Characteristics exponents of the triangular solution in the
elliptical restricted three body problem under the
radiation and oblateness of primaries
A.Narayan 1*, Amit Shrivastava 2, B. Ishwar 3
1 Department of Mathematics, Bhilai Institute of Technology, Durg,491001,India
2 Department of Mathematics, Rungta College of Engg. & Tech.,Bhilai-490020, India
3 Department of Mathematics, B.R.A. Bihar University, Muzaffarpur-842001, India
*Corresponding author E-mail: firstname.lastname@example.org
This paper studies effects of the oblateness and radiation of both the primaries on the stability of the infinitesimal motion about triangular equilibrium points (L4,5) in the elliptical restricted three body problem (ER3BP) around the binary system We have exploited analytical method for determining of characteristics exponent to the variational equations with periodic coefficients, developed by Bennet (! 965b), which is based on the Floquet's theory. The stability of the infinitesimal motion about the triangular points under the effects of radiation and oblateness of both the primaries around the binary systems Achird, Luyten726-8, Kruger 60, Alpha Centauri AB and Xi Bootis, has been studied. The stability of infinitesimal around the triangular points has been studied based on the analytical and numerical exploration is simulated by drawing transition curves bounding the region of stability in the (μ-e) plane. The region of stability changed with variations in eccentricity, oblateness and radiation pressures. It is observed that the equilibrium points stable in the shaded portion of the transition curve, whereas unstable outside the region of the transition curves.
The present paper is devoted to the analysis of the effects of the radiation and oblateness of both the primaries on the stability infinitesimal about the triangular equilibrium points of the planar ER3BP around the binary system. The ER3BP is a generalization of the classical problem. The eccentricity of the orbits plays a significant role which might not be seen in the circular case. The orbits of most celestial bodies are elliptical rather than the circular as such, the ER3BP describes the dynamical system more accurately.. We investigated the stability of triangular equilibrium points under the effects of the radiation and oblateness of both the primaries by exploiting analytical method for determining the characteristic exponents, which is based on the Floquet's theory.
The bodies of the elliptical restricted three body problem are generally considered to be spherical in shape, but in actual situations, it is observed that several heavenly bodies are either oblate spheroid or triaxial rigid bodies. The planets (Earth, Jupiter and Saturn) and stars (Archerner, Achird, Antares, Altair, Luyten etc.) are sufficiently oblate and are very significant in the study of celestial and stellar systems. The lacks of spherity of the heavenly bodies causes large perturbation. In addition to oblateness of heavenly body’s triaxiality, the radiation forces of the bodies, the atmospheric drag and the solar wind are also caused of perturbation.
This motivates our study, the stability of triangular equilibrium points under the influence of oblateness and radiation of the pri
maries in the elliptical restricted three body problem around the binary system.
The linear stability of the triangular equilibrium points of the ER3BP has been studied thoroughly by Danby (1964), Bennet (1965), Tschouner (1971), Roberts (1973), Meire (1981), Gyorgrey (1985), Kumar and Choudhary (1990), Khasan (1990, 1996), Markellos et al. (1992) with highlighting the transition curve separating region of stability in the (μ-e) plane. Conxita (1995), Jefferys (1965-1966), Selaru (1995) and Zsoft and Erdi (2003) have studied the different aspects of the same problem. Balint Erdi (2009) studied the parametric resonance stability around L4 in the ER3BP.
The influence of eccentricity of the orbits of the primaries with or without radiation pressure(s) on the existence of the equilibrium points and their stability was studied by Zimvoschikov and Thakai (2004), Markeev (2005), Ammar (2008). The influence of the eccentricity with oblateness and radiation parameters on the location and stability of collinear and triangular equilibrium points has been investigated by Narayan and Ramesh (2011-2012), Narayan and shrivastava (2012) and Jagadish and Umer (2012). Recently the linear stability of the triangular equilibrium points of the ER3BP has been studied by Narayan and Singh (2014a,2014b) and Narayan and Usha (2014).
The present study is devoted to the analysis of the stability of triangular points under radiating and oblate primaries by exploiting the analytical techniques developed by Bennet (1965). This method is mainly based on the Floquet’s theory for a system with periodic coefficients.
This paper is organized in five sections, section-1 describes introduction, section -2 provides the equations of motion, while section-3 describes the calculation of characteristic exponents and section-4 provides the graphical representation of transition curves, which are divided into stable and unstable regions. The discussion and conclusion are drawn in section-5.
2. Equations of motion
The differential equations of motion for the elliptical restricted three body problem under the oblate and radiating primaries in barycentric, pulsating and non-dimensional coordinates are represented (Jagadish and Umer, (2012) )as :
The mean of motion, n is given by
where and are the masses of the bigger and smaller primaries situated at the point and are the mass radiation factor; are the oblateness parameter of the primaries; are the distances of the infinitesimal mass from the bigger and smaller primaries respectively; while a and e are respectively the semi- major axis and eccentricity of the orbits.
The co-ordinate of triangular equilibrium point is represented (Jagadish and Umer (2012)) as:
In order to investigate the stability of the infinitesimal equilibrium point (2.5) in the first approximation, we derived the equations for the variation in the coordinates.
Let, denote the small displacement in;
Differentiating these with respect to v, we get;
Expanding equation (2.7) by Taylor’s theorem and retaining only upto the first order terms in the infinitesimal and, we get:
where and are the values of and at the equilibrium point which is given by (2.5).
At the equilibrium point we have:
Hence, the set of equation (2.1) with the help (2.8) and (2.9) is reduced to the following form:
Differentiating partially with respect to and evaluating ,and at the equilibrium point given by (2.5), which is obtained (Jagadish and Umer (2012)) as given below:
We have investigated the stability of triangular equilibrium points under the radiation and oblateness of primaries in the elliptical restricted three body problem This investigation is based on Floquet theory, which determines characteristic components in the system with periodic coefficients. The transformed variational equation of motion of elliptical restricted three body problem under the oblate and radiating primaries which is represented in matrix form, as given as:
3. Determination of characteristic exponents
In order to find solution of variational equation of motion of the system, we have exploited the Floquet’s theory of determining characteristic exponents of the system with periodic coefficients.
We seek the solution of the system of equation (2.12) in the form:
where yk is periodic with period 2π and λk are the characteristic exponents. Dropping the suffix in (3.1), we get:
Differentiating with respect to v, we get:
Using the equation (3.2), the variational equation of motion takes the form:
where is the unit matrix of the same order as that of.
Now, using the expressions, which are mentioned below:
and the corresponding matrix which is expanded as:
Substituting the equation (3.4) and (3.5) in (3.3), we get:
Equating the coefficients of terms with the same power of from (3.9) ,both the sides and using (3.3), we get:
If a constant vector is assumed for the zeroth order solution, then for the nth order equation of non-homogenous terms have frequencies up to the order of and including. we take the particular solution as:
From the equation (3.11), we have;
Substituting these values in the set of equations (3.10), we obtain a system of equations necessary for the determination of up to the order of given as follows:
From the equation (3.13), it is evident that for the existence of, it is necessary that
From the above relation, we get:
The relation for the exponent in the elliptical restricted three body problems can be obtained using the equation (3.17):
From the first equation of the system of equation (3.14), we observe that it is necessary that the determinant of the coefficient on the left with any column replaced by the non-homogenous terms on the right must be zero.
It is represented as follows:
Since enters as a factor in all elements of the replaced column, therefore
Since the determinant of the equation (3.20) is not zero, in general, we therefore conclude that
Again from the second and the third equation of (3.14), we have the solutions from are
Substituting this value of and from (3.23) in the last equation of (3.14),we get;
The matrices within the square bracket are complex conjugate so that only the real parts are considered, then equation (3.14) can be written as:
After some mathematical manipulations, we from (3.25) obtained the value of given by
Using this value of
We find the value of the parameter A, which is calculated as mentioned below;
where A is given by the equation (3.28). Hence the solution of this system becomes:
where is given by this equation (3.29).
4. Transition curves separating stable and unstable regions
The transition curves separating stable and unstable regions, which describes the stability of the triangular equilibrium points in the elliptical restricted three body problem under the stable primaries, can be found by simply equating the expression for the characteristic roots or exponents to the value of periodic solutions. In the range, the periodic solution provides
Replacing by in (3.30), we obtain:
Squaring both sides we obtain:
Now evaluating the values of A from the equation (3.28), and those of from (3.19), we evaluate easily for different values of.
The dependence of upon given by equation (4.2) is shown in graph for various values of oblateness parameters and. The triangular equilibrium points L4, and L5 are stable in the shaded region, whereas the region outside the shaded portion is unstable. The Fig 1, Fig2 and Fig 3 are depicting the stable and unstable region for the binary system Achird, similarly the Fig.4, Fig.5 Fig.6 are representing for Luyten, Fig.7, Fig.8 Fig.9 are representing for Alpha Cen-AB, Fig.10 Fig.11, Fig.12 are representing for Kruger-60 and , Fig.13 Fig.14, Fig.15 are representing for Xi-Booties.
The combined effects of the oblateness and radiation of the primaries around the binary system, introduced a visible left shift in the bifurcation points in each of the binary systems.
Fig. 1: Transition Curve of Achird-I
Fig. 2: Transition Curve of Achird-II
Fig. 3: Transition Curve of Achird-III
Fig. 4: Transition Curve of Luyten-I
Fig. 5: Transition Curve of Luyten-II
Fig. 6: Transition Curve of Luyten-III
Fig. 7: Transition Curve of Alphan Cen-AB-I
Fig. 8: Transition Curve of Alpha Cen-AB-II
Fig. 9: Transition Curve of Alpha Cen-AB-III
Fig. 10: Transition Curve of Kruger 60-I
Fig. 11: Transition Curve of Kuger 60-II
Fig. 12: Transition Curve of Kruger 60-III
Fig. 13: Transition Curve of Xi Booties-I
Fig. 14: Transition Curve of Xi Booties-II
Fig. 15: Transition Curve of Xi Booties-III
5. Discussion and conclusion
The effects of oblateness and radiation of the primaries on the stability of the triangular equilibrium points around the binary system in the elliptical restricted three body problem has been studied. The problem is studied under the assumption that the eccentricity of the orbit of the gravitating bodies is small. The oblateness and radiation of the more massive primary does not affect the motion of the smaller primary due its large mass, whereas affects the motion of infinitesimal body.
The stability of triangular equilibrium points in the elliptical restricted three body problem under the oblate and radiating primaries around the binary system has been investigated. We have exploited an analytical method for determining of characteristic exponent based on the Floquet's theory. The stability of the triangular equilibrium points under the effects of the radiation and oblateness of both the primaries around the binary system Achird, Luyten, Alpha Cen-AB, Kruger 60 and Xi-Bootis, has been studied, using simulation technique by drawing transition curves.
It is observed that the triangular equilibrium points remains stable in the shaded region of the traced transition curves and unstable in the region outside the shaded portion, around the binary system.
Fig. 16: Shifting Of Bifercation Point for Achird
Fig. 17: Shifting of Bifurcation Point for Luyten
Fig. 18: Shiftinf of Bifurcation Point Foralpha Cen-AB
Fig. 19: Shifting of Bifurcation Point for Kruger-60
Fig. 20: Shifting of Bifurcation Point for xi-Booties
It is also observed that there is a visible left shift in the bifurcation points in each of the binary system and visible right shift is due to the effects of oblateness of the primaries around the binary system, which is obvious from the fig.16, fig.17, fig.18, fig.19 and fig.20. Hence we arrived at the conclusion that the effects of oblateness and radiation of the primaries around the binary system responsible for right shift of bifurcation points.
 Ammar M. K (20008). “The effect of solar radiation pressure on the Lagrangian points in the elliptic restricted three body problems” Astrophys, space sci. Vol. 313, pp.393-408, http://dx.doi.org/10.1007/s10509-007-9709-z.
 Bennett, A. (1965). Characteristic Exponent of the five equilibrium solutions in the elliptically restricted problems. Icarus. 4:177-187. http://dx.doi.org/10.1016/0019-1035(65)90060-6.
 Conxita Pinyol, (1995) “Ejection collision orbits with the more massive primary in the planar elliptic restricted three body problem” Celest - Mech and Dyn Astro. Vol.-61,pp.315-331,
 Danby J.M.A.(1964) “Stability of the triangular points in elliptic restricted problem of three bodies”,The Astronomical Journal, vol. – 69, pp. 165-172,
 Erdi, B., Dajka, E.F., Nagy, I. and Rajnai, R. (2009). A parametric study of stability and resonances around L4 in the elliptical restricted three body problem. Celestial Mechanics and Dynamical Astronomy. DOI 1007/10569-009-9197-2.
 Gyorgrey J., (1985). “On the non-linear stability of motion’s around L5 in the elliptic restricted problem of the three bodies”, Celestial Mech., Dyn.Astro., vol. 36, no.-3, pp.281-285
 Khasan S.N. (1990) “Three dimensional periodic solutions to the radiational Hill problem” Cosmic research” Vol.34, no.5, pp.299-317.
 Khasan S.N, (1996)“Liberation Solutions to the radiational restricted three body problem”, Cosmic research” vol.34, no.-2, pp.146-151
 Kumar V. and Choudhary R.K. (1990) “Nonlinear stability of the triangular libration points for the photogravitational elliptic restricted problem of three bodies”, celestial Mech. and Dyn. Astro. Vol. 48, no. 4, pp. 299-317,
 Markeev A.P., (1978) “Libration points in celestial mechanics and cosmodynamics”, Nauk Moscow,
 Markeev A.P. (2005) “One special case of parametric resonance in problem of celestial mechanics” Astronomy letter vol 31, No. 5, pp. 300-356, 2005. http://dx.doi.org/10.1134/1.1922534.
 Markellos V.V., Papadakis K.E. and Perdios E.A. (1996). “Non-linear stability zones around triangular equilibria in the plane circular restricted three body problem with oblateness” Astrophysics and space science, vol.245.issue 1, pp 157-164). http://dx.doi.org/10.1007/BF00637811.
 Markellos V.V., Perdios, E. and Labropoulou P. (1992). “Linear Stability of the triangular equilibrium points in the radiational elliptical restricted problem” J: Astrophysics and Space Science,: pp. 207-213.
 Meire R. (1981) “The stability of the triangular points in the elliptical restricted problem.” Celestial Mechanics, Vol-23, pp. 89-95. http://dx.doi.org/10.1007/BF01228547.
 Narayan A. and Shrivastava Amit , (2012). “Effects of oblateness and radiation of primaries on the equilibrium points in the ellipted restricted three body problems.” International Journal of Mathematical Science, Vol. 32, issue-10, pp. 330-345,
 Narayan A. and Shrivastava Amit: “Existence of Resonance Stability of Triangular Equilibrium Points in Circular Case of the Planar Elliptical Restricted Three-Body Problem under the Oblate and Radiating Primaries around the Binary System” Advances in Astronomy; doi.org/10.1155/2014/287174(2014).
 Narayan A. and Singh N. (2014): “Motion and stability of triangular equilibrium points in elliptical restricted three body problem under the radiating primaries” Astrophysics Space sci. DOI10.1007/s.10509-014-1903-I.
 Narayan A. and Usha T (2014): “Effects of radiation and triaxiality of primaries on triangular equilibrium points in elliptic restricted three body problem” Astrophysics Space sci. DOI10.1007/s.10509-014-1818-x.
 Sandoor Zsoft and B. Erdi, (2003) “Sympletic mapping for the Trojan-type motion in the elliptic restricted three body problem”, Celest Mech and Dyn. Astro. vol. 86, pp.-301-319, 2003.
 Schauner T. (1971) “Die Bevegung in der Nach der Dreieckspu mkte des elliptischen eingeschrinkten Dreikorpen problems”. Celest. Mech.3, pp. 189-196, 1971. http://dx.doi.org/10.1007/BF01228032.
 Selaru D and Cucu-Dumitrescu C (1995). “Infinitesimal orbit around Lagrange points in the elliptic restricted three body problem”, Celest. Mech. Dyn. Astron. Vol. 61, no. 4, pp. 333-346, 1995. http://dx.doi.org/10.1007/BF00049514.
 Singh Jagdish and Umar Aishetu(2012) “On the stability of triangular points in the elliptical R3BP under the radiating and oblate primaries.” Astrophys Space Science DOI 10.1007/s10509-012-1109-3. http://dx.doi.org/10.1007/s10509-012-1109-3.
 Zimvoschikov A.S. And Thakai V.N. (2004) “Instability of libration points and resonance phenomena in the photogravitaional in the elliptical restricted three body problem.”Solar system Research, 38(2), 155-4. http://dx.doi.org/10.1023/B:SOLS.0000022826.31475.