The paper deals with a modification of the restricted threebody problem in which the angular velocity variation is considered in the case where the primaries are sources of radiation. In particular, the existence and stability of its equilibrium points in the plane of motion of the primaries are studied. We find that this problem admits the wellknown five planar equilibria of the classical problem with the difference that the corresponding collinear points may be stable depending on the parameters of the problem. For all planar equilibria, sufficient parametric conditions for their stability have been established which are used for the numerical determination of the stability regions in various parametric planes. Also, for certain values of the parameters of the problem for which the equilibrium points are stable, the short and long period families have been computed. To do so, semianalytical expressions have been found for the determination of appropriate initial conditions. Special attention has been given to the continuation of the long period family, in the case of the classical restricted threebody problem, where we show numerically that periodic orbits of the short period family, which are bifurcation points with the long period family, are connected through the characteristic curve of the long period family.
The restricted threebody problem is the most celebrated problem of Celestial Mechanics (see, among others, Szebehely [
The Chermnykh’s problem is a generalization of the Euler’s problem of two fixed gravitational centers and the restricted threebody problem, in which the third body of negligible mass moves in the orbital plane of a dumbbell which is rotating with constant angular velocity
On the other hand, an interesting modification of the classical problem is the photogravitational restricted threebody problem in which the repulsive force of the radiation is also considered in the potential function and it was introduced for studying the specific threebody problem of Sun, planet, and a dust particle (Radzievskii [
In this paper, the equilibrium points as well as the periodic orbits around them are studied in the framework of the photogravitational Chermnykh’s restricted threebody problem with the aim to investigate the changes which may result in these basic dynamical features due to the three parameters of this modelproblem (the angular velocity
Specifically, the paper is organized as follows. In Section
We consider a barycentric, rotating, and dimensionless coordinate system O
The rotating coordinate system (O
The Eulerian equilibrium points lie on the O
For zero velocity and acceleration components in the equations of motion (
In order to study their stability we transfer the origin at an equilibrium point
All the above analysis has been used to determine the stability regions of all equilibrium points of the model under consideration. So, in Figure
Stability regions of all equilibrium points in the
In Figures
Stability regions of all equilibrium points in the
Stability regions of all equilibrium points in the
Stability regions of all equilibrium points in the
In Figure
In conclusion, as the value of the parameter
In Figure
Stability regions of the triangular equilibrium points and the inner equilibrium point
The general solution of (
We will first focus on the determination of the long period family emanating from the Lagrangian equilibrium point
In Figure
The evolution of the long period family
In Figure
The evolution of the branch
An alternative illustration of the evolution of the long period family is also shown in Figure
(a) Projection of the characteristic curve of the long period family in the
(a) Projection of the characteristic curve of the long period family in the
In Table
Sample members of the long period family emanating from the triangular equilibrium point






 


81.965981  0.2851750  0.1855400  −0.2593142  2.9747462  1.023708  0.951895 

88.312060  0.1696324  0.1210896  −0.1151430  2.9964843  1.021820  0.929480 

94.548562  0.3197985  0.1679201  −0.3395816  2.9546957  5.590865  0.932739 

100.870333  0.2765249  0.1515870  −0.2739704  2.9733912  7.641533  0.918313 
We will now proceed to incorporate the other parameters of the problem and compute the corresponding long and short period families. The stability analysis of the equilibrium points, in Section
With regard to the triangular equilibrium points we have determined the corresponding families for the mass ratio
Projection of the characteristic curve of the long period family for the parameter values
(a) The bifurcation of the short period family with the Lyapunov family emanating from the collinear equilibrium point
In Tables
Sample members of the long period family emanating from the triangular equilibrium point






 


50.433206  0.1294818  0.1565572  −0.2214219  2.4608218  0.817029  0.958225 

54.583313  0.1250745  0.1515822  −0.2084973  2.4639318  0.634769  0.966236 

58.728922  0.1157852  0.1421300  −0.1878419  2.4669604  1.401536  0.975416 
Sample members of the short period family emanating from the triangular equilibrium point








4.206499  0.0220000  0.0239493  −0.0423981  2.4744822  1.894299  0.999646 
4.205279  0.1520000  0.1352379  −0.3200771  2.4361058  1.848432  0.999648 
4.197186  0.4156000  0.2395543  −1.0010119  2.0169264  1.607336  0.999336 
4.187397  0.2010095  0.0259248  −1.2868579  0.9614449  −0.581246  0.999994 
For the short and long period families emanating from the collinear equilibrium points when these are stable we present the corresponding diagrams in Figure
Projections, in the
With regard to the short period families, at all collinear equilibrium points, they emanate from and terminate at their corresponding equilibria. In particular, the short period family emanating from
Sample members of the short period families emanating from the stable equilibrium points
Initial conditions for representative periodic orbits of these families are presented in Table
Sample members of the short and long period families emanating from the collinear equilibrium points for





 


6.721506  0.2232037  −0.0252754  0.1032007  −0.9880  0.9610 

14.627908  0.1815907  0.0111466  0.1033552  0.7358  −0.1155 

8.087888  1.7300000  −0.3327057  1.3965491  4.4041  −0.3814 

19.595694  1.2147080  0.1069103  1.4042630  0.9273  0.8702 

13.071411  −0.7991938  0.0721319  0.4596958  −0.8108  0.9720 

15.920543  −0.6517328  −0.0378064  0.4589966  0.4090  −0.9978 
Some basic dynamical features of the photogravitational Chermnykh’s restricted threebody problem were studied. In this problem, the primaries are considered to be radiating sources with mass reduction factors
With regard to the planar equilibrium points it was found that, as in the classical case, the problem admits five stationary solutions and, in particular, three collinear and two triangular. An interesting result is that the triangular equilibrium points may fall onto the O
For all cases of stable collinear equilibrium points, the motion around them is bounded and is composed of two harmonic motions, giving rise to short and long period families, which consist of symmetric periodic orbits. Note that, for the corresponding Chermnykh’s or photogravitational problems and with respect to the specific equilibria, these families may be found only for the inner collinear equilibrium point, while for the classical problem, it is known that only the unstable Lyapunov families emanate from them. Certain examples of the short and long period families were given to show their existence and evolution. For the considered short period families of our examples we found that each one of them terminates at its corresponding equilibrium point.
Concerning the respective families emanating from the triangular equilibrium points, specific examples were drawn showing that their characteristic curves have a similar evolution with that of the corresponding families of the classical problem, which was also examined thoroughly. Specifically, the short period family terminates on the Lyapunov family emanating from the outer collinear equilibrium point to the left of the larger primary, while the long period family forms bridges connecting orbits of the short period family which are bifurcation points among these two families. The corresponding families consist of asymmetric periodic orbits. To the authors’ knowledge, related results for the Chermnykh or the photogravitational restricted threebody problem showing explicitly the evolution of the long period family do not exist in the literature.
A natural extension of the present study is to consider negative values for the radiation factors, a case for which at least new equilibrium points out of the orbital plane are allowed to exist. Another interesting case for study would be also to incorporate the PoyntingRobertson relativistic correction in the radiation force as well as to consider the belt gravitational potential as an additional term in the potential function.
The authors declare that there is no conflict of interests regarding the publication of this paper.
A. A. Nikaki acknowledges financial support under University of Patras “K. Karatheodory” research grant.