1 Introduction

The restricted three-body problem (R3BP) describes the motion of a test particle having infinitesimal mass and moving under the gravitational effects of two primaries, which move in circular orbits around their center of mass on account of their mutual attraction and the test particle does not in any way influence the motion of the primaries. The R3BP constitutes one of the most important problems in dynamical astronomy and the study in its many variants has had important implications in several scientific fields, including, among others, galactic dynamics, chaos theory, molecular physics and celestial mechanics. The model of R3BP is still an active research field amongst scientists and astronomers because of its applications in lunar theory, artificial satellites and in the dynamics of the solar and stellar systems. The R3BP possesses five equilibrium solutions: three collinear and two triangular. The collinear points are unstable, while the triangular points are stable for the mass ratio \(\mu < 0.038521\) (Szebehely 1967).

The formulation of the classical R3BP did not consider the case when the test particle gravitates in the neighborhood of one or both radiating primaries under the influence of both gravitational and light radiation forces. An example is the motion of a dust grain near a binary star system in which one or both stars are radiating. Studies when one or both primaries are radiation source have been discussed under different assumptions by Simmons et al (1985), AbdulRaheem and Singh (2006), Singh and Leke (2010, 2014a, 2014b), Singh and Amuda (2014, 2017) among several others. In estimating the light radiation force, some models of the R3BP did not take into account all the forces arising from the Doppler shift and that of the absorption and subsequent re-emission of the incident radiation. These forces constitute the so-called Poynting–Robertson (P–R) effect. The P–R effect will operate to sweep small particles of the solar system into the Sun at a cosmically rapid rate. Authors like Schuerman (1980), Murray (1994), Ragos and Zafiropoulos (1995), Das et al. (2009), and Singh and Amuda (2014) have discussed the effect of this force under different assumptions.

Studies of planetary system have revealed some disc of dust particles, which are regarded as young analogues of the Kuiper Belt in our solar system. In stellar systems, this phenomenon is also valid. Out of an observed 69 A3-F8 main sequence binary star system, nearly 60% showed dust discs surrounding binary stars. Greaves et al. (1998) found a dust ring around a star e Eridani. Olivier et al. (2001) studied dust-enshrouded AGB stars in the solar neighborhood while Vassiliadis and Wood (1994) studied the post-asymptotic giant branch evolution of, low-to intermediate-mass stars. Circumbinary disc is commonly observed around post-AGB systems and are known to play an important role in their evolution. Trilling et al. (2007) detected debris discs in many main-sequence stellar binary systems using the Spitzer Space Telescope. Van Winckel et al. (2009) observed that circumbinary discs around a post-AGB star, which should have been formed during a strong binary interaction, have a major impact on the evolution of binary system, as their effects could shorten the AGB life time, judging from the measured orbits and mass function. IRAS 114712-0800, is an example of a highly evolved post-AGB star of spectral type F, with a large infrared excess produced by thermal emission of circumstellar dust and surrounded by a stable circumbinary disc (Van Winckel et al. 2012).

The white dwarf Giclas 29–38 (WD 2326 + 049; G29-38) has garnered intense interest since Zuckerman and Becklin (1987) discovered infrared emission from this object far in excess of the photosphere and proposed that the excess could be due to a brown dwarf companion. Kuchner et al. (1998) observed that the infrared excess of G29-38 is not due to a single orbiting companion and this supports the hypothesis that the source of the near-infrared excess is not a cool companion but a dust cloud. Reach et al. (2005) investigated the dust cloud around G29-38 using powerful instruments of Spitzer Space Telescope. These observations support the idea that a relatively recent disruption of a comet or asteroid created the dust cloud around the star G29-38 and suggests that the material around G29-38 is not accredited ISM but rather is indigenous to the star.

Hence, the study of the R3BP when the primaries are surrounded by disc is of great importance and its relevance in astronomy. In this premise, Jiang and Yeh (2004) explored the possible chaotic and regular orbits for disc–star–planet systems and found that the influence from the disc can change the locations of the equilibrium points and the orbital behavior. Further, Jiang and Yeh (2006), Yeh and Jiang (2006) discovered that, in addition to Lagrange points, new equilibrium points can exist under some conditions. Kushvah (2008) and, Singh and Leke (2014a) confirmed the existence of new equilibrium points. Kishor and Kushvah (2013) studied the linear stability and resonances in the generalized photogravitational Chermnykh-like problem with a disc, while Singh and Leke (2014b) discussed an analytical and numerical treatment of motion of dust grain particles around triangular equilibrium points with post-AGB binary star enclosed by a circumbinary disc.

In this study, our focus is set on analyzing the instability of triangular equilibrium points of a test particle under the gravitational effects of two radiating stars enclosed by a circumbinary disc coupled with the Poynting-Robertson drag effect. The paper is organized as follows: In Sect. 2, the equations governing the model are described while the location of the triangular equilibrium points is discussed in Sect. 3. The stability of the triangular equilibrium points is discussed in Sect. 4. The discussion and conclusion are drawn in Sect. 5 and 6, respectively.

2 Equations of the Dynamical System

Following Kushvah (2008), and Singh and Amuda (2014); the equations of motion of a passively gravitating test particle in a barycentric rotating coordinate system \(Oxyz\) under the frame of the circular R3BP when the primaries emit radiation pressure and P–R drag, have the form:

$$ \ddot{x} - 2\,\dot{y} = \Omega_{x} $$
$$ \ddot{y} + 2\dot{x} = \Omega_{y} $$
(1)
$$ \ddot{z} = \Omega_{z} . $$

where

$$ \begin{gathered} \Omega _{x} = x - \frac{{(1 - \mu )(x + \mu )\,q_{1} }}{{r_{1}^{3} }} - \frac{{\mu q_{2} (x + \mu - 1)}}{{r_{2}^{3} }} \hfill \\ \;\;\;\;\;\;\;\;\; - \frac{{W_{1} }}{{r_{1}^{2} }}\left[ {\frac{{(x + \mu )}}{{r_{1}^{2} }}\left( {(x + \mu )\dot{x} + y\dot{y} + z\dot{z}} \right) + \dot{x} - y} \right] \hfill \\ {\text{ }} - \frac{{W_{2} }}{{r_{2}^{2} }}\left[ {\frac{{(x + \mu - 1)}}{{r_{2}^{2} }}\left( {(x + \mu - 1)\dot{x} + y\dot{y} + z\dot{z}} \right) + \dot{x} - y} \right] \hfill \\ \end{gathered} $$
$$ \begin{gathered} \Omega _{y} = y\left[ {1 - \frac{{(1 - \mu )\,q_{1} }}{{r_{1}^{3} }} - \frac{{\mu \,q_{2} }}{{r_{2}^{3} }}} \right] \hfill \\ \;\;\;\;\;\;\;\; - \frac{{W_{1} }}{{r_{1}^{2} }}\left[ {\frac{y}{{r_{1}^{2} }}\left( {(x + \mu )\dot{x} + y\dot{y} + z\dot{z}} \right) + \dot{y} + (x + \mu )} \right] \hfill \\ {\text{ }} - \frac{{W_{2} }}{{r_{2}^{2} }}\left[ {\frac{y}{{r_{2}^{2} }}\left( {(x + \mu - 1)\dot{x} + y\dot{y} + z\dot{z}} \right) + \dot{y} + (x + \mu - 1)} \right] \hfill \\ \end{gathered} $$
$$ \begin{gathered} \Omega _{z} = - z\left[ {\frac{{(1 - \mu )q_{1} }}{{r_{1}^{3} }} + \frac{{\mu q_{2} }}{{r_{2}^{3} }}} \right] \hfill \\ \;\;\;\;\;\;\;\; - \frac{{W_{1} }}{{r_{1}^{2} }}\left[ {\frac{z}{{r_{1}^{2} }}\left( {(x + \mu )\dot{x} + y\dot{y} + z\dot{z}} \right) + \dot{z}} \right] \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\, - \frac{{W_{2} }}{{r_{2}^{2} }}\left[ {\frac{z}{{r_{2}^{2} }}\left( {(x + \mu - 1)\dot{x} + y\dot{y} + z\dot{z}} \right) + \dot{z}} \right] \hfill \\ \end{gathered} $$
$$ W_{1} = \frac{{(1 - \mu )(1 - q_{1} )}}{{c_{d} }},\,\,\,W_{2} = \frac{{\mu \,(1 - q_{2} )}}{{c_{d} }}. $$
$$ r_{1}^{2} = (x + \mu )^{2} + y^{2} ,\,\,\,\,\,\,\,\,\,r_{2}^{2} = (x + \mu - 1)^{2} + y^{2} . $$
(2)

Here,\(r_{1}\) is the distance of the test particle from the bigger primary of mass \(m_{1}\),while \(r_{2}\) is the distance from the smaller primary having mass \(m_{2}\). \(\mu\)\(\left( {0 < \mu < \frac{1}{2}} \right)\) is the mass parameter of the configuration and is expressed as the ratio of \(m_{2}\) to the sum \(m_{1} + m_{2}\). \(0 < q_{i} = \left( {1 - \frac{{F_{pi} }}{{F_{gi} }}} \right) < 1\,\,(i = 1,\,2)\) are mass reduction factors of the bigger and smaller primaries, respectively, where \(F_{pi}\) and \(F_{gi}\) are their radiation and gravitational forces. Thus \(q_{i} = 1\) shows the absence of the radiation pressure. Also, \(0 < 1 - q_{i} < < 1,\) while \(0 < W_{i} < < 1\,(i = 1,\,2)\) are the P–R drag of the bigger and smaller primaries, respectively, and \(c_{d}\) is the dimensionless velocity of light.

Next, we consider the influence of the gravitational potential from the circumbinary disc of dust and we use a model that best explains a flattened potential given by Miyamoto and Nagai (1975); which describes the gravitational potential within a system, and written as

$$ \overline{V}\left( {r,z} \right) = - \frac{{GM_{b} }}{{\sqrt {r^{2} + \left( {a + \sqrt {z^{2} + b^{2} } } \right)^{2} } }} $$
(3)

where G is the gravitational constant and \(M_{b}\) is the mass of the disc. a and b are parameters which determine the density profile of the disc; a is the flatness parameter while b is the core parameter. \(r = \sqrt {x^{2} + y^{2} }\) is the radial distance of the test particle.

This flattened system reduces to the Plummer model if \(a = 0\) and the Kuzmin disc if \(b = 0\). When \(a = b = 0\), the potential equals to the one with a point mass. The behavior of the two parameters having dimension of distance determines the two limiting cases. If \(a = 0\) the axial symmetry is reduced to its special case (the spherical one) while for \(b = 0\) the system has a collapse. In our problem, we consider motion in the xy-plane and so \(z = 0\). Therefore, Eq. (3) reduces to

$$ V\left( r \right) = - \frac{{GM_{b} }}{{\sqrt {r^{2} + \left( {a + b} \right)^{2} } }} $$
(4)

Now, using Eq. (4) and following the lead of Singh and Leke (2014b), the equations of motion (1), in the \(xy -\) plane is cast into the modified form:

$$ \ddot{x} - 2n\,\dot{y} = U_{x} $$
(5)
$$ \ddot{y} + 2n\dot{x} = U_{y} $$

where

$$ \begin{gathered} U_{x} = \,n^{2} x - \frac{{(1 - \mu )(x + \mu )\,q_{1} }}{{r_{1}^{3} }} - \frac{{\mu q_{2} (x + \mu - 1)}}{{r_{2}^{3} }} \hfill \\ \;\;\;\;\;\;\; - \frac{{W_{1} }}{{r_{1}^{2} }}\left[ {\frac{{(x + \mu )}}{{r_{1}^{2} }}\left( {(x + \mu )\dot{x} + y\dot{y}} \right) + \dot{x} - ny} \right] \hfill \\ \;\;\;\;\;\;\; - \frac{{W_{2} }}{{r_{2}^{2} }}\left[ {\frac{{(x + \mu - 1)}}{{r_{2}^{2} }}\left( {(x + \mu - 1)\dot{x} + y\dot{y}} \right) + \dot{x} - ny} \right] - \frac{{M_{b} x}}{{\left( {r^{2} + T^{2} } \right)^{{3/2}} }} \hfill \\ \end{gathered} $$
$$ \begin{gathered} U_{y} = y\left[ {n^{2} - \frac{{(1 - \mu )\,q_{1} }}{{r_{1}^{3} }} - \frac{{\mu q_{2} }}{{r_{2}^{3} }}} \right] \hfill \\ \;\;\;\;\;\;\; - \frac{{W_{1} }}{{r_{1}^{2} }}\left[ {\frac{y}{{r_{1}^{2} }}\left( {(x + \mu )\dot{x} + y\dot{y}} \right) + \dot{y} + n(x + \mu )} \right] \hfill \\ \;\;\;\;\;\;\;\; - \frac{{W_{2} }}{{r_{2}^{2} }}\left[ {\frac{y}{{r_{2}^{2} }}\left( {(x + \mu - 1)\dot{x} + y\dot{y}} \right) + \dot{y} + n(x + \mu - 1)} \right] - \frac{{M_{b} y}}{{\left( {r^{2} + T^{2} } \right)^{{3/2}} }} \hfill \\ \end{gathered} $$
$$ n = \sqrt {1 + \frac{{2M_{b} r_{C} }}{{\left( {r_{C}^{2} + T^{2} } \right)^{3/2} }}} $$
(6)

\(n\) is the mean motion of the primaries and is described by the potential from the disc, while \(T\) and \(r_{C}\) are the parameters which defines the density profile of the disc and the radial distance of the test particle in the classical R3BP, respectively.

3 Locations of Triangular Equilibrium Points

The locations of equilibrium points \(L_{i} \left( {i = 1,2...5} \right)\) of the R3BP are widely used in many branches of astronomy, and are important as they mark places where particles either can be trapped (\(L_{4}\) and \(L_{5}\)) or will pass through with a minimum expenditure of energy. The locations of triangular equilibrium points are found when the acceleration and velocity component of Eq. (5) are zero. That is, they are the solutions of the equations \(U_{x} = 0,\) \(U_{y} = 0\) and \(y \ne 0\) to get

$$ \,n^{2} x - \frac{{(1 - \mu )(x + \mu )\,q_{1} }}{{r_{1}^{3} }} - \frac{{\mu \,q_{2} (x + \mu - 1)}}{{r_{2}^{3} }} - \frac{{M_{b} x}}{{\left( {r^{2} + T^{2} } \right)^{3/2} }} + \frac{{W_{1} \,ny}}{{r_{1}^{2} }} + \frac{{W_{2} n\,y}}{{r_{2}^{2} }} = 0, $$
(7)

and

$$ \begin{gathered} \hfill \\ \left[ {n^{2} - \frac{{(1 - \mu )\,q_{1} }}{{r_{1}^{3} }} - \frac{{\mu \,q_{2} }}{{r_{2}^{3} }} - \frac{{M_{b} }}{{\left( {r^{2} + T^{2} } \right)^{3/2} }}} \right]y - \frac{{W_{1} n\,(x + \mu )}}{{r_{1}^{2} }} - \frac{{W_{2} n\,(x + \mu - 1)}}{{r_{2}^{2} }} = 0. \hfill \\ \end{gathered} $$
(8)

Now, when the P–R drag and disc are not present in Eqs. (7) and (8), we have

$$ x\left[ {1 - \frac{{(1 - \mu )\,q_{1} }}{{r_{1}^{3} }} - \frac{{\mu q_{2} }}{{r_{2}^{3} }}} \right] - (1 - \mu )\mu \left[ {\frac{{\,q_{1} }}{{r_{1}^{3} }} - \frac{{q_{2} }}{{r_{2}^{3} }}} \right] = 0, $$
(9)
$$ 1 - \frac{{(1 - \mu )\,q_{1} }}{{r_{1}^{3} }} - \frac{{\mu q_{2} }}{{r_{2}^{3} }} = 0. $$

Solving the above pair of equations respectively, we get

$$ r_{1} = q_{1}^{\frac{1}{3}} ,\,\,\,\,\,r_{2} = q_{2}^{\frac{1}{3}} $$
(10)

Now, Eq. (10) are the solutions of (9) when the primaries emit radiation pressure only. Therefore, using (10) we can assume the solutions of (7) and (8), to be

$$ r_{1} = q_{1}^{\frac{1}{3}} + \chi_{1} ,\,\,\,\,\,\,r_{2} = q_{2}^{\frac{1}{3}} + \chi_{2} ,\,\,\,\,\chi_{i} \left( {i = 1,\,2} \right) < < 1, $$
(11)

However, the exact \(x\)-coordinate of the triangular points is found by subtracting the second equation of (2) from the first to get

$$ x = \frac{1}{2} - \mu + \frac{{r_{1}^{2} - r_{2}^{2} }}{2} $$
(12)

Substituting Eq. (11) in(12), and ignoring second and higher order terms, we get

$$ x = x_{0} + q_{1}^{\frac{1}{3}} \chi_{1} - q_{2}^{\frac{1}{3}} \chi_{2} $$
(13)

where \(x_{0} = \frac{1}{2} - \mu + \frac{{q_{1}^{\frac{2}{3}} - q_{2}^{\frac{2}{3}} }}{2}\).


Now from \(y^{2} = r_{1}^{2} - (x + \mu )^{2} ,\) we get

$$ y = y_{0} \left[ {1 + \frac{1}{{2y_{0}^{2} }}\left( {q_{1}^{\frac{1}{3}} \chi_{1} + q_{2}^{\frac{1}{3}} \chi_{2} } \right) - \frac{1}{{2y_{0}^{2} }}\left( {q_{1}^{\frac{2}{3}} - q_{2}^{\frac{2}{3}} } \right)\,\left( {q_{1}^{\frac{1}{3}} \chi_{1} - q_{2}^{\frac{1}{3}} \chi_{2} } \right)} \right] $$
(14)

where

$$ y_{0} = \pm \frac{1}{2}\left[ { - 1 + 2\left( {q_{1}^{\frac{2}{3}} + q_{2}^{\frac{2}{3}} } \right) - \left( {q_{1}^{\frac{2}{3}} - q_{2}^{\frac{2}{3}} } \right)^{2} } \right]^{\frac{1}{2}} . $$

Next, substituting Eqs. (6), (11), (13) and (14) into Eqs. (7) and (8), and neglecting higher order terms of small quantities, we get respectively:

$$ a_{1} \chi_{1} + b_{1} \chi_{2} = c_{1} $$

and

$$ a_{2} \chi_{1} + b_{2} \chi_{2} = c_{2} $$

where

$$ a_{1} = 3x_{0} \,q_{1}^{{ - \frac{1}{3}}} - 3\mu x_{0} \,q_{1}^{{ - \frac{1}{3}}} + 3\mu \,q_{1}^{{ - \frac{1}{3}}} - 3\mu^{2} \,q_{1}^{{ - \frac{1}{3}}} $$
$$ b_{1} = - 3\mu \,q_{2}^{{ - \frac{1}{3}}} + 3\mu^{2} \,q_{2}^{{ - \frac{1}{3}}} + 3\mu \,x_{0} \,q_{2}^{{ - \frac{1}{3}}} $$
$$ c_{1} = - \frac{{2M_{b} r_{c} \,x_{0} }}{{\left( {r_{c}^{2} + T^{2} } \right)^{3/2} }} + \frac{{M_{b} (x_{0} + \mu )}}{{\left( {r_{c}^{2} + T^{2} } \right)^{3/2} }} - y_{0} q_{1}^{{ - \frac{2}{3}}} W_{1} - y_{0} q_{2}^{{ - \frac{2}{3}}} W_{2} $$
$$ a_{2} = 3y_{0} q_{1}^{{\frac{ - 1}{3}}} - 3\mu y_{0} q_{1}^{{\frac{ - 1}{3}}} $$
$$ b_{2} = 3\mu y_{0} q_{2}^{{\frac{ - 1}{3}}} $$
$$ \begin{gathered} c_{2} = - \frac{{2M_{b} r_{c} \,y_{0} }}{{\left( {r_{c}^{2} + T^{2} } \right)^{{3/2}} }} + \frac{{M_{b} y_{0} }}{{\left( {r_{c}^{2} + T^{2} } \right)^{{3/2}} }} \hfill \\ \;\;\;\;\;\;\; + x_{0} q_{1} ^{{ - \frac{2}{3}}} \,W_{1} + \mu \,q_{1} ^{{ - \frac{2}{3}}} \,W_{1} \hfill \\ \,\,\,\,\,\,\,\,\,\,\, + x_{0} q_{2} ^{{ - \frac{2}{3}}} \,W_{2} + \mu \,q_{2} ^{{ - \frac{2}{3}}} \,W_{2} - q_{2} ^{{ - \frac{2}{3}}} \,W_{2} \hfill \\ \end{gathered} $$

The values of \(\chi_{1}\) and \(\chi_{2}\) are found using the relations:

$$\chi_{1} = \frac{{b_{2} c_{1} - b_{1} c_{2} }}{{a_{1} b_{2} - a_{2} b_{1} }},\,\,\,\,\,\chi_{2} = \frac{{a_{1} c_{2} - a_{2} c_{1} }}{{a_{1} b_{2} - a_{2} b_{1} }} $$

to get

$$ \chi_{1} = - \frac{{W_{1} }}{3\sqrt 3 (1 - \mu )} - \frac{{2\,W_{2} }}{3\sqrt 3 (1 - \mu )} - \frac{{M_{b} (2\,r_{c} - 1)}}{{3\,(r_{c}^{2} + T^{2} )^{3/2} }} $$
$$ \chi_{2} = \frac{{2W_{1} }}{3\mu \sqrt 3 } + \frac{{W_{2} }}{3\mu \sqrt 3 } - \frac{{M_{b} (2\,r_{c} - 1)}}{{3\,(r_{c}^{2} + T^{2} )^{3/2} }} $$
(15)

Substituting Eq. (15) into Eqs. (13) and (14), and simplifying, we get

$$ \begin{gathered} x = \frac{1}{2} - \mu - \frac{1}{3}\left( {1 - q_{1} } \right) + \frac{1}{3}\left( {1 - q_{2} } \right) - \frac{{W_{1} (2 - \mu )}}{3\mu \sqrt 3 (1 - \mu )} - \frac{{W_{2} (1 + \mu )}}{3\mu \sqrt 3 (1 - \mu )} \hfill \\ y = \pm \left[ {\frac{\sqrt 3 }{2}\left( {1 - \frac{2}{9}\left( {1 - q_{1} } \right) - \frac{2}{9}\left( {1 - q_{2} } \right) + \frac{{4M_{b} (2\,r_{c} - 1)}}{{9(r_{c}^{2} + T^{2} )^{3/2} }}} \right) + \frac{{W_{1} (2 - 3\mu )}}{9\mu (1 - \mu )} + \frac{{W_{2} (1 - 3\mu )}}{9\mu (1 - \mu )}\,} \right]\, \hfill \\ \end{gathered} $$
(16)

The values of \(x\) and \(y\) obtained in Eq. (16) are called the triangular equilibrium points by virtue of the two triangles they form with line joining the primaries and are denoted by \(L_{4,5} (x, \pm y)\). The locations of these points depend on the mass ratio, radiation pressures, P–R drag and the components of the disc. The expansions (16) are different from those obtained by Jiang and Yeh (2006), due to the presence of radiation pressure and P–R drag. These points are also different from those in Kushvah (2008) due to the spherical nature of the primaries, radiation pressure of the smaller primary and P–R drag. These locations are also different from the one in Singh and Leke (2014b), because of triaxiality of the bigger primary, oblateness of the smaller one, radiation pressure and P–R drag of the primaries. Finally, Eqs. (16) differ from those of Singh and Taura (2014) due to the sphericity of both primaries, radiation pressure and P–R drag, while they are different from those in Singh and Amuda (2014) due to the presence of a disc, P–R drag and sphericity of the second primary. If in Eq. (16), we ignore the P–R drag \(\left( {W_{i} = 0} \right)\) and suppose there is no circumbinary disc \(\left( {M_{b} = 0} \right)\), the triangular equilibrium points will reduce to those of the photogravitational R3BP of Radzievsky (1950, 1953). Further, if the primaries are black bodies \(\left( {q_{i} = 1} \right)\), then the triangular points of the classical R3BP (Szebehely 1967) are recovered.

Now, using the software Mathematica to evaluate the mass ratio for the binary IRAS 11472-0800-G29-38 in Appendix 1 and then numerically evaluate the locations of the triangular points given by Eq. (16) for some assumed values of the radiation pressure, P–R drag and parameters of the disc, in Appendix 2 and 3. The graphs of the locations of triangular points are presented in Fig. 1, which shifts nearer to or away from the primaries depending on the system parameters.

Fig. 1
figure 1

Locations of triangular points \(L_{4,5}\) for \(M_{b} = 0.09,\)\(c_{d} = 46393.84,\) \(q_{1} = 0.99992\),\(q_{2} = 0.99996\) when \(\mu = 0.118166\)(BLACK), \(\mu = 0.308756\)(GREEN) and \(\mu = 0.450617\) (RED)

Full size image

4 Instability of Triangular Equilibrium Points

The linear stability of an equilibrium configuration is the ability to restrain the body motion near the equilibrium point. To examine the stability of the triangular equilibrium points of our problem, we displace the test particle a little away from an equilibrium point with a small velocity. If the particle rapidly departs from the vicinity of the triangular points, then the points is said to be in an unstable location. If, however, the body keeps oscillating about the location, then such location is said to be a stable location.

Now, let the location of an equilibrium points be denoted by \((a_{0} ,b_{0} )\) and consider a small displacement \((\xi ,\eta )\) from the point such that \(x = a_{0} + \xi\) and \(y =\)\(b_{0} + \eta\). Substituting these values in the equations of motion (5), we obtain the variational equations

$$ \begin{gathered} \ddot{\xi } - 2n\dot{\eta } = (U_{{x\dot{x}}} )^{0} \dot{\xi } + (U_{{x\dot{y}}} )^{0} \dot{\eta } + (U_{xx} )^{0} \xi + (U_{xy} )^{0} \eta \hfill \\ \ddot{\eta } + 2n\dot{\xi } = (U_{{y\dot{x}}} )^{0} \dot{\xi } + (U_{{y\dot{y}}} )^{0} \dot{\eta } + (U_{yx} )^{0} \xi + (U_{yy} )^{0} \eta \hfill \\ \end{gathered} $$
(17)

Here, only linear terms in \(\xi\) and \(\eta\) have been taken. The second order partial derivatives of \(U\) are denoted by subscripts. The superscript 0 indicates that the derivatives are to be evaluated at the equilibrium point \((a_{0} ,\,b_{0} ).\)

The characteristic equation corresponding to Eq. (17) is

$$ \lambda^{4} + a\lambda^{3} + b\lambda^{2} + c\lambda + d = 0 $$
(18)

where

$$ a = - (U^{0}_{{y\dot{y}}} + U^{0}_{{x\dot{x}}} ) $$
$$ b = 4n^{2} + U^{0}_{{x\dot{x}}} U^{0}_{{y\dot{y}}} - U^{0}_{xx} - U^{0}_{yy} - (U^{0}_{{x\dot{y}}} )^{2} $$
$$ c = U^{0}_{{x\dot{x}}} U^{0}_{yy} + U^{0}_{xx} U^{0}_{{y\dot{y}}} + 2nU^{0}_{xy} - 2nU^{0}_{yx} - U^{0}_{{y\dot{x}}} U^{0}_{xy} - U^{0}_{yx} U^{0}_{{x\dot{y}}} $$
$$ d = U^{0}_{xx} U^{0}_{yy} - U^{0}_{yx} U^{0}_{xy} $$

Evaluating the second order partial derivatives at the equilibrium points and neglecting higher order terms, we get

$$ U_{{x\dot{x}}}^{0} = - (x_{0} + \mu )^{2} W_{1} - W_{1} - (x_{0} + \mu - 1)^{2} W_{2} - W_{2} $$
$$ U_{{x\dot{y}}}^{0} = - (x_{0} + \mu )y_{0} W_{1} - (x_{0} + \mu - 1)y_{0} W_{2} = U_{{y\dot{x}}}^{0} $$
$$ U_{{y\dot{y}}}^{0} = - (y_{0}^{2} + 1)W_{1} - (y_{0}^{2} + 1)W_{2} $$
$$ \begin{gathered} U_{xx}^{0} = 3x_{0}^{2} - 3\mu^{2} + 3\mu + \left( {2x_{0}^{2} + 4\mu x_{0} + 2\mu^{2} - 2\mu^{2} x_{0} - 4\mu^{2} x_{0} - 2\mu^{3} } \right)\left( {1 - q_{1} } \right) \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\, + \left( {2\mu x_{0}^{2} + 2\mu^{3} + 4\mu^{2} x_{0} - 4\mu x_{0} - 4\mu^{2} + 2\mu } \right)\left( {1 - q_{2} } \right) \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\, + \left( {3 - 3\mu - 15\mu^{2} + 15\mu^{3} + 6x_{0} - 15x_{0}^{2} - 30\mu x_{0} + 15\mu x_{0}^{2} + 30\mu^{2} x_{0} } \right)\,\chi_{1} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\, + \left( { - 12\mu + 30\mu^{2} - 15\mu^{3} - 6x_{0} - 15\mu x_{0}^{2} + 30\mu x_{0} - 30\mu^{2} x_{0} } \right)\,\chi_{2} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\, + \frac{{5M_{b} \left[ {2r_{c} - 1} \right]}}{{4(r_{c}^{2} + T^{2} )^{3/2} }} + \frac{{3M_{b} \left[ {1 - 4\mu + 4\mu^{2} } \right]}}{{4(r_{c}^{2} + T^{2} )^{5/2} }} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, - \left( {2x_{0} y_{0} + 2\mu y_{0} } \right)\,W_{1} - \left( {2x_{0} y_{0} + 2\mu y_{0} - 2y_{0} } \right)\,W_{2} \hfill \\ \end{gathered} $$
$$ \begin{gathered} U_{xy}^{0} = 3x_{0} y_{0} + \left( {2x_{0} y_{0} - 2\mu y_{0} - 2\mu \,x_{0} y_{0} - 2\mu^{2} y_{0} } \right)\left( {1 - q_{1} } \right) \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \left( {2\mu x_{0} y_{0} + 2\mu^{2} y_{0} - 2\mu y_{0} } \right)\left( {1 - q_{2} } \right) \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \left( {3y_{0} + \frac{{3x_{0} }}{{2y_{0} }} - 15x_{0} y_{0} - 15\mu y_{0} + 15\mu x_{0} y_{0} + 15\mu^{2} y_{0} } \right)\chi_{1} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \left( { - 3y_{0} + \frac{{3x_{0} }}{{2y_{0} }} + 15\mu y_{0} - 15\mu x_{0} y_{0} - 15\mu^{2} y_{0} } \right)\chi_{2} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \frac{{11\sqrt 3 M_{b} \,\left[ {1 - 2\mu } \right]\left( {1 - 2\mu } \right)}}{{12(r_{c}^{2} + T^{2} )^{3/2} }} + \frac{{3\sqrt 3 M_{b} \,\left[ {1 - 2\mu } \right]}}{{4(r_{c}^{2} + T^{2} )^{5/2} }} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + (1-2y^2_0)W_1 + (1-2y^2_0)W_2 \hfill \\ \end{gathered} $$
$$ \begin{gathered} U_{yx}^{0} = 3x_{0} y_{0} + \left( {2x_{0} y_{0} - 2\mu y_{0} - 2\mu \,x_{0} y_{0} - 2\mu^{2} y_{0} } \right)\left( {1 - q_{1} } \right) \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \left( {2\mu x_{0} y_{0} + 2\mu^{2} y_{0} - 2\mu y_{0} } \right)\left( {1 - q_{2} } \right) \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \left( {3y_{0} + \frac{{3x_{0} }}{{2y_{0} }} - 15x_{0} y_{0} - 15\mu y_{0} + 15\mu x_{0} y_{0} + 15\mu^{2} y_{0} } \right)\chi_{1} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \left( { - 3y_{0} + \frac{{3x_{0} }}{{2y_{0} }} + 15\mu y_{0} - 15\mu x_{0} y_{0} - 15\mu^{2} y_{0} } \right)\chi_{2} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \frac{{11\sqrt 3 M_{b} \,\left[ {1 - 2\mu } \right]\left( {1 - 2\mu } \right)}}{{12(r_{c}^{2} + T^{2} )^{3/2} }} + \frac{{3\sqrt 3 M_{b} \,\left[ {1 - 2\mu } \right]}}{{4(r_{c}^{2} + T^{2} )^{5/2} }} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + 2\left( {x_{0} + \mu } \right)^{2} W_{1} - W_{1} + 2\left( {x_{0} + \mu - 1} \right)^{2} W_{2} - W_{2} \hfill \\ \end{gathered} $$
$$ \begin{gathered} U_{yy}^{0} = 3y_{0}^2 + \left( {2y_{0}^2 - 2\mu y_{0}^2} \right)\left( {1 - q_{1} } \right) \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \left( {2\mu y_{0}^2} \right)\left( {1 - q_{2} } \right) \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \left( {6-3\mu-15y_{0}^2+15\mu y_{0}^2} \right)\chi_{1} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \left( {3+ 3\mu-15\mu y_0^{2}} \right)\chi_{2} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \frac{{7 M_{b} \,\left[ {2{r_c}-1} \right]}}{{4(r_{c}^{2} + T^{2} )^{3/2} }} +\frac{{9M_{b}}}{{4(r_{c}^{2} + T^{2} )^{5/2} }}\hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + 2\left( {x_{0}y_{0} + 2\mu y_0} \right) W_{1} + 2\left( {x_{0}y_{0} + 2\mu y_0}-2y_0 \right) W_{2} \hfill \\ \end{gathered} $$

Now, substituting these partial derivatives in Eq. (18) and simplifying, we have

$$ a = 3\left( {W_{1} + W_{2} } \right), $$
$$ b = 1 + \frac{{\left( {2r_{c} + 3} \right)M_{b} }}{{(r_{c}^{2} + T^{2} )^{3/2} }} - \frac{{3\left( {\mu^{2} - \mu + 1} \right)M_{b} }}{{(r_{c}^{2} + T^{2} )^{5/2} }} + \frac{{\sqrt 3 W_{1} }}{3} - \frac{{\sqrt 3 W_{2} }}{3}, $$
$$ c = \left( {3 + \frac{9}{4}\mu } \right)W_{1} + \left( {\frac{21}{4} - \frac{9}{4}\mu } \right)W_{2} , $$
$$ \begin{gathered} d = \frac{27}{4}\mu \left( {1 - \mu } \right) + \frac{3}{2}\mu \left( {1 - \mu } \right)\left( {1 - q_{1} } \right) + \frac{3}{2}\mu \left( {1 - \mu } \right)\left( {1 - q_{2} } \right) + \frac{{33\mu \left( {1 - \mu } \right)\left( {2r_{c} - 1} \right)M_{b} }}{{2(r_{c}^{2} + T^{2} )^{3/2} }} + \frac{{27\mu \left( {1 - \mu } \right)M_{b} }}{{4(r_{c}^{2} + T^{2} )^{5/2} }} \hfill \\ \begin{array}{*{20}c} {} & {} \\ \end{array} + \frac{{9\sqrt 3 \left( {3\mu - 2} \right)W_{1} }}{4} + \frac{{9\sqrt 3 \left( {3\mu - 1} \right)W_{2} }}{4} \hfill \\ \end{gathered} $$

The principal diagonal minors of the Hurwitz matrix can be written as

$$ \Delta_{1} = 3\left( {W_{1} + W_{2} } \right) > 0 $$
$$ \Delta_{2} = - \frac{3}{4}\mu \,W_{1} - \frac{3}{4}(1 - \mu )\,W_{2} < 0 $$
$$ \Delta_{3} = \left[ {3\left( {1 + \frac{3}{4}\mu } \right)\,W_{1} + \frac{3}{4}\left( {7 - 3\mu } \right)\,W_{2} } \right]\,\Delta_{2} < 0 $$
$$ \Delta_{4} = d\,\Delta_{3} < 0. $$

By Routh-Hurwitz criterion, the existence of negative minors implies that the motion around triangular points is unstable.

We perform a numerical exploration to find the roots of the characteristic Eq. (18) by adopting the system IRAS 11,472-0800-G29-38 with some assumed values of parameters. The roots \(\lambda_{k} \left( {k = 1,\,2,\,3,\,4} \right)\) in Appendix (4, 5, 6, 7, 8) under different classifications are numerically evaluated. In Appendix 4 and 5, we evaluate the roots for the range of the mass parameter \(0.118166616 \le \mu \le 0.450617\), mass of the disc \(0 \le M_{b} \le 0.09\), \(c_{d} = 46939.84\) when \(q_{1} = 0.99992\), \(q_{2} = 0.99996\) and \(q_{i} = 1\), respectively, while in Appendix 6, we evaluate when \(M_{b} = 0.04.\) In these cases, the existence of complex characteristic roots shows the motion near triangular points are unstable.

To further analyze the behavior of the system parameters: mass ratio \(\left( \mu \right)\), radiation pressure \((q_{i} )\), P–R drag \(W_{i} \left( {i = 1,\,2} \right)\) and the component of the disc \(\left( {M_{b} } \right)\); we consider the range of the mass parameter to be \(0.000003 \le \mu \le 0.450617\) in Appendix 7 and 8. The idea is to be able to discern the effects of these parameters on the instability outcome of the triangular points. Therefore, in Appendix 7 we evaluate these roots under effect of mass of the disc but in the absence of radiation pressure of both primaries. In the case when \(0.000003 \le \mu \le 0.038520\) all the roots are pure imaginary and the triangular point is stable. However, when \(\mu = 0.038521\), those roots are complex in the absence of the disc mass (\(M_{b} = 0\)) but are all imaginary quantities when the mass of the disc is present. This shows that the disc can convert the instability behavior to stability; hence it is a stabilizing force in the absence of radiation force.

Next, we introduce the radiation forces of the primaries to our analysis of the characteristic roots. The computations of the roots when \(q_{1} = 0.99992\) and \(q_{2} = 0.99996\) coupled with the P–R drag are tabulated in Appendix 8. In this case, all the roots are complex in the entire range of the mass parameter \(0.000003 \le \mu \le 0.450617\). This shows the motion is unstable weather the disc is present or absent. Thus, in the presence of radiation associated with P–R drag, the mass of disc is unable to change the instability behavior into stability. Hence, the radiation associated with P–R drag is a destabilizing force.

5 Discussion

This paper presents a study of motion of around triangular equilibrium points of the restricted problem of three bodies under the effects of circumbinary disc, radiation and drag forces. The test particle is assumed to have infinitesimal mass and moves under the gravitational and radiational influence of two massive bodies enclosed by a circumbinary disc with effective Poynting-Robertson drag forces. The governing equations of motion of the system are determined by the mass of the disc, mass ratio, radiation and P–R drag forces. The model equations differ from those derived by Jiang and Yeh (2006), Kushvah (2008), Singh and Leke (2014a, b), Singh and Taura (2014), and Singh and Amuda (2017).

The equilibrium solutions of the restricted three-body problem mark places where particles either can be trapped (L4 and L5) or will pass through with a minimum expenditure of energy. We have found the location of the triangular equilibrium points given in Eq. (16). These points are influenced by the mass of the disc, radiation and P–R effects. Using the software Mathematica, we numerically evaluate the locations of the triangular equilibrium points for the binary system IRAS11472-0800-G29-38. We also give graphical representations to demonstrate the effects of the mass of the disc mass, radiation and P–R drag effects.

In the R3BP, for all values of the free parameters, the triangular equilibrium points \(L_{4,5}\) may be stable when \(\mu \le \mu_{c} = 0.038520\) where \(\mu_{c}\) is the critical mass parameter. We linearized the equations of motion and obtained the characteristic Eq. (18). Further, we evaluated in Appendix 4, 5, 6, 7, 8, the roots of (18) under different classifications. In Appendix 4, the roots are estimated under the combined effects of the mass of the disc, radiation and P–R drag forces for the range of the mass parameter \(0.118166616 \le \mu \le 0.450617\). In this case, all the roots are complex and this induces instability at the triangular points. Hence, the triangular point is unstable for the binary IRAS11472-0800-G29-38.

In order to investigate the behavior of the system parameters: mass ratio \(\left( \mu \right)\), radiation pressure \((q_{i} )\), P–R drag \(W_{i} \left( {i = 1,\,2} \right)\) and the mass of the disc \(\left( {M_{b} } \right)\); we broaden the range of the mass parameter to be \(0.000003 \le \mu \le 0.450617\) in Appendix 7 and 8. The idea is to be able to get a better understanding of the effects the system parameters pose on the instability around the triangular equilibrium points. In Appendix 7, the roots are obtained under effect of mass of the disc but in the absence of radiation pressure and P–R drag of both primaries. In the case when \(0.000003 \le \mu \le 0.038520\) all the roots are pure imaginary and the triangular point is stable. However, when \(\mu = 0.038521\), the four roots are complex in the absence of the disc (\(M_{b} = 0\)) but are all imaginary quantities when the mass of the disc is present. This shows that the disc is a stabilizing force and increases region of stability. This is in agreement with the observations of Singh and Leke (2014b).

Further, we introduced the radiation forces of the primaries to our analysis of the characteristic roots and all the roots are complex in the entire range of the mass ratio; even when \(\mu < \mu_{c}\). Hence, the radiation pressure and P–R drag induce instability at the triangular equilibrium point and makes the location an unstable one. Hence, the component of the radiation force is strongly a destabilizing force and agrees with AbdulRaheem and Singh (2006), Singh and Leke (2010). Hence, we conclude that the triangular equilibrium points are in general an unstable location under the combined effects of radiation pressure, P–R drag and disc.

6 Conclusions

This study investigates motion of a test particle around triangular equilibrium points of the CR3BP when the primaries emit radiation pressure and drag forces, and are enclosed by a circumbinary disc. The equations of motion and locations of the triangular equilibrium points have been obtained. It is seen that they are influenced by the mass of the circumbinary disc, radiation pressure and P–R drag of the primaries. Numerically, we evaluated the locations of the triangular equilibrium points for the binary system IRAS11472-0800-G29-38, and analyzed the stability behavior. It is seen that these points are in general an unstable equilibrium point in the linear sense. The instability of the triangular points is greatly influenced by the radiation pressure and P–R drag forces as they dominate the stabilizing tendency of the disc.