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Aishetu Umar, Aminu Abubakar Hussain, "Impacts of Poynting–Robertson Drag and Dynamical Flattening Parameters on Motion around the Triangular Equilibrium Points of the Photogravitational ER3BP", Advances in Astronomy, vol. 2021, Article ID 6657500, 11 pages, 2021. https://doi.org/10.1155/2021/6657500
Impacts of Poynting–Robertson Drag and Dynamical Flattening Parameters on Motion around the Triangular Equilibrium Points of the Photogravitational ER3BP
Using an analytical and numerical study, this paper investigates the equilibrium state of the triangular equilibrium points of the Sun-Earth system in the frame of the elliptic restricted problem of three bodies subject to the radial component of Poynting–Robertson (P–R) drag and radiation pressure factor of the bigger primary as well as dynamical flattening parameters of both primary bodies (i.e., Sun and Earth). The equations of motion are presented in a dimensionless-pulsating coordinate system , and the positions of the triangular equilibrium points are found to depend on the mass ratio and the perturbing forces involved in the equations of motion. A numerical analysis of the positions and stability of the triangular equilibrium points of the Sun-Earth system shows that the perturbing forces have no significant effect on the positions of the triangular equilibrium points and their stability. Hence, this research work concludes that the motion of an infinitesimal mass near the triangular equilibrium points of the Sun-Earth system remains linearly stable in the presence of the perturbing forces.
The study of the equilibrium state of an infinitesimal mass (a test particle) with regard to the dynamical system of the restricted three-body problem (R3BP) remains one of the most important and interesting aspects in the study of celestial mechanics and dynamical astronomy. The elliptic restricted three-body problem (ER3BP) deals with the description-study of the motion of an infinitesimal mass (), in the vicinity of the gravitational fields of two dominant bodies and which are called the primaries (bigger primary and smaller primary, respectively), where . These primary bodies revolve about their common centre of mass in elliptic orbits, under the influence of their mutual gravitational attraction. The ER3BP admits five equilibrium points at which the test particle (infinitesimal mass) would remain fixed if placed there. Three of such points lie on the line joining the two dominant bodies and , called the collinear equilibrium points , and are unstable, while the other two points form equilateral triangles with the two dominant bodies, called the triangular equilibrium points and , and are stable for , where is the mass ratio defined by and is the critical mass parameter .
In an attempt to have a much more realistic description of the motion of an infinitesimal mass over the decades, the classical R3BP has been modified in the sense that additional dynamical potentials of the system were considered in different approaches [2–8] and others.
Abouelmagd  in his study of the R3BP found out that the locations of the triangular points and their linear stability are affected by the oblateness of the more massive primary, up to the second zonal harmonic . In his numerical study, he concluded that the existence of sometimes does not affect the stability of the equilibrium points as in the Earth-Moon, Saturn-Phoebe, and Uranus-Caliban systems. Also, Abouelmagd et al.  examined the effects of oblateness of the three participating bodies together with small perturbations in Coriolis and centrifugal forces. Afterwards, Abouelmagd et al.  studied the effect of the first two even zonal harmonic coefficients of both the primaries as well as the periodic orbits around the equilibrium points. They found out that the triangular equilibrium points are linearly stable for and unstable for , while the collinear equilibrium points remain linearly unstable.
Singh and Tyokyaa  examined the stability of the triangular points in the ER3BP with oblateness up to the second zonal harmonic of both primaries. They concluded that the location and linear stability of the triangular points are affected by the oblateness of the primary bodies, eccentricity of the orbits of the primaries, and the semi-major axis of the system and both destabilized the system. In the same year, Singh et al.  investigated the influence of the zonal harmonics ( and ) of the primary and the radiation pressure of the secondary on the locations and linear stability of the triangular points. They however claimed that the parameters involved in the system affect the positions of the equilibrium points and destabilize the system as well.
The Doppler shift and absorptions and subsequent re-emission of incident radiation, that is, the so-called Poynting–Robertson’s (P–R) drag, are often neglected by many researchers in the estimation of light radiation force. Poynting  while studying radiation in the Solar System stated that radiation affects temperature and small bodies. He asserted that particles such as cosmic dust grain or small meteors are affected considerably by gravitational and light radiation force as they approach luminous celestial bodies. Furthermore, infinitesimal bodies in solar orbits suffer a gradual loss of angular momentum and ultimately spiral into the Sun. Later, Robertson  in a modified theory of Poynting considered only terms of the first order in the ratio of velocity of the particles to that of light. He investigated the dynamical effects of drag in the Solar System and derived the expression for the times of fall from circular orbits. Thus at a cosmically rapid rate, the P–R effect sweeps small particles of the Solar System into the sun.
Researchers like Burns et al. , Murray , Singh and Simeon , Alhussain , Chakraborty and Narayan , Amuda et al. , and others studied the R3BP by taking into account the P–R drag in different views. Mishra et al.  examined the stability of triangular points under the assumption that the bigger primary is a source of radiation with the incident P–R drag while the smaller is an oblate spheroid in the frame of the ER3BP. They concluded that the triangular points are unstable. In a recent study, Singh and Amuda  investigated the linear stability around of a test particle in the field of post-AGB binary system with the effective P–R drag force. They asserted that P–R drag and the mass parameter contribute in shifting the locations of the triangular points and the triangular points are unstable in the linear sense due to the presence of complex conjugate roots.
The aim of the present paper is to further investigate the effects of radiation pressure and P–R drag of the bigger primary on in the ER3BP, taking into account the effects of dynamical flattening parameters of the primary bodies.
2. Equations of Motion
Let , and be the three masses, and being the dominant bodies having an elliptic orbit about their common centre of mass, while being the infinitesimal mass which moves in the same plane with the dominant bodies under the influence of their force-field without influencing their motion. Let , , and denote the coordinates of , and , respectively, in the sidereal coordinates. Using Newton’s law, the equations of motion of an infinitesimal mass in the sidereal coordinates system can be represented aswhere ; and are the distances of an infinitesimal mass from and , respectively, and are defined as ; is the Gaussian constant of gravitation and the dot indicates differentiation with respect to time .
Now, we choose and rotate the synodic coordinates uniformly with a positive unit rate, which has the same origin at the centre of mass of the two dominant bodies with the sidereal coordinates. The direction of the axis is chosen such that the two dominant bodies always lie on it. Therefore, the equations of motion in sidereal coordinates are related to the equations of motion in synodic coordinates with respect to the true anomaly. By transforming the equations of motion from true anomaly to eccentric anomaly in a dimensionless-pulsating (rotating) coordinate system , we havewhere the prime represents differentiation with respect to eccentric anomaly and , , and are the eccentricity of the orbits and mean motion of the dominant bodies, while , are the distances of an infinitesimal mass from the bigger primary and smaller primary, respectively, and are defined as , , , and is the semi-major axis of around See [3, 24, 25].
Now, integrating equations (3) with respect to the eccentric anomaly and averaging, we getwhere is the potential-like function defined by .
Thus, the defined dynamical system in equations (4) is the required equations of motion for the ER3BP.
Let be the radiation pressure factor of the bigger primary which is given by
This implies that such that , where is the gravitational force of and is the radiation pressure of .
Considering the potential theory, the external gravitational potential due to a body that has axial symmetry can be written in terms of Legendre polynomials as (see de Pater and Lissauer  for more details), where denotes the mass of the body; denotes the radial distance from the centre of the particle to the centre of any other body; denotes the angle between the body’s symmetry axis and vector to the particle; denotes the mean radius of the body; denotes the dimensionless coefficient that characterizes the degree of nonspherical components of the potential, is zero for odd , and when is even, is called a zonal harmonic coefficient; and the term denotes the Legendre polynomials of degree and is defined by . We assume that the infinitesimal mass moves in the same plane of motion as the dominant bodies and this plane coincides with the equatorial plane . Therefore, the potential energy of the infinitesimal mass under the effects of the dynamical flattening parameters of both dominant bodies can be written aswhere and () represent the dynamical flattening parameters of the bigger and smaller primaries, respectively, and are the zonal harmonics coefficients; is defined as in equation (3); and and are the mean radii of the dominant masses and , respectively.
The distance between the dominant bodies is in the elliptic orbit, where are semi-major axis between the dominant bodies, common eccentricity of the dominant bodies, and eccentric anomaly, respectively.
And the mean distance between them is as follows:
Assuming that the dominant bodies are in elliptical motion with constant angular velocity (mean motion), then the orbits of and with respect to the centre of mass, with semi-major axes, would be and , respectively, having the same eccentricity; thus, the motion of the bigger and smaller primary can be written as [1, 28]where , , and are the mean motion, distance between the dominant bodies, and Gaussian constant of gravitation, respectively. The distance between the dominant bodies is defined to be the semi-major axis of the orbit (i.e., ), since the dominant bodies are in elliptic orbits.
Hence, adding equations (8) together, we obtain
Assume that the sum of the masses of the dominant bodies and the semi-major axis between them are the units of mass and length, respectively; i.e., , and . Also, the unit of time is chosen so as to make the Gaussian constant, . Hence, equation (9) becomes
Using equations (4), (5), (6), and (10), the equations of motion of an infinitesimal mass in the frame of the ER3BP can be modified, taking into account the dynamical flattening parameters of both dominant bodies together with the radiation pressure as well as P–R drag due to the bigger primary in a dimensionless-pulsating (rotating) coordinate system aswhereand is the dimensionless speed of light [18, 22, 29].
3. Locations of the Triangular Equilibrium Points
To obtain the equilibrium positions of an infinitesimal mass, the equations must be satisfied in the equations of motion (11); i.e., they are the solutions of the equations , and thus resulting inwhich can be rewritten as
In the absence of the dynamical flattening parameters and P–R drag (i.e., ), the solutions of equations (18) and (19) are . Then, considering the above parameters, the solutions of equations (18) and (19) would change slightly bywhere .
Substituting equations (20) in equations (18) and (19) together with help of equation (10), we obtain the series equations in terms of . Solving these equations by holding the expressions which contain , (since and , then , and have the same powers of mean radii of the dominant bodies), (where ) and also by restricting ourselves only to the quadratic terms in , we have
Using defined in equations (3), then the exact solutions of the triangular points are
Hence, equations (24) are the required locations of the triangular equilibrium points denoted by .
4. Stability of the Triangular Equilibrium Points
To examine the stability of the triangular equilibrium points, we place the infinitesimal mass at one of the equilibrium points and give it a small velocity. The point is stable for the oscillatory solutions with small amplitude and unstable for exponentially diverging solutions.
Assume that are the coordinates of the equilibrium points under consideration and let be the small displacements from these coordinates of the equilibrium points.
These can be written as
Now, using equations (25) and holding only the linear terms of Taylor’s theorem as the second and higher power of being very small, equation (11) becomewhere the superscript of equations (26) indicates that the partial derivatives are evaluated at the equilibrium points At equilibrium points, . Hence, equations (26) become
Equations (28) have a nontrivial solution if
Now, the second partial derivatives of the modified potential-like function at triangular equilibrium points are
Hence, the general expressions for the roots of the characteristic equation (30) arewhere
5. Numerical Application
In this section, we study numerically the locations and stability of the triangular equilibrium points of the Sun-Earth system, by taking into account the dynamical flattening parameters of both the Sun () and Earth (), the radiation pressure factor, and P–R drag of the Sun.
The first two even zonal harmonics of the Sun are and those of the Earth are . Also, the orbital eccentricity of the Earth is . Now, the dynamical flattening parameters of the primary bodies are given by , .
The radiation pressure factor of the Sun is defined as such that can be expressed as , where is the luminosity of the Sun, is the gravitational constant, is the speed of light, and is the mass per unit area. By using Stefan–Boltzmann’s law, the luminosity of the primary can be expressed as , where is the Stefan–Boltzmann’s constant (see ). Also, the dimensionless velocity of light and the mass ratio of the Sun-Earth system are given by and , respectively .
The modified equations of motion of an infinitesimal mass in the framework of the elliptic restricted three-body problem under the effects of dynamical flattening parameters of both primaries, radiation pressure factor, and P–R drag of the bigger primary (i.e., the Sun) are given in equations (11), while the locations and characteristic equation of the triangular points are given in equations (24) and (34), respectively.
Figure 1 shows the effects of the perturbing forces involved in the problem under consideration for the three different cases and classical case as well on the locations of triangular equilibrium points . Graph (a) is the classical case, graph (b) shows the effects of radiation pressure factor together with P–R drag, and graph (c) shows the effects of dynamical flattening parameters while graph (d) shows the combine effects of radiation pressure factor, P–R drag, and dynamical flattening parameters. This clearly shows that the perturbing forces under consideration have no significant effect on the locations of the triangular equilibrium points in the vicinity of the Sun-Earth system. These effects can only be seen in the table (see Table 1).
Table 2 shows numerical roots of the characteristic equation (30) for the classical case, as well as three other cases. In all cases, the characteristic roots reveal that all the roots are purely imaginary. This shows that the perturbing forces under consideration have no significant effect on the stability of the triangular equilibrium points . Hence, the motion of an infinitesimal mass near the triangular equilibrium points of the Sun-Earth system is stable in the linear sense under the influence of these perturbations.
For and , the present results of the triangular points in the circular case are in conformity with Singh and Taura  for in their results and for , which also agrees with those of Abouelmagd et al. ; the difference in configuration of the primary bodies is responsible for the difference in sign.
Their results are [5, 11]
In the case in the present work, the obtained results of the triangular points in the circular case are in agreement with those of Singh and Simeon  by taking in their results.
Their results are
Our results for the second partial derivatives and the characteristic equation differ from those of Singh and Taura , Abouelmagd et al. , and Singh and Simeon  due to the elliptic nature of our potential-like function. However, the P–R drag parts of the partial derivatives coincide with those of Singh and Simeon , that is, for Their results for the second partial derivatives are
Thus, for in the elliptic case of the triangular points, the obtained results coincide with those of Narayan and Shrivastava ; that is, for . Their results are
In the elliptic case, the obtained results do not agree with those of Singh and Umar  and Singh and Tyokyaa . This is because we have used the modified mean motion (equation (10)) which does not contain the eccentricity and semi-major axis . However, by substituting and , their results in the triangular case coincide fully with ours upon relaxing some parameters in our problem. The same applies to the second partial derivatives.
Their results are [12, 14]
The P–R drag finds importance in the investigation of the stability of zodiacal cloud, orbital evolution of cometary meteor steams, asteroidal particles, and dust rings around planets. This model is applicable not only to the Sun-Earth system but also to other systems in both the solar and stellar systems as well.
The data used to support the findings of this study are available from the corresponding author upon request.
It is part of work/research for the partial fulfilment of Ph.D. in Mathematics by the authors, respectively.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
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