case). Bodies also emit radiation and thereby experience resulting pressure. Radiation from a material or body is the transfer of heat energy in the form of electromagnetic wave which does not require an electric medium or the emitting of energy. Any hot object radiates heat energy in form of electromagnetic wave of all wavelengths. However, the heat emitted by a surface depends on the nature of the surface, temperature of the body and the surface area. Example of a radiating body is the Sun in the solar system.
The radiation pressure force FP changes with distance by the same law as the gravitational force Fg and acts opposite to radiation pressure. It is possible that the force will lead to a reduction of the effective mass of the particle, and since this reduction depends on the properties of the particles, it is acceptable to speak about a reduced mass, thus, the resultant force on the particle is
Where is the mass reduction factor and the force of the body is given by FP = (1 - q)Fg such that 0 < (1-q) << 1
We note that;
If q=1, the radiation pressure has no effect.
If 0 < q < 1, the gravitational force is greater than the radiation force. If q = 0, the radiation force balances the gravitational force.
If q < 0 , the radiation pressure overrides the gravitational attraction.
Following these discoveries, several studies such as Hamilton and Burns (1992), Singh and Ishwar (1999), Kunitsyn (2000), Singh (2009), Singh and Leke (2010), Singh and Umar (2012 a,b), Singh and Taura (2013) have produced significant results in the study of the restricted three- body problem under different assumptions, by taking into account the radiation pressure forces.
1.6.7 Ellipsoid
An ellipsoid is a closed quadric surface that is a three-dimensional analogue of an ellipse.
Figure 1.2: Shape of an ellipsoid.
1.6.8 Oblateness.
In the figure above, if a b c , we have an oblate spheroid. In other words, an ellipsoid having a polar axis shorter than the diameter of the equatorial circle whose plane bisects it is known as an oblate spheroid, i.e., its two out of three moments of inertia are equal.
We denote
Ai (i = 1,2) for the oblateness coefficients of the bigger and smaller primary respectively such that 0 < Ai < 1 (McCuskey, 1963). The values , where AE1 and AE2 are the equatorial radii, AP1 and AP2 , the polar radii of the bigger and smaller primaries respectively and R the distance between the primaries. Oblate spheroids are contracted along a line, whereas prolate spheroids are elongated. It can be formed by rotating an ellipse about its minor axis, forming an equator with the end points of the major axis. As with all ellipsoids, it can also be described by the lengths of three mutually perpendicular principal axes, which are in this case two arbitrary equatorial semi major axes and one semi –minor axis.
The R3BP assumes that masses concerned are spherically symmetrical in homogeneous layers, but it is found that celestial bodies, such as Saturn and Jupiter are sufficiently oblate (Beatty et al.1999). The minor planets (e.g. Ceres) and meteoroids have irregular shapes (Millis et al.1978; Norton & Chitwood 2008). The oblateness or triaxiality of a body can produce perturbation deviation from the two- body motion. The orbits of the fifth satellite of Jupiter Amalthea is one of the most striking examples of perturbations arising from oblateness in the solar system (Moulton, 1914).
Rotation in stars produces an equatorial bulge due to centrifugal force, and as a result of the rapid rotation after formation of Neutron stars, white and black dwarfs, they may be considered oblate. A neutron star on formation can rotate at rate of nearly a thousand rotation per second (Du et al.2009).The millisecond pulsar PRSB1937+21, spinning about 642 times a second and the pulsar PSRJ1748-422ad, spinning 716 times a second are some of the swiftest spinning pulsar (Hessels et al. 2006).This has motivated several researchers such as Subbarao & Sharma (1975), Elipe &Ferrer (1985), Singh and Ishwar (1999), Abdurraheem & Singh (2006), Vishnu et al (2008),Singh (2011,2012), Singh & Umar (2012a,b), Singh & Leke (2013), Singh & Umar (2013a,b,c, 2014a,b,c,2015,) to include oblateness of one or both primaries in their studies. It is the approximate shape of many planets and celestial bodies, including Saturn, Jupiter and to a lesser extend the Earth. It is therefore the most used geometric figure for defining reference ellipsoids, upon which cartographic and geodetic system are based.
1.6.9 Synodic or Rotating Co-ordinate System
If the system of coordinates is such that the x -h plane rotates in the positive direction with an angular velocity equal to that of the common velocity of one primary with respect to the other keeping the origin fixed, then the coordinate system is known as a synodic co-ordinate system. The synodic co-ordinate system is accelerated, since it is rotating with a velocity.
1.6.10 Eccentricity
The orbital eccentricity of an astronomical body is the amount by which its orbit deviates from a perfect circle and is usually denoted by e. Where for e < 1 , the orbit is elliptic; for e = 0, the orbit is circle; for 𝑒 = 1 the orbit is parabolic and for e >1, the orbit is hyperbolic.
1.6.11 Equilibrium points
The equilibrium points are those points at which the velocity and the acceleration of an infinitesimal mass are zero. CR3BP admits five equilibrium points in the plane of motion of the primaries, three are the collinear points L1, L2, & L3 lying on the line connecting the primaries, while the other two are the triangular points L4 & L5 forming equilateral triangles with the primaries. The collinear equilibrium points were found by Euler in 1767 while the triangular equilibrium points were worked out by Lagrange in 1772.
Figure 1.3: The five Equilibrium Points of the Sun-Earth-Moon System
Aside from the five equilibrium points of the classical restricted three-body problem, further studies have revealed other types of equilibrium points, examples are the out – of-plane equilibrium points, which are referred to as co-planer points. These equilibrium points have been found in the studies of the restricted three-body problem with variable masses (Singh and Leke 2010,2012, 2013). In the case of problem of constant masses, these points are found by expressing the equation of motion in three- dimensional form, when radiation or oblateness is involved (see Radzievsky 1950,1953; Douskos and Markellos 2006; Shankaran et al.2011;Singh and Umar 2013a,b,c and 2015).
1.6.12 Stability of Equilibrium Points
The motion of an infinitesimal particle near one of equilibrium points is said to be stable if for a given small displacement and small velocity of the particle oscillates for considerable time around that point and when the time elapses it returns to the same point, otherwise unstable.
1.6.13 Stability of a Linear System
In ordinary differential equations, the stability of linear systems is determined by the eigen values of the coefficient matrix. The locations of the infinitesimal body would be displaced a little from the equilibrium point due to perturbations. If the resultant motion of the infinitesimal body is a rapid departure from the vicinity of the point, we call such location of equilibrium point an “unstable one” otherwise it is stable. In order to examine the stability of the orbit in the vicinity of the equilibrium points, we apply this small displacement method by shifting the origin or coordinates of the infinitesimal mass and linearizing the equations of motion around the coordinates of the equilibrium. The variational equations of motion corresponding to the dynamical system are derived, which in turn through the trial solutions are transformed to a matrix form and a characteristic equation of the variational equations of the dynamical system is obtained. The stability of the solutions depends on the nature of the characteristic roots.
(i) For complex roots; the equilibrium point is asymptotically stable when all roots have negative real parts, and unstable when some or all roots have positive real part, while multiple complex roots can either be stable or unstable.
(ii) For pure imaginary roots; equilibrium point is stable, though not asymptotically stable. If there are multiple roots, the solution contains mixed terms (i.e., periodic and secular terms), the equilibrium point is unstable.
(iii) For real roots; the equilibrium point is stable if all the roots are both real and negative, but unstable if any of the roots is positive. This statement is also true for multiple roots.
1.6.14 Kepler’s Laws of Planetary Motion
• Each planet moves, relative to the sun, in an elliptical orbit, the Sun being at one of the two foci of the ellipse.
• The rate of motion in the elliptical orbit is such that the vector pointing to the position of the planet relative to the sun spans equal areas of the orbital plane in equal times.
• The square of the orbital period T, is proportional to the cube of the semi-major axis of the orbital ellipse.
1.6.15 Newton’s Laws of Motion
• A body continues in its state of rest, or in a uniform motion in a straight line, unless it is compelled to change that state by a force impressed upon it.
• The acceleration of motion of a body is directly proportional to the force to which it is subjected, and inversely proportional to its mass, and takes place in the direction in which the force acts.
• To every action there is an equal and opposite reaction; or, the mutual actions of two bodies are always equal and oppositely directed.
1.6.16 Perturbation
In astronomy, perturbation 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.
1.6.17 Stability in the sense of Lyapunov
A dynamic system x = f (x) is Lyapunov stable or internally stable about an equilibrium point 𝑥𝑒𝑞 if state trajectories are confined to a bounded region whenever the initial condition x0 is chosen sufficiently close to 𝑥𝑒𝑞 .
Mathematically, given R > 0 there always exists r > 0 so that if IIx0 - xeqII < r , then IIx(t) - xeqII < R for all t > 0. As seen in figure 1.4, R defines a desired confinement region, while r defines the neighborhood of 𝑥𝑒𝑞 where x0 must belong so that x(t) does not exit in the confinement region.
An equilibrium point 𝑥𝑒𝑞 of 𝑥 = f (x) ( i.e., f (xeq ) = 0 ) is
i. Locally stable, if for every R > 0 there exist r > 0 , such that IIx(0) - xeqII < r implies IIx(t) - xeqII < R for all x > 0.
ii. Locally asymptotically stable, if locally stable and
iii. Globally asymptotically stable, if it is asymptotically stable for all x(0) Î R^{n}
Figure 1.4: Lyapunov Stability Region
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