^{1, 2} is used in the present studies. The characteristic roots are obtained by linearizing the equation of the motion around the Lagrangian points. _{ }^{3, 4, }which decreases radiation force, whereas it increases with oblateness when we consider the value of new mean motion. Through special choice of initial conditions, retrograde elliptical periodic orbits exist for the case µ = µcrit, whose eccentricity increases with oblateness and decreases with radiation force for non-zero oblateness, although there is slight variation in L_{2} location.

The three-body problem, in general, is about the study of three massive bodies (m1, m2, m3) in space which affect one another by their gravitational forces. In the restricted three-body problem, one of these three bodies are considered to be very small and that it has a very less mass which does not affect the motion of the other two bodies called primaries with masses m1 and m2, but it is affected by their mutual forces. The circular restricted three-body problem (CR3BP) studies the motion of a body with a negligible mass under the influence of two bigger bodies, called the primaries, which revolve around their center of mass in ‘n’ circular orbit. To make the problem more applicable to real cases, perturbations are also included. Here, time (t) is the independent variable on which the other variables depend. When the restricted three- body problem is solved, it has been observed that there are 5 points on the plane of the primaries where the gravitational forces of the primaries nullify each other and there is no gravitational

Since the Sun — the source of radiation — is almost spherical, and some of the planets like Saturn and Jupiter whose equatorial and polar radii are approximately 60400, 54600 and 71400, 67000 km, respectively, and sufficiently oblate, hence a modification is considered that the smaller primary (planet) is an oblate spheroid. The more massive primary, the Sun, is a source of radiation. It is considered that the equatorial plane of the smaller primary is coincident with the plane of motion and only the planar case has been studied.

In this paper, case considered when the gravitation prevails. As the solar radiation pressure force

Where

Many of the celestial bodies do not have an exact sphere shape. Instead, they have an oblate spheroid shape. An oblate spheroid is obtained by rotating an ellipse about its minor axis i.e., the equatorial radius becomes longer than the polar radius. This oblateness of the planets render a change to the mean motion of orbit of the primaries, because of the variation of gravitation. The effect of oblateness is quantified by the term J2, which varies for each planet. Oblateness

In^{3}. In^{ }in the frame work of the perturbed photo-gravitational restricted three-body problem, the first order exterior resonant orbits and the first, third and fifth order interior resonant periodic orders were analysed. The location, eccentricity and periodic of the first order exterior and interior resonant orbits are investigated in the unperturbed and perturbed cases for a specific value of Jacobi constant. In^{8} interior resonance periodic orbits around the Sun in the Sun-Jupiter photogravitational restricted three-body problem by including the oblateness of Jupiter was studied using the method of Poincaré surface of section. In ^{9} a passive micron size particle in the field of radiating binary stellar system in the framework of circular restricted three body problem influenced from radial radiation pressure and Poynting-Robertson drag (PR drag) on the equilibrium points and their stability in the binary stellar systems RW-Monocerotis and Krüger-60 was studied. In ^{10} the motion in the vicinity of triangular equilibrium points of the circular restricted three-body problem of a passively gravitating dust particle in the gravitational field for the binaries system (Kruger 60 and Achird) is investigated. The two bodies of the binary are both oblate radiating stars possessing P-R drag.

We have calculated the value of the critical mass parameter,

Following the terminology and notation of ^{11} and the unit of mass is considered equivalent to the sum of the primary masses, the unit of length is equivalent to their separation and the unit of time is such that the Gaussian constant of gravitation is unity. Similar to as in ^{3, 6, }the equation of motion in the dimensionless barycentric-synodic coordinate system (x, y) are

where the force function equation is given by,

where r_{1} and r_{2 }are the magnitudes of the position of the spacecraft or satellite from the massive and smaller primary, respectively; q is the mass reduction factor of the massive primary and A_{2} is the oblateness coefficient of the smaller primary. The magnitudes of the position of the satellite or spacecraft are given as

The Jacobian integral of Equations (1) and (2) is

The curves of zero-velocity are given by

The curves are symmetric with respect to the x-axis, since

i.e.,

In this study, we have derived mean motion expression including the secular perturbation effects. From ^{12, 13} for planar motion (inclination i = 0).

Using the relationship,

We get

The mean motion including the precession effect due to oblateness is calculated as:

Since our study is based on CR3BP, substituting the value of eccentricity (e) = 0 in the above equation,

or,

In dimensionless unit

_{,}

or_{2}).

When we take y=0, equation (5) determines the location of the collinear points L_{1}(x_{1}, 0), L_{2}(x_{2}, 0), L_{3}(x_{3},0) where

where,

Solving the equations (5) and (6), where y≠0 we get

By the help of equation (7), we can locate the other two points

Replacing

The Characteristic equation of Equations (9) is

At the collinear points, we get

With some calculation, it can be proved that

We can note that the roots

where

The General solution of Equations (9) can be written as

And

It is noted that ^{4}, we can choose the initial conditions

respectively, where

_{2 }(oblateness)=0,0.01.

μ | ε | A2 | ξ | x1 | Eccentricity (e) |

0.1 | 0 | 0 | 0.35969983 | -1.25969983 | 0.92543667 |

0.1 | 0.01 | 0 | 0.35889138 | -1.25889138 | 0.92572111 |

0.1 | 0.01 | 0.01 | 0.36100212 | -1.26100212 | 0.94471622 |

0.2 | 0 | 0 | 0.47104869 | -1.27104869 | 0.91600058 |

0.2 | 0.01 | 0 | 0.47035553 | -1.27035553 | 0.91619536 |

0.2 | 0.01 | 0.01 | 0.46785205 | -1.26785205 | 0.93117171 |

0.3 | 0 | 0 | 0.55673469 | -1.25673469 | 0.90787718 |

0.3 | 0.01 | 0 | 0.55613915 | -1.25613915 | 0.90802208 |

0.3 | 0.01 | 0.01 | 0.55080979 | -1.25080979 | 0.92163148 |

We can see that, eccentricity increases with oblateness and slightly increases with radiation pressure. And orbit is elliptical.

μ
ε
A2
ξ
x2
Eccentricity (e)
0.1
0
0
0.29096488
-0.60903511
0.96842307
0.1
0.01
0
0.29223645
-0.60776354
0.96813137
0.1
0.01
0.01
0.30720465
-0.59279534
0.97304460
0.2
0
0
0.36192404
-0.43807595
0.97132709
0.2
0.01
0
0.36319101
-0.43680898
0.97111656
0.2
0.01
0.01
0.37481534
-0.42518465
0.97382094
0.3
0
0
0.41387021
-0.28612978
0.97276616
0.3
0.01
0
0.41511790
-0.28488209
0.97092712
0.3
0.01
0.01
0.42467365
-0.27532634
0.97438134

We can see that, eccentricity increases with oblateness and eccentricity decreases with radiation force. And orbit is elliptical.

μ
ε
A2
ξ
x3
Eccentricity (e)
0.1
0
0
0.94160890
1.04160890
0.86835862
0.1
0.01
0
0.93840960
1.03840960
0.86836855
0.1
0.01
0.01
0.91962433
1.01962433
0.87972477
0.2
0
0
0.88283946
1.08283946
0.87750511
0.2
0.01
0
0.87979735
1.07979735
0.87333010
0.2
0.01
0.01
0.86147227
1.06147227
0.88140032
0.3
0
0
0.82320559
1.12320559
0.87932469
0.3
0.01
0
0.82033371
1.12033371
0.87936718
0.3
0.01
0.01
0.80254908
1.10254908
0.88560716

We see that, eccentricity increases with oblateness, and eccentricity increases slightly with radiation force. And orbit is elliptical.

System
μ
A2
ε
x1
Eccentricity(e)
Sun-Mars
3.2127*10-7
5211.14188981*10-16
0 0.001 0.01
-1.00475597 -1.00464773 -1.00386702
0.94985821 0.92214540 0.96819065
Sun-Jupiter
9.5333*10-4
211.71783905*10-12
0 0.001 0.01
-1.02506012 -1.02495138 -1.02401151
0.99599793 0.99604663 0.99645211
Sun-Saturn
2.85639*10-4
65.276555555*10-12
0 0.001 0.01
-1.04606463 -1.04595722 -1.04501136
0.94712293 0.94736490 0.94949755
Sun-Uranus
4.36431*10-5
6713.99789137*10-16
0 0.001 0.01
-1.02596188 -1.02585315 -1.02491275
0.94255895 0.94300822 0.94691164
Sun-Neptune
5.14805*10-5
2037.88920652*10-16
0 0.001 0.01
-1.04117099 -1.04094671 -1.03897779
0.89538651 0.89592653 0.90098599

System
μ
A2
ε
x2
Eccentricity(e)
Sun-Mars
3.2127*10-7
5211.14188981*10-16
0 0.001 0.01
-0.99525841 -0.99514427 -0.99384034
0.95049170 0.94809550 0.92227849
Sun-Jupiter
9.5333*10-4
211.71783905*10-12
0 0.001 0.01
-0.93237854 -0.93226238 -0.93119681
0.95468743 0.95452551 0.95303883
Sun-Saturn
2.85639*10-4
65.276555555*10-12
0 0.001 0.01
-0.95475377 -0.95463906 -0.95357825
0.95320893 0.95296564 0.95071447
Sun-Uranus
4.36431*10-5
6713.99789137*10-16
0 0.001 0.01
-0.97574465 -0.97563121 -0.97455947
0.95180148 0.95134427 0.94702807
Sun-Neptune
5.14805*10-5
2037.88920652*10-16
0 0.001 0.01
-0.97437855 -0.97426504 -0.97319526
0.95189331 0.95146075 0.95327614

System
μ
A2
ε
x3
Eccentricity(e)
Sun-Mars
3.2127*10-7
5211.14188981*10-16
0 0.001 0.01
1.00000013 0.99966668 0.99665562
0.86602540 0.86602540 0.86602512
Sun-Jupiter
9.5333*10-4
211.71783905*10-12
0 0.001 0.01
1.00039722 1.00006390 0.99705404
0.86602570 0.86602570 0.86602570
Sun-Saturn
2.85639*10-4
65.276555555*10-12
0 0.001 0.01
1.00011901 0.99978561 0.99677490
0.86602543 0.86602543 0.86602543
Sun-Uranus
4.36431*10-5
6713.99789137*10-16
0 0.001 0.01
1.00001818 0.99968474 0.99667373
0.86602540 0.86602540 0.86602540
Sun-Neptune
5.14805*10-5
2037.88920652*10-16
0 0.001 0.01
1.00002144 0.99968801 0.99667701
0.86602540 0.86602540 0.86602540

Periodic orbits around the Sun in Sun-Jupiter system for L_{1} is generated by MATLAB for the mass ratio of µ =0.00095333 and solar radiation pressure q = 1 and 0.99, as shown in [

Periodic orbits around the Sun in Sun-Jupiter system for L_{2} is generated by MATLAB for the mass ratio of µ =0.00095333 and solar radiation pressure q = 1 and 0.99, as shown in [

Periodic orbits around the Sun in Sun-Jupiter system for L_{3} is generated by MATLAB for the mass ratio of µ =0.00095333 and solar radiation pressure q = 1 and 0.99, as shown in [

At the triangular points

where

The characteristics Equation (10) becomes

with

We observe that the roots

are functions of

And three cases can be discussed as,

When D is positive, we note that

Shows the triangular points to be linearly stable.

The solution of Equation (9) in this case can be easily seen to consist of short - and long - period terms with angular frequencies ^{5, 8, 9 }the short-or-long-period terms can be eliminated from the solution with proper selection of initial conditions. In both cases, the motion is along a retrograde ellipse whose eccentricity and the orientation of the major axis are independent of the initial conditions.

When D is negative, the real parts of two of the four roots equation (13) are positive and equal and, hence the equilibria are unstable. However, with suitable selection of initial conditions, periodic motion can be achieved in the linear sense which approaches the equilibrium point asymptotically.

The discriminant of the quadratic Equation (12) is zero when

When ^{14}.

Solution of the Equation (14) for

Where

And

or,

The value of the Critical mass that we got from the Mean motion

And the value of the Critical mass that we got from the mean motion expression given by ^{ }

First value of oblateness in (µ_{crit}) or Critical mass is been found out for different values of mean motions (n^{2}).

n2
μcrit (only first term of A)
1+32A2
-0.06277956
1+3A2
-0.0020507319
1+4A2
0.038435157
1+5A2
0.07892104618
1+6A2
0.119406935

Graph explaining the mean motion and first value of oblateness in critical mass.

In earlier case of mean motion ‘n’ given by ^{3}, only secular effect of oblateness on the mean motion was considered ^{3}. However, when the secular effect of the oblateness on the mean motion is considered, argument of perigee and right ascension of the ascending node ^{2}=1+6A_{2} is included in the present studies.

In this study, mean motion is increasing since the effect of oblateness on mean motion, argument of perigee and right ascension of ascending node have been considered. All these three parameters are included to get the mean motion of primaries n^{2}=1+6A_{2}_{. }The value obtained of the critical mass μcrit higher than the unperturbed value of μc = 0.0385209 in ^{3} with oblateness of the smaller primary. It is a very interesting result, because the zone of mass parameter μ providing stable solutions at the triangular points increases with oblateness. It can be observed from [