Periodic Orbits of the First Kind in the Autonomous Four-body Problem with the Case of Collision ()
1. Introduction
We know that the four most popular methods of proving the existence of periodic orbits are:
(i) the method of analytic continuation,
(ii) the process of equating Fourier coefficients of equal frequencies,
(iii) the application of fixed point theorem given by Poincare,
(iv) the method of power series.
Giacaglia [1] used the method of analytic continuation to examine the existence of periodic orbits of collision in the Restricted Three-body Problem (R3BP). Bhatnagar [2] generalized the problem in elliptic case. The problem of Giacaglia [1] was further extended by Bhatnagar [3] in the R4BP by taking the primaries at the vertices of an equilateral triangle. With different perturbations like oblateness, triaxiality, photogravitation, Pointing-Robertson drag effects of the primaries, the existence of periodic orbits of collision in the R3BP and in the R4BP, have been studied by different authors in two and three-dimensional co-ordinate system during the period of last three decades of the 20th century but nobody established the proper mathematical model of the R4BP. Recently Ceccaroni and Biggs [8] has studied the autonomous coplanar CR4BP by taking the third primary of comparatively inferior mass at the triangular equilibrium point
of R3BP and with an extension to low-thrust propulsion for application to the future science mission.
In present paper, we have proposed to study the existence of periodic orbits of first kind in the Autonomous Four-body Problem by the method of analytic continuation. By using Poincare surfaces of section (PSS), the conditions for the existence of periodic orbits given by Duboshin [4] have been confirmed. For collision case, we have applied the criterion given by Levi-Civitas [6] [7] and it is satisfied by our model.
2. Equations of Motion
Let
be the three massive bodies of masses
respectively, where
and the fourth body of mass
be at
. These bodies are moving in the same plane under some restrictions as follows:
The fourth body at
of mass
is assumed to be of infinitesimal mass not influencing the motion of
but motions of
is being influenced by the motions of
. Further, we have assumed that the mass
at
is taken small enough, so that it can’t influence the motion of the dominating primaries
and
and it is placed at any one of the triangular libration points (Lagrangian Points) of the classical restricted three body problem. Since the third primary can’t influence the motions of
and
, so the centre of rotation of the system remains at the barycentre of two main primaries
and
. Also, it is supposed, all the primaries are moving in the same plane in circular orbits around the bary-centre of massive primaries
and
with the same angular velocity
and the fourth body
is moving under the gravitational field and plane of motion of three primaries
then to check the nature of motion of infinitesimal mass
.
Let the line joining
and
be taken as the x-axis and their mass centre (bary-centre) O, as the origin. Let the line through O and perpendicular to
lying in the plane of motion of the primaries be taken as the y-axis. Let the positions of masses
be
and
respectively. Let
be the position vector of
and
be the displace-
![]()
Figure 1. Configuration of four-body problem.
ments of
and
relative to
as shown in Figure 1, then
(1)
Let
be the gravitational forces exerted on
by the primaries respectively, then
(2)
where
is the gravitational constant.
The total gravitational force acting on
by the three primaries is given by
(3)
Let
be the magnitude of angular velocity
and
be the unit vector normal to the plane of motion of the primaries, then
.
The Equation of motion of the infinitesimal mass
in synodic frame is
(4)
Since the synodic frame are revolving with constant angular velocity
about the bary-centre, hence
and thus Equation (4) reduces to
(5)
In cartesian form, the equations of motion of the infinitesimal mass
in the gravitational field of three primaries, are given by
(6)
Also the linear velocity of the infinitesimal mass
on its orbit; is given by
(7)
If
are two components of
, then from Equation (7),
(8)
If mass of the infinitesimal body is supposed to be unity, then the kinetic energy of the infinitesimal mass is given by
(9)
Let
be the momenta corresponding to the co-ordinates
respectively, then
(10)
Combination of Equations ((9) and (10)) yields
(11)
The gravitational potential of the body of mass
at any point of
outside it, is given by
![]()
then, total gravitational potential at
due to three primaries is given by
(12)
The Hamiltonian of the infinitesimal body of unit mass is given by
![]()
(13)
Let
be the reduced mass of the second primary and
be the reduced mass of the third primary, then from the definition of reduced mass, we have
![]()
then ![]()
The coordinates of
are given by
![]()
Clearly
, which implies that
forms an equilateral triangle of sides of unit length. We know that
is very small in comparison of masses of the other two primaries, so we can choose
as the order of
i.e.,
. Now choosing unit of time in such a manner that
and
and taking
, then the Hamilton canonical equations of motion of the infinitesimal body
are given by
(14)
where
(15)
is the reduced Hamiltonian corresponding to canonically conjugate variables
and
.
3. Regularization at the Singularity ![]()
In our Hamiltonian
given in Equation (15), there are three singularities
. To examine the existence of periodic orbits of collision with the first primary, we have to eliminate the singularity
. For this, let us define an extended generating function S given by
, (16)
with
(17)
where
is the momenta associated with new co-ordinate
.
Clearly,
(18)
(19)
Also,
(20)
Thus the Hamiltonian
given in Equation (15), can be written in terms of new variables
, as
(21)
Let us introduce pseudo time
by the differential equation
(22)
Thus the regularized Hamilton-canonical equations of motion of the infinitesimal body corresponding to the Hamiltonian
, are given by
(23)
where the regularized Hamiltonian
is given by
![]()
(24)
Let us write
, then
(25)
(26)
4. Generating Solution (i.e., Solutions When
)
For generating solutions, we shall choose
for our Hamiltonian function, so in order to solve the Hamilton-Jacobi equation associated with
, let us write
(27)
where
is an arbitrary constant.
Since
is not involved explicitly in
: hence by using Equation (27) in Equation (25), the Hamilton-Jacobi equation may be written as
(28)
Putting
, then the Equation (27) becomes
(29)
It may be noted that this differential equation is exactly the same as in Giacaglia [1] and Bhatnagar [2] [3] and therefore the solution of Equation (29) can be written by the method of separation of variables, as
(30)
where
is an arbitrary constant.
Let us introduce a new quantity
by
then from Equation (30), we get
(31)
Combination of Equations (29) and (30) yields
![]()
(32)
where
(33)
(34)
where
is the smaller root of the roots of the equation
.
From Equation (33), we conclude that for general solution; we need only two arbitrary constants as
and
. Therefore the solution of Equation (30) may be regarded as a general solution.
Let us introduce the parameters
by the relations
(35)
where
is the semi-major axis,
is the eccentricity and
is the latus-rec- tum of the elliptic orbit of the infinitesimal body.
It may be noted that for
and
is the other root of the equation
.
We introduce a parameter
by the relation
(36)
From Equations (33), (35) and (36), we get
![]()
(37)
Again from Equation (25)
![]()
Thus the equations of motion associated with
are given as
![]()
(38)
where
denotes the differentiation with respect to
.
Now from
we get
.
Also
implies ![]()
and
[Using Equation (38)]
Thus from the above relations, we have
(39)
From Equation (32), we get
![]()
(40)
From Equation (30),
![]()
where ![]()
[Using Equation (34)]
![]()
where ![]()
If we take
and
as arbitrary constants, the solutions may be written as
(41)
From the second equation of system (41), we get the argument as
(42)
Since
(43)
hence for the problem generated by Hamiltonian
(regularized two-body problem in rotating co-ordinate system), we have
(44)
The variables
can now be expressed in terms of the canonical elements for
, as
(45)
where
is given by the first equation of system (42).
When
and
, then
(46)
where
is given by the second equation of system (42).
The original synodic cartesian co-ordinates in a non-uniformly rotating system are obtained from Equations (18) and (20), when
, as
(47)
The sidereal cartesian co-ordinates are obtained by considering the transformations
(48)
where
is given by
,
![]()
where
is a constant.
In terms of canonical variables introduced, the complete Hamiltonian may be written as
, where
can be obtained from Equation (26) after changing into canonical variables.
The equations of motion for the complete Hamiltonian are
(49)
Equation (49) forms the basis of a general perturbation theory for the present problem. The solution described by Equations ((44) and (45)) and is periodic if
and g have commensurable frequencies, i.e., if
![]()
where
and
are integers.
The periods of
are
and
respectively, so that in case of commensurability, the period of the solution is
or
.
5. Existence of Periodic Orbits When ![]()
Here we shall follow the method given by Chaudhary [9] to prove the existence of periodic orbits. Let
then from Equation (44), when
, we have
![]()
Integrating these equations with respect to
, we get
(50)
These are the generating solutions of two-body problems. The generating solution will be periodic with the period
, if
(51)
when
are integers, so that
are commensurable.
Following Poincare [5] , the general solution in the neighbourhood of the generating solution, may be given as
(52)
where
is the new independent variable given by
![]()
The necessary and sufficient conditions for the existence of periodic solution are
(53)
Restricting our solution only up to the first order infinitesimals, the equations of motion may be written as
(54)
(55)
where
![]()
![]()
Expanding
in ascending powers of
, Equation (54) may be written as
![]()
Rejecting the second order term
, integrating and putting the value of
in Equation (51), we get
(56)
and ![]()
The Equation (55) gives
![]()
![]()
Equation (45) gives
(57)
By solving the Equations (54)-(57), we can find the values of
, as analytic function of
, reducing to zero with
, if the conditions for periodic orbits given by Duboshin [4] are satisfied i.e.,
(i)
, and (58)
(ii)
, together (59)
(iii)
, (60)
where
is the zero degree terms of
given in Equation (26).
Now,
![]()
From Equation (43),
![]()
then
(61)
From the Equation (26)
![]()
where
![]()
![]()
,
Thus
(62)
Taking only zero order terms i.e., for ![]()
(63)
where
.
Now from equations of system (52)
(64)
and from Equation (63)
![]()
where
and ![]()
Here
if either
or
and
if either
or
.
But
and
don’t imply each other, so
is only the case for which
and
will be simultaneously zero.
Now choosing suitably
, then
(say)
and
![]()
![]()
Thus,
(65)
Now,
![]()
![]()
As
so from Equation (63), we have
![]()
Thus,
![]()
Using Equation (65), we get
![]()
![]()
Thus the conditions for the existence of periodic orbits given by Duboshin [4] are satisfied i.e., in the region of motion of the infinitesimal body, periodic orbits exist.
6. Poincare Surfaces of Section (PSS)
In this previous section, we have shown that Duboshin’s condition [4] for the existence of periodic orbits when
, are satisfied. So to justify the mathematical model given in Equations (58)-(60), we have applied the method of Poincare surfaces of section (PSS) to the reduced equations of motion
(66)
together with the Jacobi Integral
(67)
To study the motion of the infinitesimal body by PSS, it is necessary to know its position
and velocity
which correspond to a point in four- dimensional phase space. By defining a plane
, in the resulting three- dimensional space, the values of
and
can be plotted. Every time the particle has
, whenever the trajectory intersects the plane in a particular direction say
.
The techniques of PSS suggest to determine the regular or chaotic nature of the trajectories. If there are smooth, well-defined island then the trajectory is likely to be regular and the islands correspond to oscillation around a periodic orbit. As the curves shrink down to a point, the points represent a periodic orbit as per Kolmogorov-Arnold-Moser (KAM) theory. Any fuzzy distribution of points in surfaces of section, implies that trajectory is chaotic. In Figure 2, for
and
Poincare surfaces of section have been plotted in which atleast seven points are visible towards which the regular trajectories shrink, hence by KAM theory, periodic orbits exists. Again Figure 3 represents a Poincare surfaces of section for
and
in which atleast nine points are visible towards which the regular trajectories shrink, so we can say that the periodic orbits exist in the region of motion of infinitesimal mass. Other than the neighbourhood of these points, the quasi-periodic and chaotic regions are seen in the PSS. In Figure 4, in PSS for
and
, atleast ten shrinking regions of regular curves to a point are visible, i.e., that the degree of existence of periodic orbits increases in the region of motion of the infinitesimal mass. Thus by increasing the values of the Jacobi’s constant, the chances of existence of periodic orbits increase. Thus the Duboshin conditions and PSS both confirms the existence of periodic orbits when
. In Figure 5, regions plot of ZVC (Zero Velocity Curves) for
is shown, in which central white circle represents regions of no motion and coloured annulus represents the regions of periodic orbits. Figure 6 depicts the contour plot of ZVC for
.
7. Periodic Orbits of Collision When ![]()
Levi-Civita [6] [7] proved that the invariant relation for collision orbits can be analytically continued from the one that corresponds to the problem of two bo-
![]()
Figure 2. Poincare Surface of Section for
.
![]()
Figure 3. Poincare Surface of Section for
.
![]()
Figure 4. Poincare Surface of Section for
.
dies. Bhatnagar [2] [3] has developed this as
. For the present paper when
, the condition must be
(68)
for sufficiently small
and
.
Further, he has proved that, in particular, such relation is uniform integral of the differential equation of motion along any collision orbit. He has also proved this integral is a power series in terms of the distance from the origin and the series is convergent through the radius of convergence is generally small. In section (5), we have shown that periodicity is conserved by analytic continuation. Let us show that the condition of collision is also conserved by analytic continuation.
Figure 7 shows the geometrical configuration of collision orbits. In order to show the validity of that continuation, we shall consider orbits corresponding to the case when
. When
, the orbits starts as an ejection
from the origin and return to it after
. Bhatnagar [2] [3] and Levi-Civita [6] [7] finds the condition for collision as
(69)
where ![]()
![]()
![]()
Figure 7. Geometrical configuration of collision orbits.
Therefore, the condition of Equation (69) became,
![]()
(70)
But,
![]()
![]()
(71)
Thus from Equations ((70) and (71))
![]()
![]()
(72)
Here the Equation (71) corresponds to the Equation (68), so it is easy to say that the collision orbits exist.
8. Discussions and Conclusion
In section 1 of this paper, historical background has been sketched with original and previous contributions. In section 2, the equations of motion of the infinitesimal mass moving under the gravitational field of the three primaries situated at the vertices of an equilateral triangle taken by Ceccaroni and Biggs [8] . In this the reduced Hamiltonian
has been derived for regularization in the next section 3. In this section, the regularized Hamiltonian
has been established. In section 4, generating solutions have been found by taking
as the corresponding Hamiltonian. In this section generating solution forms a basis for general solution by the process of analytic continuation. In section 5, using Duboshin’s criterion [4] for the existence of periodic orbits has been satisfied following the method of Choudhary [9] . For confirmation of the existence of periodic orbits in section 4, we have analyzed PSS in section 6 and justified that the region of motion regular trajectory shrinking towards a point represents the periodic orbits and other region of the PSS represents quasi-periodic and chaotic belt in the region of motion. In section 7, the periodic orbits of collision for
have been shown. In our discussion, we have shown that our condition of collision orbit
has a resemblance with the condition given by Bhatnagar [2] [3] .
Definitions
Bary-Centre: It is the center of mass of two or more bodies that are orbiting each other, or the point around which they both orbit.
Synodic Co-ordinate System: The co-ordinate system, in which the xy-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, is called synodic co-ordinate system.
Reduced Mass: Mass ratio of the smaller primary to the total mass of the primaries or the non-dimensional mass of the smaller primary is known as reduced mass of the smaller primary.
Regularization: The process of elimination of the singularity from the force function is known as regularization.
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