Influence of the Zonal Harmonics of the Primary on L4,5 in the Photogravitational ER3BP

We investigate in the framework of the elliptic restricted three-body problem (ER3BP), the influence of the zonal harmonics (J2 and J4) of the primary and the radiation pressure of the secondary on the positions and stability of the triangular equilibrium points. The triangular points of the problem are affected by the parameters involved in the systems’ dynamics. The positions change with increase in the zonal harmonics, eccentricity and radiation pressure. The triangular points remain stable in the interval 0 c     as shown arbitrarily.


Introduction
The elliptic restricted three-body problem (ER3BP) describes the motion of a particle, called the third body under the gravitational attractions of two finite bodies (primaries) which revolve in elliptic orbits around their common centre of mass in accordance with the laws of motion of the two-body problem. It has not been fully explored (planar or spheroid) and has as such so far produced very few analytical results. Various studies have considered the ER3BP from various perspectives both with constant and variable coefficient (Kunitsyn and Tureshbaev [1]; Singh and Ishwar [2], Sahoo and Ishwar [3], Kuntisyn [4], Tsirogiannis et al. [5], Kumar and Ishwar [6], Tkhai [7], Kumar and Narayan [8], Singh and Umar [9][10][11][12]. Singh and Umar [13] found in considering the effects of the shapes of the primaries that the positions and stability are affected by traxiality, oblateness, semi-major axis and eccentricity of their orbits; hence the collinear points remain unstable in the lypunov sense. The orbits of most celestial bodies are elliptic; as such, the study of the ER3BP can produce significant results neglected by the CR3BP. The classical CR3BP fails when one or both primaries is/are luminous. This is called the 'photogravitational' problem formulated by Radzievskii [14], where he discovered that the locations of the infinitesimal mass strongly depends on the mass reduction factor (q 1 ). He introduced the planar (1950) and non-planar cases of the CR3BP (1953). He discussed it for the Sun, planet and a dust particle and Galaxy Kernel-Sun-Particle systems. The effect of radiation pressure(s) has attracted much attentions of several researchers over the last few decades Umar and Singh [15], Singh and Taura [16], Singh and Umar [10,11], Sharma et al. [17], Kunitsyn [18], Singh and Ishwar [3], Kumar and Choudry [19] and Simons et al. [20], have considered the R3BP under different characterizations.
When the participating bodies are considered to be strictly spherical and point masses moving under their mutual gravitational attraction in circular orbits around their common barycenter as in the CR3BP, an important aspect of these bodies is neglected. Celestial bodies are mostly irregular in shape. The Earth, Jupiter, Saturn, Regulus, Neutron stars/Pulsars and black dwarfs are oblate spheroids (Arutyunyan et al. [21], Papoyan et al. [22], Laarakkers [23], Shibata [24], Boshkayev et al. [25] and Heyl [26]. Our Moon, Pluto and its moon Charon are triaxial. The asphericity, triaxiality or oblateness of the celestial bodies causes large perturbations from a two-body orbit. This inspired several researchers such as Subbarao and Sharma [27], Elipe and Ferrer [28], EL-Shaboury [29], Sharma [30], Singh and Begha [31], Singh [32] and Umar and Singh [33] to include asphericity of the bodies in their investigation of the R3BP.
Zonal harmonics arise due to meridinal ellipticity and the typical precession rates of planetary orbits are about 20 pas/yr, as such they play an important role in high accuracy modelling of the motion of celestial bodies. Several studies, Singh and Taura [34], Singh and Omale [35] and Abouelmagd et al. [36] have included the second zonal harmonics in their investigation of the R3BP. Singh and Taura [34] examined the effects of oblateness up to J 4 of both primaries together with gravitational potential from a circular cluster of material points on the stability of the triangular points in the CR3BP. While Abouelmagd et al. [36] investigated the effects of the even zonal harmonic up to J 4 of both primaries, on the existence of libration points and their linear stability as well as analysing the existence of periodic orbits around these points.
In the present study, we endeavour to extend the work of Singh and Umar [11] by including the effect of oblateness up to the coefficient J 4 on the locations of triangular points and their stability with application to the binary pulsar PSR J1903+0327.
This paper is organised in six sections; section 1 is the introduction; section 2 deals with the equations of motion; section 3 focuses on the locations of triangular point. The linear stability of these points is examined in section4; while the numerical applications and conclusion are presented in section 5 and section 6, respectively.

Equations of motion
The equations of motion of a dust particle, in the framework of the ER3BP under the consideration that the primary is an oblate spheroid with the zonal harmonics J 2 and J 4 and the secondary a source of radiation, can be written in a dimensionless-pulsating coordinate system ( , , )    following Singh and Umar [11] and Singh and Taura [33] and the mean motion n is given by where 1 , 2 are the masses of the primaries positioned at the points ( ,0,0), i=1,2; q is the radiation pressure factor of the secondary, a and e are the semi-major axis and eccentricity of the orbits, respectively; A i = J 2i R 1 2 , A i ≪ 1 (i=1,2) characterize the zonal harmonics of the primary whose mean radii is R 1 .

Locations of triangular points.
Equilibrium points are those points at which the infinitesimal mass is at rest. Thus the equilibrium points are the solutions of the equations Ω = Ω = Ω = 0, which yield In the absence of oblateness of the primary, system (6) provides When oblateness is considered, the value of 1 2 will change slightly by 1 2 (say), respectively so that; Considering only linear terms in 1 , 2 , 2 and neglecting their higher powers and products, (3) gives Now, solving (7) and (8), we get Using (8) and (9) in (6), we get Putting the values of 1 2 from (10) in (9), we obtain Using (4) and (11) to yield International Frontier Science Letters Vol. 10 25 These points ( , ± ), presented by (12) in the − plane are denoted by L 4,5 and are known as the triangular equilibrium points.

Linear stability of triangular points.
The motion of a test particle near any of the equilibrium points is said to be stable if given a small displacement with a small velocity, the particle oscillates considerably about the point and stays around for all time, otherwise it is said to be unstable. The motion of a particle in the ξη − plane is investigated by giving the triangular points small displacement (θ, ω). Then we write ξ = ξ 0 + θ and η = η 0 + ω.
In the variational form, we have the equations of motion as The characteristic equation of this system is where the superscript 0 indicates that the partial derivatives are to be evaluated at the triangular point ( 0 , 0 ).
In the case of triangular points L 4,5 , when the primary is oblate and the secondary is luminous, we have Substituting these values in the characteristic equation (13) and considering only linear terms in 2 , 1 , 2 , , and , where = 1 − , = 1 − , , ≪ 1, we obtain For stable motion, we choose , 1 , 2 , such that 2 < 0 i.e The necessary conditions for the stability of the triangular points (12) are given by equations (16) and (18). Solving the quadratic equation D = 0 for gives the critical value of the mass parameter as   (19) This establishes the effects of the various perturbing agents of oblateness (J 2 and J 4 ), radiation pressure q = 1-β , eccentricity e and semi-major axis a = 1-α. A critical examination of equation (19) shows that the various parameters cause a reduction in the size of the region of stability as shown in figures 3, 4 and 5 for increasing semi-major axis, eccentricity and radiation pressure factor. Thus, they all have destabilizing effects.

Numerical Applications
This section locates numerically the triangular points of the problem using (12) for the binary system PSR J1903+0327. This is a millisecond pulsar in a highly eccentric orbit (e = 0.437), with a pulsar mass m 1 =1.67±0.02M sun and a main sequence companion of mass m 2 =1.1M sun with a luminosity (class V) 2.5L sun . The mass ratio is = 2 1 + 2 gives  =0.397112.
The radiation pressure factor q is computed taking =1, on the basis of Stefan-Boltzmann's law, where q = 1-(Singh and Umar 2012) and M and L are the mass and luminosity of a star, respectively; r and are the radius and density of a moving body; is the radiation pressure efficiency factor of a star; A= 3 16 is a constant. In the C.G.S. system, A=2.9838 × 10 −5 and supposing r=0.02cm and =1.4g −3 for some dust grain particles in the systems. Thus, q=0.997581 for the system PSR J1903 + 0327.
Using these data, we compute the locations of the triangular points, highlighting the effects of the parameters. The effect of the zonal harmonics A 1 (J 2 ) in the absence of A 2 and then with a constant A 2 (J 4 ) is shown in tables 1 and 2 respectively. Then, keeping A 1 constant, the effect of A 2 is examined (table 3). Subsequently, the effects of eccentricity and semi-major axis are presented in tables 4 and 5 using the software package Mathematica.
We show the effects of zonal harmonics J 4 , eccentricity e, semi major axis a and radiation pressure q on the size of the region of stability (tables 6-9) for arbitrary values of the mass ratio in the range 0< < .

Conclusions
Using the equations of motion of the ER3BP with constant coefficients, under the consideration that the primary is an oblate body and the secondary is a source of radiation, the influences of the even zonal harmonics (J 2 and J 4 ), eccentricity and radiation pressure on the positions and stability of the triangular points have been investigated. The combined effect of the zonal harmonics J 2 and J 4 is a shift away from the origin and towards the line joining the primaries. It is seen from tables 1, 2 and 3 that individually, they both have the same effect but J 2 acts faster than J 4 . Increasing the eccentricity of the orbits while keeping J 2 and J 4 constant causes a shift towards the origin and towards the line joining the primaries, as is shown in table 4 and figure 1. We observe that for high eccentricity and in the quasi-parabolic case, the triangular points cease to exist. The reverse is the case with increase in semi-major axis (table 5 and figure 2). This agrees with Singh and Umar [11] with J 4 =0.
The parameters all have destabilizing effects, causing a reduction in the size of the region of stability. As eccentricity, semi-major axis and radiation pressure increase, the size of the region of stability decreases and the system becomes unstable (tables 7-9 and figure 3-5).