The study of three-body problem dates back to centuries. The solution to this problem was infeasible until one of the bodies was consider to be of negligible mass compare to the other two which are heavier masses and problem is now known as restricted three body problem.

In this form the motion of the infinistimal particle in the vicinity of the primaries, which move in circular or elliptic orbits around their common centre of mass due to their mutual gravitational attraction is explained. There are five equilibrium points in restricted three body problem three are collinear points L_1,2,3 and two are called  the Lagrangian  equilibrium points (L_4,5).They lie on the orbital plane of  the motion of the primaries. The latter are conditionally stable, while the former are unstable.

In the classical era  of investigation the bodies are considered  to be moving in circular orbits, hence the system was called circular restricted three-body problem (CR3BP), in this system the infinestimal mass moves in cicular orbits, using a synodic reference system with the primaries fixed with respect to uniform rotating axes, with the Hamiltonian explicitly independent of time. However, it was soon discover that the bodies of most planets are not perfect sphere due to effect of gravity and rotation, but they are slightly flattened in the direction of axis of rotation. Hence CR3BP is unreliable where the long-time behavior of practically important dynamical systems are to be determined (Singh and Umar, 2012a).

Therefore, several authors prefer to study the motion of a test particle in ER3BP under different assumptions. These include (Narayan et.al. 2015; Umar and Hussain 2021; Singh and Tokyaa 2016; Abd El-Salam 2015). (Narayan et.al. 2015; Umar and Hussain 2021). They have studied the motion of a particle when the primaries are radiating and triaxial and found the equilibrium points to be unstable. The equilibrium points are determined from the eigen-values of the characteristic equation of the systems. The Eigen values explains the effect of petubations on the system. Radiation and triaxiality have been found by many researchers to induce instability at equilibrium points, particularly radiation as proved by (Singh and isah,2021).In this paper radiation is absent but Poyting-Robertson drag force and belt are present, (Singh and Taura 2014c ,Singh and Haruna 2020; Narayan et.al 2015;Vicent et.al 2022) have shown that belt is a stabilizing force. (Singh and Taura 2014c), developed a  CR3BP model consisting of a radiating- triaxial bigger primary and an oblate smaller primary  surrounded by a belt. The motion of the particle was analysed analytically and numerically around triangular equilibrium points, they observed that triangular points are stable for  and unstable for , where  is the critical mass ratio.It was also observed that the potential from the belt increases the range of stability.

Gravitational force is the dominant force in the classical circular restricted three body problem (CCR3BP).The effect of triaxiality, PR-Drag and belt can be studied when the classical potential function is generalized. Authors like have generalized the ER3BP in order to study the influence of these forces [9, 11, 14, 25,].

Circumbinary discs (belts, discs) are accretions of matter composed mainly of dust, gas, asteroids and planetesimals that orbit both the primary and secondary stars in binary systems. There is an increase in the study of   binary systems in the last decade, this is because many extra solar planetary systems show the presence of belts of dust particles which are considered as the young analogues of Kuiper belt. During investigation Trilling et. al. 2007) observe that about 60% of the observed 69 A3-F8 main sequence binary star system are surrounded by dust discs and  detected debris discs  in many main-sequence stellar  binary  system with the aid of spritzer telescope [24]. This model is tested using circumpolar Achird, also known as Eta Cassiopeiae, a binary star system situated in the northern constellation of Cassiopeia. Its primary component, Eta Cassiopeiae A, is a G-type main-sequence star of spectral type G0V, similar to the Sun. Its companion a secondary star, Eta Cassiopeiae B, is a K-type main-sequence star of spectral type K7V. It was reported by (Daniel 2017) that discs around a central binary system play an important role in star and planet formation and in the evolution of galactic discs [3].

A study conducted by Jiang and Yeh (2004), shows that the disc can influence locations of equilibrium points due to the effect of gravitational force from the belt on the infinitesimal mass. In another study by Jiang and Yeh (2006) and Yeh and Jiang (2006) they discovered the possibility of new equilibrium points in addition to the Lagragian points under certain conditions. The existence of these new equilibrium points was confirmed by (Kushvah 2008; Singh and Leke 2014a).The stability of equilibrium points under the effect of the disc in combination with other perturbing forces have been studied by many researchers.

The Poynting-Robertson (P-R) Drag force was named after John Henry Poynting and Howard Percy Robertson who discovered this force in 1903. The P-R Drag force is a component of radiation pressure force and acts tangentially to the grain’s motion. The force acts in opposite direction to the dust grain’s motion and causes a drop in its angular momentum. Many studies on P-R Drag force are always accompanied by radiating primaries this because the P-R Drag is a component of radiation force. However in this paper we assume radiation is absent so that we can observe the behaviour of the particle under the influence of the P-R Drag of the smaller primary.

  The P-R Drag force is a constituent of forces emanating from Doppler shift and absorption including the re-emitted incident radiation force. ‘This causes tiny particles to spiral into the sun at a cosmically rapid rate because the P-R drag force alone is not sufficient in giving an estimate of lifetimes for dust grain due to their large ratio of cross section to mass which could reduce the lifetimes for dust grains by enhancing the collision probabilities. Consequently the particles (grains) drift inwards on a timescale longer at a sub micrometre sizes and shorter at much larger sizes’ (Singh and Amuda 2019). Given the importance of these force  authors like: (Murray 1994;Ragos and Zafropoulus 1995;Das et al  2008)  study the effect of this force, while others such as (Singh and Amuda 2019),examined the stability of triangular points  under the combined effect of radiation, circumbinary disc and drag forces in CR3BP  and it was observed that when the mass ratio is less than the critical mass value in the presence of the disc the equilibrium points are stable in the linear sense ,but becomes unstable in the presence of radiation and P-R drag. An investigation by (Tajudeen et.al. 2021) reveals that, in the absence of radiation pressure, oblateness and centrifugal force the motion is stable, however, when the P-R drag of the bigger primary is effective and functional even by an almost negligible value the equilibrium points will be unstable. This is an extension of this work in ER3BP by studying the effect of P-R drag of the smaller primary alone. (Jaiyeola et.al. 2016; Tajudeen et.al. 2021; Singh and Simeon 2017) have studied PR-drag in R3BP with or without circumbinary disc. However our equation of motion is different from theirs, because our study is in the framework of ER3BP.Our equation of motion is a modification of (Umar and Hussain 2021) an elliptic case.

In this paper, our aim is to investigate the stability of TEPs of a test particle  under the  effect of triaxiality and P-R drag of the smaller primary surrounded by a circumbinary disc  in the ER3BP using both analytic method and energy potential of  the system. This paper is organized in 3 sections. The first section is introduction, the equations of motion, locations of equilibrium points and their linear stability can be found in section 2, and section 3 contains numerical applications, discussion and conclusion.

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