This paper investigates the motion of a test particle around the equilibrium points under the setup of the Robe’s circular restricted threebody problem in which the masses of the three bodies vary arbitrarily with time at the same rate. The first primary is assumed to be a fluid in the shape of a sphere whose density also varies with time. The nonautonomous equations are derived and transformed to the autonomized form. Two collinear equilibrium points exist, with one positioned at the center of the fluid while the other exists for the mass ratio and density parameter provided the density parameter assumes value greater than one. Further, circular equilibrium points exist and pairs of outofplane equilibrium points forming triangles with the centers of the primaries are found. The outofplane points depend on the arbitrary constant
The classical restricted threebody problem (RTBP) constitutes one of the most important problems in dynamical astronomy. The study of this problem is of great theoretical, practical, historical, and educational relevance. The investigation of this problem in its several versions has been the focus of continuous and intense research activity for more than two hundred years. The study of this problem in its many variants has had important implications in several scientific fields including, among others, celestial mechanics, galactic dynamics, chaos theory, and molecular physics. The RTBP is still a stimulating and active research field that has been receiving considerable attention of scientists and astronomers because of its applications in dynamics of the solar and stellar systems, lunar theory, and artificial satellites.
A different kind of restricted threebody problem was formulated by Robe [
In estimating buoyancy force, Robe [
A. R. Plastino and A. Plastino [
The classical restricted threebody problem assumes that the masses of celestial bodies are constant. However, the phenomenon of isotropic radiation or absorption in stars led scientists to formulate the restricted problem of three bodies with variable mass. As an example, we could mention the motion of rockets, black holes formation, motion of a satellite around a radiating star surrounded by a cloud and varying its mass due to particles of the cloud, and comets loosing part or all of their mass as a result of roaming around the Sun (or other stars) due to their interaction with the solar wind which blows off particles from their surfaces. The problem of the motion of astronomical objects with variable mass has many interesting applications in stellar, galactic, and planetary dynamics.
The study of two bodies with variable masses seems to have been first investigated by Dufour [
Besides the GyldenMeshcherskii problem, there are other different cases of two bodies with variable masses, which are classified according to the presence or absence of reactive forces, to whether the bodies move in an inertial frame or not, and so on (see [
In this paper, the existence and stability of equilibrium points under the frame of the Robe problem [
This paper is orginzed as follows: Section
Let
We adopt a rotating coordinate system
The barycentric coordinates
Now, in order to obtain useful dynamical predictions, we transform
Also, the dynamical system has the particular solution of the following type [
Finally, in addition, we assume that the densities of the fluid and the test particle vary such that
Substituting (
Equations (
Now, we choose units for the distance and time, such that at initial time
Next, without loss of generality, we introduce the mass parameter defined as
Equations (
The equilibrium points represent stationary solutions of the RTBP. These solutions are the singularities of the manifold of the components of the velocity and the coordinates and are found by setting
The solutions of first equation of (
When the first and the second equation of (
The solutions of first and the third equations of (
From first equation of (
Now, the solutions exist only for
These solutions are found by solving the first and second equations of (
Hence, we have the solution
The outofplane equilibrium points are found by solving first and third equations of (
The outofplane points for



Comments 

0.01  0.98009 

Real out of plane point do not exist 
0.5  0.4945  Complex  — 
0.999  Real and negative  Complex  — 
0.999001  —  215.369  Real outofplane points exist 
1 

1.91758  — 
1.001 

1.3988  — 
1.002 

1.12303  — 
1.003 

0.932023  Real outofplane point exist 
1.004 

0.781521  — 
1.005 

0.652458  — 
1.006 

0.533794  — 
1.007 

0.416357  — 
1.008 

0.28657  — 
1.009 

0.0755768  — 

Real and negative  Imaginary  Real outofplane point do not exist 
The outofplane points for



Comments 

0.01  0.9801  Complex  Real out of plane point do not exist 
0.5  0.9801  —  — 
0.999  0.00099  —  — 
0.999001  0.00098901  —  — 
1  0  Infinity  Infinite remote solution 
1.001 

1.9138  Real outofplane points exist 
1.002 

1.39986  — 
1.003 

1.12416  — 
1.004 

0.933202  — 
1.005 

0.782774  — 
1.006 

0.653819  — 
1.007 

0.535323  — 
1.008 

0.418177  — 
1.009 

0.289043  — 
1.0091 

0.274411  — 
1.01 

0.0839973  — 

Real and negative  Imaginary  Real outofplane point do not exist 
The outofplane points for



Comments 

0.01  0.98011  Complex  Real outofplane point do not exist 
0.5  0.4955  —  — 
0.999  0.001989  —  — 
0.999001  0.00198801  —  — 
1  0.001  —  — 
1.001  0.000011  —  — 
1.002  −0.000978  1.92083  Real outofplane point exist 
1.003  −0.001967  1.40162  — 
1.004  −0.002956  1.12573  — 
1.005  −0.003945  0.934716  — 
1.006  −0.004934  0.784303  — 
1.007  −0.005923  0.655425  — 
1.008  −0.006912  0.537083  — 
1.009  −0.007901  0.420232  — 
1.0091  −0.0079999  0.408219  — 
1.01  −0.0099  0.291786  — 
1.0111  −0.0099779  0.0393474  — 

Real and negative  Complex  Real outofplane point do not exist 
Outofplane points for,
Outofplane points for,
Outofplane points for
We summarize our numerical effort as follows. In Table
Outofplane points for,
To examine the stability of an equilibrium configuration, that is, its ability to restrain the body motion in its vicinity, we apply small displacement
Now, we linearize (
Robe [
Finding first and second derivatives of the solutions, substituting them in the first two equations of (
Now, the values of the second order partial derivatives computed at the point
where
When
Substituting (
Now, the characteristic equation (
The roots of (
Now, since
These equilibrium points exist only for
The partial derivatives at these points are
For the stability of the outofplane equilibrium points, we consider the following partial derivatives:
The roots (
In the case when
mass of test particle
density of salt water
density of test particle
In new units, we have
Similarly, when
mass of test particle
density of salt water
density of submarine
In new units, we have
Aside from these examples, we also consider the case when
Using the software package
The characteristic roots





0.999001 



1 



1.001 



1.002 



1.003 



1.004 



1.009 



1.01 



1.0111 



The characteristic roots





0.999001  0.0000547723 


1  0.0541419 


1.001  0.0756118 


1.002  0.0913455 


1.003  0.103949 


1.004  0.114444 


1.009  0.148257 


1.01  0.152474 


1.0111  0.156379 


The characteristic roots





0.999001 



1 



1.001 



1.002  0.0540899 


1.003  0.0755428 


1.004  0.0912669 


1.009  0.137552 


1.01  0.14327 


1.0111  0.148654 


The characteristic roots





0.999001 



1 



1.001 



1.002 



1.003 



1.004 



1.009 



1.01 



1.0111 



From these tables, we see that that for a specific set of values of these parameters at least one of the roots among all has a positive real part or a complex root with the existence of a positive real part. Therefore, this causes the solutions to be unbounded and consequently producing unstable equilibrium points. Hence, we conclude that the outofplane equilibrium points are unstable equilibrium points due to a positive root and positive real part in complex roots. This agrees with the result of Singh [
The equilibrium solutions of the nonautonomous system with variable coefficients are in general unstable points according to the Lyapunov’s theorem of stable solutions [
We have derived the equations of motion and established the possible equilibrium points of the third body of infinitesimal mass in a setup of Robe’s [
The equilibrium points are sought, and it is seen that the point at the center of the fluid is always an equilibrium point of the Robe problem. An equilibrium point near the center of the fluid, points on the circle (circular points), and two outofplane points on the
The linear stability of the equilibrium points of the autonomized have been studied and the outcomes are analogous with the stability results in Hallan and Rana [
In our recent paper, Singh and Leke [
In the previous study, the autonomized dynamical system with constant coefficients is gotten, only when the shell is empty or when the densities of the medium and the test particle are equal, while in the present study, such limitation do not arise. In the present study, we found two collinear equilibrium points on the line joining the centers of the fluid and the second primary with one at the center of the fluid and the other away from it. Further, circular equilibrium points exist on the
The linear stability analysis however turns out to be same as the equilibrium points on the line collinear with the centers of the primary of the autonomized system which are conditionally stable; while the equilibrium points on the
The result of our research work can be summarized as follows. The restricted problem under the framework of the Robe’s [
This study may be useful in the investigations of the dynamic problem of Earthsize planets covered completely by a water envelope (water planets), and also the study of the small oscillation of the Earth’s inner core taking into account the Moon’s attraction during the course of evolution. The problem discussed in this paper is highly idealized and therefore calls for more research.