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We analyze the periodic orbits, quasi periodic orbits and chaotic orbits in the photo gravitational Sun-Saturn system incorporating actual oblateness of Saturn in the planar circular restricted three body problem. In this paper, we study the effect of solar radiation pressure on the location of Sun centered and Saturn centered orbits, its diameter, semi major axis and eccentricity by taking different values of solar radiation pressure q and different values of Jacobi constant “C”, and by considering actual oblateness of Saturn using Poincare surface of section (PSS) method. It is ob-served that by the introduction of perturbing force due to solar radiation pressure admissible range of Jacobi constant C decreases, it is also observed that as value of C decreases the number of islands decreases and as a result the number of periodic and quasi periodic orbits decreases.Fur-ther, the periodic orbits around Saturn and Sun moves towards Sun by decreasing perturbation due to solar radiation pressure q for a specific choice of Jacobi constant C. It is also observed that due to solar radiation pressure, semi major axis and eccentricity of Sun centered periodic orbit reduces, whereas, due to solar radiation pressure uniform change in semi major axis and eccen-tricity of Saturn centered periodic orbits is observed.

Restricted three body problem (RTBP) describes the motion of an infinitesimal mass which moves under the gravitational effect of two finite masses called primaries. The primaries are supposed to move in circular orbits around their center of mass on account of their mutual attraction. Usually Sun and any one of its planets are taken as primaries. The secondary body is taken as the satellite of the primary planet or asteroid or comet or artificial satellite [

Lebedev experimentally demonstrated that the minute pressure exerted by light on bodies is inversely proportional to the square of the distance between the light source and the illuminated body. Since then many researchers have taken this force in to consideration, apart from other perturbing forces. [

The study of periodic orbits plays an important role in the understanding of the general properties of different dynamical systems. A large number of periodic orbits were generated by [

The set of stable periodic and quasi-periodic trajectories define regions of regular motion or stability “islands” that spread in a chaotic “sea” made of trajectories with high sensitivity with respect to the initial condition. As per Kolmogorov-Arnold-Moser (KAM) theory, the point represents a periodic orbit in the rotating frame, and the closed curves around the point correspond to the quasi-periodic orbits.

PSS gives a qualitative picture of stability regions in the planar problems. [_{1} and L_{2} in the Sun-Earth system. [

In this paper we have studied PSS method for Sun-Saturn system for periodic and quasi-periodic orbits. This work is mainly concentrated on two major islands, one gives Sun centered orbit and other gives Saturn centered orbit. Since these two islands are major islands they are available in each PSS corresponding to different solar radiation pressure q with different Jacobi constant C. So, effect of perturbation on two different family of periodic orbit can be analysed by obtaining PSS. This paper is organised as follows. The basic equations of motion incorporating the perturbed force due to radiation and oblateness is given in Section 2. The PSS method is described in Section 3. Computational techniques are used to obtain PSS and periodic as well as quasi-periodic orbits in Section 4 and conclusions of the study are presented in Section 5.

Restricted three-body problem describes the motion of an infinitesimal mass which moves under the gravitational effect of two finite masses called primaries. The primaries are moving in circular orbits around their centre of mass on account of their mutual attraction and the infinitesimal mass not influencing the motion of the primaries.

We consider the case when bigger primary (Sun) is source of radiation and smaller primary (Saturn) is oblate spheroid. We consider that the equatorial plane of the Saturn is coincident with the plane of the motion and study only the planar case.

As the solar radiation pressure force

Thus, the sun’s resultant force acting on the particle is

where

where

Here, AE and AP represent equatorial and polar radii of Saturn and R is the distance between Sun and Saturn.

Choose the unit of mass equal to the sum of the primary masses, the unit of length is equal to their separation and the unit of time is such that Gaussian constant of gravitation is unity. The usual dimensionless synodic coordinate system Oxy is used to express this motion. The origin of this system is positioned on the center of mass of the primaries while the bigger and smaller primaries always lie on the Ox axis at P(−μ, 0) and at Q(1 − μ, 0), respectively.

Following [

where,

Here,

and,

By integrating, we get,

where, C is the Jacobi constant of integration.

The Poincare surface of section (PSS) method is used for determining the regular or chaotic nature of the trajectory. The numerical method of PSS is used to generate orbits and to study the location and stability of orbits in various systems.

In order to determine the orbital elements of the infinitesimal mass at any instant it is necessary to know its position and velocity, which correspond to a point in a four dimensional phase space. For the PSS method, the equations of motion are integrated in (x, y) variables using a Runge-Kutta Gill fourth order variable or fixed

step-size integrator. The initial conditions are selected along the x-axis. By defining a plane, say

ing a plane in the phase space and plotting the points when the trajectory intersects this plane in a particular direction. We have constructed PSS on the x,

For Sun-Saturn system the mass of sun m_{1} = 1.9881 × 10^{30} kg, m_{2} = 568.36 × 10^{24} kg. Thus, m = m_{2}/(m_{1} + m_{2}) = 0.0002857696. Also equatorial radius of Saturn is 60268 km, polar radius of Saturn is 54,364 km. and distance

between sun and Saturn is 1433000000 km. So, according to the formula,

cient A_{2} = 6.59158 × 10^{−11}. We have explored the Sun-centered and Saturn-centered orbits in the Sun-Saturn system and variations in them due to solar radiation pressure. For different solar radiation pressure we have different range of Jacobi constant C. For each value of q, we can find maximum value of C using equation (6), such

that

is real. q = 0.9 gives maximum value of Jacobi constant C as 2.807 such that infinitesimal mass having real velocity within region between two primaries, Sun and Saturn. In other words, for q = 0.9, admissible range of C

lie in the interval [1, 2.807] for values of x in the interval [0, 1]. If C = 2.808 then for same q,

negative for x in the range [0.9290, 0.9510]. This is excluded region for infinitesimal mass as velocity becomes complex. For q = 0.9, we have analysed the PSS for 1≤ C ≤ 2.8 and DC = 0.1. PSS for C = 1.0 contains only 2 points as shown in

First quasi periodic orbit obtained at x = 0.1168 for C= 1.8 when q = 0.9. This orbit is located at the center of the island as shown in

By observing PSS for different values of Jacobi constants from C = 1.0 to C = 2.8 we conclude that as value of C increases, the number of islands increases and as a result periodic orbits and quasi periodic orbits increase.

Here, we mainly concentrate on one of the major island which gives sun centred orbit and analyze its nature for different Jacobi constant C.

Semi-major axis “a” and eccentricity “e” of the Sun centered orbit are obtained by Murraay and Dermott (1999) as

For Sun centered periodic orbits,

By reducing perturbation due to solar radiation pressure (q = 0.9845) using same procedure admissible range of Jacobi constant C is [1, 2.985]. C = 2.986 gives negative

For Sun centered periodic orbits,

For q = 1, the shifting of periodic orbits towards Saturn by increasing value of Jacobi constant C is shown in

concluded that solar radiation pressure is responsible for shifting location of periodic orbits towards Saturn which is shown in

It is observed from

By considering q = 0.9345, admissible range of Jacobi constant C is [1, 2.88]. C = 2.881 gives negative

PSS of island corresponding to Saturn centered periodic orbit for C = 2.8 with q = 0.9, 0.9345, 0.9645 and 1 are shown in Figures 19-22 respectively. Center of the island gives Saturn centered periodic orbit at x = 0.95285, x = 0.72165, x = 0.6365, and at x = 0.56455 respectively.

Figures 23-26 show PSS for island corresponding to Saturn centered periodic orbit for C = 2.79 with q = 0.9, 0.9345, 0.9645 and 1 respectively. Center of the island gives Saturn centered periodic orbit at x = 0.8957, 0.70345, 0.62315 and at 0.55435 respectively.

Figures 27-30 depict PSS of island for C = 2.78 with q = 0.9, 0.9345, 0.9645 and 1 respectively. Center of the island gives Saturn centered periodic orbit at x = 0.8429, 0.68645, 0.61065, and at x = 0.5444 respectively. From this it is observed that for a given C by decreasing perturbation due to solar radiation pressure Saturn centered periodic orbit moves towards Sun. This is the effect of solar radiation pressure on retrograde Saturn centered periodic orbits. From

Sun which is similar to Sun centered periodic orbits.

becomes slow and smooth change in semi major axis.

It is observed that as the Jacobi constant C increases, the number of islands, number of quasi periodic orbits and number of periodic orbits increases for any perturbation due to solar radiation pressure. As q tends to 1, admissible range of Jacobi constant C increases. In other words, solar radiation pressure reduces admissible range of C. For q = 1 maximum value of C is 3.018. By increasing perturbation due to solar radiation pressure up to q = 0.9845 maximum value of Jacobi constant reaches C = 2.985, by increasing more perturbation due to solar radiation pressure up to q = 0.9645 maximum value of C decreases and reach at C = 2.943. For q = 0.9345, maximum value of C reaches at C = 2.88 and for q = 0.9 maximum value of C is 2.807. It is further noticed that as Jacobi constant C increases Sun centered periodic orbit and Saturn centered periodic orbits shift towards Saturn for a given perturbation due to solar radiation pressure q. It means Jacobi constant C acting opposite to gravitational force. For a given C, as q tends to 1, location of Sun centered and Saturn centered periodic orbits moves towards Sun. That is solar radiation pressure is responsible for shifting periodic orbits towards Saturn which is expected as solar radiation pressure is opposite to gravitational attraction of Sun. Thus, by decreasing perturbation due to solar radiation pressure, effect of gravitational attraction increasing and as a result periodic orbit shift towards Sun. For a given q, as C increases semi major axis increases and eccentricity of Sun centered periodic orbits decreases. For a given q and its corresponding maximum value of Jacobi constant C Saturn centered periodic orbit showing sudden change in semi major axis and eccentricity. It is observed that semi major axis increases uniformly up to certain value of C and then shows a sharp increase, where as eccentricity decreases uniformly up to certain value of C and then shows a sharp increase. When the solar radiation pressure q tends to 0.9, this sharp change in the semi major axis and eccentricity graphs becomes smooth. Thus, it is concluded that perturbation due to solar radiation pressure reduces sudden change in semi major axis and eccentricity of Saturn centered periodic orbits. It is further observed that as q tends to 1, semi major axis and eccentricity of Sun centered periodic orbit increases for given C. In other words, due to solar radiation pressure, semi major axis and eccentricity of Sun centered periodic orbit reduces.

Niraj Pathak,R. K. Sharma,V. O. Thomas, (2016) Evolution of Periodic Orbits in the Sun-Saturn System. International Journal of Astronomy and Astrophysics,06,175-197. doi: 10.4236/ijaa.2016.62015