A Study of Albedo Effects on Libration Points in the Elliptic Restricted Three-Body Problem

Abstract

This paper deals with the libration points in the elliptic restricted three-body problem. The combined effects of the primary bodies’ orbital eccentricity and albedo on the existence and stability of libration points is analyzed. It is found that there exist five libration points, three collinear and two non-collinear. The non-collinear libration points are stable for a critical value of mass parameter μ_c, the collinear libration points are unstable for all values of μ.

Introduction

Albedo effect is one of the most interesting non-gravitational force having significant effects on the motion of an infinitesimal mass. Albedo is the fraction of solar energy reflected diffusely from the planet back into space [13]; the measure of the reflectivity of the planet’s surface. Therefore, the Albedo can be defined as the fraction of incident solar radiation returned to the space from surface of the planet [31],

$$ Albedo=\frac{radiation\kern0.17em reflected\kern0.17em back\kern0.17em to\kern0.34em space}{incident\kern0.17em radiation}. $$

Albedo is a dimensionless quantity measured on a scale from 0 to 1. A body or surface that has zero albedo indicates the body is a ‘black-body’, which absorbs all the incident radiations. An albedo value of unity represents a ‘white-body’, which is a perfect reflector that reflects all incident radiations completely and uniformly in all directions. A high-albedo surface has lower temperature because it reflects the majority of the radiation that hits it. On the other hand, a low-albedo surface has higher temperature as it absorbs more of the incoming radiation. For instance, fresh snow has a high albedo of 0.95 as it reflects 95% of incoming radiation, while water reflects about 10% of the incoming radiation, resulting in a low albedo of 0.1. On average, the albedo of Earth is 0.3 as 30% of incident solar radiation is reflected by the entire Earth. Generally, dark surfaces have a low albedo and light surfaces have a high albedo.

In previous studies, authors did not consider the effect of reflected radiations upon the spacecraft. As this effect is much less than the direct radiation effect known as the photogravitational effect, it was generally neglected by authors in recent decades. If this effect is neglected, it means the primaries are considered as black-bodies, which is a contradiction to the fact that there is no planet in our solar system whose albedo is zero. No planet in our solar system is a black-body. The planets with their respective average albedos are as follows:

Planet	Mercury	Venus	Earth	Mars	Jupiter	Saturn	Uranus	Neptune
Albedo	0.12	0.75	0.30	0.16	0.34	0.34	0.30	0.29

In the light of all above facts, we have decided to develop a new model for the elliptic restricted three-body problem in which one primary is a source of radiation and the other a non-black-body. Albedo is studied by Anselmo et al. [3]; Nuss [28]; McInnes [24]; Bhanderi and Bak [5]; Pontus [29]; MacDonald and McInnes [21], etc. Idrisi [14, 15] considered the albedo effect in the circular restricted three-body problem considering the primaries as point masses, and found that the albedo not only effects the location of libration points but also the stability. Idrisi and Shahbaz Ullah (2017) studied the same problem with one more parameter, i.e. the oblateness of the smaller primary. Again, Idrisi [14, 15] studied the albedo effect on libration points in the circular restricted three-body problem taking into account the triaxiality of the smaller primary.

In this paper we are interested in investigating the albedo effect on the libration points in the elliptic restricted three-body problem (ER3BP) when the primaries are point masses but not black-bodies. The elliptic restricted three-body problem is a generalization of the circular restricted three-body problem in which two bodies with finite masses called primaries move around their center of mass in elliptic orbits under the influence of their mutual gravitational attraction, and an infinitesimal mass moving in the plane formed by the primaries is attracted and influenced by their motion without influencing them. There exist three collinear and two non-collinear libration points. The collinear libration points L₁, L₂ and L₃ are unstable for all values of mass parameter μ (0 ≤ μ ≤ ½) while the non-collinear libration points L₄,₅ are stable for a critical value of mass parameter μ < μ_c = 0.03852 …., Szehebely (1967). The ER3BP has been described in detailed by Danby [8], Bennett [4], Szebehely [36] and Markeev [22]. The influence of the eccentricity of the orbits of the primaries with or without radiation pressure on the existence and stability of the equilibrium points was studied by Gyorgyey [12], Kumar and Choudhry [17], Markellos et al. [23], Zimovshchikov and Tkhai [38], Ammar [2], Kumar and Ishwar [18], Narayan and Kumar [25], Singh and Umar [35], Narayan and Usha [26], Abd-El-Salam [1], Narayan et al. [27].

Some of the notable researches in photogravitational restricted three-body problem (PRTBP) are carried by Radzievskii [30]; Chernikov [7]; Schuerman D.W [32]; Simmons et. al. [34]; Kunitsyn and Tureshbaev [19]; Lukyanov [20]; Sharma [33]; Xuetang et.al. [37]; Grun et.al. [11]; Douskos [9]; Ershkov [10]; Katour et.al. [16] etc. This paper is divided into six sections. In section 2, the equations of motion are derived. The mean-motion of the primaries is obtained in section 3. The existence of non-collinear and collinear libration points is shown in section 4. In section 5, the stability of non-collinear and collinear libration points is discussed. Section 6 concludes the document.

Equations of Motion

Let m₁ and m₂ be the masses of the primaries such that m₁ is a source of radiation and m₁ > m₂. The primaries move in elliptic orbits around their center of mass O. An infinitesimal mass m₃ < < 1, moves in the plane of motion of m₁ and m₂. The vectors from m₁, m₂ and O to m₃ are r₁, r₂ and r, respectively. F₁ and F₂ are the gravitational forces acting on m₃ due to m₁ and m₂, respectively. F_p is the force due to solar radiation pressure by m₁ on m₃ and F_A is the Albedo force due to solar radiation reflected by m₂ on m₃. Let the line joining m₁ and m₂ be taken as the X –axis and O, their center of mass, as origin. Let the line passing through O and perpendicular to OX and lying in the plane of motion of m₁ and m₂ be the Y –axis (Fig. 1). Let us consider a synodic system of co-ordinates Oxyz initially coincident with the inertial system OXYZ, rotating with angular velocity $\dot{f}$ about the Z –axis (the z –axis is coincide with the Z –axis). We wish to find the equations of motion of m₃ using the terminology of Szebehely [36] in the synodic co-ordinate system with dimensionless variables such that the distance between the primaries is unity, the unit of time t is such that the gravitational constant G = 1 and the sum of the masses of the primaries is unity (m₁ + m₂ = 1).

Fig. 1

The configuration of the ER3BP under albedo effect

The forces acting on m₃ due to m₁ and m₂ are F₁ (1 – F_p / F₁) = F₁ (1 – α) and F₂ (1 – F_A / F₂) = F₂ (1 – β) respectively, where α = F_p / F₁ < <1 and β = F_A / F₂ < < 1. Also, α and β can be expressed as:

$$ \alpha =\frac{\ell_1}{2\pi G{m}_1 c\sigma};\beta =\frac{\ell_2}{2\pi G{m}_2 c\sigma}; $$

where ℓ₁ is the luminosity of the larger primary m₁, ℓ₂is the luminosity of smaller primary m₂, G is the gravitational constant, c is the speed of light and σ is mass per unit area of the infinitesimal mass m₃.

Now,

$$ \frac{\beta }{\alpha }=\frac{m_1}{m_2}\frac{\ell_2}{\ell_1}\Rightarrow \beta =\alpha \left(\frac{1-\mu }{\mu}\right)k. $$

(1)

Let $k=\frac{\ell_2}{\ell_1}=\mathrm{constant},$ 0 < α < 1, 0 < β < α and 0 < k < 1.

The equations of motion of infinitesimal mass m₃ < < m₁, m₂ in terms of pulsating coordinates (ξ, η) are given by

$$ \left.\begin{array}{l}\xi \hbox{'}\hbox{'}-2\eta \hbox{'}={\varOmega}_{\xi}^{\ast },\\ {}\eta \hbox{'}\hbox{'}+2\xi \hbox{'}={\varOmega}_{\eta}^{\ast}\kern0.48em \end{array}\right\} $$

(2)

where

$$ {\displaystyle \begin{array}{l}{\varOmega}^{\ast }=\frac{1}{\sqrt{1-{e}^2}}\left[\frac{\xi^2+{\eta}^2}{2}+\frac{\varOmega }{n^2}\right],\\ {}\varOmega =\frac{\left(1-\mu \right)\left(1-\alpha \right)}{r_1}+\frac{\mu \left(1-\beta \right)}{r_2},\end{array}} $$

n :

mean-motion of the primaries,

e :

common eccentricity of elliptic orbit described by the primaries (0 < e < 1),

$$ {r}_1^2={\left(\xi -\mu \right)}^2+{\eta}^2, $$

(3)

$$ {\displaystyle \begin{array}{l}{r}_2^2={\left(\xi +1-\mu \right)}^2+{\eta}^2,\\ {}0<\mu =\frac{m_2}{m_1+{m}_2}<\frac{1}{2}\Rightarrow {m}_1=1-\mu; {m}_2=\mu .\end{array}} $$

(4)

Mean-Motion of the Primaries

In the elliptic case, the distance between the primaries is $r=\frac{a\left(1-{e}^2\right)}{1+e\cos f},$ and the mean distance between the primaries is given by $\frac{1}{2\pi}\underset{0}{\overset{2\pi }{\int }}r\; df=\frac{a\left(1-{e}^2\right)}{\sqrt{1+{e}^2}},$a is semi-major axis of the elliptic orbit of one primary around the other.

Since, the orbits of the primaries with respect to the center of mass have semi-major axes a₁ = am₂ and a₂ = am₁, and the same eccentricity [36], their equations of motion are given by $\frac{n^2a{m}_1\left(1-{e}^2\right)}{\sqrt{1+{e}^2}}=\frac{G{m}_1{m}_2}{r^2}$ and $\frac{n^2a{m}_2\left(1-{e}^2\right)}{\sqrt{1+{e}^2}}=\frac{G{m}_2{m}_1}{r^2}$.

Adding these equations yields ${n}^2=\frac{\sqrt{1+{e}^2}}{a\left(1-{e}^2\right)},$ since, m₁ + m₂ = 1 and we choose the unit of time such that the gravitational constant G = 1. Considering only terms of e² and neglect their product, we have

$$ {n}^2=1+\frac{3}{2}{e}^2,\left(a=1\right). $$

(5)

Libration Points

The libration points are the solutions of the Eqs. ${\varOmega}_{\xi}^{\ast }=0\kern0.24em \mathrm{and}\kern0.24em {\varOmega}_{\eta}^{\ast }=0,$ i.e.,

$$ \frac{1}{\sqrt{1-{e}^2}}\left[\xi -\frac{1}{n^2}\left\{\frac{\left(1-\mu \right)\left(\xi -\mu \right)\left(1-\alpha \right)}{r_1^3}+\frac{\mu \left(\xi +1-\mu \right)\left(1-\beta \right)}{r_2^3}\;\right\}\right]=0, $$

(6)

$$ \mathrm{and}\frac{\eta }{\sqrt{1-{e}^2}}\left[1-\frac{1}{n^2}\left\{\frac{\left(1-\mu \right)\left(1-\alpha \right)}{r_1^3}+\frac{\mu\;\left(1-\beta \right)}{r_2^3}\kern0.24em \right\}\right]=0. $$

(7)

In general, the libration points are the intersection of the contours ${\varOmega}_{\xi}^{\ast}\;\mathrm{and}\kern0.36em {\varOmega}_{\eta}^{\ast }$. As shown in the Fig. 2 (for μ = 0.1, α = 0.01, k = 0.001 and e = 0.01) there exist five libration points, three collinear (L₁, L₂, L₃) and two non-collinear (L₄, L₅).

Fig. 2

Libration points; μ = 0.1, k = 0.001, α = 0.01, e = 0.01

Non-collinear Libration Points

The non-collinear libration points are the solution of the Eqs. ${\varOmega}_{\xi}^{\ast }=0\kern0.24em \mathrm{and}\kern0.36em {\varOmega}_{\eta}^{\ast }=0,$η ≠ 0, i.e.,

$$ \xi -\frac{1}{n^2}\left\{\frac{\left(1-\mu \right)\left(\xi -\mu \right)\left(1-\alpha \right)}{r_1^3}+\frac{\mu \left(\xi +1-\mu \right)\left(1-\beta \right)}{r_2^3}\right\}=0, $$

(8)

$$ 1-\frac{1}{n^2}\left\{\frac{\left(1-\mu \right)\left(1-\alpha \right)}{r_1^3}+\frac{\mu\;\left(1-\beta \right)}{r_2^3}\;\right\}=0. $$

(9)

The solution of Eqs. (8) and (9) is ${r}_1^3=\frac{1-\alpha }{n^2}\kern0.36em \mathrm{and}\kern0.24em {r}_2^3=\frac{1-\beta }{n^2}$.

Substitute these values of r₁ and r₂ into Eqs. (3) and (4) and solving for ξ and η, we have

$$ {\displaystyle \begin{array}{l}\xi =\mu -\frac{1}{2}+\frac{\left(1-{e}^2\right)}{3\;}\left(\alpha -\beta \right),\\ {}\eta =\pm \frac{\sqrt{3}}{2}\left[1-\frac{2}{3}{e}^2-\frac{2}{9}\left(1-{e}^2\right)\left(\alpha +\beta \right)\right].\end{array}} $$

Using the relation (1), β = α (1− μ) k / μ, we have

$$ \left.\begin{array}{l}\xi =\mu -\frac{1}{2}+\frac{1}{3\;}\alpha \left(1-{e}^2\right)\left[1-\frac{\left(1-\mu \right)\;}{\mu }k\right],\\ {}\eta =\pm \frac{\sqrt{3}}{2}\left[1-\frac{2}{3}{e}^2-\frac{2}{9}\alpha \left(1-{e}^2\right)\left(1+\frac{\left(1-\mu \right)}{\mu }k\right)\right].\kern0.48em \end{array}\right\} $$

(10)

Thus, there exist two non-collinear libration points L_4,5 forming a scalene triangle with the primaries as r₁ ≠ r₂ and affected by Albedo and eccentricity of elliptic orbits of the primaries. From Eq. (8), the following results can be verified:

• For e = 0, the results are conform with Idrisi [14, 15].

• For e = 0 and k = 0, the results agree with Bhatnagar and Chawla [6].

• For e = 0 and α = 0, the results agree with Szebehely [36].

In the Figs. 3 and 4, it is shown that the coordinates of L₄ (ξ, η) vary with respect to e and α. From Fig. 3, it is observed that as e increases ξ changes little while η is displaced downward; the point L₄ moves toward the ξ-axis (Fig. 5). Hence, the shape of the triangle formed by L₄ with the primaries reduces. From Fig. 4, it has been observed that as α increases, ξ rapidly move towards the center of mass while the value of η decreases uniformly. Thus, the triangle formed by L₄ with the primaries reduces, and L₄ moves toward the η-axis (Fig. 6). The location of libration points L_4,5 for fixed values of k (0.001) and μ(0.1), and different values of e and α are given in Table 1.

Fig. 3

e versus ξ and η; α = 0.01, k = 0.001, μ = 0.1.

Fig. 4

α versus ξ and η; e = 0.01, k = 0.001, μ = 0.1.

Fig. 5

L₄ versus e; α = 0.01, k = 0.001, μ = 0.1

Fig. 6

L₄ versus α; e = 0.01, k = 0.001, μ = 0.1

Table 1 Non-collinear libration points L_4,5 (ξ, ± η)

Collinear Libration Points

The collinear libration points are the solutions of Eqs. ${\varOmega}_{\xi}^{\ast }=0\kern0.24em \mathrm{and}\kern0.36em {\varOmega}_{\eta}^{\ast }=0,$η = 0, i.e.,

$$ f\left(\xi \right)={n}^2\xi -\frac{\left(1-\mu \right)\left(\xi -\mu \right)\left(1-\alpha \right)}{r_1^3}-\frac{\mu \left(\xi +1-\mu \right)\left(1-\beta \right)}{r_2^3}=0, $$

(11)

where r_i = | ξ – ξ_i |, i = 1, 2. The solution is a fifth degree equation in ξ.

On plotting the function f(ξ) for μ = 0.1, α = 0.01, k = 0.001 and e = 0.01, three collinear libration points, L_i (ξ_i, 0), i = 1, 2, 3 are observed in the intervals (−∞, μ − 1), (μ − 1, μ) and (μ, ∞), respectively. For other values of μ, α, k and e, the result is same as there exist only three collinear libration points L_i (ξ_i, 0), i = 1, 2, 3 in the intervals (−∞, μ − 1), (μ − 1, μ) and (μ, ∞), respectively. Thus, there exist only three collinear libration points (Fig. 7).

Fig. 7

Collinear libration points L_i (ξ_i, 0), i = 1, 2, 3

Eq. (11) determines the location of the collinear libration points L₁(ξ₁, 0), L₂ (ξ₂, 0) and L₃ (ξ₃, 0) to lie in the intervals (−∞, μ − 1), (μ − 1, μ) and (μ, ∞), respectively, where

$$ \left.\begin{array}{l}{\xi}_1=\mu -1-{\varepsilon}_1,\\ {}{\xi}_2=\mu -1+{\varepsilon}_2,\\ {}{\xi}_3=\mu +{\varepsilon}_3.\kern1.68em \end{array}\right\} $$

(12)

Since the libration point L₁ lies in the interval (−∞, μ − 1), i.e., left of the smaller primary, we have r₁ = μ − ξ₁ and r₂ = μ − 1 − ξ₁, which when substituted in Eq. (11), give

$$ {n}^2{\xi}_1+\frac{\left(1-\mu \right)\left({\xi}_1-\mu \right)\left(1-\alpha \right)}{r_1^3}+\frac{\mu \left({\xi}_1+1-\mu \right)\left(1-\beta \right)}{r_2^3}=0 $$

(13)

Similarly, for L₂ (ξ₂, 0) and L₃ (ξ₃, 0), Eq. (11) becomes.

$$ {n}^2{\xi}_2+\frac{\left(1-\mu \right)\left({\xi}_2-\mu \right)\left(1-\alpha \right)}{r_1^3}-\frac{\mu \left({\xi}_2+1-\mu \right)\left(1-\beta \right)}{r_2^3}=0 $$

(14)

$$ {n}^2{\xi}_3-\frac{\left(1-\mu \right)\left({\xi}_3-\mu \right)\left(1-\alpha \right)}{r_1^3}-\frac{\mu \left({\xi}_3+1-\mu \right)\left(1-\beta \right)}{r_2^3}=0 $$

(15)

The solutions of Eqs. (13), (14) and (15) are given in Table 2 for on specific values of the parameters appearing in the first line.

Table 2 Collinear Libration Points L_i (ξ_i, 0), i = 1, 2, 3

In Fig. 8, the green curves are the collinear libration point values of ξ_i (i = 1, 2, 3) in the interval 0 < e < 1; α = 0.01 and red curves show the collinear libration point values of ξ_i (i = 1, 2, 3) in 0 < α < 1; e = 0.01; for k = 0.001; μ = 0.1. It is observed that the first collinear libration point L₁ always lies to the left of the primary m₂, the second libration point L₂, lies between the center of mass of the primaries O and m₁, and the third libration point L₃ lies to the right of the primary m₁. It is also observed that the libration point L₁ moves toward m₂, L₂ moves toward center of mass and L₃ moves toward m₁ in 0 < e < 1 and 0 < α < 1.

Fig. 8

Collinear libration points for k = 0.001 and μ = 0.1

Stability of Libration Points

The variational equations are obtained by substituting ξ = ξ₀+ ε and η = η₀+ δ in the equations of motion (2), where (ξ_o, η_o) are the coordinates of a particular libration point and ε, δ < <1, i.e.,

$$ \left.\begin{array}{l}\varepsilon \hbox{'}\hbox{'}-2\delta \hbox{'}=\varepsilon\;{\varOmega}_{\xi \xi}^{\ast 0}+\delta\;{\varOmega}_{\xi \eta}^{\ast 0},\\ {}\delta \hbox{'}\hbox{'}+2\varepsilon \hbox{'}=\varepsilon\;{\varOmega}_{\xi \eta}^{\ast 0}+\delta\;{\varOmega}_{\eta \eta}^{\ast 0}.\kern0.96em \end{array}\right\} $$

(16)

Here we have taken only linear terms in ε and δ. The subscript in Ω^* indicates the second partial derivative of Ω^* and superscript 0 indicates that the derivative is to be evaluated at the libration point (ξ₀, η₀). The characteristic equation corresponding to Eq. (16) is

$$ {\lambda}^4+\left(4-{\varOmega}_{\xi \xi}^{\ast 0}-{\varOmega}_{\eta \eta}^{\ast 0}\right)\;{\lambda}^2+{\varOmega}_{\xi \xi}^{\ast 0}\;{\varOmega}_{\eta \eta}^{\ast 0}-{\left({\varOmega}_{\xi \eta}^{\ast 0}\right)}^2=0. $$

(17)

Letting λ² = Π, Eq. (17) becomes

$$ {\varPi}^2+{q}_1\;\varPi +{q}_2=0, $$

(18)

which is a quadratic equation in Π, and its roots are given by

$$ {\varPi}_{1,2}=\frac{1}{2}\left(-{q}_1\pm \sqrt{D}\right) $$

(19)

where ${q}_1=4-{\varOmega}_{\xi \xi}^{\ast 0}-{\varOmega}_{\eta \eta}^{\ast 0};\kern0.36em {q}_2={\varOmega}_{\xi \xi}^{\ast 0}\;{\varOmega}_{\eta \eta}^{\ast 0}-{\left({\varOmega}_{\xi \eta}^{\ast 0}\right)}^2;\kern0.36em D={q}_1^2-4{q}_2$.

Therefore, the roots of characteristic Eq. (17) are given by, ${\lambda}_{1,2}=\pm \sqrt{\varPi_1}\kern0.24em \mathrm{and}\kern0.24em {\lambda}_{3,4}=\pm \sqrt{\varPi_2}$.

The libration point (ξ₀, η₀) is said to be stable if ∆₁ < 0 and ∆₂ < 0 or D ≥ 0.

Stability of Non-collinear Libration Points

At the non-collinear libration point L₄(ξ₀, η₀),

$$ {\displaystyle \begin{array}{l}{\varOmega}_{\xi \xi}^{\ast 0}=\frac{3}{4}\left[1-\frac{2}{3}\left(1-3\mu \right)\alpha +\frac{2}{3}\left(2-3\mu \right)\beta \right]-\frac{3}{8}\left[1-\frac{2}{3}\left(5-11\mu \right)\alpha +\frac{2}{3}\left(6-11\mu \right)\beta \right]{e}^2,\\ {}{\varOmega}_{\xi \eta}^{\ast 0}=\frac{3\sqrt{3}}{2}\left[\mu -\frac{1}{2}+\frac{1}{9}\left(1+\mu \right)\alpha -\frac{1}{9}\left(2-\mu \right)\beta \right]-\frac{13}{4\sqrt{3}}\left[\mu -\frac{1}{2}+\frac{1}{39}\left(27-5\mu \right)\alpha -\frac{1}{39}\left(22+5\mu \right)\beta \right]{e}^2,\\ {}{\varOmega}_{\eta \eta}^{\ast 0}=\frac{9}{4}+\frac{1}{2}\left(1-3\mu \right)\alpha -\frac{1}{2}\left(2-3\mu \right)\beta -\frac{1}{4}\left[\frac{17}{2}-\frac{1}{9}\left(11+51\mu \right)\alpha -\frac{1}{9}\left(62-51\mu \right)\beta \right]{e}^2.\end{array}} $$

The motion near the Libration point (ξ₀, η₀) is said to be bounded if D ≥ 0 i.e.

$$ {\displaystyle \begin{array}{l}1-27\mu +27{\mu}^2+6\left(1-\mu \right) k\alpha -3\frac{k\left(1-m\right)\alpha }{m}-6 m\alpha +\\ {}\left(5+39\mu -39{\mu}^2-\frac{28}{9}\alpha -12k\left(1-\mu \right)\alpha +\frac{31}{18}\frac{k\left(1-\mu \right)\alpha }{\mu }+2\mu \alpha \right){e}^2\ge 0\end{array}} $$

(20)

As shown in the Fig. 9, for α = 0, k = 0, e is defined in the interval μ = 0.03852 … ≤ e ≤ μ = 0.5 where μ = 0.03852 … is the value of critical mass parameter in classical case and for α = 0.01, k = 0.001, e is defined in the interval μ = 0.03457 … ≤ e ≤ μ = 0.5. Thus, the domain of e expands as α increases.

Fig. 9

μ versus e

For α = 0 and e = 0, μ_c = 0.03852 …. is the critical value of mass parameter in classical case [36]. When α ≠ 0, e ≠ 0, we suppose that μ_c = μ_o + p₁α + p₂e² as the root of the Eq. (20), where, μ_o = 0.0385208965 … and p₁, p₂ are to be determined in a manner such that D = 0. Therefore, we have

$$ {\displaystyle \begin{array}{l}{p}_1=-\frac{k-3k{\mu}_o+2{\mu}_o^2+2k{\mu}_o^2}{9{\mu}_o\left(1-2{\mu}_o\right)},\\ {}{p}_2=\frac{5+39{\mu}_o-39{\mu}_o^2}{27\left(1-2{\mu}_o\right)}.\end{array}} $$

(21)

∴ μ_c = 0.0385208965 … − (0.00891747 + 2.78224 k) α + 0.258607 e².

As shown in the Fig. 10, Π_1,2 < 0 for μ ≤ μ_c. Thus the eigenvalues of the characteristic Eq. (17) are given by ${\lambda}_{1,2}=\pm i\sqrt{\varPi_1},{\lambda}_{3,4}=\pm i\sqrt{\varPi_2}$. Therefore, the non-collinear libration points L_4,5 are periodic and bounded and hence stable for the critical mass parameter μ ≤ μ_c, where μ_c is defined in Eq. (21). It is also observed that as the eccentricity e increases the value of the critical mass parameter μ_c increases exponentially (Fig. 11) and as α increase, μ_c decreases uniformly (Fig. 12).

Fig. 10

μ versus Π; α = 0.01; e = 0.01; k = 0.0001

Fig. 11

e versus μ_c; α = 0.01; k = 0.0001

Fig. 12

α versus μ_c; e = 0.01; k = 0.0001

Stability of Collinear Libration Points

First we consider the point lying in the interval (μ − 2, μ − 1). For this point, r₂ < 1, r₁ > 1 and

$$ \left.\begin{array}{l}{\varOmega}_{\xi \xi}^{\ast 0}=\frac{1}{\sqrt{1-{e}^2}}\left[1+\frac{1}{n^2}\left\{\frac{2\left(1-\mu \right)\left(1-\alpha \right)}{r_1^3}+\frac{2\mu \left(1-\beta \right)}{r_2^3}\right\}\right]>0,\\ {}{\varOmega}_{\xi \eta}^{\ast 0}=0,\\ {}{\varOmega}_{\eta \eta}^{\ast 0}=\frac{1}{\sqrt{1-{e}^2}}\left[1-\frac{1}{n^2}\left\{\frac{\left(1-\mu \right)\left(1-\alpha \right)}{r_1^3}+\frac{\mu \left(1-\beta \right)}{r_2^3}\right\}\right]<0.\kern1.2em \end{array}\right\} $$

(22)

Similarly, for the points lying in the interval (μ − 1,0) and (μ, μ + 1), ${\varOmega}_{\xi \xi}^{\ast 0}>0,{\varOmega}_{\xi \eta}^{\ast 0}=0\kern0.24em \mathrm{and}\kern0.24em {\varOmega}_{\eta \eta}^{\ast 0}<0.$ Since the discriminant of Eq. (20) is positive and the four roots of the characteristic Eq. (17) can be written as λ_1,2 = ±s and λ_3,4 = ± it (s and t are real), the motion around the collinear libration points is unbounded, and consequently, the collinear libration points are unstable.

Conclusion

In the present paper, the existence and stability of libration points in the elliptic restricted three-body problem has been studied under albedo effect. The equations of motion including albedo effect are derived as Eq. (2). For β = 0, the problem reduces to the photogravitational elliptic restricted three-body problem where only the more massive primary is a source of radiation. For β = 0 and e = 0, the problem becomes the photogravitational circular restricted three-body problem where the more massive primary is a source of radiation. It is found that there exist five libration points, three collinear and two non-collinear. It is observed that the first collinear libration point L₁ always lies to the left of the primary m₂, the second libration point L₂ lies between the center of mass of the primaries O and m₁, and the third libration point L₃ lies to the right of the primary m₁.It is also observed that the libration point L₁ moves toward m₂, L₂ moves toward the center of mass and L₃ moves toward m₁ in 0 < e < 1 and 0 < α < 1 (Fig. 8). There exist two non-collinear libration points L_4,5, and From Fig. 3, it is observed that as e increases ξ varies little while η is displaced downward. The point L₄ moves toward the ξ-axis (Fig. 5) and hence the shape of the triangle formed by L₄ with the primaries reduces. From Fig. 4, it has been observed that as α increases, ξ rapidly move towards the center of mass, while the value of η decreases uniformly. Thus, the triangle formed by L₄ with the primaries reduces and L₄ moves toward the η-axis (Fig. 6). The non-collinear libration points L_4,5 are not periodic and bounded and hence stable for the critical mass parameter μ ≤ μ_c, where μ_c is defined by μ_c = 0.0385208965 … − (0.00891747 + 2.78224 k) α + 0.258607 e² while the collinear libration points are unstable.