stability of motions near lagrange points in the elliptic...

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MSC Computational Physics

Thesis

Stability of motions near Lagrange points in

the Elliptic Restricted Three Body Problem

Zografos Panagiotis

Supervising Professor:

Voyatzis George

Aristotle University of Thessaloniki

Faculty of Science, Department of Physics

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Contents

Abstract__________________________________________________3

1. Introduction_____________________________________________4

2. Description of the Elliptic Restricted Three Body

Problem__________________________________________________6

A. Equations of Motion in the Inertial Barycentric

system__________________________________________________11

B. Equations of Motion in the Rotating system______________15

C. The Five Lagrange points of Equilibrium________________20

D. The Fast Lyapunov Indicator (FLI)_____________________27

3. The Stability of The Lagrange points of Equilibrium and Stability

range___________________________________________________28

A. Stability of the Lagrange points_______________________28

B. Regular orbits in the neighborhood of the Lagrange

points___________________________________________________33

C. Examination of the results____________________________97

D. Appreciation for Exoplanetary systems__________________99

4. Conclusions___________________________________________102

APPENDIX_____________________________________________103

References______________________________________________143

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Abstract

The nature of stability of the Lagrange points has been

studied in the two-dimensional, elliptic restricted three body

problem. The regular regions near the neighborhoods of

Lagrange points 𝐿4 and 𝐿5 have been computed for different

values of the mass ratio μ and the orbital eccentricity e of the

primaries. It has been found that the volume of stability

regions decreases as both μ and e increase. The range of the

regular regions near 𝐿4 of 15 extrasolar planetary systems,

consisting of one star and one Jupiter-like planet, has been

calculated. Given a large enough regular region, there exists

the possibility of co-orbital companions or Trojan-like objects

in stable orbits around the triangular equilibrium points. A

system of rotating-pulsating coordinates has been used. The

equations of motion and the corresponding variational

equations have been integrated numerically using the Runge-

Kutta method of the 4th order. The nature of stability of the

Lagrange points has been calculated using the Floquet theory.

The range of the regular regions θ has been determined by

calculation of the Fast Lyapunov Indicators (FLI) for a grid of

initial conditions around 𝐿4 and 𝐿5. For the purposes of this

study several computer programs have been written using C

and C++ programming language.

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1. Introduction

The elliptic restricted three body problem is a special case

of the three body problem. This means that one of the three

bodies has mass equal to zero, and that the other two bodies

move in Keplerian elliptic orbits. The purpose here is to study

the motion of the massless body under the gravitational

influence of the other two bodies. This study can be carried out

in two or three dimensions, but here we chose the former and

the simplest.

Unlike the “circular restricted three body problem” which

is a well-documented and simple dynamical system, the elliptic

problem is considerably more complicated and presents

different challenges in its study. This is due to the fact that the

elliptic problem is non-conservative, as the absence of the

energy integral adds numerous implications.

The interest of studying the elliptic problem stems from

the fact that it is a better approximation of the motions that

occur in the solar system in comparison to the circular

problem. For instance, the motion of the moon as it is affected

by the Sun and the Earth, or the motion of a satellite under the

influence of the Earth and the moon, are better studied by

using primaries moving in elliptic orbits.

The equations of motion of the elliptic problem differ

from those in the circular problem in three important places:

1. The elliptic problem contains two parameters

instead of one, the eccentricity e and the mass ratio

of the primary bodies μ.

2. The elliptic problem has no energy (Jacobi) integral.

3. The independent variable is present in the equations

of motion, even when rotating axes are used.

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The details of the solution of the equations of motion and the

variational equations are discussed in Section 2.

This work concentrates on the nature of stability of the

triangular Lagrange equilibrium points 𝐿4 and 𝐿5, as well as

the stability of orbits in their neighboring regions for a large

number of μ and e (including cases of e=0) and different initial

conditions of the massless body. This work’s goal is to

determine the regions near the Lagrange points, where if the

massless body is left motionless (initial velocity components

equal to zero), it will follow a stable or regular orbit in the

neighborhood of the Lagrange points and not veer to infinity.

In order to accomplish this, the nature of stability of the five

Lagrange points - which according to Broucke (1969) falls into

seven categories - was calculated for different pairs of μ-e

using the Floquet theory. All of the above were made possible

by numerical integration of the equations of motion of the

elliptic problem in the rotating-pulsating frame of reference

using the Runge-Kutta method of the 4th order, as well as

integration of the variational equations.

The stability of the neighborhood of the Lagrange points

in the elliptic three-body problem is of great interest when the

problem is applied to exoplanetary systems (EPS). The vast

majority of extrasolar planetary systems that have been

discovered contain one Jupiter-like gas giant, their masses

ranging from 5𝑀⊕ up to several Jupiter masses (Schwarz,

Dvorak, Süli & Érdi, 2007). The importance of the stable

regions near the equilibrium points and especially near the

triangular Lagrange points 𝐿4 and 𝐿5 becomes apparent if one

poses the question of whether or not a Trojan-like planet may

move in a stable orbit around the aforementioned Lagrange

points. Even more importantly, can such planets be habitable?

“When astronomers look for life in EPS, they have to know the

location of the so-called habitable zone (HZ), which is defined

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as a region around the star where the planet could receive

enough radiation to maintain liquid water on its surface and to

be able to build a stable atmosphere” (Schwarz, Dvorak, Süli

& Érdi, 2007). When the gas giant of an EPS moves into the

HZ, a habitable satellite could have a stable orbit (e.g. Titan

around Saturn), or a Trojan-like planet could exist in a stable

orbit near the 𝐿4 and 𝐿5 Lagrange points (Schwarz, Dvorak,

Süli & Érdi, 2007). The results of this work concerning the

regular regions near the equilibrium points, depending on the

mass ratio μ and the eccentricity e, are cross-examined with

known EPS in order to suggest possible candidate systems for

investigation of the possible existence of Trojan-like planets in

those systems.

The computer programs needed for the calculations were

written in C programming language. The results, diagrams and

data matrices are provided and discussed in Section 3.

2. Description of the Elliptic Restricted Three Body

Problem

In this section, the equations of motion of the elliptic

problem relative to different coordinate systems are presented.

The results of this study were obtained by numerical

integration of these equations in two dimensions.

Suppose a particle called the satellite is moving under the

gravitational influence of two massive bodies according to

Newton’s laws. The massive bodies are called the primaries.

All three objects are moving on the same plane. The problem

is defined as restricted because the satellite is massless

meaning that it can’t influence the motion of the primaries,

which move in elliptic Keplerian orbits relative to each other

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or relative to their center of mass. Elliptic orbits with various

values of eccentricity e including e=0 (circular problem), are

considered in this study.

In the following, the equations of motion in an inertial

barycentric frame of reference will be presented, although the

most important calculations use rotating systems of

coordinates. At the initial value of the independent variable

(t=0), the primaries always lie on the x-axis at an apse, at

periapsis at minimum distance or apopsis at maximum distance

(Broucke, 1969). Also, a system of regular units is used in

such a way that the semi-major axis α and the mean motion n

of the motion of the primaries equals 1. In this case, the masses

of the primaries (including the gravitational constant) can be

written as

𝑚1 = 1 − 𝜇, 𝑚2 = 𝜇 <1

2. (1)

The distance between the primaries is

𝑟 = (1 − 𝑐𝑜𝑠𝐸) =𝑝

1 + 𝑒 𝑐𝑜𝑠𝑣 (2)

where E is the eccentric anomaly, v the true anomaly and

𝑝 = (1 − 𝑒2) (3)

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is the semilatus rectum. In this barycentric system of

coordinates, the coordinates of the primaries are

𝜉1 = −𝜇 𝑟 𝑐𝑜𝑠𝑣 = −𝜇(𝑐𝑜𝑠𝐸 − 𝑒) (4𝑎)

𝜂1 = −𝜇 𝑟 𝑠𝑖𝑛𝑣 = −𝜇(1 − 𝑒2)12 𝑠𝑖𝑛𝐸 (4𝑏)

𝜉2 = (1 − 𝜇) 𝑟 𝑐𝑜𝑠𝑣 = (1 − 𝜇)(𝑐𝑜𝑠𝐸 − 𝑒) (4𝑐)

𝜂2 = (1 − 𝜇)𝑟 𝑠𝑖𝑛𝑣 = (1 − 𝜇)(1 − 𝑒2)12 𝑠𝑖𝑛𝐸 (4𝑑).

Kepler’s equation will be used to relate the eccentric

anomaly E with the time t:

𝑡 + 𝑥 = 𝐸 − 𝑒 𝑠𝑖𝑛𝐸 (5)

Here, the phase constant x is either equal to 0 (periapsis) or π

(apopsis). Equation (5) can be differentiated with respect to t to

produce the differential equation

𝑑𝐸

𝑑𝑡=

1

1 − 𝑒 𝑐𝑜𝑠𝐸 (6)

which is solved numerically in order to acquire E, which in

turn is used to calculate the coordinates of the primaries from

eq. (4).

A few more formulas relating to the motion of the

primaries are proven useful. The first and second derivatives of

r, v and E are

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𝑟′ =𝑒 𝑠𝑖𝑛𝑣

𝑝12

=𝑒 𝑠𝑖𝑛𝐸

𝑟, 𝑟′′ =

𝑒 𝑐𝑜𝑠𝑣

𝑟2=𝑝 − 𝑟

𝑟3

𝑣′ =𝑝12

𝑟2, 𝑣′′ =

−2𝑒 𝑠𝑖𝑛𝑣

𝑟3 (7)

𝐸′ =1

𝑟, 𝐸′′ =

−𝑒 𝑠𝑖𝑛𝑣

𝑟2𝑝12

.

Using eq. (7), the energy integral of the two-body problem of

the primaries can be verified:

1

2(𝑟′2 + 𝑟2𝑣′2) −

1

𝑟= −

1

2. (8)

The above derivatives are notated with a prime to indicate

differentiation with respect to the time t. Later, the true

anomaly v will be used as the independent variable, and the

corresponding derivatives will be indicated by dots. The

derivatives of r with respect to v are

�̇� =𝑒𝑟2 𝑠𝑖𝑛𝑣

𝑝, �̈� =

2�̇�2

𝑟+ 𝑟 (1 −

𝑟

𝑝). (9)

The energy integral can then be written as

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𝑝

2(�̇�2

𝑟+ 𝑟) −

1

𝑟= −

1

2. (10)

By combining eq. (10) and the second eq. (9), the following

differential equation for r is obtained:

�̈� = −2

𝑝𝑟3 +

3

𝑝𝑟2 − 𝑟. (11)

Eq. (11) can be used to determine r except when p=0 or e=1, in

which case eq. (1) has to be used, after solving Kepler’s

equation eq. (5) or (6). Also, since changes in the independent

variable from t to v will be made, the relation between the t-

derivatives and the v-derivatives of any given quantity F are

𝐹′ =𝑝12

𝑟2�̇�, 𝐹′′ =

𝑝(𝑟�̈� − 2�̇��̇�)

𝑟5. (12)

In the following sections, a special set of coordinates

(𝜉̅, �̅�) will be introduced. These are called “reduced” or

pulsating coordinates and they introduce a radial scale change

such that the elliptic motion of the primaries in the system

(𝜉, 𝜂) is transformed in a circular motion in the system (𝜉̅, �̅�)

(Broucke, 1969).The relation between the two sets is

proportional to r:

𝜉 = 𝑟𝜉̅, 𝜂 = 𝑟�̅�. (13)

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The reduced coordinates of the primaries 𝑚1 and 𝑚2 are then

written as

𝜉1̅ = −𝜇 𝑐𝑜𝑠𝑣, 𝜂1̅̅̅ = −𝜇 𝑠𝑖𝑛𝑣

(14)

𝜉2̅ = (1 − 𝜇) 𝑐𝑜𝑠𝑣, 𝜂2̅̅ ̅ = (1 − 𝜇) 𝑠𝑖𝑛𝑣

and represent circular motion (with non-constant angular

momentum (Broucke, 1969).When r=0 or e=1 this

transformation cannot be used.

A. Equations of Motion in the Inertial Barycentric

system

As mentioned before, in this work most calculations were

made by using a “rotating” system of coordinates. However,

before the equations of motion of the satellite relative to this

system are presented, it is important to present the equations of

motion in the inertial barycentric system, as the transition to

the rotating system will be much easier. Theoretically, the

results of this study can be obtained by solely using the inertial

system, although the calculations are much less convenient.

In the inertial barycentric frame of reference, the equations

of motion of the satellite are derived from the Lagrangian

function

𝐿 =1

2(𝜉′2 + 𝜂′2) +

1 − 𝜇

𝑠1+𝜇

𝑠2. (15)

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The distances between the satellite and the primaries 𝑚1 and

𝑚2 respectively are

𝑠12 = (𝜉 − 𝜉1)

2 + (𝜂 − 𝜂1)2

(16)

𝑠22 = (𝜉 − 𝜉2)

2 + (𝜂 − 𝜂2)2

If 𝑞𝑗 and 𝑞�̇� denote the generalized positions and generalized

velocities respectively such that

𝑞1 = 𝜉, 𝑞2 = 𝜂

(17)

𝑞1̇ = 𝜉′, 𝑞2̇ = 𝜂′

then by using the equation

𝑑

𝑑𝑡(𝜕𝐿

𝜕𝑞�̇�) −

𝜕𝐿

𝜕𝑞𝑗= 0 (18)

(Hadjidemetriou, 2000)

the equations of motion are thus

𝜉′′ = −(1 − 𝜇)(𝜉 − 𝜉1)

𝑠13 −

𝜇(𝜉 − 𝜉2)

𝑠23

(19)

𝜂′′ = −(1 − 𝜇)(𝜂 − 𝜂1)

𝑠13 −

𝜇(𝜂 − 𝜂2)

𝑠23 .

Lagrangian eq. (15) is non-conservative because it explicitly

contains the independent variable t through 𝑠1 and 𝑠2.

According to Broucke (1969) using the true anomaly v as

the independent variable, the Lagrangian equation would

change to

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𝐿 =𝑟2

𝑝12

𝐿 =𝑝12

2𝑟2(𝜉2̇ + 𝜂2̇) +

𝑟2

𝑝12

(1 − 𝜇

𝑠1+𝜇

𝑠2). (20)

If the “reduced” or “pulsating” coordinates (𝜉,̅ �̅�) are used,

according to eq. (13), and

𝜉′ = 𝑟′𝜉̅ + 𝑟𝜉 ′̅, 𝜉̇ = �̇�𝜉̅ + 𝑟𝜉̅̇

(21)

𝜂′ = 𝑟′�̅� + 𝑟𝜂′̅, 𝜉̇ = �̇��̅� + 𝑟�̇̅�

the Lagrangian equations in eqs. (15) and (20) can be

transformed accordingly to

𝐿 =𝑟2

2(𝜉 ′̅

2+ 𝜂′̅

2) + 𝑟𝑟′(𝜉�̅� ′̅ + �̅��̅�′) +

𝑟′2

2(𝜉2̅̅ ̅ + 𝜂2̅̅ ̅)

+1

𝑟′(1 − 𝜇

𝑟1+𝜇

𝑟2) (22)

𝐿 =𝑝12

2(𝜉̅2̇ + �̅�2̇) +

𝑝12�̇�

𝑟(𝜉̅𝜉̅̇ + �̅��̇̅�) +

𝑝12

2

�̇�2

𝑟2(𝜉2̅̅ ̅ + 𝜂2̅̅ ̅)

+𝑟

𝑝12

(1 − 𝜇

𝑟1+𝜇

𝑟2) (23)

where

𝑟12 = (𝜉̅ − 𝜉1̅)

2+ (�̅� − 𝜂1̅̅̅)

2 =𝑠12

𝑟2

𝑟22 = (𝜉̅ − 𝜉2̅)

2+ (�̅� − 𝜂2̅̅ ̅)

2 =𝑠22

𝑟2. (24)

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Dividing eq. (23) by 𝑝1

2 and substituting �̇� by its value from eq.

(9), the Lagrangian transforms into

𝐿 =1

2(𝜉̅2̇ + �̅�2̇) +

𝑒 𝑟 𝑠𝑖𝑛𝑣

𝑝(𝜉�̅�̅̇ + �̅��̇̅�)

+𝑒2 𝑟2 sin2 𝑣

2𝑝2(𝜉̅2 + �̅�2) +

𝑟

𝑝(𝑚1𝑟1+𝑚2𝑟2). (25)

Eq. (25) can be replaced by a simpler Lagrangian by

subtracting the following exact differential, which by taking

into account eq. (2) and eq. (9) is

𝑑

𝑑𝑣(𝑒 𝑟 𝑠𝑖𝑛𝑣

2𝑝(𝜉2̅̅ ̅ + 𝜂2̅̅ ̅))

=𝑒 𝑟 𝑠𝑖𝑛𝑣

𝑝(𝜉̅𝜉̅̇ + �̅��̇̅�) +

𝑒2 𝑟2 sin2 𝑣

2𝑝2(𝜉̅2 + �̅�2)

+1

2(1 −

𝑟

𝑝) (𝜉̅2 + �̅�2), (26)

from eq. (25). The above exact differential can be safely

omitted from the Lagrangian without change in the equations

of motion. Thus, the simplified Lagrangian is

𝐿 =1

2(𝜉2̅̅ ̅̇ + 𝜂2̅̅ ̅̇) +

1

2(𝑟

𝑝− 1) (𝜉2̅̅ ̅ + 𝜂2̅̅ ̅)

+𝑟

𝑝(𝑚1𝑟1+𝑚2𝑟2). (27)

The equations of motion in the “inertial barycentric pulsating”

frame of reference, with the true anomaly v as independent

variable, are thus

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𝜉̅̈ = (𝑟

𝑝− 1) 𝜉̅ −

𝑟

𝑝(𝑚1(𝜉̅ − 𝜉1̅)

𝑟13 +

𝑚2(𝜉̅ − 𝜉2̅)

𝑟23 )

(28)

�̈̅� = (𝑟

𝑝− 1) �̅� −

𝑟

𝑝(𝑚1(�̅� − 𝜂1̅̅̅)

𝑟13 +

𝑚2(�̅� − 𝜂2̅̅ ̅)

𝑟23 )

The forces which are present in the last equations of

motion, eq. (28), contain three terms: an apparent radial force

that comes only from the radial scale change of the coordinate

system, and 𝑚1 and 𝑚2, which are, of course, the Newtonian

attraction potential from the two primaries.

B. Equations of motion in the Rotating system

Now, the equations of motion in a rotating barycentric

system are presented. The angle of rotation is the true anomaly

v. The angular velocity of the axes are time dependent, unless

the eccentricity e equals zero (Broucke, 1969). The equations

of motion will be written both with time t and the true anomaly

v as independent variable, in ordinary and in pulsating

coordinates. The equations of motion written in rotating

pulsating coordinates and with the true anomaly v as the

independent variable provide certain advantages, which will be

discussed later, and this is the reason they were chosen for the

calculations.

The ordinary rotating coordinates (�̅�, �̅�) are related to the

inertial coordinates (𝜉, 𝜂) by

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𝜉 = �̅� 𝑐𝑜𝑠𝑣 − �̅� 𝑠𝑖𝑛𝑣

(29)

𝜂 = �̅� 𝑠𝑖𝑛𝑣 + �̅� 𝑐𝑜𝑠𝑣

The rotating coordinates simplify things by keeping the two

primaries 𝑚1 and 𝑚2 on the x-axis permanently. Their

coordinates in this system are

𝑥1̅̅̅ = −𝜇 𝑟, 𝑦1̅̅ ̅ = 0

(30)

𝑥2̅̅ ̅ = (1 − 𝜇) 𝑟, 𝑦2̅̅ ̅ = 0

These coordinates are not constant; the two primaries are

oscillating on the x-axis (Broucke, 1969). Lagrangian eq. (15)

combined with eqs. (29) transforms into

𝐿 =1

2(𝑥′2̅̅ ̅̅ + 𝑦′2̅̅ ̅̅ ) + (�̅�𝑦 ′̅ + �̅�𝑥 ′̅)𝑣′ +

1

2(𝑥2 + 𝑦2)𝑣′2

+ (𝑚1𝑠1+𝑚2𝑠2) (31)

and the equations of motion derived from this Lagrangian are

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𝑥′′̅̅̅̅ − 2𝑦 ′̅𝑣′ − �̅�𝑣′′ − �̅�𝑣′2

= −(1 − 𝜇)(�̅� − 𝑥1̅̅̅)

𝑠13 −

𝜇(�̅� − 𝑥2̅̅ ̅)

𝑠23 (32𝑎)

𝑦′′̅̅̅̅ − 2𝑥 ′̅𝑣′ − �̅�𝑣′′ − �̅�𝑣′2 = −(1 − 𝜇)�̅�

𝑠13 −

𝜇�̅�

𝑠23 . (32𝑏)

The above equations will be transformed to the desired

form by writing them using rotating-pulsating coordinates

(x,y). These coordinates are defined in the same way as eq.

(13):

�̅� = 𝑟𝑥, �̅� = 𝑟𝑦 (33)

Lagrangian eq. (31) transforms in the new form

𝐿 =𝑟2

2(𝑥′2 + 𝑦′2) + 𝑟𝑟′(𝑥𝑥′ + 𝑦𝑦′) + 𝑝

12(𝑥𝑦′ − 𝑦𝑥′)

+1

2(𝑟′2 +

𝑝

𝑟2) (𝑥2 + 𝑦2) +

1

𝑟(𝑚1𝑟1+𝑚2𝑟2) (34)

and the corresponding equations of motion are written as

𝑟2𝑥′′ − 2𝑦′𝑝12 + 2𝑟𝑟′𝑥′ −

1

𝑟𝑥

= −1

𝑟(𝑚1(𝑥 − 𝑥1)

𝑟13 +

𝑚2(𝑥 − 𝑥2)

𝑟23 ) (35𝑎)

𝑟2𝑦′′ − 2𝑥′𝑝12 + 2𝑟𝑟′𝑦′ −

1

𝑟𝑦 = −

1

𝑟(𝑚1𝑦

𝑟13 +

𝑚2𝑦

𝑟23 ) (36𝑏)

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The above equations of motion are written with time t as

the independent variable, but they can be simplified if the true

anomaly v is used as the independent variable instead. This

change in variable is done by using the expression of 𝑣′ of eq.

(7), and in the same way it was used to obtain the Lagrangian

eq. (20). By subtracting the exact derivative, as in eqs. (26),

(27),

𝑑

𝑑𝑣(1

2

�̇�

𝑟(𝑥2 + 𝑦2)) (37)

and dividing the Lagrangian equation by 𝑝1

2, then eq. (34)

finally transforms into

𝐿 =1

2(�̇�2 + �̇�2) + (𝑥�̇� − 𝑦�̇�)

+𝑟

𝑝[1

2(𝑥2 + 𝑦2) +

𝑚1𝑟1+𝑚2𝑟2] (35).

The corresponding equations of motion are thus

�̈� − 2�̇� =𝑟

𝑝(𝑥 −

𝑚1(𝑥 − 𝑥1)

𝑟13 −

𝑚2(𝑥 − 𝑥2)

𝑟23 )

(36)

�̈� + 2�̇� =𝑟

𝑝(𝑦 −

𝑚1𝑦

𝑟13 −

𝑚2𝑦

𝑟23 ).

19

The Lagrangian eq. (35) and its equations of motion (36),

present the unique advantage that they differ only by a factor

of 𝑟

𝑝=

1

1+𝑒 𝑐𝑜𝑠𝑣 from the Lagrangian and classical equations of

motion of the circular restricted three body problem. They

have also been used in the study of orbits near the equilibrium

points in this work.

It is important to note that since the numerical integration

of eqs. (36) was achieved using the Runge-Kutta 4 method, the

system of equations (36) was altered slightly in such a way that

instead of containing two second order differential equations, it

contained four first order differential equation. This was easily

done using the following substitutions:

𝑢𝑥 = �̇�, 𝑢�̇� = �̈�

(37)

𝑢𝑦 = �̇�, 𝑢�̇� = �̈�

The system (36) using eqs. (37) transforms into

�̇� = 𝑢𝑥

𝑢�̇� − 2𝑢𝑦 =𝑟

𝑝(𝑥 −

𝑚1(𝑥 − 𝑥1)

𝑟13 −

𝑚2(𝑥 − 𝑥2)

𝑟23 )

(38)

�̇� = 𝑢𝑦

𝑢�̇� + 2𝑢𝑥 =𝑟

𝑝(𝑦 −

𝑚1𝑦

𝑟13 −

𝑚2𝑦

𝑟23 )

20

where 𝑢𝑥 , 𝑢𝑦 are, of course, the satellites velocity components.

The systems (36) and (38) of differential equations are

equivalent.

C. The five Lagrange points of Equilibrium

The five Lagrange points (equilibrium points) are present

both in the circular and in the elliptic three body problem. At

these points, the satellite remains in the same position relative

to the primaries if no other forces are applied to it (i.e. the

force field at these points is zero). The study of the five

Lagrange points, their stability as well as their neighborhoods

was made mainly by using the rotating-pulsating coordinates

(x, y), but the inertial coordinates (ξ, η) can also be used.

In the rotating-pulsating frame of reference, the Lagrange

points are fixed (Broucke, 1969). From the equations of

motion (36), it is clear that there are five particular solutions

with constant coordinates and with �̈� = �̈� = �̇� = �̇� = 0. These

constant coordinates are solutions to the equations of motion

by setting their right side equal to zero. From eqs. (36) we

obtain

𝑥 −𝑚1(𝑥 − 𝑥1)

𝑟13 −

𝑚2(𝑥 − 𝑥2)

𝑟23

= 𝑥 (1 −1 − 𝜇

𝑟13 −

𝜇

𝑟23) + 𝜇(1 − 𝜇) (

1

𝑟23 −

1

𝑟13)

= 0 (39𝑎)

𝑦 −𝑚1𝑦

𝑟13 −

𝑚2𝑦

𝑟23 = 𝑦 (1 −

1 − 𝜇

𝑟13 −

𝜇

𝑟23) = 0 (39𝑏)

21

In the above equations 𝑚1, 𝑚2 were replaced by

𝑚1 = 1 − 𝜇, 𝑚2 = 𝜇 (40)

and 𝑥1, 𝑥2 were replaced by their values in the rotating-

pulsating system:

𝑥1 = −𝜇, 𝑥2 = 1 − 𝜇. (41)

The case where the eccentricity 𝑒 = 1 has been excluded

because in the rotating-pulsating system of coordinates r must

be always different from zero. The eqs. (39) are the same

equations one would arrive at in the study of the classical

circular three-body problem (Broucke, 1969). The first two

equilibrium solutions are easily found from eqs. (39) if 𝑟1 =

𝑟2 = 1. Since the distances between the primaries and the

satellite are

𝑟12 = (𝑥 − 𝑥1)

2 + 𝑦2

(42)

𝑟22 = (𝑥 − 𝑥2)

2 + 𝑦2

solving the system (42) for 𝑟1 = 𝑟2 = 1 provides the

coordinates of the 𝐿4 and 𝐿5 Lagrange points which

correspond to the equilateral triangle configurations with the

two primaries. They are thus:

22

𝐿4: 𝑥4 =1

2(1 − 2𝜇), 𝑦4 = +

312

2

(43)

𝐿5: 𝑥5 =1

2(1 − 2𝜇), 𝑦5 = −

312

2

The other three solutions of eq. (39) are 𝐿1, 𝐿2 and 𝐿3.

These are called collinear equilibrium points because they lie

on the syzygy-axis (the line of the primaries) where y=0

(Broucke, 1969).

Their abscissa x therefore is root of equation

𝑓(𝑥) ≡ −𝑥 + (1 − 𝜇)(𝑥 − 𝑥1)

𝑟13 + 𝜇

(𝑥 − 𝑥2)

𝑟23 (44)

where

𝑟13 = |𝑥 − 𝑥1|

3

(45)

𝑟23 = |𝑥 − 𝑥2|

3

Equation (44) has one root in each of the three intervals on

both sides of 𝑚1 and 𝑚2 and between them. This is easily

verifiable if one looks at the derivative of f(x):

23

𝑑𝑓(𝑥)

𝑑𝑥= −1 −

2(1 − 𝜇)

|𝑥 − 𝑥1|3−

2𝜇

|𝑥 − 𝑥2|3< 0. (46)

Since the derivative is negative in each interval, f(x) is

monotonously decreasing from +∞ to −∞ in each interval

thus crossing the x-axis. In order to obtain the abscissae of the

𝐿1, 𝐿2, 𝐿3 Lagrange points, eq. (44) was numerically solved

using the Newton-Raphson method. The following figure show

the change in 𝑥𝐿1, 𝑥𝐿2, and 𝑥𝐿3 with respect to the mass ratio μ.

Figure 1: Change in 𝒙𝑳𝟏, 𝒙𝑳𝟐, 𝐚𝐧𝐝 𝒙𝑳𝟑 with respect to the

mass ratio μ. The mass ratio is between 𝟎. 𝟎𝟎𝟎𝟏 < 𝝁 < 𝟎. 𝟓

and the eccentricity is e=0.2.

As aforementioned, the purpose of this work is to study

the neighborhood of the Lagrange points of equilibrium, in

order to determine the stability of these points as well as the

stability of their neighborhoods. To achieve this, the equations

of motion, eqs. (36), must be linearized in the neighborhood of

0.2 0.4 0.6 0.8xL1

0.1

0.2

0.3

0.4

0.5

mu

1.05 1.10 1.15 1.20 1.25xL2

0.1

0.2

0.3

0.4

0.5

mu

1.15 1.10 1.05 1.00xL3

0.1

0.2

0.3

0.4

0.5

mu

24

the Lagrange points in order to obtain the so-called first order

variational equations (Broucke, 1969). Eqs. (36) are written in

the form

�̈� − 2�̇� −𝑟

𝑝𝑥 =

𝑟

𝑝𝑈𝑥

(47)

�̈� + 2�̇� −𝑟

𝑝𝑦 =

𝑟

𝑝𝑈𝑦

where U is the potential function

𝑈 =1 − 𝜇

𝑟1+𝜇

𝑟2. (48)

The subscripts x and y in U are used to represent the partial

derivatives of U.

The variational equations derived from eqs. (47) are

written as

𝛿�̈� − 2𝛿�̇� −𝑟

𝑝𝛿𝑥 =

𝑟

𝑝(𝑈𝑥𝑥𝛿𝑥 + 𝑈𝑥𝑦𝛿𝑦)

(49)

𝛿�̈� + 2𝛿�̇� −𝑟

𝑝𝛿𝑦 =

𝑟

𝑝(𝑈𝑦𝑥𝛿𝑥 + 𝑈𝑦𝑦𝛿𝑦)

25

Again, as with the equations of motion (36), the variational

equations (49) were numerically integrated using the Runge-

Kutta of the 4th order method. For this reason, the system (49)

was transformed to include four first-order differential

equations, in a similar way as eqs. (38). This transformed

system is written as

𝛿�̇� = 𝛿𝑢𝑥

𝛿𝑢�̇� − 2𝛿𝑢𝑦 −𝑟

𝑝𝛿𝑥 =

𝑟

𝑝(𝑈𝑥𝑥𝛿𝑥 + 𝑈𝑥𝑦𝛿𝑦)

(50)

𝛿�̇� = 𝛿𝑢𝑦

𝛿𝑢�̇� + 2𝛿𝑢𝑥 −𝑟

𝑝𝛿𝑦 =

𝑟

𝑝(𝑈𝑦𝑥𝛿𝑥 + 𝑈𝑦𝑦𝛿𝑦)

Equations (49) are a system of linear differential equations

with nonconstant periodic coefficients, due to factor r, which

depends on the cosine of the true anomaly v. In the circular

three-body problem, the respective system would have

constant coefficients since 𝑟 = 1. This principal difference

between the circular and the elliptic problem, makes the study

of the latter a more difficult proposition, thus a more complex

method is required in order to study the variational equations

in the elliptic problem. The Floquet theory was used, in order

to determine the stability of the Lagrange points. Omitting the

details of the theory, a 4x4 monodromy matrix D is obtained

by integrating the variational equations for a single period of

the true anomaly v (0 < 𝑣 < 2𝜋) using the following initial

conditions for the variational equations:

𝛿𝑥(0) = 1, 𝛿𝑦(0) = 0, 𝛿𝑢𝑥(0) = 0, 𝛿𝑢𝑦(0) = 0 (51)

26

At 𝑣 = 2𝜋 the integration of the variational equations yields

𝛿𝑥(2𝜋) = 𝛿𝑥1, 𝛿𝑦(2𝜋) = 𝛿𝑦1,

(52)

𝛿𝑢𝑥(2𝜋) = 𝛿𝑢𝑥1, 𝛿𝑢𝑦(2𝜋) = 𝛿𝑢𝑦1

The integration of system (50) was repeated three more times,

each time changing the place of unity in the initial conditions

(51). For instance, the second integration would use

𝛿𝑥(0) = 0, 𝛿𝑦(0) = 1, 𝛿𝑢𝑥(0) = 0, 𝛿𝑢𝑦(0) = 0 (53)

in order to obtain

𝛿𝑥(2𝜋) = 𝛿𝑥2, 𝛿𝑦(2𝜋) = 𝛿𝑦2,

(54)

𝛿𝑢𝑥(2𝜋) = 𝛿𝑢𝑥2, 𝛿𝑢𝑦(2𝜋) = 𝛿𝑢𝑦2

and so forth. The monodromy matrix D is thus constructed in

the following way:

𝐷 = (

𝛿𝑥1 𝛿𝑥2𝛿𝑦1 𝛿𝑦2

𝛿𝑥3 𝛿𝑥4𝛿𝑦3 .

𝛿𝑢𝑥1 𝛿𝑢𝑥2𝛿𝑢𝑦1 𝛿𝑢𝑦2

. .

. .

) (55)

27

The stability of any point is then determined by numerically

calculating the eigenvalues and their magnitudes of the

monodromy matrix D using the widely used QR algorithm.

The QR algorithm has its basis on the QR decomposition,

in which a matrix A is decomposed to a product A=QR of an

orthogonal matrix Q and an upper triangular matrix R.

Formally, let A be a real matrix of which we want to compute

the eigenvalues, and let A0:=A. At the k-th step (starting with k

= 0), we compute the QR decomposition 𝐴𝑘 = 𝑄𝑘𝑅𝑘 where 𝑄𝑘

is an orthogonal matrix (i.e., QT = Q−1) and 𝑅𝑘 is an upper

triangular matrix. We then form 𝐴𝑘+1 = 𝑅𝑘𝑄𝑘 . Note that

𝐴𝑘+1 = 𝑅𝑘𝑄𝑘 = 𝑄𝑘−1𝑄𝑘𝑅𝑘𝑄𝑘 = 𝑄𝑘

−1𝐴𝑘𝑄𝑘 = 𝑄𝑘𝑇𝐴𝑘𝑄𝑘 so all

the 𝐴𝑘 are similar and hence they have the same eigenvalues.

The algorithm is numerically stable because it proceeds by

orthogonal similarity transforms. Under certain conditions, the

matrices 𝐴𝑘converge to a triangular matrix. The eigenvalues of

a triangular matrix are listed on the diagonal, and the

eigenvalue problem is solved.

All the above calculations were made for many different

values for the mass ratio μ and the eccentricity e. The results

and the kinds of stability of the Lagrange points are discussed

in Section 3.

D. The fast Lyapunov indicator (FLI)

The regions near the Lagrange points of equilibrium are of

great interest and part of this work focuses on finding the parts

of these regions which produce regular satellite orbits. This is

again accomplished by integrating the equations of motion

(38) and the variational equations (50) together for initial

conditions that correspond to the neighboring region of the

Lagrange points, this time calculating for every single orbit the

so-called fast Lyapunov indicator or FLI.

28

The FLI is a great tool which allows the identification of

regular and chaotic orbits. The FLI accomplishes this by

measuring the mean exponential distancing of a neighboring

orbit to a control orbit (Voyatzis & Meletlidou, 2015). There

are many formulas that define FLI, but the following was used:

𝐹𝐿𝐼(𝑣)

= log

(

√𝛿𝑥(𝑣)2 + 𝛿𝑦(𝑣)2 + 𝛿𝑢𝑥(𝑣) + 𝛿𝑢𝑦(𝑣)

𝑣

)

(56)

For a regular orbit of the system, the FLI increases in value at

a slow rate as 𝑣 → ∞. On the other hand, for a chaotic orbit

the FLI increases at a much faster rate. Therefore, after a

comparatively small interval of the independent variable v, the

FLI either has a small value, indicating a regular orbit or it has

a large value which indicates a chaotic orbit (Voyatzis &

Meletlidou, 2015).

3. The Stability of The Lagrange points of Equilibrium

and their Neighborhoods

A. Stability of the Lagrange points

The stability of the Lagrange points will be discussed. As

shown in Section 2C, the type of stability of any point can be

calculated by constructing a monodromy matrix D by

29

integrating the variational equations (50) as indicated by the

Floquet theory. The stability then depends upon the

corresponding eigenvalues of the monodromy matrix D,

𝜆1, 𝜆2, 𝜆3, 𝜆4.

Of particular interest are, of course, the Lagrange points of

equilibrium and for which combinations of the mass ratio μ

and the eccentricity e they present linear stability. It is well

known that the collinear Lagrange points 𝐿1, 𝐿2, 𝐿3 found in

the circular three body problem, show instability for any

combination of the values of μ and e (Szebehely, 1967). This

fact persists in the elliptic problem and as a result, only the

triangular Lagrange points 𝐿4, 𝐿5 were tested for linear

stability.

The four eigenvalues, which can be real or complex

numbers, of the 4x4 monodromy matrix D are found not to be

independent of one another. It is proved that the eigenvalues

form reciprocal pairs. If e.g. 𝜆1 ∈ ℝ then 𝜆2 = 1/𝜆1 is also an

eigenvalue. Also if 𝜆1 = 𝑎 + 𝑖𝑏 ∈ ℂ then 𝜆2 = 𝑎 − 𝑖𝑏 is also

an eigenvalue. Depending on the place of the four eigenvalues

on the unit circle, there exist seven stability regions. The six

are unstable regions and one is stable. The stable region

corresponds to the eigenvalues that are on the unit circle.

The seven stability regions and the properties of their

eigenvalues are described in Table 1.

30

Table 1: Properties of the seven stability regions

Region Properties of

eigenvalues

𝝀,𝟏

𝝀, 𝝁,

𝟏

𝝁

Remarks Properties of

Orbits

1 �̅� = 1/𝜆,�̅� = 1/𝜇 (λ and μ

complex with

|𝜆| = |𝜇| =1)

All four on

unit circle

Stability

2 �̅� = 𝜇,�̅� = 𝜆

(λ and μ

complex)

Not on unit

circle

Complex

instability

3 λ real, μ real

𝜆𝜇 < 0

Two positive

and two

negative

Even-odd

instability

4 λ real >0

μ real >0

Four real

positive

Even-even

instability

5 λ real <0

μ real <0

Four real

negative

Odd-odd

instability

6 λ real >0

μ complex

�̅� = 1/𝜇

Two real

positive and

two complex

on unit circle

Even-semi-

instability

7 λ complex

�̅� = 1/𝜆

μ real <0

Two real

negative and

two complex

on unit circle

Odd-semi-

instability

(Broucke, 1969).

31

Figure 2: Eigenvalue configuration for stability regions

λ

1/λ

μ

1/μ

1.0 0.5 0.5 1.0x

1.0

0.5

0.5

1.0

y

Region 1

λ

μ

1/μ

1/λ

1.0 0.5 0.5 1.0 1.5x

1.5

1.0

0.5

0.5

1.0

1.5

y

Region 2

λ1/λμ1/μ

1.5 1.0 0.5 0.5 1.0 1.5x

1.0

0.5

0.5

1.0

y

Region 3

1/μ 1/λ λ μ

1.0 0.5 0.5 1.0 1.5x

1.0

0.5

0.5

1.0

y

Region 4

1/μ 1/λ λ μ

1.5 1.0 0.5 0.5 1.0x

1.0

0.5

0.5

1.0

y

Region 5

μ

1/μ

λ 1/λ

1.0 0.5 0.5 1.0x

1.0

0.5

0.5

1.0

y

Region 6

32

The results of the analysis concerning the linearized

stability of the triangular Lagrange points 𝐿4, 𝐿5 for a wide

range of values of the mass ratio μ and the eccentricity e, are

presented in Figure 2

Figure 3: Linear stability of the triangular Lagrange points in

the elliptic restricted three-body problem

The shaded areas in figure 2 represent linear stability. “The

point denoted by 𝜇∗ on the μ axis corresponds to the value of

the mass ratio at which any nonzero eccentricity introduces

λ

1/λ

1/μ μ

1.0 0.5 0.5 1.0x

1.0

0.5

0.5

1.0

y

Region 7

33

linear instability” (Szebehely, 1967, p. 599). The point denoted

by 𝜇0 represents the value at which instability presents itself

when 𝑒 = 0.

The numerical calculation of the points (μ, e) that present

linear stability, also provide the values of the points 𝜇∗ and 𝜇0.

These points are found to be

𝜇∗ = 0.0286, 𝜇0 = 0.0385.

As figure 3 shows, the eccentricity of the motion of the two

primaries may introduce stability for 𝜇 > 𝜇0. In fact, the orbits

around the triangular Lagrange points are stable up to

𝜇 = 0.04698 with the proper value of e (Szebehely, 1967).

B. Regular orbits in the neighborhood of the Lagrange

points

In this section, the results of the calculations of the regular

orbits around the Lagrange points of equilibrium are presented.

These results consist of a number of regions of initial

conditions around the equilibrium points for which the satellite

follows a regular orbit. The calculations were made for

different combinations of the mass ratio μ and the eccentricity

e. The range of the regular region is also calculated as the

angle θ of the region measured from the axes point of origin.

Figure 4 shows exactly how the range of the regular region θ is

calculated.

34

Figure 4: Range of the regular region θ. The shaded

area is the regular region.

As mentioned in Section 2D, the fast Lyapunov indicator

or FLI was used to distinguish between regular and chaotic

orbits. The initial conditions of the satellite that have been used

are 𝒙�̇� = 𝟎, 𝒚�̇� = 𝟎, which means that the satellite starts

motionless, and for 𝒙𝟎, 𝒚𝟎 a grid has been considered with its

center at the coordinates of the Lagrange point 𝐿4 (𝑥𝐿4, 𝑦𝐿4)or

𝐿5 (𝑥𝐿5, 𝑦𝐿5). Thus, the initial conditions 𝑥0 and 𝑦0 are

between 𝒙𝑳𝒊 − 𝟎. 𝟗 < 𝒙𝟎 < 𝒙𝑳𝒊 + 𝟎. 𝟗 and

𝒚𝑳𝒊 − 𝟎. 𝟒 < 𝒚𝟎 < 𝒚𝑳𝒊 + 𝟎. 𝟒. It is also important to note that

all the calculations were made whit the two primaries at

periapsis (minimum elongation). The following figure shows

the FLI as a function of the independent variable v for a

regular and a chaotic orbit.

35

Figure 5: The change in FLI with respect to the true anomaly v

for a regular and a chaotic orbit. The orbits have been calculated

for μ=0.001 and e=0.05 for 325 orbital periods (𝒗𝒎𝒂𝒙 = 𝟔𝟓𝟎𝝅).

Initial conditions of the regular orbit are 𝒙𝟎 = −𝟎. 𝟐𝟖𝟔, 𝒚𝟎 =

𝟎. 𝟗𝟓𝟖𝟓𝟐𝟓 and for the chaotic orbit 𝒙𝟎 = 𝟎. 𝟓, 𝒚𝟎 = 𝟏. 𝟎𝟒𝟑𝟔𝟕𝟖. The

coordinates of 𝑳𝟒 are (𝒙𝑳𝟒 = 𝟎. 𝟒𝟗𝟗, 𝒚𝑳𝟒 = 𝟎. 𝟖𝟔𝟔𝟎𝟐𝟓).

From figure 5 it can clearly be seen that the value of FLI for a

regular orbit stays relatively low whereas it quickly increases

for a chaotic orbit. The following two figures show the

corresponding orbits of the satellite which are taken for a

smaller interval of the independent variable v.

36

Figure 6: Regular orbit for μ=0.001 and e=0.05 for 15 orbital

periods (𝒗𝒎𝒂𝒙 = 𝟑𝟎𝝅). Initial conditions of the satellite are 𝒙𝟎 =

−𝟎. 𝟐𝟖𝟔, 𝒚𝟎 = 𝟎. 𝟗𝟓𝟖𝟓𝟐𝟓. The coordinates of 𝑳𝟒 are (𝒙𝑳𝟒 =

𝟎. 𝟒𝟗𝟗, 𝒚𝑳𝟒 = 𝟎. 𝟖𝟔𝟔𝟎𝟐𝟓).

Figure 7: Chaotic orbit for μ=0.001 and e=0.05 for 15 orbital

periods (𝒗𝒎𝒂𝒙 = 𝟑𝟎𝝅). Initial conditions of the satellite are

𝒙𝟎𝟎. 𝟓, 𝒚𝟎 = 𝟏. 𝟎𝟒𝟑𝟔𝟕𝟖. The coordinates of 𝑳𝟒 are (𝒙𝑳𝟒 =

𝟎. 𝟒𝟗𝟗, 𝒚𝑳𝟒 = 𝟎. 𝟖𝟔𝟔𝟎𝟐𝟓). Even though the initial conditions are

closer to 𝑳𝟒 in comparison to the regular orbit, a chaotic orbit is

produced.

37

However, apart from the FLI, as an added measure of

accuracy, the actual distance between the satellite and the

Lagrange point has also been taken into account. This is

calculated by

𝑑𝐿 = √(𝑥 − 𝑥𝐿)2 + (𝑦 − 𝑦𝐿)

2 (57)

where 𝑥𝐿 and 𝑦𝐿 denote the coordinates of the Lagrange point

whose neighborhood is tested. If this distance increases beyond

a certain threshold after a certain interval of the independent

variable v, then the orbit clearly veers into infinity and thus the

initial conditions which produced it are excluded.

In addition, orbits which lead to collisions of the satellite

with the two primaries are also excluded. The distance

between the satellite and the primaries is calculated by

𝑑𝑝𝑟1 = √(𝑥 − 𝑥1)2 + (𝑦 − 𝑦1)

2

(58)

𝑑𝑝𝑟2 = √(𝑥 − 𝑥2)2 + (𝑦 − 𝑦2)

2

If these distances become zero, a collision has occurred and the

corresponding initial conditions are excluded.

With all this information the so called stability maps have

been constructed around the Lagrange points. These maps

consist of the initial condition 𝑥0, 𝑦0 of the satellite, for which

it follows a regular orbit. In each such pair of initial conditions

38

a color has been added according to the value of FLI that has

been calculated for this particular orbit. For the chaotic orbits

and the collision orbits a special arbitrary value of 100 is

assigned to the FLI for the purposes of visualization. The color

ranges from black to yellow with yellow being the region that

does not produce regular orbits. It was found that the boundary

of the value of the FLI that separates the regular orbits region

from the chaotic orbits region is FLI=6.0. The implication here

is that initial conditions that produce regular orbits with FLI

value close to 6.0 (represented by orange color), could be

excluded from the regular region if the integrations were to be

made for larger intervals of the independent variable v. For this

reason the range of the regular regions have been calculated

using initial conditions which produce orbits with a value of

FLI close to 1.50 (represented by blue color). The results that

follow were obtained for a maximum value of v equal to 650π

or 325 orbital periods of the primaries. The interval of μ tested

is 0.001 < 𝜇 < 0.04 with step equal to 0.0025 starting from

0.0025. The values of e that have been tested start at 0.0 with

step equal to 0.05 and end whenever the range of the regular

regions θ reach 0.

39

i. Regular regions near equilibrium point 𝑳𝟒

μ=0.001, e=0.0, θ=1.45 rad

μ=0.001, e=0.05, θ=1.30 rad

40

μ=0.001, e=0.1, θ=1.07rad

μ=0.001, e=0.15, θ=0.96 rad

41

μ=0.001, e=0.2, θ=0.86 rad

μ=0.001, e=0.25, θ=0.80 rad

42

μ=0.001, e=0.3, θ=0.74 rad

μ=0.001, e=0.35, θ=0.65 rad

43

μ=0.001, e=0.4, θ=0.61 rad

μ=0.001, e=0.45, θ=0.41 rad

44

μ=0.001, e=0.5, θ=0.38 rad

μ=0.001, e=0.55, θ=0.18 rad

45

μ=0.001, e=0.6, θ=0.12 rad

μ=0.001, e=0.65, θ=0.08 rad

After e=0.65 there are no initial conditions that produce

regular orbits.

46

μ=0.0025, e=0.0, θ=1.43 rad

μ=0.0025, e=0.05, θ=1.29 rad

47

μ=0.0025, e=0.1, θ=1.04 rad

μ=0.0025, e=0.15, θ=0.99 rad

48

μ=0.0025, e=0.2, θ=0.75 rad

μ=0.0025, e=0.25, θ=0.72 rad

49

μ=0.0025, e=0.3, θ=0.58 rad

μ=0.0025, e=0.35, θ=0.41 rad

50

μ=0.0025, e=0.4, θ=0.38 rad

μ=0.0025, e=0.45, θ=0.21 rad

51

μ=0.0025, e=0.5, θ=0.20 rad

μ=0.0025, e=0.55, θ=0.04 rad


regular orbits.

52

μ=0.005, e=0.0, θ=1.26 rad

μ=0.005, e=0.05, θ=1.15 rad

53

μ=0.005, e=0.1, θ=0.96 rad

μ=0.005, e=0.15, θ=0.50 rad

54

μ=0.005, e=0.2, θ=0.41 rad

μ=0.005, e=0.25, θ=0.46 rad

55

μ=0.005, e=0.3, θ=0.32 rad

μ=0.005, e=0.35, θ=0.18 rad

56

μ=0.005, e=0.4, θ=0.17 rad

μ=0.005, e=0.45, θ=0.08 rad

57

μ=0.005, e=0.5, θ=0.0 rad

After e=0.5 there are no initial conditions that produce regular

orbits.

μ=0.0075, e=0.0, θ=1.24 rad

58

μ=0.0075, e=0.05, θ=0.99 rad

μ=0.0075, e=0.1, θ=0.56 rad

59

μ=0.0075, e=0.15, θ=0.28 rad

μ=0.0075, e=0.2, θ=0.19 rad

60

μ=0.0075, e=0.25, θ=0.27 rad

μ=0.0075, e=0.3, θ=0.29 rad

61

μ=0.0075, e=0.35, θ=0.03 rad


regular orbits.

μ=0.01, e=0.0, θ=0.88 rad

62

μ=0.01, e=0.05, θ=0.77 rad

μ=0.01, e=0.1, θ=0.62 rad

63

μ=0.01, e=0.15, θ=0.53 rad

μ=0.01, e=0.2, θ=0.25 rad

64

μ=0.01, e=0.25, θ=0.06 rad

μ=0.01, e=0.3, θ=0.06 rad

65

μ=0.01, e=0.35, θ=0.14 rad


regular orbits.

μ=0.0125, e=0.0, θ=0.65 rad

66

μ=0.0125, e=0.05, θ=0.31 rad

μ=0.0125, e=0.05, θ=0.16 rad

67

μ=0.0125, e=0.1, θ=0.16 rad

μ=0.0125, e=0.15, θ=0.13 rad

68

μ=0.0125, e=0.2, θ=0.04 rad

μ=0.0125, e=0.25, θ=0.09 rad

69

μ=0.0125, e=0.3, θ=0.09 rad


orbits.

μ=0.015, e=0.0, θ=0.53 rad

70

μ=0.015, e=0.05, θ=0.15 rad

μ=0.015, e=0.1, θ=0.18 rad

71

μ=0.015, e=0.15, θ=0.18 rad

μ=0.015, e=0.2, θ=0.22 rad

72

μ=0.015, e=0.25, θ=0.06 rad


regular orbits.

μ=0.0175, e=0.0, θ=0.74 rad

73

μ=0.0175, e=0.05, θ=0.69 rad

μ=0.0175, e=0.1, θ=0.48 rad

74

μ=0.0175, e=0.15, θ=0.31 rad

μ=0.0175, e=0.2, θ= 0.04 rad


orbits.

75

μ=0.02, e=0.0, θ=0.64 rad

μ=0.02, e=0.05, θ=0.60 rad

76

μ=0.02, e=0.1, θ=0.19 rad


orbits.

μ=0.0225, e=0.0, θ=0.11 rad

77

μ=0.0225, e=0.05, θ=0.07 rad

μ=0.0225, e=0.1, θ=0.03 rad


orbits.

78

μ=0.025, e=0.0, θ=0.03 rad

μ=0.025, e=0.05, θ=0.07 rad


regular orbits.

79

μ=0.0275, e=0.0, θ=0.12 rad


regular orbits.

μ=0.03, e=0.0, θ=0.19 rad


orbits.

80

μ=0.0325, e=0.0, θ=0.20 rad

μ=0.0325, e=0.05, θ=0.03 rad


regular orbits.

81

μ=0.035, e=0.0, θ=0.17 rad

μ=0.035, e=0.05, θ=0.08 rad

82

μ=0.035, e=0.1, θ=0.01 rad


orbits.

μ=0.0375, e=0.0, θ=0.14 rad

83

μ=0.0375, e=0.05, θ=0.03 rad

μ=0.0375, e=0.1, θ=0.01 rad


orbits.

84

μ=0.04, e=0.0, θ=0.06 rad

After μ=0.05, e=0.0 there are no initial conditions that produce

regular orbits.

ii. Regular regions near equilibrium point 𝑳𝟓

The regular regions near equilibrium point 𝐿5 are almost

completely symmetrical to the ones near 𝐿4 the reason being

that actually 𝐿5 becomes 𝐿4 as the initial conditions become

𝑦 → −𝑦 and the independent variable becomes 𝑣 → −𝑣. As a

result and for the sake of time, less values of the mass ratio μ

have been considered here. The results of the calculations are

as follows:

85

μ=0.005, e=0.0, θ=1.26 rad

μ=0.005, e=0.05, θ=1.09 rad

86

μ=0.005, e=0.1, θ=1.04 rad

μ=0.005, e=0.15, θ=0.54 rad

87

μ=0.005, e=0.2, θ=0.46 rad

μ=0.005, e=0.25, θ=0.53 rad

88

μ=0.005, e=0.3, θ=0.36 rad

μ=0.005, e=0.35, θ=0.21 rad

89

μ=0.005, e=0.4, θ=0.19 rad

μ=0.005, e=0.45, θ=0.10 rad

90

μ=0.005, e=0.5, θ=0.0 rad


orbits.

μ=0.01, e=0.0, θ=0.88 rad

91

μ=0.01, e=0.05, θ=0.73 rad

μ=0.01, e=0.1, θ=0.65 rad

92

μ=0.01, e=0.15, θ=0.54 rad

μ=0.01, e=0.2, θ=0.28 rad

93

μ=0.01, e=0.25, θ=0.06 rad

μ=0.01, e=0.3, θ=0.06 rad

94

μ=0.01, e=0.35, θ=0.14 rad


regular orbits.

μ=0.02, e=0.0, θ=0.64 rad

95

μ=0.02, e=0.05, θ=0.59 rad

μ=0.02, e=0.1, θ=0.19 rad


orbits.

96

μ=0.03, e=0.0, θ=0.19 rad


orbits.

μ=0.04, e=0.0, θ=0.05 rad


orbits.

97

C. Examination of the results

This examination of the results focuses on the regular

regions near the Lagrange point 𝐿4. The apparent symmetry

between the regular regions near 𝐿4 and 𝐿5 means that any

conclusions that are drawn for one region can be applied to the

other.

By examining the range of the regular regions near 𝐿4, it

becomes clear that the range θ decreases in value as the mass

ratio μ and the eccentricity of the orbits increases as shown in

figures 3-5. In fact, the rate of its decrease is rather fast as the

regular regions are virtually non-existent after μ=0.03 (save for

very low values of e) and e=0.65 in the most extreme cases

where μ is closer to zero. This suggests that the chances of the

existence of co-orbital companions, such as Trojan-like bodies,

increase in dynamical systems with low to moderate orbital

eccentricities (< 0.3) (Schwarz, Dvorak, Süli & Érdi, 2007)

where the regular area is larger. Figure 8 shows the change in

the regular range θ with the mass ratio μ for 𝑒 = 0.0 and 𝑒 =

0.1. Since the majority of extrasolar planetary systems that

have been discovered resemble a dynamical system not unlike

the Sun and Jupiter system (1M⊙ = 1047.56 MJupiter , 𝜇 =

0.0009546), the most interesting range of the mass ratio μ,

that warrants further examination, is 0.001≤ μ ≤0.005. Figure 9

shows the change in the regular range θ with the orbital

eccentricity e for 𝜇 = 0.001 and 𝜇 = 0.005.

98

Figure 8: The change in the range of the regular region θ with

the mass ratio μ for e=0.0


the eccentricity e for μ=0.001 and μ=0.005

Figure 10 shows the change of the range θ with both μ and

e. While it is clear that θ decreases as e increases, there is a

noticeable increase of θ around μ=0.018 as indicated by the

“bump”.

99


the mass ratio μ and the eccentricity e

D. Stability Regions of Exoplanetary systems

The results of Sections 3B and 3C can be applied to

existing exoplanetary systems in an attempt to suggest possible

candidate systems in which co-orbital companions could exist

in stable orbits near the Lagrange point 𝐿4. By taking into

account that the range of the regular region θ decreases as the

mass ratio μ and the orbital eccentricity e increase in value, a

list of 15 exoplanetary systems has been examined to

determine their regular regions. The criteria which the possible

candidates must fulfill are a low to moderate eccentricity e (<

0.3), and a mass ratio μ with values between 0.001 and 0.005.

100

This would resemble a system in which the planet’s mass

would be comparable to that of Jupiter, and the host star’s

mass would be comparable to that of the Sun. This is because,

as can be seen in figure 11, the majority of the known

exoplanetary systems fulfill these criteria.

Figure 11: Distribution of known exoplanetary systems with

regards to the mass of the host star (in 𝑴⊙) and the mass of the

planet (in 𝑴𝑱𝒖𝒑𝒊𝒕𝒆𝒓). The color range shows their orbital eccentricity.

The distribution is denser around 𝟏𝑴⊙and 𝟏𝑴𝑱𝒖𝒑𝒊𝒕𝒆𝒓.

Table 2 gives the list of the candidate exoplanetary systems

for which the range of the regular region θ has been calculated,

along with their characteristics.

101

Table 2

System 𝑀𝑆(𝑀⊙) 𝑀𝑃(𝑀𝐽𝑢𝑝) μ e θ (rad)

HR 810 1.11 2.26 0.001944 0.161 0.909211

HD 95872 0.95 4.6 0.0046223 0.06 1.065137

HD 73267 0.89 3.06 0.0032821 0.256 0.543495

WASP-38 1.216 2.712 0.002129 0.0321 1.357102

HD 195019 1.06 3.7 0.0033321 0.014 1.394702

HD 96127 0.91 4.0 0.004196 0.3 0.509436

Kepler-434 1.198 2.86 0.0022789 0.131 0.951384

HD 28185 1.24 5.7 0.0043881 0.07 0.887029

WASP-8 1.033 2.244 0.0020737 0.31 0.566593

HIP 8541 1.17 5.59 0.0045609 0.16 0.754938

tau Boo 1.3 5.84 0.0042884 0.0787 0.758422

Kepler-43 1.32 3.23 0.0023359 0.025 1.401148

KOI-830 0.87 1.27 0.0013935 0.22 0.799872

WASP-32 3.6 1.1 0.0031241 0.018 1.405745

HIP 109600 0.87 2.68 0.0029406 0.163 0.786861

102

4. Conclusions

In this work, the nature of stability of the Lagrange points

𝐿4 and 𝐿5 and of their neighborhoods for different values of

the mass ratio μ and the orbital eccentricity e, using the Elliptic

Restricted Three Body Problem as a theoretical basis has been

studied. The range of the regions near the equilibrium points

𝐿4 and 𝐿5, where a motionless satellite follows regular orbits,

has been determined using the Fast Lyapunov Indicators (FLI).

The extend of these regions depends on the mass parameter μ

and the eccentricity e. The range of the regular regions θ

decreases as both the mass ration μ and the eccentricity e

increase. It is possible that co-orbital companions such as

Trojan-like bodies of small mass exist in the regular regions

near 𝐿4 and 𝐿5 of extrasolar planetary systems that consists of

a Sun-like star and one Jupiter-like planet. Fifteen

exoplanetary systems have been examined and their regular

region have been determined in order to propose possible

candidates for investigation of the existence of possible co-

orbital companions in these systems.

Further works are necessary to better calculate the range

of the regular regions near the equilibrium points considering

larger intervals of the independent variable. In addition, the list

of exoplanetary systems that could be host to Trojan-like

bodies can be greatly expanded, since there are more than 2700

known exoplanetary systems, with more discovered yearly.

103

APPENDIX

The programs that have been prepared for the this study

are presented here. The programming languages C and C++

have been used.

Program A:

//INTEGRATION OF THE ELLIPCTIC RESTRICTED THREE BODY PROBLEM IN THE

BARYCENTRIC INERTIAL FRAME OF REFERENCE

#include <iostream>

#include <cstdio>

#include <cstdlib>

#include <cmath>

using namespace std;

//ODE for eccentric anomaly E

double dydx(double x, double y, double e){

double dy;

dy=1.0/(1.0-(e*cos(y)));

return dy;

}

//ODEs for satellite coordinates in the barycentric inertial frame

double dqdt(double t, double q, double p, double vq, double vp,

double m, double q1, double q2, double p1, double p2){

double dq;

dq=vq;

return dq;

}

double dvqdt(double t, double q, double p, double vq, double vp,


double dvq, s1, s2;

s1=sqrt(pow((q-q1), 2.0) + pow((p-p1), 2.0));


dvq=-(1-m)*((q-q1)/(pow(s1, 3.0))) - m*((q-q2)/(pow(s2, 3.0)));

return dvq;

}

double dpdt(double t, double q, double p, double vq, double vp,


double dp;

dp=vp;

104

return dp;

}

double dvpdt(double t, double q, double p, double vq, double vp,


double dvp, s1, s2;



dvp=-(1-m)*((p-p1)/(pow(s1, 3.0))) - m*((p-p2)/(pow(s2, 3.0)));

return dvp;

}

//function f(a) for Newton-Raphson

double func(double m, double a){

double fa;

fa=(1-a)/(pow(m, 2)) - a/(pow(1-m, 2)) + (m*a)/pow(abs(a), 3) +

((1-m)*(a-1))/pow(abs(a-1), 3);

return fa;

}

//function f'(a) for Newton-Raphson (root of a>1)

double funcdot(double m, double a){

double fdota;

fdota=-(1/pow(1-m, 2)) - (1/pow(m, 2)) + ((1-m)/pow(abs(a-1),

3)) + (m/pow(abs(a), 3)) - ((3*(a-1)*(1-m))/pow(abs(a-1), 4)) -

((3*a*m)/pow(abs(a), 4));

return fdota;

}

//function f'(a) for Newton-Raphson (roots of a<1)

double funcdot2(double m, double a){

double fdota2;

fdota2=-(1/pow(1-m, 2)) - (1/pow(m, 2)) + ((1-m)/pow(abs(a-1),

3)) + (m/pow(abs(a), 3)) + ((3*(a-1)*(1-m))/pow(abs(a-1), 4)) +

((3*a*m)/pow(abs(a), 4));

return fdota2;

}

int main(int argc, char *argv[]) {

double tn,En,k1,k2,k3,k4,E,t,h,t0,E0,e;

double m, q1, q2, p1, p2, xi0;

double ql1, pl1, ql2, pl2, ql3, pl3;

double q4, p4;

double q, p, vq, vp, q0, p0, vq0, vp0;

double qn, pn, vqn, vpn;

double limit;

double a1, a2, a3, a4, b1, b2, b3, b4, c1, c2, c3, c4, d1, d2,

d3, d4;

double a01, a02, a03, an1, an2, an3;//for Newton-Raphson

FILE * fp1;

105

FILE * fp2;

FILE * fp3;

FILE * fp4;

FILE * fp5;

FILE * fp6;

FILE * fp7;

FILE * fp8;

FILE * fp9;

/*Initial conditions and step h*/

t0=0.0; // initial time

E0=0.0; //initial E

//Initial Coordinates

q0=-2.0;

p0=0.0;

//Initial velocity components

vq0=-0.0;

vp0=-0.6;

m=0.2; //mass ratio

e=0.9; //eccentricity

h=0.005; //RK4 step

limit=t0+120; //time limit

a01=1.1; //Newton-Raphson starting values

a02=-0.01;

a03=0.5;

tn=t0;

En=E0;

qn=q0;

pn=p0;

vqn=vq0;

vpn=vp0;

/*Newton-Raphson*/

an1=a01;

if( func(m, a01)!=0 && funcdot(m, a01)!=0 ){

do{

a01=an1;

an1=a01 - ((func(m, a01)/funcdot(m, a01)));

}

while(fabs(an1 - a01) >= 0.0000001);

}

an2=a02;

if( func(m, a02)!=0 && funcdot2(m, a02)!=0 ){

do{

106

a02=an2;

an2=a02 - ((func(m, a02)/funcdot2(m, a02)));

}

while(fabs(an2 - a02) >= 0.0000001);

}

an3=a03;

if( func(m, a03)!=0 && funcdot2(m, a03)!=0 ){

do{

a03=an3;

an3=a03 - ((func(m, a03)/funcdot2(m, a03)));

}

while(fabs(an3 - a03) >= 0.0000001);

}

cout << "a1= " << an1 << endl;

cout << "a2= " << an2 << endl;

cout << "a3= " << an3 << endl;

//Coordinates of the Primaries

q1=-m*(cos(En)-e);

p1=-m*sqrt(1-pow(e,2))*sin(En);

q2=(1-m)*(cos(En)-e);

p2=(1-m)*sqrt(1-pow(e,2))*sin(En);

//Quantity q0

xi0=(1/2.0)-m;

//Coordinates of L4/L5 lagrange Point

q4=xi0*(cos(En)-e) - (sqrt(3.0)/2.0)*sqrt(1-pow(e,2))*sin(En);

p4=xi0*sqrt(1-pow(e,2))*sin(En) + (sqrt(3.0)/2.0)*(cos(En)-e);

//Coordinates of L1 (a>1) Lagrange Point

ql1=(an1 - m)*(cos(En) - e);

pl1=(an1 - m)*(sqrt(1-pow(e,2))*sin(En));

//Coordinates of L2 (a<1) Lagrange Point

ql2=(an2 - m)*(cos(En) - e);


//Coordinates of L3 (0<a<1) Lagrange Point

ql3=(an3 - m)*(cos(En) - e);


fp1=fopen("EccentricAnomaly.txt", "w+");

fp2=fopen("Primary1Coordinates.txt", "w+");


fp4=fopen("SatelliteCoordinates.txt", "w+");

fp5=fopen("L4Coordinates.txt", "w+");





107

while(tn<=limit){

//RK4 for E

k1=h*dydx(tn,En, e);

k2=h*dydx((tn+h/2.0),(En+k1/2.0), e);

k3=h*dydx((tn+h/2.0),(En+k2/2.0), e);

k4=h*dydx((tn+h),(En+k3), e);

//RK4 for our ODE system

a1=h*dqdt(tn,qn,pn,vqn,vpn,m,q1,q2,p1,p2);

b1=h*dpdt(tn,qn,pn,vqn,vpn,m,q1,q2,p1,p2);

c1=h*dvqdt(tn,qn,pn,vqn,vpn,m,q1,q2,p1,p2);

d1=h*dvpdt(tn,qn,pn,vqn,vpn,m,q1,q2,p1,p2);

a2=h*dqdt((tn+h/2.0),(qn+a1/2.0),(pn+b1/2.0),(vqn+c1/2.0),(vpn+d1/2

.0),m,q1,q2,p1,p2);

b2=h*dpdt((tn+h/2.0),(qn+a1/2.0),(pn+b1/2.0),(vqn+c1/2.0),(vpn+d1/2

.0),m,q1,q2,p1,p2);

c2=h*dvqdt((tn+h/2.0),(qn+a1/2.0),(pn+b1/2.0),(vqn+c1/2.0),(vpn+d1/

2.0),m,q1,q2,p1,p2);

d2=h*dvpdt((tn+h/2.0),(qn+a1/2.0),(pn+b1/2.0),(vqn+c1/2.0),(vpn+d1/

2.0),m,q1,q2,p1,p2);

a3=h*dqdt((tn+h/2.0),(qn+a2/2.0),(pn+b2/2.0),(vqn+c2/2.0),(vpn+d2/2

.0),m,q1,q2,p1,p2);

b3=h*dpdt((tn+h/2.0),(qn+a2/2.0),(pn+b2/2.0),(vqn+c2/2.0),(vpn+d2/2

.0),m,q1,q2,p1,p2);

c3=h*dvqdt((tn+h/2.0),(qn+a2/2.0),(pn+b2/2.0),(vqn+c2/2.0),(vpn+d2/

2.0),m,q1,q2,p1,p2);

d3=h*dvpdt((tn+h/2.0),(qn+a2/2.0),(pn+b2/2.0),(vqn+c2/2.0),(vpn+d2/

2.0),m,q1,q2,p1,p2);

a4=h*dqdt((tn+h),(qn+a3),(pn+b3),(vqn+c3),(vpn+d3),m,q1,q2,p1,p2);

b4=h*dpdt((tn+h),(qn+a3),(pn+b3),(vqn+c3),(vpn+d3),m,q1,q2,p1,p2);

c4=h*dvqdt((tn+h),(qn+a3),(pn+b3),(vqn+c3),(vpn+d3),m,q1,q2,p1,p2);

d4=h*dvpdt((tn+h),(qn+a3),(pn+b3),(vqn+c3),(vpn+d3),m,q1,q2,p1,p2);

E=((k1+2.0*k2+2.0*k3+k4)/6.0);

q=((a1+2.0*a2+2.0*a3+a4)/6.0);

p=((b1+2.0*b2+2.0*b3+b4)/6.0);

vq=((c1+2.0*c2+2.0*c3+c4)/6.0);

vp=((d1+2.0*d2+2.0*d3+d4)/6.0);

fprintf(fp1, "%f\t%f\n", tn, En);

fprintf(fp2, "%f\t%f\n", q1, p1);

108


fprintf(fp4, "%f\t%f\n", qn, pn);


fprintf(fp6, "%f\t%f\n", q4, -p4);

fprintf(fp7, "%f\t%f\n", ql1, pl1);



//cout << "t= " << tn << " E= " << En << " q1= " << q1 << "

p1= " << p1 << " q2= " << q2 << " p2= " << p2 << " q= " << qn << "

p= " << pn << endl;

tn=tn+h;

En=En+E;

qn=qn+q;

pn=pn+p;

vqn=vqn+vq;

vpn=vpn+vp;

q1=-m*(cos(En)-e);

p1=-m*sqrt(1-pow(e, 2))*sin(En);

q2=(1-m)*(cos(En)-e);

p2=(1-m)*sqrt(1-pow(e, 2))*sin(En);

q4=xi0*(cos(En)-e) - (sqrt(3.0)/2.0)*sqrt(1-

pow(e,2))*sin(En);

p4=xi0*sqrt(1-pow(e,2))*sin(En) + (sqrt(3.0)/2.0)*(cos(En)-

e);

ql1=(an1 - m)*(cos(En) - e);


ql2=(an2 - m)*(cos(En) - e);


ql3=(an3 - m)*(cos(En) - e);


}

fclose(fp1);

fclose(fp2);

fclose(fp3);

fclose(fp4);

fclose(fp5);

fclose(fp6);

fclose(fp7);

fclose(fp8);

fclose(fp9);

return 0;

}

109

Program B:

//INTEGRATION OF THE ELLIPCTIC RESTRICTED THREE BODY PROBLEM IN THE

ROTATING-PULSATING FRAME OF REFERENCE AND FLI CALCULATION FOR A

REGULAR ORBIT AND A CHAOTIC ORBIT

#include <stdio.h>

#include <stdlib.h>

#include <math.h>

#include <stdint.h>

//ODEs for satellite coordinates in the pulsating-rotating frame

double dxdv(double v, double x, double y, double ux, double uy,

double m, double e){

double dx;

dx=ux;

return dx;

}

double dydv(double v, double x, double y, double ux, double uy,


double dy;

dy=uy;

return dy;

}

double duxdv(double v, double x, double y, double ux, double uy,


double dux, rdivp, m1, m2, r1, r2;

rdivp=1/(1+e*cos(v));

m1=1-m;

m2=m;

r1=sqrt(pow((x+m), 2.0) + pow(y, 2.0));

r2=sqrt(pow((x+m-1), 2.0) + pow(y, 2.0));

dux=2*uy+rdivp*(x - m1*((x+m)/pow(r1, 3.0)) - m2*((x+m-

1)/pow(r2, 3.0)));

return dux;

}

double duydv(double v, double x, double y, double ux, double uy,


double duy, rdivp, m1, m2, r1, r2;


m1=1-m;

m2=m;

r1=sqrt(pow((x+m), 2.0) + pow(y, 2.0));

r2=sqrt(pow((x+m-1), 2.0) + pow(y, 2.0));

duy=-2*ux+rdivp*(y - m1*(y/pow(r1, 3.0)) - m2*(y/pow(r2,

3.0)));

return duy;

}

//Variational ODEs for Lagrange points stability

110

double VdDxdv(double v, double Dx, double Dy, double Dux, double

Duy, double x, double y, double m, double e){

double dDx;

dDx=Dux;

return dDx;

}

double VdDydv(double v, double Dx, double Dy, double Dux, double


double dDy;

dDy=Duy;

return dDy;

}

double VdDuxdv(double v, double Dx, double Dy, double Dux, double


double dDux, rdivp, Uxx,Uxy;


Uxx=m*( (3*pow(x+m-1, 2))/pow(pow(x+m-1, 2)+pow(y, 2), 5/2.0) -

1/pow(pow(x+m-1, 2)+pow(y, 2), 3/2.0) ) + (1-m)*( (3*pow(x+m,

2))/pow(pow(x+m, 2)+pow(y, 2), 5/2.0) - 1/pow(pow(x+m, 2)+pow(y,

2), 3/2.0) );

Uxy=(3*m*(x+m-1)*y)/pow(pow(x+m-1, 2)+pow(y, 2), 5/2.0) +

(3*(1-m)*(x+m)*y)/pow(pow(x+m, 2)+pow(y, 2), 5/2.0);

dDux=2*Duy + rdivp*Dx + rdivp*(Uxx*Dx + Uxy*Dy);

return dDux;

}

double VdDuydv(double v, double Dx, double Dy, double Dux, double


double dDuy, rdivp, Uyx,Uyy;


Uyy=m*( (3*pow(y, 2))/pow(pow(x+m-1, 2)+pow(y, 2), 5/2.0) -

1/pow(pow(x+m-1, 2)+pow(y, 2), 3/2.0) ) + (1-m)*( (3*pow(y,


2), 3/2.0) );

Uyx=(3*m*(x+m-1)*y)/pow(pow(x+m-1, 2)+pow(y, 2), 5/2.0) +


dDuy=-2*Dux + rdivp*Dy + rdivp*(Uyx*Dx + Uyy*Dy);

return dDuy;

}

//Equation f(x) for L1, l2, L3

double fx(double x, double m){

double func, r1c, r2c;

r1c=pow(fabs(x+m), 3);

r2c=pow(fabs(x+m-1), 3);

func=-x + (1.0-m)*((x+m)/r1c) + m*((x+m-1)/r2c);

return func;

}

111

//Equation f'(x) for L1, l2, l3 (root x>1)

double fdotx1(double x, double m){

double func1;

func1=-1.0 + m/pow(fabs(x+m-1.0), 3) + (1.0-m)/pow(fabs(x+m),

3) - (3.0*m*(x+m-1))/pow(fabs(x+m-1.0), 4) - (3.0*(1.0-

m)*(x+m))/pow(fabs(x+m), 4);

return func1;

}

//Equation f'(x) for L1, l2, l3 (root x<0)


double func2;


3) + (3.0*m*(x+m-1.0))/pow(fabs(x+m-1.0), 4) + (3.0*(1.0-


return func2;

}

//Equation f'(x) for L1, l2, l3 (root 0<x<1)


double func3;


3) + (3.0*m*(x+m-1.0))/pow(fabs(x+m-1.0), 4) - (3.0*(1.0-


return func3;

}


int i, j, k, colms=0, rows=0;

int i1=0,i2=1,i3=2,i4=3;

double v, v0, x, y, ux, uy, x0, y0, x0c, y0c, ux0, uy0, h;

double vn, xn, yn, uxn, uyn, limit;

double Dx, Dy, Dux, Duy, Dx0, Dy0, Dux0, Duy0;//for Variational

ODEs

double Dxn, Dyn, Duxn, Duyn;//for Variational ODEs

double stabtestx, stabtesty;

double x1, x2, y1, y2;

double xl1, yl1, xl2, yl2, xl3, yl3, xl4, yl4, xl5, yl5;//for

Lagrange points

double m, e, mass, ecc;


d3, d4;

double k1, k2, k3, k4, l1, l2, l3, l4, m1, m2, m3, m4, n1, n2,

n3, n4;

double x01, x02, x03, xn1, xn2, xn3;//for Newton-Raphson

double FLI, FLI2;

FILE * fp1;

FILE * fp2;

FILE * fp3;

FILE * fp4;

FILE * fp5;

FILE * fp6;

112

FILE * fp7;

FILE * fp8;

FILE * fp9;


v0=0.0; //true anomaly

m=0.001; //mass ratio


h=0.005; //RK4 step

limit=v0+30*3.14159265;

x01=1.1; //Newton-Raphson starting values

x02=-1.0;

x03=0.5;

/*Newton-Raphson*/

xn1=x01;

if( fx(x01, m)!=0 && fdotx1(x01, m)!=0 ){

do{

x01=xn1;

xn1=x01 - ((fx(x01, m)/fdotx1(x01, m)));

}

while(fabs(xn1 - x01) >= pow(10, -15));

}

xn2=x02;

if( fx(x02, m)!=0 && fdotx2(x02, m)!=0 ){

do{

x02=xn2;

xn2=x02 - ((fx(x02, m)/fdotx2(x02, m)));

}


}

xn3=x03;

if( fx(x03, m)!=0 && fdotx3(x03, m)!=0 ){

do{

x03=xn3;

xn3=x03 - ((fx(x03, m)/fdotx3(x03, m)));

113

}


}

printf("xl1=%f\n",xn1);


printf("xl3=%f\n\n",xn3);


x1=-m;

y1=0.0;

x2=1-m;

y2=0.0;

//L4/l5 Coordinates

xl4=0.5*(1-2*m);

yl4=sqrt(3)/2;

xl5=0.5*(1-2*m);

yl5=-sqrt(3)/2;

//L1/L2/L3 Coordinates

xl1=xn1;

yl1=0.0;

xl2=xn2;

yl2=0.0;

xl3=xn3;

yl3=0.0;




fp4=fopen("L4L5Coordinates.txt", "w+");

fp5=fopen("L1L2L3Coordinates.txt", "w+");

fp6=fopen("VariationalCoordinatesAt2pi.txt", "w+");

fp7=fopen("InitialConditionsFLI.txt", "w+");

fp8=fopen("FLI_TrueAnomalyRegular.txt", "w+");

fp9=fopen("FLI_TrueAnomalyChaotic.txt", "w+");

fprintf(fp2, "%f\t%f\n", x1, y1);


fprintf(fp4, "%f\t%f\n%f\t%f", xl4, yl4, xl5, yl5);

fprintf(fp5, "%f\t%f\n%f\t%f\n%f\t%f", xl1, yl1, xl2, yl2, xl3,

yl3);

//Initial Coordinates (Regular Orbits)

x0=-0.286;

y0=0.958525 ;

//Initial Coordinates (Chaotic Orbits)

x0c=0.50;

y0c=1.043678;


ux0=0.0;

uy0=0.0;

//Initial Variational Coordinates

Dx0=100.0;

Dy0=100.0;

Dux0=0.0;

Duy0=0.0;

114

Dxn=Dx0;

Dyn=Dy0;

Duxn=Dux0;

Duyn=Duy0;

vn=v0;

xn=x0;

yn=y0;

uxn=ux0;

uyn=uy0;

while(vn<=limit){

//RK4 for ODE system

a1=h*dxdv(vn, xn, yn, uxn, uyn, m, e);

b1=h*dydv(vn, xn, yn, uxn, uyn, m, e);

c1=h*duxdv(vn, xn, yn, uxn, uyn, m, e);

d1=h*duydv(vn, xn, yn, uxn, uyn, m, e);

a2=h*dxdv((vn+h/2.0), (xn+a1/2.0), (yn+b1/2.0),

(uxn+c1/2.0), (uyn+d1/2.0), m, e);

b2=h*dydv((vn+h/2.0), (xn+a1/2.0), (yn+b1/2.0),

(uxn+c1/2.0), (uyn+d1/2.0), m, e);

c2=h*duxdv((vn+h/2.0), (xn+a1/2.0), (yn+b1/2.0),

(uxn+c1/2.0), (uyn+d1/2.0), m, e);

d2=h*duydv((vn+h/2.0), (xn+a1/2.0), (yn+b1/2.0),

(uxn+c1/2.0), (uyn+d1/2.0), m, e);


(uxn+c2/2.0), (uyn+d2/2.0), m, e);


(uxn+c2/2.0), (uyn+d2/2.0), m, e);


(uxn+c2/2.0), (uyn+d2/2.0), m, e);


(uxn+c2/2.0), (uyn+d2/2.0), m, e);

a4=h*dxdv((vn+h), (xn+a3), (yn+b3), (uxn+c3), (uyn+d3), m,

e);

b4=h*dydv((vn+h), (xn+a3), (yn+b3), (uxn+c3), (uyn+d3), m,

e);

c4=h*duxdv((vn+h), (xn+a3), (yn+b3), (uxn+c3), (uyn+d3), m,

e);

d4=h*duydv((vn+h), (xn+a3), (yn+b3), (uxn+c3), (uyn+d3), m,

e);

x=((a1+2.0*a2+2.0*a3+a4)/6.0);

y=((b1+2.0*b2+2.0*b3+b4)/6.0);

ux=((c1+2.0*c2+2.0*c3+c4)/6.0);

uy=((d1+2.0*d2+2.0*d3+d4)/6.0);

xn=xn+x;

yn=yn+y;

uxn=uxn+ux;

uyn=uyn+uy;

fprintf(fp1, "%f\t%f\n", xn, yn);

115

//RK4 for Variational ODEs

k1=h*VdDxdv(vn, Dxn, Dyn, Duxn, Duyn, xn, yn, m, e);

l1=h*VdDydv(vn, Dxn, Dyn, Duxn, Duyn, xn, yn, m, e);

m1=h*VdDuxdv(vn, Dxn, Dyn, Duxn, Duyn, xn, yn, m, e);

n1=h*VdDuydv(vn, Dxn, Dyn, Duxn, Duyn, xn, yn, m, e);

k2=h*VdDxdv((vn+h/2.0), (Dxn+k1/2.0), (Dyn+l1/2.0),

(Duxn+m1/2.0), (Duyn+n1/2.0), xn, yn, m, e);

l2=h*VdDydv((vn+h/2.0), (Dxn+k1/2.0), (Dyn+l1/2.0),


m2=h*VdDuxdv((vn+h/2.0), (Dxn+k1/2.0), (Dyn+l1/2.0),


n2=h*VdDuydv((vn+h/2.0), (Dxn+k1/2.0), (Dyn+l1/2.0),










k4=h*VdDxdv((vn+h), (Dxn+k3), (Dyn+l3), (Duxn+m3),

(Duyn+n3), xn, yn, m, e);

l4=h*VdDydv((vn+h), (Dxn+k3), (Dyn+l3), (Duxn+m3),


m4=h*VdDuxdv((vn+h), (Dxn+k3), (Dyn+l3), (Duxn+m3),


n4=h*VdDuydv((vn+h), (Dxn+k3), (Dyn+l3), (Duxn+m3),


Dx=((k1+2.0*k2+2.0*k3+k4)/6.0);

Dy=((l1+2.0*l2+2.0*l3+l4)/6.0);

Dux=((m1+2.0*m2+2.0*m3+m4)/6.0);

Duy=((n1+2.0*n2+2.0*n3+n4)/6.0);

vn=vn+h;

Dxn=Dxn+Dx;

Dyn=Dyn+Dy;

Duxn=Duxn+Dux;

Duyn=Duyn+Duy;

FLI2=log((sqrt(pow(Dxn, 2.0)+pow(Dyn, 2.0) + Duxn +

Duyn))/vn);

fprintf(fp8, "%f\t%f\n", vn, FLI2);

}

fprintf(fp6, "%f\t%f\t%f\t%f\n", Dxn, Dyn, Duxn, Duyn);

//FLI Calculation

FLI=log((sqrt(pow(Dxn, 2.0)+pow(Dyn, 2.0) + Duxn +

Duyn))/vn);

printf("x0=%f\ty0=%f\tFLI=%f\n",x0, y0, FLI);

if(FLI<6.0){

116

fprintf(fp7, "%f\t%f\t%f\n", x0, y0, FLI);

}

Dx0=100.0;//reinitiallise initial variational conditions

Dy0=100.0;

Dux0=0.0;

Duy0=0.0;

Dxn=Dx0;

Dyn=Dy0;

Duxn=Dux0;

Duyn=Duy0;

vn=v0;//reinitiallise true anomaly and initial conditions

xn=x0c;

yn=y0c;

uxn=ux0;

uyn=uy0;

//CHAOTIC ORBIT

while(vn<=limit){







(uxn+c1/2.0), (uyn+d1/2.0), m, e);


(uxn+c1/2.0), (uyn+d1/2.0), m, e);


(uxn+c1/2.0), (uyn+d1/2.0), m, e);


(uxn+c1/2.0), (uyn+d1/2.0), m, e);


(uxn+c2/2.0), (uyn+d2/2.0), m, e);


(uxn+c2/2.0), (uyn+d2/2.0), m, e);


(uxn+c2/2.0), (uyn+d2/2.0), m, e);


(uxn+c2/2.0), (uyn+d2/2.0), m, e);


e);


e);


e);


e);

x=((a1+2.0*a2+2.0*a3+a4)/6.0);

y=((b1+2.0*b2+2.0*b3+b4)/6.0);

ux=((c1+2.0*c2+2.0*c3+c4)/6.0);

uy=((d1+2.0*d2+2.0*d3+d4)/6.0);

117

xn=xn+x;

yn=yn+y;

uxn=uxn+ux;

uyn=uyn+uy;


k1=h*VdDxdv(vn, Dxn, Dyn, Duxn, Duyn, xn, yn, m, e);

l1=h*VdDydv(vn, Dxn, Dyn, Duxn, Duyn, xn, yn, m, e);

m1=h*VdDuxdv(vn, Dxn, Dyn, Duxn, Duyn, xn, yn, m, e);

n1=h*VdDuydv(vn, Dxn, Dyn, Duxn, Duyn, xn, yn, m, e);

















k4=h*VdDxdv((vn+h), (Dxn+k3), (Dyn+l3), (Duxn+m3),


l4=h*VdDydv((vn+h), (Dxn+k3), (Dyn+l3), (Duxn+m3),


m4=h*VdDuxdv((vn+h), (Dxn+k3), (Dyn+l3), (Duxn+m3),


n4=h*VdDuydv((vn+h), (Dxn+k3), (Dyn+l3), (Duxn+m3),


Dx=((k1+2.0*k2+2.0*k3+k4)/6.0);

Dy=((l1+2.0*l2+2.0*l3+l4)/6.0);

Dux=((m1+2.0*m2+2.0*m3+m4)/6.0);

Duy=((n1+2.0*n2+2.0*n3+n4)/6.0);

vn=vn+h;

Dxn=Dxn+Dx;

Dyn=Dyn+Dy;

Duxn=Duxn+Dux;

Duyn=Duyn+Duy;

FLI2=log((sqrt(pow(Dxn, 2.0)+pow(Dyn, 2.0) + Duxn +

Duyn))/vn);

fprintf(fp9, "%f\t%f\n", vn, FLI2);

}

vn=v0;//reinitiallise true anomaly

fclose(fp1);

118

fclose(fp2);

fclose(fp3);

fclose(fp4);

fclose(fp5);

fclose(fp6);

fclose(fp7);

fclose(fp8);

fclose(fp9);

return 0;

}

Program C:

//CALCULATION OF THE NATURE OF STABILLITY OF THE LAGRANGE POINTS

FOR A GRID OF MASS RATIO AND ECCENTRICITY e USING THE FLOQUET

THEORY VIA THE QR METHOD AND HESSENBERG ELIMINATION

#include <stdio.h>

#include <stdlib.h>

#include <math.h>

#include <stdint.h>

#define SWAP(g,h) {y=(g);(g)=(h);(h)=y;}

#define SIGN(a,b) ((b) > 0 ? fabs(a) : -fabs(a))

void nrerror(char error_text[]){

void exit();

fprintf(stderr,"Numerical Recipes run-time error...\n");

fprintf(stderr,"%s\n",error_text);

fprintf(stderr,"...now exiting to system...\n");

exit(1);

}

//Offset vector allocation

float *vector(int nl,int nh){

double *v;

v=(double *)malloc((unsigned) (nh-nl+1)*sizeof(double));

if (!v) nrerror("allocation failure in vector()");

return v-nl;

}

//Offset matrix conversion

double **convert_matrix(double *a,int nrl,int nrh,int ncl,int nch){

int i,j,nrow,ncol;

double **mat;

nrow=nrh-nrl+1;

ncol=nch-ncl+1;

mat = (double **) malloc((unsigned) (nrow)*sizeof(double*));

if (!mat) nrerror("allocation failure in convert_matrix()");

mat-= nrl;

for(i=0,j=nrl;i<=nrow-1;i++,j++) mat[j]=a+ncol*i-ncl;

return mat;

}

//Hessenberg Elimination

119

void elmhes(double **a, int n){

int m,j,i;

double y,x;

for (m=2;m<n;m++){

x=0.0;

i=m;

for (j=m;j<=n;j++) {

if (fabs(a[j][m-1]) > fabs(x)) {

x=a[j][m-1];

i=j;

}

}

if (i != m) {

for (j=m-1;j<=n;j++) SWAP(a[i][j],a[m][j])

for (j=1;j<=n;j++) SWAP(a[j][i],a[j][m])

}

if(x){

for (i=m+1;i<=n;i++) {

if (y=a[i][m-1]) {

y /= x;

a[i][m-1]=y;

for (j=m;j<=n;j++)a[i][j] -= y*a[m][j];

for (j=1;j<=n;j++)a[j][m] += y*a[j][i];

}

}

}

}

}

//QR algorithm for Eigenvalues

void hqr(double **a, int n, double wr[], double wi[]){

int nn,m,l,k,j,its,i,mmin;

double z,y,x,w,v,u,t,s,r,q,p,anorm;

void nrerror();

anorm=fabs(a[1][1]);

for (i=2;i<=n;i++)

for (j=(i-1);j<=n;j++)

anorm += fabs(a[i][j]);

nn=n;

t=0.0;

while (nn >= 1) {

its=0;

do {

for (l=nn;l>=2;l--) {

s=fabs(a[l-1][l-1])+fabs(a[l][l]);

if (s == 0.0) s=anorm;

if (fabs(a[l][l-1]) + s == s) break;

}

x=a[nn][nn];

if (l == nn) {

wr[nn]=x+t;

wi[nn--]=0.0;

} else {

y=a[nn-1][nn-1];

w=a[nn][nn-1]*a[nn-1][nn];

if (l == (nn-1)) {

p=0.5*(y-x);

q=p*p+w;

z=sqrt(fabs(q));

120

x += t;

if (q >= 0.0) {

z=p+SIGN(z,p);

wr[nn-1]=wr[nn]=x+z;

if (z) wr[nn]=x-w/z;

wi[nn-1]=wi[nn]=0.0;

} else {

wr[nn-1]=wr[nn]=x+p;

wi[nn-1]= -(wi[nn]=z);

}

nn -= 2;

} else {

if (its == 30) nrerror("Too many iterations in

HQR");

if (its == 10 || its == 20) {

t += x;

for (i=1;i<=nn;i++) a[i][i] -= x;

s=fabs(a[nn][nn-1])+fabs(a[nn-1][nn-2]);

y=x=0.75*s;

w = -0.4375*s*s;

}

++its;

for (m=(nn-2);m>=l;m--) {

z=a[m][m];

r=x-z;

s=y-z;

p=(r*s-w)/a[m+1][m]+a[m][m+1];

q=a[m+1][m+1]-z-r-s;

r=a[m+2][m+1];

s=fabs(p)+fabs(q)+fabs(r);

p /= s;

q /= s;

r /= s;

if (m == l) break;

u=fabs(a[m][m-1])*(fabs(q)+fabs(r));

v=fabs(p)*(fabs(a[m-1][m-

1])+fabs(z)+fabs(a[m+1][m+1]));

if (u+v == v) break;

}

for (i=m+2;i<=nn;i++) {

a[i][i-2]=0.0;

if (i != (m+2)) a[i][i-3]=0.0;

}

for (k=m;k<=nn-1;k++) {

if (k != m) {

p=a[k][k-1];

q=a[k+1][k-1];

r=0.0;

if (k != (nn-1)) r=a[k+2][k-1];

if (x=fabs(p)+fabs(q)+fabs(r)) {

p /= x;

q /= x;

r /= x;

}

}

if (s=SIGN(sqrt(p*p+q*q+r*r),p)) {

if (k == m) {

if (l != m)

a[k][k-1] = -a[k][k-1];

} else

a[k][k-1] = -s*x;

121

p += s;

x=p/s;

y=q/s;

z=r/s;

q /= p;

r /= p;

for (j=k;j<=nn;j++) {

p=a[k][j]+q*a[k+1][j];

if (k != (nn-1)) {

p += r*a[k+2][j];

a[k+2][j] -= p*z;

}

a[k+1][j] -= p*y;

a[k][j] -= p*x;

}

mmin = nn<k+3 ? nn : k+3;

for (i=l;i<=mmin;i++) {

p=x*a[i][k]+y*a[i][k+1];

if (k != (nn-1)) {

p += z*a[i][k+2];

a[i][k+2] -= p*r;

}

a[i][k+1] -= p*q;

a[i][k] -= p;

}

}

}

}

}

} while (l < nn-1);

}

}




double dx;

dx=ux;

return dx;

}



double dy;

dy=uy;

return dy;

}





m1=1-m;

m2=m;

r1=sqrt(pow((x+m), 2.0) + pow(y, 2.0));

122

r2=sqrt(pow((x+m-1), 2.0) + pow(y, 2.0));


1)/pow(r2, 3.0)));

return dux;

}





m1=1-m;

m2=m;

r1=sqrt(pow((x+m), 2.0) + pow(y, 2.0));

r2=sqrt(pow((x+m-1), 2.0) + pow(y, 2.0));


3.0)));

return duy;

}




double dDx;

dDx=Dux;

return dDx;

}



double dDy;

dDy=Duy;

return dDy;

}








2), 3/2.0) );




return dDux;

}




123





2), 3/2.0) );




return dDuy;

}







return func;

}



double func1;


3) - (3.0*m*(x+m-1))/pow(fabs(x+m-1.0), 4) - (3.0*(1.0-


return func1;

}



double func2;


3) + (3.0*m*(x+m-1.0))/pow(fabs(x+m-1.0), 4) + (3.0*(1.0-


return func2;

}



double func3;


3) + (3.0*m*(x+m-1.0))/pow(fabs(x+m-1.0), 4) - (3.0*(1.0-


return func3;

}



int i1=0,i2=1,i3=2,i4=3;

double v, v0, x, y, ux, uy, x0, y0, ux0, uy0, h;

124


double Dx, Dy, Dux, Duy, Dx0[4]={1,0,0,0}, Dy0[4]={0,1,0,0},

Dux0[4]={0,0,1,0}, Duy0[4]={0,0,0,1};//for Variational ODEs

double Dxn[4], Dyn[4], Duxn[4], Duyn[4];//for Variational ODEs

double stabtestx, stabtesty, stabtestx2, stabtesty2;


double xl1, yl1, xl2, yl2, xl3, yl3, xl4, yl4, xl5, yl5,

truexl4, trueyl4, truexl5, trueyl5;//for Lagrange points



d3, d4;


n3, n4;


double A[4][4], D[4][4], D2[4][4], **DD, **mat;

double massecc[100000][2], massecc2[10000][2];

double *eigmag;

double *wr, *wi;//Real and Imaginary parts for Eigenvalues

eigmag=vector(1,4);

wr=vector(1,4);

wi=vector(1,4);

int alpha;

DD=(double **) malloc((unsigned) 4*sizeof(double*));

for(alpha=0;alpha<=4;alpha++) DD[alpha]=D[alpha];

FILE * fp1;

FILE * fp2;

FILE * fp3;

FILE * fp4;

FILE * fp5;

FILE * fp6;

FILE * fp7;

FILE * fp8;

FILE * fp9;





h=0.005; //RK4 step

limit=v0+2*3.1415;


x02=-1.0;

x03=0.5;

/*Newton-Raphson*/

xn1=x01;

if( fx(x01, m)!=0 && fdotx1(x01, m)!=0 ){

125

do{

x01=xn1;

xn1=x01 - ((fx(x01, m)/fdotx1(x01, m)));

}


}

xn2=x02;

if( fx(x02, m)!=0 && fdotx2(x02, m)!=0 ){

do{

x02=xn2;

xn2=x02 - ((fx(x02, m)/fdotx2(x02, m)));

}


}

xn3=x03;

if( fx(x03, m)!=0 && fdotx3(x03, m)!=0 ){

do{

x03=xn3;

xn3=x03 - ((fx(x03, m)/fdotx3(x03, m)));

}


}


x1=-m;

y1=0.0;

x2=1-m;

y2=0.0;

//L4/l5 Coordinates

xl4=0.5*(1-2*m);

yl4=sqrt(3)/2;

xl5=0.5*(1-2*m);

yl5=-sqrt(3)/2;


xl1=xn1;

yl1=0.0;

xl2=xn2;

yl2=0.0;

xl3=xn3;

yl3=0.0;

//Initial Coordinates

x0=xl5;

y0=yl5;

126


ux0=0.0;

uy0=0.01;

//Lagrange point to be tested for stability (LEAVE AS IS)

stabtestx=xl4;

stabtesty=yl4;

vn=v0;

xn=x0;

yn=y0;

uxn=ux0;

uyn=uy0;





fp5=fopen("L1L2L3Coordinates.txt","w+");


fp7=fopen("HessenbergD.txt", "w+");

fp8=fopen("e_mStableL4.txt", "w+");

fp9=fopen("e_m_StabReg.txt", "w+");

while(vn<=limit){







(uxn+c1/2.0), (uyn+d1/2.0), m, e);


(uxn+c1/2.0), (uyn+d1/2.0), m, e);


(uxn+c1/2.0), (uyn+d1/2.0), m, e);


(uxn+c1/2.0), (uyn+d1/2.0), m, e);


(uxn+c2/2.0), (uyn+d2/2.0), m, e);


(uxn+c2/2.0), (uyn+d2/2.0), m, e);


(uxn+c2/2.0), (uyn+d2/2.0), m, e);


(uxn+c2/2.0), (uyn+d2/2.0), m, e);


e);


e);


e);


e);

x=((a1+2.0*a2+2.0*a3+a4)/6.0);

y=((b1+2.0*b2+2.0*b3+b4)/6.0);

127

ux=((c1+2.0*c2+2.0*c3+c4)/6.0);

uy=((d1+2.0*d2+2.0*d3+d4)/6.0);

vn=vn+h;

xn=xn+x;

yn=yn+y;

uxn=uxn+ux;

uyn=uyn+uy;

fprintf(fp1, "%f\t%f\n", xn, yn);

}






yl3);

for(mass=0.0; mass<=0.05; mass=mass+0.0001){

for(ecc=0.0; ecc<=1.0; ecc=ecc+0.001){

//L4/l5 Coordinates

truexl4=0.5*(1-2*mass);

trueyl4=sqrt(3)/2;

truexl5=0.5*(1-2*mass);

trueyl5=-sqrt(3)/2;

//Lagrange point to be tested for stability (CHANGE 4 TO 5 FOR

L5)

stabtestx2=truexl4;

stabtesty2=trueyl4;

for(i=0; i<=3; i++){

Dxn[i]=Dx0[i];

Dyn[i]=Dy0[i];

Duxn[i]=Dux0[i];

Duyn[i]=Duy0[i];

while(vn<=limit){


k1=h*VdDxdv(vn, Dxn[i], Dyn[i], Duxn[i], Duyn[i],

stabtestx2, stabtesty2, mass, ecc);

l1=h*VdDydv(vn, Dxn[i], Dyn[i], Duxn[i], Duyn[i],


m1=h*VdDuxdv(vn, Dxn[i], Dyn[i], Duxn[i], Duyn[i],


n1=h*VdDuydv(vn, Dxn[i], Dyn[i], Duxn[i], Duyn[i],


k2=h*VdDxdv((vn+h/2.0), (Dxn[i]+k1/2.0), (Dyn[i]+l1/2.0),

(Duxn[i]+m1/2.0), (Duyn[i]+n1/2.0), stabtestx2, stabtesty2, mass,

ecc);

128

l2=h*VdDydv((vn+h/2.0), (Dxn[i]+k1/2.0), (Dyn[i]+l1/2.0),


ecc);

m2=h*VdDuxdv((vn+h/2.0), (Dxn[i]+k1/2.0), (Dyn[i]+l1/2.0),


ecc);

n2=h*VdDuydv((vn+h/2.0), (Dxn[i]+k1/2.0), (Dyn[i]+l1/2.0),


ecc);

k3=h*VdDxdv((vn+h/2.0), (Dxn[i]+k2/2.0), (Dyn[i]+l2/2.0),


ecc);

l3=h*VdDydv((vn+h/2.0), (Dxn[i]+k2/2.0), (Dyn[i]+l2/2.0),


ecc);

m3=h*VdDuxdv((vn+h/2.0), (Dxn[i]+k2/2.0), (Dyn[i]+l2/2.0),


ecc);

n3=h*VdDuydv((vn+h/2.0), (Dxn[i]+k2/2.0), (Dyn[i]+l2/2.0),


ecc);

k4=h*VdDxdv((vn+h), (Dxn[i]+k3), (Dyn[i]+l3), (Duxn[i]+m3),

(Duyn[i]+n3), stabtestx2, stabtesty2, mass, ecc);

l4=h*VdDydv((vn+h), (Dxn[i]+k3), (Dyn[i]+l3), (Duxn[i]+m3),

(Duyn[i]+n3), stabtestx2, stabtesty2, mass, ecc);

m4=h*VdDuxdv((vn+h), (Dxn[i]+k3), (Dyn[i]+l3),

(Duxn[i]+m3), (Duyn[i]+n3), stabtestx2, stabtesty2, mass, ecc);

n4=h*VdDuydv((vn+h), (Dxn[i]+k3), (Dyn[i]+l3),

(Duxn[i]+m3), (Duyn[i]+n3), stabtestx2, stabtesty2, mass, ecc);

Dx=((k1+2.0*k2+2.0*k3+k4)/6.0);

Dy=((l1+2.0*l2+2.0*l3+l4)/6.0);

Dux=((m1+2.0*m2+2.0*m3+m4)/6.0);

Duy=((n1+2.0*n2+2.0*n3+n4)/6.0);

vn=vn+h;

Dxn[i]=Dxn[i]+Dx;

Dyn[i]=Dyn[i]+Dy;

Duxn[i]=Duxn[i]+Dux;

Duyn[i]=Duyn[i]+Duy;

}

A[i][0]=Dxn[i];

A[i][1]=Dyn[i];

A[i][2]= Duxn[i];

A[i][3]=Duyn[i];


}

//Transpose matrix A to ontain matrix D

for(i=0; i<=3; i++){

for(j=0; j<=3; j++){

D[i][j]=A[j][i];

}

129

}

/* //Print matrix D

printf("D=\n");

for(i=0; i<=3; i++){

printf("%f\t%f\t%f\t%f\n", DD[i][0], DD[i][1], DD[i][2],

DD[i][3] );

fprintf(fp6, "%f\t%f\t%f\t%f\n", D[i][0], D[i][1], D[i][2],

D[i][3] );

}

printf("\n");*/

//Eigenvalues with QR method via Hessenberg Elimination

//Conversion of matrix to D to offset form

mat=convert_matrix(&DD[0][0],1,4,1,4);

//Hessenberg form of Matrix D

elmhes(mat,4);

/* //Print matrix m Hessenberg D

printf("Hessenberg D (elements below the subdiagonal are

considered 0)\n");

for(i=1; i<=4; i++){

printf("%f\t%f\t%f\t%f\n", mat[i][1], mat[i][2], mat[i][3],

mat[i][4]);

fprintf(fp7, "%f\t%f\t%f\t%f\n", mat[i][1], mat[i][2],

mat[i][3], mat[i][4] );

}

printf("\n");*/

//QR method for Eigenvalues

hqr(mat,4,wr,wi);

/* //printf("Eigenvalues\n");

for(i=1; i<=4; i++){

printf("l%i=%f+(%f)*i\n", i, wr[i], wi[i]);

}*/

//Eigenvalues Magnitude calculation

for(i=1; i<=4; i++){

eigmag[i]=sqrt(pow(wr[i], 2) + pow(wi[i], 2));

//printf("Magnitude of eigenvalue l%i=%f\n", i, eigmag[i]);

}

if(eigmag[1]>0.999999 && eigmag[1]<1.000001 &&

eigmag[2]>0.999999 && eigmag[2]<1.000001 && eigmag[3]>0.999999 &&

eigmag[3]<1.000001 && eigmag[4]>0.999999 &&

eigmag[4]<1.000001){//UPDATE CRITIRIA FOR STABILITY ACCORDING TO

p.29

fprintf(fp8,"%f\t%f\n", mass, ecc);

}

130

//Stability Region 1-7 Calculation

printf("-------------------------------------------------------

-------------------------\n");

printf("m=%f\te=%f\n", mass, ecc);

fprintf(fp9, "%f\t%f\t", mass, ecc);

for(i=1; i<=4; i++){

printf("l%i=%f+(%f)*i\t", i, wr[i], wi[i]);

printf("Mag l%i=%f\n", i, eigmag[i]);

}

if( wr[2]==wr[1] && wi[2]==-wi[1] && wr[4]==wr[3] &&

wi[4]==-wi[3] && eigmag[1]>0.999999 && eigmag[1]<1.000001 &&

eigmag[2]>0.999999 && eigmag[2]<1.000001 && eigmag[3]>0.999999 &&

eigmag[3]<1.000001 && eigmag[4]>0.999999 && eigmag[4]<1.000001 ){

fprintf(fp9, "%i\n", 1);

printf("Reg.1\n");

}

else if( wr[3]==wr[1] && wi[3]==-wi[1] && wr[4]==wr[2] &&

wi[4]==-wi[2] ){


printf("Reg.2\n");

}

else if( wr[2]==wr[1] && wi[2]==-wi[1] && wr[4]==wr[3] &&

wi[4]==-wi[3] ){


printf("Reg.2\n");

}

else if( wi[1]==0.0 && wi[2]==0.0 && wi[3]==0.0 &&

wi[4]==0.0 && wr[1]*wr[3]<0.0 ){


printf("Reg.3\n");

}

else if( wi[1]==0.0 && wi[2]==0.0 && wi[3]==0.0 &&

wi[4]==0.0 && wr[1]>0.0 && wr[3]>0.0 ){


printf("Reg.4\n");

}

else if( wi[1]==0.0 && wi[2]==0.0 && wi[3]==0.0 &&

wi[4]==0.0 && wr[1]<0.0 && wr[3]<0.0 ){


printf("Reg.5\n");

}

else if( wi[3]==0.0 && wi[4]==0.0 && wr[3]>0.0 && wr[4]>0.0

&& wr[2]==wr[1] && wi[2]==-wi[1] ){


printf("Reg.6\n");

}

else if( wi[1]==0.0 && wi[2]==0.0 && wr[1]>0.0 && wr[2]>0.0

&& wr[4]==wr[3] && wi[4]==-wi[3] ){


printf("Reg.6\n");

}

else if( wi[1]==0.0 && wi[4]==0.0 && wr[1]>0.0 && wr[4]>0.0

&& wr[3]==wr[2] && wi[3]==-wi[2] ){


printf("Reg.6\n");

}

else if( wr[4]==wr[3] && wi[4]==-wi[3] && wi[1]==0.0 &&

wi[2]==0.0 && wr[1]<0.0 && wr[2]<0.0 ){

131


printf("Reg.7\n");

}


wi[4]==0.0 && wr[3]<0.0 && wr[4]<0.0 ){


printf("Reg.7\n");

}


wi[4]==0.0 && wr[1]<0.0 && wr[4]<0.0 ){


printf("Reg.7\n");

}

printf("\n");

}

}

fclose(fp1);

fclose(fp2);

fclose(fp3);

fclose(fp4);

fclose(fp5);

fclose(fp6);

fclose(fp7);

fclose(fp8);

fclose(fp9);

return 0;

}

Program D:

//CALCULATION OF THE REGULAR REGIONS NEAR L4 OR L5 VIA THE FLI

METHOD

#include <stdio.h>

#include <stdlib.h>

#include <math.h>

#include <stdint.h>

#include <omp.h>

#define bool char

#define true 1

#define false 0

//Grid of Initial Conditions Around the Lagrange Point to be tested

#define UPPER_BOUNDARY_X 0.9

#define LOWER_BOUNDARY_X 0.9

#define UPPER_BOUNDARY_Y 0.4

#define LOWER_BOUNDARY_Y 0.4

#define AREA_STEP_X 0.0025

#define AREA_STEP_Y 0.0025

//Number of threads to be used by the computer(USE EVEN NUMBER, IF

THREADS = 8 CHANGE THE BOUNDARIES OF X(above) TO BE DIVISIBLE BY 8)

#define NUMBER_OF_THREADS 6

132

#define min(a,b) a >= b ? b : a




double dx;

dx=ux;

return dx;

}



double dy;

dy=uy;

return dy;

}





m1=1-m;

m2=m;

r1=sqrt(pow((x+m), 2.0) + pow(y, 2.0));

r2=sqrt(pow((x+m-1), 2.0) + pow(y, 2.0));


1)/pow(r2, 3.0)));

return dux;

}





m1=1-m;

m2=m;

r1=sqrt(pow((x+m), 2.0) + pow(y, 2.0));

r2=sqrt(pow((x+m-1), 2.0) + pow(y, 2.0));


3.0)));

return duy;

}




double dDx;

dDx=Dux;

return dDx;

133

}



double dDy;

dDy=Duy;

return dDy;

}








2), 3/2.0) );




return dDux;

}








2), 3/2.0) );




return dDuy;

}







return func;

}



double func1;


3) - (3.0*m*(x+m-1))/pow(fabs(x+m-1.0), 4) - (3.0*(1.0-


134

return func1;

}



double func2;


3) + (3.0*m*(x+m-1.0))/pow(fabs(x+m-1.0), 4) + (3.0*(1.0-


return func2;

}



double func3;


3) + (3.0*m*(x+m-1.0))/pow(fabs(x+m-1.0), 4) - (3.0*(1.0-


return func3;

}



int i1=0,i2=1,i3=2,i4=3;

double v, v0, x, y, ux, uy, x0, y0, ux0, uy0, h;


double Dx, Dy, Dux, Duy, Dx0, Dy0, Dux0, Duy0;//for Variational

ODEs

double Dxn, Dyn, Duxn, Duyn;//for Variational ODEs

double stabtestx, stabtesty;


double xl1, yl1, xl2, yl2, xl3, yl3, xl4, yl4, xl5, yl5;//for

Lagrange points



d3, d4;


n3, n4;


double FLI;

double OmegaA, OmegaB, Theta;

double distance0, distancePr1, distancePr2;

bool FLIcheck;

int firstNormalAreaPoint_k = -1 , firstNormalAreaPoint_i = -1;

bool firstNormalAreaPoint = true;

FILE * fp1;

FILE * fp2;

FILE * fp3;

FILE * fp4;

FILE * fp5;

FILE * fp6;

FILE * fp7;

FILE * fp8;

135

FILE * fp9;

int thread_ID;

int number_Of_Threads;

int RECORDS;

double threadStart;

double threadEnd;

double ***res_mat, *temp;

int *res_records; //for each thread

double progress = 0.0;





h=0.005; //RK4 step

limit=v0+650*3.14159265;//Interval of Integration


x02=-1.0;

x03=0.5;

/*Newton-Raphson*/

xn1=x01;

if( fx(x01, m)!=0 && fdotx1(x01, m)!=0 ){

do{

x01=xn1;

xn1=x01 - ((fx(x01, m)/fdotx1(x01, m)));

}


}

xn2=x02;

if( fx(x02, m)!=0 && fdotx2(x02, m)!=0 ){

do{

x02=xn2;

xn2=x02 - ((fx(x02, m)/fdotx2(x02, m)));

}


}

136

xn3=x03;

if( fx(x03, m)!=0 && fdotx3(x03, m)!=0 ){

do{

x03=xn3;

xn3=x03 - ((fx(x03, m)/fdotx3(x03, m)));

}


}



printf("xl3=%f\n\n",xn3);


x1=-m;

y1=0.0;

x2=1-m;

y2=0.0;

//L4/l5 Coordinates

xl4=0.5*(1-2*m);

yl4=sqrt(3)/2;

xl5=0.5*(1-2*m);

yl5=-sqrt(3)/2;


xl1=xn1;

yl1=0.0;

xl2=xn2;

yl2=0.0;

xl3=xn3;

yl3=0.0;





fp5=fopen("L1L2L3Coordinates.txt", "w+");


fp7=fopen("InitialConditionsFLI.txt", "w+");

fp8=fopen("FLI.txt", "w+");

fp9=fopen("x0y0FLIAll.txt", "w+");





yl3);

RECORDS = ( ( UPPER_BOUNDARY_X + LOWER_BOUNDARY_X ) /

AREA_STEP_X ) * ( ( UPPER_BOUNDARY_Y + LOWER_BOUNDARY_Y ) /

AREA_STEP_Y );

number_Of_Threads = NUMBER_OF_THREADS;

137

omp_set_num_threads( number_Of_Threads );

res_mat = (double***)malloc( number_Of_Threads *

sizeof(double**) );

res_records = ( int* )malloc( number_Of_Threads * sizeof(

int ) );

for( i = 0; i < number_Of_Threads; ++i )

{

res_mat[ i ] = (double**)malloc( 3 * sizeof(double*) );

// 3: for x0,y0,FLI

res_records[ i ] = 0;

for( j = 0; j < 3; ++j )

{

res_mat[ i ][ j ] = (double*)malloc( RECORDS *

sizeof(double) );

}

}

#pragma omp parallel

private(x0,y0,ux0,uy0,Dx0,Dy0,Dux0,Duy0,Dxn,Dyn,Duxn,Duyn,vn,xn,yn,

uxn,uyn,x,y,ux,uy,Dx,Dy,Dux,Duy,\

a1,b1,c1,d1,a2,b2,c2,d2,a3,b3,c3,d3,a4,b4,c4,d4,k1,l1,m1,n1,k2,l2,m

2,n2,k3,l3,m3,n3,k4,l4,m4,n4,\

FLI,thread_ID,threadStart,threadEnd,temp, FLIcheck, distance0,

distancePr1, distancePr2) \

firstprivate(

limit,h,m,e,v0,number_Of_Threads ) \

shared( res_mat,res_records,progress )

{

thread_ID = omp_get_thread_num();

number_Of_Threads = omp_get_num_threads();

threadStart = ( thread_ID*( ( UPPER_BOUNDARY_X +

LOWER_BOUNDARY_X ) / number_Of_Threads ) ) + xl4-LOWER_BOUNDARY_X;

//Change xl1-5 here for different Lagrange point

threadEnd = ( ( (thread_ID+1)*( ( UPPER_BOUNDARY_X +

LOWER_BOUNDARY_X ) / number_Of_Threads ) ) + xl4-LOWER_BOUNDARY_X

);

threadEnd = ( thread_ID == number_Of_Threads - 1 ?

threadEnd + AREA_STEP_X : threadEnd ); // last thread should

include upper limit too.

for(x0= threadStart; x0 < threadEnd;

x0=x0+AREA_STEP_X){

for(y0=yl4-LOWER_BOUNDARY_Y;

y0<=yl4+UPPER_BOUNDARY_Y; y0=y0+AREA_STEP_Y){ //Change yl1-5 here

for different Lagrange point

138


ux0=0.0;

uy0=0.0;

//Initial Variational Coordinates

Dx0=100.0;

Dy0=100.0;

Dux0=0.0;

Duy0=0.0;

Dxn=Dx0;

Dyn=Dy0;

Duxn=Dux0;

Duyn=Duy0;

vn=v0;

xn=x0;

yn=y0;

uxn=ux0;

uyn=uy0;

FLIcheck = true;

while(vn<=limit){






a2=h*dxdv((vn+h/2.0), (xn+a1/2.0),

(yn+b1/2.0), (uxn+c1/2.0), (uyn+d1/2.0), m, e);

b2=h*dydv((vn+h/2.0), (xn+a1/2.0),

(yn+b1/2.0), (uxn+c1/2.0), (uyn+d1/2.0), m, e);

c2=h*duxdv((vn+h/2.0), (xn+a1/2.0),

(yn+b1/2.0), (uxn+c1/2.0), (uyn+d1/2.0), m, e);

d2=h*duydv((vn+h/2.0), (xn+a1/2.0),

(yn+b1/2.0), (uxn+c1/2.0), (uyn+d1/2.0), m, e);

a3=h*dxdv((vn+h/2.0), (xn+a2/2.0),

(yn+b2/2.0), (uxn+c2/2.0), (uyn+d2/2.0), m, e);

b3=h*dydv((vn+h/2.0), (xn+a2/2.0),

(yn+b2/2.0), (uxn+c2/2.0), (uyn+d2/2.0), m, e);

c3=h*duxdv((vn+h/2.0), (xn+a2/2.0),

(yn+b2/2.0), (uxn+c2/2.0), (uyn+d2/2.0), m, e);

d3=h*duydv((vn+h/2.0), (xn+a2/2.0),

(yn+b2/2.0), (uxn+c2/2.0), (uyn+d2/2.0), m, e);

a4=h*dxdv((vn+h), (xn+a3), (yn+b3),

(uxn+c3), (uyn+d3), m, e);

b4=h*dydv((vn+h), (xn+a3), (yn+b3),

(uxn+c3), (uyn+d3), m, e);

c4=h*duxdv((vn+h), (xn+a3), (yn+b3),

(uxn+c3), (uyn+d3), m, e);

d4=h*duydv((vn+h), (xn+a3), (yn+b3),

(uxn+c3), (uyn+d3), m, e);

x=((a1+2.0*a2+2.0*a3+a4)/6.0);

139

y=((b1+2.0*b2+2.0*b3+b4)/6.0);

ux=((c1+2.0*c2+2.0*c3+c4)/6.0);

uy=((d1+2.0*d2+2.0*d3+d4)/6.0);

xn=xn+x;

yn=yn+y;

uxn=uxn+ux;

uyn=uyn+uy;


k1=h*VdDxdv(vn, Dxn, Dyn, Duxn, Duyn, xn,

yn, m, e);

l1=h*VdDydv(vn, Dxn, Dyn, Duxn, Duyn, xn,

yn, m, e);

m1=h*VdDuxdv(vn, Dxn, Dyn, Duxn, Duyn, xn,

yn, m, e);

n1=h*VdDuydv(vn, Dxn, Dyn, Duxn, Duyn, xn,

yn, m, e);

k2=h*VdDxdv((vn+h/2.0), (Dxn+k1/2.0),

(Dyn+l1/2.0), (Duxn+m1/2.0), (Duyn+n1/2.0), xn, yn, m, e);

l2=h*VdDydv((vn+h/2.0), (Dxn+k1/2.0),


m2=h*VdDuxdv((vn+h/2.0), (Dxn+k1/2.0),


n2=h*VdDuydv((vn+h/2.0), (Dxn+k1/2.0),


k3=h*VdDxdv((vn+h/2.0), (Dxn+k2/2.0),


l3=h*VdDydv((vn+h/2.0), (Dxn+k2/2.0),


m3=h*VdDuxdv((vn+h/2.0), (Dxn+k2/2.0),


n3=h*VdDuydv((vn+h/2.0), (Dxn+k2/2.0),


k4=h*VdDxdv((vn+h), (Dxn+k3), (Dyn+l3),

(Duxn+m3), (Duyn+n3), xn, yn, m, e);

l4=h*VdDydv((vn+h), (Dxn+k3), (Dyn+l3),


m4=h*VdDuxdv((vn+h), (Dxn+k3), (Dyn+l3),


n4=h*VdDuydv((vn+h), (Dxn+k3), (Dyn+l3),


Dx=((k1+2.0*k2+2.0*k3+k4)/6.0);

Dy=((l1+2.0*l2+2.0*l3+l4)/6.0);

Dux=((m1+2.0*m2+2.0*m3+m4)/6.0);

Duy=((n1+2.0*n2+2.0*n3+n4)/6.0);

vn=vn+h;

Dxn=Dxn+Dx;

Dyn=Dyn+Dy;

Duxn=Duxn+Dux;

Duyn=Duyn+Duy;

140

//manually set FLI=100 if the satellite's

trajectory is not near lagrange point based on its distance from

Lagrange point, or if there is a collision with the primaries

distance0 = sqrt( pow(xn - xl4 , 2.0) +

pow(yn - yl4, 2.0) ); //Change xl1-5, yl1-5 here for different

Lagrange point

distancePr1 = sqrt( pow((xn - x1), 2.0) +

pow((yn - y1), 2.0) );

distancePr2 = sqrt( pow((xn - x2), 2.0) +

pow((yn - y2), 2.0) );

if ( distance0 > 1.0 || distancePr1 < 0.001

|| distancePr2 < 0.001 ){

FLI = 100;

FLIcheck = false;

break;

}

}

//FLI Calculation

if (FLIcheck){

FLI=log((sqrt(pow(Dxn,

2.0)+pow(Dyn, 2.0) + Duxn + Duyn))/vn);

}

//fprintf(fp9, "%f\t%f\t%f\n", x0, y0,

FLI);

res_mat[ thread_ID ][ 0 ][ res_records[

thread_ID ] ] = x0;


thread_ID ] ] = y0;


thread_ID ] ] = FLI;

++(res_records[ thread_ID ]);

}

printf("\rProgress\t[");

for(i = 0; i < 100; i+=5 )

{

if( ( progress / ( ( UPPER_BOUNDARY_X +

LOWER_BOUNDARY_X ) ) ) * 100 > i )

printf("|");

else

printf("-");

// printf("Progress: %.2f%%\n",(

progress / ( ( UPPER_BOUNDARY + LOWER_BOUNDARY ) ) ) * 100 );

}

141

printf("]\t%.2f%%\r", min( progress / ( (

UPPER_BOUNDARY_X + LOWER_BOUNDARY_X ) ) * 100, 100 ) );

#pragma omp critical

progress += AREA_STEP_X;

}

}

for( i = 0; i < number_Of_Threads; ++i )

{

for( k = 0; k < res_records[ i ]; ++k )

{

//Results of Initial Conditions and FLI for the

whole grid

fprintf(fp9, "%f\t%f\t%f\n", res_mat[ i ][

0/*x0*/ ][ k ], res_mat[ i ][ 1/*y0*/ ][ k ], res_mat[ i ][

2/*FLI*/ ][ k ] );

//If FLI < 6.0 write initial conditions and FLI

in seperate files

if( res_mat[ i ][ 2/*FLI*/ ][ k ] < 6.0 )

{

if( firstNormalAreaPoint )

{

firstNormalAreaPoint = false;

firstNormalAreaPoint_i = i;

firstNormalAreaPoint_k = k;

}

fprintf(fp7, "%f\t%f\n", res_mat[ i ][

0/*x0*/ ][ k ], res_mat[ i ][ 1/*y0*/ ][ k ]);

fprintf(fp8, "%f\n", res_mat[ i ][ 2/*FLI*/

][ k ]);

OmegaB =

atan(res_mat[i][1/*y0*/][k]/res_mat[i][0/*x0*/][k]);

}

}

}

if( !firstNormalAreaPoint )

{

if(res_mat[ firstNormalAreaPoint_i ][0][

firstNormalAreaPoint_k ]>0.0) {

OmegaA = atan(res_mat[ firstNormalAreaPoint_i ][1][

firstNormalAreaPoint_k ]/res_mat[ firstNormalAreaPoint_i ][0][

firstNormalAreaPoint_k ]);

}

else if(res_mat[ firstNormalAreaPoint_i ][0][

firstNormalAreaPoint_k ]<0.0){

OmegaA = 3.14159265 - atan(res_mat[

firstNormalAreaPoint_i ][1][ firstNormalAreaPoint_k ]/(-res_mat[

firstNormalAreaPoint_i ][0][ firstNormalAreaPoint_k ]));

}

}

142

//Range of the Regular Region Calculation(CAUTION: THE

PROGRAM USES THE FIRST AND LAST POINTS OF THE REGION WITHOUT TAKING

INTO ACCOUNT FLI VALUES THAT ARE CLOSE TO 6.0)

Theta = OmegaA-OmegaB;

printf("\nmu=%f\te=%f\tTheta = %f rad\n", m, e, Theta);

fclose(fp1);

fclose(fp2);

fclose(fp3);

fclose(fp4);

fclose(fp5);

fclose(fp6);

fclose(fp7);

fclose(fp8);

return 0;

}

143

References

Broucke, R. A. (1967). On Changes of Independent Variable in

Dynamical Systems and Applications to Regularization. Icarus,

Vol. 7, No. 2, pp. 221-231

Broucke, R. A. (1969). Periodic Orbits in the Elliptic Restricted

Three-Body Problem. Pasadena, California, Jet Propulsion

Laboratory, California Institute of Technology.

Hadjidemetriou, I. (2000). Theoretical Mechanics: Analytic

Dynamics, Special Theory of Relativity Vol. 2. Thessaloniki,

Giahoudi Publications.

Schwarz, R., Dvorak, R., Süli, Á., & Érdi, B. (2007). Survey of

the stability region of hypothetical habitable Trojan planets.

DOI: 10.1051/0004-6361:20077994

Szebehely, V. (1967). Theory of Orbits: The Restricted Problem

of Three Bodies. New York and London, Academic Press.

Voyatzis, G., & Meletlidou, E. (2015). Introduction to non-linear

dynamical systems. Athens, Kallipos Publications.

Exoplanet.eu, The Extrasolar Planets Encyclopedia (updated:

Oct. 11, 2017). Diagrams retrieved October 6, 2017, from

http://exoplanet.eu/

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